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PLEASE CAN YOU TELL ME HOW THE COORDINATES (3,3) DON'T SATISFY THE EQUATION x+2 0 ALSO TELL ME THAT IF - Maths - Linear Inequalities - 10392203 | Meritnation.com
PLEASE CAN YOU TELL ME HOW THE COORDINATES (3,3) DON'T SATISFY THE EQUATION x+2>0.ALSO TELL ME THAT IF AN EQUATION LIKE x-2y<0 COMES THEN HOW WILL WE DECIDE THE DIRECTION OF LINE AS ITS COORDINATES ARE WHEN X=Y=0.
PLEASE ANSWER IT SOON.
For the first part of your question please provide with complete question.
For second part of your question the answer is as follows:
Choose a point \left(0,0\right) if possible, not lying on the line ax+by<0 or ax+by>0:\phantom{\rule{0ex}{0ex}}Substitute its coordinates in the inequation. If the inequation is satisfied, then shade the portion of the\phantom{\rule{0ex}{0ex}}plane which contains the chosen point; otherwise shade the portion which does not contain the chosen point. |
Rewrite the expression using rational exponents: \sqrt[4]{x^{3}}
\sqrt[4]{{x}^{3}}
Given radical expression:
\sqrt[4]{{x}^{3}}
To write Radicals as Exponents:
\sqrt[n]{{x}^{m}}={\sqrt[n]{x}}^{m}={x}^{\frac{m}{n}}
\sqrt[4]{{x}^{3}}={x}^{\frac{3}{4}}
Hence, the radical expression
\sqrt[4]{{x}^{3}}
is written in rational exponents as
{x}^{\frac{3}{4}}
Simplify and solve the given algebraic expression.
\frac{\left(9{a}^{3}{b}^{4}{\right)}^{\frac{1}{2}}}{15{a}^{2}b}
\sqrt[5]{{x}^{2}{y}^{2}}\text{ }\cdot \text{ }\sqrt[4]{x}
To calculate: The simplified value of the radical expression
5÷\sqrt{x}
The tax saving due to federal deduction on interest paid for the student loan.
Given: The modified adjusted gross income is $70000.
The interest paid on student loan is $3000.
The parabola shown is the graph of
f\left(x\right)=A{x}^{2}\text{ }+\text{ }2x\text{ }+\text{ }C.
The x-intercepts of the graph are at
-4\text{ }and\text{ }-3.
Find the exact value of the y-intercept and the coordinates of the vertex of the graph (expressed in terms of rational numbers and radicals). |
Calculate the Laplace transform L\{f\} of the function f(t)=e^{4t+2}-8t^6+8 \sin
Calculate the Laplace transform L\{f\} of the function f(t)=e^{4t+2}-8t^6+8 \sin (2t)+9 using the basic formulas
L\left\{f\right\}
f\left(t\right)={e}^{4t+2}-8{t}^{6}+8\mathrm{sin}\left(2t\right)+9
using the basic formulas and the linearity of the Laplace transform
f\left(t\right)={e}^{4t+2}-8{t}^{6}+8\mathrm{sin}\left(2t\right)+9
F\left(t\right)={e}^{2}{e}^{4t}-8{t}^{6}+8\mathrm{sin}\left(2t\right)+9
L\left\{f\left(t\right)\right\}=L\left\{{e}^{2}{e}^{4t}-8{t}^{6}+8\mathrm{sin}\left(2t\right)+9\right\}
Apply linearity
L\left\{f\left(t\right)\right\}={e}^{2}L\left\{{e}^{4t}\right\}-8L\left\{{t}^{6}\right\}-8L\left\{\mathrm{sin}\left(2t\right)\right\}+9L\left\{1\right\}
using the basic formulas
L\left\{{e}^{at}\right\}=\frac{1}{s-a},L\left\{1\right\}=\frac{1}{s}
L\left\{{t}^{n}\right\}=\frac{n!}{{s}^{n+1}}
L\left\{\mathrm{sin}\left(at\right)\right\}=\frac{a}{{s}^{2}+{a}^{2}}
L\left\{f\left(t\right)\right\}={e}^{2}\frac{1}{s-4}-8\frac{6!}{{s}^{7}}-8\frac{2}{{s}^{2}+{2}^{2}}+9\frac{1}{s}
L\left\{f\right\}=\frac{{e}^{2}}{s-4}-\frac{5760}{{s}^{7}}-\frac{16}{{s}^{2}+{2}^{2}}+\frac{9}{s}
How to find the Laplace transform of
\frac{1-\mathrm{cos}\left(t\right)}{{t}^{2}}
F(S)=?
\frac{dy}{dt}={t}^{3}+2{t}^{2}-8t
Also, part 8B. asks: Show that the constant function
y\left(t\right)=0
For the following exercises, use logarithmic differentiation to find
\frac{dy}{dx}
y={\left({x}^{2}-1\right)}^{\mathrm{ln}x}
dy/dt-y=z,\text{ }y\left(0\right)=0
For each of the following differential equations, determine the general or particular solution:
2x{y}^{\prime }+y={y}^{2}\mathrm{log}x
Find Laplace of given fn:
f\left(t\right)={\int }_{0}^{t}\mathrm{sin}\left(t-\tau \right)\mathrm{cos}\tau d\tau |
Characterizing Fundamental Resonance Peaks on Flat‐Lying Sediments Using Multiple Spectral Ratio Methods: An Example from the Atlantic Coastal Plain, Eastern United States | Bulletin of the Seismological Society of America | GeoScienceWorld
Characterizing Fundamental Resonance Peaks on Flat‐Lying Sediments Using Multiple Spectral Ratio Methods: An Example from the Atlantic Coastal Plain, Eastern United States
Lisa S. Schleicher;
Lisa S. Schleicher *
U.S. Geological Survey, National Strong Motion Project, Menlo Park, California, U.S.A.
Corresponding author: [email protected]
U.S. Geological Survey, Earthquake Hazards Program, Reston, Virginia, U.S.A.
Lisa S. Schleicher, Thomas L. Pratt; Characterizing Fundamental Resonance Peaks on Flat‐Lying Sediments Using Multiple Spectral Ratio Methods: An Example from the Atlantic Coastal Plain, Eastern United States. Bulletin of the Seismological Society of America 2021;; 111 (4): 1824–1848. doi: https://doi.org/10.1785/0120210017
Damaging ground motions from the 2011
Mw
5.8 Virginia earthquake were likely increased due to site amplification from the unconsolidated sediments of the Atlantic Coastal Plain (ACP), highlighting the need to understand site response on these widespread strata along the coastal regions of the eastern United States. The horizontal‐to‐vertical spectral ratio (HVSR) method, using either earthquake signals or ambient noise as input, offers an appealing method for measuring site response on laterally extensive sediments, because it requires a single seismometer rather than requiring a nearby bedrock site to compute a horizontal sediment‐to‐bedrock spectral ratio (SBSR). Although previous studies show mixed results when comparing the two methods, the majority of these studies investigated site responses in confined sedimentary basins that can generate substantial 3D effects or have relatively small reflection coefficients at their base. In contrast, the flat‐lying ACP strata and the underlying bedrock reflector should cause 1D resonance effects to dominate site response, with amplification of the fundamental resonance peaks controlled by the strong impedance contrast between the base of the sediments and the underlying bedrock. We compare site‐response estimates on the ACP strata derived using the HVSR and SBSR methods from teleseismic signals recorded by regional arrays and observe a close match in the frequencies of the fundamental resonance peak (
f0
) determined by both methods. We find that correcting the HVSR amplitude using source term information from a bedrock site and multiplying the peak by a factor of 1.2 results in amplitude peaks that, on average, match SBSR results within a factor of 2. We therefore conclude that the HVSR method may successfully estimate regional linear weak‐motion site‐response amplifications from the ACP, or similar geologic environments, when appropriate region‐specific corrections to the amplitude ratios are used.
HVSR analysis
DCShake
ENAM Array |
The perimeter of a rectangle is 48 cm It would remain unchanged if - Maths - Introduction to Graphs - 11553307 | Meritnation.com
The perimeter of a rectangle is 48 cm. It would remain unchanged if
Your question appears to be incomplete. However if the question is
" The perimeter of a rectangle is 48 cm. It would remain unchanged if the length of rectangle is twice the breadth. find the length and breadth of rectangle."
Let the breadth of rec\mathrm{tan}gle be x\phantom{\rule{0ex}{0ex}}then the length of rec\mathrm{tan}gle be 2x\phantom{\rule{0ex}{0ex}}Since the perimeter of rec\mathrm{tan}gle remained unchanged \phantom{\rule{0ex}{0ex}}\therefore perimeter of rec\mathrm{tan}gle=48\phantom{\rule{0ex}{0ex}}⇒2\left(L+B\right)=48\phantom{\rule{0ex}{0ex}}⇒2\left(2x+x\right)=48\phantom{\rule{0ex}{0ex}}⇒2×3x=48\phantom{\rule{0ex}{0ex}}⇒x=\frac{48}{6}\phantom{\rule{0ex}{0ex}}⇒x=8\phantom{\rule{0ex}{0ex}}thus, the breadth of rec\mathrm{tan}gle=8\phantom{\rule{0ex}{0ex}}and length of rec\mathrm{tan}gle=2x=2×8=16\phantom{\rule{0ex}{0ex}}
Shomipandey answered this
I think the question is incomplete. "It would remain unchanged if..."
Incomplete Question !! |
The three forms of linear equations you have studied are slope-intercept form, point-slope
The three forms of linear equations you have studied are slope-intercept form, point-slope form, and standard form. Explain when each form is most use
If we have the slope m and the y-intercept b, then we should use the slope-intercept form
y=mx+6
If we have the slope m and the point
\left({x}_{1},{y}_{1}\right)
, then we should use the point-slope form
y-{y}_{1}=m\left(x-{x}_{1}\right)
If we have the x-intercept and the y-intercept , then we should use the standard form
Ax+By=C
Where x-intercept is - and y-intercept is
\frac{C}{B}
f\left(x\right)=\sqrt{4-x}
a=0
\sqrt{3.9}
\sqrt{3.99}
This question from linear algebra
Suppose you [ have a consistent system of linear equations, with coefficients in R, which are homogeneous - that is, all the
{b}_{i}
are 0. Explain why the set of solutions to this system forms a vector space over
\mathbb{R}
. Then, explain why if the system was not homogeneous (i.e. if at least one of the
{b}_{i}
is nonzero) the set of solutions would definitely NOT form a vector space over
\mathbb{R}
Fill in the blanks. The process used to write a system of linear equations in row-echelon form is called ________ elimination.
My HW asks me to solve the following Linear Recurrence:
f\left(0\right)=3
f\left(1\right)=1
f\left(n\right)=4f\left(n-1\right)+21f\left(n-2\right)
Unfortunately my professor ran through the concept of Linear Recurrence rather quickly so I'm stuck. But this is what I've done so far:
1). Assuming
{x}_{n}=f\left(n\right)
, I rewrote the equation as
{x}^{n}=4{x}^{n-1}+21{x}^{n-2}
2). I then divided each part of the equation using the common factor
{x}^{n-2}
{x}^{2}=4x+21
, a quadratic.
3). I then used the quadratic formula to get two values, 6 and 2.
From here I don't know how to proceed. I know I'm trying to write a closed form of the above equation, right? How do the values I've found figure into that? I'm also not sure what the salience of 'the boundary conditions' are (are those
f\left(0\right)=3
f\left(1\right)=1
Determine if (1,3) is a solution to the given system of linear equations.
5x+y=8
x+2y=5
Consider a simple linear equation of the form:
n=\frac{2x+2}{3}
Let n and x represent something that comes in whole positive quantities (for example physical objects).
1.Define the equation only for n and x that are a part of natural numbers (whole numbers >0)
2.Solve the equation satisfying the above restriction (without for instace graphing it and looking for n and x that work) |
Error, invalid input: f expects its 1st argument, a, to be of type integer, but received 2.5 - Maple Help
Home : Support : Online Help : Error, invalid input: f expects its 1st argument, a, to be of type integer, but received 2.5
\mathrm{sin}\left(\left[1,2, 3\right]\right);
\mathrm{whattype}\left(\left[1,2,3\right]\right);
\textcolor[rgb]{0,0,1}{\mathrm{list}}
\left[1,2,3\right]
\mathrm{Describe}\mathit{}\left(\mathrm{sin}\right)
x
\mathrm{sin}
\mathrm{sin}\left(1\right);\mathrm{sin}\left(2\right);\mathrm{sin}\left(3\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)
\mathrm{sin}~\left(\left[1,2,3\right]\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)\right] |
Pulling Out the Stops: Designing a Muscle for a Tug-of-War Competition - OpenSim Documentation - Global Site
The force-producing properties of muscle are complex, highly nonlinear, and can have substantial effects on movement (see McMahon, 1984 for a review). For simplicity, lumped-parameter, dimensionless muscle models that are capable of representing a variety of muscles with different architectures are commonly used in the dynamic simulation of movement (Zajac, 1989). In this exercise, you will explore the differential equations that describe muscle activation and muscle–tendon contraction dynamics when using a Hill-type muscle model. You will use OpenSim to implement a simple muscle–tendon model and conduct simulations to investigate how various model parameters affect the dynamic response of the actuator. The lab will conclude with a Virtual Muscle Tug-of-War in which you will design an optimal muscle and compete against other user-generated muscles. May the best muscle win!
Become familiar with the differential equations describing muscle activation and muscle–tendon contraction dynamics.
Learn how to model and simulate muscle–tendon dynamics using OpenSim.
Become comfortable with modifying existing code that models muscle activation and muscle–tendon mechanics.
Explore the effect of various model parameters and simulation conditions on the dynamic response of muscle.
Design your own optimal muscle–tendon actuator to compete in a virtual muscle tug-of-war.
The original lab was designed by Jeff Reinbolt, B.J. Fregly, Kate Saul, Darryl Thelen, Silvia Blemker, Clay Anderson, and Scott Delp. The lab was refined by Hoa Hoang and Daniel Jacobs.
The model in this exercise consists of a cube with a single translational degree of freedom along the Z-axis. A Tug_of_War model has been included in the OpenSim distribution (Models/Tug_of_War/) that uses two Thelen 2003 Muscles to move the cube. In this exercise, we will use two Millard 2012 Muscles instead of the Thelen muscles. The muscles are arranged to pull on opposite sides of the cube. The cube has a mass of 20 kg and sides of length 0.1 m, and the distance between the fixed ground supports is 0.7 m. Thus, each muscle–tendon actuator is 0.3 m long when the block is centered.
