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Boundary Data Smoothness for Solutions of Three Point Boundary Value Problems for Second Order Ordinary Differential Equations | EMS Press
Boundary Data Smoothness for Solutions of Three Point Boundary Value Problems for Second Order Ordinary Differential Equations
Under certain conditions, solutions of the boundary value problem,
y'' = f(x,y,y')
a < x < b
y(x_1) = y_1
y(x_3) - y(x_2) = y_2
a< x_1 < x_2 < x_3 < b
, are differentiated with respect to the boundary conditions.
Johnny Henderson, Christopher C. Tisdell, Boundary Data Smoothness for Solutions of Three Point Boundary Value Problems for Second Order Ordinary Differential Equations. Z. Anal. Anwend. 23 (2004), no. 3, pp. 631–640 |
To Horace Darwin 1 November [1877]1
I have had a very civil letter from the son of Ld Eldon’s agent, but it is clear he has got no information worth anything, but says he will observe again.—
He tells me that at Cirencester there is a Roman pavement, exposed 50 years ago, & that this is not level, & this is a point about which I am curious.—2 Thinking over whole case, it seems to me that it wd be very desirable for my work, that you shd. inspect Chedworth in the big woods & Cirencester.3 You wd have to go to the latter to visit Chedworth.— Will it interfere seriously with your Cambridge plans, for if so I shd. not at all like you to go?— Frank says he shd. like to go with you. I cd afterwards tell you the several points— A whole day at Chedworth or rather at Foss Bridge4 would suffice, &
\frac{1}{2}
day at Cirencester. I have asked Mr Hall to give me letter of introduction for you to person in charge. The job wd be to get a labourer, & I shd like one hole dug beneath foundations of one of the old walls to a depth of 1 foot or 18 inches.—
I hope that your water-works progress well.—5
Pick the surgeon was here yesterday & say Lennys knee is going on as well as possible—6 —No inflammation.— & that in a fortnights time he is with knee-caps on to try a kind of walking first with 2 crutches & then with one.— By next summer he thinks knee will be quite strong.—
Fairly good account today of Litchfield—7
The year is established from a reference in Earthworms, p. 198, to Horace and Francis Darwin’s visit to the Roman villa at Chedworth, Gloucestershire, in November 1877.
The agent of John Scott (Lord Eldon) and his son have not been identified, although their surname was presumably Hall (see below in the letter). The letter from the agent’s son has not been found. CD did not mention the pavement at Cirencester, Gloucestershire, in Earthworms, but it may have been the one at Barton Farm, Cirencester (see Sewell and Powell 1910). See also letter from T. H. Farrer, 5 September 1877 and n. 1.
CD gave an account of Horace and Francis’s findings at Chedworth in Earthworms, pp. 197–9.
Fossebridge is the location of an inn near Chedworth.
Earlier in the year, Horace had been working on the waterworks at Brighton (letter from Horace Darwin to Emma Darwin, [February 1877?] (DAR 258: 817)).
Leonard Darwin had hurt his knee playing lawn tennis (letter from Emma Darwin to H. E. Litchfield, [23 September 1877] (DAR 219.9: 159)). Pick the surgeon: Thomas Pickering Pick, assistant surgeon at St George’s Hospital, London.
The letter has not been found. Richard Buckley Litchfield, CD’s son-in-law, had been taken ill with appendicitis in Switzerland in September; he and CD’s daughter Henrietta Emma Litchfield returned to England on 8 November (letter from Emma Darwin to H. E. Litchfield, [13 September 1877] (DAR 219.9: 155), postcard from H. E. Litchfield to G. H. Darwin, [8 November 1877] (DAR 245: 314), Emma Darwin (1915) 2: 227). |
The 360-day calendar is a method of measuring durations used in financial markets, in computer models, in ancient literature, and in prophetic literary genres.
It is based on merging the three major calendar systems into one complex clock, with the 360-day year derived from the average year of the lunar and the solar: (365.2425 (solar) + 354.3829 (lunar))/2 = 719.6254/2 = 359.8127 days, rounding to 360.
For example, the 27th of June (Gregorian calendar) would be the 4th of July in the USA.
1 Ancient Calendars
2 Financial use
Ancient CalendarsEdit
Ancient calendars around the world initially used a 360 day calendar. [1]
Romans initially used a calendar which had 360 days, with varying length of months.[2][3]
The Rig Veda describes a calendar with twelve months and 360 days.[4]
In the Mayan Long Count Calendar, the equivalent of the year, the tun, was 360 days.[5]
Ancient Egyptians also used a 360 day calendar. One myth tells of how the extra 5 days were added.
A long time ago, Ra, who was god of the sun, ruled the earth. During this time, he heard of a prophecy that Nut, the sky goddess, would give birth to a son who would depose him. Therefore Re cast a spell to the effect that Nut could not give birth on any day of the year, which was then itself composed of precisely 360 days. To help Nut to counter this spell, the wisdom god Thoth devised a plan.
Thoth went to the moon god Khonsu and asked that he play a game known as Senet, requesting that they play for the very light of the moon itself. Feeling confident that he would win, Khonsu agreed. However, in the course of playing he lost the game several times in succession, such that Thoth ended up winning from the moon a substantial measure of its light, equal to about five days.
With this in hand, Thoth then took this extra time, and gave it to Nut. In doing so this had the effect of increasing the earth’s number of days per year, allowing Nut to give birth to a succession of children; one upon each of the extra 5 days that were added to the original 360. And as for the moon, losing its light had quite an effect upon it, for it became weaker and smaller in the sky. Being forced to hide itself periodically to recuperate; it could only show itself fully for a short period of time before having to disappear to regain its strength.
Financial useEdit
A duration is calculated as an integral number of days between startdate and enddate B. The difference in years, months and days are usually calculated separately:
{\displaystyle duration(A,B)=(B_{y}-A_{y})\times 360+(B_{m}-A_{m})\times 30+(B_{d}-A_{d});A\leq B}
There are several methods commonly available which differ in the way that they handle the cases where the months are not 30 days long, i.e. how they adjust dates:
European method (30E/360)[8][9][10][11][12]
If either date A or B falls on the 31st of the month, that date will be changed to the 30th.
All months are considered to last 30 days and hence a full year has 360 days, but another source says that February has its actual number of days.
US/NASD method (30US/360)[13]
ISDA method[14][10][11][12]
If date A falls on the 31st of a month, then date A will be changed to the 30th.
If date A falls on the 30th of the month after applying the rule above, and date B falls on the 31st of the month, then date B will be changed to the 30th.
All months are considered to last 30 days except February which has its actual length. Any full year, however, always counts for 360 days.
BMA/PSA method[10][11][12]
Alternative European method (30E+/360)[10][12]
If date B falls on the 31st of a month, then date B will be changed to the 1st of the following month.
360-day calendar implementation in spreadsheet functions
Microsoft Excel and StarOffice/OpenOffice/LibreOffice DAYS360 NASD, but not SIA-compliant[15][16][17]
YEARFRAC NASD and European
SQL Server 2000 Analysis Services Days360
Mathworks Financial Toolbox days360 US/NASD
days360e European
days360isda ISDA
days360psa PSA
Gnumeric DAYS360
Apple Numbers DAYS360 NASD and European
365-day calendar, another accounting calendar with fixed year length
Day count convention, standards for counting days
French Republican Calendar, a calendar with twelve 30-day months and five or six appended holidays
Iranian calendars, where the Old Persian calendar had 360 days with an extra month added every 6 years
^ "An Original 360-Day Year". 360dayyear.com. Retrieved 2021-06-17.
^ Stern, Sacha (2012). Calendars in Antiquity: Empires, States, and Societies. Oxford University Press. p. 208. ISBN 978-0-19-958944-9.
^ Plutarch. "Life of Numa". The University of Chicago.
^ B.G., Sidharth (1999). The Celestial Key to the Vedas. p. 86. ISBN 9780892817535.
^ "Maya Calendar Converter". Smithsonian National Museum of the American Indian. 2015. Retrieved September 10, 2015.
^ "Earth's Original 360-Day Year and Calendar". 360dayyear.com. 2015. Archived from the original on 2021-12-22. Retrieved 31 January 2019.
^ "An 'Ideal' Earth Year of 360 Days?". Keith Hunter, Ancient World Mysteries. 2015. Retrieved 31 January 2019.
^ ISMA book “Bond Markets: Structures and Yield Calculations”, ISBN 1-901912-02-7, and ISMA’s Circular 14 of 1997
^ 2006 ISDA Definitions, Sec. 4.16 (g)
^ a b c d "Derivatives Risk Management Software & Pricing Analytics | FINCAD".
^ a b c d Accrual & Discounting Conventions
^ Standard Securities Calculation Methods, Fixed Income Securities Formulas for Price, Yield, and Accrued Interest: Volume 1, 1993, Jan Mayle, New York, NY: Securities Industry Association, ISBN 1882936019
^ 2006 ISDA Definitions, Sec. 4.16 (f)
^ See Microsoft Kb Article 916004. This bug is present in Excel versions 97 through 2007. This can be demonstrated by evaluating DAYS360(DATE(2006,2,28), DATE(2007,2,28)); here years starting and ending on the last day in February only have 358 days.
^ Open Office is also not SIA-compliant, to maintain Excel compatibilitySee Issue 84934 — ODFF: DAYS360 compliance
^ OpenOffice Documentation
Retrieved from "https://en.wikipedia.org/w/index.php?title=360-day_calendar&oldid=1080844394" |
Electric Dipole Moment and Radiation Power - MATLAB & Simulink Example - MathWorks Deutschland
Common Center of Mass
Radiation Power per Unit of Time
Parameters of the Elliptical Orbit
Averaged Radiation Power
If One Particle is Much Heavier Than the Other
This example finds the average radiation power of two attracting charges moving in an elliptical orbit (an electric dipole).
The two opposite charges, e1 and e2, form an electric dipole. The masses of the charged particles are m1 and m2, respectively. For the common center of mass m1*r1 + m2*r2 = 0, where r1 and r2 are distance vectors to the charged particles. The distance between charged particles is r = r1 - r2.
syms m1 m2 e1 e2 r1 r2 r
[r1,r2] = solve(m1*r1 + m2*r2 == 0, r == r1 - r2, r1, r2)
\frac{{m}_{2} r}{{m}_{1}+{m}_{2}}
-\frac{{m}_{1} r}{{m}_{1}+{m}_{2}}
Find the dipole moment of this system:
d = e1*r1 + e2*r2;
simplify(d)
\frac{r \left({e}_{1} {m}_{2}-{e}_{2} {m}_{1}\right)}{{m}_{1}+{m}_{2}}
According to the Larmor formula, the total power radiated in a unit of time is
J=\frac{2}{3{c}^{3}}{\underset{}{\overset{¨}{d}}}^{2}
, or, in terms of the distance between the charged particles,
J=\frac{2}{3{c}^{3}}\frac{m1m2}{m1+m2}{\left(\frac{e1}{m1}-\frac{e2}{m2}\right)}^{2}{\underset{}{\overset{¨}{r}}}^{2}
. Here dot means a time derivative. Coulomb's law
m\underset{}{\overset{¨}{r}}=-\frac{\alpha }{{r}^{2}}
lets you find the values of acceleration
\underset{}{\overset{¨}{r}}
in terms of the reduced mass of the system,
m=\frac{m1m2}{m1+m2}
, and the product of the charges of the particles,
\alpha =|e1e2|
syms m c
m = m1*m2/(m1 + m2);
r2 = -alpha/(m*r^2);
J = simplify(subs(2/(3*c^3)*d^2, r, r2))
\frac{2 {\alpha }^{2} {\left({e}_{1} {m}_{2}-{e}_{2} {m}_{1}\right)}^{2}}{3 {c}^{3} {{m}_{1}}^{2} {{m}_{2}}^{2} {r}^{4}}
The major semiaxis a and eccentricity
ϵ
of an elliptical orbit are given by the following expressions, where E is the total orbital energy, and
L=m{r}^{2}\underset{}{\overset{˙}{\varphi }}
is the angular momentum.
syms E L phi
a = alpha/(2*E)
\frac{\alpha }{2 \text{E}}
eccentricity = sqrt(1-2*E*L^2/(m*alpha^2))
\sqrt{1-\frac{2 \text{E} {L}^{2} \left({m}_{1}+{m}_{2}\right)}{{\alpha }^{2} {m}_{1} {m}_{2}}}
The equation of an elliptical orbit,
1+ϵ\mathrm{cos}\varphi =a\left(1-{ϵ}^{2}\right)/r
, lets you express the distance r in terms of the angle phi.
r = a*(1 - eccentricity^2)/(1 + eccentricity*cos(phi));
The average radiation power of two charged particles moving in an elliptical orbit is an integral of the radiation power over one full cycle of motion, normalized by the period of motion,
{J}_{avg}=1/T{\int }_{0}^{T}J\phantom{\rule{0.16666666666666666em}{0ex}}dt
. The period of motions T is
T = 2*pi*sqrt(m*a^3/alpha);
Changing the integration variable t to phi, you get the following result. Use the simplify function to get a shorter integration result. Here, use subs to evaluate J.
J = subs(J);
Javg = simplify(1/T*int(J*m*r^2/L, phi, 0, 2*pi))
Javg =
-\frac{2 \sqrt{2} {\alpha }^{2} {\left({e}_{1} {m}_{2}-{e}_{2} {m}_{1}\right)}^{2} \left(2 \text{E} {L}^{2} {m}_{1}+2 \text{E} {L}^{2} {m}_{2}-3 {\alpha }^{2} {m}_{1} {m}_{2}\right)}{3 {L}^{5} {c}^{3} {\left({m}_{1}+{m}_{2}\right)}^{3} \sqrt{\frac{{\alpha }^{2} {m}_{1} {m}_{2}}{{\text{E}}^{3} \left({m}_{1}+{m}_{2}\right)}}}
Estimate the average radiation power of the electric dipole with one particle much heavier than the over, m1>>m2. For this, compute the limit of the expression for radiation power, assuming that m1 tends to infinity.
limJ = limit(Javg, m1, Inf);
simplify(limJ)
-\frac{2 \sqrt{2} {\alpha }^{2} {{e}_{2}}^{2} \left(2 \text{E} {L}^{2}-3 {\alpha }^{2} {m}_{2}\right)}{3 {L}^{5} {c}^{3} \sqrt{\frac{{\alpha }^{2} {m}_{2}}{{\text{E}}^{3}}}} |
Electromagnetic spectrum - zxc.wiki
The electromagnetic spectrum - EM spectrum for short and more precisely called the electromagnetic wave spectrum - is the entirety of all electromagnetic waves of different wavelengths . The light spectrum , also color spectrum , is the part of the electromagnetic spectrum that is visible to humans.
The spectrum is divided into different areas. This classification is arbitrary and, for historical reasons, is based on the wavelength in the low-energy range. Wavelength ranges over several orders of magnitude with similar properties are grouped into categories such as light , radio waves , etc. A subdivision can also be made according to the frequency or the energy of the individual photon (see below). In the case of very short wavelengths, correspondingly high quantum energy, a classification according to energy is common.
Arranged according to decreasing frequency and thus increasing wavelength, the short-wave and therefore high-energy gamma rays are located at the beginning of the spectrum , the wavelength of which extends into atomic orders of magnitude. At the end there are the longest waves , the wavelengths of which are many kilometers.
The wavelength is converted into a frequency using the formula . Where is the speed of light .
{\ displaystyle f}
{\ displaystyle f = c / \ lambda}
{\ displaystyle c}
Overview of the electromagnetic spectrum
Overview with visible spectrum in detail
The areas of the electromagnetic spectrum
Photons -
Generation / excitation
Low frequency Extremely Low Frequency (ELF) 10 mm 100 mm 3 Hz 30 Hz > 2.0 · 10 −33 J
> 12 feV Floor dipole , antenna systems Traction current
Super Low Frequency (SLF) 1 mm 10 mm 30 Hz 300 Hz > 2.0 · 10 −32 J
> 120 feV Mains frequency , (formerly) submarine communication
0.3 kHz 3000 Hz
3 kHz > 2.0 · 10 −31 J
> 1.2 peV
waves Longitudinal waves (SLW) 10 km 100 km 3 kHz 30 kHz > 2.0 · 10 −30 J
> 12 peV Submarine communication ( DHO38 , ZEVS , Sanguine , SAQ ), radio navigation , heart rate monitors
Radio waves Long wave (LW) 1 km 10 km 30 kHz 300 kHz > 2.0 · 10 −29 J
> 120 peV Oscillator circuit + antenna Long wave radio , DCF77 , induction hob
Medium wave (MW) 100 m 1000 m 300 kHz 3 MHz > 2 · 10 −28 J
> 1.2 neV Medium wave broadcasting , HF surgery , (1.7 MHz-3 MHz boundary wave , short wave broadcasting )
Short wave (KW) 10 m 100 m 3 MHz 30 MHz > 1.1 · 10 −27 J
> 12 neV Boundary wave , shortwave broadcasting , HAARP , diathermy , RC model making
Ultra short wave (VHF) 1 m 10 m 30 MHz 300 MHz > 2.0 · 10 −26 J
> 120 neV Oscillator circuit + antenna Radio , television , radar , magnetic resonance imaging
Microwaves Decimeter waves 10 centimeters 1 m 300 MHz 3 GHz > 2.0 · 10 −25 J
> 1.2 µeV Magnetron , klystron , burl , cosmic background radiation
Excitation of nuclear magnetic resonance and electron spin resonance , molecular rotations
Radar , magnetic resonance imaging , cellular communications , television , microwave oven , WiFi , Bluetooth , GPS , 5G
Centimeter waves 1 cm 10 centimeters 3 GHz 30 GHz > 2.0 · 10 −24 J
> 12 µeV Radar , radio astronomy , directional radio , satellite broadcasting , WLAN , 5G
Millimeter waves 1 mm 1 cm 30 GHz 300 GHz
0.3 THz > 2.0 · 10 −23 J
> 120 µeV Radar , radio astronomy , directional radio
Terahertz radiation 30 µm 3 mm 0.1 THz 10 THz > 6.6 · 10 −23 J
> 0.4 meV Synchrotron , free-electron laser Radio astronomy , spectroscopy , imaging techniques
Infrared radiation (heat radiation) Far infrared 50 µm 1 mm 300 GHz 6 THz > 2.0 · 10 −22 J
> 1.2 meV Heat radiator , synchrotron
Molecular vibrations Infrared spectroscopy , Raman spectroscopy , infrared astronomy
Mid infrared 3.0 µm 50 µm 6 THz 100 THz > 4.0 · 10 −21 J
> 25 meV Carbon dioxide laser , quantum cascade laser Thermography
Near infrared 780 nm 3.0 µm 100 THz 385 THz > 8.0 10 −20 J
> 500 meV Nd: YAG laser , laser diode , light emitting diode Remote control , data communication ( IRDA ), CD
light red 640 nm 780 nm 384 THz 468 THz 1.59-1.93 eV Radiant heaters ( incandescent lamp ), gas discharge ( neon tube ), dye and other lasers , synchrotron , light-emitting diode
Excitation of valence electrons DVD , laser pointer , data transmission ( optical fiber )
Red, green: laser level ,
colorimetry ,
photometry ,
red, yellow, green: traffic light system ,
purple: Blu-ray disc
orange 600 nm 640 nm 468 THz 500 THz 1.93-2.06 eV
yellow 570 nm 600 nm 500 THz 526 THz 2.06-2.17 eV
green 490 nm 570 nm 526 THz 612 THz 2.17-2.53 eV
blue 430 nm 490 nm 612 THz 697 THz 2.53-2.88 eV
violet 380 nm 430 nm 697 THz 789 THz 2.88-3.26 eV
UV rays Near UV (" black light ") 315 nm 380 nm 789 THz 952 THz 3.26-3.94 eV Gas discharge , synchrotron , excimer laser , light emitting diode Black light fluorescence , phosphorescence , bank note checking , photolithography , disinfection , UV light , spectroscopy
Medium UV (" Dorno radiation") 280 nm 315 nm 952 THz 1071 THz
1 PHz 3.94-4.43 eV
Far UV 200 nm 280 nm 1 PHz 1.5 PHz 4.43-6.2 eV
Vacuum UV 100 nm 200 nm 1.5 PHz 3 PHz > 9.9 · 10 -19 J
6.2-12 eV XUV tube , synchrotron , nanoplasm EUV lithography , X-ray microscopy , nanoscopy
EUV 10 nm 121 nm 2.5 PHz 30 PHz > 5.0 · 10 −18 years
10.2-120 eV
X-rays 10 pm 10 nm 30 PHz 30 EHz > 2.0 · 10 −16 J
> 120 eV X-ray tube , synchrotron
Excitation of internal electrons , Auger electrons
medical diagnostics , security technology, X-ray structure analysis , X-ray diffraction , photoelectron spectroscopy , X-ray absorption spectroscopy
gamma rays 10 pm 30 EHz > 2.0 · 10 −14 J
> 120 keV Radioactivity , annihilation
Excitation of core states medical radiation therapy , Mössbauer spectroscopy
DIN 5031 Part 7: Radiation physics in the optical field and lighting technology; Designation of the wavelength ranges. January 1984 (IR, VIS and UV).
Commons : Electromagnetic Spectrum - collection of images, videos, and audio files
Poster "Electromagnetic Radiation Spectrum" (PDF, English; 992 kB)
The electromagnetic spectrum in the world of physics
↑ are as defined in the Radio Regulations, 2012 edition, Article 1.5 also to the radio waves.
↑ German Institute for Standardization (Ed.): Radiation physics in the optical field and lighting technology; Designation of the wavelength ranges. DIN 5031 part 7, January 1984.
This page is based on the copyrighted Wikipedia article "Elektromagnetisches_Spektrum" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA. |
𝔽
be a filtration and τ be a random time. Let
𝔾
be the progressive enlargement of
𝔽
with τ. We study the following formula, called the optional splitting formula: For any
𝔾
-optional process Y, there exists an
𝔽
-optional process Y′ and a function Y′′ defined on [0,∞] × (ℝ+ × Ω) being
ℬ\left[0,\infty \right]\otimes 𝒪\left(𝔽\right)
measurable, such that
Y={Y}^{\text{'}}{\mathbb{1}}_{\left[0,\tau \right)}+{Y}^{\text{'}\text{'}}\left(\tau \right){\mathbb{1}}_{\left[\tau ,\infty \right)}
. (This formula can also be formulated for multiple random times τ1,...,τk). We are interested in this formula because of its fundamental role in many recent papers on credit risk modeling, and also because of the fact that its validity is limited in scope and this limitation is not sufficiently underlined. In this paper we will determine the circumstances in which the optional splitting formula is valid. We will then develop practical sufficient conditions for that validity. Incidentally, our results reveal a close relationship between the optional splitting formula and several measurability questions encountered in credit risk modeling. That relationship allows us to provide simple answers to these questions.
Classification : 60G07, 60G44, 91G40, 97M30
Mots clés : optional process, progressive enlargement of filtration, credit risk modeling, conditional density hypothesis
author = {Song, Shiqi},
title = {Optional splitting formula in a progressively enlarged filtration},
TI - Optional splitting formula in a progressively enlarged filtration
Song, Shiqi. Optional splitting formula in a progressively enlarged filtration. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 829-853. doi : 10.1051/ps/2014003. http://archive.numdam.org/articles/10.1051/ps/2014003/
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[4] A. Bélanger and S. Shreve and D. Wong, A unified model for credit derivatives. Working paper (2002).
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[35] S. Song, Drift operator in a market affected by the expansion of information flow: a case study (2012). Preprint arXiv:1207.1662v1.
[36] S. Song, Local solution method for the problem of enlargement of filtration (2013). Preprint arXiv:1302.2862.
[37] C. Stricker and M. Yor, Calcul stochastique dépendant d'un paramètre. Probab. Theory Relat. Fields 45 (1978) 109-133. | MR 510530 | Zbl 0388.60056
[38] H. Von Weizsäcker, Exchanging the order of taking suprema and countable intersections of σ-algebras. Ann. Inst. Henri Poincaré Section B, Tome 19 1 (1983) 91-100. | Numdam | MR 699981 | Zbl 0509.60002
[39] D. Wu, Dynamized copulas and applications to counterparty credit risk. Ph.D. Thesis, University of Evry (2012).
[40] K. Yano and M. Yor, Around Tsirelson's equation, or: The evolution process may not explain everything (2010). Preprint arXiv:0906.3442. |
Polymer chemistry - Wikipedia
Polymer chemistry is a sub-discipline of chemistry that focuses on the chemical synthesis, structure, and chemical and physical properties of polymers and macromolecules. The principles and methods used within polymer chemistry are also applicable through a wide range of other chemistry sub-disciplines like organic chemistry, analytical chemistry, and physical chemistry. Many materials have polymeric structures, from fully inorganic metals and ceramics to DNA and other biological molecules, however, polymer chemistry is typically referred to in the context of synthetic, organic compositions. Synthetic polymers are ubiquitous in commercial materials and products in everyday use, commonly referred to as plastics, and rubbers, and are major components of composite materials. Polymer chemistry can also be included in the broader fields of polymer science or even nanotechnology, both of which can be described as encompassing polymer physics and polymer engineering.[1][2][3][4]
Portion of backbone of nylon 6,6.
The work of Henri Braconnot in 1777 and the work of Christian Schönbein in 1846 led to the discovery of nitrocellulose, which, when treated with camphor, produced celluloid. Dissolved in ether or acetone, it becomes collodion, which has been used as a wound dressing since the U.S. Civil War. Cellulose acetate was first prepared in 1865. In years 1834-1844 the properties of rubber (polyisoprene) were found to be greatly improved by heating with sulfur, thus founding the vulcanization process.
In 1884 Hilaire de Chardonnet started the first artificial fiber plant based on regenerated cellulose, or viscose rayon, as a substitute for silk, but it was very flammable.[5] In 1907 Leo Baekeland invented the first polymer made independent of the products of organisms, a thermosetting phenol-formaldehyde resin called Bakelite. Around the same time, Hermann Leuchs reported the synthesis of amino acid N-carboxyanhydrides and their high molecular weight products upon reaction with nucleophiles, but stopped short of referring to these as polymers, possibly due to the strong views espoused by Emil Fischer, his direct supervisor, denying the possibility of any covalent molecule exceeding 6,000 daltons.[6] Cellophane was invented in 1908 by Jocques Brandenberger who treated sheets of viscose rayon with acid.[7]
Leading figures in polymer chemistry
Hermann Staudinger, father of polymer chemistry
Wallace Carothers, inventor of nylon.
Structures of some electrically conductive polymers: polyacetylene; polyphenylene vinylene; polypyrrole (X = NH) and polythiophene (X = S); and polyaniline (X = NH/N) and polyphenylene sulfide (X = S).
Structure of polydimethylsiloxane, illustrating a polymer with an inorganic backbone.
The chemist Hermann Staudinger first proposed that polymers consisted of long chains of atoms held together by covalent bonds, which he called macromolecules. His work expanded the chemical understanding of polymers and was followed by an expansion of the field of polymer chemistry during which such polymeric materials as neoprene, nylon and polyester were invented. Before Staudinger, polymers were thought to be clusters of small molecules (colloids), without definite molecular weights, held together by an unknown force. Staudinger received the Nobel Prize in Chemistry in 1953. Wallace Carothers invented the first synthetic rubber called neoprene in 1931, the first polyester, and went on to invent nylon, a true silk replacement, in 1935. Paul Flory was awarded the Nobel Prize in Chemistry in 1974 for his work on polymer random coil configurations in solution in the 1950s. Stephanie Kwolek developed an aramid, or aromatic nylon named Kevlar, patented in 1966. Karl Ziegler and Giulio Natta received a Nobel Prize for their discovery of catalysts for the polymerization of alkenes. Alan J. Heeger, Alan MacDiarmid, and Hideki Shirakawa were awarded the 2000 Nobel Prize in Chemistry for the development of polyacetylene and related conductive polymers.[8] Polyacetylene itself did not find practical applications, but organic light-emitting diodes (OLEDs) emerged as one application of conducting polymers.[9]
Teaching and research programs in polymer chemistry were introduced in the 1940s. An Institute for Macromolecular Chemistry was founded in 1940 in Freiburg, Germany under the direction of Staudinger. In America, a Polymer Research Institute (PRI) was established in 1941 by Herman Mark at the Polytechnic Institute of Brooklyn (now Polytechnic Institute of NYU).
Polymers and their propertiesEdit
The viscosity of polymer solutions is a valued parameter. Viscometers such as this are employed in such measurements.
Polymers are high molecular mass compounds formed by polymerization of monomers. The simple reactive molecule from which the repeating structural units of a polymer are derived is called a monomer. A polymer can be described in many ways: its degree of polymerisation, molar mass distribution, tacticity, copolymer distribution, the degree of branching, by its end-groups, crosslinks, crystallinity and thermal properties such as its glass transition temperature and melting temperature. Polymers in solution have special characteristics with respect to solubility, viscosity, and gelation. Illustrative of the quantitative aspects of polymer chemistry, particular attention is paid to the number-average and weight-average molecular weights
{\displaystyle M_{n}}
{\displaystyle M_{w}}
{\displaystyle M_{n}={\frac {\sum M_{i}N_{i}}{\sum N_{i}}},\quad M_{w}={\frac {\sum M_{i}^{2}N_{i}}{\sum M_{i}N_{i}}},\quad }
The formation and properties of polymers have been rationalized by many theories including Scheutjens–Fleer theory, Flory–Huggins solution theory, Cossee-Arlman mechanism, Polymer field theory, Hoffman Nucleation Theory, Flory-Stockmayer Theory, and many others.
Segments of polypropylene, showing the slightly different structures of isotactic (above) and syndiotactic (below) polymers.
The study of polymer thermodynamics helps improve the material properties of various polymer-based materials such as polystyrene (styrofoam) and polycarbonate. Common improvements include toughening, improving impact resistance, improving biodegradability, and altering a material’s solubility.[10]
As polymers get longer and their molecular weight increases, their viscosity tend to increase. Thus, the measured viscosity of polymers can provide valuable information about the average length of the polymer, the progress of reactions, and in what ways the polymer branches.[11]
Composites are formed by combining polymeric materials to form an overall structure with properties that differ from the sum of the individual components.
Polymers can be classified in many ways. Polymers, strictly speaking, comprise most solid matter: minerals (i.e. most of the earth's crust) are largely polymers, metals are 3-d polymers, organisms, living and dead, are composed largely of polymers and water. Often polymers are classified according to their origin: biopolymers vs synthetic polymers vs inorganic polymers.
A strand of cellulose showing the hydrogen bonds (dashed) within and between the chains.
Biopolymers are the structural and functional materials that comprise most of the organic matter in organisms. One major class of biopolymers are proteins, which are derived from amino acids. Polysaccharides, such as cellulose, chitin, and starch, are biopolymers derived from sugars. The polynucleic acids DNA and RNA are derived from phosphorylated sugars with pendant nucleotides that carry genetic information.
Synthetic polymers are the structural materials manifested in plastics, synthetic fibers, paints, building materials, furniture, mechanical parts, and adhesives. Synthetic polymers may be divided into thermoplastic polymers and thermoset plastics. Thermoplastic polymers include polyethylene, teflon, polystyrene, polypropylene, polyester, polyurethane, Poly(methyl methacrylate), polyvinyl chloride, nylons, and rayon. Thermoset plastics include vulcanized rubber, bakelite, Kevlar, and polyepoxide. Almost all synthetic polymers are derived from petrochemicals.
Self-RepairingEdit
Some polymers can be characterized as self-healing: designed to automatically repair themselves after damage. Thermodynamic principles have guided formation of these polymers. Subjecting polymers to specific heat and temperature regimes can overcome the activation energy of the forward polymerization reaction, encouraging the polymer to reform through the Diels-Alder reaction.[12]
Polymer chemistry and polymer thermodynamics is widely studied for sustainability and life cycle assessments (LCA). Polymer electrolyte membranes are considered important for sustainability because of their widespread us in fuel cells. The polymer electrolyte membrane (PEM) fuel cell is a hydrogen fuel cell that directly converts hydrogen and oxygen into electrical energy, water, and waste heat, without generating harmful gases emitted by conventional internal combustion engines. In general, PEM to sustain fuel cells have the advantages of compact structure, low operating temperature, fast startup speed, long working life and zero pollution.
Polymer topology
^ "The Macrogalleria: A Cyberwonderland of Polymer Fun". www.pslc.ws. Retrieved 2018-08-01.
^ Odian, George G. Principles of polymerization (Fourth ed.). Hoboken, N.J. ISBN 9780471478751. OCLC 54781987.
^ Hans-Heinrich Moretto, Manfred Schulze, Gebhard Wagner (2005) "Silicones" in Ullmann's Encyclopedia of Industrial Chemistry, Wiley-VCH, Weinheim. doi:10.1002/14356007.a24_057
^ "The Early Years of Artificial Fibres". The Plastics Historical Society. Retrieved 2011-09-05.
^ Kricheldorf, Hans, R. (2006), "Polypeptides and 100 Years of Chemistry of α-Amino Acid N-Carboxyanhydrides", Angewandte Chemie International Edition, 45 (35): 5752–5784, doi:10.1002/anie.200600693, PMID 16948174
^ "History of Cellophane". about.com. Retrieved 2011-09-05.
^ Friend, R. H.; Gymer, R. W.; Holmes, A. B.; Burroughes, J. H.; Marks, R. N.; Taliani, C.; Bradley, D. D. C.; Santos, D. A. Dos; Brdas, J. L.; Lgdlund, M.; Salaneck, W. R. (1999). "Electroluminescence in conjugated polymers". Nature. 397 (6715): 121–128. Bibcode:1999Natur.397..121F. doi:10.1038/16393. S2CID 4328634.
^ X Zhang, X Peng, SW Zhang. “7 - Synthetic biodegradable medical polymers: Polymer blends” Science and Principles of Biodegradable and Bioresorbable Medical Polymers, 2017. 217-254.
^ "Viscosity of Polymer Solutions". polymerdatabase.com. Retrieved 2019-03-05.
^ Zhong, N. & Post, W. . “Self-repair of structural and functional composites with intrinsically self-healing polymer matrices: A review”. Composites part A: applied science and manufacturing, 69, s. 226–239. 2015
Retrieved from "https://en.wikipedia.org/w/index.php?title=Polymer_chemistry&oldid=1082469507" |
Phosphoglucokinase - Wikipedia
In enzymology, a phosphoglucokinase (EC 2.7.1.10) is an enzyme that catalyzes the chemical reaction
ATP + alpha-D-glucose 1-phosphate
{\displaystyle \rightleftharpoons }
ADP + alpha-D-glucose 1,6-bisphosphate
Thus, the two substrates of this enzyme are ATP and alpha-D-glucose 1-phosphate, whereas its two products are ADP and alpha-D-glucose 1,6-bisphosphate.
This enzyme belongs to the family of transferases, specifically those transferring phosphorus-containing groups (phosphotransferases) with an alcohol group as acceptor. The systematic name of this enzyme class is ATP:alpha-D-glucose-1-phosphate 6-phosphotransferase. Other names in common use include glucose-phosphate kinase, phosphoglucokinase (phosphorylating), and ATP:D-glucose-1-phosphate 6-phosphotransferase. This enzyme participates in starch and sucrose metabolism.
Paladini AC, Caputto R (August 1949). "The enzymatic synthesis of glucose-1,6-diphosphate". Archives of Biochemistry. 23 (1): 55–66. PMID 18135764.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Phosphoglucokinase&oldid=996258028" |
Glycine transaminase - Wikipedia
In enzymology, a glycine transaminase (EC 2.6.1.4) is an enzyme that catalyzes the chemical reaction
glycine + 2-oxoglutarate
{\displaystyle \rightleftharpoons }
glyoxylate + L-glutamate
Thus, the two substrates of this enzyme are glycine and 2-oxoglutarate, whereas its two products are glyoxylate and L-glutamate.
This reactions strongly favours synthesis of glycine.[1] This enzyme belongs to the family of transferases, specifically the transaminases, which transfer nitrogenous groups. The systematic name of this enzyme class is glycine:2-oxoglutarate aminotransferase. Other names in common use include glutamic-glyoxylic transaminase, glycine aminotransferase, glyoxylate-glutamic transaminase, L-glutamate:glyoxylate aminotransferase, and glyoxylate-glutamate aminotransferase. This enzyme participates in glycine, serine and threonine metabolism. It employs one cofactor, pyridoxal phosphate.
^ Textbook of Biochemistry for Medical Students, by DM Vasudevan, Sreekumari S, Kannan Vaidyanathan, 9th edition, page 283.
Nakada HI (1964). "Glutamic-glycine transaminase from rat liver". J. Biol. Chem. 239: 468–471. PMID 14169146.
Thompson JS, Richardson KE (1966). "Isolation and characterization of a glutamate-glycine transaminase from human liver". Arch. Biochem. Biophys. 117 (3): 599–603. doi:10.1016/0003-9861(66)90101-9.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Glycine_transaminase&oldid=983912417" |
Antenna Diversity Analysis for 800 MHz MIMO - MATLAB & Simulink Example - MathWorks América Latina
Frequency Band Parameters
Create Two Identical Dipoles
Plot Input Reflection Coefficient of Isolated Dipole
Create Two-Element Array
Vary spatial orientation of dipole
Vary Spacing between Antennas
Check Correlation Frequency Response
This example analyzes a 2-antenna diversity scheme to understand the effect that position, orientation and frequency have on received signals. The analysis is performed under the assumptions that impedance matching is not achieved and mutual coupling is taken into account [1].
Define the operating frequency, analysis bandwidth and calculate the wavelength in free space.
BW_frac = .1;
fmin = freq - BW_frac*freq;
fmax = freq + BW_frac*freq;
Use the dipole antenna element from the Antenna Toolbox™ library and create 2 identical thin dipoles of length
\lambda /2
d1 = dipole('Length',lambda/2,'Width',lambda/200);
Calculate the input reflection coefficient of an isolated dipole and plot it to confirm the lack of impedance match at 800MHz.
Numfreq = 101;
f = linspace(fmin,fmax,Numfreq);
S = sparameters(d1,f);
DipoleS11Fig = figure;
title('Reflection Coefficient')
Create the two-element antenna diversity system and position the 2 antennas apart by 5
\lambda
range = 5*lambda;
l.Element = [d1 d2];
l.ElementSpacing = range;
Calculate and plot the power transfer function (S21 in dB) for two antennas. To do so, calculate the scattering parameters for the system and plot S21 over the entire frequency range.
S = sparameters(l,f);
ArrayS21Fig = figure;
title('Power Transfer Function')
The response peak is clearly not at 800 MHz. In addition, note the loss in signal strength due to attenuation in free space.
Power transfer between the two antennas can now be investigated as a function of antenna orientation. A correlation coefficient is used in MIMO systems to quantify the system performance. Two approaches to calculate the correlation coefficient exist; using the far-field behavior and using the S-parameters. The field-based approach involves numerical integration. The calculation suggested in this example uses the function correlation available in the Antenna Toolbox™ and based on the S-parameters approach [1]. By rotating one antenna located on the positive x-axis, we change its polarization direction and find the correlation
numpos = 101;
orientation = linspace(0,90,numpos);
S21_TiltdB = nan(1,numel(orientation));
Corr_TiltdB = nan(1,numel(orientation));
for i = 1:numel(orientation)
d2.Tilt = orientation(i);
l.Element(2) = d2;
S = sparameters(l,freq);
Corr = correlation(l,freq,1,2);
S21_TiltdB = 20*log10(abs(S.Parameters(2,1,1)));
Corr_TiltdB(i) = 20*log10(Corr);
plot(orientation,S21_TiltdB,orientation,Corr_TiltdB,'LineWidth',2)
axis([min(orientation) max(orientation) -65 -20]);
xlabel('Tilt variation on 2nd dipole (deg.)')
title('Correlation, S_2_1 Variation with Polarization')
legend('S_2_1','Correlation');
We observe that the power transfer function and the correlation function between the two antennas are identical as the antenna orientation changes for one of the dipoles.
Restore both dipoles so that they are parallel to each other. Run a similar analysis by changing the spacing between the 2 elements.
d2.Tilt = 0;
Nrange = 201;
Rmin = 0.001*lambda;
Rmax = 2.5*lambda;
range = linspace(Rmin,Rmax,Nrange);
S21_RangedB = nan(1,Nrange);
Corr_RangedB = nan(1,Nrange);
for i = 1:Nrange
l.ElementSpacing = range(i);
S21_RangedB(i)= 20*log10(abs(S.Parameters(2,1,1)));
Corr_RangedB(i)= 20*log10(Corr);
plot(range./lambda,S21_RangedB,range./lambda,Corr_RangedB,'--','LineWidth',2)
axis([min(range./lambda) max(range./lambda) -50 0]);
xlabel('Distance of separation, d/\lambda')
title('Correlation, S_2_1 Variation with Range')
The 2 curves are clearly different in their behavior when the separation distance between two antennas increases. This plot reveals the correlation valleys that exist at specific separations such as approximately 0.75
\lambda
, 1.25
\lambda
\lambda
, and 2.25
\lambda
Pick the separation to be 1.25
\lambda
, which is one of the correlation valleys. Analyze the correlation variation for 10% bandwidth centered at 800 MHz.
Rpick = 1.25*lambda;
l.ElementSpacing = Rpick;
Corr_PickdB = 20.*log10(correlation(l,f,1,2));
plot(f./1e9,Corr_TiltdB,'LineWidth',2)
axis([min(f./1e9) max(f./1e9) -65 0]);
title('Correlation Variation with Frequency')
The results of the analysis reveals that the two antennas have a correlation below 30 dB over the band specified.
[1] S. Blanch, J. Romeu, and I. Corbella, "Exact representation of antenna system diversity performance from input parameter description," Electron. Lett., vol. 39, pp. 705-707, May 2003. Online at: http://upcommons.upc.edu/e-prints/bitstream/2117/10272/4/ExactRepresentationAntenna.pdf |
Home : Support : Online Help : Connectivity : Web Features : Worksheet Package : Convert
Convert(worksheet, format=output_format)
Maple XML data structure; valid Maple worksheet
The Convert command converts a worksheet given as a Maple XML data structure to the specified format. The two output formats supported are maple8_xml and mw. The result is converted to an XML data structure that can be used with the XMLTools package. The XML document structure returned by this procedure is of type Worksheet:-worksheet.
\mathrm{with}\left(\mathrm{Worksheet}\right):
\mathrm{dir}≔\mathrm{kernelopts}\left(\mathrm{mapledir}\right):
\mathrm{doc}≔\mathrm{ReadFile}\left(\mathrm{cat}\left(\mathrm{dir},"/examplesclassic/obj.mws"\right),\mathrm{format}="maple8_xml"\right):
\mathrm{mws}≔\mathrm{Convert}\left(\mathrm{doc},\mathrm{format}="mw"\right):
Worksheet[ReadFile] |
The Split Common Fixed Point Problem for ϱ -Strictly Pseudononspreading Mappings
2013 The Split Common Fixed Point Problem for
\varrho
-Strictly Pseudononspreading Mappings
Shubo Cao
We introduce and analyze the viscosity approximation algorithm for solving the split common fixed point problem for the strictly pseudononspreading mappings in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related results.
Shubo Cao. "The Split Common Fixed Point Problem for
\varrho
-Strictly Pseudononspreading Mappings." J. Appl. Math. 2013 (SI11) 1 - 9, 2013. https://doi.org/10.1155/2013/241789
Shubo Cao "The Split Common Fixed Point Problem for
\varrho
-Strictly Pseudononspreading Mappings," Journal of Applied Mathematics, J. Appl. Math. 2013(SI11), 1-9, (2013) |
H. Wilcox, D. Bacon, R. C. Nichol, P. J. Rooney, A. Terukina, A. K. Romer, K. Koyama, G.-B. Zhao, R. Hood, R. Mann, M. Hilton, M. Manolopoulou, M. Sahlén, C. A. Collins, A. R. Liddle, J. A. Mayers, N. Mehrtens, C. J. Miller, J. P. Stott, P. T. P. Viana
The chameleon gravity model postulates the existence of a scalar field that couples with matter to mediate a fifth force. If it exists, this fifth force would influence the hot X-ray emitting gas filling the potential wells of galaxy clusters. However, it would not influence the clusters weak lensing signal. Therefore, by comparing X-ray and weak lensing profiles, one can place upper limits on the strength of a fifth force. This technique has been attempted before using a single, nearby cluster (Coma, z = 0.02). Here we apply the technique to the stacked profiles of 58 clusters at higher redshifts (0.1 < z < 1.2), including 12 new to the literature, using X-ray data from the XMM Cluster Survey and weak lensing data from the Canada–France–Hawaii–Telescope Lensing Survey. Using a multiparameter Markov chain Monte Carlo analysis, we constrain the two chameleon gravity parameters (β and ϕ∞). Our fits are consistent with general relativity, not requiring a fifth force. In the special case of f(R) gravity (where
\beta =\sqrt{1/6}
), we set an upper limit on the background field amplitude today of |fR0| < 6 × 10−5 (95 per cent CL). This is one of the strongest constraints to date on |fR0| on cosmological scales. We hope to improve this constraint in future by extending the study to hundreds of clusters using data from the Dark Energy Survey.
gravitation - gravitational lensing: weak - X-rays: galaxies: clusters |
linalg(deprecated)/stackmatrix - Maple Help
Home : Support : Online Help : linalg(deprecated)/stackmatrix
join two or more matrices together vertically
stackmatrix(A, B,...)
Important: The linalg package has been deprecated. Use the superseding command LinearAlgebra[Matrix], instead.
The function stackmatrix joins two or more matrices or vectors together vertically, where a vector is interpreted as a row vector. The matrices and vectors must have the same number of columns.
The command with(linalg,stackmatrix) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{linalg}\right):
a≔\mathrm{matrix}\left(2,2,[1,2,3,4]\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\end{array}]
b≔\mathrm{matrix}\left(2,2,[5,6,7,8]\right)
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{8}\end{array}]
\mathrm{stackmatrix}\left(a,b\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{8}\end{array}]
v≔\mathrm{vector}\left(2,[1,2]\right)
\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
\mathrm{stackmatrix}\left(v,b,v\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{6}\\ \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{8}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
linalg(deprecated)[augment]
linalg(deprecated)[extend] |
GreatestFactorialFactorization - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Polynomials : PolynomialTools : GreatestFactorialFactorization
compute a greatest factorial factorization of a univariate polynomial
GreatestFactorialFactorization(f,x)
The GreatestFactorialFactorization command computes a greatest factorial factorization
[c,[[\mathrm{g1},\mathrm{e1}],[\mathrm{g2},\mathrm{e2}],...]]
of f w.r.t. x. It satisfies the following properties.
f=c\mathrm{g1}\left(x\right)\mathrm{g1}\left(x-1\right)...\mathrm{g1}\left(x-\mathrm{e1}+1\right)\mathrm{g2}\left(x\right)\mathrm{g2}\left(x-1\right)...\mathrm{g2}\left(x-\mathrm{e2}+1\right)...
\mathrm{gcd}\left(\mathrm{gi}\left(x\right)\mathrm{gi}\left(x-1\right)...\mathrm{gi}\left(x-\mathrm{e1}+1\right),\mathrm{gj}\left(x+1\right)\mathrm{gj}\left(x+\mathrm{ej}\right)\right)=1
1<=i<=j
c
\mathrm{g1},...
are nonconstant primitive polynomials w.r.t. x, and
0<\mathrm{e1}<\mathrm{e2}<\mathrm{...}
The greatest factorial factorization is unique up to multiplication by units.
GreatestFactorialFactorization can handle the same types of coefficients as the Maple function gcd.
[f,[]]
\mathrm{with}\left(\mathrm{PolynomialTools}\right):
\mathrm{GreatestFactorialFactorization}\left(-{x}^{8}+{x}^{2},x\right)
[\textcolor[rgb]{0,0,1}{-1}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]]
\mathrm{GreatestFactorialFactorization}\left(\mathrm{expand}\left(\mathrm{pochhammer}\left(x,3\right)\mathrm{pochhammer}\left(x,5\right)\right),x\right)
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]]]
Paule, Peter. "Greatest factorial factorization and symbolic summation." Journal of Symbolic Computation Vol. 20, (1995): 235-268.
Gerhard, Juergen. "Modular algorithms for polynomial basis conversion and greatest factorial factorization." Proceedings of the Seventh Rhine Workshop on Computer Algebra, RWCA pp. 125-141 ed. T. Mulders, 2000. |
A person is suffering from myopic defect. he is able to see cle-Turito
Answer:The correct answer is: Concave lens of 20 cm focal lengthFor viewing far objects, concave lenses are used and for concave lens
{\int }_{-\pi /4}^{\pi /4} \frac{{e}^{x}\left(x\mathrm{sin}x\right)}{{e}^{2x}-1}dx
\frac{l}{3}=\frac{m}{-3}=\frac{n}{-9}
\frac{l}{-1}=\frac{m}{+1}=\frac{n}{3}
\frac{x-2}{-1}=\frac{y+1}{1}=\frac{z+1}{3}
{\int }_{-\pi /4}^{\pi /4} \frac{{e}^{x}\left(x\mathrm{sin}x\right)}{{e}^{2x}-1}dx
\frac{l}{3}=\frac{m}{-3}=\frac{n}{-9}
\frac{l}{-1}=\frac{m}{+1}=\frac{n}{3}
\frac{x-2}{-1}=\frac{y+1}{1}=\frac{z+1}{3}
f:R\to R,f\left(x\right)=\left\{\begin{array}{c}|x-\left[x\right]|,\left[x\right]\\ |x-\left[x+1\right]|,\left[x\right]\end{array}\right\
\begin{array}{r}\text{ is odd }\\ 1\text{ is even where [.] }\end{array}
{\int }_{-2}^{4} f\left(x\right)dx
f:R\to R,f\left(x\right)=\left\{\begin{array}{c}|x-\left[x\right]|,\left[x\right]\\ |x-\left[x+1\right]|,\left[x\right]\end{array}\right\
\begin{array}{r}\text{ is odd }\\ 1\text{ is even where [.] }\end{array}
{\int }_{-2}^{4} f\left(x\right)dx
{\int }_{0}^{1} |\mathrm{sin} 2\pi x|\mid dx
{\int }_{0}^{1} |\mathrm{sin} 2\pi x|\mid dx
{\int }_{0}^{100} \left\{\sqrt{x}\right\}dx
{\int }_{0}^{100} \left\{\sqrt{x}\right\}dx
\frac{x-4/3}{2}=\frac{y+6/5}{3}=\frac{z-3/2}{4}
\frac{5y+6}{8}=\frac{2z-3}{9}=\frac{3x-4}{5}
\frac{x-4/3}{2}=\frac{y+6/5}{3}=\frac{z-3/2}{4}
\frac{5y+6}{8}=\frac{2z-3}{9}=\frac{3x-4}{5} |
Are the following expressions equivalent? Explain how you know. Try to use at least two different methods to justify your answer.
(x+2)^{1/2}+(x+2)^{3/2}
(x+3)(x+2)^{1/2}
Try a few numbers. Do you get the same value for both expressions if:
x=2
x=7
x=−1
Checking a few numbers does not prove that the expressions are equivalent.