Millard 2012 Muscle Model
Musculotendon actuators consist of an active contractile element, a passive elastic element, and an elastic tendon. The maximum active force a muscle can develop varies nonlinearly with its length, represented by the active-force–length curve \mathbf{f}^{\mathrm{L}} \big( \tilde{\ell}^{\,\mathrm{M}} \big)
//<![CDATA[ \begin{array}{l}\mathbf{f}^{\mathrm{L}} \big( \tilde{\ell}^{\,\mathrm{M}} \big)\end{array} //]]>
, peaking at a force of f_o^{\,\mathrm{M}}
//<![CDATA[ \begin{array}{l}f_o^{\,\mathrm{M}}\end{array} //]]>
at a length of \ell_o^{\,\mathrm{M}}
//<![CDATA[ \begin{array}{l}\ell_o^{\,\mathrm{M}}\end{array} //]]>
(the tilde is used to denote forces, velocities, muscle lengths, and tendon lengths that are normalized by f_o^{\,\mathrm{M}}
//<![CDATA[ \begin{array}{l}f_o^{\,\mathrm{M}}\end{array} //]]>
, v_{\mathrm{max}}^{\,\mathrm{M}}
//<![CDATA[ \begin{array}{l}v_{\mathrm{max}}^{\,\mathrm{M}}\end{array} //]]>
, \ell_o^{\,\mathrm{M}}
//<![CDATA[ \begin{array}{l}\ell_o^{\,\mathrm{M}}\end{array} //]]>
, and \ell_\mathrm{s}^{\mathrm{T}}
//<![CDATA[ \begin{array}{l}\ell_\mathrm{s}^{\mathrm{T}}\end{array} //]]>
, respectively). During non-isometric contractions, the force developed by muscle varies nonlinearly with its rate of lengthening, which is represented by the force–velocity curve \mathbf{f}^{\mathrm{V}} \big( \tilde{v}^{\,\mathrm{M}} \big)
//<![CDATA[ \begin{array}{l}\mathbf{f}^{\mathrm{V}} \big( \tilde{v}^{\,\mathrm{M}} \big)\end{array} //]]>
. Force is also developed when muscle is stretched beyond a threshold length, regardless of whether the muscle is activated, which is represented by the passive-force–length curve \mathbf{f}^{\mathrm{PE}} \big( \tilde{l}^{\,\mathrm{M}} \big)
//<![CDATA[ \begin{array}{l}\mathbf{f}^{\mathrm{PE}} \big( \tilde{l}^{\,\mathrm{M}} \big)\end{array} //]]>
. Muscle force (f^{\,\mathrm{M}}
//<![CDATA[ \begin{array}{l}f^{\,\mathrm{M}}\end{array} //]]>
) is computed using these curves as follows:
f^{\,\mathrm{M}} = f_o^{\,\mathrm{M}} \left( a \mathbf{f}^{\mathrm{L}} \big( \tilde{\ell}^{\,\mathrm{M}} \big) \mathbf{f}^{\mathrm{V}} \big( \tilde{v}^{\,\mathrm{M}} \big) + \mathbf{f}^{\mathrm{PE}} \big( \tilde{\ell}^{\,\mathrm{M}} \big) \right)
where a
//<![CDATA[ \begin{array}{l}a\end{array} //]]>
is the muscle activation, which ranges from a_{\mathrm{min}}
//<![CDATA[ \begin{array}{l}a_{\mathrm{min}}\end{array} //]]>
Muscle attaches to bone through tendon. Since a long tendon may stretch appreciably beyond its slack length (\ell_{\mathrm{s}}^{\mathrm{T}}
//<![CDATA[ \begin{array}{l}\ell_{\mathrm{s}}^{\mathrm{T}}\end{array} //]]>
) when under tension, tendon is modeled as a nonlinear elastic element developing force according to the tendon-force–length curve \mathbf{f}^{\mathrm{T}} \big( \tilde{\ell}^{\,\mathrm{T}} \big)
//<![CDATA[ \begin{array}{l}\mathbf{f}^{\mathrm{T}} \big( \tilde{\ell}^{\,\mathrm{T}} \big)\end{array} //]]>
. Muscle fibers attach to tendon at a pennation angle (\alpha
//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>
), scaling the force they transmit to the tendon. If the tendon is assumed to be elastic and the mass of the muscle is assumed to be negligible, then the muscle–tendon force equilibrium equation (i.e., f^{\,\mathrm{M}} \cos \alpha - f^{\,T} = 0
//<![CDATA[ \begin{array}{l}f^{\,\mathrm{M}} \cos \alpha - f^{\,T} = 0\end{array} //]]>
) takes the following form:
f_o^{\,\mathrm{M}} \left( a \mathbf{f}^{\mathrm{L}} \big( \tilde{\ell}^{\,\mathrm{M}} \big) \mathbf{f}^{\mathrm{V}} \big( \tilde{v}^{\,\mathrm{M}} \big) + \mathbf{f}^{\mathrm{PE}} \big( \tilde{\ell}^{\,\mathrm{M}} \big) \right) \cos \alpha - f_o^{\,\mathrm{M}} \mathbf{f}^{\mathrm{T}} \big( \tilde{\ell}^{\,\mathrm{T}} \big) = 0\qquad(1)
We have developed default force curves for the musculotendon model that have been fit to experimental data (see Millard et al., 2013 for details). These curves can be adjusted to model muscle and tendon whose characteristics deviate from these default patterns.
In a forward-dynamic simulation, the force generated by a musculotendon actuator is calculated from the length (\ell^{\mathrm{M}}
//<![CDATA[ \begin{array}{l}\ell^{\mathrm{M}}\end{array} //]]>
), velocity (v^{\mathrm{M}}
//<![CDATA[ \begin{array}{l}v^{\mathrm{M}}\end{array} //]]>
), and activation (a
//<![CDATA[ \begin{array}{l}a\end{array} //]]>
) of the muscle. The length \ell^{\mathrm{M}}
//<![CDATA[ \begin{array}{l}\ell^{\mathrm{M}}\end{array} //]]>
is obtained by integrating the velocity v^{\mathrm{M}}
//<![CDATA[ \begin{array}{l}v^{\mathrm{M}}\end{array} //]]>
, which is obtained by solving Eq. (1) for the normalized muscle velocity (\tilde{v}^{\,\mathrm{M}}
//<![CDATA[ \begin{array}{l}\tilde{v}^{\,\mathrm{M}}\end{array} //]]>
), thereby providing an ordinary differential equation (see Zajac, 1989):
\tilde{v}^{\,\mathrm{M}} = \mathbf{f}_{\mathrm{inv}}^{\mathrm{V}} \left( \frac{ \mathbf{f}^{\mathrm{T}} \big( \tilde{\ell}^{\,\mathrm{T}} \big) / \cos \alpha \ -\ \mathbf{f}^{\mathrm{PE}} \big( \tilde{\ell}^{\,\mathrm{M}} \big) }{ a \mathbf{f}^{\mathrm{L}} \big( \tilde{\ell}^{\,\mathrm{M}} \big) } \right)\qquad(2)
where \mathbf{f}_{\mathrm{inv}}^{\mathrm{V}}
//<![CDATA[ \begin{array}{l}\mathbf{f}_{\mathrm{inv}}^{\mathrm{V}}\end{array} //]]>
is the inverse of the force–velocity curve. Although Eq. (1) is devoid of numerical singularities, Eq. (2) has four: as \alpha \rightarrow 90^{\circ}
//<![CDATA[ \begin{array}{l}\alpha \rightarrow 90^{\circ}\end{array} //]]>
, as a \rightarrow 0
//<![CDATA[ \begin{array}{l}a \rightarrow 0\end{array} //]]>
, as \mathbf{f}^{\mathrm{L}} \big( \tilde{\ell}^{\,\mathrm{M}} \big) \rightarrow 0
//<![CDATA[ \begin{array}{l}\mathbf{f}^{\mathrm{L}} \big( \tilde{\ell}^{\,\mathrm{M}} \big) \rightarrow 0\end{array} //]]>
, and as \partial \mathbf{f}^{\mathrm{V}} \big( \tilde{v}^{\,\mathrm{M}} \big) / \partial \tilde{v}^{\,\mathrm{M}} \rightarrow 0
//<![CDATA[ \begin{array}{l}\partial \mathbf{f}^{\mathrm{V}} \big( \tilde{v}^{\,\mathrm{M}} \big) / \partial \tilde{v}^{\,\mathrm{M}} \rightarrow 0\end{array} //]]>
. Since these conditions are often encountered during a simulation, the quantities causing singularities in Eq. (2) are altered so that the singularities are approached but never reached: \alpha < 90^{\circ}
//<![CDATA[ \begin{array}{l}\alpha < 90^{\circ}\end{array} //]]>
, a > 0
//<![CDATA[ \begin{array}{l}a > 0\end{array} //]]>
, \mathbf{f}^{\mathrm{L}} \big( \tilde{\ell}^{\,\mathrm{M}} \big) > 0
//<![CDATA[ \begin{array}{l}\mathbf{f}^{\mathrm{L}} \big( \tilde{\ell}^{\,\mathrm{M}} \big) > 0\end{array} //]]>
, and \partial \mathbf{f}^{\mathrm{V}} \big( \tilde{v}^{\,\mathrm{M}} \big) / \partial \tilde{v}^{\,\mathrm{M}} > 0
//<![CDATA[ \begin{array}{l}\partial \mathbf{f}^{\mathrm{V}} \big( \tilde{v}^{\,\mathrm{M}} \big) / \partial \tilde{v}^{\,\mathrm{M}} > 0\end{array} //]]>
Without modifying the formulation of the equilibrium model, the muscle is able to reach unrealistically short lengths (Thelen, 2003) and cannot be simulated when fully deactivated. We use a unilateral constraint on muscle length to prevent the muscle from becoming unrealistically short:
\begin{align} \tilde{v}^{\,\mathrm{M}} = \begin{cases} 0 &: \tilde{\ell}^{\,\mathrm{M}} \leq \tilde{\ell}_{\mathrm{min}}^{\,\mathrm{M}} \;\;\text{and}\;\; \tilde{v}^{\,\mathrm{M*}} < 0\\ \tilde{v}^{\,\mathrm{M*}} &: \text{otherwise} \end{cases} \end{align}
where \tilde{v}^{\,\mathrm{M*}}
//<![CDATA[ \begin{array}{l}\tilde{v}^{\,\mathrm{M*}}\end{array} //]]>
is a candidate value for \tilde{v}^{\,\mathrm{M}}
//<![CDATA[ \begin{array}{l}\tilde{v}^{\,\mathrm{M}}\end{array} //]]>
computed using Eq. (2).
A muscle can neither generate force nor relax instantaneously. The development of force is a complex sequence of events that begins with the firing of motor units and culminates in the formation of actin–myosin cross-bridges within the myofibrils of the muscle. When the motor units of a muscle depolarize, action potentials are elicited in the fibers of the muscle and cause calcium ions to be released from the sarcoplasmic reticulum. The increase in calcium ion concentrations then initiates the cross-bridge formation between the actin and myosin filaments. In isolated muscle twitch experiments, the delay between a motor unit action potential and the development of peak force has been observed to vary from as little as 5 milliseconds for fast ocular muscles to as much as 40 or 50 milliseconds for muscles comprised of higher percentages of slow-twitch fibers. The relaxation of muscle depends on the re-uptake of calcium ions into the sarcoplasmic reticulum. This re-uptake is a slower process than the calcium ion release, so the time required for muscle force to fall can be considerably longer than the time for it to develop.
The activation dynamics of muscle can be modeled with a first-order differential equation. This equation relates the rate of change of muscle activation (i.e., the concentration of calcium ions within the muscle) to the muscle excitation (i.e., the firing of motor units):
\frac{da}{dt} = \frac{u-a}{\tau(a,u)}\qquad(1)
where u
//<![CDATA[ \begin{array}{l}u\end{array} //]]>
//<![CDATA[ \begin{array}{l}a\end{array} //]]>
are the excitation and activation signals, respectively. In the model, activation is allowed to vary continuously between 0 (no contraction) and 1 (full contraction). In the body, the activation of a muscle is a function of the number of motor units recruited and the firing frequency of these motor units. Some models of excitation–contraction coupling distinguish these two control mechanisms (Hatze, 1976), but it is often not computationally feasible to use such models when conducting complex dynamic simulations. In a simulation, the muscle excitation signal is assumed to represent the net effect of both motor neuron recruitment and firing frequency. Like muscle activation, the excitation signal is also allowed to vary continuously between 0 (no excitation) and 1 (full excitation). The activation and deactivation time constants can be assumed to be 10 and 40 ms, respectively (Zajac, 1989; Winters, 1990).
The activation model presented by Thelen (2003) closely follows the activation dynamic model found in Winters (1995 – Eq. 2, line 2 and Eq. 3), where the time derivative of activation (da/dt
//<![CDATA[ \begin{array}{l}da/dt\end{array} //]]>
) is equal to the difference between excitation (u
//<![CDATA[ \begin{array}{l}u\end{array} //]]>
) and activation (a
//<![CDATA[ \begin{array}{l}a\end{array} //]]>
) scaled by a variable time constant (\tau(a,u)
//<![CDATA[ \begin{array}{l}\tau(a,u)\end{array} //]]>
). The primary difference between the activation model presented by Thelen and the one presented by Winters lies in their expressions for \tau(a,u)
//<![CDATA[ \begin{array}{l}\tau(a,u)\end{array} //]]>
, which differ only in the values of the two coefficients shown in parentheses in Eqs. 2 and 3:
\tau(a,u) = \begin{cases} t_{\mathrm{act}} \left( 0.5+1.5a \right) &: u > a\qquad(2)\\ t_{\mathrm{deact}}\ / \left( 0.5+1.5a \right) &: u \leq a\qquad(3)\end{cases}
Note that the model of activation dynamics presented in Eq. 1 does not respect a lower bound for activation. Equilibrium muscle models (commonly used to model muscle in lumped-parameter musculoskeletal simulations) have a singularity in their state equations when activation is zero, making the above activation dynamic model unsuitable for simulations using equilibrium muscle models. Equation 1 can be made to respect a lower bound on activation by first constraining both activation and excitation to remain between the minimum activation level (0.01 by default) and the maximum activation level (1.0 by default).
The competition is a single-elimination tournament between pairs of muscles. Each match is a one-second forward dynamic simulation where each muscle starts with minimal activation (a = 0.01). The match is won by the muscle that has moved the block to its side of the arena at the end of the simulation. Your challenge is to specify the muscle–tendon parameters and excitation time history necessary to overcome the other challengers.
Anderson, F.C. and Pandy, M.G. (1999). A dynamic optimization solution for vertical jumping in three dimensions. Computer Methods in Biomechanics and Biomedical Engineering, 2(3):201–231.
Hatze, H. (1976). The complete optimization of a human motion. Mathematical Biosciences, 28(1–2):99–135.
McMahon, T.A. (1984). Muscles, Reflexes, and Locomotion. Princeton University Press, Princeton, New Jersey.
Millard, M., Uchida, T., Seth, A., Delp, S.L. (2013). Flexing computational muscle: modeling and simulation of musculotendon dynamics. ASME Journal of Biomechanical Engineering, 135(2):021005.