Can you algebraically rewrite one expression so that it is the same as the other expression? |
Aconitate decarboxylase - Wikipedia
In enzymology, an aconitate decarboxylase (EC 4.1.1.6) is an enzyme that catalyzes the chemical reaction
{\displaystyle \rightleftharpoons }
itaconate + CO2
Hence, this enzyme has one substrate, cis-aconitate, and two products, itaconate and CO2.
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is cis-aconitate carboxy-lyase (itaconate-forming). Other names in common use include cis-aconitic decarboxylase, CAD, cis-aconitate carboxy-lyase, and cis-aconitate carboxy-lyase. This enzyme participates in c5-branched dibasic acid metabolism.
BENTLEY R, THIESSEN CP (1957). "Biosynthesis of itaconic acid in Aspergillus terreus. III. The properties and reaction mechanism of cis-aconitic acid decarboxylase". J. Biol. Chem. 226 (2): 703–20. PMID 13438855.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Aconitate_decarboxylase&oldid=917333323" |
Ingress of hot gas through the rim seals of gas turbines can be modeled theoretically using the so-called orifice equations. In Part I of this two-part paper, the orifice equations were derived for compressible and incompressible swirling flows, and the incompressible equations were solved for axisymmetric rotationally induced (RI) ingress. In Part II, the incompressible equations are solved for nonaxisymmetric externally induced (EI) ingress and for combined EI and RI ingress. The solutions show how the nondimensional ingress and egress flow rates vary with
Θ0
, the ratio of the flow rate of sealing air to the flow rate necessary to prevent ingress. For EI ingress, a “saw-tooth model” is used for the circumferential variation of pressure in the external annulus, and it is shown that
ε
Θ0
; the theoretical variation of
ε
Θ0
is similar to that found in Part I for RI ingress. For combined ingress, the solution of the orifice equations shows the transition from RI to EI ingress as the amplitude of the circumferential variation of pressure increases. The predicted values of
ε
for EI ingress are in good agreement with the available experimental data, but there are insufficient published data to validate the theory for combined ingress. |
Depositional impact on the elastic characteristics of the organic shale reservoir and its seismic application: A case study of the Longmaxi-Wufeng Shale in the Fuling gas field, Sichuan BasinDepositional elastic effects | Geophysics | GeoScienceWorld
. E-mail: [email protected]; [email protected]; [email protected].
. E-mail: [email protected].
Xiwu Liu;
Xiwu Liu
. E-mail: [email protected]; [email protected]; [email protected].
Jinqiang Zhang;
Yuwei Liu;
Kejian Li;
Luanxiao Zhao, Yang Wang, Xiwu Liu, Jinqiang Zhang, Yuwei Liu, Xuan Qin, Kejian Li, Jianhua Geng; Depositional impact on the elastic characteristics of the organic shale reservoir and its seismic application: A case study of the Longmaxi-Wufeng Shale in the Fuling gas field, Sichuan Basin. Geophysics 2020;; 85 (2): B23–B33. doi: https://doi.org/10.1190/geo2019-0326.1
Seismic characterization of the depositional evolution history of the organic shale reservoir is essential for reservoir quality evaluation and geologic model building in unconventional plays. However, a direct link between the depositional environment and seismic elastic responses in organic-rich shales remains unclear. By combining the depositional history and rock-physics analysis, we have determined how the depositional environment affects the elastic characteristics of the Longmaxi-Wufeng Shale in the Fuling gas field, Sichuan Basin, Southwestern China. Sedimentological control on the elastic properties mainly lies in two aspects: First, the distinct elastic features of the overlying turbidity mudstone and the underlying deepwater shelf Longmaxi Formation are primarily caused by the rock structure difference due to water energy; second, within the deepwater shelf siliceous shale formation, the elastic property variations are primarily controlled by the progradation tract system and water depths. We evaluate the effect of two types of quartz (biogenic quartz and detrital quartz) in conjunction with organic matter on the elasticity of organic shale. Furthermore, we determine that the two most commonly used seismic inversion attributes, P-impedances and the
VP/VS
ratio, can be used to indicate the depositional facies evolution. This also gives insights into using geophysical attributes to directly characterize depositional facies for unconventional shale reservoirs.
Fuling Field
Evolution and migration of shale facies and their control on shale gas: A case study from the Wufeng-Longmaxi Formations in the Sichuan Basin and its surroundings
Quantitative prediction of total organic carbon content in shale-gas reservoirs using seismic data: A case study from the Lower Silurian Longmaxi Formation in the Chang Ning gas field of the Sichuan Basin, China |
Succinyl-CoA hydrolase - Wikipedia
In enzymology, a succinyl-CoA hydrolase (EC 3.1.2.3) is an enzyme that catalyzes the chemical reaction
succinyl-CoA + H2O
{\displaystyle \rightleftharpoons }
CoA + succinate
Thus, the two substrates of this enzyme are succinyl-CoA and H2O, whereas its two products are CoA and succinate.
This enzyme belongs to the family of hydrolases, specifically those acting on thioester bonds. The systematic name of this enzyme class is succinyl-CoA hydrolase. Other names in common use include succinyl-CoA acylase, succinyl coenzyme A hydrolase, and succinyl coenzyme A deacylase. This enzyme participates in citrate cycle (tca cycle).
Gergely J, Hele P, Ramakrishnan CV (1952). "Succinyl and acetyl coenzyme A deacylases". J. Biol. Chem. 198 (1): 323–334. PMID 12999747.
Thioesterases (EC 3.1.2)
Acetyl-CoA thioesterases
Acyl-CoA thioesterases
Formyl-CoA thioesterases
Palmitoyl protein thioesterases
Succinyl-CoA thioesterases
Ubiquitin C-terminal hydrolases
Retrieved from "https://en.wikipedia.org/w/index.php?title=Succinyl-CoA_hydrolase&oldid=918617972" |
3.3 Output File Format (Structural Segmentation)
3.4 Algorithm Calling Format
The aim of the MIREX structural segmentation evaluation is to identify the key structural sections in musical audio. The segment structure (or form) is one of the most important musical parameters. It is furthermore special because musical structure -- especially in popular music genres (e.g. verse, chorus, etc.) -- is accessible to everybody: it needs no particular musical knowledge. This task was first run in 2009.
The MIREX 2009 Collection: 297 pieces, most of it derived from the work of the Beatles.
MIREX 2010 RWC collection. 100 pieces of popular music. There are two ground truths. The first is the one originally included with the RWC dataset. The explanation of the second set of annotations can be found at http://hal.inria.fr/docs/00/47/34/79/PDF/PI-1948.pdf. The second set of annotations contains no labels for segments, but rather provides an annotation of segment boundaries.
MIREX 2012 dataset. The new data set contains over 1,000 annotated pieces covering a range of musical styles. The majority of the pieces have been annotated by two independent annotators.
Submissions to this task will have to conform to a specified format detailed below. Submissions should be packaged and contain at least two files: The algorithm itself and a README containing contact information and detailing, in full, the use of the algorithm.
The structural segmentation algorithms will return the segmentation in an ASCII text file for each input .wav audio file. The specification of this output file is immediately below.
Output File Format (Structural Segmentation)
The Structural Segmentation output file format is a tab-delimited ASCII text format. This is the same as Chris Harte's chord labelling files (.lab), and so is the same format as the ground truth as well. Onset and offset times are given in seconds, and the labels are simply letters: 'A', 'B', ... with segments referring to the same structural element having the same label.
<onset_time(sec)>\t<offset_time(sec)>\t<label>\n
where \t denotes a tab, \n denotes the end of line. The < and > characters are not included. An example output file would look something like:
0.000 5.223 A
5.223 15.101 B
15.101 20.334 A
Algorithm Calling Format
The submitted algorithm must take as arguments a SINGLE .wav file to perform the structural segmentation on as well as the full output path and filename of the output file. The ability to specify the output path and file name is essential. Denoting the input .wav file path and name as %input and the output file path and name as %output, a program called foobar could be called from the command-line as follows:
foobar %input %output
foobar -i %input -o %output
Moreover, if your submission takes additional parameters, foobar could be called like:
foobar .1 %input %output
foobar -param1 .1 -i %input -o %output
If your submission is in MATLAB, it should be submitted as a function. Once again, the function must contain String inputs for the full path and names of the input and output files. Parameters could also be specified as input arguments of the function. For example:
foobar('%input','%output')
foobar(.1,'%input','%output')
A README file accompanying each submission should contain explicit instructions on how to to run the program (as well as contact information, etc.). In particular, each command line to run should be specified, using %input for the input sound file and %output for the resulting text file.
For instance, to test the program foobar with a specific value for parameter param1, the README file would look like:
matlab -r "foobar(.1,'%input','%output');quit;"
Both the result and the ground truth are handled in short frames (e.g., beat or fixed 100ms). All frame pairs in a structure description are handled. The pairs in which both frames are assigned to the same cluster (i.e., have the same label) form the sets
{\displaystyle P_{E}}
(for the system result) and
{\displaystyle P_{A}}
(for the ground truth). The pairwise precision rate can be calculated by
{\displaystyle P={\frac {|P_{E}\cap P_{A}|}{|P_{E}|}}}
, pairwise recall rate by
{\displaystyle R={\frac {|P_{E}\cap P_{A}|}{|P_{A}|}}}
, and pairwise F-measure by
{\displaystyle F={\frac {2PR}{P+R}}}
. (Levy & Sandler TASLP2008)
{\displaystyle p_{i,j}=C_{i,j}/F}
{\displaystyle p_{i}^{a}=\sum _{j=1}^{|L_{E}|}C{i,j}/F}
{\displaystyle p_{j}^{e}=\sum _{i=1}^{|L_{A}|}C{i,j}/F}
{\displaystyle p_{i,j}^{a|e}=C_{i,j}/\sum _{i=1}^{|L_{A}|}C{i,j}}
{\displaystyle p_{i,j}^{e|a}=C_{i,j}/\sum _{j=1}^{|L_{E}|}C{i,j}}
{\displaystyle H(E|A)=-\sum _{i=1}^{|L_{A}|}p_{i}^{a}\sum _{j=1}^{|L_{E}|}p_{i,j}^{e|a}\log _{2}(p_{i,j}^{e|a})}
{\displaystyle H(A|E)=-\sum _{j=1}^{|L_{E}|}p_{j}^{e}\sum _{i=1}^{|L_{A}|}p_{i,j}^{a|e}\log _{2}(p_{i,j}^{a|e})}
{\displaystyle S_{O}=1-{\frac {H(E|A)}{\log _{2}(|L_{E}|)}}}
, and the undersegmentation score
{\displaystyle S_{U}=1-{\frac {H(A|E)}{\log _{2}(|L_{A}|)}}}
These public corpora give a combined 220 songs.
A hard limit of 24 hours will be imposed on analysis times. Submissions exceeding this limit may not receive a result.
Ju-Chiang Wang / [email protected]
Axel Marmoret / [email protected]
Retrieved from "https://www.music-ir.org/mirex/w/index.php?title=2021:Structural_Segmentation&oldid=13546" |
Starch synthase - Wikipedia
Crystal structure of glycogen synthase 1 from Agrobacterium tumefaciens
In enzymology, a starch synthase (EC 2.4.1.21) is an enzyme that catalyzes the chemical reaction
ADP-glucose + (1,4-alpha-D-glucosyl)n
{\displaystyle \rightleftharpoons }
ADP + (1,4-alpha-D-glucosyl)n+1
Thus, the two substrates of this enzyme are ADP-glucose and a chain of D-glucose residues joined by 1,4-alpha-glycosidic bonds, whereas its two products are ADP and an elongated chain of glucose residues. Plants use these enzymes in the biosynthesis of starch.
This enzyme belongs to the family of hexosyltransferases, specifically the glycosyltransferases. The systematic name of this enzyme class is ADP-glucose:1,4-alpha-D-glucan 4-alpha-D-glucosyltransferase. Other names in common use include ADP-glucose-starch glucosyltransferase, adenosine diphosphate glucose-starch glucosyltransferase, adenosine diphosphoglucose-starch glucosyltransferase, ADP-glucose starch synthase, ADP-glucose synthase, ADP-glucose transglucosylase, ADP-glucose-starch glucosyltransferase, ADPG starch synthetase, and ADPG-starch glucosyltransferase
Five isoforms seems to be present. GBSS which is linked to amylose synthesis. The others are SS1, SS2, SS3 and SS4. These have different roles in amylopectin synthesis. New work implies that SS4 is important for granule initiation. (Szydlowski et al., 2011)
As of late 2007, 4 structures have been solved for this class of enzymes, with PDB accession codes 1RZU, 1RZV, 2BFW, and 2BIS.
Chambers J, Elbein AD (1970). "Biosynthesis of glucans in mung bean seedlings. Formation of beta-(1,4)-glucans from GDP-glucose and beta-(1,3)-glucans from UDP-glucose". Arch. Biochem. Biophys. 138 (2): 620–31. doi:10.1016/0003-9861(70)90389-9. PMID 4317490.
FRYDMAN RB, CARDINI CE (1965). "Studies on adenosine diphosphate d-glucose: α-1,4-glucan α-4-glucosyltransferase of sweet-corn endosperm". Biochim. Biophys. Acta. 96 (2): 294–303. doi:10.1016/0926-6593(65)90013-5. PMID 14298833.
Greenberg E; Preiss J (1965). "Biosynthesis of bacterial glycogen. II. Purification and properties of the adenosine diphosphoglucose:glycogen transglucosylase of arthrobacter species NRRL B1973". J. Biol. Chem. 240: 2341–2348. PMID 14304835.
Leloir LF, de Fekete MA, Cardini CE (1961). "Starch and oligosaccharide synthesis from uridine diphosphate glucose". J. Biol. Chem. 236: 636–641. PMID 13760681.
Whelan, W.J. and Schultz, J. (Eds.), Miami Winter Symposia, vol. 1, North Holland, Utrecht, 1970, p. 122-138.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Starch_synthase&oldid=1049332593" |
Planetary gear set of carrier, worm planet, and sun wheels with adjustable gear ratio, worm thread type, and friction losses - MATLAB - MathWorks í•œêµ
Sun-Planet Worm Gear
Friction parameterization
Worm-carrier and sun-carrier viscous friction coefficients
Planetary gear set of carrier, worm planet, and sun wheels with adjustable gear ratio, worm thread type, and friction losses
The Sun-Planet Worm Gear block represents a two-degree-of-freedom planetary gear built from a carrier and sun and planet gears. By type, the sun and planet gears are crossed helical spur gears arranged as a worm drive, in which the planet gear is the worm. Such transmissions are used in the Torsen® T-1 differential. When transmitting power, the sun gear can be independently rotated by the worm (planet) gear, by the carrier, or by both.
You specify a fixed gear ratio, which signifies the worm angular velocity divided by the sun gear angular velocity. You control the direction by setting the worm thread type to left-hand or right-hand. Rotation of the right-hand worm in the positive direction causes the sun gear to rotate in the positive direction. The positive directions of the sun gear and the carrier are the same.
Equation variables are:
RWG Gear ratio that signifies the worm angular velocity divided by the sun gear angular velocity: The ratio is positive for the right-hand worm and negative for the left-hand worm
ωS Angular velocity of the sun gear
ωP Angular velocity of the worm gear
ωC Angular velocity of the carrier
ωSC Angular velocity of the sun gear with respect to the carrier
α Normal pressure angle
λ Worm lead angle
Ï„S Torque applied to sun shaft
Ï„P Torque applied to planet shaft
Ï„C Torque applied to carrier shaft
τ Torque due to meshing friction: The loss depends on the device efficiency and the power flow direction. To avoid an abrupt change of the friction torque at ωS = 0, the friction torque is introduced via the hyperbolic function.
Ï„instfr Instantaneous value of the friction torque used to simulate friction losses
Ï„fr Torque due to friction in steady-state
ηWG Efficiency of worm to gear power transfer
ηGW Efficiency of gear to worm power transfer
μSC Viscous friction coefficient for the sun-carrier interface
μWC Viscous friction coefficient for the worm-carrier interface
Ideal Gear Constraints and Gear Ratio
The Sun-Planet Worm Gear block imposes one kinematic constraint on the three connected axes:
{\mathrm{Ï}}_{\text{S}}=\frac{{\mathrm{Ï}}_{\text{P}}}{{R}_{\text{WG}}}+{\mathrm{Ï}}_{\text{C}}.
The gear has two independent degrees of freedom. The gear pair is (1,2) = (S,P).
{R}_{\text{WG}}{\mathrm{Ï}}_{\text{P}}+{\mathrm{Ï}}_{\text{S}}–{\mathrm{Ï}}_{\text{loss}}=\text{ }0
{\mathrm{Ï}}_{\text{C}}=\text{ }–{\mathrm{Ï}}_{\text{S}}
Nonideal Gear Constraints
In a nonideal gear, the angular velocity and geometric constraints are unchanged, but the transferred torque and power are reduced by:
Coulomb friction due to worm-sun gear meshing, which is characterized by the friction coefficient k or constant efficiencies [ηWG, ηGW]
Viscous couplings of driveshafts with bearings, which are parametrized by viscous friction coefficients μSC and μWC
Because the transmission incorporates a worm gear, the efficiencies are different for the direct and reverse power transfer. The table shows the value of the efficiency for all combinations of the power transfer.
Driving Shaft Driven Shaft
Planet Sun Carrier
Planet N/A ηWG ηWG
Sun ηGW N/A No loss
Carrier ηGW No loss N/A
When you set Friction model to Constant efficiency and leave Friction parameterization set to Friction coefficient and geometrical parameters, the model considers geometric surface contact friction. In this case, ηWG and ηGW rely on:
The worm-gear threading geometry, specified by lead angle λ and normal pressure angle α.
{\mathrm{η}}_{\text{WG}}=\text{ }\frac{\left(\text{cos}\mathrm{α}–k·\text{tan}\mathrm{λ}\right)}{\left(\text{cos}\mathrm{α}+\frac{k}{\text{tan}\mathrm{λ}}\right)}
{\mathrm{η}}_{\text{GW}}=\text{ }\frac{\left(\text{cos}\mathrm{α}–\frac{k}{\text{tan}\mathrm{λ}}\right)}{\left(\text{cos}\mathrm{α}+k·\text{tan}\mathrm{α}\right)}
When you set Friction model to Constant efficiency and set Friction parameterization to Efficiencies, or when you set Friction model to Temperature-dependent efficiency, the model treats the efficiencies as constant. In this case, you specify ηWG and ηGW independently of geometric details.
You can enable self-locking behavior by making the efficiency negative. Power cannot be transmitted from the sun gear to the worm or from the carrier to the worm unless some torque is applied to the worm to release the train. In this case, the absolute value of the efficiency specifies the ratio at which the train is released. The smaller the train lead angle, the smaller the reverse efficiency.
The efficiencies, η, of meshing between the worm gear and planet gear are fully active only if the transmitted power is greater than the power threshold.
The viscous friction coefficients of the worm-carrier and sun-carrier bearings control the viscous friction torque experienced by the carrier from lubricated, nonideal gear threads. For details, see Nonideal Gear Constraints.
W — Worm gear
Rotational mechanical conserving port associated with the worm gear.
Gear ratio — Gear ratio
Gear or transmission ratio, RWG, which signifies the worm gear angular velocity divided by the sun gear angular velocity. This gear ratio must be strictly positive.
Worm thread type — Thread rotation direction
Direction of positive worm rotation. If you select Left-hand, positive worm rotation results in negative gear rotation.
The table shows how the options that you choose for Friction model affect the visibility of other parameters in the Meshing Losses tab. To learn how to read the table, see Parameter Dependencies.
Meshing Losses Parameter Dependencies
Meshing Losses Setting Parameters and Values
No meshing losses - Suitable for HIL simulation Constant efficiency Temperature-dependent efficiency
Friction parameterization Temperature
Friction coefficient and geometrical parameters Efficiencies
Normal pressure angle Worm-gear efficiency Worm-gear efficiency
Lead angle Gear-worm efficiency Gear-worm efficiency
Power threshold Power threshold Power threshold
Friction parameterization — Friction parameterization method
Characterization of the friction between gear threads:
Friction coefficient and geometrical parameters — Friction is determined by geometric surface contact friction.
Efficiencies — Friction is determined by constant efficiencies, where 0 < η < 1.
17.5 deg (default) | scalar
Thread pressure angle, α, in the normal plane. The value must be greater than zero and less than 90 degrees.
Thread helix angle, where λ = arctan[L/(πd)]. L is the worm lead and d is the worm pitch diameter. This value must be greater than zero.
Friction coefficient — Thread friction
Dimensionless coefficient of normal friction in the thread. Must be greater than zero.
Worm-gear efficiency — Torque transfer efficiency from the worm gear to the sun gear
[.75, .65, .6] or 0.74 (default) | vector or scalar
Vector of output-to-input power ratios that describe the power flow from the worm gear to the sun gear, ηWG. When you set Friction model to Constant efficiency, specify the value as a scalar. When you set Friction model to Temperature-dependent efficiency, specify the value as a vector. The block uses the vector values to construct a 1-D temperature-efficiency lookup table.
Friction model to Constant efficiency and Friction parameterization to Efficiencies — In this case, specify the value as a scalar.
Friction model to Temperature-dependent efficiency — In this case, specify the value as a vector.
Gear-worm efficiency — Torque transfer efficiency from the sun gear to the worm gear
[.5, .45, .4] or 0.65 (default) | vector or scalar
Vector of output-to-input power ratios that describe the power flow from the sun gear to the worm gear, ηGW. When you set Friction model to Constant efficiency, specify the value as a scalar. When you set Friction model to Temperature-dependent efficiency, specify the value as a vector. The block uses the vector values to construct a 1-D temperature-efficiency lookup table.
Power threshold — Minimum efficiency power threshold
When you set Friction model to Constant efficiency, the block lowers the efficiency losses to zero when no power is transmitted. When you set Friction model to Temperature-dependent efficiency, the function smooths the efficiency factors between zero when at rest and the values provided by the temperature-efficiency lookup tables at the power thresholds.
Worm-carrier and sun-carrier viscous friction coefficients — Gear viscous friction
Vector of viscous friction coefficients [μWC μSC], for the worm-carrier and sun-carrier shafts, respectively.
Leadscrew | Sun-Planet | Sun-Planet Bevel | Worm Gear |
$\Gamma$--convergence of the Allen-Cahn energy with an oscillating forcing term | EMS Press
\Gamma
--convergence of the Allen-Cahn energy with an oscillating forcing term
N. Dirr
We consider a standard functional in the mesoscopic theory of phase transitions, consisting of a gradient term with a double-well potential, and we add to it a bulk term modelling the interaction with a periodic mean zero external field. This field is amplified and dilated with a power of the transition layer thickness
\eps
leading to a nontrivial interaction of forcing and concentration when
\eps \to 0
. We show that the functionals
\Gamma
-converge after additive renormalization to an anisotropic surface energy, if the period of the oscillation is larger than the interface thickness. Difficulties arise from the fact that the functionals have non constant absolute minimizers and are not uniformly bounded from below.
Matteo Novaga, N. Dirr, M. Lucia,
\Gamma
--convergence of the Allen-Cahn energy with an oscillating forcing term. Interfaces Free Bound. 8 (2006), no. 1, pp. 47–78 |
Gluconokinase - Wikipedia
Gluconokinase homodimer, E.Coli
In enzymology, a gluconokinase (EC 2.7.1.12) is an enzyme that catalyzes the chemical reaction
ATP + D-gluconate
{\displaystyle \rightleftharpoons }
ADP + 6-phospho-D-gluconate
Thus, the two substrates of this enzyme are ATP and D-gluconate, whereas its two products are ADP and 6-phospho-D-gluconate.
This enzyme belongs to the family of transferases, specifically those transferring phosphorus-containing groups (phosphotransferases) with an alcohol group as acceptor. The systematic name of this enzyme class is ATP:D-gluconate 6-phosphotransferase. Other names in common use include gluconokinase (phosphorylating), and gluconate kinase. This enzyme participates in pentose phosphate pathway.
As of late 2007, 6 structures have been solved for this class of enzymes, with PDB accession codes 1KNQ, 1KO1, 1KO4, 1KO5, 1KO8, and 1KOF.
COHEN SS (1951). "Gluconokinase and the oxidative path of glucose-6-phosphate utilization". J. Biol. Chem. 189 (2): 617–28. PMID 14832279.
LEDER IG (1957). "Hog kidney gluconokinase". J. Biol. Chem. 225 (1): 125–36. PMID 13416223.
Narrod SA; Wood WA (1956). "Carbohydrate oxidation by Pseudomonas fluorescens. V. Evidence for gluconokinase and 2-ketogluconokinase". J. Biol. Chem. 220 (1): 45–55. PMID 13319325.
SABLE HZ, GUARINO AJ (1952). "Phosphorylation of gluconate in yeast extracts". J. Biol. Chem. 196 (1): 395–402. PMID 12980980.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Gluconokinase&oldid=1049565274" |
Cross Wigner-Ville distribution and cross smoothed pseudo Wigner-Ville distribution - MATLAB xwvd - MathWorks í•œêµ
Cross Wigner-Ville Distribution of Signals
Cross Wigner-Ville Distribution of Chirps
Use Cross Wigner-Ville Distribution to Estimate Instantaneous Frequency
Cross Wigner-Ville Distribution
d = xwvd(x,y) returns the cross Wigner-Ville distribution of x and y.
d = xwvd(x,y,fs) returns the cross Wigner-Ville distribution when x and y are sampled at a rate fs.
d = xwvd(x,y,ts) returns the cross Wigner-Ville distribution when x and y are sampled with a time interval ts between samples.
d = xwvd(___,'smoothedPseudo') returns the cross smoothed pseudo Wigner-Ville distribution of x and y. The function uses the length of the input signals to choose the lengths of the windows used for time and frequency smoothing. This syntax can include any combination of input arguments from previous syntaxes.
d = xwvd(___,'smoothedPseudo',twin,fwin) specifies the time window, twin, and the frequency window, fwin, used for smoothing. To use the default window for either time or frequency smoothing, specify the corresponding argument as empty, [].
d = xwvd(___,'smoothedPseudo','NumFrequencyPoints',nf) computes the cross smoothed pseudo Wigner-Ville distribution using nf frequency points. You can specify twin and fwin in this syntax, or you can omit them.
d = xwvd(___,'MinThreshold',thresh) sets to zero those elements of d whose amplitude is less than thresh. This syntax applies to both the cross Wigner-Ville distribution and the cross smoothed pseudo Wigner-Ville distribution.
[d,f,t] = xwvd(___) also returns a vector of frequencies, f, and a vector of times, t, at which d is computed.
xwvd(___) with no output arguments plots the real part of the cross Wigner-Ville or cross smoothed pseudo Wigner-Ville distribution in the current figure.
Generate two signals sampled at 1 kHz for 1 second and embedded in white noise. One signal is a sinusoid of frequency 150 Hz. The other signal is a chirp whose frequency varies sinusoidally between 200 Hz and 400 Hz. The noise has a variance of
0.{1}^{2}
Compute the Wigner-Ville distribution of the sum of the signals.
Compute and plot the cross Wigner-Ville distribution of the signals. The cross-distribution corresponds to the cross-terms of the Wigner-Ville distribution.
Generate a two-channel signal that consists of two chirps. The signal is sampled at 3 kHz for one second. The first chirp has an initial frequency of 400 Hz and reaches 800 Hz at the end of the sampling. The second chirp starts at 500 Hz and reaches 1000 Hz at the end. The second chirp has twice the amplitude of the first chirp.
Store the signal as a timetable. Compute and plot the cross Wigner-Ville distribution of the two channels.
Compute the instantaneous frequency of a signal by using a known reference signal and the cross Wigner-Ville distribution.
Create a reference signal consisting of a Gaussian atom sampled at 1 kHz for 1 second. A Gaussian atom is a sinusoid modulated by a Gaussian. Specify a sinusoid frequency of 50 Hz. The Gaussian is centered at 64 milliseconds and has a variance of
0.{01}^{2}
Create the "unknown" signal to analyze, consisting of a chirp. The signal starts suddenly at 0.4 second and ends suddenly half a second later. In that lapse, the frequency of the chirp decreases linearly from 400 Hz to 100 Hz.
Create a two-component signal consisting of the sum of the unknown and reference signals. The smoothed pseudo Wigner-Ville distribution of the result provides an "ideal" time-frequency representation.
Compute and display the smoothed pseudo Wigner-Ville distribution.
Compute the cross Wigner-Ville distribution of the unknown and reference signals. Take the absolute value of the distribution and set to zero the elements with amplitude less than 10. The cross Wigner-Ville distribution is equal to the cross-terms of the two-component signal.
Plot the real part of the cross Wigner-Ville distribution.
Enhance the Wigner-Ville cross-terms by adding the ideal time-frequency representation to the cross Wigner-Ville distribution. The cross-terms of the Wigner-Ville distribution occur halfway between the reference signal and the unknown signal.
Identify and plot the high-energy ridge corresponding to the cross-terms. To isolate the ridge, find the time values where the cross-distribution has nonzero energy.
Reconstruct the instantaneous frequency of the unknown signal by using the ridge and the reference function. Plot the instantaneous frequency as a function of time.
vectors | timetables
Input signals, specified as vectors or MATLAB® timetables each containing a single vector variable. x and y must both be vectors or both be timetables and must have the same length.
If x and y are timetables, then they must contain increasing finite row times.
If the input signals have odd length, the function appends a zero to make the length even.
Example: cos(pi/8*(0:159))'+randn(160,1)/10 specifies a sinusoid embedded in white noise.
Example: timetable(seconds(0:5)',rand(6,1)) specifies a random variable sampled at 1 Hz for 4 seconds.
2*pi (default) | positive numeric scalar
twin, fwin — Time and frequency windows
vectors of odd length
Time and frequency windows used for smoothing, specified as vectors of odd length. By default, xwvd uses Kaiser windows with shape factor β = 20.
The default length of twin is the smallest odd integer greater than or equal to round(length(x)/10).
The default length of fwin is the smallest odd integer greater than or equal to nf/4.
Each window must have a length smaller than or equal to 2*ceil(length(x)/2).
Example: kaiser(65,0.5) specifies a 65-sample Kaiser window with a shape factor of 0.5.
nf — Number of frequency points
2*ceil(length(x)/2) (default) | integer
Number of frequency points, specified as an integer. This argument controls the degree of oversampling in frequency. The number of frequency points must be at least (length(fwin)+1)/2 and cannot be greater than the default.
thresh — Minimum nonzero value
Minimum nonzero value, specified as a real scalar. The function sets to zero those elements of d whose amplitudes are less than thresh.
d — Cross Wigner-Ville distribution
Cross Wigner-Ville distribution, returned as a matrix. Time increases across the columns of d, and frequency increases down the rows. The matrix is of size Nf × Nt, where Nf is the length of f and Nt is the length of t.
If the input has time information, then f contains frequencies expressed in Hz.
If the input does not have time information, then f contains normalized frequencies expressed in rad/sample.
If the input has time information, then t contains time values expressed in seconds.
If the input does not have time information, then t contains sample numbers.
The number of time points is fixed as 4*ceil(length(x)/2).
For continuous signals x(t) and y(t), the cross Wigner-Ville distribution is defined as
{\mathrm{XWVD}}_{x,y}\left(t,f\right)={â«}_{â\infty }^{\infty }x\left(t+\frac{\mathrm{Ï}}{2}\right){y}^{*}\left(tâ\frac{\mathrm{Ï}}{2}\right){e}^{âj2\mathrm{Ï}f\mathrm{Ï}}\text{â}d\mathrm{Ï}.
For a discrete signal with N samples, the distribution becomes
{\text{XWVD}}_{x,y}\left(n,k\right)=\underset{m=âN}{\overset{N}{â}}x\left(n+m/2\right)\text{â}{y}^{*}\left(nâm/2\right)\text{â}{e}^{âj2\mathrm{Ï}km/N}.
For odd values of m, the definition requires evaluation of the signal at half-integer sample values. It therefore requires interpolation, which makes it necessary to zero-pad the discrete Fourier transform to avoid aliasing.
The cross Wigner-Ville distribution contains interference terms that often complicate its interpretation. To sharpen the distribution, one can filter the definition with lowpass windows. The cross smoothed pseudo Wigner-Ville distribution uses independent windows to smooth in time and frequency:
{\text{XSPWVD}}_{x,y}^{g,H}\left(t,f\right)={â«}_{â\infty }^{\infty }g\left(t\right)\text{â}H\left(f\right)\text{â}x\left(t+\frac{\mathrm{Ï}}{2}\right){y}^{*}\left(tâ\frac{\mathrm{Ï}}{2}\right){e}^{âj2\mathrm{Ï}f\mathrm{Ï}}\text{â}d\mathrm{Ï}. |
Modern Photography/Format - Wikibooks, open books for an open world
Modern Photography/Format
The format of an image is said to be the size of the recording medium or its digital output.
Format as aspect ratioEdit
Wikipedia has related information at Aspect ratio (image)
Essentially all cameras have a rectangular (or square) frame, the shape of which is expressed in terms of the 'aspect ratio', or width divided by height.
{\displaystyle {\textrm {AspectRatio}}={\textrm {Width}}\div {\textrm {Height}}}
Typically aspect ratios vary from 1:1 for square-format cameras to 16:9 for some mobile phones, and as much as 4:1 for panoramic cameras. For most cameras, this is a set quantity, although many modern cameras allow the photographer to set the aspect ratio by cropping the image in camera.
Common aspect ratiosEdit
For example, the most common still image photography format in the world is 35mm, which was the dominant film size at the close of the analog film era. It is still produced today, but more importantly its aspect ratio has been inherited by digital camera image sensor formats. The 35mm format is actually, somewhat confusingly, 36mm in width by 24mm in height. The aspect ratio of this format (dividing by twelve to obtain the simplest whole-number ratio) is therefore said to be 3:2. Many common formats (APS family of formats, for example) are equal to or closely approximate this aspect ratio.
A notable exception to the 3:2 still image format aspect ratio is the Four Thirds System, with an aspect ratio of 4:3 as seen in some compact digital cameras.
From analog to digitalEdit
Chart of common sensor sizes.
In the analog era, it was common to express formats in terms of the physical size of the recording medium, for example 4×3 inches. These days it is more common to express the format in terms of the accepted short form for the standardized sensor size. For example '35mm' or 'APS-C'.
In terms of composition, these format names are arguably more usefully expressed in terms of aspect ratio, ie. 'the shape of the frame', such as 3:2 or 4:3.
In terms of the resolution available for post-processing digital images, the format alone cannot tell us, we must look at the sensor. Sensor sizes are expressed in terms of pixels, eg. 8688×5792 pixels. It is not possible to determine the pixel count from the sensor format, and vice versa, though there is some relation: larger sensor sizes tend to have larger pixel counts. Because large modern pixel numbers can be hard to remember, the industry has standardized on megapixels, or the number of total pixels divided by 1,000,000 (one million). For example, the Canon 5DS has a resolution of 8688×5792 pixels, which is equal to 50,320,896 pixels in total, or roughly 50.3 megapixels.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Modern_Photography/Format&oldid=3719590" |
S-Adenosylmethionine synthetase enzyme - Wikipedia
S-Adenosylmethionine synthetase enzyme
(Redirected from S-adenosylmethionine synthetase enzyme)
S-adenosylmethionine synthase 2, tetramer, Human
S-Adenosylmethionine synthetase (EC 2.5.1.6), also known as methionine adenosyltransferase (MAT), is an enzyme that creates S-adenosylmethionine (also known as AdoMet, SAM or SAMe) by reacting methionine (a non-polar amino acid) and ATP (the basic currency of energy).[1]
2 Conserved motifs in the 3'UTR of MAT2A mRNA
3 Protein overview
4 S-adenosylmethionine synthetase N terminal domain
4.1 N terminal domain function
4.2 N terminal domain structure
5 S-adenosylmethionine synthetase Central domain
5.1 Central terminal domain function
5.2 Central domain Structure
6 S-adenosylmethionine synthetase, C terminal domain
6.1 C terminal domain function
6.2 C terminal domain Structure
AdoMet is a methyl donor for transmethylation. It gives away its methyl group and is also the propylamino donor in polyamine biosynthesis. S-adenosylmethionine synthesis can be considered the rate-limiting step of the methionine cycle.[2]
As a methyl donor SAM allows DNA methylation. Once DNA is methylated, it switches the genes off and therefore, S-adenosylmethionine can be considered to control gene expression.[3]
SAM is also involved in gene transcription, cell proliferation, and production of secondary metabolites.[4] Hence SAM synthetase is fast becoming a drug target, in particular for the following diseases: depression, dementia, vacuolar myelopathy, liver injury, migraine, osteoarthritis, and as a potential cancer chemopreventive agent.[5]
This article discusses the protein domains that make up the SAM synthetase enzyme and how these domains contribute to its function. More specifically, this article explores the shared pseudo-3-fold symmetry that makes the domains well-adapted to their functions.[6]
ATP + L-methionine + H2O
{\displaystyle \rightleftharpoons }
phosphate + diphosphate + S-adenosyl-L-methionine
Conserved motifs in the 3'UTR of MAT2A mRNA[edit]
A computational comparative analysis of vertebrate genome sequences have identified a cluster of 6 conserved hairpin motifs in the 3'UTR of the MAT2A messenger RNA (mRNA) transcript.[7] The predicted hairpins (named A-F) have strong evolutionary conservation and 3 of the predicted RNA structures (hairpins A, C and D) have been confirmed by in-line probing analysis. No structural changes were observed for any of the hairpins in the presence of metabolites SAM, S-adenosylhomocysteine or L-Methionine. They are proposed to be involved in transcript stability and their functionality is currently under investigation.[7]
Protein overview[edit]
The S-adenosylmethionine synthetase enzyme is found in almost every organism bar parasites which obtain AdoMet from their host. Isoenzymes are found in bacteria, budding yeast and even in mammalian mitochondria. Most MATs are homo-oligomers and the majority are tetramers. The monomers are organised into three domains formed by nonconsecutive stretches of the sequence, and the subunits interact through a large flat hydrophobic surface to form the dimers.[8]
S-adenosylmethionine synthetase N terminal domain[edit]
S-adenosylmethionine synthetase N terminal domain
S-AdoMet_synt_N
1mxa / SCOPe / SUPFAM
In molecular biology the protein domain S-adenosylmethionine synthetase N terminal domain is found at the N-terminal of the enzyme.
N terminal domain function[edit]
The N terminal domain is well conserved across different species. This may be due to its important function in substrate and cation binding. The residues involved in methionine binding are found in the N-terminal domain.[8]
N terminal domain structure[edit]
The N terminal region contains two alpha helices and four beta strands.[6]
S-adenosylmethionine synthetase Central domain[edit]
S-adenosylmethionine synthetase Central domain
Central terminal domain function[edit]
The precise function of the central domain has not been fully elucidated, but it is thought to be important in aiding catalysis.
Central domain Structure[edit]
The central region contains two alpha helices and four beta strands.[6]
S-adenosylmethionine synthetase, C terminal domain[edit]
Methionine adenosyltransferase in a complex ADP and l-methionine.
In molecular biology, the protein domain S-adenosylmethionine synthetase, C-terminal domain refers to the C terminus of the S-adenosylmethionine synthetase
C terminal domain function[edit]
The function of the C-terminal domain has been experimentally determined as being important for cytoplasmic localisation. The residues are scattered along the C-terminal domain sequence however once the protein folds, they position themselves closely together.[3]
C terminal domain Structure[edit]
The C-terminal domains contains two alpha-helices and four beta-strands.[6]
^ Horikawa S, Sasuga J, Shimizu K, Ozasa H, Tsukada K (August 1990). "Molecular cloning and nucleotide sequence of cDNA encoding the rat kidney S-adenosylmethionine synthetase". J. Biol. Chem. 265 (23): 13683–6. doi:10.1016/S0021-9258(18)77403-6. PMID 1696256.
^ Markham GD, Pajares MA (2009). "Structure-function relationships in methionine adenosyltransferases". Cell Mol Life Sci. 66 (4): 636–48. doi:10.1007/s00018-008-8516-1. PMC 2643306. PMID 18953685.
^ a b Reytor E, Pérez-Miguelsanz J, Alvarez L, Pérez-Sala D, Pajares MA (2009). "Conformational signals in the C-terminal domain of methionine adenosyltransferase I/III determine its nucleocytoplasmic distribution". FASEB J. 23 (10): 3347–60. doi:10.1096/fj.09-130187. hdl:10261/55151. PMID 19497982. S2CID 25548921.
^ Yoon S, Lee W, Kim M, Kim TD, Ryu Y (2012). "Structural and functional characterization of S-adenosylmethionine (SAM) synthetase from Pichia ciferrii". Bioprocess Biosyst Eng. 35 (1–2): 173–81. doi:10.1007/s00449-011-0640-x. PMID 21989639. S2CID 40318843.
^ Kamarthapu V, Rao KV, Srinivas PN, Reddy GB, Reddy VD (2008). "Structural and kinetic properties of Bacillus subtilis S-adenosylmethionine synthetase expressed in Escherichia coli". Biochim Biophys Acta. 1784 (12): 1949–58. doi:10.1016/j.bbapap.2008.06.006. PMID 18634909.
^ a b c d Takusagawa F, Kamitori S, Misaki S, Markham GD (1996). "Crystal structure of S-adenosylmethionine synthetase". J Biol Chem. 271 (1): 136–47. doi:10.1074/jbc.271.1.136. PMID 8550549.
^ a b Parker BJ, Moltke I, Roth A, Washietl S, Wen J, Kellis M, Breaker R, Pedersen JS (November 2011). "New families of human regulatory RNA structures identified by comparative analysis of vertebrate genomes". Genome Res. 21 (11): 1929–43. doi:10.1101/gr.112516.110. PMC 3205577. PMID 21994249.
^ a b Garrido F, Estrela S, Alves C, Sánchez-Pérez GF, Sillero A, Pajares MA (2011). "Refolding and characterization of methionine adenosyltransferase from Euglena gracilis". Protein Expr Purif. 79 (1): 128–36. doi:10.1016/j.pep.2011.05.004. hdl:10261/55441. PMID 21605677.
Methionine+adenosyltransferase at the US National Library of Medicine Medical Subject Headings (MeSH)
Retrieved from "https://en.wikipedia.org/w/index.php?title=S-Adenosylmethionine_synthetase_enzyme&oldid=1077168468" |
1 October 2011 Equivalence of formalities of the little discs operad
Pavol Ševera, Thomas Willwacher
There is a remarkable Drinfeld associator given by Kontsevich’s integrals over configuration spaces of points in the plane. Using this Drinfeld associator we show that Kontsevich’s and Tamarkin’s formalities of the little discs operad are homotopic. The basic technical tool for this result is an
{L}_{\infty }
-algebra of graphs whose cohomology is the Drinfeld-Kohno Lie algebra of infinitesimal pure braids.
Pavol Ševera. Thomas Willwacher. "Equivalence of formalities of the little discs operad." Duke Math. J. 160 (1) 175 - 206, 1 October 2011. https://doi.org/10.1215/00127094-1443502
Pavol Ševera, Thomas Willwacher "Equivalence of formalities of the little discs operad," Duke Mathematical Journal, Duke Math. J. 160(1), 175-206, (1 October 2011) |
Ruby bakriminel bolts (e) - The RuneScape Wiki
Enchanted ruby tipped bakriminel bolts. Item bonus: Blood Forfeit - 5% chance to hit your opponent for 20% of their lifepoints and hit yourself for 10% of your lifepoints.
Item JSON: {"edible":"no","members":"yes","stackable":"yes","stacksinbank":"yes","death":"reclaimable","name":"Ruby bakriminel bolts (e)","bankable":"yes","gemw":{"name":"Ruby bakriminel bolts (e)","limit":1000},"equipable":"yes","disassembly":"yes","release_date":"22 January 2018","id":"41629","release_update_post":"Auras - Bakriminel Bolts","lendable":"no","destroy":"Drop","highalch":78,"weight":0,"tradeable":"yes","examine":"Enchanted ruby tipped bakriminel bolts. Item bonus: Blood Forfeit - 5% chance to hit your opponent for 20% of their lifepoints and hit yourself for 10% of your lifepoints.","noteable":"no"}Buy limit: 1000
Ruby bakriminel bolts (e) are ruby bakriminel bolts that have been enchanted via the Enchant Bakriminel Bolt (Ruby) spell with level 95 Magic.
Enchanted ruby bakriminel bolts have a 5% chance of triggering the Blood Forfeit effect, which hits the target for up to 20% of their remaining life points, while dealing the player 10% of their own life points as recoil. In player killing situations, the damage is capped at 3,000; against monsters, it is capped at at 10,000 for regular hits. Critical hits are capped at 12,000 (15,000 with Erethdor's grimoire). For monsters with over 1,000,000 life points remaining, the chance of triggering is lowered by 1% for every 1,000,000.
2 Blood Forfeit
2.1 Chance of occurring
2.3 Damage done to the player
Blood Forfeit[edit | edit source]
The actual effect of these bolts is somewhat complex. Because of its power level, blood forfeit has several restrictions placed on it:
Chance of occurring[edit source]
The chance the effect occurs is at most:
For every 1 million health the chance of the effect occurring decreases by 1%. For this reason, the chance of the effect occurring at Seiryu the Azure Serpent is always 1% except during story mode.
When using a bleed ability like Fragmentation shot, the effect cannot occur. During combo abilities like Rapid fire the effect can occur on every hit.
When a monster is hit by the effect, they get a 6 seconds invisible cooldown during which the effect cannot occur on them again. However, if a player hits another target with an area of effect ability like Ricochet, additional targets do not share this cooldown and have their own chance to activate, unaffected by the primary target.
To calculate the damage, let
{\displaystyle C}
be the current life points of the target and let
{\displaystyle M}
be the maximum life points of the target.
The damage of blood forfeit starts at 20% of the enemy's health but the percentage goes down as their life points decrease (to a minimum of 1%). The percentage of current health the bolts will do is
{\displaystyle {\text{max}}\left(1,\left\lfloor 20\cdot {\frac {C}{M}}\right\rfloor \right)\%}
Therefore, the damage done from this would be
{\displaystyle \left\lfloor {\text{max}}\left(1,\left\lfloor 20\cdot {\frac {C}{M}}\right\rfloor \right)\%\times C\right\rfloor }
Furthermore, the bolts have a soft damage cap. The soft damage cap scales down linearly with
{\displaystyle C}
The soft damage cap,
{\displaystyle S_{\text{cap}}}
, is then:
{\displaystyle S_{\text{cap}}=\left\lfloor 10{,}000\cdot {\frac {2C}{M}}\right\rfloor }
The actual damage before boosts,
{\displaystyle D}
, will then be the minimum of these:
{\displaystyle D={\text{min}}\left(\left\lfloor {\text{max}}\left(1,\left\lfloor 20\cdot {\frac {C}{M}}\right\rfloor \right)\%\times C\right\rfloor ,S_{\text{cap}}\right)}
For example, when fighting against Nex in her blood phase and she has 100,000 health left, the player will deal 10% of her current health as damage, with a damage cap of 10,000, resulting in 10,000 damage when the effect occurs without any modifiers. When she falls to 80,000 health and is in her ice phase, the effect will only deal 8% of her current health resulting in 6,400 damage without any modifiers. If the effect occurs against Nex: Angel of Death with 750,000 health remaining, the player will deal 5% of her maximum health (well over 30,000 damage) but with a damage cap of 5,000, resulting in 5,000 damage before modifiers.
Any effect which modifies the damage blood forfeit replaces, such as Needle strike, Prayer, boosted levels or Scrimshaw of cruelty will modify the damage after the above has been calculated, although still subject to the regular damage caps. Non-critical hits have a damage cap of 10,000.[exception 1] The damage cap for critical hits vary depending on if the player is using an active Erethdor's grimoire and/or if their target is affected by Smoke Cloud.
Critical hit damage caps
Critical hit cap
Anything that modifies Ability damage directly, such as weapon and equipment bonus or level, will not boost the damage as blood forfeit replaces the ability damage. A list of possible boosts can be found on the ability damage page.
For example, in the previous situation where Nex is at 80,000 health points and the effect triggers but the player is using the Desolation curse, boosted by 19 levels using a Supreme overload and under the effects of the remnants of the Stone of Jas from the Glacor Cave, blood forfeit would deal roughly 7,564-7,640 damage. The precise perk guarantees around 7,640 damage.
This formula is similar in PvP situations. The damage is always capped at 3,000 damage but scales down when the opponent's life points fall below half:
{\displaystyle D={\begin{cases}3{,}000&{\text{if}}\quad C>M/2\\\left\lfloor 3{,}000\cdot {\frac {2C}{M}}\right\rfloor &{\text{if}}\quad C\leq M/2\end{cases}}}
^ An exception to this is Tuska's Wrath with an active Erethdor's grimoire - the maximum damage is 15,000 (even for non-critical hits).
Damage done to the player[edit source]
The damage done to the player is not affected by any modifiers and will always deal 10% of their current health. Onslaught can trigger the effect (but damage bonuses that do not affect onslaught can also not affect blood forfeit during an onslaught). Onslaught's own self-damage would be taken as normal for that hit. Ruby bolts (e)'s self-damage appears to be 10% of the life points before the previous self-damage hit is taken into account, e.g. if the player had 6,000 life points, Onslaught deals 1,000 self-damage and then the next hit triggers ruby bolts (e), they would take 600 additional damage instead of 500.
Ruby bakriminel bolts (e) ×10
Ruby bakriminel bolts 10 2,073 20,730
Retrieved from ‘https://runescape.wiki/w/Ruby_bakriminel_bolts_(e)?oldid=35633896’ |
Lenny is going on capitally.2 He took a little tea & powdered meat this evening! His poor dear little face is something like itself again.
Any time that you can send me a few valerian flowers & of Erythræa, I shd. like to look at them.—3 But if the cases hold good you shall draw up a little account, which I will look over, & send it to the Linnean Socy.
I don’t quite understand your scale of measurement of pistil in Valerian: you speak of it as
\frac{3}{4}
&c & yet it projects beyond the corolla, if I understand rightly.—4
The Erythræa almost seems a more curious case: is there any difference in nectary or in secretion of nectar or position of flowers in the two forms?5
Good Night— I am tired.— | C. Darwin
Dated by the relationship to the letter from W. E. Darwin, 14 July 1862, and to the letter to W. E. Darwin, [after 14 July 1862], and by reference to a postscript, written in pencil, in Emma Darwin’s hand: ‘Enquire a little about quiet sea places near Southampton I expect G & F will come to you at the end of next week’. George Howard Darwin and Francis stayed with William in Southampton from 2 August (see letter from G. H. Darwin, [after 5 August 1862] and n. 3). The letter was written on a Thursday; 24 July was the Thursday prior to the week ending 2 August 1862.
Leonard Darwin was recovering from scarlet fever (Emma Darwin’s diary (DAR 242); see also letter to Asa Gray, 23[–4] July [1862], letters to W. E. Darwin, 4 [July 1862], 9 July [1862], and [after 14 July 1862], and letter to Asa Gray, 14 July [1862]).
See letter from W. E. Darwin, 14 July 1862, and letter to W. E. Darwin, [after 14 July 1862].
See letter from W. E. Darwin, 14 July 1862.
See letter from W. E. Darwin, 14 July 1862. There is a series of observations and drawings of the parts of Erythraea centaurium, made in July, in William’s botanical notebook (DAR 117: 16–31).
Discusses dimorphic plants, valerian and Erythraea. Would like to look at them; suggests WED draw up a paper on them. |
H-minimal graphs of low regularity in <em><strong>H</strong><sup>1</sup></em> | EMS Press
H-minimal graphs of low regularity in <em><strong>H</strong><sup>1</sup></em>
In this paper we investigate H-minimal graphs of lower regularity. We show that noncharacteristic
C^1
H-minimal graphs whose components of the unit horizontal Gauss map are in
W^{1,1}
are ruled surfaces with
C^2
seed curves. Moreover, in light of a structure theorem of Franchi, Serapioni and Serra Cassano, we see that any H-minimal graph is, up to a set of perimeter zero, composed of such pieces. Along these lines, we investigate ways in which patches of
C^1
H-minimal graphs can be glued together to form continuous piecewise
C^1
H-minimal surfaces. We apply this description of H-minimal graphs to the question of the existence of smooth solutions to the Dirichlet problem with smooth data. We find a necessary and sufficient condition for the existence of smooth solutions and produce examples where the conditions are satisfied and where they fail. In particular we illustrate the failure of the smoothness of the data to force smoothness of the solution to the Dirichlet problem by producing a class of curves whose H-minimal spanning graphs cannot be
C^2
Scott D. Pauls, H-minimal graphs of low regularity in <em><strong>H</strong><sup>1</sup></em>. Comment. Math. Helv. 81 (2006), no. 2, pp. 337–381 |
Specifies the orbit of an object in space
The argument of periapsis (also called argument of perifocus or argument of pericenter), symbolized as ω, is one of the orbital elements of an orbiting body. Parametrically, ω is the angle from the body's ascending node to its periapsis, measured in the direction of motion.
For specific types of orbits, words such as perihelion (for heliocentric orbits), perigee (for geocentric orbits), periastron (for orbits around stars), and so on may replace the word periapsis. (See apsis for more information.)
{\displaystyle \omega =\arccos ((\mathbf {n} \cdot \mathbf {e} } \over {\mathbf {\left|n\right|} \mathbf {\left|e\right|} ))}
If ez < 0 then ω → 2π − ω.
n is a vector pointing towards the ascending node (i.e. the z-component of n is zero),
e is the eccentricity vector (a vector pointing towards the periapsis).
{\displaystyle \omega =\arctan 2\left(e_{y},e_{x}\right)}
If the orbit is clockwise (i.e. (r × v)z < 0) then ω → 2π − ω.
ex and ey are the x- and y-components of the eccentricity vector e.
In the case of circular orbits it is often assumed that the periapsis is placed at the ascending node and therefore ω = 0. However, in the professional exoplanet community, ω = 90° is more often assumed for circular orbits, which has the advantage that the time of a planet's inferior conjunction (which would be the time the planet would transit if the geometry were favorable) is equal to the time of its periastron.[1][2][3]
^ Iglesias-Marzoa, Ramón; López-Morales, Mercedes; Jesús Arévalo Morales, María (2015). "Thervfit Code: A Detailed Adaptive Simulated Annealing Code for Fitting Binaries and Exoplanets Radial Velocities". Publications of the Astronomical Society of the Pacific. 127 (952): 567–582. doi:10.1086/682056.
^ Kreidberg, Laura (2015). "Batman: BAsic Transit Model cAlculatioN in Python". Publications of the Astronomical Society of the Pacific. 127 (957): 1161–1165. arXiv:1507.08285. Bibcode:2015PASP..127.1161K. doi:10.1086/683602. S2CID 7954832.
^ Eastman, Jason; Gaudi, B. Scott; Agol, Eric (2013). "EXOFAST: A Fast Exoplanetary Fitting Suite in IDL". Publications of the Astronomical Society of the Pacific. 125 (923): 83. arXiv:1206.5798. Bibcode:2013PASP..125...83E. doi:10.1086/669497. S2CID 118627052.
Argument Of Perihelion in Swinburne University Astronomy Website |
Direct product of groups - Wikipedia
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Z} }
In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted
{\displaystyle G\oplus H}
. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.
(g1, h1) · (g2, h2) = (g1 * g2, h1 ∆ h2)
The binary operation on G × H is associative.
Let R+ be the group of positive real numbers under multiplication. Then the direct product R+ × R+ is the group of all vectors in the first quadrant under the operation of component-wise multiplication
(x1, y1) × (x2, y2) = (x1 × x2, y1 × y2).
{\displaystyle G}
{\displaystyle H}
{\displaystyle G\times H}
(1,1) (a,1) (1,b) (a,b)
(a,1) (1,1) (a,b) (1,b)
(1,b) (a,b) (1,1) (a,1)
(a,b) (1,b) (a,1) (1,1)
The direct product is commutative and associative up to isomorphism. That is, G × H ≅ H × G and (G × H) × K ≅ G × (H × K) for any groups G, H, and K.
|G × H| = |G| |H|.
|(g, h)| = lcm(|g|, |h|).
Algebraic structure[edit]
G′ = { (g, 1) : g ∈ G } and H′ = { (1, h) : h ∈ H }.
Every element of P can be expressed uniquely as the product of an element of G and an element of H.
Then V is the internal direct product of the two-element subgroups {1, a} and {1, b}.
{\displaystyle \langle a\rangle }
be a cyclic group of order mn, where m and n are relatively prime. Then
{\displaystyle \langle a^{n}\rangle }
{\displaystyle \langle a^{m}\rangle }
are cyclic subgroups of orders m and n, respectively, and
{\displaystyle \langle a\rangle }
is the internal direct product of these subgroups.
Similarly, when n is odd the orthogonal group O(n, R) is the internal direct product of the special orthogonal group SO(n, R) and the two-element subgroup {−I, I}, where I denotes the identity matrix.
The symmetry group of a cube is the internal direct product of the subgroup of rotations and the two-element group {−I, I}, where I is the identity element and −I is the point reflection through the center of the cube. A similar fact holds true for the symmetry group of an icosahedron.
{\displaystyle D_{4n}=\langle r,s\mid r^{2n}=s^{2}=1,sr=r^{-1}s\rangle .}
Then D4n is the internal direct product of the subgroup
{\displaystyle \langle r^{2},s\rangle }
(which is isomorphic to D2n) and the two-element subgroup {1, rn}.
{\displaystyle G=\langle S_{G}\mid R_{G}\rangle \ \ }
{\displaystyle \ \ H=\langle S_{H}\mid R_{H}\rangle ,}
{\displaystyle S_{G}}
{\displaystyle S_{H}}
are (disjoint) generating sets and
{\displaystyle R_{G}}
{\displaystyle R_{H}}
are defining relations. Then
{\displaystyle G\times H=\langle S_{G}\cup S_{H}\mid R_{G}\cup R_{H}\cup R_{P}\rangle }
{\displaystyle R_{P}}
is a set of relations specifying that each element of
{\displaystyle S_{G}}
commutes with each element of
{\displaystyle S_{H}}
{\displaystyle G=\langle a\mid a^{3}=1\rangle \ \ }
{\displaystyle \ \ H=\langle b\mid b^{5}=1\rangle }
{\displaystyle G\times H=\langle a,b\mid a^{3}=1,b^{5}=1,ab=ba\rangle .}
Normal structure[edit]
Conjugacy and centralizers[edit]
Automorphisms and endomorphisms[edit]
{\displaystyle {\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}}
Finite direct products[edit]
{\displaystyle \prod _{i=1}^{n}G_{i}\;=\;G_{1}\times G_{2}\times \cdots \times G_{n}}
The elements of G1 × ⋯ × Gn are tuples (g1, ..., gn), where gi ∈ Gi for each i.
(g1, ..., gn)(g1′, ..., gn′) = (g1g1′, ..., gngn′).
Infinite direct products[edit]
More generally, given an indexed family { Gi }i∈I of groups, the direct product Πi∈I Gi is defined as follows:
The elements of Πi∈I Gi are the elements of the infinite Cartesian product of the sets Gi; i.e., functions ƒ: I → ⋃i∈I Gi with the property that ƒ(i) ∈ Gi for each i.
Unlike a finite direct product, the infinite direct product Πi∈I Gi is not generated by the elements of the isomorphic subgroups { Gi }i∈I. Instead, these subgroups generate a subgroup of the direct product known as the infinite direct sum, which consists of all elements that have only finitely many non-identity components.
Semidirect products[edit]
Free products[edit]
Subdirect products[edit]
Fiber products[edit]
^ Gallian, Joseph A. (2010). Contemporary Abstract Algebra (7 ed.). Cengage Learning. p. 157. ISBN 9780547165097.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Direct_product_of_groups&oldid=1043741382" |
Erg (unit) - zxc.wiki
Erg (unit)
The title of this article is ambiguous. Further meanings are listed under Erg .
Unit name erg
{\ displaystyle \ mathrm {erg}}
Physical quantity (s) Energy (e.g. work , internal energy , heat )
{\ displaystyle E; \, W; \, U; \, Q}
{\ displaystyle {\ mathsf {M \; L ^ {2} \; T ^ {- 2}}}}
system CGS system of units
{\ displaystyle \ mathrm {1 \, erg = 0 {,} 1 \; \ mu J}}
{\ displaystyle \ mathrm {= 1 \ cdot 10 ^ {- 7} \; {\ frac {kg \, m ^ {2}} {s ^ {2}}}}}
{\ displaystyle \ mathrm {1 \, erg = 1 \; {\ frac {cm ^ {2} \, g} {s ^ {2}}}}}
Derived from dyn , centimeters
The erg ( unit symbol : erg; from Greek ἔργον , ergon , work ) is a unit of measurement in the CGS system of units for energy . The unit is widely used in astrophysics and sometimes mechanics .
Except for a power of ten, the erg corresponds to the unit joule , which is common in the MBS- based International System of Units (SI) :
{\ displaystyle {\ begin {aligned} 1 \ \ mathrm {erg} & = \ mathrm {0 {,} 1 \ \ mu J = 1 \ cdot 10 ^ {- 7} J} \\\ Leftrightarrow 10 ^ {7 } \ \ mathrm {erg} & = 1 \ \ mathrm {J} \ end {aligned}}}
Considering the dimensions of the energy provides a dimension
energy = Force x length
= Mass · acceleration · length
= Mass (length / time) 2
and therefore the unit in the cgs system
{\ displaystyle {\ begin {aligned} 1 \ \ mathrm {erg} & = \ mathrm {1 \ g \ cdot \ left ({\ frac {cm} {s}} \ right) ^ {2}} \\ & = \ mathrm {1 \ dyn \ cdot \ cm} \ end {aligned}}}
with the Dyn .
In Germany, the Erg is no longer a legal unit since January 1, 1978 .
Centimeter | Gram | second
Gal | Dyn | Barye | Erg | Stokes | Poise | Kayser | Darcy • Perm | Rayl
Biot (Abampere) | Buckingham | Debye | Franklin (Statcoulomb) | Gamma • Gauss | Gilbert | Maxwell | Oersted | Statvolt | Townsend
This page is based on the copyrighted Wikipedia article "Erg_%28Einheit%29" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA. |
Pyruvate dehydrogenase (cytochrome) - Wikipedia
In enzymology, a pyruvate dehydrogenase (cytochrome) (EC 1.2.2.2) is an enzyme that catalyzes the chemical reaction
pyruvate + ferricytochrome b1 + H2O
{\displaystyle \rightleftharpoons }
acetate + CO2 + ferrocytochrome b1
The 3 substrates of this enzyme are pyruvate, ferricytochrome b1, and H2O, whereas its 3 products are acetate, CO2, and ferrocytochrome b1.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with a cytochrome as acceptor. The systematic name of this enzyme class is pyruvate:ferricytochrome-b1 oxidoreductase. Other names in common use include pyruvate dehydrogenase, pyruvic dehydrogenase, pyruvic (cytochrome b1) dehydrogenase, pyruvate:ubiquinone-8-oxidoreductase, and pyruvate oxidase (ambiguous). This enzyme participates in pyruvate metabolism. It has 2 cofactors: FAD, and Thiamin diphosphate.
Williams FR; Hager LP (1961). "A crystalline flavin pyruvate oxidase". J. Biol. Chem. 236: PC36–PC37.
Koland JG, Gennis RB (1982). "Identification of an active site cysteine residue in Escherichia coli pyruvate oxidase". J. Biol. Chem. 257 (11): 6023–7. PMID 7042705.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Pyruvate_dehydrogenase_(cytochrome)&oldid=918614892" |
Home : Support : Online Help : Mathematics : Discrete Mathematics : Ordinals : degree
leading exponent of an ordinal number
trailing exponent of an ordinal number
leading coefficient of an ordinal number
trailing coefficient of an ordinal number
leading term of an ordinal number
trailing term of an ordinal number
degree(o, formal)
tdegree(o, formal)
lcoeff(o, formal)
tcoeff(o, formal)
lterm(o, formal)
tterm(o, formal)
The degree and tdegree commands return the leading and trailing exponents of the ordinal number o, respectively, that is, either an ordinal or a nonnegative integer.
The lcoeff and tcoeff commands return the leading and trailing coefficients of the ordinal number o, respectively, that is, either a nonnegative integer or a polynomial with positive integer coefficients.
The lterm and tterm commands return the leading and trailing terms of the ordinal number o, respectively. The result is returned in form of a list
[e,c]
e
is either an ordinal or a nonnegative integer, and the coefficient
c
is either a nonnegative integer or a polynomial with positive integer coefficients.
The degree(o), lcoeff(o), and lterm(o) calling sequences return the leading exponent, coefficient, and term, respectively, of the ordinal number
o
The degree (leading exponent) of
o
is the largest exponent in
o
with respect to the ordering of ordinals, and lcoeff and lterm return the coefficient and term, respectively, corresponding to the largest exponent.
The tdegree(o), tcoeff(o), and tterm(o) calling sequences return the trailing exponent, coefficient, and term, respectively, of the ordinal number
o
The trailing exponent of
o
is the smallest exponent in
o
with respect to the ordering of ordinals, and tcoeff and tterm return the coefficient and term, respectively, corresponding to the smallest exponent.
lterm is equivalent to [degree,lcoeff] and tterm is equivalent to [tdegree,tcoeff].
o
is a nonnegative integer or a polynomial with positive integer coefficients (representing a nonnegative integer), then degree and ldegree both return
0
, and lcoeff and tcoeff both return
o
o
is a parametric ordinal and it cannot be determined whether the leading or trailing coefficient is nonzero, all six commands return an error, unless the option formal is given.
The degree, lcoeff and tcoeff commands overload the corresponding top-level routines degree, lcoeff, and tcoeff, respectively. The top-level commands are still accessible via the :- qualifier, that is, :-degree, :-lcoeff, and :-tcoeff, respectively.
o
is a nonconstant polynomial with positive integer coefficients, then the results of degree and :-degree differ: the former returns the leading exponent of
o
as an ordinal, namely
0
, while the latter returns the total degree of
o
as a polynomial, that is, a positive integer. Similarly, the results of lcoeff and :-lcoeff also differ in this case, as do the results of tcoeff and :-tcoeff.
\mathrm{with}\left(\mathrm{Ordinals}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{`+`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`.`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`<`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{<=}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Add}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Base}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Dec}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Decompose}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Div}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Eval}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Factor}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Gcd}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Lcm}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LessThan}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Log}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Max}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Min}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Mult}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Ordinal}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Power}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Split}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Sub}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`^`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{degree}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{lcoeff}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{log}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{lterm}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\omega }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{quo}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{rem}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tcoeff}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tdegree}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tterm}}]
\mathrm{o1}≔\mathrm{Ordinal}\left([[\mathrm{\omega },3],[2,x],[1,2],[0,5]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{o1}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}
[\mathrm{degree}\left(\mathrm{o1}\right),\mathrm{lcoeff}\left(\mathrm{o1}\right)]=\mathrm{lterm}\left(\mathrm{o1}\right)
[\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
[\mathrm{tdegree}\left(\mathrm{o1}\right),\mathrm{tcoeff}\left(\mathrm{o1}\right)]=\mathrm{tterm}\left(\mathrm{o1}\right)
[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]
lterm and tterm always return a list.
\mathrm{degree}\left(5\right)=\mathrm{tdegree}\left(5\right),\mathrm{lcoeff}\left(5\right)=\mathrm{tcoeff}\left(5\right),\mathrm{lterm}\left(5\right)=\mathrm{tterm}\left(5\right)
\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]
Nonconstant polynomial input.
\mathrm{degree}\left(2{x}^{2}+1\right)\ne :-\mathrm{degree}\left(2{x}^{2}+1\right),\mathrm{lcoeff}\left(2{x}^{2}+1\right)\ne :-\mathrm{lcoeff}\left(2{x}^{2}+1\right)
\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{\ne }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{\ne }\textcolor[rgb]{0,0,1}{2}
An error is returned in the parametric case if the proper exponent or coefficient cannot be determined.
\mathrm{o2}≔\mathrm{Ordinal}\left([[2,x],[1,y],[0,z]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{o2}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}
\mathrm{degree}\left(\mathrm{o2}\right)
Error, (in degree) cannot determine if x is nonzero
\mathrm{degree}\left(\mathrm{o2},\mathrm{formal}\right)
\textcolor[rgb]{0,0,1}{2}
{\mathrm{\omega }}^{2}+\mathrm{o2}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}
\mathrm{degree}\left(\right),\mathrm{lcoeff}\left(\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}
\mathrm{tcoeff}\left(\mathrm{o2}\right)
Error, (in tcoeff) cannot determine if z is nonzero
\mathrm{tcoeff}\left(\mathrm{o2},\mathrm{formal}\right)
\textcolor[rgb]{0,0,1}{z}
\mathrm{o2}+3
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)
\mathrm{tdegree}\left(\right),\mathrm{tcoeff}\left(\right)
\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
The Ordinals[degree], Ordinals[tdegree], Ordinals[lcoeff], Ordinals[tcoeff], Ordinals[lterm] and Ordinals[tterm] commands were introduced in Maple 2015. |
Compute Joint Torques To Balance An Endpoint Force and Moment - MATLAB & Simulink - MathWorks France
Spatially transforming a wrench from the end-effector frame to the base frame means to exert a new wrench in a frame that happens to collocate with the base frame in space, but is still fixed to the end-effector body; this new wrench has the same effect as the original wrench exerted at the ee origin. In the figure below,
{\mathbf{f}}_{\mathbf{ext}}
{\mathbf{n}}_{\mathbf{ext}}\text{\hspace{0.17em}}
are the endpoint linear force and moment respectively, and the
{\mathbf{f}}_{\mathbf{ee}}^{\mathbf{base}}\text{\hspace{0.17em}}
{\mathbf{n}}_{\mathbf{ee}}^{\mathbf{base}\text{\hspace{0.17em}}}
are the spatially transformed forces and moments, respectively. In the snippet below, fbase_ee is the spatially transformed wrench. |
Aurora is starting a fitness program at her school and needs to find an easy measure of fitness to assign workouts and gauge improvement. She decides that using resting heart rate would be helpful. Assume that the resting heart rate of young women is approximately normally distributed with a mean of
74
beats per minute (bpm) with a standard deviation of
5
bpm.
Aurora found a chart that said resting heart rates between
70
79
(inclusive) are considered average for women her age. What proportion of her classmates would she expect to find in the average range?
70
79
, mean, standard deviation)
The chart also claims that resting heart rate of
66
or less indicates excellent fitness in otherwise healthy young women. What portion of her classmates would she expect to find in excellent shape?
-10
99
66
If Aurora’s fitness program lowers the mean heart rate of her female classmates by
4
bpm, what will be the increase in the proportion of women now classified as being in excellent shape?
What would the new mean be?
Repeat your steps in part (b) with the new mean. |
Difference between revisions of "Lower bounds on APN-distance for all known APN functions in dimension 8" - Boolean Functions
Difference between revisions of "Lower bounds on APN-distance for all known APN functions in dimension 8"
m (Nikolay moved page Lower bounds on APN-distance for all known APN functions to Lower bounds on APN-distance for all known APN functions in dimension 8 without leaving a redirect: Separating dimension 8 from the main table due to large size)
{\displaystyle \Pi _{F}^{0}}
{\displaystyle m_{F}}
4217 93, 994, 10510, 11119, 11728, 12350, 12958, 13544, 141 |
M ¯ 0 , n is not a Mori dream space
{\overline{M}}_{0,n}
is not a Mori dream space
Ana-Maria Castravet, Jenia Tevelev
Building on the work of Goto, Nishida, and Watanabe on symbolic Rees algebras of monomial primes, we prove that the moduli space of stable rational curves with
n
punctures is not a Mori dream space for
n>133
. This answers a question posed by Hu and Keel.
Ana-Maria Castravet. Jenia Tevelev. "
{\overline{M}}_{0,n}
is not a Mori dream space." Duke Math. J. 164 (8) 1641 - 1667, 1 June 2015. https://doi.org/10.1215/00127094-3119846
Received: 15 December 2013; Revised: 25 August 2014; Published: 1 June 2015
Secondary: 14J60 , 14M25 , 14N20
Keywords: elementary transformations , moduli of rational curves , Mori dream spaces , symbolic Rees algebras , toric varieties , weighted projective planes
Ana-Maria Castravet, Jenia Tevelev "
{\overline{M}}_{0,n}
is not a Mori dream space," Duke Mathematical Journal, Duke Math. J. 164(8), 1641-1667, (1 June 2015) |
Experiment (probability theory) - Wikipedia
Procedure that can be infinitely repeated, with a well-defined set of outcomes
This article is about the probabilistic model used in actual experiments. For a discussion about actual experiments, see experiment.
1 Experiments and trials
Experiments and trials[edit]
Main article: Probability space
{\displaystyle \scriptstyle {\mathcal {F}}}
{\displaystyle \scriptstyle {\mathcal {F}}}
{\displaystyle \scriptstyle {\mathcal {F}}}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Experiment_(probability_theory)&oldid=1072741677" |
1 Samuel 6 - Wikipedia
1 Samuel 6 is the sixth chapter of the First Book of Samuel in the Old Testament of the Christian Bible or the first part of the Books of Samuel in the Hebrew Bible.[1] According to Jewish tradition the book was attributed to the prophet Samuel, with additions by the prophets Gad and Nathan,[2] but modern scholars view it as a composition of a number of independent texts of various ages from c. 630–540 BCE.[3][4] This chapter describes how the Ark of Covenant was returned to Israel by the Philistines, a part of the "Ark Narrative" (1 Samuel 4:1–7:1) within a section concerning the life of Samuel (1 Samuel 1:1–7:17).[5]
The pages containing the Books of Samuel (1 & 2 Samuel) in Leningrad Codex (1008 CE).
Hebrew Bible part
Order in the Hebrew part
4 The Ark returned to Israel (6:1–19)
5 The Ark at Kirjath Jearim (6:20–21)
9.1 Commentaries on Samuel
Some early manuscripts containing the text of this chapter in Hebrew are of the Masoretic Text tradition, which includes the Codex Cairensis (895), Aleppo Codex (10th century), and Codex Leningradensis (1008).[6] Fragments containing parts of this chapter in Hebrew were found among the Dead Sea Scrolls including 4Q51 (4QSama; 100–50 BCE) with extant verses 1–13, 16–18, 20–21.[7][8][9][10]
Extant ancient manuscripts of a translation into Koine Greek known as the Septuagint (originally was made in the last few centuries BCE) include Codex Vaticanus (B;
{\displaystyle {\mathfrak {G}}}
B; 4th century) and Codex Alexandrinus (A;
{\displaystyle {\mathfrak {G}}}
A; 5th century).[11][a]
Places mentioned in this chapter
The event in this chapter happened at the end of judges period in Israel, about 1100 BC.
The Ark returned to Israel (6:1–19)Edit
The Philistines realized that the ark had to be returned to Israel to stop the plagues (verse 2, cf. 1 Samuel 5:11), so they consulted their priests and diviners to avoid further humiliation (verses 1–9). Two issues were raised in verse 3:
What was the appropriate offering to accompany the ark?
Was it really YHWH who had humiliated them?[13]
The answer for the first concern is to send gifts (cf. Exodus 3:21) on the basis of value ('gold'), corresponding to the victims ('five' for the five lords of the Philistines) and representing the plagues ('tumors' and 'mice').[13] The gifts are called 'guilt offering (ʼašām), serving a double function: as a sacrifice to ensure that YHWH would 'lighten his hand' and as a compensatory tribute to YHWH.[13] They learned from the Exodus tradition 'not to be obstinate and prevent the return of the ark' (verse 6).[13]
The answer to the second concern was sought by the use of divination (verses 7–9), utilizing untrained cows, separated from their young calves (therefore inclined to return home), and released unguided, so when the cows went straight to the territory of Israel (in the direction of border city Beth-shemesh; verses 10–18), the Philistines were convinced that the plagues came from YHWH and their gifts were acceptable (verses 16–18).[13] The Israelites celebrated the return of the ark, and utilized the cows to be an appropriate sacrifice for the removal of ritual 'contamination', as the animals and the cart were new, unused, and therefore ritually clean (cf. Numbers 19:2).[13] The sacrifice was performed on a 'large stone of Abel' in the field of an unknown Joshua (verse 18), which afterward also became the resting place for the ark (verse 15).[13]
"The Ark is returned to Beth-shemesh" (1 Samuel 6:1-16). Doré's English Bible. 1866
And the ark of the Lord was in the country of the Philistines seven months.[14]
"In the country": or "in the field"[15] denoting a place outside 'under the open air', because apparently none of the Philistine cities dared to host the ark and the people thought it would deliver them from their calamity.[16] However, Targum states "in the cities of the Philistines", that is, from one city to the other.[16]
"Philistines": a group of people coming from the northeastern Mediterranean area (whih includes the island of Crete; cf. Amos 9:7) and entering "Palestine" (which is derived from "Philistine") around 1200 BCE.[17]
"Seven months": the ark was returned during the wheat harvest (1 Samuel 6:13), so the battles between Israel and the Philistines which led to the capture of the ark happened at the end of autumn or the beginning of winter.[16] Josephus states that the ark was with the Philistines four months only.[18]
Therefore you will make images of your tumors and images of your mice that ravage the land. And you will give glory to the God of Israel. Perhaps He will lighten His hand from off you, even from off your gods and from off your land.[19]
"Tumors" (KJV: emerods): or "bleeding piles", or more probably "boils".[20] The mention of "mice" together with "tumors" indicates that the outbreak (cf. 1 Samuel 5:6) could be bubonic plague.[21]
The Ark at Kirjath Jearim (6:20–21)Edit
Similar to what happened to the Philistines, the ark caused plagues for Israelites when they did not show due respect to it, so the ark was moved from Beth-shemesh to Kiriath Jearim ("city of the forests"), probably due to its previous connection with Baal-worship (cf. 'city of Baal', Joshua 18:14, and 'Baalah', Joshua 15:9, 10).[13] The custodian of the city was Eleazar, son of Abinadab, both had names that often appear in levitical lists.[13]
And they sent messengers to the inhabitants of Kiriath Jearim, saying, "The Philistines have brought back the ark of the LORD. Come down, and take it up to you."[22]
"Kiriath Jearim": now identified with Deir el-Azar (Tel Qiryat Yearim),[23][24] a place near Abu Ghosh on a hill about 7 miles west of Jerusalem.[25]
Related Bible parts: 1 Samuel 4, 1 Samuel 5
^ The whole book of 1 Samuel is missing from the extant Codex Sinaiticus.[12]
^ Würthwein 1995, pp. 35–37.
^ Ulrich 2010, pp. 266–267.
^ Dead sea scrolls - 1 Samuel
This article incorporates text from a publication now in the public domain: Herbermann, Charles, ed. (1913). "Codex Sinaiticus". Catholic Encyclopedia. New York: Robert Appleton Company.
^ a b c d e f g h i Jones 2007, p. 203.
^ 1 Samuel 6:1 KJV
^ Septuagint in agro, cf. Pagninus, Montanus; apud Gill, "1 Samuel 6"
^ a b c Gill, John. Exposition of the Entire Bible. "1 Samuel 6". Published in 1746-1763.
^ Coogan 2007, p. 405 Hebrew Bible.
^ Josephus, Antiquities book 6, chapter 1, section 4.
^ 1 Samuel 6:5 MEV
^ Cambridge Bible for Schools and Colleges. 1 Samuel 5. Accessed 28 April 2019.
^ 1 Samuel 6:21 MEV
^ Cooke, Francis T. (1925). "The Site of Kirjath-Jearim". Annual of the American Schools of Oriental Research. 5: 105–120. doi:10.2307/3768522. JSTOR 3768522.
^ Robinson, E. (1856). Later Biblical Researches in Palestine and in the Adjacent Regions - A Journal of Travels in the Year 1852. Boston: Crocker & Brewster. p. 156.
^ "The Shmunis Family Foundation Excavations at Kiriath-Jearim". The Shmunis Family Foundation Excavations at Kiriath-Jearim. Retrieved 2019-12-29.
Commentaries on SamuelEdit
Chapman, Stephen B. (2016). 1 Samuel as Christian Scripture: A Theological Commentary. Wm. B. Eerdmans Publishing Company. ISBN 978-1467445160.
Evans, Paul (2018). Longman, Tremper (ed.). 1-2 Samuel. The Story of God Bible Commentary. Zondervan Academic. ISBN 978-0310490944.
Coogan, Michael David (2007). Coogan, Michael David; Brettler, Marc Zvi; Newsom, Carol Ann; Perkins, Pheme (eds.). The New Oxford Annotated Bible with the Apocryphal/Deuterocanonical Books: New Revised Standard Version, Issue 48 (Augmented 3rd ed.). Oxford University Press. ISBN 978-0195288810.
Halley, Henry H. (1965). Halley's Bible Handbook: an abbreviated Bible commentary (24th (revised) ed.). Zondervan Publishing House. ISBN 0-310-25720-4.
Hayes, Christine (2015). Introduction to the Bible. Yale University Press. ISBN 978-0300188271.
Jones, Gwilym H. (2007). "12. 1 and 2 Samuel". In Barton, John; Muddiman, John (eds.). The Oxford Bible Commentary (first (paperback) ed.). Oxford University Press. pp. 196–232. ISBN 978-0199277186. Retrieved February 6, 2019.
Knight, Douglas A (1995). "Chapter 4 Deuteronomy and the Deuteronomists". In James Luther Mays, David L. Petersen and Kent Harold Richards (ed.). Old Testament Interpretation. T&T Clark. ISBN 9780567292896.
Ulrich, Eugene, ed. (2010). The Biblical Qumran Scrolls: Transcriptions and Textual Variants. Brill.
Würthwein, Ernst (1995). The Text of the Old Testament. Translated by Rhodes, Erroll F. Grand Rapids, MI: Wm. B. Eerdmans. ISBN 0-8028-0788-7. Retrieved January 26, 2019.
Shmuel I - I Samuel - Chapter 6 (Judaica Press). Hebrew text and English translation [with Rashi's commentary] at Chabad.org
1 Samuel chapter 6. Bible Gateway
Retrieved from "https://en.wikipedia.org/w/index.php?title=1_Samuel_6&oldid=1067112045" |
Attenuation - SEG Wiki
Attenuation is a reduction in the energy of a traveling wave as it propagates through a medium. Attenuation — the falloff of a wave’s energy with distance — has three main causes: (1) transmission loss at interfaces because of reflection, diffraction, mode conversion, and scattering (Bowman, 1955[1]); (2) geometric divergence effects as waves spread out from a source; and (3) absorption, which is the conversion of kinetic energy into heat by friction (note that kinetic energy is the energy of motion). Writers do not always distinguish between the terms attenuation and absorption. Here, we refer to attenuation in the general sense of energy loss to any cause, and we use the term absorption in the special sense of energy loss to heat.
Transmission loss is a wave’s energy loss as the wave travels through an interface. In transmission loss, the energy that is lost is diverted from the traveling wave of interest. There is no loss of total kinetic energy because the lost energy merely travels somewhere else. For example, when a wave meets an interface, some energy is reflected back from the interface, and only part of the wave’s energy is transmitted though the interface.
Mode conversion is the conversion of P-wave energy into S-wave energy or vice versa. Mode conversion occurs when a wave arrives at an interface at an obliquely incident angle to that interface. Converted waves divert energy away from the given wave.
The energy of a wave in a homogeneous material is proportional to the square of its amplitude (which can be expressed in terms of pressure, particle velocity, or particle acceleration). A point source produces a spherical wave that spreads out from the source. The energy is distributed over the area of the sphere, which increases as the square of the sphere’s radius. Thus, the wave’s energy per unit area varies inversely as the square of the distance from the source. The wave’s amplitude is proportional to the square root of the energy per unit area. Thus, the wave’s amplitude is inversely proportional to the distance the wave has traveled. Spherical spreading — that is, the loss of amplitude because a wave spreads out — is one cause of attenuation. Such energy loss is called geometric divergence.
The third cause of attenuation is energy loss from absorption. Absorption is the result of frictional dissipation of elastic energy into heat. The loss from absorption turns out to be approximately exponential with distance. The second and third mechanisms of attenuation] (i.e., spherical spreading and absorption) can be combined in the equation
{\displaystyle {\begin{aligned}A=A_{0}{\frac {x_{0}}{x}}e^{-\alpha x}.\end{aligned}}}
Here, A is the amplitude at distance x from the source,
{\displaystyle A_{0}}
is the amplitude of distance zero, and
{\displaystyle \alpha }
is the absorption coefficient. The amplitude of a seismic wave falls off with distance x from the source, in accordance with spherical spreading. In equation 1 , the factor
{\displaystyle x_{0}/x}
indicates spherical spreading, whereas the coefficient
{\displaystyle \alpha }
indicates the absorption.
The value of the absorption coefficient depends on the material. Absorption of elastic waves in rocks represents an important problem in seismic exploration. Absorption constants have been measured for a variety of earth materials. However, the mechanism for absorption in many types of rocks, particularly in softer sedimentary rocks or in fluid-filled porous rocks, is still a subject of investigation.
Extensive laboratory work with dry rocks has shown that the absorption coefficient
{\displaystyle \alpha }
tends to be proportional to the first power of frequency f. Such a relation indicates that the mechanism of absorption is solid friction associated with the particle motion in the wave. The logarithmic decrement
{\displaystyle \delta }
is the logarithm of the ratio of amplitude of any cycle to that of the following one in a train of damped waves. The quality factor Q is related to
{\displaystyle \delta }
{\displaystyle Q=\pi /\delta }
. The absorption coefficient
{\displaystyle \alpha }
is related to the logarithmic decrement by
{\displaystyle \alpha =\delta f/V}
. Let V denote the P-wave velocity. The coefficient
{\displaystyle \alpha }
is related to the quality factor Q by
{\displaystyle {\begin{aligned}\alpha ={\frac {\pi f}{QV}}.\end{aligned}}}
{\displaystyle \delta }
and Q are used to describe the attenuation characteristics of rocks. From equation 2 , we see that waves with higher frequencies tend to be absorbed more than waves with lower frequencies are absorbed. This property gives rise to a progressive lowering of the apparent frequencies as the wave travels. In addition, we see that waves with higher velocities tend to be absorbed less than are waves with lower velocities.
Norman Ricker did important work on determining the shape that the seismic pulse acquires as it propagates through rock (Ricker, 1940[2], 1941[3], 1953[4]). His equations predicted the waveform that would be observed after an impulsive signal has traveled a given distance through an absorbing material. Ricker’s work included first-power, second-power, and fourth-power frequency dependence of the absorption coefficient. He found that the wave’s shape for second-power frequency dependence most closely resembled what could be measured at the time in the field. Such a second-power frequency dependence suggested to him a “viscoelastic” frictional-loss mechanism of a type that usually is associated with viscous liquids. From that attenuation law, Ricker gave equations that generated waveforms for ground displacement and for particle velocity. At large distances from the source, such a waveform becomes symmetric. In his honor, this waveform is now called the Ricker wavelet.
Ricker observed such predicted waveforms in the Pierre Shale of Colorado. However, later, more refined experiments conducted in the Pierre Shale indicated that an absorption mechanism that was proportional to the first power of the frequency fit the experimental data far better. This first-power dependence implies solid friction, and today, the solid-friction hypothesis generally is accepted. Even so, the Ricker wavelet continues to be used as a convenient representation of the basic seismic pulse.
There is a large variation of absorption characteristics not only among different rock types but also among members of the same rock type. Let us compute the absorption coefficients using equation 2 with a frequency of 40 Hz. For example, one shale might have a velocity of 3 km/s and an absorption coefficient of 0.70. Another shale might have a velocity of 2 km/s and an absorption coefficient of 2.20. Two sandstones, each with a velocity of 4 km/s, might have absorption coefficients of 0.75 and 1.75, respectively. Two limestones, each with a velocity of 6 km/s, might have absorption coefficients of 0.04 and 0.30, respectively. Although overlaps exist in the ranges of absorption-coefficient values for sedimentary and igneous rocks, sedimentary rocks generally are more absorptive than igneous rocks are.
Schuster (2005)[5] derived an interferometric form of Fermat’s principle that helps us to perform high-resolution estimation of the velocity distribution among deep interfaces. He used Fermat’s interferometric principle to redatum the surface sources and receivers to interface A kinematically. The velocity model above interface A does not need to be known, so the distorting effects of the overburden and statics are eliminated by his target-oriented approach.
none Useful attenuation mechanisms
Phase Input-output Models
↑ Bowman, R., 1955, Scattering of seismic waves by small inhomogeneities: Ph.D. thesis, Department of Geology and Geophysics, MIT.
↑ Ricker, N., 1940, The form and nature of seismic waves and the structure of seismograms: Geophysics, 5, 348–366.
↑ Ricker, N., 1941, A note on the determination of the viscosity of shale from the measurement of wavelet breadth: Geophysics, 6, 254–258.
↑ Ricker, N., 1953, The form and laws of propagation of seismic wavelets: Geophysics, 18, 10–40.
↑ Schuster, G. T., 2005, Fermat’s interferometric principle for target-oriented traveltime tomography: Geophysics, 70, no. 4, U47–U50.
Retrieved from "https://wiki.seg.org/index.php?title=Attenuation/en&oldid=172064" |
Division algebras and noncommensurable isospectral manifolds
1 November 2006 Division algebras and noncommensurable isospectral manifolds
A. W. Reid [R, Theorem 2.1] showed that if
{\Gamma }_{1}
{\Gamma }_{2}
are arithmetic lattices in
G={\mathrm{PGL}}_{2}\left(\mathbb{R}\right)
{\mathrm{PGL}}_{2}\left(\mathbb{C}\right)
which give rise to isospectral manifolds, then
{\Gamma }_{1}
{\Gamma }_{2}
are commensurable (after conjugation). We show that for
d\ge 3
\mathcal{S}={\mathrm{PGL}}_{d}\left(\mathbb{R}\right)/{\mathrm{PO}}_{d}\left(\mathbb{R}\right)
\mathcal{S}={\mathrm{PGL}}_{d}\left(\mathbb{C}\right)/{\mathrm{PU}}_{d}\left(\mathbb{C}\right)
, the situation is quite different; there are arbitrarily large finite families of isospectral noncommensurable compact manifolds covered by
\mathcal{S}
. The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants
Alexander Lubotzky. Beth Samuels. Uzi Vishne. "Division algebras and noncommensurable isospectral manifolds." Duke Math. J. 135 (2) 361 - 379, 1 November 2006. https://doi.org/10.1215/S0012-7094-06-13525-1
Alexander Lubotzky, Beth Samuels, Uzi Vishne "Division algebras and noncommensurable isospectral manifolds," Duke Mathematical Journal, Duke Math. J. 135(2), 361-379, (1 November 2006) |
Glycine—tRNA ligase - Wikipedia
Glycine—tRNA ligase
2PME, 2PMF, 2Q5H, 2Q5I, 2ZT5, 2ZT6, 2ZT7, 2ZT8, 2ZXF, 4KQE, 4KR2, 4QEI, 5E6M
GARS1, CMT2D, DSMAV, GlyRS, HMN5, SMAD1, glycyl-tRNA synthetase, GARS, glycyl-tRNA synthetase 1, HMN5A, SMAJI
OMIM: 600287 MGI: 2449057 HomoloGene: 1547 GeneCards: GARS1
bis(5'-nucleosyl)-tetraphosphatase (asymmetrical) activity
Glycine—tRNA ligase also known as glycyl–tRNA synthetase is an enzyme that in humans is encoded by the GARS1 gene.[5][6][7]
This gene encodes glycyl-tRNA synthetase, one of the aminoacyl-tRNA synthetases that charge tRNAs with their cognate amino acids. The encoded enzyme is an (alpha)2 dimer which belongs to the class II family of tRNA synthetases.[7]
In enzymology, a glycine-tRNA ligase (EC 6.1.1.14) is an enzyme that catalyzes the chemical reaction
ATP + glycine + tRNAGly
{\displaystyle \rightleftharpoons }
AMP + diphosphate + glycyl-tRNAGly
The 3 substrates of this enzyme are ATP, glycine, and tRNA(Gly), whereas its 3 products are AMP, diphosphate, and glycyl-tRNA(Gly).
This enzyme belongs to the family of ligases, to be specific those forming carbon-oxygen bonds in aminoacyl-tRNA and related compounds. The systematic name of this enzyme class is glycine:tRNAGly ligase (AMP-forming). Other names in common use include glycyl-tRNA synthetase, glycyl-transfer ribonucleate synthetase, glycyl-transfer RNA synthetase, glycyl-transfer ribonucleic acid synthetase, and glycyl translase. This enzyme participates in glycine, serine and threonine metabolism and aminoacyl-trna biosynthesis.
Glycyl-tRNA synthetase has been shown to interact with EEF1D.[8] Mutant forms of the protein associated with peripheral nerve disease have been shown to aberrantly bind to the transmembrane receptor proteins neuropilin 1[9] and Trk receptors A-C.[10]
Glycyl-tRNA synthetase has been shown to be a target of autoantibodies in the human autoimmune diseases, polymyositis or dermatomyositis.[7]
The peripheral nerve diseases Charcot-Marie-Tooth disease type 2D (CMT2D) and distal spinal muscular atrophy type V (dSMA-V) have been liked to dominant mutations in GARS.[11][12] CMT2D usually manifests during the teenage years, and results in muscle weakness predominantly in the hands and feet.[13] Two mouse models of CMT2D have been used to better understand the disease, identifying that the disorder is caused by a toxic gain-of-function of the mutant glycine-tRNA ligase protein.[14] The CMT2D mice display peripheral nerve axon degeneration [15][16] and defective development[17] and function[18]> of the neuromuscular junction.
^ Nichols RC, Pai SI, Ge Q, Targoff IN, Plotz PH, Liu P (Nov 1995). "Localization of two human autoantigen genes by PCR screening and in situ hybridization--glycyl-tRNA synthetase locates to 7p15 and alanyl-tRNA synthetase locates to 16q22". Genomics. 30 (1): 131–2. doi:10.1006/geno.1995.0028. PMID 8595897.
^ Ionasescu V, Searby C, Sheffield VC, Roklina T, Nishimura D, Ionasescu R (Sep 1996). "Autosomal dominant Charcot-Marie-Tooth axonal neuropathy mapped on chromosome 7p (CMT2D)". Human Molecular Genetics. 5 (9): 1373–5. CiteSeerX 10.1.1.588.4502. doi:10.1093/hmg/5.9.1373. PMID 8872480.
^ a b c "Entrez Gene: GARS glycyl-tRNA synthetase".
^ Sang Lee J, Gyu Park S, Park H, Seol W, Lee S, Kim S (Feb 2002). "Interaction network of human aminoacyl-tRNA synthetases and subunits of elongation factor 1 complex". Biochemical and Biophysical Research Communications. 291 (1): 158–64. doi:10.1006/bbrc.2002.6398. PMID 11829477.
^ He W, Bai G, Zhou H, Wei N, White NM, Lauer J, Liu H, Shi Y, Dumitru CD, Lettieri K, Shubayev V, Jordanova A, Guergueltcheva V, Griffin PR, Burgess RW, Pfaff SL, Yang XL (2015). "CMT2D neuropathy is linked to the neomorphic binding activity of glycyl-tRNA synthetase". Nature. 526 (7575): 710–4. Bibcode:2015Natur.526..710H. doi:10.1038/nature15510. PMC 4754353. PMID 26503042.
^ Sleigh JN, Dawes JM, West SJ, Wei N, Spaulding EL, Gómez-Martín A, Zhang Q, Burgess RW, Cader MZ, Talbot K, Yang XL, Bennett DL, Schiavo G (2017). "Trk receptor signaling and sensory neuron fate are perturbed in human neuropathy caused by Gars mutations". Proc Natl Acad Sci U S A. 114 (16): E3324–E3333. doi:10.1073/pnas.1614557114. PMC 5402433. PMID 28351971.
^ Motley WW, Talbot K, Fischbeck KH (Feb 2010). "GARS axonopathy: not every neuron's cup of tRNA". Trends in Neurosciences. 33 (2): 59–66. doi:10.1016/j.tins.2009.11.001. PMC 2822721. PMID 20152552.
^ Antonellis A, Ellsworth RE, Sambuughin N, Puls I, Abel A, Lee-Lin SQ, Jordanova A, Kremensky I, Christodoulou K, Middleton LT, Sivakumar K, Ionasescu V, Funalot B, Vance JM, Goldfarb LG, Fischbeck KH, Green ED (2003). "Glycyl tRNA synthetase mutations in Charcot-Marie-Tooth disease type 2D and distal spinal muscular atrophy type V". American Journal of Human Genetics. 72 (5): 1293–9. doi:10.1086/375039. PMC 1180282. PMID 12690580.
^ Sivakumar K, Kyriakides T, Puls I, Nicholson GA, Funalot B, Antonellis A, Sambuughin N, Christodoulou K, Beggs JL, Zamba-Papanicolaou E, Ionasescu V, Dalakas MC, Green ED, Fischbeck KH, Goldfarb LG (Oct 2005). "Phenotypic spectrum of disorders associated with glycyl-tRNA synthetase mutations". Brain. 128 (Pt 10): 2304–14. doi:10.1093/brain/awh590. PMID 16014653.
^ Motley WW, Seburn KL, Nawaz MH, Miers KE, Cheng J, Antonellis A, Green ED, Talbot K, Yang XL, Fischbeck KH, Burgess RW (Dec 2011). "Charcot-Marie-Tooth-linked mutant GARS is toxic to peripheral neurons independent of wild-type GARS levels". PLOS Genetics. 7 (12): e1002399. doi:10.1371/journal.pgen.1002399. PMC 3228828. PMID 22144914.
^ Seburn KL, Nangle LA, Cox GA, Schimmel P, Burgess RW (Sep 2006). "An active dominant mutation of glycyl-tRNA synthetase causes neuropathy in a Charcot-Marie-Tooth 2D mouse model". Neuron. 51 (6): 715–26. doi:10.1016/j.neuron.2006.08.027. PMID 16982418. S2CID 11035583.
^ Achilli F, Bros-Facer V, Williams HP, Banks GT, AlQatari M, Chia R, Tucci V, Groves M, Nickols CD, Seburn KL, Kendall R, Cader MZ, Talbot K, van Minnen J, Burgess RW, Brandner S, Martin JE, Koltzenburg M, Greensmith L, Nolan PM, Fisher EM (Jul–Aug 2009). "An ENU-induced mutation in mouse glycyl-tRNA synthetase (GARS) causes peripheral sensory and motor phenotypes creating a model of Charcot-Marie-Tooth type 2D peripheral neuropathy". Disease Models & Mechanisms. 2 (7–8): 359–73. doi:10.1242/dmm.002527. PMC 2707104. PMID 19470612.
^ Sleigh JN, Grice SJ, Burgess RW, Talbot K, Cader MZ (May 2014). "Neuromuscular junction maturation defects precede impaired lower motor neuron connectivity in Charcot-Marie-Tooth type 2D mice". Human Molecular Genetics. 23 (10): 2639–50. doi:10.1093/hmg/ddt659. PMC 3990164. PMID 24368416.
^ Spaulding EL, Sleigh JN, Morelli KH, Pinter MJ, Burgess RW, Seburn KL (2016). "Synaptic Deficits at Neuromuscular Junctions in Two Mouse Models of Charcot-Marie-Tooth Type 2d". The Journal of Neuroscience. 36 (11): 3254–67. doi:10.1523/JNEUROSCI.1762-15.2016. PMC 4792937. PMID 26985035.
Fraser MJ (May 1963). "Glycyl-RNA synthetase of rat liver: partial purification and effects of some metal ions on its activity". Canadian Journal of Biochemistry and Physiology. 41 (5): 1123–33. doi:10.1139/o63-128. PMID 13959340.
Niyomporn B, Dahl JL, Strominger JL (Feb 1968). "Biosynthesis of the peptidoglycan of bacterial cell walls. IX. Purification and properties of glycyl transfer ribonucleic acid synthetase from Staphylococcus aureus". The Journal of Biological Chemistry. 243 (4): 773–8. doi:10.1016/S0021-9258(19)81732-5. PMID 4295604.
Hipps D, Shiba K, Henderson B, Schimmel P (Jun 1995). "Operational RNA code for amino acids: species-specific aminoacylation of minihelices switched by a single nucleotide". Proceedings of the National Academy of Sciences of the United States of America. 92 (12): 5550–2. Bibcode:1995PNAS...92.5550H. doi:10.1073/pnas.92.12.5550. PMC 41733. PMID 7539919.
Williams J, Osvath S, Khong TF, Pearse M, Power D (Apr 1995). "Cloning, sequencing and bacterial expression of human glycine tRNA synthetase". Nucleic Acids Research. 23 (8): 1307–10. doi:10.1093/nar/23.8.1307. PMC 306854. PMID 7753621.
Ge Q, Trieu EP, Targoff IN (Nov 1994). "Primary structure and functional expression of human Glycyl-tRNA synthetase, an autoantigen in myositis". The Journal of Biological Chemistry. 269 (46): 28790–7. doi:10.1016/S0021-9258(19)61975-7. PMID 7961834.
Shiba K, Schimmel P, Motegi H, Noda T (Nov 1994). "Human glycyl-tRNA synthetase. Wide divergence of primary structure from bacterial counterpart and species-specific aminoacylation". The Journal of Biological Chemistry. 269 (47): 30049–55. doi:10.1016/S0021-9258(18)43986-5. PMID 7962006.
Rho SB, Lee KH, Kim JW, Shiba K, Jo YJ, Kim S (Sep 1996). "Interaction between human tRNA synthetases involves repeated sequence elements". Proceedings of the National Academy of Sciences of the United States of America. 93 (19): 10128–33. Bibcode:1996PNAS...9310128R. doi:10.1073/pnas.93.19.10128. PMC 38348. PMID 8816763.
Mudge SJ, Williams JH, Eyre HJ, Sutherland GR, Cowan PJ, Power DA (Mar 1998). "Complex organisation of the 5'-end of the human glycine tRNA synthetase gene". Gene. 209 (1–2): 45–50. doi:10.1016/S0378-1119(98)00007-9. PMID 9524218.
Kneussel M, Hermann A, Kirsch J, Betz H (Mar 1999). "Hydrophobic interactions mediate binding of the glycine receptor beta-subunit to gephyrin". Journal of Neurochemistry. 72 (3): 1323–6. doi:10.1046/j.1471-4159.1999.0721323.x. PMID 10037506. S2CID 24302707.
Sang Lee J, Gyu Park S, Park H, Seol W, Lee S, Kim S (Feb 2002). "Interaction network of human aminoacyl-tRNA synthetases and subunits of elongation factor 1 complex". Biochemical and Biophysical Research Communications. 291 (1): 158–64. doi:10.1006/bbrc.2002.6398. PMID 11829477.