Schutte, L.M. (1993). Using Musculoskeletal Models to Explore Strategies for Improving Performance in Electrical Stimulation-Induced Leg Cycle Ergometry. PhD Dissertation, Mechanical Engineering Department, Stanford University.
Thelen, D.G. (2003). Adjustment of muscle mechanics model parameters to simulate dynamic contractions in older adults. ASME Journal of Biomechanical Engineering, 125(1):70–77.
Winters, J.M. (1990). Hill-based muscle models: a systems engineering perspective, in Multiple Muscle Systems: Biomechanics and Movement Organization, edited by Winters, J.M. and Woo, S.L., Springer-Verlag, New York.
Zajac, F.E. (1989). Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. Critical Reviews in Biomedical Enginering, 17(4):359–411.
Next: Designing a Muscle for a Tug-of-War Competition
Home: Examples and Tutorials |
Wilks coefficient - Wikipedia
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The Wilks coefficient or Wilks formula is a mathematical coefficient that can be used to measure the relative strengths of powerlifters despite the different weight classes of the lifters. Robert Wilks, CEO of Powerlifting Australia, is the author of the formula.
The following equation is used to calculate the Wilks coefficient:
{\displaystyle {\text{Coeff}}={\frac {500}{a+bx+cx^{2}+dx^{3}+ex^{4}+fx^{5}}}}
where x is the body weight of the lifter in kilograms.
The total weight lifted (in kg) is multiplied by the coefficient to find the standard amount lifted, normalised across all body weights.
b 16.2606339 −27.23842536447
d -0.00113732 −0.00930733913
e 7.01863 × 10−6 4.731582 × 10−5
f −1.291 × 10−8 −9.054 × 10−8
The formula was updated in March 2020 to allow for a rebalancing of coefficients, with men and women's performances better aligned and the extreme bodyweight classes brought into better balance with the middle bodyweight classes.[1]
{\displaystyle {\text{Coeff}}={\frac {600}{a+bx+cx^{2}+dx^{3}+ex^{4}+fx^{5}}}}
a 47.46178854 -125.4255398
b 8.472061379 13.71219419
c 0.07369410346 -0.03307250631
d -0.001395833811 -0.001050400051
e 7.07665973070743 × 10−6 9.38773881462799 × 10−6
f -1.20804336482315 x 10−8 -2.3334613884954 × 10−8
One journal article has been written on the topic of Wilks formula validation.[2] Based on the men's and women's world record holders and the top two performers for each event in the IPF's 1996 and 1997 World Championships (a total of 30 men and 27 women for each lift), it concluded:
There is no bias for men's or women's bench press and total.
There is a favorable bias toward intermediate weight class lifters in the women's squat with no bias for men's squat.
There is a linear unfavorable bias toward heavier men and women in the deadlift.
The main function of the Wilks formula is involved in powerlifting contests. It is used to identify the best lifters across the different body-weight categories and can also be used to compare male and female lifters as there are formulas for both sexes. First, second and third places on the winner's podium within their own age, bodyweight and gender classes are awarded to the competitors who lift the most weight respectively. Where two lifters in a class achieve the same combined total lifted weight, the lighter lifter is determined the winner.
The Wilks formula comes into play when comparing and determining overall champions across the different categories. The formula can also be used in team and handicap competitions where the team includes lifters of significantly varying bodyweights. The Wilks formula, like its predecessors (the O'Carroll[3] and Schwartz [4] formulas), was set up to address the imbalances whereby lighter lifters tend to have a greater Power-to-weight ratio, with lighter lifters tending to lift more weight in relation to their own body-weight. This occurs for a number of reasons relating to simple physics, the nature of the makeup and limitations of the human skeletal and muscular system as well as the shorter leverages of smaller people.[5] Note the totals section and that lighter lifters below 100 kg body-weight achieve totals in excess of ten times bodyweight whereas heavier lifters do not. The Wilks system is primarily a handicapping process that provides an adjusted statistical method to compare all lifters of varying classes and groups on an equal standing and makes allowances for the disparities.
According to this setup, a male athlete weighing 320 pounds and lifting a total of 1400 pounds would have a normalised lift weight of 353.0, and a lifter weighing 200 pounds and lifting a total of 1000 pounds (the sum of their highest successful attempts at the squat, bench, and deadlift) would have a normalised lift weight of 288.4. Thus the 320-pound lifter would win this competition. Notably, the lighter lifter is actually stronger for his body-weight, with a total of 5 times his own weight, while the heavier lifter could only manage 4.375 times his own bodyweight. In this way, the Wilks Coefficient places a greater emphasis on absolute strength, rather than ranking lifters solely based on the relative strength of the lifter compared to body-weight. This creates an even playing field between light and heavyweight lifters—the lighter lifters tend to have a higher relative strength level in comparison to the heavyweight lifters, who tend to have a greater amount of absolute strength.
While Wilks coefficient was used in the IPF up until the end of 2018,[6][7] other federations use other coefficients or they even made one of their own like NASA. The switch by IPF comes at a time where Olympic Weightlifting Federation (IWF)[8] decided in June 2018 to change from the existing Sinclair coefficient to Robi Points. Former IWF Technology Director Robert Nagy developed the Robi Points system. The Robi Points are calculated based on the actual World Records in the category and the point value of a result equal to a World Record is the same (1000 points) in all bodyweight categories
Alternatives are Glossbrenner coefficient[9] (WPC), Reshel coefficient[10] (GPC, GPA, WUAP, IRP), Outstanding Lifter (OL) or NASA coefficient[11] (NASA), Schwartz/Malone coefficient and Siff coefficient.
While all coefficients take into account gender and body weight difference there are some that also take into account age difference. For cadet and junior age group Foster coefficient is used and for master age group (40yo and above) McCulloch or Reshel coefficient.
^ "Wilks Formula 2 Released". Powerlifting Australia. 2020-02-28. Retrieved 2021-01-19.
^ Vanderburgh, Paul M; Batterham, Alan M (1999). "Validation of the Wilks powerlifting formula". Medicine & Science in Sports & Exercise. 31 (12): 1869–75. doi:10.1097/00005768-199912000-00027. PMID 10613442.
^ Myers, Al (2010-07-13). "O'Carroll Formula". USAWA. Retrieved 2021-01-19.
^ "Reflections on Strength, Gender, and Lifting Formulas" (PDF). www.starkcenter.org. Retrieved 2021-01-19.
^ "International Powerlifting Federation IPF". International Powerlifting Federation IPF. 2021-01-09. Retrieved 2021-01-19.
^ "The IPF Is Getting Rid of the Wilks Formula". BarBend. 2018-10-08. Retrieved 2021-01-19.
^ "Robi / Sinclair". International Weightlifting Federation. 2021-01-19. Retrieved 2021-01-19.
^ "APF Best Lifter Formula". worldpowerliftingcongress.com. Retrieved 2019-04-08.
^ "Reshel Formula the coefficient f". globalpowerliftingalliance.com. Retrieved 2019-04-08.
^ "Coefficient System". NASA Powerlifting. 2014-08-03. Retrieved 2019-04-08.
Online wilks calculator with lbs- and kg-based calculation
1RM & Wilks Points Calculator
Comparison of all used powerlifting coefficients
Retrieved from "https://en.wikipedia.org/w/index.php?title=Wilks_coefficient&oldid=1059454863" |
Power transferred per unit area
In physics, the intensity or flux of radiant energy is the power transferred per unit area, where the area is measured on the plane perpendicular to the direction of propagation of the energy. In the SI system, it has units watts per square metre (W/m2), or kg⋅s−3 in base units. Intensity is used most frequently with waves such as acoustic waves (sound) or electromagnetic waves such as light or radio waves, in which case the average power transfer over one period of the wave is used. Intensity can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the kinetic energy carried by drops of water from a garden sprinkler.
Intensity can be found by taking the energy density (energy per unit volume) at a point in space and multiplying it by the velocity at which the energy is moving. The resulting vector has the units of power divided by area (i.e., surface power density).
{\displaystyle P=\int \mathbf {I} \,\cdot d\mathbf {A} ,}
where P is the net power radiated, I is the intensity vector as a function of position, the magnitude |I| is the intensity as a function of position, and dA is a differential element of a closed surface that contains the source.
If one integrates a uniform intensity, |I| = constant, over a surface that is perpendicular to the intensity vector, for instance over a sphere centered around the point source, the equation becomes
{\displaystyle P=|I|\cdot A_{\mathrm {surf} }=|I|\cdot 4\pi r^{2}\,,}
where |I| is the intensity at the surface of the sphere, r is the radius of the sphere, and
{\displaystyle A_{\mathrm {surf} }=4\pi r^{2}}
is the expression for the surface area of a sphere.
Solving for |I| gives
{\displaystyle |I|={\frac {P}{A_{\mathrm {surf} }}}={\frac {P}{4\pi r^{2}}}.}
Anything that can transmit energy can have an intensity associated with it. For a monochromatic propagating electromagnetic wave, such as a plane wave or a Gaussian beam, if E is the complex amplitude of the electric field, then the time-averaged energy density of the wave, travelling in a non-magnetic material, is given by:
{\displaystyle \left\langle U\right\rangle ={\frac {n^{2}\varepsilon _{0}}{2}}|E|^{2},}
{\displaystyle I={\frac {\mathrm {c} n\varepsilon _{0}}{2}}|E|^{2},}
where n is the refractive index, c is the speed of light in vacuum and
{\displaystyle \varepsilon _{0}}
^ Paschotta, Rüdiger. "Optical Intensity". Encyclopedia of Laser Physics and Technology. RP Photonics.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Intensity_(physics)&oldid=1087140862" |
The Pathway To Euler's Formula Practice Problems Online | Brilliant
e^ {i\pi} +1 = 0
In Euler's identity, which value is not actually a number (real or complex)?
Euler's identity combines the five values:
e, i, \pi, 1,
0
in an elegant and entirely non-obvious way. There is something beautiful and powerful going on here, and our goal in this course is for you to come to fully understand it!
This introduction quiz covers just the tip of the iceberg that is Euler's formula, just enough so that you can start to get an idea of why it's interesting and how it can be useful.
e
i
\pi
1
0
They are all numbers
e^{i\pi}
When most math students (even those who have seen the imaginary constant
i
before) first run into Euler’s identity, their initial response is typically something like:
2
seconds: “Wow, it’s so small and simple looking.”
2
seconds: “...and it combines so many important constants so elegantly.”
4
years-lifetime: “...wait. That actually makes no sense whatsoever. It’s like saying the square root of five apples is equal to blue--it’s just completely nonsensical.”
e
\pi i
times Multiply
ie
\pi
times None of the above
But Euler’s formula is not mystical. For the most part, it works like any other algebra you may have encountered. For example, exponential rules like the power rule still apply.
e^{2\pi i} - 1 = 0
Hint: Take a look at the proof attempt below. Do you believe it, or does it contain a mistake?
e^{\pi i} + 1 = 0
e^{\pi i} = -1
(e^{\pi i})^2 = (-1)^2
e^{2\pi i} = 1
Euler’s formula is just as elegant with
\tau
as it is with
\pi.
(\tau = 2\pi
and is pronounced “tau.”
)
Euler's full equation is
e^{i\theta} = \cos(\theta) + i \sin(\theta),
of which Euler’s identity,
e^{i\pi} = -1,
is just one case where
\theta = \pi:
e^{i\pi} = \cos(\pi) + i\sin(\pi) = -1.
Using this fact as a hint, what is the value of
\sin(\pi) ?
Looking at the equation above, it shouldn't be too surprising that, in order to really understand Euler's formula, you'll need to build up a very high level of confidence working with
e
0
i
1
\pi
Using Euler’s formula, you’ll be able to derive the most annoying of the trigonometry identities in seconds instead of needing to memorize them!
Below is an example of using Euler’s full formula to derive two trig identities at the same time.
Based on the math below, which of the answer options is a correct trig identity?
e^{\theta i} = \cos(\theta) + i \sin(\theta)
e^{2\theta i} = \cos(2\theta) + i \sin(2\theta)
\big(e^{\theta i}\big)^2 = \big(\cos(\theta) + i\sin(\theta)\big)^2
\big(e^{\theta i}\big)^2 = e^{2\theta i}
\big(\cos(\theta) + i\sin(\theta)\big)^2 = \cos(2\theta) + i\sin(2\theta).
Expanding the left hand side gives
\cos^2(\theta) - \sin^2(\theta)+ 2i\cos(\theta)\sin(\theta) = \cos(2\theta) + i\sin(2\theta).
Hint: Remember that, if two complex numbers are equal, then their real and imaginary parts must both be equal.
\begin{array} {l} \sin(2\theta) \\ = 2\cos(\theta)\sin(\theta) \end{array}
\begin{array} {l} \cos(2\theta) \\ = \cos^2(\theta) - \sin^2(\theta) \end{array}
Both are correct Neither is correct
In this course, you’ll also learn to apply Euler’s formula to some geometry/trigonometry that you may have seen before, and also to some physics that you probably haven’t seen before.
Not only is Euler’s formula beautiful, it’s also an incredibly powerful tool! |
Compare linear mixed-effects models - MATLAB - MathWorks América Latina
p
-value of 0 indicates that model altlme is significantly better than the simpler model lme.
p
-value suggests that the larger model, lme is not significantly better than the smaller model, lme2. The smaller values of Akaike and Bayesian Information Criteria for lme2 also support this.
p
-value suggests that the larger model, lme is not significantly better than the smaller model, lme2. The smaller values of AIC and BIC for lme2 also support this.
p-value=\frac{\left[\sum _{j=1}^{nsim}I\left({T}_{{H}_{0}}\left(j\right)\ge T\right)\right]+1}{nsim+1}.
-2*\mathrm{log}{L}_{M}.
Dev=De{v}_{1}-De{v}_{2}=2\left(\mathrm{log}L{M}_{2}-\mathrm{log}L{M}_{1}\right).
D=\left(\begin{array}{cc}{D}_{11}& 0\\ 0& 0\end{array}\right), |
y=5{x}^{3}+7{x}^{2}-3x+1
{y}^{\prime }{=}^{1}{\left(5{x}^{3}\right)}^{\prime }+{\left(7{x}^{2}\right)}^{\prime }-{\left(3x\right)}^{\prime }+{1}^{\prime }
{=}^{2}5\stackrel{˙}{3}{x}^{2}+7\stackrel{˙}{2}x-3\stackrel{˙}{1}+0
=15{x}^{2}+14x-3
g\left(x\right)=\frac{1}{{e}^{x}+{e}^{-x}}
{s}^{\prime }\left(t\right){=}^{1}{\left(6{t}^{-2}\right)}^{\prime }+{\left(3{t}^{3}\right)}^{\prime }-{\left(4{t}^{\frac{1}{2}}\right)}^{\prime }
Simplify Trig Identity in
\mathrm{sin}\theta
\mathrm{cos}\theta
I just need to simplify in terms of
\mathrm{sin}\theta
\mathrm{cos}\theta
I did this one earlier today but forgot how I did it. I just need help on the other two.
\frac{\mathrm{cos}\theta }{\mathrm{sin}\theta }+\frac{1}{\mathrm{cos}\theta }=\frac{{\mathrm{cos}}^{2}\theta +\mathrm{sin}\theta }{\mathrm{sin}\theta \mathrm{cos}\theta }
\mathrm{sin}\theta +\frac{1}{\mathrm{cos}\theta }
\frac{\mathrm{sin}\theta }{\mathrm{cos}\theta }+\frac{1}{\mathrm{sin}\theta }
Fill in the blanks. When evaluating limits at infinity for complicated rational functions, divide the numerator and denominator by the ________ term in the denominator.
\frac{\left(\mathrm{arctan}\frac{1}{2}+\mathrm{arctan}\frac{1}{3}\right)}{\left(\mathrm{arcot}\frac{1}{2}+\mathrm{arcot}\frac{1}{3}\right)}
The exponential growth models describe the population of the indicated country, A, in millions, t years after 2006. Canada A=33.1e0.009t Uganda A=28.2e0.034t
Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The models indicate that in 2013, Ugandas |
Custom Static Optimization in MATLAB - OpenSim Documentation - Global Site
Custom Static Optimization in MATLAB
The tutorial below is designed for use with OpenSim version 4.0 and later.
Methods for estimating muscle forces from experimental data are an essential part of the biomechanist's simulation toolkit. Static optimization is a widely used tool for estimating muscle activity due to its speed and ease-of-use, and many OpenSim users rely on the Static Optimization Tool as a key part of their simulation research pipelines. Despite its popularity, OpenSim's implementation of this method only permits one type of cost function and a fixed set of constraints, and therefore is only applicable to research questions that can accept these limitations. In this lab, you will write your own code to better understand the elements required to solve the static optimization problem, draw comparisons to OpenSim's implementation, and learn how extend the basic tool to fit new research applications.
By working through this lab, you will:
Learn the basics of the static optimization problem, including OpenSim's implementation and alternative implementations.
Become familiar with using the OpenSim API through MATLAB.
Learn how to write your own static optimization code.
The goal of static optimization is to solve for muscle activations that produce the dynamics of an observed motion. Since there are more muscles than degrees-of-freedom in the human body, this problem is "non-unique" (i.e., many possible solutions exist), hence the need for optimization. We often describe this as the "muscle redundancy problem". We use the term "static" since no dynamics appear in the optimization problem: activation dynamics are ignored, tendons are assumed rigid, and multibody dynamics are prescribed via motion data. Most static optimization problems take the following form:
\begin{align*} \text{minimize } \text{ } & \sum_{i=1}^{N_{m}} (a_i)^p \\ \text{subject to } \text{ } & M_j = \sum_{i=1}^{N_m} r_{i,j} * f_i^m & \text{ for } j = 1 \ldots N_q \\ & f_i^m = F_i^{max} (a_i f_l(\tilde{l}_i^m) f_v(\tilde{v}_i^m) + f_p(\tilde{l}_i^m)) & \text{ for } i = 1 \ldots N_m \\ & 0 \leq a_i \leq 1 & \text{ for } i = 1 \ldots N_m \end{align*}
\begin{align*} a_i & - \text{activation in } i^{\text{th}} \text{ muscle} \\ p & - \text{positive integer} \\ M_j & - \text{net joint moment about } j^{\text{th}} \text{ joint} \\ r_{i,j} & - \text{moment arm for } i^{\text{th}} \text{ muscle about } j^{\text{th}} \text{ joint} \\ f_i^m & - \text{force in } i^{\text{th}} \text{ muscle} \\ F_i^{max} & - \text{max isometric force in } i^{\text{th}} \text{ muscle} \\ \tilde{l}_i^m & - \text{normalized fiber length in } i^{\text{th}} \text{ muscle} \\ \tilde{v}_i^m & - \text{normalized fiber velocity in } i^{\text{th}} \text{ muscle} \\ f_l & - \text{active fiber force-length curve} \\ f_v & - \text{active fiber force-velocity curve} \\ f_p & - \text{passive fiber force-length curve} \\ N_m & - \text{number of muscles} \\ N_q & - \text{number of coordinates} \end{align*}
The labels "minimize" and "subject to" denote the problem cost function and constraints, respectively. The muscle activations are the only problem "design variables", or the values that the optimizer can change to minimize the cost and satisfy the constraints. Net joint moments are typically computed from inverse dynamics, and inverse kinematics is used to compute all muscle kinematic states.
OpenSim's Static Optimization Tool is similar to the problem definition above (see How Static Optimization Works). OpenSim uses a similar cost function where activation raised to a user-defined power is minimized. Typically, an activations-squared cost function (p = 2
//<![CDATA[ \begin{array}{l}p = 2\end{array} //]]>
) is used for most research applications. However, OpenSim's implementation does not include passive muscle contributions when computing total muscle force, f^m
//<![CDATA[ \begin{array}{l}f^m\end{array} //]]>
. Lastly, rather than matching muscle-generated moments to net joint moments from inverse dynamics, OpenSim's StaticOptimizationTool requires that joint accelerations generated from muscle forces match joint accelerations computed from motion data. In this lab, we will use the moment-matching approach, but think about how you would solve the acceleration-matching problem as you work on your code.
(Optional) Optimization Theory
One important property of static optimization is that the problem is linear in its design variables. While the characteristic curves that define muscle forces are non-linear, muscle activations are the only design variables in the problem and appear linearly in the muscle-generated joint moment equations. If possible, it is beneficial to write optimization problems with linear constraints only, as these problems can usually be solved more efficiently than problems with non-linear constraints. Linear constraints are so advantageous that entire subfields in optimization theory are dedicated to solving problems with this form (see Linear Programming and Quadratic Programming to learn more).
We've provided materials to help you get started writing your own optimization code. You may click on the links provided here, or click the menu icon, , in the top right of the page and "Attachments" to download all files at once. Place all the materials in a common working directory, preferably something not write-protected (e.g., C:\Users\<account_name>\CustomStaticOptimization).
Files from the Rajagopal model distribution:
subject_walk_adjusted.osim – RRA-adjusted Rajagopal2016 model.
grf_walk.mot – Ground reaction force and torque data.
grf_walk.xml – ExternalLoads to be added to the model to apply ground reaction loads.
coordinates.mot – Coordinate values from an inverse kinematics solution.
loadFilterCropArray.m – Utility to load STO files into MATLAB array, filter, and crop to a specified time range. This function relies on the osimTableToStruct MATLAB utility that comes with OpenSim.
CustomStaticOptimization.m – Main assignment file. Write your code here.
If you don't have MATLAB scripting with OpenSim set up already, refer to Scripting with Matlab.
IV. Coding Static Optimization
In this section, you will code your own version of the static optimization problem described in the Background section. Write your problem in CustomStaticOptimization.m included in the materials. To guide you, we've structured CustomStaticOptimization.m to be completed in individual parts, which are labeled via comments in the starter code (e.g., Part 1). The sections you need to fill in will be denoted with commented brackets like the following:
% TODO {
Before you start coding, first look through the whole file and the parts you need to fill in to get a high-level view of the assignment. You may then continue reading through this section for more guidance on the code sections you need to complete.
Part 1: Fill in the missing InverseDynamicsTool commands.
To get started, you will need to fill in the missing OpenSim API commands to compute the net joint moments. Refer to the Doxygen page for the InverseDynamicsTool to find the missing commands you'll need.
If you have never used Doxygen or the API, refer to Guide to Using Doxygen and Introduction to the OpenSim API. You can also type 'methodsview InverseDynamicsTool' into the MATLAB command window to get a list of available commands.
Part 2: Plot and inspect the generalized forces from inverse dynamics.
Before you continue on to the next steps, compare the generalized forces (i.e., the net joint moments, M_j
//<![CDATA[ \begin{array}{l}M_j\end{array} //]]>
) from inverse dynamics to the plot below to make sure you have computed them correctly. Note that these results are specific to the model and data included with this example.
Click on this plot to enlarge and download.
Part 3: Remove any unneeded generalized force data columns.
Your static optimization solution should provide a reasonable estimate of muscle activity for walking. Therefore, we should only solve the problem across degrees-of-freedom with reliable experimental measurements that also contribute to the muscle activity solution. Inspect the Rajagopal2016 model included with the lab. Since we're only interested in lower extremity muscle activity, we can exclude matching generalized forces for certain model coordinates from the optimization problem. In addition, you may consider removing some muscle-actuated degrees-of-freedom that may have poor inverse kinematics solutions. Fill in the 'forcesToRemove' array as instructed in the CustomStaticOptimization.m file to remove unnecessary forces.
Part 4: Store max isometric force values and disable muscle dynamics
The model provided for you includes 80 lower-extremity Millard2012EquilibriumMuscles. We need to store an array of max isometric force values so we can compute total muscle force in the optimization. We will also disable activation and tendon compliance dynamics in each muscle, since static optimization does not include muscle dynamics. For a list of the available API commands for the muscles to complete these tasks, see the Millard2012EquilibriumMuscle API docs.
Since OpenSim returns muscles retrieved from the model ForceSet as the base type Muscle, we need to "downcast" each muscle object to the Millard2012EquilibriumMuscle type. To visualize downcasting, open the inheritance diagram at the top of the Millard2012EquilibriumMuscle API page and you'll see the Millard muscle below the base Muscle class. This shows that Millard2012EquilibriumMuscle "inherits" from Muscle, meaning that Millard2012EquilbriumMuscle can use all the commands and properties defined in Muscle (*mostly, this is a simplification, see Inheritanceand Inheritance in C++). You need to downcast to the Millard2012EquilibriumMuscle type if you want to use the commands specific to that type in other places in your code.
Part 5: Perform static optimization.
Use FMINCON, MATLAB's constrained optimization solver, to solve the static optimization problem with an activations-squared cost function (i.e., \sum_{i=1}^{N_{M}} (a_i)^2
//<![CDATA[ \begin{array}{l}\sum_{i=1}^{N_{M}} (a_i)^2\end{array} //]]>
). This is the main part of the coding assignment and will likely take the most time to complete. There is some starter code to select which data points to solve the problem on and to construct the FMINCON options structure; the rest is up to you!
Try outlining your problem in pseudocode before writing any actual MATLAB code.
Which elements of the problem are the same across all time steps and which elements need to be computed at each time step?
Similarly, which computations need to happen "inside" the optimization, and which can be pre-computed?
How will you implement the problem cost function? The first argument to FMINCON accepts a MATLAB function handle, which is a MATLAB variable type that represents a function. Function handles can be the typical named functions or anonymous functions. Note that the function handle passed to FMINCON can only be a function of the problem design variables.
How will you implement the problem constraints? Consider the discussion from the Background section.
Part 6: Plot results.
Plot your muscle activation solution and any other relevant outputs.
V. Follow-up Questions
Describe the muscle activation solution you plotted in Part 6. Does your solution match muscle activity levels for normal walking? Compare your solution to electromyography (EMG) data from the literature. What are potential explanations for any discrepancies?
Provide 3 research questions that you could use your new static optimization code to answer. For each question, briefly describe any additions or changes to the data, model or code necessary.
Approximately how long did it take you to complete this assignment?
Was the assignment too hard/too easy?
Do you think the skills learned were mostly worth the time invested in this assignment?
If you could make one improvement to this assignment for future classes, what would it be? |
The graph of a polynomial function has the following characteristics: a) Its domain and ra
The graph of a polynomial function has the following characteristics: a) Its domain and range are the set of all real numbers. b) There are turning po
The graph of a polynomial function has the following characteristics: a) Its domain and range are the set of all real numbers. b) There are turning points at
x=-2
, 0, and 3. a) Draw the graphs of two different polynomial functions that have these three characteristics. b) What additional characteristics would ensure that only one graph could be drawn?
The other equation that works, is
f\left(x\right)=\frac{1}{4}{x}^{4}-\frac{1}{3}{x}^{3}-3{x}^{2}-1
a.) there are two polynomials that work for this situation.
The answer is a.)
f\left(x\right)=\frac{1}{4}{x}^{4}-\frac{1}{3}{x}^{3}-3{x}^{2}
Write an equation for the polynomial graph:
y\left(x\right)=?
x=-2,
x=1
x=3
. y-intercept at
\left(0,-4\right)
y={e}^{x}\text{ }\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\text{ }y=1+x+\frac{{x}^{2}}{2}+\frac{{x}^{3}}{6}+\frac{{x}^{4}}{24}
y=f\left(x\right)
f\left(x\right)={x}^{4}+{x}^{3}-3{x}^{2}-x+2
A=limn\to \infty Rn=limn\to \infty \left[f\left({x}_{1}\right)\Delta x+f\left({x}_{2}\right)\Delta x+...+f\left(xn\right)\Delta x\right]
f\left(x\right)=7x\mathrm{cos}\left(7x\right),0\le x\le \pi 2
f\left(x\right)=3{x}^{3}+{x}^{2}-10x-8 |
Wave function - Simple English Wikipedia, the free encyclopedia
In quantum mechanics, the Wave function, usually represented by Ψ, or ψ, describes the probability of finding an electron somewhere in its matter wave. To be more precise, the square of the wave function gives the probability of finding the location of the electron in the given area, since the normal answer for the wave function is usually a complex number. The wave function concept was first introduced in the Schrödinger equation.
Mathematical Interpretation[change | change source]
The formula for finding the wave function (i.e., the probability wave), is below:
{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {x} ,\,t)={\hat {H}}\Psi (\mathbf {x} ,\,t)}
where i is the imaginary number, ψ (x,t) is the wave function, ħ is the reduced Planck constant, t is time, x is position in space, Ĥ is a mathematical object known as the Hamilton operator. The reader will note that the symbol
{\displaystyle {\frac {\partial }{\partial t}}}
denotes that the partial derivative of the wave function is being taken.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Wave_function&oldid=7842700" |
Rectification_(geometry) Knowpia
In Euclidean geometry, rectification, also known as critical truncation or complete-truncation is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.[1] The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.
A rectified cube is a cuboctahedron – edges reduced to vertices, and vertices expanded into new faces
A birectified cube is an octahedron – faces are reduced to points and new faces are centered on the original vertices.
A rectified cubic honeycomb – edges reduced to vertices, and vertices expanded into new cells.
A rectification operator is sometimes denoted by the letter r with a Schläfli symbol. For example, r{4,3} is the rectified cube, also called a cuboctahedron, and also represented as
{\displaystyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}
. And a rectified cuboctahedron rr{4,3} is a rhombicuboctahedron, and also represented as
{\displaystyle r{\begin{Bmatrix}4\\3\end{Bmatrix}}}
Conway polyhedron notation uses a for ambo as this operator. In graph theory this operation creates a medial graph.