Antonellis A, Ellsworth RE, Sambuughin N, Puls I, Abel A, Lee-Lin SQ, Jordanova A, Kremensky I, Christodoulou K, Middleton LT, Sivakumar K, Ionasescu V, Funalot B, Vance JM, Goldfarb LG, Fischbeck KH, Green ED (May 2003). "Glycyl tRNA synthetase mutations in Charcot-Marie-Tooth disease type 2D and distal spinal muscular atrophy type V". American Journal of Human Genetics. 72 (5): 1293–9. doi:10.1086/375039. PMC 1180282. PMID 12690580.
Del Bo R, Locatelli F, Corti S, Scarlato M, Ghezzi S, Prelle A, Fagiolari G, Moggio M, Carpo M, Bresolin N, Comi GP (Mar 2006). "Coexistence of CMT-2D and distal SMA-V phenotypes in an Italian family with a GARS gene mutation". Neurology. 66 (5): 752–4. doi:10.1212/01.wnl.0000201275.18875.ac. PMID 16534118. S2CID 1143879.
James PA, Cader MZ, Muntoni F, Childs AM, Crow YJ, Talbot K (Nov 2006). "Severe childhood SMA and axonal CMT due to anticodon binding domain mutations in the GARS gene". Neurology. 67 (9): 1710–2. doi:10.1212/01.wnl.0000242619.52335.bc. PMID 17101916. S2CID 16491162.
Xie W, Schimmel P, Yang XL (Dec 2006). "Crystallization and preliminary X-ray analysis of a native human tRNA synthetase whose allelic variants are associated with Charcot-Marie-Tooth disease". Acta Crystallographica Section F. 62 (Pt 12): 1243–6. doi:10.1107/S1744309106046434. PMC 2225372. PMID 17142907.
Cader MZ, Ren J, James PA, Bird LE, Talbot K, Stammers DK (Jun 2007). "Crystal structure of human wildtype and S581L-mutant glycyl-tRNA synthetase, an enzyme underlying distal spinal muscular atrophy". FEBS Letters. 581 (16): 2959–64. doi:10.1016/j.febslet.2007.05.046. PMID 17544401. S2CID 24584635.
Xie W, Nangle LA, Zhang W, Schimmel P, Yang XL (Jun 2007). "Long-range structural effects of a Charcot-Marie-Tooth disease-causing mutation in human glycyl-tRNA synthetase". Proceedings of the National Academy of Sciences of the United States of America. 104 (24): 9976–81. Bibcode:2007PNAS..104.9976X. doi:10.1073/pnas.0703908104. PMC 1891255. PMID 17545306.
GeneReviews/NCBI/NIH/UW entry on Charcot-Marie-Tooth Neuropathy Type 2
GeneReviews/NCBI/NIH/UW entry on GARS-Associated Axonal Neuropathy, Charcot-Marie-Tooth Neuropathy Type 2D, Distal Spinal Muscular Atrophy V
2pme: The Apo crystal Structure of the glycyl-tRNA synthetase
2pmf: The crystal structure of a human glycyl-tRNA synthetase mutant
Retrieved from "https://en.wikipedia.org/w/index.php?title=Glycine—tRNA_ligase&oldid=1069490026" |
Xinzhi Liu, Piyapong Niamsup, Qiru Wang, Yi Zhang
Delay-Dependent Synchronization for Complex Dynamical Networks with Interval Time-Varying and Switched Coupling Delays
T. Botmart, P. Niamsup
We investigate the local exponential synchronization for complex dynamical networks with interval time-varying delays in the dynamical nodes and the switched coupling term simultaneously. The constraint on the derivative of the time-varying delay is not required which allows the time delay to be a fast time-varying function. By using common unitary matrix for different subnetworks, the problem of synchronization is transformed into the stability analysis of some linear switched delay systems. Then, when subnetworks are synchronizable and nonsynchronizable, a delay-dependent sufficient condition is derived and formulated in the form of linear matrix inequalities (LMIs) by average dwell time approach and piecewise Lyapunov-Krasovskii functionals which are constructed based on the descriptor model of the system and the method of decomposition. The new stability condition is less conservative and is more general than some existing results. A numerical example is also given to illustrate the effectiveness of the proposed method.
p
p
-Stabilizability of Stochastic Nonlinear and Bilinear Hybrid Systems under Stabilizing Switching Rules
Ewelina Seroka, Lesław Socha
The problem of
p
th mean exponential stability and stabilizability of a class of stochastic nonlinear and bilinear hybrid systems with unstable and stable subsystems is considered. Sufficient conditions for the
p
th mean exponential stability and stabilizability under a feedback control and stabilizing switching rules are derived. A method for the construction of stabilizing switching rules based on the Lyapunov technique and the knowledge of regions of decreasing the Lyapunov functions for subsystems is given. Two cases, including single Lyapunov function and a a single Lyapunov-like function, are discussed. Obtained results are illustrated by examples.
Kexue Zhang, Xinzhi Liu
A New Series of Three-Dimensional Chaotic Systems with Cross-Product Nonlinearities and Their Switching
Xinquan Zhao, Feng Jiang, Zhigang Zhang, Junhao Hu
This paper introduces a new series of three-dimensional chaotic systems with cross-product nonlinearities. Based on some conditions, we analyze the globally exponentially or globally conditional exponentially attractive set and positive invariant set of these chaotic systems. Moreover, we give some known examples to show our results, and the exponential estimation is explicitly derived. Finally, we construct some three-dimensional chaotic systems with cross-product nonlinearities and study the switching system between them. |
Glucose-1-phosphate guanylyltransferase - Wikipedia
In enzymology, a glucose-1-phosphate guanylyltransferase (EC 2.7.7.34) is an enzyme that catalyzes the chemical reaction
GTP + alpha-D-glucose 1-phosphate
{\displaystyle \rightleftharpoons }
diphosphate + GDP-glucose
Thus, the two substrates of this enzyme are GTP and alpha-D-glucose 1-phosphate, whereas its two products are diphosphate and GDP-glucose.
This enzyme belongs to the family of transferases, specifically those transferring phosphorus-containing nucleotide groups (nucleotidyltransferases). The systematic name of this enzyme class is GTP:alpha-D-glucose-1-phosphate guanylyltransferase. Other names in common use include GDP glucose pyrophosphorylase, and guanosine diphosphoglucose pyrophosphorylase. This enzyme participates in starch and sucrose metabolism.
Danishefsky I, Heritier-Watkins O (1967). "Nucleoside diphosphate glucose pyrophosphorylases in mast cell tumors". Biochim. Biophys. Acta. 139 (2): 349–57. doi:10.1016/0005-2744(67)90038-1. PMID 6034677.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Glucose-1-phosphate_guanylyltransferase&oldid=917403943" |
Algebraic_number Knowpia
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio,
{\displaystyle (1+{\sqrt {5}})/2}
, is an algebraic number, because it is a root of the polynomial x2 − x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number
{\displaystyle 1+i}
is algebraic because it is a root of x4 + 4.
The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1.
All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as π and e, are called transcendental numbers.
The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental.
All rational numbers are algebraic. Any rational number, expressed as the quotient of an integer a and a (non-zero) natural number b, satisfies the above definition, because x = a/b is the root of a non-zero polynomial, namely bx − a.[1]
Quadratic irrational numbers, irrational solutions of a quadratic polynomial ax2 + bx + c with integer coefficients a, b, and c, are algebraic numbers. If the quadratic polynomial is monic (a = 1), the roots are further qualified as quadratic integers.
Gaussian integers, complex numbers a + bi for which both a and b are integers, are also quadratic integers. This is because a + bi and a - bi are the two roots of the quadratic x2 - 2ax + a2 + b2.
A constructible number can be constructed from a given unit length using a straightedge and compass. It includes all quadratic irrational roots, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots. (By designating cardinal directions for 1, −1, i, and −i, complex numbers such as
{\displaystyle 3+i{\sqrt {2}}}
are considered constructible.)
Any expression formed from algebraic numbers using any combination of the basic arithmetic operations and extraction of nth roots gives another algebraic number.
Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of nth roots (such as the roots of x5 − x + 1). That happens with many but not all polynomials of degree 5 or higher.
Values of trigonometric functions of rational multiples of π (except when undefined): for example, cos π/7, cos 3π/7, and cos 5π/7 satisfy 8x3 − 4x2 − 4x + 1 = 0. This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers. Likewise, tan 3π/16, tan 7π/16, tan 11π/16, and tan 15π/16 satisfy the irreducible polynomial x4 − 4x3 − 6x2 + 4x + 1 = 0, and so are conjugate algebraic integers.
Some but not all irrational numbers are algebraic:
{\displaystyle {\sqrt {2}}}
{\displaystyle {\frac {\sqrt[{3}]{3}}{2}}}
are algebraic since they are roots of polynomials x2 − 2 and 8x3 − 3, respectively.
The golden ratio φ is algebraic since it is a root of the polynomial x2 − x − 1.
The numbers π and e are not algebraic numbers (see the Lindemann–Weierstrass theorem).[2]
Algebraic numbers on the complex plane colored by degree (bright orange/red = 1, green = 2, blue = 3, yellow = 4)
If a polynomial with rational coefficients is multiplied through by the least common denominator, the resulting polynomial with integer coefficients has the same roots. This shows that an algebraic number can be equivalently defined as a root of a polynomial with either integer or rational coefficients.
Given an algebraic number, there is a unique monic polynomial with rational coefficients of least degree that has the number as a root. This polynomial is called its minimal polynomial. If its minimal polynomial has degree n, then the algebraic number is said to be of degree n. For example, all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational.
The algebraic numbers are dense in the reals. This follows from the fact they contain the rational numbers, which are dense in the reals themselves.
The set of algebraic numbers is countable (enumerable),[3][4] and therefore its Lebesgue measure as a subset of the complex numbers is 0 (essentially, the algebraic numbers take up no space in the complex numbers). That is to say, "almost all" real and complex numbers are transcendental.
All algebraic numbers are computable and therefore definable and arithmetical.
For real numbers a and b, the complex number a + bi is algebraic if and only if both a and b are algebraic.[5]
The sum, difference, product and quotient (if the denominator is nonzero) of two algebraic numbers is again algebraic, as can be demonstrated by using the resultant, and algebraic numbers thus form a field
{\displaystyle {\overline {\mathbb {Q} }}}
{\displaystyle \mathbb {A} }
, but that usually denotes the adele ring). Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. That can be rephrased by saying that the field of algebraic numbers is algebraically closed. In fact, it is the smallest algebraically-closed field containing the rationals and so it is called the algebraic closure of the rationals.
The set of real algebraic numbers itself forms a field.[6]
Numbers defined by radicalsEdit
Any number that can be obtained from the integers using a finite number of additions, subtractions, multiplications, divisions, and taking (possibly complex) nth roots where n is a positive integer are algebraic. The converse, however, is not true: there are algebraic numbers that cannot be obtained in this manner. These numbers are roots of polynomials of degree 5 or higher, a result of Galois theory (see Quintic equations and the Abel–Ruffini theorem). For example, the equation:
{\displaystyle x^{5}-x-1=0}
has a unique real root that cannot be expressed in terms of only radicals and arithmetic operations.
Closed-form numberEdit
Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "closed-form numbers", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "elementary numbers", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers explicitly defined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such as e or ln 2.
Algebraic integersEdit
An algebraic integer is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are
{\displaystyle 5+13{\sqrt {2}},}
{\displaystyle 2-6i,}
{\textstyle {\frac {1}{2}}(1+i{\sqrt {3}}).}
Therefore, the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monic polynomials x − k for all
{\displaystyle k\in \mathbb {Z} }
. In this sense, algebraic integers are to algebraic numbers what integers are to rational numbers.
The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring. The name algebraic integer comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any number field are in many ways analogous to the integers. If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as OK. These are the prototypical examples of Dedekind domains.
Special classesEdit
^ Some of the following examples come from Hardy and Wright 1972: 159–160 and pp. 178–179
^ Also, Liouville's theorem can be used to "produce as many examples of transcendental numbers as we please," cf. Hardy and Wright p. 161ff
^ Niven 1956, Corollary 7.3.
Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics, vol. 84 (Second ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2103-4, ISBN 0-387-97329-X, MR 1070716 |
Prime number/Citable Version - Citizendium
< Prime number
The prime number 11 illustrated with square tiles. 12 squares can be arranged into a rectangle with sides of length 3 and 4, so 12 is not a prime number. There is no way to form a full rectangle more than one square wide with 11 squares, so 11 is a prime number.
A prime number is a number that can be evenly divided by exactly two positive whole numbers, namely 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, and 17. A prime number
{\displaystyle p}
cannot be factored as the product of two numbers
{\displaystyle \scriptstyle p=a\times b,}
except for the trivial factorizations
{\displaystyle \scriptstyle p=1\times p=p\times 1}
that all numbers possess.
With the exception of 2, all prime numbers are odd numbers, but not every odd number is prime. For example,
{\displaystyle \scriptstyle 9=3\times 3}
{\displaystyle \scriptstyle 15=3\times 5,}
so neither 9 nor 15 is prime. By the strict mathematical definition, the number 1 is not considered to be prime (although this is a matter of definition, and mathematicians in the past often did consider 1 to be a prime).
The importance of prime numbers in arithmetic comes in large part from the unique factorization of numbers. Every number
{\displaystyle \scriptstyle N>1}
can be written as a product of prime factors, and any two such expressions for
{\displaystyle N}
will differ only in the order of the factors. For example, we can write
{\displaystyle \scriptstyle 5040=7\times 5\times 3\times 3\times 2\times 2\times 2\times 2}
, where all of the factors in the product are prime numbers; there is no other way of writing 5040 as a product of prime numbers except by rearranging the prime factors we already have, such as
{\displaystyle \scriptstyle 5040=2\times 3\times 2\times 7\times 2\times 3\times 2\times 5}
. Because of the unique factorization of numbers into prime numbers, an analogy can be made between the role prime numbers play in arithmetic and the role atoms play in chemistry.
The study of prime numbers has a long history, going back to ancient times, and it remains an active part of number theory today. Although the study of prime numbers used to be an interesting but not terribly useful area of mathematical research, today it has important applications. Understanding properties of prime numbers and their generalizations is essential to modern cryptography—in particular to public key ciphers that are crucial to Internet commerce, wireless networks, and military applications. Less well-known is that other computer algorithms also depend on properties of prime numbers.
1 There are infinitely many primes
2 Locating primes
3 Distribution of prime numbers
4 Some unsolved problems
4.1 The twin prime conjecture
4.3 Primes of special forms
One basic fact about the prime numbers is that there are infinitely many of them. In other words, the list of prime numbers 2, 3, 5, 7, 11, 13, 17, ... never stops. There are a number of ways of showing that there are infinitely many primes, but one of the oldest and most familiar proofs goes back go Euclid.
Euclid proved that for any finite set of prime numbers, there is always another prime number which is not in that set. Choose any finite set of prime numbers
{\displaystyle \scriptstyle \{p_{1},p_{2},p_{3},\ldots ,p_{n}\}}
{\displaystyle N=p_{1}p_{2}\cdots p_{n}+1}
presents a problem. On the one hand, the number
{\displaystyle \scriptstyle N-1=p_{1}p_{2}\cdots p_{n}}
{\displaystyle p_{1}}
, and multiples of the same prime number are never next to each other, so N itself can't be a multiple of
{\displaystyle \scriptstyle p_{1}}
. The same argument actually shows that
{\displaystyle N}
itself cannot be a multiple of any of the prime numbers
{\displaystyle \scriptstyle p_{1},p_{2},p_{3},\dots ,p_{n}}
. On the other hand, every number
{\displaystyle \scriptstyle N>1}
is divisible by some prime. (For example, its smallest divisor greater than 1 must be a prime.[1])
This shows that there is at least one prime number (namely, the smallest divisor of N greater than 1) that is excluded from our initial finite set. Since any finite set of prime numbers can thus be extended to a larger finite set of prime numbers, we conclude that there are infinitely many prime numbers.
Although most other proofs of the infinitude of prime numbers are more complicated, they can also provide more information. One mathematical milestone known as the Prime Number Theorem estimates how many of the numbers between 1 and x are prime numbers (see below). Another such proof is Euler's demonstration that the sum of the reciprocals of the primes diverges to ∞.
Locating primes
How can we tell which numbers are prime and which are not? It is sometimes possible to tell that a number is not prime by looking at its digits: for example, any number whose last digit is even is divisble by 2, and any number ending with 5 or 0 is divisible by 5. Therefore, except for the prime numbers 2 and 5, any prime number must end with the digit 1, 3, 7, or 9. This check can be used to rule out the possibility of a randomly chosen number being prime more than half of the time, but numbers that end with 1, 3, 7, or 9 often have divisors that are harder to spot.
To find large prime numbers, we must use a systematic procedure — an algorithm. Nowadays, prime-finding calculations are performed by computers, often using very complicated algorithms, but there are simple algorithms that can be carried out by hand if the numbers are small. In fact, the simplest methods for locating prime numbers are some of the oldest algorithms, known since antiquity. Two classical algorithms are called trial division and the sieve of Eratosthenes.
Trial division consists of systematically searching the list of numbers
{\displaystyle \scriptstyle 2,3,\dots ,n-1}
for a divisor; if none is found, the number is prime. If n has a small divisor, we can quit as soon as we've found it, but in the worst case — if n is prime — we have to test all
{\displaystyle \scriptstyle n-2}
numbers to be sure. This algorithm can be improved by realizing the following: if n has a divisor a that is larger than
{\displaystyle \scriptstyle {\sqrt {n}}}
, there must be another divisor b that is smaller than
{\displaystyle \scriptstyle {\sqrt {n}}}
. Thus, it is sufficient to look for a divisor up to
{\displaystyle \scriptstyle {\sqrt {n}}}
. This makes a significant difference: for example, we only need to try dividing by 2, 3, ..., 31 to verify that 997 is prime, rather than all the numbers 2, 3, ..., 996. Trial division might be described as follows using pseudocode:
Algorithm: trial division
Give{\displaystyle n}
{\displaystyle i}
= 2, 3, ... less than or equal to
{\displaystyle \scriptstyle {\sqrt {n}}}
{\displaystyle i}
{\displaystyle n}
Return "
{\displaystyle n}
is not prime"
Continue with the next
{\displaystyle i}
{\displaystyle i}
have been checked:
{\displaystyle n}
is prime"
The Sieve of Eratosthenes not only provides a method for testing a number to see if it is prime, but also for enumerating the (infinite) set of prime numbers. The idea of the method to write down a list of numbers starting from 2 ranging up to some limit, say:
The first number (2) is prime, so we mark it, and cross out all of its multiples:
The smallest unmarked number is 3, so we mark it and cross out all its multiples (some of which may already have been crossed out):
The smallest unmarked number (5) is the next prime, so we mark it and cross out all of its multiples:
Notice that there are no multiples of 5
{\displaystyle \scriptstyle (\leq 20)}
that have not already been crossed out, but that doesn't matter at this stage. Proceeding as before, we add 7, 11, 13, 17 and 19 to our list of primes:
We have now found all prime numbers up to 20.
It is evident that the prime numbers are randomly distributed but, unfortunately, we don't know what 'random' means. - R. C. Vaughan
The list of prime numbers suggests that they thin out the further you go: 44% of the one-digit numbers are prime, but only 23% of the two-digit numbers and 16% of the three-digit numbers. The trial division method explained above provides an intuitive explanation. To test whether a number
{\displaystyle n}
is prime, you have to try whether it can be divided by all numbers between 2 and
{\displaystyle \scriptstyle {\sqrt {n}}}
. Large numbers have to undergo more tests, so fewer of them will be prime.
The Prime Number Theorem explains how fast the prime numbers thin out. It says that if you are looking around the number
{\displaystyle n}
, about one in every
{\displaystyle \scriptstyle \log \,n}
numbers is prime (here,
{\displaystyle \scriptstyle \log \,n}
denotes the natural logarithm of
{\displaystyle n}
). The formal statement of the prime number theorem is
{\displaystyle \lim _{x\to \infty }{\frac {\pi (x)\log x}{x}}=1}
{\displaystyle \pi (x)}
{\displaystyle \scriptstyle \leq x.}
There are many unsolved problems concerning prime numbers. A few such problems (posed as conjectures) are:
Twin primes are pairs of prime numbers differing by 2. Examples of twin primes include 5 and 7, 11 and 13, and 41 and 43. The Twin Prime Conjecture states that there are an infinite number of these pairs. It remains unproven.
The Goldbach conjecture is that every even number greater than 2 can be expressed as the sum of two primes. For example, if you choose the even number 48, you can find
{\displaystyle 48=41+7}
where 41 and 7 are prime numbers.
Primes of special forms
It is not known whether there are infinitely many primes of the form
{\displaystyle 2^{n}-1}
(called Mersenne primes). These primes arise in the study of perfect numbers, and factors of numbers of the form
{\displaystyle 2^{n}-1}
(sometimes called 'Mersenne numbers') are a fruitful source of large prime numbers.
{\displaystyle 2^{n}+1}
(called 'Fermat primes'). Fermat primes arise in elementary geometry because if
{\displaystyle p}
is a Fermat prime, it is possible to construct a regular
{\displaystyle p}
-gon with a ruler and compass. In particular, it is possible to construct a regular 17-sided polygon (or 17-gon, for short) with a ruler and compass.
{\displaystyle n^{2}+1}
A prime number is usually defined as a positive integer other than 1 that is (evenly) divisible only by 1 and itself.
There is another way of defining prime numbers, and that is that a number is prime if whenever it divides the product of two numbers, it must divide one of those numbers. A nonexample (if you will) is that 4 divides 12 (i.e. is a factor of 12), but 4 does not divide 2 and 4 does not divide 6 even though 12 is 2 times 6. This means that 4 is not a prime number. We may express this second possible definition in mathematical notation as follows: A number
{\displaystyle \scriptstyle p\in \mathbb {N} }
(natural number) is prime if for any
{\displaystyle \scriptstyle a,b\in \mathbb {N} }
{\displaystyle \scriptstyle p|ab}
(read p divides ab), either
{\displaystyle p|a}
{\displaystyle p|b}
If the first characterization of prime numbers is taken as the definition, the second is derived from it as a theorem (known as Euclid's lemma), and vice versa. The equivalence of these two definitions (in the integers
{\displaystyle \scriptstyle \mathbb {Z} }
) is not immediately obvious. In fact, it is a significant result.[2]
↑ The number N has at least one divisor greater than 1, because N is a divisor of itself. Let q denote the smallest divisor of N greater than 1. To see why q must be prime: Any divisor of q is also a divisor of N, and since q is the smallest, then q has no divisors smaller than itself except 1, therefore it is prime.
↑ The Euclidean algorithm may be used to show that
{\displaystyle \scriptstyle \mathbb {Z} }
is a principal ideal domain, and this implies that irreducibles are prime.
Apostol, Tom M. (1976). Introduction to Analytic Number Theory. Springer-Verlag. ISBN 0-387-90163-9.
Ribenboim, Paulo (2004). The Little Book of Bigger Primes, second edition. Springer-Verlag. ISBN 0-387-20169-6.
Scharlau, Winfried; Hans Opolka (1985). From Fermat to Minkowski: Lectures on the Theory of Numbers and its Historical Development. Springer-Verlag. ISBN 0-387-90942-7.
(Note that Scharlau and Opolka was originally published as: Scharlau, Winfried; Hans Opolka (1980). Von Fermat bis Minkowski: Eine Vorlesung über Zahlentheorie und ihre Entwicklung. Springer-Verlag. ).
Retrieved from "https://citizendium.org/wiki/index.php?title=Prime_number/Citable_Version&oldid=22908" |
Capillary wave - Wikipedia
(Redirected from Capillary waves)
Wave on the surface of a fluid, dominated by surface tension
This article is about ripples on fluid interfaces. For ripples in electricity, see Ripple (electrical). For other uses, see Ripple (disambiguation).
Capillary wave (ripple) in water
Ripples on Lifjord in Øksnes, Norway
Capillary waves produced by droplet impacts on the interface between water and air.
A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension.
Capillary waves are common in nature, and are often referred to as ripples. The wavelength of capillary waves on water is typically less than a few centimeters, with a phase speed in excess of 0.2–0.3 meter/second.
A longer wavelength on a fluid interface will result in gravity–capillary waves which are influenced by both the effects of surface tension and gravity, as well as by fluid inertia. Ordinary gravity waves have a still longer wavelength.
When generated by light wind in open water, a nautical name for them is cat's paw waves. Light breezes which stir up such small ripples are also sometimes referred to as cat's paws. On the open ocean, much larger ocean surface waves (seas and swells) may result from coalescence of smaller wind-caused ripple-waves.
1.1 Capillary waves, proper
1.2 Gravity–capillary waves
1.2.1 Gravity wave regime
1.2.2 Capillary wave regime
1.2.3 Phase velocity minimum
Dispersion relation[edit]
The dispersion relation describes the relationship between wavelength and frequency in waves. Distinction can be made between pure capillary waves – fully dominated by the effects of surface tension – and gravity–capillary waves which are also affected by gravity.
Capillary waves, proper[edit]
The dispersion relation for capillary waves is
{\displaystyle \omega ^{2}={\frac {\sigma }{\rho +\rho '}}\,|k|^{3},}
{\displaystyle \omega }
{\displaystyle \sigma }
the surface tension,
{\displaystyle \rho }
the density of the heavier fluid,
{\displaystyle \rho '}
the density of the lighter fluid and
{\displaystyle k}
the wavenumber. The wavelength is
{\displaystyle \lambda ={\frac {2\pi }{k}}.}
For the boundary between fluid and vacuum (free surface), the dispersion relation reduces to
{\displaystyle \omega ^{2}={\frac {\sigma }{\rho }}\,|k|^{3}.}
Gravity–capillary waves[edit]
Dispersion of gravity–capillary waves on the surface of deep water (zero mass density of upper layer,
{\displaystyle \rho '=0}
). Phase and group velocity divided by
{\displaystyle \scriptstyle {\sqrt[{4}]{g\sigma /\rho }}}
as a function of inverse relative wavelength
{\displaystyle \scriptstyle {\frac {1}{\lambda }}{\sqrt {\sigma /(\rho g)}}}
• Blue lines (A): phase velocity, Red lines (B): group velocity.
• Drawn lines: dispersion relation for gravity–capillary waves.
• Dashed lines: dispersion relation for deep-water gravity waves.
• Dash-dotted lines: dispersion relation valid for deep-water capillary waves.
In general, waves are also affected by gravity and are then called gravity–capillary waves. Their dispersion relation reads, for waves on the interface between two fluids of infinite depth:[1][2]
{\displaystyle \omega ^{2}=|k|\left({\frac {\rho -\rho '}{\rho +\rho '}}g+{\frac {\sigma }{\rho +\rho '}}k^{2}\right),}
{\displaystyle g}
{\displaystyle \rho }
{\displaystyle \rho '}
are the mass density of the two fluids
{\displaystyle (\rho >\rho ')}
{\displaystyle (\rho -\rho ')/(\rho +\rho ')}
in the first term is the Atwood number.
Gravity wave regime[edit]
Further information: Airy wave theory
For large wavelengths (small
{\displaystyle k=2\pi /\lambda }
), only the first term is relevant and one has gravity waves. In this limit, the waves have a group velocity half the phase velocity: following a single wave's crest in a group one can see the wave appearing at the back of the group, growing and finally disappearing at the front of the group.
Capillary wave regime[edit]
Shorter (large
{\displaystyle k}
) waves (e.g. 2 mm for the water–air interface), which are proper capillary waves, do the opposite: an individual wave appears at the front of the group, grows when moving towards the group center and finally disappears at the back of the group. Phase velocity is two thirds of group velocity in this limit.
Phase velocity minimum[edit]
Between these two limits is a point at which the dispersion caused by gravity cancels out the dispersion due to the capillary effect. At a certain wavelength, the group velocity equals the phase velocity, and there is no dispersion. At precisely this same wavelength, the phase velocity of gravity–capillary waves as a function of wavelength (or wave number) has a minimum. Waves with wavelengths much smaller than this critical wavelength
{\displaystyle \lambda _{m}}
are dominated by surface tension, and much above by gravity. The value of this wavelength and the associated minimum phase speed
{\displaystyle c_{m}}
{\displaystyle \lambda _{m}=2\pi {\sqrt {\frac {\sigma }{(\rho -\rho ')g}}}\quad {\text{and}}\quad c_{m}={\sqrt {\frac {2{\sqrt {(\rho -\rho ')g\sigma }}}{\rho +\rho '}}}.}
For the air–water interface,
{\displaystyle \lambda _{m}}
is found to be 1.7 cm (0.67 in), and
{\displaystyle c_{m}}
is 0.23 m/s (0.75 ft/s).[1]
If one drops a small stone or droplet into liquid, the waves then propagate outside an expanding circle of fluid at rest; this circle is a caustic which corresponds to the minimal group velocity.[3]
As Richard Feynman put it, "[water waves] that are easily seen by everyone and which are usually used as an example of waves in elementary courses [...] are the worst possible example [...]; they have all the complications that waves can have."[4] The derivation of the general dispersion relation is therefore quite involved.[5]
There are three contributions to the energy, due to gravity, to surface tension, and to hydrodynamics. The first two are potential energies, and responsible for the two terms inside the parenthesis, as is clear from the appearance of
{\displaystyle g}
{\displaystyle \sigma }
. For gravity, an assumption is made of the density of the fluids being constant (i.e., incompressibility), and likewise
{\displaystyle g}
(waves are not high enough for gravitation to change appreciably). For surface tension, the deviations from planarity (as measured by derivatives of the surface) are supposed to be small. For common waves both approximations are good enough.
The third contribution involves the kinetic energies of the fluids. It is the most complicated and calls for a hydrodynamic framework. Incompressibility is again involved (which is satisfied if the speed of the waves is much less than the speed of sound in the media), together with the flow being irrotational – the flow is then potential. These are typically also good approximations for common situations.
The resulting equation for the potential (which is Laplace equation) can be solved with the proper boundary conditions. On one hand, the velocity must vanish well below the surface (in the "deep water" case, which is the one we consider, otherwise a more involved result is obtained, see Ocean surface waves.) On the other, its vertical component must match the motion of the surface. This contribution ends up being responsible for the extra
{\displaystyle k}
outside the parenthesis, which causes all regimes to be dispersive, both at low values of
{\displaystyle k}
, and high ones (except around the one value at which the two dispersions cancel out.)
Dispersion relation for gravity–capillary waves on an interface between two semi–infinite fluid domains
Consider two fluid domains, separated by an interface with surface tension. The mean interface position is horizontal. It separates the upper from the lower fluid, both having a different constant mass density,
{\displaystyle \rho }
{\displaystyle \rho '}
for the lower and upper domain respectively. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. Then the flows are potential, and the velocity in the lower and upper layer can be obtained from
{\displaystyle \nabla \phi }
{\displaystyle \nabla \phi '}
{\displaystyle \phi (x,y,z,t)}
{\displaystyle \phi '(x,y,z,t)}
are velocity potentials.
Three contributions to the energy are involved: the potential energy
{\displaystyle V_{g}}
due to gravity, the potential energy
{\displaystyle V_{st}}
due to the surface tension and the kinetic energy
{\displaystyle T}
of the flow. The part
{\displaystyle V_{g}}
due to gravity is the simplest: integrating the potential energy density due to gravity,
{\displaystyle \rho gz}
{\displaystyle \rho 'gz}
) from a reference height to the position of the surface,
{\displaystyle z=\eta (x,y,t)}
{\displaystyle V_{\mathrm {g} }=\iint dx\,dy\;\int _{0}^{\eta }dz\;(\rho -\rho ')gz={\frac {1}{2}}(\rho -\rho ')g\iint dx\,dy\;\eta ^{2},}
assuming the mean interface position is at
{\displaystyle z=0}
An increase in area of the surface causes a proportional increase of energy due to surface tension:[7]
{\displaystyle V_{\mathrm {st} }=\sigma \iint dx\,dy\;\left[{\sqrt {1+\left({\frac {\partial \eta }{\partial x}}\right)^{2}+\left({\frac {\partial \eta }{\partial y}}\right)^{2}}}-1\right]\approx {\frac {1}{2}}\sigma \iint dx\,dy\;\left[\left({\frac {\partial \eta }{\partial x}}\right)^{2}+\left({\frac {\partial \eta }{\partial y}}\right)^{2}\right],}
where the first equality is the area in this (Monge's) representation, and the second applies for small values of the derivatives (surfaces not too rough).
The last contribution involves the kinetic energy of the fluid:[8]
{\displaystyle T={\frac {1}{2}}\iint dx\,dy\;\left[\int _{-\infty }^{\eta }dz\;\rho \,\left|\mathbf {\nabla } \Phi \right|^{2}+\int _{\eta }^{+\infty }dz\;\rho '\,\left|\mathbf {\nabla } \Phi '\right|^{2}\right].}
Use is made of the fluid being incompressible and its flow is irrotational (often, sensible approximations). As a result, both
{\displaystyle \phi (x,y,z,t)}
{\displaystyle \phi '(x,y,z,t)}
must satisfy the Laplace equation:[9]
{\displaystyle \nabla ^{2}\Phi =0}
{\displaystyle \nabla ^{2}\Phi '=0.}
These equations can be solved with the proper boundary conditions:
{\displaystyle \phi }
{\displaystyle \phi '}
must vanish well away from the surface (in the "deep water" case, which is the one we consider).
Using Green's identity, and assuming the deviations of the surface elevation to be small (so the z–integrations may be approximated by integrating up to
{\displaystyle z=0}
{\displaystyle z=\eta }
), the kinetic energy can be written as:[8]
{\displaystyle T\approx {\frac {1}{2}}\iint dx\,dy\;\left[\rho \,\Phi \,{\frac {\partial \Phi }{\partial z}}\;-\;\rho '\,\Phi '\,{\frac {\partial \Phi '}{\partial z}}\right]_{{\text{at }}z=0}.}
To find the dispersion relation, it is sufficient to consider a sinusoidal wave on the interface, propagating in the x–direction:[7]
{\displaystyle \eta =a\,\cos \,(kx-\omega t)=a\,\cos \,\theta ,}
{\displaystyle a}nd wave phase
{\displaystyle \theta =kx-\omega t}
. The kinematic boundary condition at the interface, relating the potentials to the interface motion, is that the vertical velocity components must match the motion of the surface:[7]
{\displaystyle {\frac {\partial \Phi }{\partial z}}={\frac {\partial \eta }{\partial t}}}
{\displaystyle {\frac {\partial \Phi '}{\partial z}}={\frac {\partial \eta }{\partial t}}}
{\displaystyle z=0}
To tackle the problem of finding the potentials, one may try separation of variables, when both fields can be expressed as:[7]
{\displaystyle {\begin{aligned}\Phi (x,y,z,t)&=+{\frac {1}{|k|}}{\text{e}}^{+|k|z}\,\omega a\,\sin \,\theta ,\\\Phi '(x,y,z,t)&=-{\frac {1}{|k|}}{\text{e}}^{-|k|z}\,\omega a\,\sin \,\theta .\end{aligned}}}
Then the contributions to the wave energy, horizontally integrated over one wavelength
{\displaystyle \lambda =2\pi /k}
in the x–direction, and over a unit width in the y–direction, become:[7][10]
{\displaystyle {\begin{aligned}V_{\text{g}}&={\frac {1}{4}}(\rho -\rho ')ga^{2}\lambda ,\\V_{\text{st}}&={\frac {1}{4}}\sigma k^{2}a^{2}\lambda ,\\T&={\frac {1}{4}}(\rho +\rho '){\frac {\omega ^{2}}{|k|}}a^{2}\lambda .\end{aligned}}}
The dispersion relation can now be obtained from the Lagrangian
{\displaystyle L=T-V}
{\displaystyle V}
the sum of the potential energies by gravity
{\displaystyle V_{g}}
{\displaystyle V_{st}}
{\displaystyle L={\frac {1}{4}}\left[(\rho +\rho '){\frac {\omega ^{2}}{|k|}}-(\rho -\rho ')g-\sigma k^{2}\right]a^{2}\lambda .}
For sinusoidal waves and linear wave theory, the phase–averaged Lagrangian is always of the form
{\displaystyle L=D(\omega ,k)a^{2}}
, so that variation with respect to the only free parameter,
{\displaystyle a}
, gives the dispersion relation
{\displaystyle D(\omega ,k)=0}
.[11] In our case
{\displaystyle D(\omega ,k)}
is just the expression in the square brackets, so that the dispersion relation is:
{\displaystyle \omega ^{2}=|k|\left({\frac {\rho -\rho '}{\rho +\rho '}}\,g+{\frac {\sigma }{\rho +\rho '}}\,k^{2}\right),}
As a result, the average wave energy per unit horizontal area,
{\displaystyle (T+V)/\lambda }
{\displaystyle {\bar {E}}={\frac {1}{2}}\,\left[(\rho -\rho ')\,g+\sigma k^{2}\right]\,a^{2}.}
As usual for linear wave motions, the potential and kinetic energy are equal (equipartition holds):
{\displaystyle T=V}
Thermal capillary wave
Wave-formed ripple
Ripples on water created by water striders
Light breeze ripples in the surface water of a lake
^ a b c Lamb (1994), §267, page 458–460.
^ Dingemans (1997), Section 2.1.1, p. 45.
Phillips (1977), Section 3.2, p. 37.
^ Falkovich, G. (2011). Fluid Mechanics, a short course for physicists. Cambridge University Press. Section 3.1 and Exercise 3.3. ISBN 978-1-107-00575-4.
^ R.P. Feynman, R.B. Leighton, and M. Sands (1963). The Feynman Lectures on Physics. Addison-Wesley. Volume I, Chapter 51-4.
^ See e.g. Safran (1994) for a more detailed description.
^ Lamb (1994), §174 and §230.
^ a b c d e Lamb (1994), §266.
^ a b Lamb (1994), §61.
^ Lamb (1994), §20
^ Lamb (1994), §230.
^ a b Whitham, G. B. (1974). Linear and nonlinear waves. Wiley-Interscience. ISBN 0-471-94090-9. See section 11.7.
^ Lord Rayleigh (J. W. Strutt) (1877). "On progressive waves". Proceedings of the London Mathematical Society. 9: 21–26. doi:10.1112/plms/s1-9.1.21. Reprinted as Appendix in: Theory of Sound 1, MacMillan, 2nd revised edition, 1894.
Longuet-Higgins,M. S. (1963). "The generation of capillary waves by steep gravity waves". Journal of Fluid Mechanics. 16 (1): 138–159. Bibcode:1963JFM....16..138L. doi:10.1017/S0022112063000641. ISSN 1469-7645.
Lamb, H. (1994). Hydrodynamics (6th ed.). Cambridge University Press. ISBN 978-0-521-45868-9.
Phillips, O. M. (1977). The dynamics of the upper ocean (2nd ed.). Cambridge University Press. ISBN 0-521-29801-6.
Dingemans, M. W. (1997). Water wave propagation over uneven bottoms. Advanced Series on Ocean Engineering. Vol. 13. World Scientific, Singapore. pp. 2 Parts, 967 pages. ISBN 981-02-0427-2.
Safran, Samuel (1994). Statistical thermodynamics of surfaces, interfaces, and membranes. Addison-Wesley.
Tufillaro, N. B.; Ramshankar, R.; Gollub, J. P. (1989). "Order-disorder transition in capillary ripples". Physical Review Letters. 62 (4): 422–425. Bibcode:1989PhRvL..62..422T. doi:10.1103/PhysRevLett.62.422. PMID 10040229.
Wikimedia Commons has media related to Ripples (water waves).
Capillary waves entry at sklogwiki
Retrieved from "https://en.wikipedia.org/w/index.php?title=Capillary_wave&oldid=1000083497" |
The Rational(f, k) command computes a closed form of the indefinite sum of
k
f\left(k\right)
s\left(k\right)
t\left(k\right)
f\left(k\right)=s\left(k+1\right)-s\left(k\right)+t\left(k\right)
t\left(k\right)
k
\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{k}t\left(k\right)
g,[p,q]
g
is the closed form of the indefinite sum of
k
p
is a list containing the integer poles of
q
s
that are not poles of
\mathrm{with}\left(\mathrm{SumTools}[\mathrm{IndefiniteSum}]\right):
f≔\frac{1}{{n}^{2}+\mathrm{sqrt}\left(5\right)n-1}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\sqrt{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}
g≔\mathrm{Rational}\left(f,n\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{5}}}{\textcolor[rgb]{0,0,1}{2}}\right)}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{5}}}{\textcolor[rgb]{0,0,1}{2}}\right)}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\sqrt{\textcolor[rgb]{0,0,1}{5}}}{\textcolor[rgb]{0,0,1}{2}}\right)}
\mathrm{evala}\left(\mathrm{Normal}\left(\mathrm{eval}\left(g,n=n+1\right)-g\right),\mathrm{expanded}\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\sqrt{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}
f≔\frac{13-57x+2y+20{x}^{2}-18xy+10{y}^{2}}{15+10x-26y-25{x}^{2}+10xy+8{y}^{2}}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{20}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{18}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{57}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{13}}{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{25}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{26}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{15}}
g≔\mathrm{Rational}\left(f,x\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{25}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{34}}{\textcolor[rgb]{0,0,1}{25}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Psi }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{5}}\right)\textcolor[rgb]{0,0,1}{+}\left(\frac{\textcolor[rgb]{0,0,1}{17}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{25}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{5}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Psi }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)
\mathrm{simplify}\left(\mathrm{combine}\left(f-\left(\mathrm{eval}\left(g,x=x+1\right)-g\right),\mathrm{\Psi }\right)\right)
\textcolor[rgb]{0,0,1}{0}
f≔\frac{1}{n}-\frac{2}{n-3}+\frac{1}{n-5}
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}}
g,\mathrm{fp}≔\mathrm{Rational}\left(f,n,'\mathrm{failpoints}'\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{fp}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]]
f
n=0,3,5
g
n=1,2,4 |
Developing an Algorithm for Undistorting an Image - MATLAB & Simulink Example - MathWorks Deutschland
Define Distortion Model
Calculate Inverse Distortion Model
Function for Drawing the Lens Distortion
Any real world point
\mathit{P}\left(\mathit{X},\mathit{Y},\mathit{Z}\right)
can be defined with respect to some 3-D world origin.
Relative to a camera lens, this 3-D point can be defined as
{\mathit{p}}_{0}
, which is obtained by rotating and translating
\mathit{P}
{p}_{0}=\left({x}_{0},{y}_{0},{z}_{0}\right)
R\phantom{\rule{0.16666666666666666em}{0ex}}P+t
The 3-D point
{\mathit{p}}_{0}
is then projected into the camera's image plane as a 2D point, (
{\mathit{x}}_{1}
{\mathit{y}}_{1}
{x}_{1}=\frac{{x}_{0}}{{z}_{0}}
{y}_{1}=\frac{{y}_{0}}{{z}_{0}}
When a camera captures an image, it does not precisely capture the real points, but rather a slightly distorted version of the real points which can be denoted (
{\mathit{x}}_{2}
{\mathit{y}}_{2}
). The distorted points can be described using the following function:
{x}_{2}={x}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}\left(1+{k}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}{r}^{2}+{k}_{2}\phantom{\rule{0.16666666666666666em}{0ex}}{r}^{4}\right)+2\phantom{\rule{0.16666666666666666em}{0ex}}{p}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}{x}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}{y}_{1}+{p}_{2}\phantom{\rule{0.16666666666666666em}{0ex}}\left({r}^{2}+2\phantom{\rule{0.16666666666666666em}{0ex}}{{x}_{1}}^{2}\right)
{y}_{2}={y}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}\left(1+{k}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}{r}^{2}+{k}_{2}\phantom{\rule{0.16666666666666666em}{0ex}}{r}^{4}\right)+2\phantom{\rule{0.16666666666666666em}{0ex}}{p}_{2}\phantom{\rule{0.16666666666666666em}{0ex}}{x}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}{y}_{1}+{p}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}\left({r}^{2}+2\phantom{\rule{0.16666666666666666em}{0ex}}{{y}_{1}}^{2}\right)
{\mathit{k}}_{1}
{\mathit{k}}_{2}
= radial distortion coefficients of the lens
{\mathit{p}}_{1}
{\mathit{p}}_{2}
= tangential distortion coefficients of the lens
r=\sqrt{{{x}_{1}}^{2}+{{y}_{1}}^{2}}
An example of lens distortion is shown below; original distorted image (left) and undistorted image (right).
Note the curvature of the lines toward the edges of the first image. For applications such as image reconstruction and tracking, it is important to know the real world location of points. When we have a distorted image, we know the distorted pixel locations (
{\mathit{x}}_{2}
{\mathit{y}}_{2}
). It's our goal to determine the undistorted pixel locations (
{\mathit{x}}_{1}
{\mathit{y}}_{1}
) given (
{\mathit{x}}_{2}
{\mathit{y}}_{2}
) and the distortion coefficients of the particular lens.
While otherwise straightforward, the nonlinear nature of the lens distortion makes the problem challenging.
We begin by defining our distortion model:
syms k_1 k_2 p_1 p_2 real
syms r x y
distortionX = subs(x * (1 + k_1 * r^2 + k_2 * r^4) + 2 * p_1 * x * y + p_2 * (r^2 + 2 * x^2), r, sqrt(x^2 + y^2))
distortionX =
{p}_{2} \left(3 {x}^{2}+{y}^{2}\right)+x \left({k}_{2} {\left({x}^{2}+{y}^{2}\right)}^{2}+{k}_{1} \left({x}^{2}+{y}^{2}\right)+1\right)+2 {p}_{1} x y
distortionY = subs(y * (1 + k_1 * r^2 + k_2 * r^4) + 2 * p_2 * x * y + p_1 * (r^2 + 2 * y^2), r, sqrt(x^2 + y^2))
distortionY =
{p}_{1} \left({x}^{2}+3 {y}^{2}\right)+y \left({k}_{2} {\left({x}^{2}+{y}^{2}\right)}^{2}+{k}_{1} \left({x}^{2}+{y}^{2}\right)+1\right)+2 {p}_{2} x y
{\mathit{k}}_{1}=0
We plot a grid of pixel locations assuming our lens has a radial distortion coefficient
{\mathit{k}}_{1}=0
. Note that distortion is smallest near the center of the image and largest near the edges.
parameters = [k_1 k_2 p_1 p_2];
parameterValues = [0 0 0 0];
plotLensDistortion(distortionX,distortionY,parameters,parameterValues)
x
y
{\mathit{k}}_{1}=0.15
Explore the sensitivity to changes in
{\mathit{k}}_{1}
parameterValues = [0.15 0 0 0];
x \left(\frac{3 {x}^{2}}{20}+\frac{3 {y}^{2}}{20}+1\right)
y \left(\frac{3 {x}^{2}}{20}+\frac{3 {y}^{2}}{20}+1\right)
Given a camera's lens distortion coefficients and a set of distorted pixel locations (
{\mathit{x}}_{2}
{\mathit{y}}_{2}
), we want to be able to calculate the undistorted pixel locations (
{\mathit{x}}_{1}
{\mathit{y}}_{1}
). We will look at the specific case where all distortion coefficients are zero except for
{\mathit{k}}_{1}
which equals 0.2.
We begin by defining the distortion coefficients.
syms X Y positive
eq1 = X == distortionX
X={p}_{2} \left(3 {x}^{2}+{y}^{2}\right)+x \left({k}_{2} {\left({x}^{2}+{y}^{2}\right)}^{2}+{k}_{1} \left({x}^{2}+{y}^{2}\right)+1\right)+2 {p}_{1} x y
eq2 = Y == distortionY
Y={p}_{1} \left({x}^{2}+3 {y}^{2}\right)+y \left({k}_{2} {\left({x}^{2}+{y}^{2}\right)}^{2}+{k}_{1} \left({x}^{2}+{y}^{2}\right)+1\right)+2 {p}_{2} x y
We define the distortion equations for given distortion coefficients, and solve for the undistorted pixel locations (
{\mathit{x}}_{1}
{\mathit{y}}_{1}
parameterValues = [0.2 0 0 0];
eq1 = expand(subs(eq1, parameters, parameterValues))
X=\frac{{x}^{3}}{5}+\frac{x {y}^{2}}{5}+x
Y=\frac{{x}^{2} y}{5}+\frac{{y}^{3}}{5}+y
Result = solve([eq1, eq2], [x,y], 'MaxDegree', 3,'Real',true)
x: (X*(((5*Y^3)/(2*(X^2 + Y^2)) + ((25*Y^6)/(4*(X^2 + Y^2)^2) + (125*...
y: ((5*Y^3)/(2*(X^2 + Y^2)) + ((25*Y^6)/(4*(X^2 + Y^2)^2) + (125*Y^6)...
Since element 1 is the only real solution, we will extract that expression into its own variable.
[Result.x Result.y]
\begin{array}{l}\left(\begin{array}{cc}\frac{X {\sigma }_{1}}{Y}& {\sigma }_{1}\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}={\sigma }_{2}-\frac{5 {Y}^{2}}{3 \left({X}^{2}+{Y}^{2}\right) {\sigma }_{2}}\\ \\ \mathrm{ }{\sigma }_{2}={\left(\frac{5 {Y}^{3}}{2 \left({X}^{2}+{Y}^{2}\right)}+\sqrt{\frac{25 {Y}^{6}}{4 {\left({X}^{2}+{Y}^{2}\right)}^{2}}+\frac{125 {Y}^{6}}{27 {\left({X}^{2}+{Y}^{2}\right)}^{3}}}\right)}^{1/3}\end{array}
Now we have analytical expressions for the pixel locations X and Y which we can use to undistort our images.
function plotLensDistortion(distortionX,distortionY,parameters,parameterValues)
% distortionX is the expression describing the distorted x coordinate
% distortionY is the expression describing the distorted y coordinate
% k1 and k2 are the radial distortion coefficients
% p1 and p2 are the tangential distortion coefficients
% This is the grid spacing over the image
% Inspect and parametrically substitute in the values for k_1 k_2 p_1 p_2
distortionX = subs(distortionX,parameters,parameterValues)
distortionY = subs(distortionY,parameters,parameterValues)
% Loop over the grid
for x_i = -1:spacing:1
for y_j = -1:spacing:1
% Compute the distorted location
xout = subs(distortionX, {x,y}, {x_i,y_j});
yout = subs(distortionY, {x,y}, {x_i,y_j});
% Plot the original point
plot(x_i,y_j, 'o', 'Color', [1.0, 0.0, 0.0])
% Draw the distortion direction with Quiver
p1 = [x_i,y_j]; % First Point
p2 = [xout,yout]; % Second Point
dp = p2-p1; % Difference
quiver(p1(1),p1(2),dp(1),dp(2),'AutoScale','off','MaxHeadSize',1,'Color',[0 0 1]) |
Electrocardiogram - Citizendium
1 Performance of an ECG
2 The basic waveform
2.1 PR interval
2.2.1 QRS complex
2.2.2 ST segment
2.2.3 T wave
3 Common abnormalities
3.1 Early repolarization
3.3 ST segment elevation
3.4 ST segment depression
3.5 Q waves
3.6 Long QT interval
3.7 Partial AV block
3.8 Bigeminy and trigeminy
3.10 T wave inversion
4 Electrolyte effects on the ECG
4.2.1 Hyperkalemia
ECG stands for electrocardiogram. It is also known as EKG or ECG, either to reflect the German or Dutch Elektrokardiogramm, or to avoid confusion with EEG for electroencephalogram. While there had been earlier measurements of the electric activity of the heart, the Dutch physician Willem Einthoven developed the first medically useful device in 1903, for which he received the 1924 Nobel Prize in Physiology or Medicine.