The rectification of any regular self-dual polyhedron or tiling will result in another regular polyhedron or tiling with a tiling order of 4, for example the tetrahedron {3,3} becoming an octahedron {3,4}. As a special case, a square tiling {4,4} will turn into another square tiling {4,4} under a rectification operation.
Example of rectification as a final truncation to an edgeEdit
Rectification is the final point of a truncation process. For example, on a cube this sequence shows four steps of a continuum of truncations between the regular and rectified form:
Higher degree rectificationsEdit
Higher degree rectification can be performed on higher-dimensional regular polytopes. The highest degree of rectification creates the dual polytope. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points, and so on.
Example of birectification as a final truncation to a faceEdit
This sequence shows a birectified cube as the final sequence from a cube to the dual where the original faces are truncated down to a single point:
In polygonsEdit
The dual of a polygon is the same as its rectified form. New vertices are placed at the center of the edges of the original polygon.
In polyhedra and plane tilingsEdit
Each platonic solid and its dual have the same rectified polyhedron. (This is not true of polytopes in higher dimensions.)
The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriated scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual:
The rectified tetrahedron, whose dual is the tetrahedron, is the tetratetrahedron, better known as the octahedron.
The rectified octahedron, whose dual is the cube, is the cuboctahedron.
The rectified icosahedron, whose dual is the dodecahedron, is the icosidodecahedron.
A rectified square tiling is a square tiling.
A rectified triangular tiling or hexagonal tiling is a trihexagonal tiling.
Tetrapentagonal tiling
In nonregular polyhedraEdit
If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a polyhedral graph as its 1-skeleton, and from that graph one may form the medial graph by placing a vertex at each edge midpoint of the original graph, and connecting two of these new vertices by an edge whenever they belong to consecutive edges along a common face. The resulting medial graph remains polyhedral, so by Steinitz's theorem it can be represented as a polyhedron.
The Conway polyhedron notation equivalent to rectification is ambo, represented by a. Applying twice aa, (rectifying a rectification) is Conway's expand operation, e, which is the same as Johnson's cantellation operation, t0,2 generated from regular polyhedral and tilings.
In 4-polytopes and 3D honeycomb tessellationsEdit
Each Convex regular 4-polytope has a rectified form as a uniform 4-polytope.
A regular 4-polytope {p,q,r} has cells {p,q}. Its rectification will have two cell types, a rectified {p,q} polyhedron left from the original cells and {q,r} polyhedron as new cells formed by each truncated vertex.
A rectified {p,q,r} is not the same as a rectified {r,q,p}, however. A further truncation, called bitruncation, is symmetric between a 4-polytope and its dual. See Uniform 4-polytope#Geometric derivations.
(Dual rectification)
Trirectification
r{p,q,r}
2r{p,q,r}
Order-4 dodecahedral
Rectified order-4 dodecahedral
Rectified order-5 cubic
Order-5 cubic
Degrees of rectificationEdit
A first rectification truncates edges down to points. If a polytope is regular, this form is represented by an extended Schläfli symbol notation t1{p,q,...} or r{p,q,...}.
A second rectification, or birectification, truncates faces down to points. If regular it has notation t2{p,q,...} or 2r{p,q,...}. For polyhedra, a birectification creates a dual polyhedron.
Higher degree rectifications can be constructed for higher dimensional polytopes. In general an n-rectification truncates n-faces to points.
If an n-polytope is (n-1)-rectified, its facets are reduced to points and the polytope becomes its dual.
Notations and facetsEdit
There are different equivalent notations for each degree of rectification. These tables show the names by dimension and the two type of facets for each.
Facets are edges, represented as {2}.
Vertical Schläfli symbol
t0{p} {p} {2}
Regular polyhedra and tilingsEdit
Facets are regular polygons.
t0{p,q} {p,q} {p}
t1{p,q} r{p,q} =
{\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}
{p} {q}
t2{p,q} {q,p} {q}
Regular Uniform 4-polytopes and honeycombsEdit
Facets are regular or rectified polyhedra.
t0{p,q,r} {p,q,r} {p,q}
{\displaystyle {\begin{Bmatrix}p\ \ \\q,r\end{Bmatrix}}}
= r{p,q,r}
{\displaystyle {\begin{Bmatrix}p\\q\end{Bmatrix}}}
= r{p,q} {q,r}
(Dual rectified)
{\displaystyle {\begin{Bmatrix}q,p\\r\ \ \end{Bmatrix}}}
= r{r,q,p} {q,r}
{\displaystyle {\begin{Bmatrix}q\\r\end{Bmatrix}}}
= r{q,r}
t3{p,q,r} {r,q,p} {r,q}
Regular 5-polytopes and 4-space honeycombsEdit
Facets are regular or rectified 4-polytopes.
t0{p,q,r,s} {p,q,r,s} {p,q,r}
{\displaystyle {\begin{Bmatrix}p\ \ \ \ \ \\q,r,s\end{Bmatrix}}}
= r{p,q,r,s}
{\displaystyle {\begin{Bmatrix}p\ \ \\q,r\end{Bmatrix}}}
= r{p,q,r} {q,r,s}
(Birectified dual)
{\displaystyle {\begin{Bmatrix}q,p\\r,s\end{Bmatrix}}}
= 2r{p,q,r,s}
{\displaystyle {\begin{Bmatrix}q,p\\r\ \ \end{Bmatrix}}}
= r{r,q,p}
{\displaystyle {\begin{Bmatrix}q\ \ \\r,s\end{Bmatrix}}}
= r{q,r,s}
{\displaystyle {\begin{Bmatrix}r,q,p\\s\ \ \ \ \ \end{Bmatrix}}}
= r{s,r,q,p} {r,q,p}
{\displaystyle {\begin{Bmatrix}r,q\\s\ \ \end{Bmatrix}}}
= r{s,r,q}
Quadrirectified
t4{p,q,r,s} {s,r,q,p} {s,r,q}
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
Olshevsky, George. "Rectification". Glossary for Hyperspace. Archived from the original on 4 February 2007. |
Substance or procedure that ends a medical condition
A cure is a substance or procedure that ends a medical condition, such as a medication, a surgical operation, a change in lifestyle or even a philosophical mindset that helps end a person's sufferings; or the state of being healed, or cured. The medical condition could be a disease, mental illness, genetic disorder, or simply a condition a person considers socially undesirable, such as baldness or lack of breast tissue.
An incurable disease may or may not be a terminal illness; conversely, a curable illness can still result in the patient's death.
The simplest cure rate model was published by Joseph Berkson and Robert P. Gage in 1952.[7] In this model, the survival at any given time is equal to those that are cured plus those that are not cured, but who have not yet died or, in the case of diseases that feature asymptomatic remissions, have not yet re-developed signs and symptoms of the disease. When all of the non-cured people have died or re-developed the disease, only the permanently cured members of the population will remain, and the DFS curve will be perfectly flat. The earliest point in time that the curve goes flat is the point at which all remaining disease-free survivors are declared to be permanently cured. If the curve never goes flat, then the disease is formally considered incurable (with the existing treatments).
{\displaystyle S(t)=p+[(1-p)\times S^{*}(t)]}
{\displaystyle S(t)}
{\displaystyle p}
{\displaystyle S^{*}(t)}
is a partial reduction in symptoms after treatment.
is a restoration of health or functioning. A person who has been cured may not be fully recovered, and a person who has recovered may not be cured, as in the case of a person in a temporary remission or who is an asymptomatic carrier for an infectious disease.
is a way to avoid an injury, sickness, disability, or disease in the first place, and generally it will not help someone who is already ill (though there are exceptions). For instance, many babies and young children are vaccinated against polio and other infectious diseases, which prevents them from contracting polio. But the vaccination does not work on patients who already have polio. A treatment or cure is applied after a medical problem has already started.
treats a problem, and may or may not lead to its cure. In incurable conditions, a treatment ameliorates the medical condition, often only for as long as the treatment is continued or for a short while after treatment is ended. For example, there is no cure for AIDS, but treatments are available to slow down the harm done by HIV and extend the treated person's life. Treatments don't always work. For example, chemotherapy is a treatment for cancer, but it may not work for every patient. In easily cured forms of cancer, such as childhood leukemias, testicular cancer and Hodgkin lymphoma, cure rates may approach 90%.[9] In other forms, treatment may be essentially impossible. A treatment need not be successful in 100% of patients to be considered curative. A given treatment may permanently cure only a small number of patients; so long as those patients are cured, the treatment is considered curative.
Cures can take the form of natural antibiotics (for bacterial infections), synthetic antibiotics such as the sulphonamides, or fluoroquinolones, antivirals (for a very few viral infections), antifungals, antitoxins, vitamins, gene therapy, surgery, chemotherapy, radiotherapy, and so on. Despite a number of cures being developed, the list of incurable diseases remains long.
Scurvy became curable (as well as preventable) with doses of vitamin C (for example, in limes) when James Lind published A Treatise on the Scurvy (1753).[10]
Antitoxins to diphtheria and tetanus toxins were produced by Emil Adolf von Behring and his colleagues from 1890 onwards. The use of diphtheria antitoxin for the treatment of diphtheria was regarded by The Lancet as the "most important advance of the [19th] Century in the medical treatment of acute infectious disease".[11][12]
Sulphonamides become the first widely available cure for bacterial infections.[citation needed]
Antimalarials were first synthesized,[13][14][15] making malaria curable.[16]
Bacterial infections became curable with the development of antibiotics.[17]
Hepatitis C, a viral infection, became curable through treatment with antiviral medications.[18][19]
^ Fuller, Arlan F.; Griffiths, C. M. (1983). Gynecologic oncology. The Hague: M. Nijhoff. ISBN 0-89838-555-5.
^ Lambert PC, Thompson JR, Weston CL, Dickman PW (2007). "Estimating and modeling the cure fraction in population-based cancer survival analysis". Biostatistics. 8 (3): 576–594. doi:10.1093/biostatistics/kxl030. PMID 17021277.
^ a b Smoll NR, Schaller K, Gautschi OP (2012). "The Cure Fraction of Glioblastoma Multiforme". Neuroepidemiology. 39 (1): 63–9. doi:10.1159/000339319. PMID 22776797.
^ "Nearing a Cancer Cure?". Harvard Health Commentaries. 21 August 2006.
^ "What's the Difference Between a Treatment and a Cure?". TeensHealth. Nemours. May 2018. Archived from the original on 2008-04-13.
^ a b c Barnes E (December 2007). "Between remission and cure: patients, practitioners and the transformation of leukaemia in the late twentieth century". Chronic Illn. 3 (4): 253–64. doi:10.1177/1742395307085333. PMID 18083680. S2CID 13259230.
^ a b c d e Friis, Robert H.; Chernick, Michael L. (2003). Introductory biostatistics for the health sciences: modern applications including bootstrap. New York: Wiley-Interscience. pp. 348–349. ISBN 0-471-41137-X.
^ Tobias, Jeffrey M.; Souhami, Robert L. (2003). Cancer and its management. Oxford: Blackwell Science. p. 11. ISBN 0-632-05531-6.
^ Saltus, Richard (Fall–Winter 2008). "What is a Cure?" (PDF). Paths of Progress. Vol. 17, no. 2. Boston: Dana-Farber Cancer Institute. p. 8. Retrieved August 2, 2020.
^ Bartholomew, M (2002-11-01). "James Lind's Treatise of the Scurvy (1753)". Postgraduate Medical Journal. BMJ. 78 (925): 695–696. doi:10.1136/pmj.78.925.695. ISSN 0032-5473. PMC 1742547. PMID 12496338.
^ (Report) (1896). "Report of the Lancet special commission on the relative strengths of diphtheria antitoxic antiserums". Lancet. 148 (3803): 182–95. doi:10.1016/s0140-6736(01)72399-9. PMC 5050965.
^ Krafts K, Hempelmann E, Skórska-Stania A (2012). "From methylene blue to chloroquine: a brief review of the development of an antimalarial therapy". Parasitol Res. 111 (1): 1–6. doi:10.1007/s00436-012-2886-x. PMID 22411634. S2CID 54526057.
^ Hempelmann E. (2007). "Hemozoin biocrystallization in Plasmodium falciparum and the antimalarial activity of crystallization inhibitors". Parasitol Res. 100 (4): 671–76. doi:10.1007/s00436-006-0313-x. PMID 17111179. S2CID 30446678.
^ Jensen M, Mehlhorn H (2009). "Seventy-five years of Resochin in the fight against malaria". Parasitol Res. 105 (3): 609–27. doi:10.1007/s00436-009-1524-8. PMID 19593586. S2CID 8037461.
^ "Fact sheet about Malaria". World Health Organization. 14 January 2020. Retrieved 2020-08-04.
^ "Battle of the Bugs: Fighting Antibiotic Resistance". U.S. Food and Drug Administration. 2016-05-04. Retrieved 2020-07-25. Just a few years after the first antibiotic, penicillin, became widely used in the late 1940s
^ Wheeler, Regina Boyle (2018-10-15). "Is Hep C Curable?". WebMD. Retrieved 2019-02-12.
^ "Hepatitis C - Symptoms and causes". Mayo Clinic. Retrieved 2020-07-25. |
(→CFD Methods)
== CFD Methods ==
The key physical features of the UFR (see [[UFR_2-12_Description#Description|Description section]])
The key physical features of the UFR (see [[UFR_2-12_Description#Description|Description]])
present significant difficulties for all the existing approaches
to turbulence representation, whether from the standpoint of solution fidelity (for the conventional (U)RANS models)
{\displaystyle {\theta }}
{\displaystyle U_{0}D/\nu }
{\displaystyle L/D}
{\displaystyle L_{z}/D}
{\displaystyle D}
{\displaystyle U_{0}}
{\displaystyle K}
{\displaystyle C_{p}=\langle (p-p_{0})\rangle /(1/2\rho _{0}U_{0}^{2})}
{\displaystyle {\left.\langle u\rangle /U_{0}\right.}}
{\displaystyle \theta }
{\displaystyle \theta }
{\displaystyle {\text{TKE}}={\frac {1}{2}}\left(\langle u'u'\rangle +\langle v'v'\rangle \right)/U_{0}^{2}}
2D TKE distribution along* y = 0;
2D TKE distribution along* x = 1.5 D (in the gap between the cylinders);
2D TKE distribution along* x = 4.45 D (0.75 D downstream of the centre of the rear cylinder).
All these and some other data are available on the web site of the BANC-I Workshop.
A detailed description of the experimental facility and measurement techniques is given in the original publications [2-4] and available on the web site of the BANC-I Workshop. So here we present only concise information about these aspects of the test case. |
I have the following equation: \sin(x-\frac{\pi}{6})+\cos(x+\frac{\pi}{4})=0
\mathrm{sin}\left(x-\frac{\pi }{6}\right)+\mathrm{cos}\left(x+\frac{\pi }{4}\right)=0
\mathrm{cos}\left(x+\frac{\pi }{4}\right)=\mathrm{sin}\left(\frac{\pi }{4}-x\right)
and the equation turns to
\mathrm{sin}\left(x-\frac{\pi }{6}\right)=\mathrm{sin}\left(x-\frac{\pi }{4}\right)
2x-\frac{5\pi }{12}=\left(2k+1\right)\pi
\mathrm{sin}x+\mathrm{sin}y=a\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{cos}x+\mathrm{cos}y=b
\mathrm{tan}\left(x-\frac{y}{2}\right)
\mathrm{cos}45=\frac{1}{\sqrt{2}}
\mathrm{cos}315,\mathrm{sin}270,\mathrm{sin}210,\mathrm{tan}210
\mathrm{csc}\left(x\right)
\alpha
\beta
\gamma
\delta
\lambda
4\mathrm{sin}\frac{\alpha }{2}+3\mathrm{sin}\frac{\beta }{2}+2\mathrm{sin}\frac{\gamma }{2}+\mathrm{sin}\frac{\delta }{2}=2\sqrt{1+\lambda }
\lambda
\mathrm{sin}\alpha
\mathrm{sin}\beta
\mathrm{sin}\gamma
\mathrm{sin}\delta
\mathrm{arcsin}\frac{1}{2}
Calculate the limit of
\underset{x\to 1}{lim}\frac{1-{x}^{2}}{\mathrm{sin}\pi x}
only with the fundamental limits |
In physics and geometry: conjectured relation between pairs of Calabi–Yau manifolds
For other uses, see Mirror symmetry.
Early cases of mirror symmetry were discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that it could be used as a tool in enumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions. Candelas and his collaborators showed that mirror symmetry could be used to count rational curves on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of its mathematical predictions have since been proven rigorously.
Today, mirror symmetry is a major research topic in pure mathematics, and mathematicians are working to develop a mathematical understanding of the relationship based on physicists' intuition. Mirror symmetry is also a fundamental tool for doing calculations in string theory, and it has been used to understand aspects of quantum field theory, the formalism that physicists use to describe elementary particles. Major approaches to mirror symmetry include the homological mirror symmetry program of Maxim Kontsevich and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow.
1.1 Strings and compactification
1.2 Calabi–Yau manifolds
3.1 Enumerative geometry
4.1 Homological mirror symmetry
4.2 Strominger–Yau–Zaslow conjecture
8.1 Popularizations
Strings and compactification[edit]
Calabi–Yau manifolds[edit]
The mirror symmetry relationship is a particular example of what physicists call a physical duality. In general, the term physical duality refers to a situation where two seemingly different physical theories turn out to be equivalent in a nontrivial way. If one theory can be transformed so it looks just like another theory, the two are said to be dual under that transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.[10] Such dualities play an important role in modern physics, especially in string theory.[11]
The idea of mirror symmetry can be traced back to the mid-1980s when it was noticed that a string propagating on a circle of radius
{\displaystyle R}
is physically equivalent to a string propagating on a circle of radius
{\displaystyle 1/R}
in appropriate units.[19] This phenomenon is now known as T-duality and is understood to be closely related to mirror symmetry.[20] In a paper from 1985, Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten showed that by compactifying string theory on a Calabi–Yau manifold, one obtains a theory roughly similar to the standard model of particle physics that also consistently incorporates an idea called supersymmetry.[21] Following this development, many physicists began studying Calabi–Yau compactifications, hoping to construct realistic models of particle physics based on string theory. Cumrun Vafa and others noticed that given such a physical model, it is not possible to reconstruct uniquely a corresponding Calabi–Yau manifold. Instead, there are two Calabi–Yau manifolds that give rise to the same physics.[22]
Enumerative geometry[edit]
Homological mirror symmetry[edit]
Mathematically, branes can be described using the notion of a category.[49] This is a mathematical structure consisting of objects, and for any pair of objects, a set of morphisms between them. In most examples, the objects are mathematical structures (such as sets, vector spaces, or topological spaces) and the morphisms are functions between these structures.[50] One can also consider categories where the objects are D-branes and the morphisms between two branes
{\displaystyle \alpha }
{\displaystyle \beta }
are states of open strings stretched between
{\displaystyle \alpha }
{\displaystyle \beta }
Strominger–Yau–Zaslow conjecture[edit]
One can choose an auxiliary circle
{\displaystyle B}
(the pink circle in the figure) such that each of the infinitely many circles decomposing the torus passes through a point of
{\displaystyle B}
. This auxiliary circle is said to parametrize the circles of the decomposition, meaning there is a correspondence between them and points of
{\displaystyle B}
{\displaystyle B}
is more than just a list, however, because it also determines how these circles are arranged on the torus. This auxiliary space plays an important role in the SYZ conjecture.[53]
The idea of dividing a torus into pieces parametrized by an auxiliary space can be generalized. Increasing the dimension from two to four real dimensions, the Calabi–Yau becomes a K3 surface. Just as the torus was decomposed into circles, a four-dimensional K3 surface can be decomposed into two-dimensional tori. In this case the space
{\displaystyle B}
is an ordinary sphere. Each point on the sphere corresponds to one of the two-dimensional tori, except for twenty-four "bad" points corresponding to "pinched" or singular tori.[53]
The Calabi–Yau manifolds of primary interest in string theory have six dimensions. One can divide such a manifold into 3-tori (three-dimensional objects that generalize the notion of a torus) parametrized by a 3-sphere
{\displaystyle B}
(a three-dimensional generalization of a sphere). Each point of
{\displaystyle B}
corresponds to a 3-torus, except for infinitely many "bad" points which form a grid-like pattern of segments on the Calabi–Yau and correspond to singular tori.[60]
Once the Calabi–Yau manifold has been decomposed into simpler parts, mirror symmetry can be understood in an intuitive geometric way. As an example, consider the torus described above. Imagine that this torus represents the "spacetime" for a physical theory. The fundamental objects of this theory will be strings propagating through the spacetime according to the rules of quantum mechanics. One of the basic dualities of string theory is T-duality, which states that a string propagating around a circle of radius
{\displaystyle R}
is equivalent to a string propagating around a circle of radius
{\displaystyle 1/R}
in the sense that all observable quantities in one description are identified with quantities in the dual description.[61] For example, a string has momentum as it propagates around a circle, and it can also wind around the circle one or more times. The number of times the string winds around a circle is called the winding number. If a string has momentum
{\displaystyle p}
and winding number
{\displaystyle n}
in one description, it will have momentum
{\displaystyle n}
{\displaystyle p}
in the dual description.[61] By applying T-duality simultaneously to all of the circles that decompose the torus, the radii of these circles become inverted, and one is left with a new torus which is "fatter" or "skinnier" than the original. This torus is the mirror of the original Calabi–Yau.[62]
T-duality can be extended from circles to the two-dimensional tori appearing in the decomposition of a K3 surface or to the three-dimensional tori appearing in the decomposition of a six-dimensional Calabi–Yau manifold. In general, the SYZ conjecture states that mirror symmetry is equivalent to the simultaneous application of T-duality to these tori. In each case, the space
{\displaystyle B}
provides a kind of blueprint that describes how these tori are assembled into a Calabi–Yau manifold.[63]
^ For an accessible introduction to string theory, see Greene 2000.
^ a b Yau and Nadis 2010, Ch. 6
^ This analogy is used for example in Greene 2000, p. 186
^ Yau and Nadis 2010, p. ix
^ Dixon 1988; Lerche, Vafa, and Warner 1989
^ The shape of a Calabi–Yau manifold is described mathematically using an array of numbers called Hodge numbers. The arrays corresponding to mirror Calabi–Yau manifolds are different in general, reflecting the different shapes of the manifolds, but they are related by a certain symmetry. For more information, see Yau and Nadis 2010, p. 160–3.
^ Aspinwall et al. 2009, p. 13
^ Hori et al. 2003, p. xvi
^ Other dualities that arise in string theory are S-duality, T-duality, and the AdS/CFT correspondence.
^ Zaslow 2008, p. 523
^ Yau and Nadis 2010, p. 168
^ a b Hori and Vafa 2000
^ a b Witten 1990
^ Givental 1996, 1998; Lian, Liu, Yau 1997, 1999, 2000
^ a b Zaslow 2008, p. 531
^ a b Hori et al. 2003, p. xix
^ This was first observed in Kikkawa and Yamasaki 1984 and Sakai and Senda 1986.
^ a b Strominger, Yau, and Zaslow 1996
^ Candelas et al. 1985
^ This was observed in Dixon 1988 and Lerche, Vafa, and Warner 1989.
^ Green and Plesser 1990; Yau and Nadis 2010, p. 158
^ Candelas, Lynker, and Schimmrigk 1990; Yau and Nadis 2010, p. 163
^ a b Yau and Nadis 2010, p. 165
^ Yau and Nadis 2010, pp. 169–170
^ Vafa 1992; Witten 1992
^ Hori et al. 2003, p. xviii
^ Kontsevich 1995b
^ Kontsevich 1995a
^ Givental 1996, 1998
^ Lian, Liu, Yau 1997, 1999a, 1999b, 2000
^ Aspinwall et al. 2009, p. vii
^ Zaslow 2008, pp. 533–4
^ Zaslow 2008, sec. 10
^ Hori et al. 2003, p. 677
^ Intriligator and Seiberg 1996
^ Aspinwall et al. 2009
^ A basic reference on category theory is Mac Lane 1998.
^ a b c Aspinwal et al. 2009, p. 575
^ a b c Yau and Nadis 2010, p. 175
^ Yau and Nadis 2010, pp. 180–1
^ Aspinwall et al. 2009, p. 616
^ Yau and Nadis 2010, p. 175–6
^ Yau and Nadis 2010, pp. 175–7.
Aspinwall, Paul; Bridgeland, Tom; Craw, Alastair; Douglas, Michael; Gross, Mark; Kapustin, Anton; Moore, Gregory; Segal, Graeme; Szendröi, Balázs; Wilson, P.M.H., eds. (2009). Dirichlet Branes and Mirror Symmetry. Clay Mathematics Monographs. Vol. 4. American Mathematical Society. ISBN 978-0-8218-3848-8.
Candelas, Philip; Lynker, Monika; Schimmrigk, Rolf (1990). "Calabi–Yau manifolds in weighted
{\displaystyle \mathbb {P} _{4}}
". Nuclear Physics B. 341 (1): 383–402. Bibcode:1990NuPhB.341..383C. doi:10.1016/0550-3213(90)90185-G.
Givental, Alexander (1996). "Equivariant Gromov-Witten invariants". International Mathematics Research Notices. 1996 (13): 613–663. doi:10.1155/S1073792896000414. S2CID 554844.
Givental, Alexander (1998). "A mirror theorem for toric complete intersections". Topological Field Theory, Primitive Forms and Related Topics: 141–175. arXiv:alg-geom/9701016. Bibcode:1998tftp.conf..141G. doi:10.1007/978-1-4612-0705-4_5. ISBN 978-1-4612-6874-1. S2CID 2884104.
Hori, Kentaro; Katz, Sheldon; Klemm, Albrecht; Pandharipande, Rahul; Thomas, Richard; Vafa, Cumrun; Vakil, Ravi; Zaslow, Eric, eds. (2003). Mirror Symmetry (PDF). Clay Mathematics Monographs. Vol. 1. American Mathematical Society. ISBN 0-8218-2955-6. Archived from the original on 2006-09-19. {{cite book}}: CS1 maint: bot: original URL status unknown (link)
Hori, Kentaro; Vafa, Cumrun (2000). "Mirror Symmetry". arXiv:hep-th/0002222.
Intriligator, Kenneth; Seiberg, Nathan (1996). "Mirror symmetry in three-dimensional gauge theories". Physics Letters B. 387 (3): 513–519. arXiv:hep-th/9607207. Bibcode:1996PhLB..387..513I. doi:10.1016/0370-2693(96)01088-X. S2CID 13985843.
Kontsevich, Maxim (1995a), "Enumeration of Rational Curves Via Torus Actions", The Moduli Space of Curves, Birkhäuser, p. 335, arXiv:hep-th/9405035, doi:10.1007/978-1-4612-4264-2_12, ISBN 978-1-4612-8714-8, S2CID 16131978
Kontsevich, Maxim (1995b). "Homological algebra of mirror symmetry". Proceedings of the International Congress of Mathematicians: 120–139. arXiv:alg-geom/9411018. Bibcode:1994alg.geom.11018K. doi:10.1007/978-3-0348-9078-6_11. ISBN 978-3-0348-9897-3. S2CID 16733945.
Lerche, Wolfgang; Vafa, Cumrun; Warner, Nicholas (1989). "Chiral rings in
{\displaystyle {\mathcal {N}}=2}
superconformal theories" (PDF). Nuclear Physics B. 324 (2): 427–474. Bibcode:1989NuPhB.324..427L. doi:10.1016/0550-3213(89)90474-4.
Lian, Bong; Liu, Kefeng; Yau, Shing-Tung (1997). "Mirror principle, I". Asian Journal of Mathematics. 1 (4): 729–763. arXiv:alg-geom/9712011. Bibcode:1997alg.geom.12011L. doi:10.4310/ajm.1997.v1.n4.a5. S2CID 8035522.
Lian, Bong; Liu, Kefeng; Yau, Shing-Tung (1999a). "Mirror principle, II". Asian Journal of Mathematics. 3: 109–146. arXiv:math/9905006. Bibcode:1999math......5006L. doi:10.4310/ajm.1999.v3.n1.a6. S2CID 17837291.
Lian, Bong; Liu, Kefeng; Yau, Shing-Tung (1999b). "Mirror principle, III". Asian Journal of Mathematics. 3 (4): 771–800. arXiv:math/9912038. Bibcode:1999math.....12038L. doi:10.4310/ajm.1999.v3.n4.a4.
Lian, Bong; Liu, Kefeng; Yau, Shing-Tung (2000). "Mirror principle, IV". Surveys in Differential Geometry. 7: 475–496. arXiv:math/0007104. Bibcode:2000math......7104L. doi:10.4310/sdg.2002.v7.n1.a15. S2CID 1099024.
Strominger, Andrew; Yau, Shing-Tung; Zaslow, Eric (1996). "Mirror symmetry is T-duality". Nuclear Physics B. 479 (1): 243–259. arXiv:hep-th/9606040. Bibcode:1996NuPhB.479..243S. doi:10.1016/0550-3213(96)00434-8. S2CID 14586676.
Vafa, Cumrun (1992). "Topological mirrors and quantum rings". Essays on Mirror Manifolds: 96–119. arXiv:hep-th/9111017. Bibcode:1991hep.th...11017V. ISBN 978-962-7670-01-8.
Popularizations[edit]
Zaslow, Eric (2005). "Physmatics". arXiv:physics/0506153.