The ECG refers to the small voltages (~1mv) found on the skin as a result of electrical activity of the heart. These electrical actions trigger various electrical and muscular activity in the heart. The health and function of the heart can be measured by the shape of the ECG waveform. Typical heart problems are leaking valves and blocked coronary arteries.[1]
Performance of an ECG
ECG electrodes usually come as 3,5 or 12 lead. These are:
3 lead - Left ARM or LA, Right Arm or RA, and Left Leg or LL
5 lead - additional leads of Right Leg or RL and V for Chest
12 lead - additional leads of V1-V6 replacing the V lead, with each V lead being placed between the ribs[2]
The ECG paper:
Normal speed is 25 mm/sec. At this speed
Each 1 mm horizontal box corresponds to 0.04 second (40 ms)
Heavier lines are placed at 0.20 sec (200 ms) intervals.
At the normal vertical standard, 10 mm equals 1 mV.
The ECG can be combined with other cardiac tests, such as ultrasonography (i.e., echocardiography), will show the ECG actions slightly preceding muscular action.
The basic waveform
Normal cardiac complex from electrocardiogram.
Although the appearance of the waveform should be different on the various leads, the repeating waveform has several common points of inflection or "waves", as well as some components that are present only in disease. More discussion is available online.[3]
Hexaxial reference system.
A prolonged PR interval is associated with increased mortality.[4]
P wave: caused by the discharge of the sinoatrial node
For more information, see: QT interval.
The normal duration of the QT interval is corrected for heart rate with Bazett's formula:
{\displaystyle QT_{c}={\frac {\text{QT interval}}{\sqrt {\text{RR interval}}}}}
The QRS complex is due to ventricular activity.
The normal angle is -30º to + 100º.
The normal duration is ≤0.10 sec (21/2 small boxes)
Q: may or may not be present, and if present, may or may not indicate disease. In combination with other indications of myocardial infarction (MI), there can be useful clinical distinctions between MIs where a Q wave is present, and non-Q-wave MIs.
R: caused by the first electrical signal arriving, via the atrioventricular node (AV node) and Purkinke fibers, at the ventricles.
S: caused by the completion of depolarization of the ventricles after they contract
The "segments", or sections between these above points also are clinically significant. In particular, the ST segment, which is normally isoelectric, tends to be elevated in the presence of myocardial infarction and depressed in the presence of myocardial ischemia. Even at rest, abnormal ST segments geriatric patients are a risk factor for coronary death.[5] By examining the degree of elevation or depression from the different leads, which essentially are different angles of view of the heart, it is possible to localize the damage.
In the presence of symptoms suggestive of heart disease, myocardial infarction cannot be excluded if the ECG is normal. It is possible to have severe heart disease and a normal ECG. Other tests are needed for a conclusive diagnosis.
T wave: repolarization of the ventricles
One of the challenges in learning to interpret electrocardiograms is that while textbooks usually show one abnormality at a time, many can coexist.
Early repolarization, defined as "elevation of the junction between the end of the QRS complex and the beginning of the ST segment (J point) from baseline," when present in the inferior leads, may be a risk factor for cardiac death.[6]
A clinical prediction rule helps distinguish early repolarization from anterior ST-segment elevation myocardial infarction (STEMI):[7]
If the following value is less than or equal to 23.4, then early repolarization is diagnosed with a sensitivity of 86% and specificity of 91%.
([1.196 x ST-segment elevation 60 ms after the J point in lead V3 in mm]+[0.059 x QTc in ms]-[0.326 x R-wave amplitude in lead V4 in mm]
For more information, see: Left ventricular hypertrophy.
Left ventricular hypertrophy can be assessed with the Sokolow–Lyon criteria.
Bigeminy and trigeminy
Electrolyte effects on the ECG
Peaked T waves; however, there is not a clear definition of peaking and even cardiologists have difficulty agreeing when peaking is present.[8] However, if the T-wave amplitude in lead with maximum R-wave deflection is over 5 or 10 mm, this correlates with expert assessment of peaking.
PR interval lengthens
QRS duration increases
Decrease in the amplitude of the T wave
U waves, especially in the lateral precordial leads V4-V6
↑ Noble RJ, Hillis JS, Rothbaum DA (1990). “Electrocardiography”, Walker HK, Hall WD, Hurst JW: Clinical methods: the history, physical, and laboratory examinations (in English), 3rd. London: Butterworths, 164. LCC RC71 .C63. ISBN 0-409-90077-X. Library of Congress
↑ Dale Dubin, Rapid Interpretation of EKG's (Third ed.), C.O.V.E.R , pp. 21-38
↑ Noble RJ, Hillis JS, Rothbaum DA (1990). “Electrocardiography”, Walker HK, Hall WD, Hurst JW: Clinical methods: the history, physical, and laboratory examinations (in English), 3rd. London: Butterworths, 164. LCC RC71 .C63. ISBN 0-409-90077-X. Library of Congress PDF
↑ Cheng S, Keyes MJ, Larson MG, et al. (June 2009). "Long-term outcomes in individuals with prolonged PR interval or first-degree atrioventricular block". JAMA 301 (24): 2571–7. DOI:10.1001/jama.2009.888. PMID 19549974. Research Blogging.
↑ Kumar A, Prineas RJ, Arnold AM, et al (December 2008). "Prevalence, prognosis, and implications of isolated minor nonspecific ST-segment and T-wave abnormalities in older adults: cardiovascular health study". Circulation 118 (25): 2790–6. DOI:10.1161/CIRCULATIONAHA.108.772541. PMID 19064684. Research Blogging.
↑ Tikkanen JT, Anttonen O, Junttila MJ, Aro AL, Kerola T, Rissanen HA et al. (2009). "Long-Term Outcome Associated with Early Repolarization on Electrocardiography.". N Engl J Med. DOI:10.1056/NEJMoa0907589. PMID 19917913. Research Blogging.
↑ Smith SW, Khalil A, Henry TD, Rosas M, Chang RJ, Heller K et al. (2012). "Electrocardiographic Differentiation of Early Repolarization From Subtle Anterior ST-Segment Elevation Myocardial Infarction.". Ann Emerg Med. DOI:10.1016/j.annemergmed.2012.02.015. PMID 22520989. Research Blogging.
↑ Montague BT, Ouellette JR, Buller GK (2008). "Retrospective review of the frequency of ECG changes in hyperkalemia.". Clin J Am Soc Nephrol 3 (2): 324-30. DOI:10.2215/CJN.04611007. PMID 18235147. PMC PMC2390954. Research Blogging.
Retrieved from "https://citizendium.org/wiki/index.php?title=Electrocardiogram&oldid=37200" |
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Phosphoglycerate dehydrogenase - Wikipedia
Metabolic enzyme PHGDH
PHGDH, 3-PGDH, 3PGDH, HEL-S-113, NLS, PDG, PGAD, PGD, PGDH, PHGDHD, SERA, NLS1, Phosphoglycerate dehydrogenase
OMIM: 606879 MGI: 1355330 HomoloGene: 39318 GeneCards: PHGDH
Posphoglycerate dehydrogenase {PHGDH} is an enzyme that catalyzes the chemical reactions
3-phospho-D-glycerate + NAD+
{\displaystyle \rightleftharpoons }
3-phosphonooxypyruvate + NADH + H+
2-hydroxyglutarate + NAD+
{\displaystyle \rightleftharpoons }
2-oxoglutarate + NADH + H+
The two substrates of this enzyme are 3-phospho-D-glycerate and NAD+, whereas its 3 products are 3-phosphohydroxypyruvate, NADH, and H+
It is also possible that two substrates of this enzyme are 2-hydroxyglutarate and NAD+, whereas its 3 products are 2-oxoglutarate, NADH, and H+.
As of 2012, the most widely studied variants of PHGDH are from the E. coli and M. tuberculosis genomes.[5] In humans, this enzyme is encoded by the PHGDH gene.[6]
3-Phosphoglycerate dehydrogenase catalyzes the transition of 3-phosphoglycerate into 3-phosphohydroxypyruvate, which is the committed step in the phosphorylated pathway of L-serine biosynthesis. It is also essential in cysteine and glycine synthesis, which lie further downstream.[7] This pathway represents the only way to synthesize serine in most organisms except plants, which uniquely possess multiple synthetic pathways. Nonetheless, the phosphorylated pathway that PHGDH participates in is still suspected to have an essential role in serine synthesis used in the developmental signaling of plants.[8][9]
Because of serine and glycine's role as neurotrophic factors in the developing brain, PHGDH has been shown to have high expression in glial and astrocyte cells during neural development.[10]
Mechanism and regulation[edit]
3-phosphoglycerate dehydrogenase works via an induced fit mechanism to catalyze the transfer of a hydride from the substrate to NAD+, a required cofactor. In its active conformation, the enzyme's active site has multiple cationic residues that likely stabilize the transition state of the reaction between the negatively charged substrate and NAD+. The positioning is such that the substrate's alpha carbon and the C4 of the nicotinamide ring are brought into a proximity that facilitates the hydride transfer producing NADH and the oxidized substrate.[5][11]
Active site of human PHGDH. Key residues (two Arg and one His) and substrates shown. The 4.2 Å distance is between the carbons undergoing hydride transfer. From 2G76 rendering of PHGDH crystallized with NAD+ and D-malate.
PHGDH is allosterically regulated by its downstream product, L-serine. This feedback inhibition is understandable considering that 3-phosphoglycerate is an intermediate in the glycolytic pathway. Given that PHGDH represents the committed step in the production of serine in the cell, flux through the pathway must be carefully controlled.
L-serine binding has been shown to exhibit cooperative behavior. Mutants that decreased this cooperativity also increased in sensitivity to serine's allosteric inhibition, suggesting a separation of the chemical mechanisms that result in allosteric binding cooperativity and active site inhibition.[12] The mechanism of inhibition is Vmax type, indicating that serine affects the reaction rate rather than the binding affinity of the active site.[11][13]
Although L-serine's allosteric effects are usually the focus of regulatory investigation, it has been noted that in some variants of the enzyme, 3-phosphoglycerate dehydrogenase is inhibited at separate positively charged allosteric site by high concentrations of its own substrate.[5][14]
3-Phosphoglycerate dehydrogenase is a tetramer, composed of four identical, asymmetric subunits. At any time, only a maximum of two adjacent subunits present a catalytically active site; the other two are forced into an inactive conformation. This results in half-of-the-sites activity with regard to both active and allosteric sites, meaning that only the two sites of the active subunits must be bound for essentially maximal effect with regard to catalysis and inhibition respectively.[15] There is some evidence that further inhibition occurs with the binding of the third and fourth serine molecules, but it is relatively minimal.[13]
The subunits from the E. coli PHGDH have three distinct domains, whereas those from M. tuberculosis have four. It is noted that the human enzyme more closely resembles that of M. tuberculosis, including the site for allosteric substrate inhibition. Concretely, three general types of PHGDH have been proposed: Type I, II, and III. Type III has two distinct domains, lacks both allosteric sites, and is found in various unicellular organisms. Type II has serine binding sites and encompasses the well-studied E. coli PHGDH. Type I possesses both the serine and substrate allosteric binding sites and encompasses M. tuberculosis and mammalian PHGDHs.[5]
The regulation of catalytic activity is thought to be a result of the movement of rigid domains about flexible “hinges.” When the substrate binds to the open active site, the hinge rotates and closes the cleft. Allosteric inhibition thus likely works by locking the hinge in a state that produces the open active site cleft.[13][16]
Crystal structure of inhibited PHGDH from M. tuberculosis due to allosterically bound serine. From 3DC2 rendering.
The variant from M. tuberculosis also exhibits an uncommon dual pH optimum for catalytic activity.[14]
3-Phosphoglycerate dehydrogenase possesses less than 20% homology to other NAD-dependent oxidoreductases and exhibits significant variance between species. There does appear to be conservation in specific binding domain residues, but there is still some variation in the positively charged active site residues between variants. For example, Type III PHGDH enzymes can be broken down into two subclasses where the key histidine residue is replaced with a lysine residue.[5][17]
Homozygous or compound heterozygous mutations in 3-phosphoglycerate dehydrogenase cause Neu-Laxova syndrome[18][19] and phosphoglycerate dehydrogenase deficiency.[20] In addition significantly shortening lifespan, PHGDH deficiencies are known to cause congenital microcephaly, psychomotor retardation, and intractable seizures in both humans and rats, presumably due to the essential signaling within the nervous system that serine, glycine, and other downstream molecules are intimately involved with. Treatment typically involves oral supplementation of serine and glycine and has been shown most effective when started in utero via oral ingestion by the mother.[21][22]
Mutations that result in increased PHGDH activity are also associated with increased risk of oncogenesis, including certain breast cancers.[23] This finding suggests that pathways providing an outlet for diverting carbon out of glycolysis may be beneficial for rapid cell growth.[24]
It has been reported that PHGDH can also catalyze the conversion of alpha-ketoglutarate to 2-Hydroxyglutaric acid in certain variants. Thus, a mutation in the enzyme is hypothesized to contribute to 2-Hydroxyglutaric aciduria in humans, although there is debate as to whether or not this catalysis is shared by human PHGDH.[5][25]
PHGDH is blood test biomarker of AD with strong evidence. https://neurosciencenews.com/brain-supplement-alzheimers-20544/
^ a b c d e f Grant GA (March 2012). "Contrasting catalytic and allosteric mechanisms for phosphoglycerate dehydrogenases". Archives of Biochemistry and Biophysics. 519 (2): 175–85. doi:10.1016/j.abb.2011.10.005. PMC 3294004. PMID 22023909.
^ "PHGDH phosphoglycerate dehydrogenase [Homo sapiens (human)] - Gene - NCBI". www.ncbi.nlm.nih.gov. Retrieved 1 March 2016.
^ "MetaCyc L-serine biosynthesis". biocyc.org. Retrieved 1 March 2016.
^ Ros R, Muñoz-Bertomeu J, Krueger S (September 2014). "Serine in plants: biosynthesis, metabolism, and functions". Trends in Plant Science. 19 (9): 564–9. doi:10.1016/j.tplants.2014.06.003. PMID 24999240.
^ Ho CL, Noji M, Saito M, Saito K (January 1999). "Regulation of serine biosynthesis in Arabidopsis. Crucial role of plastidic 3-phosphoglycerate dehydrogenase in non-photosynthetic tissues". The Journal of Biological Chemistry. 274 (1): 397–402. doi:10.1074/jbc.274.1.397. PMID 9867856.
^ Yamasaki M, Yamada K, Furuya S, Mitoma J, Hirabayashi Y, Watanabe M (October 2001). "3-Phosphoglycerate dehydrogenase, a key enzyme for l-serine biosynthesis, is preferentially expressed in the radial glia/astrocyte lineage and olfactory ensheathing glia in the mouse brain". The Journal of Neuroscience. 21 (19): 7691–704. doi:10.1523/JNEUROSCI.21-19-07691.2001. PMC 6762884. PMID 11567059. S2CID 3547638.
^ a b Grant GA, Kim SJ, Xu XL, Hu Z (February 1999). "The contribution of adjacent subunits to the active sites of D-3-phosphoglycerate dehydrogenase". The Journal of Biological Chemistry. 274 (9): 5357–61. doi:10.1074/jbc.274.9.5357. PMID 10026144.
^ Grant GA, Hu Z, Xu XL (January 2001). "Specific interactions at the regulatory domain-substrate binding domain interface influence the cooperativity of inhibition and effector binding in Escherichia coli D-3-phosphoglycerate dehydrogenase". The Journal of Biological Chemistry. 276 (2): 1078–83. doi:10.1074/jbc.M007512200. PMID 11050089.
^ a b c Grant GA, Schuller DJ, Banaszak LJ (January 1996). "A model for the regulation of D-3-phosphoglycerate dehydrogenase, a Vmax-type allosteric enzyme". Protein Science. 5 (1): 34–41. doi:10.1002/pro.5560050105. PMC 2143248. PMID 8771194.
^ a b Burton RL, Chen S, Xu XL, Grant GA (October 2007). "A novel mechanism for substrate inhibition in Mycobacterium tuberculosis D-3-phosphoglycerate dehydrogenase". The Journal of Biological Chemistry. 282 (43): 31517–24. doi:10.1074/jbc.M704032200. PMID 17761677.
^ Grant GA, Xu XL, Hu Z (April 2004). "Quantitative relationships of site to site interaction in Escherichia coli D-3-phosphoglycerate dehydrogenase revealed by asymmetric hybrid tetramers". The Journal of Biological Chemistry. 279 (14): 13452–60. doi:10.1074/jbc.M313593200. PMID 14718528.
^ Al-Rabiee R, Lee EJ, Grant GA (May 1996). "The mechanism of velocity modulated allosteric regulation in D-3-phosphoglycerate dehydrogenase. Cross-linking adjacent regulatory domains with engineered disulfides mimics effector binding". The Journal of Biological Chemistry. 271 (22): 13013–7. doi:10.1074/jbc.271.22.13013. PMID 8662776. S2CID 28327405.
^ Tobey KL, Grant GA (September 1986). "The nucleotide sequence of the serA gene of Escherichia coli and the amino acid sequence of the encoded protein, D-3-phosphoglycerate dehydrogenase". The Journal of Biological Chemistry. 261 (26): 12179–83. doi:10.1016/S0021-9258(18)67220-5. PMID 3017965.
^ Shaheen R, Rahbeeni Z, Alhashem A, Faqeih E, Zhao Q, Xiong Y, Almoisheer A, Al-Qattan SM, Almadani HA, Al-Onazi N, Al-Baqawi BS, Saleh MA, Alkuraya FS (June 2014). "Neu-Laxova syndrome, an inborn error of serine metabolism, is caused by mutations in PHGDH". American Journal of Human Genetics. 94 (6): 898–904. doi:10.1016/j.ajhg.2014.04.015. PMC 4121479. PMID 24836451.
^ Acuna-Hidalgo R, Schanze D, Kariminejad A, Nordgren A, Kariminejad MH, Conner P, Grigelioniene G, Nilsson D, Nordenskjöld M, Wedell A, Freyer C, Wredenberg A, Wieczorek D, Gillessen-Kaesbach G, Kayserili H, Elcioglu N, Ghaderi-Sohi S, Goodarzi P, Setayesh H, van de Vorst M, Steehouwer M, Pfundt R, Krabichler B, Curry C, MacKenzie MG, Boycott KM, Gilissen C, Janecke AR, Hoischen A, Zenker M (September 2014). "Neu-Laxova syndrome is a heterogeneous metabolic disorder caused by defects in enzymes of the L-serine biosynthesis pathway". American Journal of Human Genetics. 95 (3): 285–93. doi:10.1016/j.ajhg.2014.07.012. PMC 4157144. PMID 25152457.
^ Jaeken J, Detheux M, Van Maldergem L, Foulon M, Carchon H, Van Schaftingen E (June 1996). "3-Phosphoglycerate dehydrogenase deficiency: an inborn error of serine biosynthesis". Archives of Disease in Childhood. 74 (6): 542–5. doi:10.1136/adc.74.6.542. PMC 1511571. PMID 8758134.
^ de Koning TJ, Duran M, Dorland L, Gooskens R, Van Schaftingen E, Jaeken J, Blau N, Berger R, Poll-The BT (August 1998). "Beneficial effects of L-serine and glycine in the management of seizures in 3-phosphoglycerate dehydrogenase deficiency". Annals of Neurology. 44 (2): 261–5. doi:10.1002/ana.410440219. PMID 9708551. S2CID 46565109.
^ de Koning TJ, Klomp LW, van Oppen AC, Beemer FA, Dorland L, van den Berg I, Berger R (18 December 2004). "Prenatal and early postnatal treatment in 3-phosphoglycerate-dehydrogenase deficiency". Lancet. 364 (9452): 2221–2. doi:10.1016/S0140-6736(04)17596-X. PMID 15610810. S2CID 40121728.
^ Possemato R, Marks KM, Shaul YD, Pacold ME, Kim D, Birsoy K, Sethumadhavan S, Woo HK, Jang HG, Jha AK, Chen WW, Barrett FG, Stransky N, Tsun ZY, Cowley GS, Barretina J, Kalaany NY, Hsu PP, Ottina K, Chan AM, Yuan B, Garraway LA, Root DE, Mino-Kenudson M, Brachtel EF, Driggers EM, Sabatini DM (August 2011). "Functional genomics reveal that the serine synthesis pathway is essential in breast cancer". Nature. 476 (7360): 346–50. Bibcode:2011Natur.476..346P. doi:10.1038/nature10350. PMC 3353325. PMID 21760589.
^ Locasale JW, Grassian AR, Melman T, Lyssiotis CA, Mattaini KR, Bass AJ, Heffron G, Metallo CM, Muranen T, Sharfi H, Sasaki AT, Anastasiou D, Mullarky E, Vokes NI, Sasaki M, Beroukhim R, Stephanopoulos G, Ligon AH, Meyerson M, Richardson AL, Chin L, Wagner G, Asara JM, Brugge JS, Cantley LC, Vander Heiden MG (September 2011). "Phosphoglycerate dehydrogenase diverts glycolytic flux and contributes to oncogenesis" (PDF). Nature Genetics. 43 (9): 869–74. doi:10.1038/ng.890. PMC 3677549. PMID 21804546.
^ Zhao G, Winkler ME (January 1996). "A novel alpha-ketoglutarate reductase activity of the serA-encoded 3-phosphoglycerate dehydrogenase of Escherichia coli K-12 and its possible implications for human 2-hydroxyglutaric aciduria". Journal of Bacteriology. 178 (1): 232–9. doi:10.1128/JB.178.1.232-239.1996. PMC 177644. PMID 8550422.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Phosphoglycerate_dehydrogenase&oldid=1087201107" |
Rhombitrihexagonal_tiling Knowpia
{\displaystyle r{\begin{Bmatrix}6\\3\end{Bmatrix}}}
Bowers acronym Rothat
Dual Deltoidal trihexagonal tiling
In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.
John Conway calls it a rhombihexadeltille.[1] It can be considered a cantellated by Norman Johnson's terminology or an expanded hexagonal tiling by Alicia Boole Stott's operational language.
There is only one uniform coloring in a rhombitrihexagonal tiling. (Naming the colors by indices around a vertex (3.4.6.4): 1232.)
With edge-colorings there is a half symmetry form (3*3) orbifold notation. The hexagons can be considered as truncated triangles, t{3} with two types of edges. It has Coxeter diagram
, Schläfli symbol s2{3,6}. The bicolored square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, a triangular tiling results, constructed as a snub triangular tiling,
Cantic snub triangular
Snub triangular
Uniform face coloring
Uniform edge coloring
rr{3,6} s2{3,6} s{3,6}
The game Kensington
Floor tiling, Archeological Museum of Seville, Sevilla, Spain
The Temple of Diana in Nîmes, France
Roman floor mosaic in Castel di Guido
The tiling can be replaced by circular edges, centered on the hexagons as an overlapping circles grid. In quilting it is called Jacks chain.[2]
There is one related 2-uniform tiling, having hexagons dissected into 6 triangles.[3][4]
3.3.4.3.4 & 36
The rhombitrihexagonal tiling is related to the truncated trihexagonal tiling by replacing some of the hexagons and surrounding squares and triangles with dodecagons:
The rhombitrihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing (kissing number).[5] The translational lattice domain (red rhombus) contains 6 distinct circles.
There are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).
63 3.122 (3.6)2 6.6.6 36 3.4.6.4 4.6.12 3.3.3.3.6 3.3.3.3.3.3
V63 V3.122 V(3.6)2 V63 V36 V3.4.6.4 V.4.6.12 V34.6 V36
Deltoidal trihexagonal tilingEdit
It has been suggested that this section be split out into another article. (Discuss) (October 2021)
The deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. Conway calls it a tetrille.[1] The edges of this tiling can be formed by the intersection overlay of the regular triangular tiling and a hexagonal tiling. Each kite face of this tiling has angles 120°, 90°, 60° and 90°. It is one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.[6]
The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling.[7] Its faces are deltoids or kites.
It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals.
This tiling has face transitive variations, that can distort the kites into bilateral trapezoids or more general quadrilaterals. Ignoring the face colors below, the fully symmetry is p6m, and the lower symmetry is p31m with 3 mirrors meeting at a point, and 3-fold rotation points.[8]
Isohedral variations
p31m, [6,3+], (3*3)
Faces Kite Half regular hexagon Quadrilaterals
This tiling is related to the trihexagonal tiling by dividing the triangles and hexagons into central triangles and merging neighboring triangles into kites.
The deltoidal trihexagonal tiling is a part of a set of uniform dual tilings, corresponding to the dual of the rhombitrihexagonal tiling.
This tiling is topologically related as a part of sequence of tilings with face configurations V3.4.n.4, and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
Other deltoidal (kite) tilingEdit
Other deltoidal tilings are possible.
Point symmetry allows the plane to be filled by growing kites, with the topology as a square tiling, V4.4.4.4, and can be created by crossing string of a dream catcher. Below is an example with dihedral hexagonal symmetry.
Another face transitive tiling with kite faces, also a topological variation of a square tiling and with face configuration V4.4.4.4. It is also vertex transitive, with every vertex containing all orientations of the kite face.
pmg, [∞,(2,∞)+], (22*)
Wikimedia Commons has media related to Uniform tiling 3-4-6-4 (rhombitrihexagonal tiling).
^ a b Conway, 2008, p288 table
^ Ring Cycles a Jacks Chain variation
^ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern B
^ Kirby, Matthew; Umble, Ronald (2011), "Edge tessellations and stamp folding puzzles", Mathematics Magazine, 84 (4): 283–289, arXiv:0908.3257, doi:10.4169/math.mag.84.4.283, MR 2843659 .
^ Weisstein, Eric W. "Dual tessellation". MathWorld. (See comparative overlay of this tiling and its dual)
^ Tilings and Patterns
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings.
Klitzing, Richard. "2D Euclidean tilings x3o6x - rothat - O8".
Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern N, Dual p. 77-76, pattern 2 |
The Cauchy Problem for a Strongly Degenerate Quasilinear Equation | EMS Press
The Cauchy Problem for a Strongly Degenerate Quasilinear Equation
We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation
u_t = \div \, \a(u,Du)
\a(z,\xi) = \nabla_\xi f(z,\xi)
f
\xi
with linear growth as
\Vert \xi\Vert \to\infty
, satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.
Vicent Caselles, Fuensanta Andreu, José M. Mazón, The Cauchy Problem for a Strongly Degenerate Quasilinear Equation. J. Eur. Math. Soc. 7 (2005), no. 3, pp. 361–393 |
Brian has the following sums to find:
\begin{array}{l} 1 + 3 \\ 1 + 3 + 5 \\ 1 + 3 + 5 + 7 \\ \qquad \quad \; . \\ \qquad \quad \; . \\ \qquad \quad \; . \\ 1 + 3 + 5 + 7 + \dots + 101 \end{array}
Find a shortcut so that he will not have to add the last sum by hand.
1+3=4
2^2
(How many terms are in the sum?)
1+3+5=9
3^2
1+3+5+7=16
4^2
(How many terms are in the final sum?) |
Tuomo KuusiGiuseppe MingioneYannick Sire
The supercritical Lane–Emden equation and its gradient flow
Heat and mass transfer by convection in multicomponent Navier–Stokes mixtures: absence of subcritical instabilities and global nonlinear stability via the Auxiliary System Method
Newton’s <i>Philosophiae Naturalis Principia Mathematica</i> "Jesuit" Edition: The Tenor of a Huge Work
Paolo BussottiRaffaele Pisano
p
-Laplace type equation in a limiting case
Fernando FarroniLuigi GrecoGioconda Moscariello |
Wavelet packet spectrum - MATLAB wpspectrum - MathWorks 日本
wpspectrum
Wavelet Packet Spectrum for Sinusoids
Wavelet Packet Spectrum of Chirp Signal
[spec,times,freq] = wpspectrum(wpt,fs)
[___] = wpspectrum(wpt,fs,'plot')
[___,tinfo] = wpspectrum(___)
[spec,times,freq] = wpspectrum(wpt,fs) returns a matrix of wavelet packet spectrum estimates, spec, for the binary wavelet packet tree object, wpt. fs is the sampling frequency in hertz. times is a vector of times and freq is a vector of frequencies.
[___] = wpspectrum(wpt,fs,'plot') displays the wavelet packet spectrum.
[___,tinfo] = wpspectrum(___) returns the terminal nodes of the wavelet packet tree in frequency order.
Create a signal consisting of two sinusoids with disjoint support. The sinusoids have frequencies of 16 Hz and 64 Hz. Sample the signal at 500 Hz for 4 seconds.
frA = 16;
frB = 64;
sig = sin(frA*2*pi*t).*(t<2) + sin(frB*2*pi*t).*(t>=2);
title('Analyzed Signal')
Obtain the wavelet packet tree object corresponding to the level 6 wavelet packet decomposition of the signal using the sym6 wavelet.
wpt = wpdec(sig,level,wname);
Obtain and plot the wavelet packet spectrum.
[S,T,F] = wpspectrum(wpt,fs,'plot');
Generate a chirp signal sampled at 1000 Hz for 2 seconds.
sig = sin(256*pi*t.^2);
Obtain the wavelet packet tree object corresponding to the level 6 wavelet packet decomposition of the signal using the sym8 wavelet. Plot the wavelet packet spectrum.
wpt = wpdec(sig,level,'sym8');
wpt — Binary wavelet packet tree
Binary wavelet packet tree, specified as a wavelet packet tree object.
spec — Wavelet packet spectrum estimates
Wavelet packet spectrum estimates, returned as a matrix. spec is a 2J-by-N matrix, where J is the level of the wavelet packet transform, and N is the length of the time series. N is equal to the length of node 0 in the wavelet packet tree object.
The frequency spacing between the rows of spec is fs/2J+1.
Times, returned as a 1-by-N vector, where N is the length of the time series. The time spacing between elements is 1/fs.
Frequencies, returned as a 1-by-2J vector, where J is the level of the wavelet packet transform. The frequency spacing in freq is fs/2J+1.
tinfo — Terminal nodes
Terminal nodes of the wavelet packet tree object in frequency order.
The wavelet packet spectrum contains the absolute values of the coefficients from the frequency-ordered terminal nodes of the input binary wavelet packet tree. The terminal nodes provide the finest level of frequency resolution in the wavelet packet transform.
If J denotes the level of the wavelet packet transform and Fs is the sampling frequency, the terminal nodes approximate bandpass filters of the form:
\left[\frac{nFs}{{2}^{J+1}},\frac{\left(n+1\right)Fs}{{2}^{J+1}}\right)\text{â}n=0,1,2,3,â¦{2}^{J}â1
At the terminal level of the wavelet packet tree, the transform divides the interval from 0 to the Nyquist frequency into bands of approximate width
Fs/{2}^{J+1}.
wpspectrum computes the wavelet packet spectrum as follows:
Extract the wavelet packet coefficients corresponding to the terminal nodes. Take the absolute value of the coefficients.
Order the wavelet packet coefficients by frequency ordering.
Determine the time extent on the original time axis corresponding to each wavelet packet coefficient. Repeat each wavelet packet coefficient to fill in the time gaps between neighboring wavelet packet coefficients and create a vector equal in length to node 0 of the wavelet packet tree object.
[1] Wickerhauser, M.V. Lectures on Wavelet Packet Algorithms, Technical Report, Washington University, Department of Mathematics, 1992.
otnodes | wpdec | dwpt | modwpt |
homogeneity - zxc.wiki
This article deals with homogeneity in a scientific context; for mathematical and other meanings, see homogeneity (disambiguation) .
Homogeneity (from ὁμός homόs "equal" and γένεσις genesis "creation, birth", thus roughly: same quality) denotes the equality of a physical property over the entire extent of a system or the similarity of elements of a system. The term has a wide scope and can contain different meanings in detail. A measure or method with which a material or system is made homogeneous or its homogeneity is increased is called homogenization.
1 Opposites to homogeneity
3 Dependence on the size scale
4 Importance of homogeneous substances
5 consequences of chemical homogeneity
Opposites to homogeneity
Homogeneous, heterogeneous, inhomogeneous
What is not homogeneous is called inhomogeneous or heterogeneous .
A distinction is usually made between these two terms, the use of the word varies somewhat.
A body made of a uniform material but with a density that varies from place to place is, for example, referred to as inhomogeneous.
Heterogeneous (two or more phases), on the other hand, is a body made of macroscopically different components, such as a concrete slab with steel reinforcement.
In the figure, the differences in homogeneity, heterogeneity and inhomogeneity are illustrated from left to right.
In physics , matter , viewed atomically, is fundamentally not homogeneous, since the building blocks of matter do not have a uniform spatial filling . Even in the atom itself, the mass and charge distribution is not homogeneous, since it is distributed unevenly between the atomic nucleus and the atomic shell . If the atoms or molecules are distributed approximately evenly (not necessarily with the regularity of a crystal lattice , but without macroscopic fluctuations from place to place), the matter is homogeneous from a practical point of view .
The term is also applied to fields . A field, e.g. B. a magnetic field is called homogeneous if the field strength is the same at every location, otherwise inhomogeneous. Homogeneous fields are characterized by straight, parallel and evenly distributed field lines . When it comes to gradient fields, the equipotential surfaces are parallel planes which are penetrated at right angles by the field lines. While dipoles are aligned and attracted by inhomogeneous fields, homogeneous fields exert aligning moments on dipoles, but not attractive forces. Examples of approximately homogeneous fields are:
The electric field in a plate capacitor .
The magnetic field in a long coil .
The gravitational field on the earth's surface, provided the dimensions of the experimental setup are very small compared to the earth.
Finally, in theoretical physics , one speaks of the homogeneity of space when one wants to express that physical laws are invariant to translation . From this it follows, according to Noether's theorem , that the momentum is a conserved quantity .
Dependence on the size scale
An example of matter that is heterogeneous on a microscopic level but appears homogeneous on a macroscopic level is milk . A microscopic distinction can be made between areas in milk that contain fat and those that contain water . And although the two cannot mix, both areas are so small that, viewed macroscopically, they appear homogeneously distributed. Nevertheless, it can happen in such mixtures that their components separate over time and, in the case of milk, it no longer appears macroscopically homogeneous, since its watery areas are clearly different from their high-fat areas (cream). To prevent this segregation or separation, you can z. B. with the help of homogenization for an even distribution of fat and water even after a long time.
In chemistry , homogeneous substances are either pure substances or homogeneous mixtures , which also include solutions .
Meaning of homogeneous substances
The extraction of sufficiently homogeneous starting materials and / or intermediate products for industry, e.g. B. in the manufacture of the various semiconductor components of the modern electronics and computer industry, is one of the key problems of scientific and technical development. It often requires a great deal of effort (especially when extracting pure substances and / or reducing their error tolerances).
Consequences of chemical homogeneity
Homogeneous matter has the same density and composition everywhere . When in a large container with a homogeneous substance, e.g. B. with a gas , a subset V 1 is considered at one point , it contains the same amount of substance as a subset with the same volume V 1 at another point. If you divide the total amount of substance into two volumes of equal size, they each contain the same amount of substance (in this case half of the original). It follows:
The amount of substance is proportional to the volume for homogeneous substances with constant pressure and constant temperature , or vice versa:
The volume of homogeneous substances is proportional to the amount of substance at constant pressure p and constant temperature T.
For T = const and p = const the following applies:
{\ displaystyle V \ sim n \ qquad {\ frac {V} {n}} = {\ text {const}} \ qquad {\ frac {V_ {1}} {n_ {1}}} = {\ frac { V_ {2}} {n_ {2}}}}
These laws apply to all homogeneous substances as long as temperature and pressure remain unchanged, including ideal gases for which the thermal equation of state of ideal gases applies. The quotient is called the molar volume , the quotient is the concentration . The aforementioned relationships are also the basis of volumetry .
{\ displaystyle {\ tfrac {V} {n}} = V_ {m}}
{\ displaystyle {\ tfrac {n} {V}} = c}
The relationships also apply to homogeneous substances
{\ displaystyle V \ sim m \ qquad {\ frac {m} {V}} = \ rho}
Brockhaus Encyclopedia, 19th edition, Mannheim 1988
Wiktionary: homogeneous - explanations of meanings, word origins, synonyms, translations
This page is based on the copyrighted Wikipedia article "Homogenit%C3%A4t" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA. |
Ideal gas law - Citizendium
0.082057 L·atm·K-1·mol-1
8.205745 × 10-5 m3·atm·K-1·mol-1
62.36367 L·mmHg·K-1·mol-1
62.36367 L·torr·K-1·mol-1
0.73024 ft3·atm·°R-1·lb-mol-1
The ideal gas law is the equation of state of an ideal gas (also known as a perfect gas) that relates its absolute pressure p to its absolute temperature T. Further parameters that enter the equation are the volume V of the container holding the gas and the amount n (in moles) of gas contained in there. The law reads
{\displaystyle pV=nRT\,}
where R is the molar gas constant, defined as the product of the Boltzmann constant kB and Avogadro's constant NA
{\displaystyle R\equiv N_{\mathrm {A} }k_{\mathrm {B} }}
Currently, the most accurate value of R is:[1] 8.314472 ± 0.000015 J·K-1·mol-1.
The law applies to ideal gases which are hypothetical gases that consist of molecules[2] that do not interact, i.e., that move through the container independently of each other. In contrast to what is sometimes stated (see, e.g., Ref.[3]) an ideal gas does not necessarily consist of point particles without internal structure, but may be formed by polyatomic molecules with internal rotational, vibrational, and electronic degrees of freedom. The ideal gas law describes the motion of the centers of mass of the molecules and, indeed, mass centers may be seen as structureless point masses. However, for other properties of ideal gases, such as entropy, the internal structure may play a role.
The ideal gas law is a useful approximation for calculating temperatures, volumes, pressures or amount of substance for many gases over a wide range of values, as long as the temperatures and pressures are far from the values where condensation or sublimation occur.
Real gases deviate from ideal gas behavior because the intermolecular attractive and repulsive forces cause the motions of the molecules to be correlated. The deviation is especially significant at low temperatures or high pressures, i.e., close to condensation. A conventional measure for this deviation is the compressibility factor.
There are many equations of state available for use with real gases, the simplest of which is the van der Waals equation.
The early work on the behavior of gases began in pre-industrialized Europe in the latter half of the 17th century by Robert Boyle who formulated Boyle's law in 1662 (independently confirmed by Edme Mariotte at about the same time).[4] Their work on air at low pressures established the inverse relationship between pressure and volume, V = constant / p at constant temperature and a fixed amount of air. Boyle's Law is often referred to as the Boyles-Mariotte Law.
In 1699, Guillaume Amontons formulated what is now known as Amontons' law, p = constant / T.
Almost a century later, Jacques Alexandre César Charles experimented with hot-air balloons (around 1780), and additional contributions by John Dalton (1801) and Joseph Louis Gay-Lussac (1808) showed that a sample of gas, at a fixed pressure, increases in volume linearly with the temperature, i.e. V / T is constant.
In 1811, Amedeo Avogadro re-interpreted Gay-Lussac's law of combining volumes to state what is now commonly called Avogadro's law: equal volumes of any two gases at the same temperature and pressure contain the same number of molecules.
In 1834, Benoît Paul Émile Clapeyron combined the work of Boyle, Mariotte, Charles and Gay-Lussac into an equation of state of a perfect (i.e., ideal) gas: PV = RoT where Ro was a gas-dependent constant.[4][5] In 1845, Victor Regnault cast Clapeyron's perfect gas equation into the familiar ideal gas equation form by applying Avagadro's hypothesis on the volume of one mole of an ideal gas, i.e. PV = nRT.[4]
Sometimes, the ideal gas law is referred to as the Boyle-Gay-Lussac law. However, with hindsight, the Boyle-Gay-Lussac law, Amontons' law and Avogadro's law are all special cases of the ideal gas law.
Extrapolation of the volume/temperature relationship of ideal and many real gases to zero volume crosses the temperature axis at about −273 °C. This temperature is defined as the absolute zero temperature. Since any real gas would liquefy before reaching it, this temperature region remains a theoretical minimum.
Statistical mechanics derivation
The statistical mechanics[6][7] derivation of the ideal gas law provides the most precise insight into the microscopic conditions that a gas must satisfy in order to be called an ideal gas. In the derivation below, we will make the following two assumptions[8]:
The molecules constituting the gas are practically independent systems, each pursuing its own motion;
Exchange of energy between molecules occasionally takes place, so that the system can achieve a thermal equilibrium.[9]
Starting with the second assumption, we recall from equilibrium statistical mechanics that the canonical partition function is a function of N ≡ nNA, V, and T, defined as
{\displaystyle Q(N,V,T)=\sum _{I}e^{-{\mathcal {E}}_{I}/(k_{\mathrm {B} }T)}}
{\displaystyle {\mathcal {E}}_{I}}
is the I-th energy of the total gas (i.e. the energy of all N molecules). From quantum mechanics follows that a gas in a finite-size container has discrete energies; I is the discrete index labeling the different energies. The sum is over all eigenstates of the energy operator including degeneracies. Further, we recall that the Helmholtz free energy is given by
{\displaystyle A=-k_{\mathrm {B} }T\,\ln Q}
. The following expression for the absolute pressure p will be the starting point in the derivation
{\displaystyle p=-\left({\frac {\partial A}{\partial V}}\right)_{N,T}=k_{\mathrm {B} }T\left({\frac {\partial \ln Q}{\partial V}}\right)_{N,T}}
The only approximation (but a very drastic one) that must be made is assumption 1 from above, i.e. that the energies
{\displaystyle {\mathcal {E}}_{I}}
are sums of one-molecule energies
{\displaystyle \varepsilon _{i}}
. These one-molecule energies are those of a single molecule moving by itself in the vessel, an approximation that is common in many branches of physics and known as the independent particle approximation. Thus,
{\displaystyle {\mathcal {E}}_{I}=\varepsilon _{i_{1}}+\varepsilon _{i_{2}}+\cdots }
The total partition function Q will factorize into one-molecule partition functions q given by,
{\displaystyle q(N,V,T)=\sum _{i}e^{-\varepsilon _{i}/(k_{\mathrm {B} }T)}}
In absence of interactions Q becomes (assuming that the gas consists of one type of molecules only),
{\displaystyle Q=\sum _{i_{1},i_{2},\ldots }e^{-(\varepsilon _{i_{1}}+\varepsilon _{i_{2}}+\cdots )/(k_{\mathrm {B} }T)}=\sum _{i_{1}}e^{-\varepsilon _{i_{1}}/(k_{\mathrm {B} }T)}\sum _{i_{2}}e^{-\varepsilon _{i_{2}}/(k_{\mathrm {B} }T)}\cdots =q^{N}.}
This form of Q would be correct if the non-interacting gas molecules were non-identical. However, in the early days of quantum mechanics it was discovered that gas molecules of the same type are identical particles, just like electrons, and that a factorial 1/N! must be inserted to avoid overcounting. Later it was shown that this factorial arises from Bose-Einstein statistics (obeyed by the majority of stable molecules). Complete Bose-Einstein statistics itself needs only to be applied at temperatures close to the absolute zero. For higher temperatures Bose-Einstein statistics goes over into Boltzmann statistics, which requires the simple factor 1/N! that we will insert now ad hoc[10] into the expression for Q. In summary, from the additivity of the molecular energies and the application of Boltzmann statistics follows
{\displaystyle Q={\frac {q^{N}}{N!}}}
The application of Boltzmann statistics is of no consequence to the equation of state, but modifies expressions for other properties of the gas, such as the entropy. The factorization of Q would be exact if (i) the molecules would not interact and if (ii) every molecule had the whole volume V of the container to its disposal, or in other words, if the molecules themselves had zero volume.
{\displaystyle p=k_{\mathrm {B} }T\left({\frac {\partial \ln Q}{\partial V}}\right)=k_{\mathrm {B} }T\left({\frac {\partial (N\ln q-\ln N!)}{\partial V}}\right)=Nk_{\mathrm {B} }T\left({\frac {\partial \ln q}{\partial V}}\right)}
where we used the rules ln(a/b) = lna - lnb and lnan = n lna.
It follows from both classical mechanics and quantum mechanics that the molecular energy
{\displaystyle \varepsilon _{i}}
can be exactly separated as
{\displaystyle \varepsilon _{i}=\varepsilon _{i}^{\mathrm {transl} }+\varepsilon _{i}^{\mathrm {internal} }\quad \Longrightarrow \quad q=q^{\mathrm {transl} }\;q^{\mathrm {internal} }}
{\displaystyle \varepsilon _{i}^{\mathrm {transl} }}
is the translational energy of the center of mass of the molecule and
{\displaystyle \varepsilon _{i}^{\mathrm {internal} }}
is the internal (rotational, vibrational, electronic) energy of the molecule. This factorization of the one-molecule partition function into a translational and an internal factor proceeds in the same way as the factorization of the N-molecule partition function Q into one-molecule partition functions.
The internal energy of the molecule does not depend on the volume V (this is an exact result), but the translational energy does, hence
{\displaystyle p=Nk_{\mathrm {B} }T\left({\frac {\partial \ln q^{\mathrm {transl} }}{\partial V}}+{\frac {\partial \ln q^{\mathrm {internal} }}{\partial V}}\right)=Nk_{\mathrm {B} }T\;{\frac {\partial \ln q^{\mathrm {transl} }}{\partial V}}}
The determination of the translational energy of one molecule moving in a box of volume V is one of the few problems in quantum mechanics that can be solved analytically. That is, the energies
{\displaystyle \varepsilon _{i}^{\mathrm {transl} }}
are known exactly. To a very good approximation, one may replace the sum appearing in
{\displaystyle q^{\mathrm {transl} }}
by an integral, finding
{\displaystyle q^{\mathrm {transl} }\equiv \sum _{i}e^{-\varepsilon _{i}^{\mathrm {transl} }/(k_{\mathrm {B} }T)}={\frac {V}{\Lambda ^{3}}}\quad {\hbox{with}}\quad \Lambda =\left({\frac {h^{2}}{2\pi Mk_{\mathrm {B} }T}}\right)^{1/2}}
where h is Planck's constant and M is the total mass of the molecule. Note that Λ, the thermal de Broglie wavelength, does not depend on the volume V, so that
{\displaystyle p=Nk_{\mathrm {B} }T\left({\frac {\partial (\ln V-3\ln \Lambda )}{\partial V}}\right)={\frac {Nk_{\mathrm {B} }T}{V}}}
Here we applied that
{\displaystyle {\frac {\partial \ln V}{\partial V}}={\frac {1}{V}}\quad {\hbox{and}}\quad {\frac {\partial \ln \Lambda }{\partial V}}=0}
Using that N = nNA and NAkB = R (see introduction), we have
{\displaystyle p\,V=nN_{\mathrm {A} }\,k_{\mathrm {B} }\,T=nR\,T}
and that completes the proof of the ideal gas law.
↑ Molar gas constant Obtained from the NIST website. (Archived by WebCite® at http://www.webcitation.org/5dZ3JDcYN on Jan 3, 2009)
↑ Atoms may be seen as monatomic molecules.
↑ Wikipedia: Ideal gas law Version of January 2, 2009
↑ 4.0 4.1 4.2 Compressibility of Natural Gas Jeffrey L. Savidge, 78th International School for Hydrocarbon Measurement (Class 1040), 2003. From the website of the Colorado Engineering Experiment Station, Inc. (CEESI).
↑ Emile Clapeyron (1834). "Mémoire sur la puissance motrice de la chaleur (The motive power of heat)". Journal de l'École Polytechnique 14 (23): 153-190.
↑ T.L. Hill (1987). An Introduction to Statistical Thermodynamics. Dover Publications. ISBN 0-486-65242-4.
↑ D.A. McQuarrie (2000). Statistical Mechanics. University Science Books. ISBN 1-891389-15-7.
↑ R. H. Fowler (1966). Statistical Mechanics, 2nd Edition (Reprinted). Cambridge University Press, page 31. ISBN 0-521-09377-5.
↑ Such occasional exchange of energy can proceed via collisions with the walls, through interaction with a radiation field, or sporadic molecule-molecule collisions. This energy exchange is not explicitly included in the following formalism.
↑ This ad hoc insertion cuts short the application of the Bose-Einstein statistics and the proof that it leads to a multiplication by 1/N! for higher temperatures
Retrieved from "https://citizendium.org/wiki/index.php?title=Ideal_gas_law&oldid=37314" |
Formate dehydrogenase (cytochrome) - Wikipedia
Formate dehydrogenase-N hetero9mer, E.Coli
In enzymology, a formate dehydrogenase (cytochrome) (EC 1.2.2.1) is an enzyme that catalyzes the chemical reaction
formate + 2 ferricytochrome b1
{\displaystyle \rightleftharpoons }
CO2 + 2 ferrocytochrome b1 + 2 H+
Thus, the two substrates of this enzyme are formate and ferricytochrome b1, whereas its 3 products are CO2, ferrocytochrome b1, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with a cytochrome as acceptor. The systematic name of this enzyme class is formate:ferricytochrome-b1 oxidoreductase. Other names in common use include formate dehydrogenase, and formate:cytochrome b1 oxidoreductase. This enzyme participates in glyoxylate and dicarboxylate metabolism.
Gale EF (June 1939). "Formic dehydrogenase of Bacterium coli: its inactivation by oxygen and its protection in the bacterial cell". The Biochemical Journal. 33 (6): 1012–27. doi:10.1042/bj0331012. PMC 1264479. PMID 16746983.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Formate_dehydrogenase_(cytochrome)&oldid=1050657250" |
Analytical Study of Nonlinear Fractional-Order Integrodifferential Equation: Revisit Volterra's Population Model
2012 Analytical Study of Nonlinear Fractional-Order Integrodifferential Equation: Revisit Volterra's Population Model
Najeeb Alam Khan, Amir Mahmood, Nadeem Alam Khan, Asmat Ara
This paper suggests two component homotopy method to solve nonlinear fractional integrodifferential equations, namely, Volterra's population model. Padé approximation was effectively used in this method to capture the essential behavior of solutions for the mathematical model of accumulated effect of toxins on a population living in a closed system. The behavior of the solutions and the effects of different values of fractional-order
\alpha
are indicated graphically. The study outlines significant features of this method as well as sheds some light on advantages of the method over the other. The results show that this method is very efficient, convenient, and can be adapted to fit a larger class of problems.