Hori, Kentaro; Katz, Sheldon; Klemm, Albrecht; Pandharipande, Rahul; Thomas, Richard; Vafa, Cumrun; Vakil, Ravi; Zaslow, Eric, eds. (2003). Mirror Symmetry (PDF). American Mathematical Society. ISBN 0-8218-2955-6. Archived from the original on 2006-09-19. {{cite book}}: CS1 maint: bot: original URL status unknown (link)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mirror_symmetry_(string_theory)&oldid=1073178014" |
The Land Beyond the Sunset - Wikisource, the free online library
An American silent film, selected for preservation in the United States National Film Registry by the Library of Congress.
3450451The Land Beyond the Sunset1912Harold M. Shaw
REG. U . S. PAT. OFF.
Copyright 1912 Thomas A. Edison Inc. Patented (Re-Issue) Jan. 12. 1904
JOE, the newsboy MARTIN FULLER
Joe's home.
HIS GRANDMOTHER MRS. WILLIAM BECHTEL
An outing arranged for poor children.
MANAGER OF THE FRESH AIR FUND WALTER EDWIN
{\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}}
ETHEL JEWETT
MRS. WALLACE ERSKINE
THE MINISTER BIGELOW COOPER
TRAIN LEAVES CENTRAL STATION AT 7 A. M.
The morning of the picnic.
His first sight of the world beyond the slums.
He hears a fairy story.
"Jack ran and escaped from the wicked old witch. The fairies guided him to the shore and sent him a boat into which he climbed."
"He needed no oars because his little fairy friends were guiding the boat out to sea, along the path of shining light, to the Land Beyond the Sunset, where he lived happily ever after."
His boat on the shore.
And he drifted to the land beyond the sunset.
Retrieved from "https://en.wikisource.org/w/index.php?title=The_Land_Beyond_the_Sunset&oldid=12084712" |
Area of a Semicircle: Formula, Definition & Perimeter
A semicircle is a half circle. That means a semicircle will have half of the area of a circle. You might think that means it will have half the perimeter of a circle, but that is not true.
To make a semicircle, take any diameter of the circle. Remove one half of the circle along that diameter. You have a semicircle (half of a circle).
A semicircle is half the circumference of a full circle plus the diameter of a cricle,
\left(d\right)
Learn about the radius, diameter, and circumference of a circle in this lesson.
The area of a semicircle is the space contained by the circle. The area is the number of square units enclosed by the sides of the shape.
The area of a semicircle is always expressed in square units, based on the units used for the radius of a circle.
The formula for the area,
A
, of a circle is built around its radius. You find the area of a semicircle by plugging the given radius of the semicircle into the area of a semicircle formula.
The area formula is:
A = \frac{\pi {r}^{2}}{2}
To find the area of a semicircle with diameter, divide the diameter by
2
to find the radius, and then apply the area of a semicircle formula.
For example, the semicircle below has a radius of
19 cm
. What is the area of the semicircle?
To find its area, we replace
r
with the actual value:
\mathbf{A} \mathbf{=} \frac{{\mathbf{\pi r}}^{\mathbf{2}}}{\mathbf{2}}
A = \frac{\pi \left({19}^{2}\right)}{2}
A = \frac{\pi \left(361\right)}{2}
A = \frac{1134.114947}{2}
A = 567.057 c{m}^{2}
The Roman aqueduct of Barcelona in Spain is very old, dating from the first century of the Common Era. The aqueduct is very nearly gone, but it has semicircular arches still visible on a wall in Barcelona.
The arches measure
2.96 meters
in diameter. What is the perimeter and area of each arch?
\mathbf{P} \mathbf{=} \mathbf{\pi r} \mathbf{+} \mathbf{2}\mathbf{r}
P = \pi \left(1.48 m\right) + 2.96 m
P = 4.649557 m + 2.96 m
P = 7.609557 m
Now, we find the area:
\mathbf{A} \mathbf{=} \frac{{\mathbf{\pi r}}^{\mathbf{2}}}{\mathbf{2}}
A = \frac{\pi \left(1.48{m}^{2}\right)}{2}
A = \frac{6.881344 {m}^{2}}{2}
A = 3.440672 {m}^{2}
The perimeter of a semicircle is half the original circle's circumference,
C
, plus the diameter,
d
. Since the semicircle includes a straight side, its diameter, we cannot describe the distance around the shape as the circumference of a semicircle; it is a perimeter.
Recall that the formula for the perimeter (circumference),
C
, of a circle of radius,
r
C = 2\pi r
C = \pi d
To find the perimeter,
P
, of a semicircle, you need half of the circle's circumference, plus the semicircle's diameter:
P = \frac{1}{2}\left(2\pi r\right) + d
\frac{1}{2}
2
cancel each other out, so you can simplify to get this perimeter of a semicircle formula.
P = \pi r + d
Using the substitution property of equality, you can also replace diameter with radius throughout:
P = \frac{1}{2}\left(2\pi r\right) + 2r
P = \pi r + 2r
Let's try an example. A semicircle that has a diameter of
100 meters
. What is the perimeter?
P = \frac{1}{2}\left(\pi d\right) + d
P = \frac{1}{2}\left(\pi × 100\right) + 100
P = \frac{1}{2}\left(314.159265\right) + 100
P = 157.079632 + 100
P = 257.08 meters
It is fine to round the decimal places as we did here.
Let's try an example using the radius of a semicircle. A semicircle has a radius of
365 inches
. What is its perimeter?
P = \pi r + 2r
P = \pi \left(365\right) + 2\left(365\right)
P = 1,146.681318 + 730
P = 1,876.68 inches
If the question asks you to convert your answer to units like feet or yards, convert it; otherwise leave it in the original linear units. Round your answer to whatever decimal value the problem requires.
The semicircles at both ends of an NBA basketball court indicate the restricted areas beneath each basket. The semicircles have four-foot radii. What is the perimeter of one semicircle in one restricted area?
P = \pi r + 2r
P = \pi \left(4\text{'}\right) + 2\left(4\text{'}\right)
P = 12.56637\text{'} + 8\text{'}
P = 20.56637\text{'}
In this case, having a measurement to 100,000ths of a foot is unnecessary;
20.57\text{'}
is a reasonably accurate answer.
The angle inscribed in a semicircle is always
90°
. The inscribed angle is formed by drawing a line from each end of the diameter to any point on the semicircle. It doesn't matter which point on the length of the arc, the angle created where your two lines meet the arc will always be
90°
The two endpoints of the semicircle's diameter and the inscribed angle will always form a right triangle inside the semicircle. |
(Redirected from UT1)
"UT1" redirects here. For other uses, see UT1 (disambiguation).
1.1 Adoption in various countries
{\displaystyle UT2=UT1+0.022\cdot \sin(2\pi t)-0.012\cdot \cos(2\pi t)-0.006\cdot \sin(4\pi t)+0.007\cdot \cos(4\pi t)\;{\mbox{seconds}}}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Universal_Time&oldid=1089504434#Versions" |
Find sources: "Mass flow meter" – news · newspapers · books · scholar · JSTOR (July 2017) (Learn how and when to remove this template message)
A mass flow meter of the Coriolis type
A mass flow meter, also known as an inertial flow meter, is a device that measures mass flow rate of a fluid traveling through a tube. The mass flow rate is the mass of the fluid traveling past a fixed point per unit time.
The mass flow meter does not measure the volume per unit time (e.g. cubic meters per second) passing through the device; it measures the mass per unit time (e.g. kilograms per second) flowing through the device. Volumetric flow rate is the mass flow rate divided by the fluid density. If the density is constant, then the relationship is simple. If the fluid has varying density, then the relationship is not simple. For example, the density of the fluid may change with temperature, pressure, or composition. The fluid may also be a combination of phases such as a fluid with entrained bubbles. Actual density can be determined due to dependency of sound velocity on the controlled liquid concentration.[1]
Operating principle of a Coriolis flow meter[edit]
Density and volume measurements[edit]
{\displaystyle Q_{m}={\frac {K_{u}-I_{u}\omega ^{2}}{2Kd^{2}}}\tau }
^ Naumchik I.V.; Kinzhagulov I.Yu.; Kren А.P.; Stepanova К.А. (2015). "Mass flow meter for liquids". Scientific and Technical Journal of Information Technologies, Mechanics and Optics. 15 (5): 900–906.
Lecture slides on flow measurement, University of Minnesota
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mass_flow_meter&oldid=1063909698" |
Figurate number - Wikipedia
3 Triangular numbers and their analogs in higher dimensions
Triangular numbers and their analogs in higher dimensions[edit]
These are the binomial coefficients
{\displaystyle \textstyle {\binom {n+1}{2}}}
. This is the case r = 2 of the fact that the rth diagonal of Pascal's triangle for r ≥ 0 consists of the figurate numbers for the r-dimensional analogs of triangles (r-dimensional simplices).
{\displaystyle P_{1}(n)={\frac {n}{1}}={\binom {n+0}{1}}={\binom {n}{1}}}
(linear numbers),
{\displaystyle P_{2}(n)={\frac {n(n+1)}{2}}={\binom {n+1}{2}}}
(triangular numbers),
{\displaystyle P_{3}(n)={\frac {n(n+1)(n+2)}{6}}={\binom {n+2}{3}}}
(tetrahedral numbers),
{\displaystyle P_{4}(n)={\frac {n(n+1)(n+2)(n+3)}{24}}={\binom {n+3}{4}}}
(pentachoric numbers, pentatopic numbers, 4-simplex numbers),
{\displaystyle \qquad \vdots }
{\displaystyle P_{r}(n)={\frac {n(n+1)(n+2)\cdots (n+r-1)}{r!}}={\binom {n+(r-1)}{r}}}
(r-topic numbers, r-simplex numbers).
Gnomon[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Figurate_number&oldid=1081911165" |
Why do cut vegetables take up a brown colouration when exposed in air - Science - Physical and Chemical Changes - 9744867 | Meritnation.com
Why do cut vegetables take up a brown colouration when exposed in air?
Vegetables and fruits contain an enzyme polyphenol oxidase in the cells which get exposed to air when fruits and vegetables are cut and left open. On exposure to air, an oxidation reaction takes place which turns the fruits and vegetables brown.
When fruits that contain large amount of iron eg. apple are cut and left open, then the iron present in them is oxidised to iron oxide by the atmospheric oxygen which turns the fruit brown.
4\mathrm{Fe} + 3{\mathrm{O}}_{2} \to 2{\mathrm{Fe}}_{2}{\mathrm{O}}_{3}\phantom{\rule{0ex}{0ex}} \left(\mathrm{brown} \mathrm{colour}\right) |
Waves/Waves in One Dimension - Wikibooks, open books for an open world
Waves/Waves in One Dimension
< Waves(Redirected from Waves/Waves in one Dimension)
A Wikibookian suggests that Waves/Sine Waves be merged into this chapter because:Not much difference in content
The Mathematics Of Waves[edit | edit source]
We start our discussion of waves by taking the equation for a very simple wave and describing its characteristics. The basic equation for such a wave is
{\displaystyle y=a\ \sin \left({\frac {2\pi x}{\lambda }}-2\pi ft+\alpha \right)}
{\displaystyle y}
is the height of the wave at position
{\displaystyle x}
{\displaystyle t}
. This equation describes a fairly simple wave, but most complex waves are just sums of simpler ones. If we freeze this equation in time at
{\displaystyle t=0}
{\displaystyle y=a\ \sin \left({\frac {2\pi x}{\lambda }}+\alpha \right)}
which looks like this: [TODO - Add a Graph]
From the graph we can see that each of the three parameters has a meaning.
{\displaystyle a}
is the amplitude of the wave, how high it is.
{\displaystyle \lambda }
is the wavelength, the distance from a part of the wave in one cycle to the same part of the wave in the next cycle.
{\displaystyle \alpha }
is the phase of the wave, which shifts the wave to the left or right. The wavelength is a distance, and is usually measured in meters, millimeters or even nanometers depending on the wave. Phase is an angle, measured in radians.
Now that we have mapped out the wave in space, let's instead set
{\displaystyle x=0}
and see how the wave changes over time
{\displaystyle y=a\ \sin(-2\pi ft+\alpha )}
{\displaystyle a}nd phase
{\displaystyle \alpha }
remain, but the wavelength is gone and a new quantity has appeared:
{\displaystyle f}
, which is the frequency, or how rapidly the wave moves up and down. Frequency is measured in units of inverse time: in a fixed period of time, how many times does the wave move up and down? The unit usually used for this is the hertz, or inverse second.
Now let's combine these two pictures and see how the wave moves. Figure 3 is a diagram of how the wave looks when you plot it in both space and time. The straight lines are the places where the simple wave reaches a maximum, minimum, or zero (where it crosses the x axis).
We can look at the zeros to determine the phase velocity of the wave. The phase velocity is how fast a part of the wave moves. We can think of it as the speed of the wave, but for more complicated waves it is only one type of speed - more on that in later sections.
We can get an equation for the zeros by setting our equation to zero.
{\displaystyle 0=a\ \sin \left({\frac {2\pi x}{\lambda }}-2\pi ft+\alpha \right)}
{\displaystyle 0={\frac {2\pi x}{\lambda }}-2\pi ft+\alpha }
{\displaystyle x=f\lambda t-{\frac {\alpha \lambda }{2\pi }}}
You see here that we have the equation for a straight line, describing a point that is moving at velocity
{\displaystyle f\lambda }
. This gives us the equation for the phase velocity of the wave, which is
{\displaystyle {\mbox{velocity}}={\mbox{frequency}}\times {\mbox{wavelength}}\quad v=f\lambda }
Retrieved from "https://en.wikibooks.org/w/index.php?title=Waves/Waves_in_One_Dimension&oldid=3224364" |
Revision as of 21:50, 25 January 2022 by TehPerson (talk | contribs) (→Pokémon GO)
{\displaystyle HP={\Biggl \lfloor }{{\Biggl (}(Base+DV)\times 2+{\biggl \lfloor }{\tfrac {{\bigl \lceil }{\sqrt {STATEXP}}{\bigr \rceil }}{4}}{\biggr \rfloor }{\Biggr )}\times Level \over 100}{\Biggr \rfloor }+Level+10}
{\displaystyle OtherStat={\Biggl \lfloor }{{\Biggl (}(Base+DV)\times 2+{\biggl \lfloor }{\tfrac {{\bigl \lceil }{\sqrt {STATEXP}}{\bigr \rceil }}{4}}{\biggr \rfloor }{\Biggr )}\times Level \over 100}{\Biggr \rfloor }+5}
{\displaystyle HP={\Bigl \lfloor }{(2\times Base+IV+\lfloor {\tfrac {EV}{4}}\rfloor )\times Level \over 100}{\Bigr \rfloor }+Level+10}
{\displaystyle OtherStat={\Biggl \lfloor }{\biggl (}{\Bigl \lfloor }{(2\times Base+IV+\lfloor {\tfrac {EV}{4}}\rfloor )\times Level \over 100}{\Bigr \rfloor }+5{\biggr )}\times Nature{\Biggr \rfloor }}
{\displaystyle Stat=(base+IV)\times cpMult}
{\displaystyle 2\times IV_{HP}+1}
{\displaystyle 2\times IV_{Attack}+1}
{\displaystyle 2\times IV_{Defense}+1} |
[[File:Poisoned Pokémon.png|thumb|250px|{{AP|Pikachu}} poisoned in the {{pkmn|anime}}]]
[[File:Bad Poison Effect.png|thumb|250px|{{AP|Leavanny}} badly poisoned in the anime]]
[[File:Bad Poison Effect.png|thumb|250px|{{p|Bruxish}} badly poisoned in the anime]]
'''Poison''' (Japanese: '''{{tt|毒|どく}}''' ''poison'') is a non-volatile [[status condition]] that causes a Pokémon to take damage over time. In the games, it is often abbreviated as PSN.