Najeeb Alam Khan. Amir Mahmood. Nadeem Alam Khan. Asmat Ara. "Analytical Study of Nonlinear Fractional-Order Integrodifferential Equation: Revisit Volterra's Population Model." Int. J. Differ. Equ. 2012 (SI2) 1 - 8, 2012. https://doi.org/10.1155/2012/845945
Najeeb Alam Khan, Amir Mahmood, Nadeem Alam Khan, Asmat Ara "Analytical Study of Nonlinear Fractional-Order Integrodifferential Equation: Revisit Volterra's Population Model," International Journal of Differential Equations, Int. J. Differ. Equ. 2012(SI2), 1-8, (2012) |
(R)-aminopropanol dehydrogenase - Wikipedia
In enzymology, a (R)-aminopropanol dehydrogenase (EC 1.1.1.75) is an enzyme that catalyzes the chemical reaction
(R)-1-aminopropan-2-ol + NAD+
{\displaystyle \rightleftharpoons }
aminoacetone + NADH + H+
Thus, the two substrates of this enzyme are (R)-1-aminopropan-2-ol and NAD+, whereas its 3 products are aminoacetone, NADH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-OH group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is (R)-1-aminopropan-2-ol:NAD+ oxidoreductase. Other names in common use include L-aminopropanol dehydrogenase, 1-aminopropan-2-ol-NAD+ dehydrogenase, L(+)-1-aminopropan-2-ol:NAD+ oxidoreductase, 1-aminopropan-2-ol-dehydrogenase, DL-1-aminopropan-2-ol: NAD+ dehydrogenase, and L(+)-1-aminopropan-2-ol-NAD+/NADP+ oxidoreductase. This enzyme participates in glycine, serine and threonine metabolism. It requires potassium as a cofactor.
Dekker EE, Swain RR (1968). "Formation of Dg-1-amino-2-propanol by a highly purified enzyme from Escherichia coli" (PDF). Biochim. Biophys. Acta. 158 (2): 306–7. doi:10.1016/0304-4165(68)90150-5. hdl:2027.42/33173. PMID 4385233.
Tuner JM (1966). "Microbial metabolism of amino ketones. Aminoacetone formation from 1-aminopropan-2-ol by a dehydrogenase in Escherichia coli". Biochem. J. 99 (2): 427–33. PMC 1265012. PMID 5329339.
Turner JM (1967). "Microbial metabolism of amino ketones: l-1-Aminopropan-2-ol dehydrogenase and l-threonine dehydrogenase in Escherichia coli". Biochem. J. 104 (1): 112–21. PMC 1270551. PMID 5340733.
Retrieved from "https://en.wikipedia.org/w/index.php?title=(R)-aminopropanol_dehydrogenase&oldid=1021842132" |
I was in London for the last Geological,1 & found at my Brother’s two pamphlets from you, one most useful to me, & a packet of Maple Sugar for Lizzie, which pleased her much; & by a great effort she agreed to send you her many thanks;—poor little dear, graciosity, as yet, is not her forte.2 I make the above facts an excuse for writing to you, for I have very little to say. I was extremely sorry to hear of the delay in the opening of the Exhibition;3 but I hope it will not interfere with the Canary Islands.—4
I went up for a Paper by the Arctic Dr. Sutherland on ice-action, read only in abstract, but I shd. think with much good matter.5 It was very pleasant to hear, that it was written owing to the admiralty manual.6 There was also a paper by Trimmer partly on the superficial deposits of Kent,7 Murchison urged his catastrophe view to account for the flints,8 so I gave your view of sub-glacial action & urged where on earth the flood, which divided France & England, could have found so vast a pile of almost clean flints. I stated that one of the arctic navigators, had informed you that the stones on the beach were angular in those countries: & on this head I asked Dr.. Sutherland, & he most strongly confirmed this statement; & I thought you would like to hear this.9 Hopkins10 spoke, he admitted to a considerable extent, the force of my notion of (plastic) icebergs being driven by their momentum over considerable inequalities in an almost straight course.11 Chambers also spoke at length: Have you seen his long & I must say interesting Paper on glaciation in Eding. New. Phil Journal: he actually reproduces Agassiz’s notion of one continuous sheet of ice over the whole northern world, & treats all Icebergians with the most supercilious contempt.—12
I do not know whether you will care to hear the above Report of our meeting; but I do not at all expect you to answer this.—
I did not stay for the battle royal at the Royal Socy. but I see Murchison & Beaufort gained the day, & Capt Inglefield was elected making one more than the proper number of admissions.13
We are all well: I am alone at present; Emma having gone for a few days to her sisters.14 On July 1st. we go for a month, the whole posse comitatus of us, to the Isle of Wight:15 & on our return I hope to go to press with my weariful cirripedes.
My kindest remembrances to Lady Lyell. Ever most truly Your’s | Charles Darwin
The meeting of the Geological Society on 1 June 1853.
Elizabeth Darwin, nearly 6 years old, was slow in developing as a child and was not able to live a fully independent adult life. Very little is known about her, but in the notebook in which observations on the Darwin children are recorded, Emma noted that Elizabeth’s speech at 4
\frac{1}{2}
years old was confused; her pronunciation was strange and her phraseology peculiar. By the time she was 5 years old, she had developed ‘a great habit of abstraction going by herself & talking to herself for an hour. She does not like to be interrupted’. (Correspondence vol. 4, Appendix III, MS pp. 42–42v.) In his letter to W. E. Darwin, 24 [February 1852], CD reported that Elizabeth ‘shivers & makes as many extraordinary grimaces as ever’. Elizabeth lived with CD and Emma in Down and, after CD’s death, remained with Emma in Cambridge and Down (Wedgwood and Wedgwood 1980, pp. 321–2; Emma Darwin (1904), p. 338). Following Emma’s death, she settled in Cambridge. Her sister Henrietta Litchfield, included letters concerning Elizabeth’s voluntary work in 1891 at the infirmary of the workhouse in Cambridge in Emma Darwin (1904), 2: 404–5. In the later edition of Emma Darwin (1915), she noted only Elizabeth’s birth. Gwen Raverat, CD’s granddaughter, who knew Elizabeth towards the end of her life in the 1920s, observed: ‘She was not good at practical things … and she could not have managed her own life without a little help and direction now and then; but she was shrewd enough in her own way, and a very good judge of character.’ (Raverat 1952, pp. 146–7). See also Darwin 1955, pp. 58–9.
Lyell, a permanent member of the Royal Commission for the Exhibition of 1851, had been asked by the Government to represent science at the New York Industrial Exhibition of 1853 (K. M. Lyell ed. 1881, 2: 187–8). This letter is addressed to Lyell as ‘Commissioner to the Great Exhibition | New York | U.S.’
Lyell intended to visit the islands in the autumn of 1853 (K. M. Lyell ed. 1881, 2: 188). However, the trip was not made until February 1854 (see letter to J. D. Dana, 6 December [1853], n. 13).
Peter Cormack Sutherland served as surgeon on several Arctic expeditions. His paper, Sutherland 1853, was communicated to the Geological Society by Andrew Crombie Ramsay.
A manual of scientific enquiry (Herschel ed. 1849), intended to direct naval officers in their scientific investigations, to which CD had contributed the chapter on geology.
Trimmer 1853, the third part of Joshua Trimmer’s work on the origin of the soils which cover the Chalk of Kent.
The flints are described in Trimmer 1853, pp. 289–90. For Roderick Impey Murchison’s views on the catastrophic origin of the flint drift, see Murchison 1851 which refers to Trimmer 1851, the first part of Trimmer’s work on the origin of the soils which cover the Chalk of Kent.
According to Lyell (and CD), flints and erratic boulders were dropped by icebergs or by local glaciers: flints transported by floods or by surf action on beaches would be rounded. For Murchison’s argument against the agency of ice in the formation of the drift deposits, see Murchison 1851, p. 395. The angular nature of the flints, according to Murchison, indicated ‘a much greater intensity of fracture in former stages of the planet than now’ and that ‘such dislocations must have been accompanied by torrents of water’; ‘ordinary tidal action’, he argued, would have produced ‘water-worn pebbles’ (Murchison 1851, p. 394).
CD eventually published a paper in 1855 on the plasticity of icebergs and their power to make rectilinear grooves across a submarine undulatory surface (Collected papers 1: 252–5).
Chambers 1853a, in which Robert Chambers stated: ‘If any man were to say, that because he can with some difficulty smooth a rough surface of wood with his thumb-nail, therefore his dining-tables must have been fashioned and polished by the joiner with that little instrument alone, I would consider him as advancing a theory fully as tenable as that which consists in attributing all the so-called glacial phenomena to ice-bergs.’ (p. 230).
The Royal Society council meeting for the nomination of fellows took place on 2 June 1853 (Abstracts of the papers communicated to the Royal Society of London 6 (1850–4): 311–12). On this day, sixteen scientific men, including Captain Edward Augustus Inglefield, were elected FRS instead of the fifteen stipulated as the maximum in the rules of the society (Hall 1984, pp. 80–2). Murchison and Francis Beaufort presumably had proposed Inglefield.
Elizabeth Wedgwood and Charlotte Langton. Both lived in Hartfield, Sussex. Emma noted in her diary that she went to Hartfield on 6 June and returned home on 10 June.
This plan was changed. Instead, on 14 July, the family took a house in Eastbourne, Sussex (‘Journal’; Correspondence vol. 5, Appendix I).
Darwin, Bernard. 1955. The world that Fred made: an autobiography. London: Chatto & Windus.
Raverat, Gwendolen Mary. 1952. Period piece; a Cambridge childhood. London: Faber & Faber.
Sutherland, P. C. 1853. On the geological and glacial phenomena of the coasts of Davis’ Strait and Baffin’s Bay. Quarterly Journal of the Geological Society of London 9: 296-312. [Vols. 5,9]
Trimmer, Joshua. 1851. On the origin of the soils which cover the Chalk of Kent. Pt 1. Quarterly Journal of the Geological Society of London 7: 31–8.
Trimmer, Joshua. 1853. On the origin of the soils which cover the Chalk of Kent. Pt 3. Quarterly Journal of the Geological Society of London 9: 286–96. [Vols. 5,8] |
Maximal overlap discrete wavelet packet transform details - MATLAB modwptdetails - MathWorks 日本
MODWPT Details Using Default Wavelet
MODWPT Details for Two Sine Waves
MODWPT Details for Noisy Sine Wave
MODWPT Details Using Scaling and Wavelet Filters
MODWPT Details for Full Packet Tree
Maximal overlap discrete wavelet packet transform details
w = modwptdetails(x)
w = modwptdetails(x,wname)
w = modwptdetails(x,lo,hi)
w = modwptdetails(___,lev)
[w,packetlevs] = modwptdetails(___)
[w,packetlevs,cfreq] = modwptdetails(___)
[___] = modwptdetails(___,'FullTree',tf)
w = modwptdetails(x) returns the maximal overlap discrete wavelet packet transform (MODWPT) details for the 1-D real-valued signal, x. The MODWPT details provide zero-phase filtering of the signal. By default, modwptdetails returns only the terminal nodes, which are at level 4 or at level floor(log2(numel(x))), whichever is smaller.
To decide whether to use modwptdetails or modwpt, consider the type of data analysis you need to perform. For applications that require time alignment, such as nonparametric regression analysis, use modwptdetails. For applications where you want to analyze the energy levels in different packets, use modwpt. For more information, see Algorithms.
w = modwptdetails(x,wname) uses the orthogonal wavelet filter specified by wname.
w = modwptdetails(x,lo,hi) uses the orthogonal scaling filter, lo, and wavelet filter, hi.
w = modwptdetails(___,lev) returns the terminal nodes of the wavelet packet tree at positive integer level lev.
[w,packetlevs] = modwptdetails(___) returns a vector of transform levels corresponding to the rows of w.
[w,packetlevs,cfreq] = modwptdetails(___) returns cfreq, the center frequencies of the approximate passbands corresponding to the MODWPT details in w.
[___] = modwptdetails(___,'FullTree',tf), where tf is false, returns details about only the terminal (final-level) wavelet packet nodes. If you specify true, then modwptdetails returns details about the full wavelet packet tree down to the default or specified level. The default for tf is false.
Obtain the MODWPT of an electrocardiogram (ECG) signal using the default length 18 Fejer-Korovkin ('fk18') wavelet and the default level, 4.
wptdetails = modwptdetails(wecg);
Demonstrate that summing the MODWPT details over each sample reconstructs the signal. The largest absolute difference between the original signal and the reconstruction is on the order of
1{0}^{-11}
xrec = sum(wptdetails);
Obtain the MODWPT details for a signal containing 100 Hz and 450 Hz sine waves. Each row of the modwptdetails output corresponds to a separate frequency band.
x = (sin(2*pi*100*t)+sin(2*pi*450*t));
wptdetails = modwptdetails(x,lo,hi);
Use modwpt to obtain the energy and center frequencies of the signal. Plot the energy in the wavelet packets. The fourth and fifteenth frequency bands contain most of the energy. Other frequency bands have significantly less energy. The frequency ranges of fourth and fifteenth bands are approximately 94-125 Hz and 438-469 Hz, respectively.
[wpt,~,cfreqs,energy] = modwpt(x,lo,hi);
bar(1:16,energy);
xlabel('Packet')
ylabel('Packet Energy')
title('Energy by Wavelet Packet')
Plot the power spectral density of the input signal.
pwelch(x,[],[],[],fs,'onesided');
title('Power Spectral Density of Input Signal')
Show that the MODWPT details have zero-phase shift from the 100 Hz input sine.
p4 = wptdetails(4,:);
plot(t,sin(2*pi*100*t).*(t>0.3 & t<0.7))
plot(t,p4.*(t>0.3 & t<0.7),'r')
legend('Sine Wave','MODWPT Details')
Obtain the MODWPT details for a 100 Hz time-localized sine wave in noise. The sampling rate is 1000 Hz. Obtain the MODWPT at level 4 using the length 22 Fejer-Korovkin ('fk22') wavelet.
x = cos(2*pi*100*t).*(t>0.3 & t<0.7)+0.25*randn(size(t));
wptdetails = modwptdetails(x,'fk22');
Plot the MODWPT details for level 4, packet number 4. The MODWPT details represent zero-phase filtering of the input signal with an approximate passband of
\left[3Fs/{2}^{5},4Fs/{2}^{5}\right)
Fs
is the sampling frequency.
plot(t,cos(2*pi*100*t).*(t>0.3 & t<0.7));
plot(t,p4,'r')
Obtain the MODWPT details of an ECG waveform using the length 18 Fejer-Korovkin scaling and wavelet filters.
wpt = modwptdetails(wecg,lo,hi);
Obtain the MODWPT details for the full wavelet packet tree of an ECG waveform. Use the default length 18 Fejer-Korovkin ('fk18') wavelet. Extract and plot the node coefficients at level 3, node 2.
[w,packetlevels] = modwptdetails(wecg,'FullTree',true);
p3 = w(packetlevels==3,:);
title('Level 3, Node 2 MODWPT Details')
'fk18' (default) | 'dbN' | 'coifN' | 'haar' | 'fkN' | 'symN'
'fkN' — Fejér-Korovkin wavelet with N coefficients, where N = 4, 6, 8, 14, 18 and 22.
Scaling filter, specified as an even-length real-valued vector. lo must satisfy the conditions necessary to generate an orthogonal scaling function. You can specify the lo and hi scaling-wavelet filter pair only if you do not specify wname.
Wavelet filter, specified as an even-length real-valued vector. hi must satisfy the conditions necessary to generate an orthogonal wavelet. You can specify the lo and hi scaling-wavelet filter pair only if you do not specify wname.
tf — Return tree option
Return tree option, specified as false or true. If tf is false, then modwptdetails returns details about only the terminal (final-level) wavelet packet nodes. If you specify true, then modwptdetails returns details about the full wavelet packet tree down to the default or specified level.
For the full wavelet packet tree, w is a 2j+1-2-by-numel(x) matrix. Each level j has 2j wavelet packet details.
w — Wavelet packet tree details
Wavelet packet tree details, returned as a matrix with each row containing the sequency-ordered wavelet packet details for the terminal nodes. The terminal nodes are at level 4 or at level floor(log2(numel(x))), whichever is smaller. The MODWPT details are zero-phase-filtered projections of the signal onto the subspaces corresponding to the wavelet packet nodes. The sum of the MODWPT details over each sample reconstructs the original signal.
For the default terminal nodes, w is a 2j-by-numel(x) matrix. For the full packet table, at level j, w is a 2j+1-2-by-numel(x) matrix of sequency-ordered wavelet packet coefficients by level and index. The approximate passband for the nth row of w at level j is
\left[\frac{nâ1}{{2}^{\left(j+1\right)}},\frac{n}{{2}^{\left(j+1\right)}}\right)
cycles per sample, where n = 1,2,...,2j.
Transform levels, returned as a vector. The levels correspond to the rows of w. If w contains only the terminal level coefficients, packetlevs is a vector of constants equal to the terminal level. If w contains the full wavelet packet tree of details, packetlevs is a vector with 2j-1 elements for each level, j. To select all the MODWPT details at a particular level, use packetlevs with logical indexing.
Center frequencies of the approximate passbands in the w rows, returned as a vector. The center frequencies are in cycles per sample. To convert the units to cycles per unit time, multiply cfreq by the sampling frequency.
The MODWPT details (modwptdetails) are the result of zero-phase filtering of the signal. The features in the MODWPT details align exactly with features in the input signal. For a given level, summing the details for each sample returns the exact original signal.
The output of the MODWPT (modwpt) is time delayed compared to the input signal. Most filters used to obtain the MODWPT have a nonlinear phase response, which makes compensating for the time delay difficult. All orthogonal scaling and wavelet filters have this response, except the Haar wavelet. It is possible to time align the coefficients with the signal features, but the result is an approximation, not an exact alignment with the original signal. The MODWPT partitions the energy among the wavelet packets at each level. The sum of the energy over all the packets equals the total energy of the input signal.
[2] Walden, A.T., and A. Contreras Cristan. “The phase-corrected undecimated discrete wavelet packet transform and its application to interpreting the timing of events.†Proceedings of the Royal Society of London A. Vol. 454, Issue 1976, 1998, pp. 2243-2266.
modwpt | imodwpt | waveinfo | wavemngr |
Citramalate lyase - Wikipedia
In enzymology, a citramalate lyase (EC 4.1.3.22) is an enzyme that catalyzes the chemical reaction
(2S)-2-hydroxy-2-methylbutanedioate
{\displaystyle \rightleftharpoons }
acetate + pyruvate
Hence, this enzyme has one substrate, (2S)-2-hydroxy-2-methylbutanedioate, and two products, acetate and pyruvate.
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is (2S)-2-hydroxy-2-methylbutanedioate pyruvate-lyase (acetate-forming). Other names in common use include citramalate pyruvate-lyase, citramalate synthase, citramalic-condensing enzyme, citramalate synthetase, citramalic synthase, (S)-citramalate lyase, (+)-citramalate pyruvate-lyase, citramalate pyruvate lyase, (3S)-citramalate pyruvate-lyase, and (2S)-2-hydroxy-2-methylbutanedioate pyruvate-lyase. This enzyme participates in c5-branched dibasic acid metabolism.
Barker HA (1967). "Citramalate lyase of Clostridium tetanomorphum". Arch. Mikrobiol. 59 (1): 4–12. doi:10.1007/BF00406311. PMID 4301387.
Dimroth P, Buckel W, Loyal R, Eggerer H (1977). "Isolation and function of the subunits of citramalate lyase and formation of hybrids with the subunits of citrate lyase". Eur. J. Biochem. 80 (2): 469–77. doi:10.1111/j.1432-1033.1977.tb11902.x. PMID 923590.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Citramalate_lyase&oldid=917351906" |
Alpha–beta transformation - Wikipedia
In electrical engineering, the alpha-beta (
{\displaystyle \alpha \beta \gamma }
) transformation (also known as the Clarke transformation) is a mathematical transformation employed to simplify the analysis of three-phase circuits. Conceptually it is similar to the dq0 transformation. One very useful application of the
{\displaystyle \alpha \beta \gamma }
transformation is the generation of the reference signal used for space vector modulation control of three-phase inverters.
2.1 Power invariant transformation
2.2 Simplified transformation
3.1 dq0 transform
In 1937 and 1938, Edith Clarke published papers with modified methods of calculations on unbalanced three-phase problems, that turned out to be particularly useful.[1]
{\displaystyle \alpha \beta \gamma }
transform applied to three-phase currents, as used by Edith Clarke, is[2]
{\displaystyle i_{\alpha \beta \gamma }(t)=Ti_{abc}(t)={\frac {2}{3}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}\\\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}}}
{\displaystyle i_{abc}(t)}
is a generic three-phase current sequence and
{\displaystyle i_{\alpha \beta \gamma }(t)}
is the corresponding current sequence given by the transformation
{\displaystyle T}
. The inverse transform is:
{\displaystyle i_{abc}(t)=T^{-1}i_{\alpha \beta \gamma }(t)={\begin{bmatrix}1&0&1\\-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}&1\\-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}&1\end{bmatrix}}{\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\\i_{\gamma }(t)\end{bmatrix}}.}
The above Clarke's transformation preserves the amplitude of the electrical variables which it is applied to. Indeed, consider a three-phase symmetric, direct, current sequence
{\displaystyle {\begin{aligned}i_{a}(t)=&{\sqrt {2}}I\cos \theta (t),\\i_{b}(t)=&{\sqrt {2}}I\cos \left(\theta (t)-{\frac {2}{3}}\pi \right),\\i_{c}(t)=&{\sqrt {2}}I\cos \left(\theta (t)+{\frac {2}{3}}\pi \right),\end{aligned}}}
{\displaystyle I}
is the RMS of
{\displaystyle i_{a}(t)}
{\displaystyle i_{b}(t)}
{\displaystyle i_{c}(t)}
{\displaystyle \theta (t)}
is the generic time-varying angle that can also be set to
{\displaystyle \omega t}
without loss of generality. Then, by applying
{\displaystyle T}
to the current sequence, it results
{\displaystyle {\begin{aligned}i_{\alpha }=&{\sqrt {2}}I\cos \theta (t),\\i_{\beta }=&{\sqrt {2}}I\sin \theta (t),\\i_{\gamma }=&0,\end{aligned}}}
where the last equation holds since we have considered balanced currents. As it is shown in the above, the amplitudes of the currents in the
{\displaystyle \alpha \beta \gamma }
reference frame are the same of that in the natural reference frame.
Power invariant transformation[edit]
The active and reactive powers computed in the Clarke's domain with the transformation shown above are not the same of those computed in the standard reference frame. This happens because
{\displaystyle T}
is not unitary. In order to preserve the active and reactive powers one has, instead, to consider
{\displaystyle i_{\alpha \beta \gamma }(t)=Ti_{abc}(t)={\sqrt {\frac {2}{3}}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\\{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}\\\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}},}
which is a unitary matrix and the inverse coincides with its transpose.[3] In this case the amplitudes of the transformed currents are not the same of those in the standard reference frame, that is
{\displaystyle {\begin{aligned}i_{\alpha }=&{\sqrt {3}}I\cos \theta (t),\\i_{\beta }=&{\sqrt {3}}I\sin \theta (t),\\i_{\gamma }=&0.\end{aligned}}}
Finally, the inverse transformation in this case is
{\displaystyle i_{abc}(t)={\sqrt {\frac {2}{3}}}{\begin{bmatrix}1&0&{\frac {1}{\sqrt {2}}}\\-{\frac {1}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {1}{\sqrt {2}}}\\-{\frac {1}{2}}&-{\frac {\sqrt {3}}{2}}&{\frac {1}{\sqrt {2}}}\\\end{bmatrix}}{\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\\i_{\gamma }(t)\end{bmatrix}}.}
Simplified transformation[edit]
Since in a balanced system
{\displaystyle i_{a}(t)+i_{b}(t)+i_{c}(t)=0}
{\displaystyle i_{\gamma }(t)=0}
one can also consider the simplified transform[4]
{\displaystyle i_{\alpha \beta }(t)={\frac {2}{3}}{\begin{bmatrix}1&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&{\frac {\sqrt {3}}{2}}&-{\frac {\sqrt {3}}{2}}\end{bmatrix}}{\begin{bmatrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{bmatrix}}}
which is simply the original Clarke's transformation with the 3rd equation excluded, and
{\displaystyle i_{abc}(t)={\frac {3}{2}}{\begin{bmatrix}{\frac {2}{3}}&0\\-{\frac {1}{3}}&{\frac {\sqrt {3}}{3}}\\-{\frac {1}{3}}&-{\frac {\sqrt {3}}{3}}\end{bmatrix}}{\begin{bmatrix}i_{\alpha }(t)\\i_{\beta }(t)\end{bmatrix}}.}
{\displaystyle \alpha \beta \gamma }
transformation can be thought of as the projection of the three phase quantities (voltages or currents) onto two stationary axes, the alpha axis and the beta axis. However, no information is lost if the system is balanced, as the equation Ia + Ib + Ic = 0 is equivalent to the equation for
{\displaystyle I_{\gamma }}
in the transform. If the system is not balanced, then the
{\displaystyle I_{\gamma }}
term will contain the error component of the projection. Thus, a
{\displaystyle I_{\gamma }}
of zero indicates that the system is balanced (and thus exists entirely in the alpha-beta coordinate space), and can be ignored for two coordinate calculations that operate under this assumption that the system is balanced. This is the elegance of the clarke transform as it reduces a three component system into a two component system thanks to this assumption.
Another way to understand this is that the equation Ia + Ib + Ic = 0 defines a plane in a euclidean three coordinate space. The alpha-beta coordinate space can be understood as the two coordinate space defined by this plane, i.e. the alpha-beta axes lie on the plane defined by Ia + Ib + Ic = 0.
This also means that in order the use the clarke transform, one must ensure the system is balanced, otherwise subsequent two coordinate calculations will be erroneous. This is a practical consideration in applications where the three phase quantities are measured and can possibly have measurement error.
Shown above is the
{\displaystyle \alpha \beta \gamma }
transform as applied to three symmetrical currents flowing through three windings separated by 120 physical degrees. The three phase currents lag their corresponding phase voltages by
{\displaystyle \delta }
{\displaystyle \alpha }
{\displaystyle \beta }
axis is shown with the
{\displaystyle \alpha }
axis aligned with phase 'A'. The current vector
{\displaystyle I_{\alpha \beta \gamma }}
rotates with angular velocity
{\displaystyle \omega }
{\displaystyle \gamma }
component since the currents are balanced.
dq0 transform[edit]
Main article: dq0 transformation
{\displaystyle dq0}
transform is conceptually similar to the
{\displaystyle \alpha \beta \gamma }
transform. Whereas the
{\displaystyle dq0}
transform is the projection of the phase quantities onto a rotating two-axis reference frame, the
{\displaystyle \alpha \beta \gamma }
transform can be thought of as the projection of the phase quantities onto a stationary two-axis reference frame.
^ O'Rourke, Colm J. (December 2019). "A Geometric Interpretation of Reference Frames and Transformations: dq0, Clarke, and Park". IEEE Transactions on Energy Conversion. 34, 4: 2070–2083. doi:10.1109/TEC.2019.2941175. hdl:1721.1/123557 – via MIT Open Access Articles.
^ W. C. Duesterhoeft; Max W. Schulz; Edith Clarke (July 1951). "Determination of Instantaneous Currents and Voltages by Means of Alpha, Beta, and Zero Components". Transactions of the American Institute of Electrical Engineers. 70 (2): 1248–1255. doi:10.1109/T-AIEE.1951.5060554. ISSN 0096-3860.
^ S. CHATTOPADHYAY; M. MITRA; S. SENGUPTA (2008). "Area Based Approach for Three Phase Power Quality Assessment in Clarke Plane". Journal of Electrical Systems. 04 (01): 62. Retrieved 2020-11-26.
^ F. Tahri, A.Tahri, Eid A. AlRadadi and A. Draou Senior, "Analysis and Control of Advanced Static VAR compensator Based on the Theory of the Instantaneous Reactive Power," presented at ACEMP, Bodrum, Turkey, 2007.
C.J. O'Rourke et al. "A Geometric Interpretation of Reference Frames and Transformations: dq0, Clarke, and Park," in IEEE Transactions on Energy Conversion, vol. 34, no. 4, pp. 2070-2083, Dec. 2019.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Alpha–beta_transformation&oldid=1054459301" |
Assessing Dow Jones Methodology
Investors may own a handful of stocks within their investment portfolio in which they track each stock's individual performance. However, the performance of a small portfolio is not indicative of the overall market. Investors also need information about market sentiment, which is where a stock index can be helpful.
An index can provide a measurable and traceable number that represents the overall market, a selected group of stocks, or a sector. A stock index can also serve as a benchmark for investment comparisons. For example, let's say your individual portfolio of stocks (or your mutual fund) returned 15%, but the market index returned 20% during the same period. As a result, your portfolio's performance (or your fund manager's performance) would be lagging behind the market.
The Dow Jones Industrial Average (DJIA) is an indicator of how 30 large, U.S.-listed companies have traded during a standard trading session.
To better understand how the Dow changes value, let's start at its beginnings. When Dow Jones & Co. first introduced the index in the 1890s, it was a simple average of the prices of all constituents. For example, let's say there were 12 stocks in the Dow index; in that instance, the Dow's value would have been calculated by simply taking the sum of closing prices of all 12 stocks and dividing it by 12 (the number of companies or "constituents of the Dow index"). Hence, the Dow started as a simple price average index.
\begin{aligned} &\text{DJIA Index Value} = \frac{\sum_{i=0}^n{P_i}}{n} \\ &\textbf{where:}\\ &P_i = \text{The price of the } i^{th} \text{ stock}\\ &n = \text{The number of stocks in the index} \end{aligned}
DJIA Index Value=n∑i=0nPiwhere:Pi=The price of the ith stockn=The number of stocks in the index
To explain the concept better with other scenarios and twists, let's build our own simple hypothetical index along the lines of the Dow.
To keep it simple, assume that there is a stock market in a country that has only two stocks trading (Ally Inc. and Belly Inc.—A & B). How do we measure the performance of this overall stock market on a daily basis, as the stock prices are changing each moment and with every price tick? Instead of tracking each stock separately, it would be much easier to get and track a single number representing the overall market constituting both the stocks. The changes in that single number (let's call it "AB index") will reflect how the overall market is performing.
Let's assume that the exchange constructs a mathematical number represented by "AB Index," which is being measured on the performance of the two stocks (A and B). Assume that stock A is trading at $20 per share and stock B is trading at $80 per share on day 1.
\begin{aligned} \frac{\sum_{i=0}^n{P_i}}{n} &= \frac{\left(\$20 + \$80 \right ) }{2}\\ &=50 \end{aligned}
n∑i=0nPi=2($20+$80)=50
\begin{aligned} \frac{\sum_{i=0}^n{P_i}}{n} &= \frac{\left(\$25 + \$75 \right ) }{2}\\ &=50 \end{aligned}
I.e., the positive price movement in one stock has canceled the equal value but negative price movement of another stock. Therefore, the index value remains unchanged.
\begin{aligned} \frac{\sum_{i=0}^n{P_i}}{n} &= \frac{\left(\$30 + \$85 \right ) }{2}\\ &=57.5 \end{aligned}
n∑i=0nPi=2($30+$85)=57.5
In the case of (2), the net sum price change was ZERO (stock A had +5 change, while stock B has -5 change, making the net sum change zero).
Even though stock A had a higher percentage price change of 20% ($30 from $25), and stock B had a lower percentage change of 13.33% ($85 from $75), the impact of stock B's $10 change contributed to a bigger change in the overall index value. This indicates that price-weighted indices (like Dow Jones and Nikkei 225) depend on the absolute values of prices rather than relative percentage changes. This has also been one of the criticizing factors of price-weighted indexes, as they don't take into account the industry size or market capitalization value of the constituents.
From the perspective of the AB index, a new stock's coming onboard should not lead to a sudden jump or drop in its value. If it continues with its usual formula, then:
\begin{aligned} \frac{\sum_{i=0}^n{P_i}}{n} &= \frac{\left(\$30 + \$85 + \$10 \right ) }{3}\\ &=41.67 \end{aligned}
n∑i=0nPi=3($30+$85+$10)=41.67
This is a sudden dip in index value from the previous 57.5 to 41.67, just because a new constituent is getting added to it. (Assuming that stocks A and B maintain their earlier day prices of $30 and $85). This would not be a very useful reflection of the overall health of the market.
\begin{aligned} &\text{Index Value} = \frac{\sum_{i=0}^{n_{old}}{P_i}}{n_{old}}\\ &\;= \frac{\sum_{i=0}^{n_{new}}{P_i}}{n_{new}}\end{aligned}
Index Value=nold∑i=0noldPi=nnew∑i=0nnewPi,
That is, assuming the stock prices from the old index are held constant, the addition of a new stock price should not affect the index.
\begin{aligned} &\text{New Index Value} = \frac{\sum_{i=0}^{n_{new}}{P_i}}{D} \\ &\textbf{where:}\\ &P_i = \text{The price of the } i^{th} \text{ stock}\\ &n_{new} = \text{The updated number of stocks in the index}\\ &D = \frac{\sum_{i=0}^{n_{new}}{P_i}}{\text{The previous index value}} \end{aligned}
New Index Value=D∑i=0nnewPiwhere:Pi=The price of the ith stocknnew=The updated number of stocks in the indexD=The previous index value∑i=0nnewPi
So on the day when stock C is included in the AB index, its correct (and continuous value) becomes:
\begin{aligned} &\frac{\sum_{i=0}^{n_{new}}{P_i}}{D}\\ &=\frac{\$30+\$85+\$10}{2.1739} = 57.5 \end{aligned}
D∑i=0nnewPi=2.1739$30+$85+$10=57.5
On the fifth day, suppose the prices of stocks A, B, and C are respectively $32, $90, and $9, then:
\begin{aligned} &\frac{\sum_{i=0}^{n_{new}}{P_i}}{D}\\ &=\frac{\$32+\$90+\$9}{2.1739} = 60.26 \end{aligned}
D∑i=0nnewPi=2.1739$32+$90+$9=60.26
Let's continue further with calculation variations. Suppose that stock B takes a corporate action that changes the stock's price without changing the company valuation. Say it is trading at $90, and the company undertakes a 3-for-1 stock split, tripling the number of available shares and reducing the price by a factor of three, i.e., from $90 to $30.
\frac{\$32+\$30+\$9}{2.1739} = 32.66
2.1739$32+$30+$9=32.66
This is way below the earlier index value of 60.26 (at step 5).
\begin{aligned} &\text{Index Value} = \frac{\sum_{i=0}^{n_{old}}{P_i}}{n_{old}}\\ &\;= \frac{\sum_{i=0}^{n_{new}}{P_i}}{n_{new}}\\ \end{aligned}
Index Value=nold∑i=0noldPi=nnew∑i=0nnewPi
New Price summation = $71 (3 stocks).
Last known good value of index = 60.26 (step 5 above), which leads to n-new or divisor value = 71/60.26 = 1.17822.
Using this new divisor value:
\frac{\$32+\$30+\$9}{1.17822} = 60.26
1.17822$32+$30+$9=60.26
(Assuming that stocks A and C maintain their earlier day prices of $32 and $9.)
Suppose stock A is delisted and needs to be removed from the AB index, leaving only stocks B and C.
\begin{aligned} &\text{New price summation} = \$30 + \$9 = \$39\\ &\text{Previous index value} = 60.26\\ &\text{New} D = 39 \div 60.26 = 0.64719\\ &\text{New index value} = 39 \div 0.64719 = 60.26 \end{aligned}
New price summation=$30+$9=$39Previous index value=60.26NewD=39÷60.26=0.64719New index value=39÷0.64719=60.26
Dow calculations and value changes work in a similar way. The above cases cover many possible scenarios for changes for price-weighted indexes like the Dow or the Nikkei. The Dow divisor is adjusted to ensure events such as stock splits don't change the numerical value of the DJIA. Over the years, the Dow divisor has been modified to keep pace with changing market conditions.
The divisor for DJIA is 0.15172752595384 as of January 2022. However, each Dow Jones index has a divisor of its own, such as the Dow Jones 15 Utilities and Dow Jones 20 Transport.
No mathematical model is perfect—each comes with its merits and demerits. Price weighting with regular divisor adjustments does enable the Dow to reflect the market sentiments at a broader level, but it does come with a few criticisms. Sudden price increments or reductions in individual stocks can lead to big jumps or drops in DJIA. For a real-life example, an AIG stock price dip from around $292 to $45 within a month’s time led to a fall of almost 3,000 points in the Dow in 2008.
The Dow Jones index has been around since 1896, despite all of its known challenges and mathematical dependencies, the DJIA remains the most followed and recognized index globally. Investors and traders looking at using DJIA as the benchmark should consider the mathematical dependencies. Additionally, indices based on other methodologies should also consider efficient index-based investments.
The Library of Congress. "Dow Jones Industrial Average First Published." Accessed Feb. 1, 2022.
Nikkei Indexes. "Nikkei Stock Average Monthly Factsheet," Page 1. Accessed Feb. 1, 2022.
S&P Dow Jones Indices. "Dow Jones Averages Methodology," Pages 7-8. Accessed Feb. 1, 2022.
Barron's "Market Lab." Accessed Feb. 1, 2022.
Yahoo! Finance. "Dow Jones Industrial Average (^DJI): Historical Data," Select Time Period, "Dec. 31, 2007-Dec. 30, 2008," Frequency, "Monthly." Accessed Feb. 1, 2022.
Yahoo! Finance. "American International Group, Inc. (AIG): Historical Data," Select Time Period, "Dec. 31, 2007-Dec. 30, 2008," Frequency, "Monthly." Accessed Feb. 1, 2022. |
Mathematical finance - Wikipedia
(Redirected from Quantitative trading)
History: Q versus P[edit]
Derivatives pricing: the Q world[edit]
{\displaystyle \mathbb {Q} }
{\displaystyle P_{0}=\mathbf {E} _{0}(P_{t})}
A process satisfying (1) is called a "martingale". A martingale does not reward risk. Thus the probability of the normalized security price process is called "risk-neutral" and is typically denoted by the blackboard font letter "
{\displaystyle \mathbb {Q} }
Risk and portfolio management: the P world[edit]
{\displaystyle \mathbb {P} }
{\displaystyle \mathbb {P} }
{\displaystyle \mathbb {Q} }
Mathematical tools[edit]
Derivatives pricing[edit]
Portfolio modelling[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mathematical_finance&oldid=1078802622#Risk_and_portfolio_management:_the_P_world" |
(glycogen-synthase-D) phosphatase - Wikipedia
(glycogen-synthase-D) phosphatase
[glycogen-synthase-D] phosphatase
In enzymology, a [glycogen-synthase-D] phosphatase (EC 3.1.3.42) is an enzyme that catalyzes the chemical reaction
[glycogen-synthase D] + H2O
{\displaystyle \rightleftharpoons }
[glycogen-synthase I] + phosphate
Thus, the two substrates of this enzyme are glycogen-synthase D and H2O, whereas its two products are glycogen-synthase I and phosphate.
This enzyme belongs to the family of hydrolases, specifically those acting on phosphoric monoester bonds. The systematic name of this enzyme class is [UDP-glucose:glycogen 4-alpha-D-glucosyltransferase-D] phosphohydrolase. Other names in common use include uridine diphosphoglucose-glycogen glucosyltransferase phosphatase, UDP-glycogen glucosyltransferase phosphatase, UDPglucose-glycogen glucosyltransferase phosphatase, glycogen glucosyltransferase phosphatase, glycogen synthetase phosphatase, glycogen synthase phosphatase, glycogen synthase D phosphatase, Mg2+ dependent glycogen synthase phosphatase, and phosphatase type 2_degree_C.
Abe N, Tsuiki S (1974). "Studies on glycogen synthase D phosphatase of rat liver--multiple nature". Biochim. Biophys. Acta. 350 (2): 383–91. doi:10.1016/0005-2744(74)90512-9. PMID 4367978.
Retrieved from "https://en.wikipedia.org/w/index.php?title=(glycogen-synthase-D)_phosphatase&oldid=917309905" |
The table below shows the lower bounds computed for some of the known APN functions for dimensions between 4 and 11. The functions in dimensions between 5 and 8 are indexed according to the table of [[Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8]]. The ones between 9 and 11 are index according to the table of [[CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n between 6 and 11)]]. A separate table listing these results for the (more than 8000) known APN functions in dimension 8 is available as [[Lower bounds on APN-distance for all known APN functions in dimension 8]].
The table below shows the lower bounds computed for some of the known APN functions for dimensions between 4 and 11. The functions in dimensions between 5 and 8 are indexed according to the table of [[Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8]]. The ones between 9 and 11 are index according to the table of [[CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n between 6 and 11)]]. A separate table listing these results for the (more than 8000) known APN functions in dimension 8 is available as [[Lower bounds on APN-distance for all known APN functions in dimension 8]]. Note that all known APN functions in dimension 7 from [[Known quadratic APN polynomial functions over GF(2^7)]] have the same value of the lower bound as e.g. <math>x^3</math> over <math>\mathbb{F}_{2^7}</math> as given in the table below.
{\displaystyle l(F)=\lceil {\frac {m_{F}}{3}}\rceil +1}
{\displaystyle l(F)}
{\displaystyle (n,n)}
{\displaystyle F}
{\displaystyle m_{F}}
{\displaystyle m_{F}=\min _{b,\beta \in \mathbb {F} _{2^{n}}}|\{a\in \mathbb {F} _{2^{n}}:(\exists x\in \mathbb {F} _{2^{n}})(F(x)+F(a+x)+F(a+\beta )=b)\}|}
{\displaystyle m_{F}}
The table below shows the lower bounds computed for some of the known APN functions for dimensions between 4 and 11. The functions in dimensions between 5 and 8 are indexed according to the table of Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8. The ones between 9 and 11 are index according to the table of CCZ-inequivalent APN functions from the known APN classes over GF(2^n) (for n between 6 and 11). A separate table listing these results for the (more than 8000) known APN functions in dimension 8 is available as Lower bounds on APN-distance for all known APN functions in dimension 8. Note that all known APN functions in dimension 7 from Known quadratic APN polynomial functions over GF(2^7) have the same value of the lower bound as e.g.
{\displaystyle x^{3}}
{\displaystyle \mathbb {F} _{2^{7}}}
as given in the table below.
{\displaystyle m_{F}} |
A recommendation engine for blog posts
Giving us some direction on where to go when we finish
Imagine my utter surprise when I look for something that can display recommended reading after a blog post, and I turn up nothing. It's always possible I haven't found the right keyword, or I've committed the sin of not using Wordpress or Gatsby. Nothing out there really fits my circumstance and need, even the few that exist for Gatsby and other platforms, so I'll take a weekend shot at building one. Chances are, there's enough copies of my situation out there that this helps someone else.
§What we want
The overall need is simple. I need something that can lead readers into posts they're likely to read after they finish reading one. No pause, no time with family, keep reading. This site - two weeks old now - isn't getting enough reads (about 50K a month) that a perceptron would notice me if I showed up with a boombox. That's fine, I didn't need one anyway. I'd also like to not go broke, so no-cold-start-lambda-functions are a no-go in case I get hugged into poverty by reddit or HN. Let's break it down.
We need a system that will:
Deliver related posts for each blog post.
Apply some order to each related post that connects to the likelihood that you'll read it.
Do as much as possible at build time, and the rest client-side. One is cheap, and the other is free real-estate.
Be open to analysis so we can realize it's broken and fix it.
Be adjustable so we can feel like we're fixing it.
Degrade gracefully in the absence of data.
Be simple enough to reason about, and with.
That's it. How can we build one?
§What we need
Now that we have a set of constraints, we can impose those on the system that doesn't exist yet.
For 1 (deliver related posts), given a post A, we want a Set of posts (or ids)
{B,C,D}
that are a subset of all posts.
2 (measure likelihood) means we need an order, so let's use an array [B, C, D]. We can use Typescript here to make things simpler.
4 (be investigable) means we need to be able to understand why B is first. We can do this by giving it a value from 0 to 1 as a confidence indicator, and an enum for why it was chosen in the first place. Now we get
type RelatedPost = {
reason: "best-looking" | "cute";
type RelatedPosts = RelatedPost[];
3 (split-computation) means we need to actually export our reasoning client-side so we can finish any computations remaining. If we simply had a list of related posts with an order, it would be very hard to add an additional criterion or make modifications.
There are a number of solutions here, but it feels elegant to choose a Symmetric Matrix (a matrix that is symmetric across a diagonal), which should give us some benefits down the line. First, computers understand matrices really well. Most operations are cheap or hardware-accelerated - which doesn't matter to us now, but it might someday when I finish a million posts. Second, we can reason about operations much more easily. With how little I write, I'm going to be a little wasteful and store the entire matrix, but we can always fold and unfold it later if we need to.
This way, we can store our likelihood expectations from all posts to all posts, like so:
type DistanceMatrix = {
identifiers: string[];
matrix: number[][];
Now, we need 5, which means adjustability. We also need 6, which means we need to know what we have data for and what we don't. My solution to this has been to generate matrices for each reason/property that a post would become related to another, and combine these likelihood matrices with some multiplier. That way, we can add a minimum likelihood threshold so weaker reasons will take themselves out of the equation. We can also over and under-weight reasons in a way that makes sense to us, and make things adjustable.
type RelatedPostMetric = {
type Metrics = RelatedPostMetric[];
We also added a vector in there so we can take into account reasons that a post should be displayed that has nothing to do with any other post. We'll see an example as we go on.
Next, let's consider these reasons.
§What we have
We can list the properties of a post that would be relevant for computing the likelihood that it's related to another post.
(YMMV, but the system we build should be adjustable so you can add your own.)
Category/Metadata: If you use tags and categories, similarity here implies your personal opinion that two things belong in the same box. It's the only real opinion you've got, so we're putting it at the top.
Title similarity: If two posts have the same proper nouns, chances are they're related.
Text similarity: Prone to errors unless fed a mountain of training data, this is still a good fallback.
Recency: This is our current final fallback. If we can't find anything related, we hope you'll be interested in the latest of things.
§Putting it together
Now that we have these, the overall logic is simple. We go through the categories in order or priority, generate the matrix or vector, cut off all values below a certain threshold, and boost it with some multiplier. Once that's done, we add it to the a global matrix.
Depending on how the pipeline is separated, we can generate this matrix at build-time, add more properties at serve time, and then again client-side. The matrices themselves are human-readable json and easy to reason about. I've linked the matrices that currently power this blog here.
Finally, we'll add a simple function to keep track of the strongest category for each likelihood (so we know why a post is being recommended).
And we're done! In < 300 lines of typescript, we have our recommendation engine.
§But what about X?
There are a number of reasons for recommending a post that I haven't included here for simplicity - and I'm always open to ideas. The core of this work has been to build something that can stand up to repeated extension and tuning, with minimal data to work from and is simple enough to be used elsewhere.
I've hosted the file in this gist if you'd like to try it out. The salient parts are below:
slugs: string[],
categories?: string[],
frontmatters: (FrontMatter | null)[],
export type RelatedPostMatrices = {
[index: string]: RelatedPosts
identifiers: string[],
matrix: number[][],
strongestCategories?: number[][],
type StrengthVector = {
vector: number[]
const metrics: RelatedPostMetric[] = [
id: 'titles',
description: 'Similar',
id: 'tagCats',
description: 'Same Category',
multiplier: 10,
id: 'recency',
description: 'New',
function applyMatrixThreshold(distanceMatrixVector: DistanceMatrix | StrengthVector, threshold: number): DistanceMatrix | StrengthVector {
if('matrix' in distanceMatrixVector)
identifiers: distanceMatrixVector.identifiers,
matrix: distanceMatrixVector.matrix.map(col => col.map(row => !isNaN(row) && row <= threshold ? row : 1))
identifiers: distanceMatrixVector.identifiers,
vector: distanceMatrixVector.vector.map(v => !isNaN(v) && v <= threshold ? v : 1 )
function normalizeVector(vector: number[]) {
const normalizeVal = (val: number, max: number, min: number): number => {
return(val - min) / (max - min);
const max = Math.max.apply(null, vector);
const min = Math.min.apply(null, vector);
const normalizedVector = vector.map(val => normalizeVal(val, max, min));
function applyFunc(incoming: DistanceMatrix | StrengthVector, func: (inp: number) => number) {
if('matrix' in incoming) {
matrix: incoming.matrix.map(r => r.map(c => func(c)))
vector: incoming.vector.map(r => func(r))
function customSigmoid(inp: number): number {
return 1./(1.+Math.exp(-8.*(inp-0.5)))
function invertNormalized(inp: number): number {
return 1-inp;
function convoluteAddWithMultiplier(baseMatrix: DistanceMatrix, incoming: DistanceMatrix | StrengthVector, multiplier: number): DistanceMatrix {
if('vector' in incoming) {
incoming.vector = incoming.vector.map(val => val*multiplier);
baseMatrix.matrix = baseMatrix.matrix.map((r, rowIndex) => r.map((c, colIndex) => incoming.vector[colIndex]+c));
incoming.matrix = incoming.matrix.map(row => row.map(val => val*multiplier));
baseMatrix.matrix = baseMatrix.matrix.map((r, rowIndex) => r.map((c, colIndex) => incoming.matrix[rowIndex][colIndex]+c));
return baseMatrix;
export function checkBuildRelatedPosts() {
if(!fs.existsSync(relatedPostsFile))
generateRelatedPosts();
export const generateRelatedPosts = () => {
console.log("Building related posts");
const posts = getAllPosts(true);
console.time("Generating corpus");
const postTitleCorpus = new Corpus(posts.map(post => post.slug), posts.map(post => post.frontmatter.title + post.frontmatter.description));
const postTagCategoryCorpus = new Corpus(posts.map(post => post.slug),
(((post.frontmatter.categories && post.frontmatter.categories.join(" ")) || "")+
((post.frontmatter.tags && post.frontmatter.tags.join(" ")) || "")) || (post.frontmatter.title + post.frontmatter.description)
const postContentCorpus = new Corpus(posts.map(post => post.slug), posts.map(post => post.markdown));
console.timeEnd("Generating corpus");
console.time("Computing metric data");
const metricData: {
[index: string]: DistanceMatrix | StrengthVector
for(let i=0;i<metrics.length;i++) {
switch(metrics[i].id) {
case 'titles':
metricData[metrics[i].id] = new Similarity(postTitleCorpus).getDistanceMatrix() as DistanceMatrix;
metricData[metrics[i].id] = new Similarity(postContentCorpus).getDistanceMatrix() as DistanceMatrix;
case 'tagCats':
metricData[metrics[i].id] = new Similarity(postTagCategoryCorpus).getDistanceMatrix() as DistanceMatrix;
case 'recency':
metricData[metrics[i].id] = applyFunc(
identifiers: posts.map(post => post.slug),
normalizeVector(
posts.map(post => post.frontmatter.date ? (new Date().getTime()-new Date(post.frontmatter.date).getTime()) : 0),
}, customSigmoid);
throw new Error("Metric detected without any data generation code.");
// Fix this when you land
metricData[metrics[i].id] = applyMatrixThreshold(metricData[metrics[i].id], metrics[i].threshold);
metricData[metrics[i].id] = applyFunc(metricData[metrics[i].id], invertNormalized);
console.timeEnd("Computing metric data");
console.time("Convoluting...");
let finalMatrix: DistanceMatrix = {
identifiers: JSON.parse(JSON.stringify(metricData.titles.identifiers)),
matrix: (metricData.titles as DistanceMatrix).matrix.map(r => r.map(c => 0)),
let strongestCategories = (metricData.titles as DistanceMatrix).matrix.map(r => r.map(c => null as null | {category: number, val: number}));
const metric = metrics[i];
let incoming = metricData[metric.id];
finalMatrix = convoluteAddWithMultiplier(finalMatrix, incoming, metric.multiplier);
incoming.matrix.map((r, rIndex) => r.map((c, cIndex) => {
const newVal = c * metric.multiplier;
const existingVal = strongestCategories[rIndex][cIndex];
if(!existingVal || existingVal.val < newVal)
strongestCategories[rIndex][cIndex] = {
category: i,
val: newVal
incoming.vector.map((r, rIndex) => {
const newVal = r * metric.multiplier;
strongestCategories.map((sc, scIndex) => {
const existingVal = strongestCategories[rIndex][scIndex];
strongestCategories[rIndex][scIndex] = {
finalMatrix.strongestCategories = strongestCategories.map(r => r.map(c => c && c.category || 0));
console.timeEnd("Convoluting...");
console.time("Writing to file...");
fs.writeFileSync(relatedPostsFile, JSON.stringify({...metricData, final: finalMatrix}));
console.timeEnd("Writing to file...");
export function getRelatedPosts(slug: string): RelatedPostMatrices {
checkBuildRelatedPosts();
const matrices: {
} = JSON.parse(fs.readFileSync(relatedPostsFile));
let slugIndex = matrices.final.identifiers.findIndex(val => val === slug);
if(slugIndex === -1)
slugIndex = 0;
let relatedPosts: RelatedPostMatrices = {};
Object.keys(matrices).map(type => {
if('matrix' in matrices[type]) {
const scores = (matrices[type] as DistanceMatrix).matrix[slugIndex]
.map((score, index) => ({score, index}))
return b.score-a.score
.filter(score => score.score > 0 && score.index !== slugIndex);
relatedPosts[type] = {
slugs: scores.map(score => matrices[type].identifiers[score.index]),
frontmatters: scores.map(score => {
return getCondensedPost(getFileNameFromSlug(matrices[type].identifiers[score.index])).frontmatter
console.error("Error reading frontmatter for slug ",getFileNameFromSlug(matrices[type].identifiers[score.index])," - ",err);
if((matrices[type] as DistanceMatrix).strongestCategories) {
relatedPosts[type].categories = scores
.map(score => (matrices[type] as DistanceMatrix).strongestCategories![slugIndex][score.index])
.map(catIndex => metrics[catIndex].description)
const scores = (matrices[type] as StrengthVector).vector
.sort((a,b) => b.score-a.score);
slugs: scores.map(c => matrices[type].identifiers[c.index]),
return relatedPosts;
Same Category (7 min)ProvenanceHow I'm losing mine, and why we shouldn't be injecting it into ourselves |
Drops - The RuneScape Wiki
The Old School RuneScape Wiki also has an article on: osrsw:Drops
"Loot" redirects here. For the friends chat feature, see LootShare.
Remains (drops) of an iron dragon.
Drops, also known as Loot, are the items that monsters leave behind for the player that killed them when they die, or when they are defeated. These items may then be picked up by players. Drops often include bones, coins, or other items. Most monsters have "100%-chance drops", which is an item or items that are always dropped by that monster upon defeat or death. 100%-chance drops are most commonly bones or demonic ashes. Certain monsters may have more than one type of 100%-chance drops, however; a common example of this are metal dragons, who drop dragon bones as well as metal salvage corresponding to their composite metal.
Typically, the player who attacks the monster first will see the drop before other players, and the attacked NPC is marked with an asterisk (*) to denote such "ownership". This does not apply to specific monsters, however, such as bosses.
Large monsters (those that take up more than one square) will always drop their drops in the south-westernmost square. This also applies to any Ranged ammunition that falls to the ground when ranging those monsters.
Drops will remain on the ground for 200 game ticks (2 minutes or 120 seconds), after which they will disappear. Drops are invisible to other players for the first 100 game ticks (1 minute or 60 seconds), after which time, anyone will be able to see (and take) the item if it is tradeable.
1 Drops tables
2.1 Binomial model
4 Loot interface
4.1 Area loot
4.2 Disabled areas
4.4 Loot beams
Drops tables[edit | edit source]
When a monster dies, it will roll most of its drop tables to see if a player should obtain an item, and then which item they should obtain.
These are the items that a monster is guaranteed to drop when it dies. Items on this table are usually remains such as bones and ashes, but there is no strict limitation to what can appear on this drop table. While all items on this table have a drop chance of "Always", not every item with an "Always" drop chance is a 100% drop; i.e., some items, such as the First dragonkin journal, are guaranteed as a drop on the first kill, but are not obtainable afterwards. This distinction keeps one-off items from being classed here; such items are usually tertiary drops.
Charms are dropped by a large selection of monsters. Charms are dropped alongside the other drops.
Generally most monsters possess a drop table when they die, which in turn rolls a drop in the main drop table. Bosses are generally guaranteed to have one roll on the main drop table, especially chest-style bosses such as Araxxi. A few select monsters are capable of rolling multiple main drops at once.
Secondary drops are another separate table also rolled alongside a main drop if the monster possesses one. This table is different from the main drop table and generally contains less rewarding items, and unless specified, cannot roll unique items. Examples of monsters with secondary drop tables include the elite dungeon bosses and Solak.
Tertiary drops are a separate table that is rolled for alongside the main drop. Unlike the main drop, multiple tertiary drops can be obtained in a single kill; however, similarly classed items cannot be obtained at the same time; e.g., if a spirit sapphire is obtained as a tertiary drop, it is impossible to also obtain a spirit ruby, but this roll will have no effect on obtaining a clue scroll.
Universal drops are a class of items that can be obtained by nearly every monster. Usually, this drops table only includes the Key token; however, promotional items can be added during Treasure Hunter promotions. Drops on this table usually follow a time release mechanic; e.g. if a key token was obtained as a drop, it will not be possible to obtain another key token for several minutes. This time gate is universal as well, being closed by, for example, skilling and obtaining a key token. Drops on this table may or may not be added directly to a player's inventory/bank.
All items have a chance of being dropped that is expressible as a number, their drop rate. Drop rates are not necessarily a guarantee; an item with a drop rate of "1 in 5" does not equate to "This item will be dropped after 5 kills." While each kill does nothing to increase the drop rate itself, it is trivial to state that more kills gives rise to more chance overall.
Binomial model[edit | edit source]
Given a known value of
{\displaystyle {\frac {1}{x}}}
, the chance of receiving such an item
{\displaystyle k}
times i{\displaystyle n}
kills can be calculated using binomial distribution.
The probability of receiving an item
{\displaystyle k}
times i{\displaystyle n}
kills with a drop rate of
{\displaystyle {\frac {1}{x}}=p}
{\displaystyle {\binom {n}{k}}p^{k}(1-p)^{n-k}}
{\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}}
For finding the probability of obtaining an item at least once, rather than a specified number of times, we can drop the binomial coefficient and simplify the equation to:
{\displaystyle 1-(1-p)^{n}}
{\displaystyle (1-p)^{n}}
is calculating the probability of not receiving the item, and we use that to calculate the inverse.
For example, it is known that the drop rate of the abyssal whip is
{\displaystyle {\frac {1}{1024}}\approx 0.000977}
. If we want to know the probability of receiving at least one abyssal whip in a task of 234 demons, we would plug into the equation:
{\displaystyle {\begin{aligned}&1-(1-0.000977)^{234}\\=&\ 1-0.999023^{234}\\\approx &\ 1-0.795543\\\approx &\ 0.204457\end{aligned}}}
Giving us the answer, we have approximately a 20.45% chance of receiving an abyssal whip during this task. See Drop Rate Calculator for the drop rate calculator.
There are two basic types of monsters: taggable and untaggable. Taggable monsters will give their drops to the first player who attacked it. Most untaggable monsters will give their drops to the player who dealt the most damage. Some monsters, such as Vorago, have unique mechanics that distribute the drops with a more complex algorithm, yet others, such as Araxxi and the Raids bosses, are looted with a coffer system, giving drops to all players who participated independently. For Ironman and Hardcore ironman mode, players must deal the most damage on the monster to receive the drop, even if they tagged it first.
Loot interface[edit | edit source]
The loot tab is a way to quickly pickup a stack of items in one go. Items that are dropped by the player, or appears for the player will appear in this interface. Certain items however, such as Pet droppings can also appear on the interface despite not being dropped by the player. The most valuable items based on Grand Exchange price will appear at the top of the interface followed by lesser valued items. Untradeables follow their own assigned value that can be found using the Wealth evaluator.
Area loot[edit | edit source]
The loot tab for quick pickups.
Options for customised pickups
This picks up all items around the player in a 7x7 grid (a three-tile radius around the player). Having the Lorehound pet active with 125 reward points spent in the Pet track increases this to 9x9 (four-tile radius). Since this has the chance of exceeding the space for the loot tab, players wishing to pick up drops in areas filled with junk should either customise what to pickup or disable area looting/loot tab. You can also open Area Loot from the Loot section of RuneMetrics without needing to click on items on the floor, or simply assign it to a key under your keybinds.
Area loot can be toggled on or off in Settings under Gameplay > Item Drops. Members can also set a loot filter to only pick up certain types of drops with the custom loot keybind.
Cursed energy does not appear in the area loot interface as an anti-luring measure.
Disabled areas[edit | edit source]
There are certain areas where the loot tab will not open even if enabled. The following areas have the loot interface disabled:
Castle Wars in game
Prifddinas Waterfall Fishing area
While the loot interface is active:
ESC: dismiss loot interface
Spacebar: Loot All
Shift + Spacebar: Loot custom
Loot beams[edit | edit source]
Main article: Loot beam
Since an update on 19 November 2013, valuable and various other rare drops will be highlighted by a beam of light when they're dropped. These beams will only shine for a short amount of time, however. If multiple items drop with each other to set off a loot beam, the displayed loot beam size will correspond to the highest-valued item within the dropped item stack.
Drops with a Grand Exchange value (total value of given drops, not counting 100%-chance drops) of 1,000–1,000,000 coins or more may be given a loot beam. A player may select the minimum Grand Exchange value of an item at which a loot beam is deployed, within the aforementioned threshold, by going into the Settings under Gameplay under Loot Settings. If the player does not select a minimum value, it will default to 500,000 until changed.
Items in the Loot window can now be examined to view their examine text plus the Grand Exchange or High Alchemy values, where applicable.
An option has been added to the Item Drops - > Loot System menu which will allow players to collect loot via the Loot Inventory when only a single item is on the floor (This feature was previously available on mobile devices but not the PC client).
The naming and descriptions of the Looting options have been adjusted for clarity following the above changes.
Added a new setting which enabled the Multiple Items Drop interface to appear for single item drops on mobile.
To enable, navigate to the 'Loot System' menu found within the 'Item Drops' menu and check 'Always Use Loot Inventory'.
Area loot now accounts for a spirit gem bag.
The ability to drag objects in the Loot interface has been removed, as it's not possible to reorder this inventory.
Lil' Tuzzy is no longer lost when disconnecting whilst the loot interface is open.
The following keybinds have been added for the loot interface:
Space to "loot all".
Shift & space to "loot custom".
Esc to close interface.
Dungeoneering keys can now be looted from the loot interface with a full inventory.
The "right-click to open loot window" setting now works correctly.
Players can no longer pick up the chocolate bar in Zanaris using the Improved Looting system.
Loot could not be picked up if dropped by trolls on a certain map tile.
Area loot (improved looting) is added to the game.
Dropped item change reversed.
Whilst it wasn't intended that dropped items didn't become visible to other players for 1 minute, it seems a lot of people considered it more of a feature than a bug. So this 'feature' has been put back into the game.
The kill stealing problem has been fixed, the treasure now goes to whoever damaged the monster most overall, rather than the person who got the last hit.
Experience is split proportionally between all people who helped kill the monster.
Dropped items now appear to other players after 1 minute, as opposed to instantly appearing to other players.
Loot from other monsters or players can no longer be stolen within the timer.
Retrieved from ‘https://runescape.wiki/w/Drops?oldid=35748446’ |
Electromagnetic Induction, Revision Notes: Kerala Class 10 SCIENCE, Science Part I - Meritnation
Permanent magnets: The substances that retain their ferromagnetic property at room temperature for a long period of time are called permanent magnets.
Electromagnets: Electromagnet is a magnet consisting of soft iron core with a coil of insulated wire wound round it. When a current flows through the wire, the core becomes magnetized.
Solenoid: It is consist of an insulating long wire wound in the form of helix. Its length is very large as compare to the its diameter.
A current carrying the rod experiences a force when placed between two poles of strong magnets. The direction of force exerted on the rod is related with the direction of current.
Magnitude of magnetic force depends upon three factors:
\left(1\right) F\propto I \left(\dots |
Glyoxylate dehydrogenase (acylating) - Wikipedia
In enzymology, a glyoxylate dehydrogenase (acylating) (EC 1.2.1.17) is an enzyme that catalyzes the chemical reaction
glyoxylate + CoA + NADP+
{\displaystyle \rightleftharpoons }
oxalyl-CoA + NADPH + H+
The 3 substrates of this enzyme are glyoxylate, CoA, and NADP+, whereas its 3 products are oxalyl-CoA, NADPH, and H+.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is glyoxylate:NADP+ oxidoreductase (CoA-oxalylating). This enzyme participates in glyoxylate and dicarboxylate metabolism.
Quayle JR, Taylor GA (1961). "Carbon assimilation by Pseudomonas oxalaticus (OX1). 5. Purification and properties of glyoxylic dehydrogenase". Biochem. J. 78: 611–615.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Glyoxylate_dehydrogenase_(acylating)&oldid=917406322" |
An Imaging Study of Compression Ignition Phenomena of Iso-Octane, Indolene, and Gasoline Fuels in a Single-Cylinder Research Engine | J. Eng. Gas Turbines Power | ASME Digital Collection
Bradley T. Zigler,
Bradley T. Zigler
e-mail: [email protected]
Stephen M. Walton,
Stephen M. Walton
Dimitris Assanis,
Margaret S. Wooldridge,
Steven T. Wooldridge
Zigler, B. T., Walton, S. M., Assanis, D., Perez, E., Wooldridge, M. S., and Wooldridge, S. T. (June 6, 2008). "An Imaging Study of Compression Ignition Phenomena of Iso-Octane, Indolene, and Gasoline Fuels in a Single-Cylinder Research Engine." ASME. J. Eng. Gas Turbines Power. September 2008; 130(5): 052803. https://doi.org/10.1115/1.2898720
High-speed imaging combined with the optical access provided by a research engine offer the ability to directly image and compare ignition and combustion phenomena of various fuels. Such data provide valuable insight into the physical and chemical mechanisms important in each system. In this study, crank-angle resolved imaging data were used to investigate homogeneous charge compression ignition (HCCI) operation of a single-cylinder four-valve optical engine fueled using gasoline, indolene, and iso-octane. Lean operating limits were the focus of the study with the primary objective of identifying different modes of reaction front initiation and propagation for each fuel. HCCI combustion was initiated and maintained over a range of lean conditions for various fuels, from
ϕ=0.69
to 0.27. The time-resolved imaging and pressure data show that high rates of heat release in HCCI combustion correlate temporally to simultaneous, intense volumetric blue emission. Lower rates of heat release are characteristic of spatially resolved blue emission. Gasoline supported leaner HCCI operation than indolene. Iso-octane showed a dramatic transition into misfire. Similar regions of preferential ignition were identified for each of the fuels considered using the imaging data.
ignition, imaging, internal combustion engines, homogeneous charge compression ignition, imaging, optical engine
Cylinders, Engines, Fuels, Gasoline, Homogeneous charge compression ignition engines, Ignition, Imaging, Pressure, Emissions, Compression, Combustion, Heat
Homogeneous Charge Compression Ignition (HCCI) Using Iso-octane, Ethanol and Natural Gas—A Comparison to Spark Ignition Operation
An Experimental Investigation of Iso-Octane Ignition Phenomena
Demonstration of Distinct Ignition Regimes Using High-Speed Digital Imaging of Iso-Octane Mixtures
Proceedings Fourth Joint Meeting of the U. S. Sections of The Combustion Institute
High-Speed Digital Imaging of Iso-Octane Mixtures at Homogeneous Charge Compression Ignition Operating Conditions
The Central States Meeting of the Combustion Institute
, Mar. 21–23, Paper No. C.1-5, pp.
Homogeneous Charge Compression Ignition Engine-Out Emissions—Does Flame Propagation Occur in Homogeneous Charge Compression Ignition?
Engine-Out Emissions From a Direct-Injection Spark-Ignition (DISI) Engine
,” 2005, SAE Paper No. 2005-01-3749.
Sunnaborg
, 1997, Design Specifications Summary for the Sandia National Laboratories Optical Engine, May.
Chemiluminescence Emission of C2, CH and OH Radicals from Opposed Jet Burner Flames
Experimental and Computational Study of CH, CH*, and OH* in an Axisymmetric Laminar Diffusion Flame |
Correlation function - Wikipedia
Correlation as a function of distance
For other uses, see Correlation function (disambiguation).
Find sources: "Correlation function" – news · newspapers · books · scholar · JSTOR (December 2009) (Learn how and when to remove this template message)
Visual comparison of convolution, cross-correlation and autocorrelation.
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables representing the same quantity measured at two different points, then this is often referred to as an autocorrelation function, which is made up of autocorrelations. Correlation functions of different random variables are sometimes called cross-correlation functions to emphasize that different variables are being considered and because they are made up of cross-correlations.
Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are no observations.
Correlation functions used in astronomy, financial analysis, econometrics, and statistical mechanics differ only in the particular stochastic processes they are applied to. In quantum field theory there are correlation functions over quantum distributions.
For possibly distinct random variables X(s) and Y(t) at different points s and t of some space, the correlation function is
{\displaystyle C(s,t)=\operatorname {corr} (X(s),Y(t)),}
{\displaystyle \operatorname {corr} }
is described in the article on correlation. In this definition, it has been assumed that the stochastic variables are scalar-valued. If they are not, then more complicated correlation functions can be defined. For example, if X(s) is a random vector with n elements and Y(t) is a vector with q elements, then an n×q matrix of correlation functions is defined with
{\displaystyle i,j}
{\displaystyle C_{ij}(s,t)=\operatorname {corr} (X_{i}(s),Y_{j}(t)).}
When n=q, sometimes the trace of this matrix is focused on. If the probability distributions have any target space symmetries, i.e. symmetries in the value space of the stochastic variable (also called internal symmetries), then the correlation matrix will have induced symmetries. Similarly, if there are symmetries of the space (or time) domain in which the random variables exist (also called spacetime symmetries), then the correlation function will have corresponding space or time symmetries. Examples of important spacetime symmetries are —
translational symmetry yields C(s,s') = C(s − s') where s and s' are to be interpreted as vectors giving coordinates of the points
rotational symmetry in addition to the above gives C(s, s') = C(|s − s'|) where |x| denotes the norm of the vector x (for actual rotations this is the Euclidean or 2-norm).
Higher order correlation functions are often defined. A typical correlation function of order n is (the angle brackets represent the expectation value)
{\displaystyle C_{i_{1}i_{2}\cdots i_{n}}(s_{1},s_{2},\cdots ,s_{n})=\langle X_{i_{1}}(s_{1})X_{i_{2}}(s_{2})\cdots X_{i_{n}}(s_{n})\rangle .}
If the random vector has only one component variable, then the indices
{\displaystyle i,j}
are redundant. If there are symmetries, then the correlation function can be broken up into irreducible representations of the symmetries — both internal and spacetime.
Properties of probability distributions[edit]
With these definitions, the study of correlation functions is similar to the study of probability distributions. Many stochastic processes can be completely characterized by their correlation functions; the most notable example is the class of Gaussian processes.
Probability distributions defined on a finite number of points can always be normalized, but when these are defined over continuous spaces, then extra care is called for. The study of such distributions started with the study of random walks and led to the notion of the Itō calculus.
The Feynman path integral in Euclidean space generalizes this to other problems of interest to statistical mechanics. Any probability distribution which obeys a condition on correlation functions called reflection positivity leads to a local quantum field theory after Wick rotation to Minkowski spacetime (see Osterwalder-Schrader axioms). The operation of renormalization is a specified set of mappings from the space of probability distributions to itself. A quantum field theory is called renormalizable if this mapping has a fixed point which gives a quantum field theory.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Correlation_function&oldid=1085342211" |
Evaluation of Nanocrystalline Coatings for Coal-Fired Ultrasupercritical Boiler Tubes | J. Pressure Vessel Technol. | ASME Digital Collection
R. Wei,
M. R. Govindaraju,
SAI Global Technologies
, P.O. Box 690313, San Antonio, TX 78249
, 1300 W. T. Harris Boulevard, Charlotte, NC 28262
Cheruvu, N. S., Wei, R., Govindaraju, M. R., and Gandy, D. W. (October 20, 2010). "Evaluation of Nanocrystalline Coatings for Coal-Fired Ultrasupercritical Boiler Tubes." ASME. J. Pressure Vessel Technol. December 2010; 132(6): 061403. https://doi.org/10.1115/1.4001548
Cyclic oxidation behavior and microstructural degradation of nanocrystalline Ni–20Cr–xAl (where
x=4 wt %
7 wt %
10 wt %
) coatings have been investigated. The coatings were deposited on Haynes 230 samples using a magnetron sputtering technique. Cyclic oxidation tests were conducted on the uncoated and coated samples at peak temperatures of
750°C
1010°C
for up to 2070 thermal cycles between the peak and room temperatures. The results showed that a dense
Al2O3
scale was formed on the external surface of all coatings after exposure at both temperatures. All three coatings showed no evidence of internal oxidation after exposure at
750°C
. Among the three coatings, only the coating containing
4 wt %
Al showed evidence of internal oxidation along the columnar grain boundaries after exposure at
1010°C
Al2O3
scale exhibited good spallation resistance during cyclic oxidation tests at both temperatures. As the Al content in the coating increased from
4 wt %
7 wt %
10 wt %
, thermal exposure led to precipitation of coarse Al-rich particles at the coating/substrate interface. In addition, thermal exposure at both temperatures led to rapid depletion of Al in the coating and grain coarsening of the coatings. The improvement in oxide scale spallation resistance and accelerated depletion of aluminum are attributed to the ultrafine grain structure of the coating and oxide scale.
boilers, grain boundaries, oxidation, sputtered coatings
Coatings, Oxidation, Cycles
Coatings in the Electricity Supply Industry: Past, Present, and Opportunities for the Future
Material Solutions for Waterwall Wastage—An Update
,” EPRI, Palo Alto, CA, EPRI Report No. TR-1009618.
Weld Overlay of Waterwall Tubing, Alternative Materials and Distortion
,” EPRI, Palo Alto, CA, EPRI Report No. TR-112643.
Long-Term Testing of Protective Coatings and Claddings at Allegheny Energy Supply at Hatfield’s Ferry #2 Boiler
,” EPRI, Palo Alto, CA, EPRI Report No. 1000186.
High Temperature Corrosion of Coatings and Boiler Steel in Reducing Chlorine-Containing Atmosphere
State-of-Knowledge Assessment for Accelerated Waterwall Corrosion With Low NOx Burners
Effects of Al Additions on the Sufidation Behavior of Iron
Tororelli
The Oxidation Sulfidation Behavior of Iron Alloy Containing 16–40 at% aluminum
Evaluation of Low Aluminum Fe–Al Alloys for Use as Weld Overlay Coatings in Reducing Environments
Proceedings from Materials the Solution Conference ‘99 on Joining of Advanced and Specialty Materials
, ASM International,
Fe–Al Weld Overlay Oxy-Fuel Thermal Spray Coatings for Corrosion Protection of Waterwalls in Fossil Fired Plants With Low NOx Burners
,” ORNL Final Report No. ORNL/Sub/95-SU604/04 & 05.
High Temperature Corrosion Resistance of Candidate FeAlCr Coatings in Low NOx Environments
Hot Corrosion Resistance of a Sputtered K38G Nanocrystalline Coating in Molten Sulfate at 900°C
Hot Corrosion of an Electrodeposited Ni-11 wt %Cr Nanocomposite Under Molten Na2SO4–NaCl
Oxidation Behavior of Nanocrystalline Fe–Ni–Cr–Al Alloy Coatings
Cyclic Oxidation of Sputter Deposited Nanocrystalline Fe–Cr–N–Al Alloy Coatings
The Mechanism Oxidation of Sputtered Ni–Cr–Al Nanocrystalline Coatings
Oxidation Behavior of Sputtered Ni–3Cr–20Al Alloy
The Effect of Coating Grain Size on the Selective Oxidation Behaviour of Ni–Cr–Al Alloy
Oxidation Behavior of Sputter-Deposited Ni–Cr–Al Microcrystalline Coatings
The Mechanism of Scale Adhesion on Sputtered Micro-Crystallized CoCrAl Films
Cyclic Oxidation Behavior and Microstructure of Nanocrystalline Ni–20Cr–4Al Coating
Degradation Mechanism Characterization and Remaining Life Prediction for NiCoCrAlY Coatings
Influence of Aluminum Depletion Effects on the Calculation of Oxidation Life of FeCrAl Alloys
Sixth International Symposium on High Temperature Corrosion and Protection of Materials
Les Embiez, France
Mennron
Long-Term Oxidation of FeCrAl ODS Alloys at High Temperature
In-Service Degradation and Life Prediction of Coatings for Advanced Land-Based Gas Turbine Buckets
Oxidation Induced Lifetime Limits of Thin Walled, Iron Based, Alumina Forming, Oxide Dispersion Strengthened Alloy Components
Evaluation of Nanocrystalline Coatings for Coal-Fired Ultra-Supercritical Boiler Tubes |
Lactate 2-monooxygenase - Wikipedia
In enzymology, a lactate 2-monooxygenase (EC 1.13.12.4) is an enzyme that catalyzes the chemical reaction
(S)-lactate + O2
{\displaystyle \rightleftharpoons }
acetate + CO2 + H2O
Thus, the two substrates of this enzyme are (S)-lactate and O2, whereas its 3 products are acetate, CO2, and H2O.
This enzyme belongs to the family of oxidoreductases, specifically those acting on single donors with O2 as oxidant and incorporation of two atoms of oxygen into the substrate (oxygenases). The oxygen incorporated need not be derived from O with incorporation of one atom of oxygen (internal monooxygenases o internal mixed-function oxidases). The systematic name of this enzyme class is (S)-lactate:oxygen 2-oxidoreductase (decarboxylating). Other names in common use include lactate oxidative decarboxylase, lactate oxidase, lactic oxygenase, lactate oxygenase, lactic oxidase, L-lactate monooxygenase, lactate monooxygenase, and L-lactate-2-monooxygenase. This enzyme participates in pyruvate metabolism. It employs one cofactor, FMN.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 2DU2.
Hayaishi O, Sutton WB (1957). "Enzymatic oxygen fixation into acetate concomitant with the enzymatic decarboxylation of L-lactate". J. Am. Chem. Soc. 79 (17): 4809–4810. doi:10.1021/ja01574a060.
Sutton WB (May 1957). "Mechanism of action and crystallization of lactic oxidative decarboxylase from Mycobacterium phlei". The Journal of Biological Chemistry. 226 (1): 395–405. PMID 13428772.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Lactate_2-monooxygenase&oldid=917454949" |
Fireworks for the annual Fourth of July show are launched straight up from a steel platform. The launch of the entire show is computer controlled. The height of a particular firework in meters off ground level is given by
h = -4.9t^2 + 49t +11.27
, where time,
t
, is in seconds.
What was the height of the platform? What is the maximum height the firework reached? How many seconds until it hit the ground?
t = 0
to find the height of the platform.
Use the Quadratic Formula to find the roots. Average them to find the line of symmetry.
You have now found the parameter
h
in the general equation
y = a(x - h)^² + k
The given equation has the same a value,
-4.9
k
by substituting the point you found in part (a).
h(x) = -4.9(t - 5)^² + 133.77
Rewrite the equation in factored form. Why might factored form of the equation be useful?
The factored form reveals the intercepts. Remember to include the a value in your factored equation. |
(S)-2-methylmalate dehydratase - Wikipedia
In enzymology, a (S)-2-methylmalate dehydratase (EC 4.2.1.34) is an enzyme that catalyzes the chemical reaction:
(S)-2-methylmalate
{\displaystyle \rightleftharpoons }
2-methylfumarate + H2O
Hence, this enzyme has one substrate, (S)-2-methylmalate, and two products, 2-methylfumarate and H2O.
This enzyme belongs to the family of lyases, specifically the hydro-lyases, which cleave carbon-oxygen bonds. The systematic name of this enzyme class is (S)-2-methylmalate hydro-lyase (2-methylfumarate-forming). Other names in common use include mesaconate hydratase, (+)-citramalate hydro-lyase, L-citramalate hydrolase, citramalate dehydratase, (+)-citramalic hydro-lyase, mesaconate mesaconase, mesaconase, and (S)-2-methylmalate hydro-lyase.
This enzyme participates in c5-branched dibasic acid metabolism. In addition, the family of lyases which is also an enzyme catalyzes the breaking the elimination reaction of the variety of amounts of chemical bonds from hydrolysis (a substitution reaction ) and oxidation, which forms a new double bond or a new ring structure.[1]
^ "Blair AH, Barker HA (1966). , Wang CC, Barker HA (1969). There are also other enzymes in international Union of Biochemistry and Molecular Biology.
Blair AH, Barker HA (1966). "Assay and purification of (+)-citramalate hydro-lyase components from Clostridium tetanomorphum". J. Biol. Chem. 241 (2): 400–8. PMID 5903732.
Wang CC, Barker HA (1969). "Purification and properties of L-citramalate hydrolyase". J. Biol. Chem. 244 (10): 2516–26. PMID 5769987.
Retrieved from "https://en.wikipedia.org/w/index.php?title=(S)-2-methylmalate_dehydratase&oldid=1072336989" |
Different techniques for studying oscillatory behavior of solution of differential equations
February 2021 Different techniques for studying oscillatory behavior of solution of differential equations
Omar Bazighifan, Rami Ahmad El-Nabulsi
The aim of this work is to study oscillatory behavior of solutions for a fourth-order neutral nonlinear differential equation
{\left(b\left(x\right){\left({w}^{m-1}\left(x\right)\right)}^{\gamma }\right)}^{\prime }+{\sum }_{i=1}^{j}{q}_{i}\left(x\right)f\left(w\left({g}_{i}\left(x\right)\right)\right)=0
x\ge {x}_{0}
. The results obtained are based on the Riccati transformation, integral averaging technique and the theory of comparison with second-order delay equations. The obtained results complements and generalize the earlier ones. Some examples are illustrated to show the applicability of the obtained results.
Omar Bazighifan. Rami Ahmad El-Nabulsi. "Different techniques for studying oscillatory behavior of solution of differential equations." Rocky Mountain J. Math. 51 (1) 77 - 86, February 2021. https://doi.org/10.1216/rmj.2021.51.77
Received: 28 March 2020; Revised: 13 July 2020; Accepted: 13 July 2020; Published: February 2021
Keywords: advanced differential equations , even-order , nonoscillatory solutions , oscillation
Omar Bazighifan, Rami Ahmad El-Nabulsi "Different techniques for studying oscillatory behavior of solution of differential equations," Rocky Mountain Journal of Mathematics, Rocky Mountain J. Math. 51(1), 77-86, (February 2021) |
Expression (mathematics) - Wikipedia
Find sources: "Expression" mathematics – news · newspapers · books · scholar · JSTOR (January 2012) (Learn how and when to remove this template message)
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.
Many authors distinguish an expression from a formula, the former denoting a mathematical object, and the latter denoting a statement about mathematical objects.[citation needed] For example,
{\displaystyle 8x-5}
{\displaystyle 8x-5\geq 5x-8}
is a formula. However, in modern mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to true or false, depending on the values that are given to the variables occurring in the expressions. For example
{\displaystyle 8x-5\geq 5x-8}
2.3 Formal languages and lambda calculus
{\displaystyle 3+8}
{\displaystyle 8x-5}
{\displaystyle 7{{x}^{2}}+4x-10}
{\displaystyle {\frac {x-1}{{{x}^{2}}+12}}}
(rational fraction)
{\displaystyle f(a)+\sum _{k=1}^{n}\left.{\frac {1}{k!}}{\frac {d^{k}}{dt^{k}}}\right|_{t=0}f(u(t))+\int _{0}^{1}{\frac {(1-t)^{n}}{n!}}{\frac {d^{n+1}}{dt^{n+1}}}f(u(t))\,dt.}
Syntax versus semanticsEdit
An expression is a syntactic construct. It must be well-formed: the allowed operators must have the correct number of inputs in the correct places, the characters that make up these inputs must be valid, have a clear order of operations, etc. Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions.
For example, in the usual notation of arithmetic, the expression 1 + 2 × 3 is well-formed, but the following expression is not:
{\displaystyle \times 4)x+,/y}
In algebra, an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression. The determination of this value depends on the semantics attached to the symbols of the expression. The choice of semantics depends on the context of the expression. The same syntactic expression 1 + 2 × 3 can have different values (mathematically 7, but also 9), depending on the order of operations implied by the context (See also Operations § Calculators).
{\displaystyle \oplus }
to designate an internal direct sum.
Formal languages and lambda calculusEdit
For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of the free variables, the value of the expression may be undefined. Thus an expression represents a function whose inputs are the values assigned to the free variables and whose output is the resulting value of the expression.[citation needed]
{\displaystyle x/y}
{\displaystyle \sum _{n=1}^{3}(2nx)}
Redden, John (2011). "Elementary Algebra". Flat World Knowledge. Archived from the original on 2014-11-15. Retrieved 2012-03-18.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Expression_(mathematics)&oldid=1089771087" |
Multisignal 1-D wavelet packet transform - MATLAB dwpt - MathWorks 日本
{2}^{4}
{H}_{0}\left(z\right)=1/8\left(-{z}^{2}+2z+6+2{z}^{-1}-{z}^{-2}\right)
{H}_{1}\left(z\right)=1/2\left(z+2+{z}^{-1}\right)
\sqrt{2}
If returning the terminal nodes of a level N decomposition, wpt is a 1-by-2N cell array. If returning the full wavelet packet tree, wpt is a 1-by-(2N+1−2) cell array.
\stackrel{Ë}{G}\left(f\right)
\stackrel{Ë}{H}\left(f\right)
represents the wavelet (highpass) analysis filter. The labels at the bottom show the partition of the frequency axis [0, ½]. |
Compute three-phase active and reactive powers using three-phase voltage and current phasors - Simulink - MathWorks Nordic
Power (3ph, Phasor)
Compute three-phase active and reactive powers using three-phase voltage and current phasors
The Power (3ph, Phasor) block outputs total active power P and reactive power Q of a three-phase voltage-current phasor signal pair. P and Q are computed as follows:
P+jQ=\frac{1}{2}\left(Va×I{a}^{\ast }+Vb×I{b}^{\ast }+Vc×I{c}^{\ast }\right)
Ia*, Ib*, Ic* are the complex conjugates of Ia, Ib, and Ic.
The figure shows sign conventions:
The three-phase phasor (three complex signals) voltage signal, in volts peak.
The three-phase phasor (three complex signals) current signal, in amperes peak.
Returns the total active power, in watts.
Returns the total active power, in vars.
The power_PhasorPowerMeasurements example shows the use of the Power (3ph, Phasor) block. |
Real Effective Exchange Rate (REER) Definition
What Is the REER?
How to Calculate the REER
What Does the REER Tell You?
Example of REER
REER vs. Spot Exchange Rate
Limitations of the REER
REER FAQs
What Is the Real Effective Exchange Rate (REER)?
The real effective exchange rate (REER) is the weighted average of a country's currency in relation to an index or basket of other major currencies. The weights are determined by comparing the relative trade balance of a country's currency against that of each country in the index.
An increase in a nation's REER is an indication that its exports are becoming more expensive and its imports are becoming cheaper. It is losing its trade competitiveness.
The real effective exchange rate (REER) compares a nation's currency value against the weighted average of the currencies of its major trading partners.
It is an indicator of the international competitiveness of a nation in comparison with its trade partners.
The formula is weighted to take into account the relative importance of each trading partner to the home country.
An increasing REER indicates that a country is losing its competitive edge.
A nation's nominal effective exchange rate (NEER), adjusted for inflation in the home country, equals its real effective exchange rate (REER).
How to Calculate the Real Effective Exchange Rate (REER)
A nation's currency may be considered undervalued, overvalued, or in equilibrium with those of other nations that it trades with. A state of equilibrium means that demand and supply are equally balanced and prices will remain stable.
A country's REER measures how well that equilibrium is being held.
REER is determined by taking the average of the bilateral exchange rates between one nation and its trading partners and then weighting it to take into account the trade allocation of each partner.
The Bank for International Settlements website provides updated effective exchange rate indices on a daily and monthly basis.
\begin{aligned} &\text{REER}=\text{CER}^n\times\text{CER}^n\times\text{CER}^n\times100\\ &\textbf{where:}\\ &\text{CER = Country exchange rate}\\ \end{aligned}
REER=CERn×CERn×CERn×100where:CER = Country exchange rate
Breaking down the formula:
The average of the exchange rates is calculated after assigning the weightings for each rate. For example, if a currency had a 60% weighting, the exchange rate would be raised to the power by 0.60. The same is done for each exchange rate and its respective weighting.
Multiply all of the exchange rates.
Then multiply the final result by 100 to create the scale or index.
Some calculations use bilateral exchange rates while other models use real exchange rates. The latter adjusts the exchange rate for inflation.
Regardless of the way in which REER is calculated, it is an average that indicates when a currency is overvalued in relation to one trading partner or undervalued in relation to another partner.
What Does the Real Effective Exchange Rate (REER) Tell You?
A country's REER is an important measure when assessing its trade capabilities.
REER can be used to measure the equilibrium value of a country's currency, identify the underlying factors of a country's trade flow, and analyze the impact that other factors, such as competition and technological changes, have on a country and ultimately on the trade-weighted index.
For example, if the U.S. dollar exchange rate weakens against the euro, U.S. exports to Europe will become cheaper. European businesses or consumers buying U.S. goods need to convert their euros to dollars to buy our exports. If the dollar is weaker than the euro, it means Europeans can get more dollars for each euro. As a result, U.S. goods get cheaper due solely to the exchange rate between the euro and the U.S. dollar.
The U.S. has a substantial trading relationship with Europe. Because of this, the euro to U.S. dollar exchange would have a larger weighting in the index. A big move in the euro exchange rate would impact the REER more than if another currency with a smaller weighting strengthened or weakened against the dollar.
Example of Real Effective Exchange Rate (REER)
Let's say the U.S. had a foreign trading relationship with only three parties: the eurozone, Great Britain, and Australia. That means the U.S. dollar has a trading relationship with the euro, the British pound, and the Australian dollar.
In this hypothetical example, the U.S. does 70% of its trading with the eurozone, 20% with Great Britain, and 10% with Australia. The basket of currencies in this case would also hold the same percentages, with the euro at 70%, the British pound at 20%, and the Australian dollar at 20%.
A move in the euro would have a greater impact on the basket than a move in the Australian dollar. If one of the exchange rates moved significantly but the weighted average of the basket didn't change, it could mean that the other currencies moved in the opposite direction, offsetting the move of the first currency.
A spot exchange rate is the current price to exchange one currency for another for delivery on the earliest possible value date. (The value date is the effective date for a financial transaction involving an asset that fluctuates in price.)
Although the spot exchange rate is for delivery on the earliest date, the standard settlement date for most spot transactions is two business days after the transaction date.
The spot exchange rate, therefore, is a current market price. The REER is an indicator of the value of a currency in relation to its trading partners.
Limitations of the Real Effective Exchange Rate (REER)
Factors besides trade can impact the REER. The real effective exchange rate doesn't take into account price changes, tariffs, or other factors that may affect trade between nations. If prices are higher in one country compared with another, trade might decrease in the country with higher prices, impacting its REER.
The weighting used in the REER calculation then has to be adjusted to reflect any changes in trade.
In addition, the central bank of each nation adjusts its monetary policy, which can lower or raise interest rates in the home country. The flow of money could increase to the countries with higher rates as investors chase yield, thus strengthening the currency exchange rate.
The REER would be impacted, but it would have little to do with trade and more to do with the interest rate markets.
Economists use REER to evaluate a country's trade flow and analyze the impact that factors such as competition and technological changes are having on a country and its economy.
Real Effective Exchange Rate (REER) FAQs
Here are the answers to some commonly-asked questions about REER.
What Is the Real Effective Change Rate?
The real effective exchange rate is a measure of the relative strength of a nation's currency in comparison with those of the nations it trades with. It is used to judge whether the nation's currency is undervalued or overvalued or, ideally, fairly valued.
How Do You Calculate Real Effective Exchange Rate?
First, weigh each nation's exchange rate to reflect its share of the home country's foreign trade. Multiply all of the weighted exchange rates. Then multiply the total by 100. That is its REER.
Or, skip the mathematics and go to the Bank for International Settlements website for its updated effective exchange rate indices.
What Is the Difference Between Real Exchange Rate and Real Effective Exchange Rate?
When Americans exchange dollars for pounds, the amount they receive is based on the real exchange rate.
What Is the Difference Between NEER and REER?
The nominal effective exchange rate (NEER) and the real effective exchange rate (REER) are both indicators of a nation's competitiveness in relation to its trading partners.
NEER is the average rate at which one nation's currency is valued in comparison with a basket of other currencies, weighted for the percentage of trade that each currency represents to that nation.
The NEER can be adjusted to compensate for the inflation rate in the home country. That adjusted number is the REER.
What Does a High REER Mean?
An increase in a nation's REER means businesses and consumers have to pay more for the products they export, while their own people are paying less for the products that it imports. It is losing its trade competitiveness.
Bank for International Settlements. "Effective exchange rate indices." Accessed April 22, 2021.
International Monetary Fund. "What is real effective exchange rate?" Accessed April 22, 2021.
Nominal Effective Exchange Rate (NEER) is the unadjusted weighted average value of a currency relative to other major currencies traded within an index. |
Hearty thanks for your note just received.1 I am very glad Mrs. Hooker feels so well & that you are off to Scotland so soon.2 I can give a good account of my patients.3 Poor Leonards kidneys are certainly in some degree organically injured; & it will be months before he will be strong. I hope we shall get to Bournemouth, where we must take separate House in 10 days or fortnight.— We are staying here at William’s house.—4
My chief object in writing is to ask for Mr Mann’s address.—5 My Bee friend, Mr Woodbury—a good man in his way, wants to offer him £8 or £10 to bring home live Bees, sending him instructions.6 It is quite hopeless. But I suppose there wd. be no impropriety in making Mr Mann the offer.? Please answer this, as soon as you are established in Scotland.— Is Oliver at Kew?7 when I am established at Bournemouth; I am completely mad to examine any fresh flowers of any Lythraceous plant & I would write & ask him if any are in bloom. Hardly any case has interested me so much as Lythrum salicaria.—8 You must let me some time examine this wonderful Vanda.9 Good Heavens what work you have had over Wellwitschia!10 How mortal man can work 5 hours with high power passes my understanding. By the way, perhaps you did not know the fact, but a young man at Ross’ told me that no one can dissect with a
\frac{1}{10}
th inch focal glass!!11 I saw your microscope, & was a little disappointed at it for zoological purposes.12 It seems to me an easily remedied, but great fault that the wheel for bringing lens nearer & further is on the cheek-side.—
Ross does not give a huge weak doublet, which, I find, almost the most useful glass. With all necessary apparatus Smith & Beck charge 11£ for my microscope!!13
But they are going to improve & I daresay spoil it. I find that slips of glass held by spring on stage of simple microscope, invaluable for quick transference of dissected object to compound. You see, God help you, by my scribbling that I am idle & am amusing myself; as my patients want nothing.—
One other question.— Can you think of plants, which have differently coloured anthers or pollen in same flowers, as in Melastomas or on same & in different plants as in Lythrum. It would be safe guide to dimorphism.—14 Do just think of this.—
Did I tell you of one curious observation which I have made on action of pollen in Linum grandiflorum: the long-styled form is sterile with its own pollen & by Jove the pollen does not even emit tubes:15 it is very curious to put pollen of long-styled & of short-styled on separate divisions of same stigma of long-styled, & after about 10 hours, to mark the wonderful difference both in state of pollen & stigma. In function, but not in appearance, the pollen of these two forms, as tested by their action may be said to be generically distinct.16
Now I have driven all care for half-an-hour out of my head; so farewell my dear old friend.— Yours affect | C. Darwin
I heartily hope that Huxley’s book will be very successful; he will be well abused.—17
Letter from J. D. Hooker, 20 August 1862.
Having recently returned from a holiday in Switzerland, intended to improve Frances Harriet Hooker’s health, the Hookers were planning to leave for a three-week trip to Scotland on 23 August 1862 (see letter from J. D. Hooker, 20 August 1862).
Emma and Leonard Darwin were both recovering from scarlet fever (see letter to John Lubbock, 21 August [1862]).
While travelling to Bournemouth for a holiday, CD, Emma, and Leonard had been obliged to remain at William Erasmus Darwin’s house in Southampton, following the onset of Emma’s scarlet fever (see letter to John Lubbock, 21 August [1862]). They did not continue their journey until 1 September (see ‘Journal’ (Correspondence vol. 10, Appendix II)).
Gustav Mann was botanical collector for the Royal Botanic Gardens, Kew, on the Niger expedition led by William Balfour Baikie (see R. Desmond 1995, p. 433).
See letter from T. W. Woodbury, 9 August 1862.
For CD’s interest in this species, see, for example, the letter to Daniel Oliver, 29 [July 1862], the letter to W. E. Darwin, [2–3 August 1862], and the letter to Asa Gray, 9 August [1862].
See letter from J. D. Hooker, 20 August 1862 and n. 12.
See letter from J. D. Hooker, 20 August 1862 and n. 6.
CD refers to the establishment of the optical instrument maker, Thomas Ross, at 2 and 3 Featherstone Buildings, High Holborn, London (Post Office London directory 1861).
In the letter to J. D. Hooker, 15 [May 1862], CD asked for details of ‘the simple microscope made by Ross’ that Hooker recommended for young surgeons. See also letter from J. D. Hooker, [17 May 1862].
CD refers to Smith, Beck & Beck, instrument makers of 6 Coleman Street and Pear Tree Cottage, Holloway Road, London. In 1847, the company (then trading as Smith & Beck) had built for CD a simple microscope to his own design. They subsequently sold copies of the original model under the name ‘Darwin’s Single Microscope’ (see Correspondence vol. 4, letter to Richard Owen, [26 March 1848] and n. 2). A full description of the microscope is given in Beck 1865, pp. 102–4.
In October 1861, CD had begun to investigate the occurrence of what he considered might be a novel form of dimorphism in the Melastomataceae, the structure and colour of the stamens facing the petals differing from that of the stamens facing the sepals in the same flower (see Correspondence vol. 9, letter to J. D. Hooker, 17 November [1861], and this volume, letter to George Bentham, 3 February [1862]). In Lythrum salicaria, the filaments and anthers of the long-styled form are different in colour from those in the other two forms (see letter to W. E. Darwin, [2–3 August 1862]). In Forms of flowers, p. 244, CD noted that he had been ‘often deceived’ by judging the occurrence of heterostyly on the basis of the length of stamens and pistils alone, and had decided that ‘the more prudent course’ was not to rank any species as heterostyled unless there was ‘evidence of more important differences between the forms’. He concluded that ‘absolutely conclusive evidence’ of heterostyly could only be derived from experiments demonstrating differences in fertility (p. 245).
CD had told Hooker that the long-styled pollen of Linum grandiflorum was sterile with its own stigma, in the letter to J. D. Hooker, 28 September [1861] (Correspondence vol. 9). See also ‘Dimorphic condition in Primula’, p. 96 (Collected papers 2: 63). CD also mentioned his more recent observations on the failure of the pollen to emit pollen-tubes, in the letters to Asa Gray, 14 July [1862] and 28 July [1862]. He described these observations in detail in ‘Two forms in species of Linum’, pp. 73–5 (Collected papers 2: 96–8).
In ‘Two forms in species of Linum’, p. 75 (Collected papers 2: 98), CD stated: Taking fertility as the criterion of distinctness, it is no exaggeration to say that the pollen of the long-styled Linum grandiflorum (and conversely of the other form) has been differentiated, with respect to the stigmas of all the flowers of the same form, to a degree correspending with that of distinct species of the same genus, or even of species of distinct genera. For CD’s interest in this question, see also the letter to Asa Gray, 9 August [1862] and n. 13, the letter to Asa Gray, 21 August [1862] and n. 5, and Correspondence vol. 10, Appendix VI.
See letter from J. D. Hooker, 20 August 1862. The reference is to T. H. Huxley 1863a.
Beck, Richard. 1865. A treatise on the construction, proper use, and capabilities of Smith, Beck, and Beck’s achromatic microscopes. London: John Van Voorst.
Lythrum. Wants to examine fresh flowers of Lythraceae. Lythrum salicaria has interested him very much.
Asks whether JDH can think of plants that have different coloured anthers or pollen in same flowers (as in Melastoma) or on same and in different plants as in Lythrum. Would be a safe guide to dimorphism.
Observation of action of pollen in Linum grandiflorum. |
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\begin{array}{l}\left[\begin{array}{c}{\stackrel{¨}{x}}_{a}\\ {\stackrel{¨}{y}}_{a}\\ {\stackrel{¨}{z}}_{a}\end{array}\right]=\frac{1}{{M}_{a}}\left[\begin{array}{c}{F}_{xa}\\ {F}_{ya}\\ {F}_{za}\end{array}\right]+\left[\begin{array}{c}{\stackrel{˙}{x}}_{a}\\ {\stackrel{˙}{y}}_{a}\\ {\stackrel{˙}{z}}_{a}\end{array}\right]×\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\frac{1}{{M}_{a}}\left[\begin{array}{c}0\\ 0\\ {F}_{za}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ {\stackrel{˙}{z}}_{a}\end{array}\right]×\left[\begin{array}{c}p\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ g\end{array}\right]=\left[\begin{array}{c}0\\ p{\stackrel{˙}{z}}_{a}\\ \frac{{F}_{za}}{{M}_{a}}+g\end{array}\right]\\ \\ \left[\begin{array}{c}\stackrel{˙}{p}\\ \stackrel{˙}{q}\\ \stackrel{˙}{r}\end{array}\right]=\left[\left[\begin{array}{c}{M}_{x}\\ {M}_{y}\\ {M}_{z}\end{array}\right]-\left[\begin{array}{c}p\\ q\\ r\end{array}\right]×\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\right]{\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]}^{-1}\\ =\left[\left[\begin{array}{c}{M}_{x}\\ 0\\ 0\end{array}\right]-\left[\begin{array}{c}p\\ q\\ 0\end{array}\right]×\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]\left[\begin{array}{c}p\\ 0\\ 0\end{array}\right]\right]{\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]}^{-1}=\left[\begin{array}{c}\frac{{M}_{x}}{{I}_{xx}}\\ 0\\ 0\end{array}\right]\end{array}
{F}_{za}=\sum _{t=1}^{Nta}\left({F}_{w{z}_{a,t}}+{F}_{z{0}_{a}}+{k}_{{z}_{a}}\left({z}_{{v}_{a,t}}-{z}_{{s}_{a,t}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{c}_{{z}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)\right)
{M}_{x}=\sum _{t=1}^{Nta}\left({F}_{w{z}_{a,t}}{y}_{{w}_{t}}+\left({F}_{z{0}_{a}}+{k}_{{z}_{a}}\left({z}_{{v}_{a,t}}-{z}_{{s}_{a,t}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{c}_{{z}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)\right){y}_{{s}_{t}}+{M}_{w{x}_{a,t}}\frac{{I}_{xx}}{{I}_{xx}+{M}_{a}{y}_{{w}_{t}}}\right)
\begin{array}{l} T{c}_{t}=\left[\begin{array}{ccc}{x}_{{w}_{1}}& {x}_{{w}_{2}}& \dots \\ {y}_{{w}_{1}}& {y}_{{w}_{2}}& \dots \\ {z}_{{w}_{1}}& {z}_{{w}_{2}}& \dots \end{array}\right]\\ S{c}_{t}=\left[\begin{array}{ccc}{x}_{{s}_{1}}& {x}_{{s}_{2}}& \dots \\ {y}_{{s}_{1}}& {y}_{{s}_{2}}& \dots \\ {z}_{{s}_{1}}& {z}_{{s}_{2}}& \dots \end{array}\right]\end{array}
{F}_{v{z}_{a,t}=-}\left({F}_{z{0}_{a}}+{k}_{{z}_{a}}\left({z}_{{v}_{a,t}}-{z}_{{s}_{a,t}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{c}_{{z}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)+{F}_{zhsto{p}_{a,t}}\right)
\begin{array}{l}{F}_{v{x}_{a,t}}={F}_{w{x}_{a,t}}\\ {F}_{v{y}_{a,t}}={F}_{w{y}_{a,t}}\\ {F}_{v{z}_{a,t}}=-{F}_{w{z}_{a,t}}\\ \\ {M}_{v{x}_{a,t}}={M}_{w{x}_{a,t}}+{F}_{w{y}_{a,t}}\left(R{e}_{w{y}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{y}_{a,t}}={M}_{w{y}_{a,t}}+{F}_{w{x}_{a,t}}\left(R{e}_{w{x}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{z}_{a,t}}={M}_{w{z}_{a,t}}\end{array}
{F}_{w{z}_{a,t}}=-Fw{a}_{z0}-kw{a}_{z}\left({z}_{{w}_{a,t}}-{z}_{{s}_{a,t}}\right)-cw{a}_{z}\left({\stackrel{˙}{z}}_{{w}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)
\begin{array}{l}{\xi }_{a,t}={\xi }_{0a}+{m}_{hcambe{r}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{camberstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\\ {\eta }_{a,t}={\eta }_{0a}+{m}_{hcaste{r}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{casterstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\\ {\zeta }_{a,t}={\zeta }_{0a}+{m}_{hto{e}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{toestee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\\ \end{array}
{\delta }_{whlstee{r}_{a,t}}={\delta }_{stee{r}_{a,t}}+{m}_{hto{e}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{toestee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|
{P}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)
{E}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)
{H}_{a,t}=-\left({z}_{{v}_{a,t}}-{z}_{{w}_{a,t}}+\frac{{F}_{z{0}_{a}}}{{k}_{{z}_{a}}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)
{z}_{wt{r}_{a,t}}=R{e}_{{w}_{a,t}}+{H}_{a,t}
\mathrm{WhlPz}={z}_{w}=\left[\begin{array}{cccc}{z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
\mathrm{Whl}\mathrm{Re}=R{e}_{w}=\left[\begin{array}{cccc}R{e}_{{w}_{1,1}}& R{e}_{{w}_{1,2}}& R{e}_{{w}_{2,1}}& R{e}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlVz}={\stackrel{˙}{z}}_{w}=\left[\begin{array}{cccc}{\stackrel{˙}{z}}_{{w}_{1,1}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlFx}={F}_{wx}=\left[\begin{array}{cccc}{F}_{w{x}_{1,1}}& {F}_{w{x}_{1,2}}& {F}_{w{x}_{2,1}}& {F}_{w{x}_{2,2}}\end{array}\right]
\mathrm{WhlFy}={F}_{wy}=\left[\begin{array}{cccc}{F}_{w{y}_{1,1}}& {F}_{w{y}_{1,2}}& {F}_{w{y}_{2,1}}& {F}_{w{y}_{2,2}}\end{array}\right]
\mathrm{WhlM}={M}_{w}=\left[\begin{array}{cccc}{M}_{w{x}_{1,1}}& {M}_{w{x}_{1,2}}& {M}_{w{x}_{2,1}}& {M}_{w{x}_{2,2}}\\ {M}_{w{y}_{1,1}}& {M}_{w{y}_{1,2}}& {M}_{w{y}_{2,1}}& {M}_{w{y}_{2,2}}\\ {M}_{w{z}_{1,1}}& {M}_{w{z}_{1,2}}& {M}_{w{z}_{2,1}}& {M}_{w{z}_{2,2}}\end{array}\right]
\mathrm{VehP}=\left[\begin{array}{c}{x}_{v}\\ {y}_{v}\\ {z}_{v}\end{array}\right]=\left[\begin{array}{cccc}{x}_{v}{}_{{}_{1,1}}& {x}_{v}{}_{{}_{1,2}}& {x}_{v}{}_{{}_{2,1}}& {x}_{v}{}_{{}_{2,2}}\\ {y}_{v}{}_{{}_{1,1}}& {y}_{v}{}_{{}_{1,2}}& {y}_{v}{}_{{}_{2,1}}& {y}_{v}{}_{{}_{2,2}}\\ {z}_{v}{}_{{}_{1,1}}& {z}_{v}{}_{{}_{1,2}}& {z}_{v}{}_{{}_{2,1}}& {z}_{v}{}_{{}_{2,2}}\end{array}\right]
\mathrm{VehV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{v}\\ {\stackrel{˙}{y}}_{v}\\ {\stackrel{˙}{z}}_{v}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{v}_{1,1}}& {\stackrel{˙}{x}}_{{v}_{1,2}}& {\stackrel{˙}{x}}_{{v}_{2,1}}& {\stackrel{˙}{x}}_{{v}_{2,2}}\\ {\stackrel{˙}{y}}_{{v}_{1,1}}& {\stackrel{˙}{y}}_{{v}_{1,2}}& {\stackrel{˙}{y}}_{{v}_{2,1}}& {\stackrel{˙}{y}}_{{v}_{2,2}}\\ {\stackrel{˙}{z}}_{{v}_{1,1}}& {\stackrel{˙}{z}}_{{v}_{1,2}}& {\stackrel{˙}{z}}_{{v}_{2,1}}& {\stackrel{˙}{z}}_{{v}_{2,2}}\end{array}\right]
\mathrm{StrgAng}={\delta }_{steer}=\left[\begin{array}{cc}{\delta }_{stee{r}_{1,1}}& {\delta }_{stee{r}_{1,2}}\end{array}\right]
\mathrm{WhlAng}\left[1,...\right]=\xi =\left[{\xi }_{a,t}\right]
\mathrm{WhlAng}\left[2,...\right]=\eta =\left[{\eta }_{a,t}\right]
\mathrm{WhlAng}\left[3,...\right]=\zeta =\left[{\zeta }_{a,t}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlP}=\left[\begin{array}{c}{x}_{w}\\ {y}_{w}\\ {z}_{w}\end{array}\right]=\left[\begin{array}{cccc}{x}_{w}{}_{{}_{1,1}}& {x}_{w}{}_{{}_{1,2}}& {x}_{w}{}_{{}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{w}{}_{{}_{1,1}}& {y}_{w}{}_{{}_{1,2}}& {y}_{w}{}_{{}_{2,1}}& {y}_{w}{}_{{y}_{2,2}}\\ {z}_{wtr}{}_{{}_{1,1}}& {z}_{wtr}{}_{{}_{1,2}}& {z}_{wtr}{}_{{}_{2,1}}& {z}_{wt{r}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{w}\\ {\stackrel{˙}{y}}_{w}\\ {\stackrel{˙}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{w}_{1,1}}& {\stackrel{˙}{x}}_{{w}_{1,2}}& {\stackrel{˙}{x}}_{{w}_{2,1}}& {\stackrel{˙}{x}}_{{w}_{2,2}}\\ {\stackrel{˙}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{y}}_{{w}_{1,2}}& {\stackrel{˙}{y}}_{{w}_{2,1}}& {\stackrel{˙}{y}}_{{w}_{2,2}}\\ {\stackrel{˙}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]=\left[\begin{array}{cccc}{\xi }_{1,1}& {\xi }_{1,2}& {\xi }_{2,1}& {\xi }_{2,2}\\ {\eta }_{1,1}& {\eta }_{1,2}& {\eta }_{2,1}& {\eta }_{2,2}\\ {\zeta }_{1,1}& {\zeta }_{1,2}& {\zeta }_{2,1}& {\zeta }_{2,2}\end{array}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{w}\\ {\stackrel{˙}{y}}_{w}\\ {\stackrel{˙}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{w}_{1,1}}& {\stackrel{˙}{x}}_{{w}_{1,2}}& {\stackrel{˙}{x}}_{{w}_{2,1}}& {\stackrel{˙}{x}}_{{w}_{2,2}}\\ {\stackrel{˙}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{y}}_{{w}_{1,2}}& {\stackrel{˙}{y}}_{{w}_{2,1}}& {\stackrel{˙}{y}}_{{w}_{2,2}}\\ {\stackrel{˙}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]=\left[\begin{array}{cccc}{\xi }_{1,1}& {\xi }_{1,2}& {\xi }_{2,1}& {\xi }_{2,2}\\ {\eta }_{1,1}& {\eta }_{1,2}& {\eta }_{2,1}& {\eta }_{2,2}\\ {\zeta }_{1,1}& {\zeta }_{1,2}& {\zeta }_{2,1}& {\zeta }_{2,2}\end{array}\right]
\mathrm{StrgAng}={\delta }_{steer}=\left[\begin{array}{cc}{\delta }_{stee{r}_{1,1}}& {\delta }_{stee{r}_{1,2}}\end{array}\right]
T{c}_{t}=\left[\begin{array}{cccc}{x}_{{w}_{1,1}}& {x}_{{w}_{1,2}}& {x}_{{w}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{{w}_{1,1}}& {y}_{{w}_{1,2}}& {y}_{{w}_{2,1}}& {y}_{{w}_{2,2}}\\ {z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
S{c}_{t}=\left[\begin{array}{cccc}{x}_{{s}_{1,1}}& {x}_{{s}_{1,2}}& {x}_{{s}_{2,1}}& {x}_{{s}_{2,2}}\\ {y}_{{s}_{1,1}}& {y}_{{s}_{1,2}}& {y}_{{s}_{2,1}}& {y}_{{s}_{2,2}}\\ {z}_{{s}_{1,1}}& {z}_{{s}_{1,2}}& {z}_{{s}_{2,1}}& {z}_{{s}_{2,2}}\end{array}\right]
Solid Axle Suspension - Coil Spring | Solid Axle Suspension - Leaf Spring | Solid Axle Suspension - Mapped |
Ultraparallel_theorem Knowpia
In hyperbolic geometry, two lines may intersect, be ultraparallel, or be limiting parallel.
Poincaré disc model: The pink line is ultraparallel to the blue line and the green lines are limiting parallel to the blue line.
The ultraparallel theorem states that every pair of (distinct) ultraparallel lines has a unique common perpendicular (a hyperbolic line which is perpendicular to both lines).
Hilbert's constructionEdit
Let r and s be two ultraparallel lines.
From any two distinct points A and C on s draw AB and CB' perpendicular to r with B and B' on r.
If it happens that AB = CB', then the desired common perpendicular joins the midpoints of AC and BB' (by the symmetry of the Saccheri quadrilateral ACB'B).
If not, we may suppose AB < CB' without loss of generality. Let E be a point on the line s on the opposite side of A from C. Take A' on CB' so that A'B' = AB. Through A' draw a line s' (A'E') on the side closer to E, so that the angle B'A'E' is the same as angle BAE. Then s' meets s in an ordinary point D'. Construct a point D on ray AE so that AD = A'D'.
Then D' ≠ D. They are the same distance from r and both lie on s. So the perpendicular bisector of D'D (a segment of s) is also perpendicular to r.[1]
(If r and s were asymptotically parallel rather than ultraparallel, this construction would fail because s' would not meet s. Rather s' would be asymptotically parallel to both s and r.)
Proof in the Poincaré half-plane modelEdit
{\displaystyle a<b<c<d}
be four distinct points on the abscissa of the Cartesian plane. Let
{\displaystyle p}
{\displaystyle q}
be semicircles above the abscissa with diameters
{\displaystyle ab}
{\displaystyle cd}
respectively. Then in the Poincaré half-plane model HP,
{\displaystyle p}
{\displaystyle q}
represent ultraparallel lines.
Compose the following two hyperbolic motions:
{\displaystyle x\to x-a}
{\displaystyle {\mbox{inversion in the unit semicircle.}}}
{\displaystyle a\to \infty ,\quad b\to (b-a)^{-1},\quad c\to (c-a)^{-1},\quad d\to (d-a)^{-1}.}
Now continue with these two hyperbolic motions:
{\displaystyle x\to x-(b-a)^{-1}}
{\displaystyle x\to \left[(c-a)^{-1}-(b-a)^{-1}\right]^{-1}x}
{\displaystyle a}
stays at
{\displaystyle \infty }
{\displaystyle b\to 0}
{\displaystyle c\to 1}
{\displaystyle d\to z}
(say). The unique semicircle, with center at the origin, perpendicular to the one on
{\displaystyle 1z}
must have a radius tangent to the radius of the other. The right triangle formed by the abscissa and the perpendicular radii has hypotenuse of length
{\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}(z+1)}
{\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}(z-1)}
is the radius of the semicircle on
{\displaystyle 1z}
, the common perpendicular sought has radius-square
{\displaystyle {\frac {1}{4}}\left[(z+1)^{2}-(z-1)^{2}\right]=z.}
The four hyperbolic motions that produced
{\displaystyle z}
above can each be inverted and applied in reverse order to the semicircle centered at the origin and of radius
{\displaystyle {\sqrt {z}}}
to yield the unique hyperbolic line perpendicular to both ultraparallels
{\displaystyle p}
{\displaystyle q}
Proof in the Beltrami-Klein modelEdit
In the Beltrami-Klein model of the hyperbolic geometry:
two ultraparallel lines correspond to two non-intersecting chords.
The poles of these two lines are the respective intersections of the tangent lines to the boundary circle at the endpoints of the chords.
Lines perpendicular to line l are modeled by chords whose extension passes through the pole of l.
Hence we draw the unique line between the poles of the two given lines, and intersect it with the boundary circle ; the chord of intersection will be the desired common perpendicular of the ultraparallel lines.
If one of the chords happens to be a diameter, we do not have a pole, but in this case any chord perpendicular to the diameter it is also perpendicular in the Beltrami-Klein model, and so we draw a line through the pole of the other line intersecting the diameter at right angles to get the common perpendicular.
The proof is completed by showing this construction is always possible:
If both chords are diameters, they intersect.(at the center of the boundary circle)
If only one of the chords is a diameter, the other chord projects orthogonally down to a section of the first chord contained in its interior, and a line from the pole orthogonal to the diameter intersects both the diameter and the chord.
If both lines are not diameters, then we may extend the tangents drawn from each pole to produce a quadrilateral with the unit circle inscribed within it.[how?] The poles are opposite vertices of this quadrilateral, and the chords are lines drawn between adjacent sides of the vertex, across opposite corners. Since the quadrilateral is convex,[why?] the line between the poles intersects both of the chords drawn across the corners, and the segment of the line between the chords defines the required chord perpendicular to the two other chords.
Alternatively, we can construct the common perpendicular of the ultraparallel lines as follows: the ultraparallel lines in Beltrami-Klein model are two non-intersecting chords. But they actually intersect outside the circle. The polar of the intersecting point is the desired common perpendicular.[2]
^ H. S. M. Coxeter (17 September 1998). Non-euclidean Geometry. pp. 190–192. ISBN 978-0-88385-522-5.
^ W. Thurston, Three-Dimensional Geometry and Topology, page 72
Karol Borsuk & Wanda Szmielew (1960) Foundations of Geometry, page 291. |
Tartronate-semialdehyde synthase - Wikipedia
Tartronate-semialdehyde synthase homotetramer, E.Coli
In enzymology, a tartronate-semialdehyde synthase (EC 4.1.1.47) is an enzyme that catalyzes the chemical reaction
2 glyoxylate
{\displaystyle \rightleftharpoons }
tartronate semialdehyde + CO2
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is glyoxylate carboxy-lyase (dimerizing tartronate-semialdehyde-forming). Other names in common use include tartronate semialdehyde carboxylase, glyoxylate carbo-ligase, glyoxylic carbo-ligase, hydroxymalonic semialdehyde carboxylase, tartronic semialdehyde carboxylase, glyoxalate carboligase, and glyoxylate carboxy-lyase (dimerizing). This enzyme participates in glyoxylate and dicarboxylate metabolism. It has 2 cofactors: FAD, and Thiamin diphosphate.
GUPTA NK, VENNESLAND B (1964). "GLYOXYLATE CARBOLIGASE OF ESCHERICHIA COLI: A FLAVOPROTEIN". J. Biol. Chem. 239: 3787–9. PMID 14257608.
BARKULIS SS, KRAKOW G (1956). "Conversion of glyoxylate to hydroxypyruvate by extracts of Escherichia coli". Biochim. Biophys. Acta. 21 (3): 593–4. doi:10.1016/0006-3002(56)90208-6. PMID 13363977.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Tartronate-semialdehyde_synthase&oldid=1079207174"
Thiamin diphosphate enzymes |
Sterilization (or sterilisation), referring to any process that eliminates, removes, kills, or deactivates all forms of life and other biological agents (such as fungi, bacteria, viruses, spore forms, prions, unicellular eukaryotic organisms such as Plasmodium, etc.) present in a specified region, such as a surface, a volume of fluid, medication, or in a compound such as biological culture media.[1][2] Sterilization can be achieved through various means, including: heat, chemicals, irradiation, high pressure, and filtration. Sterilization is distinct from disinfection, sanitization, and pasteurization in that sterilization kills, deactivates, or eliminates all forms of life and other biological agents which are present.
One of the first steps toward sterilization was made by Nicolas Appert who discovered that thorough application of heat over a suitable period slowed the decay of foods and various liquids, preserving them for safe consumption for a longer time than was typical. Canning of foods is an extension of the same principle, and has helped to reduce food borne illness ("food poisoning"). Other methods of sterilizing foods include food irradiation[3] and high pressure
(pascalization).[4]
Most medical and surgical devices used in healthcare facilities are made of materials that are able to go under steam sterilization. However, since 1950, there has been an increase in medical devices and instruments made of materials (e.g., plastics) that require low-temperature sterilization. Ethylene oxide gas has been used since the 1950s for heat- and moisture-sensitive medical devices. Within the past 15 years, a number of new, low-temperature sterilization systems (e.g., hydrogen peroxide gas plasma, peracetic acid immersion, ozone) have been developed and are being used to sterilize medical devices.][5]
Steam sterilization is the most widely used and the most dependable. Steam sterilization is nontoxic, inexpensive, rapidly microbicidal, sporicidal, and rapidly heats and penetrates fabrics.[6]
The aim of sterilization is the reduction of initially present microorganisms or other potential pathogens. The degree of sterilization is commonly expressed by multiples of the decimal reduction time, or D-value, denoting the time needed to reduce the initial number
<span class="style225">{\displaystyle N_{0}}</span>
to one tenth (
<span class="style225">{\displaystyle 10^{-1}}</span>
) of its original value.[8] Then the number of microorganisms
<span class="style225">{\displaystyle N}</span>
after sterilization time
<span class="style225">{\displaystyle t}</span>
<span class="style225"> <br class="style262"> <span class="style262">{\displaystyle {\frac {N}{N_{0}}}=10^{\left(-{\frac {t}{D}}\right)}}</span></span>
The D-value is a function of sterilization conditions and varies with the type of microorganism, temperature, water activity, pH etc.. For steam sterilization (see below) typically the temperature (in °Celsius) is given as index.
For high-risk applications such as medical devices and injections, a sterility assurance level of at least 10−6 is required by the United States Food and Drug Administration (FDA).[9]
Steam(Moist heat sterilization)
https://en.wikipedia.org/wiki/Sterilization_(microbiology)#Heat
https://en.wikipedia.org/wiki/Moist_heat_sterilization
A widely used method for heat sterilization is the Autoclave, sometimes called a converter or steam sterilizer.
Autoclaves are used in medical applications to perform sterilization and in the chemical industry to cure coatings and vulcanize rubber and for hydrothermal synthesis. They are also used in industrial applications, especially regarding composites, see autoclave (industrial).
Cutaway illustration of a jacketed rectangular-chamber autoclave
Moist heat causes destruction of micro-organisms by denaturation of macromolecules, primarily proteins. Destruction of cells by lysis may also play a role. While "sterility" implies the destruction of free-living organisms which may grow within a sample, sterilization does not necessarily entail destruction of infectious matter. Prions are an example of an infectious agent that can survive sterilization by moist heat, depending on conditions.
Many autoclaves are used to sterilize equipment and supplies by subjecting them to high-pressure saturated steam at 121 °C (249 °F) for around 15–20 minutes depending on the size of the load and the contents.[1]
The autoclave was invented by Charles Chamberland in 1879,[2] although a precursor known as the steam digester was created by Denis Papin in 1679.[3] The name comes from Greek auto-, ultimately meaning self, and Latin clavis meaning key, thus a self-locking device.[4]
It is very important to ensure that all of the trapped air is removed from the autoclave before activation, as trapped air is a very poor medium for achieving sterility.
Steam at 134 °C can achieve in three minutes the same sterility that hot air at 160 °C can take two hours to achieve.[8] Methods of air removal include:
Downward displacement (or gravity-type):
As steam enters the chamber, it fills the upper areas first as it is less dense than air. This process compresses the air to the bottom, forcing it out through a drain which often contains a temperature sensor. Only when air evacuation is complete does the discharge stop. Flow is usually controlled by a steam trap or a solenoid valve, but bleed holes are sometimes used, often in conjunction with a solenoid valve. As the steam and air mix, it is also possible to force out the mixture from locations in the chamber other than the bottom.
Steam pulsing:
air dilution by using a series of steam pulses, in which the chamber is alternately pressurized and then depressurized to near atmospheric pressure
vacuum pump sucks air or air/steam mixtures from the chamber. (Explained here).
Superatmospheric cycles:
achieved with a vacuum pump. It starts with a vacuum followed by a steam pulse followed by a vacuum followed by a steam pulse. The number of pulses depends on the particular autoclave and cycle chosen.
Subatmospheric cycles:
similar to the superatmospheric cycles, but chamber pressure never exceeds atmospheric pressure until they pressurize up to the sterilizing temperature.
Uses : In medicine
A medical autoclave is a device that uses steam to sterilize equipment and other objects. This means that all bacteria, viruses, fungi, and spores are inactivated. However, prions, such as those associated with Creutzfeldt–Jakob disease, may not be destroyed by autoclaving at the typical 134 °C for three minutes or 121 °C for 15 minutes.[citation needed] Although a wide range species of archaea, including Geogemma barosii, can survive at temperatures above 121 °C, no archaea are known to be infectious or pose a health risk to humans; in fact their biochemistry is so vastly different from our own and their multiplication rate is far too slow for microbiologists to worry about them.
Autoclaves are found in many medical settings, laboratories, and other places that need to ensure the sterility of an object. Many procedures today employ single-use items rather than sterilizable, reusable items. This first happened with hypodermic needles, but today many surgical instruments (such as forceps, needle holders, and scalpel handles) are commonly single-use rather than reusable items (see waste autoclave). Autoclaves are of particular importance in poorer countries due to the much greater amount of equipment that is re-used. Providing stove-top or solar autoclaves to rural medical centers has been the subject of several proposed medical aid missions.[citation needed]
Bio-Hazards:
Autoclaving is often used to sterilize medical waste prior to disposal in the standard municipal solid waste stream. This application has become more common as an alternative to incineration due to environmental and health concerns raised because of the combustion by-products emitted by incinerators, especially from the small units which were commonly operated at individual hospitals. Incineration or a similar thermal oxidation process is still generally mandated for pathological waste and other very toxic and/or infectious medical waste.
In dentistry, autoclaves provide sterilization of dental instruments according to health technical memorandum 01-05 (HTM01-05). According to HTM01-05, instruments can be kept, once sterilized using a vacuum autoclave for up to 12 months using sealed pouches.[9]
To facilitate efficient sterilization by steam and pressure, there are several methods of verification and indication used; these include color-changing indicator tapes and biological indicators. When using biological indicators, samples containing spores of heat-resistant microbes such as Geobacillus stearothermophilis are sterilized alongside a standard load, and are then incubated in sterile media (often contained within the sample in a glass ampule to be broken after sterilization). A color change in the media (indicating acid production by bacteria; requires the medium to be formulated for this purpose), or the appearance of turbidity (cloudiness indicating light scattering by bacterial cells) indicates that sterilization was not achieved and the sterilization cycle may need revision or improvement.
Directives / Standards / Certifications
CE PED: Pressure Equipment Directive
https://en.wikipedia.org/wiki/Pressure_Equipment_Directive
EN ISO 9001: Quality management systems standards
BS OHSAS 18001:Occupational Health and Safety Assessment Series
https://en.wikipedia.org/wiki/ASME
UNI EN ISO 14001: Environmental audit
https://en.wikipedia.org/wiki/Environmental_audit
EN ISO 13485 : Medical devices -- Quality management systems
EN 285: Sterilization - Steam sterilizers - Large sterilizers
http://www.din.de/en/meta/search/61764!search?query=en285
DIN 58951-2: Sterilization - Steam sterilizers for laboratory use - Part 2: Apparatus requirements, requirements on services and installation
http://www.din.de/en/meta/search/61764!search?query=DIN+58951-2
VDI 6300 Blatt 1: Genetic engineering operations in genetic engineering facilities - Guidance on safe operation of genetic engineering facilities
http://www.din.de/en/wdc-beuth:din21:249728861
HTM 2010: Health technical memorandum 2010 :Part 3: Validation and Varification - Sterilization
https://www.gov.uk/government/collections/health-technical-memorandum-disinfection-and-sterilization
TRB402/DIN EN 61010-2-040 : Thermolock for the sterilization of liquids Safety requirements
http://www.din.de/en/
Bio-pharmaceutical Packaging Food Cosmetics Microbiology
Machine parts, Large volume parenterals, Syringes, Blood bags, Glassware, Sealed & unseald containers, Garments, Wrapped tools, Culture media, Tools, Canned food.
Microbiology and analytical labs
Research institutes, examination agencies, universities and high schools
Bio-Technologies and life sciences institutions
Clinical diagnostic labs, Hospital operating theaters and medical care, clinical diagnostic labs
Agriculture, environmental and veterinary labs, Animal facility
Quality control labs in pharma, food/beverage, chemistry/cosmetics and other industrial sectors.
High Performance Autoclaves Key features:
Standard & Certifications:
cGLP compliance, compatible data management.
Comply BSL3–BSL4 laboratory risk category and for operating theater.
The operating software is fully validated and documented.
Optimized loading for Waste with high pathogen risk, and sealed containers.
Single or double doors for Pass-Through regulation and applications.
Engineered and pre-validated according to GAMP5.
Fully validated and documented.
Experience in the pharmaceutical industry across 60 years of history.
Industrial microprocessor, graphic LCD, key pad.
Close loop feedback PID algorithm, significant energy saving.
Continuous and accurate control of chamber temperature and pressure.
Industrial lab process controller, high process reliability.
Exclusive pressure and temperature dual sensor system, double regulation and control of the sterilization process.
High level, fully programmability and control versatility, 30 cycles easy to customize in a multi-user environment.
Sterilizing Chambers & Lids:
Volume ranges: 30L, 45, 50, 75, 140, 147, 210, 325, 456, 481, 590, 615, 700, 730 Liters.
Top AISI 316L Stainless Steel, electro, mirror polished sanitary finishing.
AISI 304 Stainless Steel steam recovery tank.
Temperature ranges and High-pressure: 3,5bar, 100 °C-144 °C, customizable.
Internal 316Lss heat-exchanger plates for chamber pre-heating by steam, and cool down by cold softened water.
Optionally these plates can be used for drying purposes at the end of the cycle.
Energy and water cost-saving. Small foot print with superior loading capacity.
Chamber Lids:
Automatic independent operation and control.
Patented pneumatic lid sealing, assuring maximum safety.
Dedicate buffer air compressor for lid closing and final drying.
Lid with quick closing and safety interlocks.
Exclusive patented swiveling pneumatic gasket"rotate-and-seal"for horizontal lid.
Guarantees perfect airtight, maximum reliability.
Improved safety and easy maintenance.
Can be equipped with safety device in compliance with TRB402/DIN EN 61010-2-040 and thermal interlock to prevent door open Internal heat exchanger for lid rapid cooling.
Pneumatically activated Sliding doors for space efficiency.
Single/double Stailess Steel sealing flange (BIOSEAL)
Built-in with aut- water feed pump.
Heat recovering system for water preheating.
Assuring utilities cost saving.
Steam heat recovery exchanger external to chamber for preheating and reusing water.
Filetrs:
Vertical position, avoiding frequent rupture.
Sanitary pneumatic diaphragm valve.
AISI 316L stainless steel components.
Electropolished mirror finished chamber.
AISI 316Lss internal heat exchanger for Chamber Lid.
Chemical inert insulation, both thermal and acoustic smoother, quieter and safer operations.
User flexibility and interfaces:
Ergonomic features for easy handling and daily use.
Ideal positioning of operator's panel and printer.
Conceptual design and modular construction with exclusive optional kits.
Easy to configure, 14 free configurable automatic programs.
For remote monitoring via Ethernet protocol.
Special test programs for routine check of sterilizer efficiency
Technical service area:
Access by a wide front door with a special safety lock for easy maintenance.
Easy installation and quick connection to the utilities.
Wide selection and configurations of options on a modular system.
External trolleys fully compatible with Glassware Washer assure cleaning integration.
Integrated software and Process controller for glassware washers steam sterilizers, making training and maintenance easier.
Glassware Washer:
The power of steam: a cost-effective cleaning:
Optimized emollient effect on greasy and sticky dirt.
True eco-friendly solution, minimize detergents, water consumption, Lowering the running costs per cycle.
Able to access hard-to-reach areas and therefore clean thoroughly.
Continous monitoring:
Drain water conductivity purity set-point further reducing water and other utilities consumption.
The cleaning process is constantly supervised.
Dedicated probes monitoring the temperature of the air/water and of the steam in chamber.
A dedicated trasducer controls the pressure of the circulating water.
An internal LED lamp operates and signals alarm evidents by changing color during the whole cycle.
Design and technical features:
All compliant with cGLP standards.
Process controller, engineered and pre-validated according to GAMP5.
Rigorous sanitary finishing of piping and all equipments and CIP to ensure a perfect washing.
Standard or customized modular racks for loads configurations and pipe connection.
Easy and friendly ergonomic loading height, process controller.
Interchangeable external trolleys compatible with paralleled lab sterilizers.
Global distribution locations:
Technical Report No. 29 (Revised 2012) Points to Consider for Cleaning Validation
https://store.pda.org/tableofcontents/tr2912_toc.pdf
https://en.wikipedia.org/wiki/Continued_process_verification
Guidance on aspects of cleaning validation in active pharmaceutical ingredient plants
http://www.chromnet.net/Taiwan/Cleaning Strategy/APIC_Cleaning_Validation_2014.pdf |
ParseURL - Maple Help
Home : Support : Online Help : Connectivity : Web Features : Network Communication : Sockets Package : ParseURL
string; URL to be parsed
The procedure ParseURL is used to break a URL into its component parts. The argument url is a Maple string which represents the URL that is to be parsed.
If the URL can be parsed successfully, then a record describing the URL is returned.
The record used to describe the URL has the following slots:
Not all components have meaning for all supported URL schema. The meaning of the resource path, in particular, is subject to interpretation according to the URL scheme. Scheme-specific interpretation of resource paths is not currently provided for in this procedure.
When a component is either meaningless, or not present and has no default value, then the corresponding slot has the value NULL. An empty string distinguishes an empty (zero length) slot value from one that is absent entirely in the URL. (For example, telnet://joe:@host.com/path/ versus telnet://[email protected]/path/)
The URL parser is based on the specification for URLs given in RFC 1738 Uniform Recourse Locators (URL). All the registered URL schema from that specification are supported. URLs whose scheme is unregistered have the following generic syntax:
<scheme>://<username>:<password>@<hostname>:<port>/<resource-path>
\mathrm{with}\left(\mathrm{Sockets}\right):
\mathrm{url}≔\mathrm{ParseURL}\left("http://www.maplesoft.com/applications/"\right)
\textcolor[rgb]{0,0,1}{\mathrm{url}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Record}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{scheme}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"http"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{hostname}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"www.maplesoft.com"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{port}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{80}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{path}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"applications/"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{user}}\textcolor[rgb]{0,0,1}{=}\left(\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{passwd}}\textcolor[rgb]{0,0,1}{=}\left(\right)\right)
\mathrm{url}:-\mathrm{scheme},\mathrm{url}:-\mathrm{hostname},\mathrm{url}:-\mathrm{port},\mathrm{url}:-\mathrm{path}
\textcolor[rgb]{0,0,1}{"http"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"www.maplesoft.com"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{80}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"applications/"}
\mathrm{url}≔\mathrm{ParseURL}\left("ftp://user:[email protected]:2002/path/to/file.txt"\right):
\mathrm{url}:-\mathrm{scheme},\mathrm{url}:-\mathrm{hostname},\mathrm{url}:-\mathrm{port},\mathrm{url}:-\mathrm{path},\mathrm{url}:-\mathrm{user},\mathrm{url}:-\mathrm{passwd}
\textcolor[rgb]{0,0,1}{"ftp"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"host.com"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2002}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"path/to/file.txt"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"user"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"passwd"}
\mathrm{url}≔\mathrm{ParseURL}\left("telnet://www.maplesoft.com:80"\right):
\mathrm{url}:-\mathrm{scheme},\mathrm{url}:-\mathrm{hostname},\mathrm{url}:-\mathrm{port},\mathrm{url}:-\mathrm{path}
\textcolor[rgb]{0,0,1}{"telnet"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"www.maplesoft.com"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{80}
\mathrm{url}≔\mathrm{ParseURL}\left("unknownScheme://user:[email protected]:2002/some/resource?with=search"\right):
\mathrm{url}:-\mathrm{scheme},\mathrm{url}:-\mathrm{hostname},\mathrm{url}:-\mathrm{port},\mathrm{url}:-\mathrm{path},\mathrm{url}:-\mathrm{user},\mathrm{url}:-\mathrm{passwd}
\textcolor[rgb]{0,0,1}{"unknownscheme"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"host.com"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2002}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"some/resource?with=search"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"user"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"passwd"}
Information Sciences Institute, "RFC 1738 Uniform Resource Locators (URL)," ISI Home Page, http://www.isi.edu/in-notes/rfc1738.txt; accessed 17 November 2005. |
Hydroxyacylglutathione hydrolase - Wikipedia
In enzymology, a Hydroxyacylglutathione hydrolase (EC 3.1.2.6) is an enzyme that catalyzes the chemical reaction
(R)-S-lactoylglutathione + water
{\displaystyle \rightleftharpoons }
glutathione + a 2-hydroxy carboxylate
Hence, this enzyme has two substrates, (R)-S-lactoylglutathione and water, and two products, glutathione and a 2-hydroxy carboxylate. With the common substrate methylglyoxal, the product is D-lactate.[1]
This enzyme belongs to the family of hydrolases, specifically the class of thioester lyases. This enzyme is commonly known as glyoxalase II. This enzyme participates in pyruvate metabolism.
^ Vander Jagt DL (1993). "Glyoxalase II: molecular characteristics, kinetics and mechanism". Biochem. Soc. Trans. 21 (2): 522–7. doi:10.1042/bst0210522. PMID 8359524.
Ball JC, Vander Jagt DL (1979). "Purification of S-2-hydroxyacylglutathione hydrolase (glyoxalase II) from rat erythrocytes". Anal. Biochem. 98 (2): 472–7. doi:10.1016/0003-2697(79)90169-6. PMID 496013.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Hydroxyacylglutathione_hydrolase&oldid=955953773" |
Volume 71 Issue 1 | Michigan Mathematical Journal
Home > Journals > Michigan Math. J. > Volume 71 > Issue 1
Borcherds’ Method for Enriques Surfaces
Simon Brandhorst, Ichiro Shimada
Michigan Math. J. 71 (1), 3-18, (March 2022) DOI: 10.1307/mmj/20195769
KEYWORDS: 14J28, 14Q10
We classify all primitive embeddings of the lattice of numerical equivalence classes of divisors of an Enriques surface with the intersection form multiplied by 2 into an even unimodular hyperbolic lattice of rank 26. These embeddings have a property that facilitates the computation of the automorphism group of an Enriques surface by Borcherds’ method.
Components of Brill–Noether Loci for Curves with Fixed Gonality
Kaelin Cook-Powell, David Jensen
Michigan Math. J. 71 (1), 19-45, (March 2022) DOI: 10.1307/mmj/1600329610
KEYWORDS: 14H51, 14T05
We describe a conjectural stratification of the Brill–Noether variety for general curves of fixed genus and gonality. As evidence for this conjecture, we show that this Brill–Noether variety has at least as many irreducible components as predicted by the conjecture and that each of these components has the expected dimension. Our proof uses combinatorial and tropical techniques. Specifically, we analyze containment relations between the various strata of tropical Brill–Noether loci identified by Pflueger in his classification of special divisors on chains of loops.
Multiplicity Along Points of a Radicial Covering of a Regular Variety
D. Sulca, O. E. Villamayor U.
Michigan Math. J. 71 (1), 47-104, (March 2022) DOI: 10.1307/mmj/20195775
We study the maximal multiplicity locus of a variety X over a field of characteristic
\mathit{p}>0
that is provided with a finite surjective radicial morphism
\mathit{\delta }:\mathit{X}\to \mathit{V}
, where V is regular, for example, when
\mathit{X}\subset {\mathbb{A}}^{\mathit{n}+1}
is a hypersurface defined by an equation of the form
{\mathit{T}}^{\mathit{q}}-\mathit{f}\left({\mathit{x}}_{1},\dots ,{\mathit{x}}_{\mathit{n}}\right)=0
and δ is the projection onto
\mathit{V}:=\mathrm{Spec}\left(\mathit{k}\left[{\mathit{x}}_{1},\dots ,{\mathit{x}}_{\mathit{n}}\right]\right)
. The multiplicity along points of X is bounded by the degree, say d, of the field extension
\mathit{K}\left(\mathit{V}\right)\subset \mathit{K}\left(\mathit{X}\right)
{\mathit{F}}_{\mathit{d}}\left(\mathit{X}\right)\subset \mathit{X}
the set of points of multiplicity d. Our guiding line is the search for invariants of singularities
\mathit{x}\in {\mathit{F}}_{\mathit{d}}\left(\mathit{X}\right)
with a good behavior property under blowups
{\mathit{X}}^{\prime }\to \mathit{X}
along regular centers included in
{\mathit{F}}_{\mathit{d}}\left(\mathit{X}\right)
, which we call invariants with the pointwise inequality property.
A finite radicial morphism
\mathit{\delta }:\mathit{X}\to \mathit{V}
as above will be expressed in terms of an
{\mathcal{O}}_{\mathit{V}}^{\mathit{q}}
\mathcal{M}\subseteq {\mathcal{O}}_{\mathit{V}}
. A blowup
{\mathit{X}}^{\prime }\to \mathit{X}
along a regular equimultiple center included in
{\mathit{F}}_{\mathit{d}}\left(\mathit{X}\right)
induces a blowup
{\mathit{V}}^{\prime }\to \mathit{V}
along a regular center and a finite morphism
{\mathit{\delta }}^{\prime }:{\mathit{X}}^{\prime }\to {\mathit{V}}^{\prime }
. A notion of transform of the
{\mathcal{O}}_{\mathit{V}}^{\mathit{q}}
\mathcal{M}\subset {\mathcal{O}}_{\mathit{V}}
{\mathcal{O}}_{{\mathit{V}}^{\prime }}^{\mathit{q}}
{\mathcal{M}}^{\prime }\subset {\mathcal{O}}_{{\mathit{V}}^{\prime }}
will be defined in such a way that
{\mathit{\delta }}^{\prime }:{\mathit{X}}^{\prime }\to {\mathit{V}}^{\prime }
is the radicial morphism defined by
{\mathcal{M}}^{\prime }
. Our search for invariants relies on techniques involving differential operators on regular varieties and also on logarithmic differential operators. Indeed, the different invariants we introduce and the stratification they define will be expressed in terms of ideals obtained by evaluating differential operators of V on
{\mathcal{O}}_{\mathit{V}}^{\mathit{q}}
-submodules
\mathcal{M}\subset {\mathcal{O}}_{\mathit{V}}
A Self-Affine Property of Evolutional Type Appearing in a Hamilton–Jacobi Flow Starting from the Takagi Function
Yasuhiro Fujita, Nao Hamamuki, Norikazu Yamaguchi
Michigan Math. J. 71 (1), 105-120, (March 2022) DOI: 10.1307/mmj/20195782
KEYWORDS: 35F21, 26A27, 47N10
In this paper, we study a Hamilton–Jacobi flow
{\left\{{\mathit{H}}_{\mathit{t}}\mathit{\tau }\right\}}_{\mathit{t}>0}
starting from the Takagi function τ. The Takagi function is well known as a pathological function that is everywhere continuous and nowhere differentiable on
\mathbb{R}
. As the first result of this paper, we derive an explicit representation of
\left\{{\mathit{H}}_{\mathit{t}}\mathit{\tau }\right\}
{\mathit{H}}_{\mathit{t}}\mathit{\tau }
is a piecewise quadratic function at any time and that the points of intersection between the parabolas are given in terms of binary expansion of real numbers. Applying the representation formula, we next give the main result, which asserts that
\left\{{\mathit{H}}_{\mathit{t}}\mathit{\tau }\right\}
has a self-affine property of evolutional type involving a time difference in the functional equality. Furthermore, we determine the optimal time until when the self-affine property is valid.
Multiplicative p-Adic Approximation
Dzmitry Badziahin, Yann Bugeaud
Let p be a prime number. We give several results towards a particular instance of a conjecture of Einsiedler and Kleinbock asserting that every p-adic number x satisfies
\underset{\mathit{a},\mathit{b}\in \mathbb{Z}\setminus \left\{0\right\}}{inf}|\mathit{a}\mathit{b}|·|\mathit{a}\mathit{x}-\mathit{b}{|}_{\mathit{p}}=0.
We highlight a close relationship between this conjecture and the (still open) p-adic Littlewood conjecture, according to which every real number ξ satisfies
\underset{\mathit{q}\in \mathbb{Z},\mathit{q}\ge 1}{inf}\mathit{q}·‖\mathit{q}\mathit{\xi }‖·|\mathit{q}{|}_{\mathit{p}}=0.
Furthermore, we discuss the analogues of these conjectures over fields of power series.
Lagrangian Cobordisms and Legendrian Invariants in Knot Floer Homology
John A. Baldwin, Tye Lidman, C.-M. Michael Wong
We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on
{\mathbb{R}}^{3}
. Our results give new, computable, and effective obstructions to the existence of such cobordisms.
Quantization and Isotropic Submanifolds
KEYWORDS: 53D50, 53D12, 32A25, 11F41
We introduce the notion of an isotropic quantum state associated with a Bohr–Sommerfeld manifold in the context of Berezin–Toeplitz quantization of general prequantized symplectic manifolds, and we study its semiclassical properties using the off-diagonal expansion of the Bergman kernel. We then show how these results extend to the case of noncompact orbifolds and give an application to relative Poincaré series in the theory of automorphic forms.
Retraction of: Linear Representations of Hyperelliptic Mapping Class Groups
Michigan Math. J. 71 (1), 221, (March 2022) DOI: 10.1307/mmj/20195783
KEYWORDS: 14H10, 14H37, 57M60, 14H40
The paper was withdrawn by the author after its first appearance online, upon learning of a gap in the proof of Theorem 3.13 and an issue with a published result on which Theorem 3.2 was based. A complete proof of Theorem 3.13, with slightly more restrictive hypotheses, can now be found in: Marco Boggi, “Notes on hyperelliptic mapping class groups”, arXiv:2110.13534. A revised version of this preprint will include all of the correct results in the retracted paper. |
Humidity/Temperature Sensor - 1125_0 at Phidgets
Mature: This product (or a similar replacement with a compatible form, fit and function) will be produced as long as the parts and components required to make it are available.
Other Humidity Sensors
Measures Ambient Temperature in the range of -30°C to +80° C with a typical error of ±0.75°C in the 0°C to 80°C range.
The temperature sensor component is rated at -40°C to +100°C, but the other components on the board, the connector and the cable are rated at -30°C to +80°C. In a fast prototyping environment the temperature sensor board can be pushed to the ratings of the sensor component, but you should use the lower temperature ratings if you plan to use the 1125 in a commercial application.
2x 3002 – 60cm Phidget Cable
This sensor is intended for use in non-condensing environments. Allowing the sensor to come in direct contact with water vapor or condensation could damage the circuitry.
Interface Boards and Hubs
This sensor can be read by any Phidget with an Analog Input or VINT Hub port. It will connect to either one using the included Phidget cable. VINT Hub ports can behave just like Analog Inputs, but have the added flexibility of being able to be used as digital inputs, digital outputs, or ports to communicate with VINT devices. For more information about VINT, see the VINT Primer.
1010_0 $80.00 8 10 bit
1018_2B $80.00 8 10 bit
1019_1B $110.00 8 10 bit
DAQ1000_0 $20.00 8 12 bit
HUB0001_0 $30.00 6 (Shared) * 16 bit
SBC3003_0 $120.00 6 (Shared) * 16 bit
This sensor comes with its own Phidget cable to connect it to an InterfaceKit or Hub, but if you need extras we have a full list down below. You can solder multiple cables together in order to make even longer Phidget cables, but you should be aware of the effects of having long wires in your system.
3.2 Phidget Cable
Any Phidget with a Analog Input or VINT port, here are some compatible products. We will be using the HUB0000 for this guide.
Connect the 1125 temperature output to the HUB0000 using one of the Phidget cables.
Connect the 1125 humidity output to the HUB0000 using the second Phidget cable.
Connect the HUB0000 to your computer with the USB cable.
In order to demonstrate the functionality of the 1125, we will connect it to the HUB0000, and then run an example using the Phidget Control Panel on a Windows machine.
The Phidget Control Panel is available for use on both macOS and Windows machines. If you would like to follow along, first take a look at the getting started guide for your operating system:
After plugging in the 1125 into the HUB0000, and the HUB0000 into your computer, open the Phidget Control Panel. You will see something like this:
Select the 1125 from the Sensor Type drop-down menu. The example will now convert the voltage into temperature (°C) and humidity (% RH) automatically. Converting the voltage to temperature (°C) and humidity (% RH) is not specific to this example, it is handled by the Phidget libraries, with functions you have access to when you begin developing!
This sensor produces an analog signal between 0-5V depending on the input temperature or humidity. This signal is measured by the Phidget it is plugged into, such as the HUB0000 or the 1018, and interpreted by the Phidget library. You will need to reference the User Guide and API for that Phidget for more information on how to use this type of device in your program. For more information about the signal produced by this device, check out the Voltage Ratio Input Primer.
You are now ready to start writing your own code for the device. The best way to do that is to start from our examples:
When used with a Phidget that measures the signal from this sensor, this Phidget is compatible with our VoltageRatioInput Examples. They outline the use of the VoltageRatioInput API, which is used to interpret the signal from this device.
Once you have your example, you will need to follow the instructions on the page for your programming language to get it running. To find these instructions, select your programming language from the Software Overview page.
The 1125 is a precision temperature to voltage converter that outputs a voltage that is directly proportional to temperature. The temperature sensor component is rated at -40°C to +100°C, but the other components on the board, the connector and the cable are rated at -30°C to +80°C. In a fast prototyping environment the 1125 can be pushed to the ratings of the sensor component, but you should use the lower temperature ratings if you plan to use the 1125 in a commercial application.
The 1125 measures the relative humidity of the environment around the sensor. Built in temperature compensation produces a linear output ranging from 10% to 95% relative humidity. Values outside of this range may be usable but will have increased error.
The Phidget libraries can automatically convert sensor voltage into temperature (°C) and humidity (% RH) by selecting the appropriate SensorType. See the Phidget22 API for more details. The formulas used to translate voltage ratio from the sensor into temperature and relative humidity are as follows:
{\displaystyle {\text{Temperature (°C)}}=({\text{VoltageRatio}}\times 222.2)-61.111}
{\displaystyle {\text{RH (}}\%{\text{)}}={\text{(VoltageRatio}}\times {\text{190.6) - 40.2}}}
The Phidget Cable is a 3-pin, 0.100 inch pitch locking connector. Pictured here is a plug with the connections labelled. The connectors are commonly available - refer to the Analog Input Primer for manufacturer part numbers.
Sensor Type Temperature/Humidity (Ambient)
Ambient Temperature Resolution 0.2 °C
Ambient Temperature Error Max ± 2 °C
Humidity Min 10 %RH
Humidity Max 95 %RH
Humidity Error Typical ± 3 %RH
May 2008 0 N/A Product Release
This device doesn't have an API of its own. It is controlled by opening two VoltageRatioInput channels on the Phidget that it's connected to. For a list of compatible Phidgets with VoltageRatio Inputs, see the Connection & Compatibility tab.
You can find details for the VoltageRatioInput API on the API tab for the Phidget that this sensor connects to.
Have a look at our temperature/humidity sensors:
1125_0 $50.00 Temperature/Humidity (Ambient) VoltageRatio Input -30 °C 80 °C 10 %RH 95 %RH
HUM1001_0 $20.00 Humidity / Temperature VINT -40 °C 85 °C 0 %RH 100 %RH |
Planetary gear set of carrier and beveled planet and sun wheels with adjustable gear ratio, assembly orientation, and friction losses - MATLAB - MathWorks í•œêµ
{r}_{C}{\mathrm{Ï}}_{C}={r}_{S}{\mathrm{Ï}}_{S}±{r}_{P}{\mathrm{Ï}}_{P}
{r}_{C}={r}_{S}±{r}_{P}
ωP is the angular velocity of the planet gear.
{g}_{PS}=\frac{{r}_{P}}{{r}_{S}}=\frac{{N}_{P}}{{N}_{S}},
gPS is the planet-sun gear ratio. As
{r}_{P}>{r}_{S}
{g}_{PS}>1
{\mathrm{Ï}}_{S}={g}_{PS}{\mathrm{Ï}}_{P}â{\mathrm{Ï}}_{C}
for a left-oriented bevel assembly
{\mathrm{Ï}}_{S}={g}_{PS}{\mathrm{Ï}}_{P}+{\mathrm{Ï}}_{C}
for a right-oriented bevel assembly
{\mathrm{Ï}}_{P}={\mathrm{Ï}}_{loss}â{g}_{PS}{\mathrm{Ï}}_{S},
Ï„loss is the torque loss.
Ï„s is the torque for the sun gear.
Ï„p is the torque for the planet gear.
In the ideal case where there is no torque loss, Ï„loss = 0. The resulting torque transfer equation is
{\mathrm{Ï}}_{P}={g}_{PS}{\mathrm{Ï}}_{S}
Torque transfer efficiency, ηPS, for the planet gear to the sun gear pair meshing. This value must be greater than 0 and less than or equal to 1.
Vector of output-to-input power ratios that describe the power flow from the outer planet gear to the inner planet gear, ηPS. The block uses the values to construct a 1-D temperature-efficiency lookup table.
Viscous friction coefficient, μS, for the sun-carrier gear motion. |
Subsets and Splits