{\textstyle 15\times \left\lfloor {\tfrac {HP_{max}}{16}}\right\rfloor } |
The art club has made a scale drawing of the new mural they will paint on the wall of the school. The area of the actual mural is
36
times larger than the area of the drawing. What is the scale factor between the side length of the drawing and the side length of the mural?
Recall that area is always written in units
^{2}
while side lengths are always written in units.
This means that the ratio of of the two areas is
36
^{2}
To find the ratio of the sides, find the square root of the ratio of the two areas.
This is the ratio of the side lengths of the larger mural to that of the scale drawing. The scale factor is therefore
6
\sqrt{\text{36 units^2}} = 6\text{ units} |
Critically Sampled and Oversampled Wavelet Filter Banks - MATLAB & Simulink - MathWorks India
2-D Dual-Tree Wavelet Transforms
Dual-Tree Double-Density Wavelet Transforms
Wavelet filter banks are special cases of multirate filter banks called tree-structured filter banks. In a filter bank, two or more filters are applied to an input signal and the filter outputs are typically downsampled. The following figure illustrates two stages, or levels, of a critically sampled two-channel tree-structured analysis filter bank. The filters are depicted in the z domain.
The filter system functions,
{\stackrel{˜}{H}}_{0}\left(z\right)
{\stackrel{˜}{H}}_{1}\left(z\right)
, are typically designed to approximately partition the input signal, X, into disjoint subbands. In wavelet tree-structured filter banks, the filter
{\stackrel{˜}{H}}_{0}\left(z\right)
is a lowpass, or scaling, filter, with a non-zero frequency response on the interval [-π/2, π/2] radians/sample or [-1/4, 1/4] cycles/sample. The filter
{\stackrel{˜}{H}}_{1}\left(z\right)
is a highpass, or wavelet, filter, with a non-zero frequency response on the interval [-π, -π/2] ∪ [π/2, π] radians/sample or [-1/2, -1/4] ∪ [1/4, 1/2] cycles/sample. The filter bank iterates on the output of the lowpass analysis filter to obtain successive levels resulting into an approximate octave-band filtering of the input. The two analysis filters are not ideal, which results in aliasing that must be canceled by appropriately designed synthesis filters for perfect reconstruction. For an orthogonal filter bank, the union of the scaling filter and its even shifts and the wavelet filter and its even shifts forms an orthonormal basis for the space of square-summable sequences,
{\ell }^{2}\left(ℤ\right)
. The synthesis filters are the time-reverse and conjugates of the analysis filters. For biorthogonal filter banks, the synthesis filters and their even shifts form the reciprocal, or dual, basis to the analysis filters. With two analysis filters, downsampling the output of each analysis filter by two at each stage ensures that the total number of output samples equals the number of input samples. The case where the number of analysis filters is equal to the downsampling factor is referred to as critical sampling. An analysis filter bank where the number of channels is greater than the downsampling factor is an oversampled filter bank.
The following figure illustrates two levels of an oversampled analysis filter bank with three channels and a downsampling factor of two. The filters are depicted in the z domain.
Assume the filter
{\stackrel{˜}{H}}_{0}\left(z\right)
, is a lowpass half-band filter and the filters
{\stackrel{˜}{H}}_{1}\left(z\right)
{\stackrel{˜}{H}}_{2}\left(z\right)
are highpass half-band filters.
Assume the three filters together with the corresponding synthesis filters form a perfect reconstruction filter bank. If additionally,
{\stackrel{˜}{H}}_{1}\left(z\right)
{\stackrel{˜}{H}}_{2}\left(z\right)
generate wavelets that satisfy the following relation
{\psi }_{1}\left(t\right)={\psi }_{2}\left(t-1/2\right),
the filter bank implements the double-density wavelet transform. The preceding condition guarantees that the integer translates of one wavelet fall halfway between the integer translates of the second wavelet. In frame-theoretic terms, the double-density wavelet transform implements a tight frame expansion.
The following code illustrates the two wavelets used in the double-density wavelet transform.
wt1 = dddtree('ddt',x,5,df,df);
wt1.cfs{5}(5,1,1) = 1;
wav1 = idddtree(wt1);
plot(wav1); hold on;
plot(wav2,'r'); axis tight;
legend('\psi_1(t)','\psi_2(t)')
You cannot choose the two wavelet filters arbitrarily to implement the double-density wavelet transform. The three analysis and synthesis filters must satisfy the perfect reconstruction (PR) conditions. For three real-valued filters, the PR conditions are
\begin{array}{l}{H}_{0}\left(z\right){H}_{0}\left(1/z\right)+{H}_{1}\left(z\right){H}_{1}\left(1/z\right)+{H}_{2}\left(z\right){H}_{2}\left(1/z\right)=2\\ {H}_{0}\left(z\right){H}_{0}\left(-1/z\right)+{H}_{1}\left(z\right){H}_{1}\left(-1/z\right)+{H}_{2}\left(z\right){H}_{2}\left(-1/z\right)=0\end{array}
You can obtain wavelet analysis and synthesis frames for the double-density wavelet transform with 6 and 12 taps using dtfilters.
[df1,sf1] = dtfilters('filters1');
df1 and df2 are three-column matrices containing the analysis filters. The first column contains the scaling filter and columns two and three contain the wavelet filters. The corresponding synthesis filters are in sf1 and sf2.
See [4] and [5] for details on how to generate wavelet frames for the double-density wavelet transform.
The main advantages of the double-density wavelet transform over the critically sampled discrete wavelet transform are
Reduced shift sensitivity
Reduced rectangular artifacts in the 2-D transform
Smoother wavelets for a given number of vanishing moments
Increased computational costs
Non-orthogonal transform
Additionally, while exhibiting less shift sensitivity than the critically sampled DWT, the double-density DWT is not shift-invariant like the complex dual-tree wavelet transform. The double-density wavelet transform also lacks the directional selectivity of the oriented dual-tree wavelet transforms.
The critically sampled discrete wavelet transform (DWT) suffers from a lack of shift invariance in 1-D and directional sensitivity in N-D. You can mitigate these shortcomings by using approximately analytic wavelets. An analytic wavelet is defined as
{\psi }_{c}\left(t\right)={\psi }_{r}\left(t\right)+j{\psi }_{i}\left(t\right)
where j denotes the unit imaginary. The imaginary part of the wavelet, ψi(t), is the Hilbert transform of the real part, ψr(t). In the frequency domain, the analytic wavelet has support on only one half of the frequency axis. This means that the analytic wavelet ψc(t) has only one half the bandwidth of the real-valued wavelet ψr(t).
It is not possible to obtain exactly analytic wavelets generated by FIR filters. The Fourier transforms of compactly supported wavelets cannot vanish on any set of nonzero measure. This means that the Fourier transform cannot be zero on the negative frequency axis. Additionally, the efficient two-channel filter bank implementation of the DWT derives from the following perfect reconstruction condition for the scaling filter,
{H}_{0}\left({e}^{j\omega }\right)
, of a multiresolution analysis (MRA)
|{H}_{0}\left({e}^{j\omega }\right){|}^{2}+|{H}_{0}\left({e}^{j\left(\omega +\pi \right)}\right){|}^{2}=2.
If the wavelet associated with an MRA is analytic, the scaling function is also analytic. This implies that
{H}_{0}\left({e}^{j\omega }\right)=0\text{ }-\pi \le \omega <0,
|{H}_{0}\left({e}^{j\omega }\right){|}^{2}=2\text{ }0\le \omega \le \pi .
The result is that the scaling filter is allpass.
The preceding results demonstrate that you cannot find a compactly support wavelet determined by FIR filters that is exactly analytic. However, you can obtain wavelets that are approximately analytic by combining two tree-structured filter banks as long as the filters in the dual-tree transform are carefully constructed to satisfy certain conditions [1],[6].
The dual-tree complex wavelet transform is implemented with two separate two-channel FIR filter banks. The output of one filter bank is considered to be the real part, while the output of the other filter bank is the imaginary part. Because the dual-tree complex wavelet transform uses two critically sampled filter banks, the redundancy is 2d for a d-dimensional signal (image). There are a few critical considerations in implementing the dual-tree complex wavelet transform. For convenience, refer to the two trees as: Tree A and Tree B.
The analysis filters in the first stage of each filter bank must differ from the filters used at subsequent stages in both trees. It is not important which scaling and wavelet filters you use in the two trees for stage 1. You can use the same first stage scaling and wavelet filters in both trees.
The scaling filter in Tree B for stages ≥ 2 must approximate a 1/2 sample delay of the scaling filter in Tree A. The one-half sample delay condition is a necessary and sufficient condition for the corresponding Tree B wavelet to be the Hilbert transform of the Tree A wavelet.[3].
The following figure illustrates three stages of the analysis filter bank for the 1-D dual-tree complex wavelet transform. The FIR scaling filters for the two trees are denoted by
\left\{{h}_{0}\left(n\right),\text{ }\text{\hspace{0.17em}}{g}_{0}\left(n\right)\right\}
. The FIR wavelet filters for the two trees are denoted by
\left\{{h}_{1}\left(n\right),{g}_{1}\left(n\right)\right\}
. The two scaling filters are designed to approximately satisfy the half-sample delay condition
{g}_{0}\left(n\right)={h}_{0}\left(n-1/2\right)
The superscript (1) denotes that the first-stage filters must differ from the filters used in subsequent stages. You can use any valid scaling-wavelet filter pair for the first stage. The filters
\left\{{h}_{0}\left(n\right),\text{ }\text{\hspace{0.17em}}{g}_{0}\left(n\right)\right\}
cannot be arbitrary scaling filters and provide the benefits of using approximately analytic wavelets.
The dual-tree wavelet transform with approximately analytic wavelets offers substantial advantages over the separable 2-D DWT for image processing. The traditional separable 2-D DWT suffers from checkerboard artifacts due to symmetric frequency support of real-valued (non-analytic) scaling functions and wavelets. Additionally, the critically sampled separable 2-D DWT lacks shift invariance just as the 1-D critically sampled DWT does. The Wavelet Toolbox™ software supports two variants of the dual-tree 2-D wavelet transform, the real oriented dual-tree wavelet transform and the oriented 2-D dual-tree complex wavelet transform. Both are described in detail in [6].
The real oriented dual-tree transform consists of two separable (row and column filtering) wavelet filter banks operating in parallel. The complex oriented 2-D wavelet transform requires four separable wavelet filter banks and is therefore not technically a dual-tree transform. However, it is referred to as a dual-tree transform because it is the natural extension of the 1-D complex dual-tree transform. To implement the real oriented dual-tree wavelet transform, use the 'realdt' option in dddtree2. To implement the oriented complex dual-tree transform, use the 'cplxdt' option.
Both the real oriented and oriented complex dual-tree transforms are sensitive to directional features in an image. Only the oriented complex dual-tree transform is approximately shift invariant. Shift invariance is not a feature possessed by the real oriented dual-tree transform.
The dual-tree double-density wavelet transform combines the properties of the double-density wavelet transform and the dual-tree wavelet transform [2].
In 1-D, the dual-tree double-density wavelet transform consists of two three-channel filter banks. The two wavelets in each tree satisfy the conditions described in Double-Density Wavelet Transform. Specifically, the integer translates of one wavelet fall halfway between the integer translates of the second wavelet. Additionally, the wavelets in Tree B are the approximate Hilbert transform of the wavelets in Tree A. To implement the dual-tree double-density wavelet transform for 1-D signals, use the 'cplxdddt' option in dddtree. Similar to the dual-tree wavelet transform, the dual-tree double-density wavelet transform provides both real oriented and complex oriented wavelet transforms in 2-D. To obtain the real oriented dual-tree double-density wavelet transform, use the 'realdddt' option in dddtree2. To obtain the complex oriented dual-tree double-density wavelet transform, use the 'cplxdddt' option.
[1] Kingsbury, N.G. “Complex Wavelets for Shift Invariant Analysis and Filtering of Signals”. Journal of Applied and Computational Harmonic Analysis. Vol 10, Number 3, May 2001, pp. 234-253.
[2] Selesnick, I. “The Double-Density Dual-Tree Wavelet Transform”. IEEE® Transactions on Signal Processing. Vol. 52, Number 5, May 2004, pp. 1304–1314.
[3] Selesnick, I. “The Design of Approximate Hilbert Transform Pairs of Wavelet Bases.” IEEE Transactions on Signal Processing, Vol. 50, Number 5, pp. 1144–1152.
[4] Selesnick, I. “The Double Density DWT” Wavelets in Signal and Image Analysis: From Theory to Practice (A.A Petrosian, F.G. Meyer, eds.). Norwell, MA: Kluwer Academic Publishers:, 2001.
[5] Abdelnour, F. “Symmetric Wavelets Dyadic Siblings and Dual Frames” Signal Processing, Vol. 92, Number 5, 2012, pp. 1216–1225.
[6] Selesnick, I,. R.G Baraniuk, and N.G. Kingsbury. “The Dual-Tree Complex Wavelet Transform.” IEEE Signal Processing Magazine. Vol. 22, Number 6, November, 2005, pp. 123–151.
[7] Vetterli, M. “Wavelets, Approximation, and Compression”. IEEE Signal Processing Magazine, Vol. 18, Number 5, September, 2001, pp. 59–73.
dualtree | dualtree2 | dddtree | dddtree2 |
Estimated Earnings - LittleGhosts
We can simulate a rough overview for your estimated earnings based on your multiplier and your performance in game.
The earnings are based on your multiplier and how much money the play to earn pool generated the day before.
For this scenario we are going to assume the following things to estimate earnings per hour:
$1M 24 hour ECTO volume
$50k in 24 hour Microtransaction volume
You can complete 2 Dark Dungeon's in 1 hour.
100x xECTO NFT Multiplier (Worth 1 billion ECTO)
5% from all microtransactions goes into play to earn pools
5% from all ECTO volume goes into play to earn pools.
Assuming these numbers you would own 1/240 of the circulating supply. Using this formula we can estimate how much you would earn in one hour:
x= Your xECTO multiplier (ex 100)
v= 24hr ECTO USD volume (ex 1000000)
e= 24hr Microtransaction USD volume (ex 50000)
c= Circulating supply(excluding staking, locked, and burned tokens)
f(x)=(((.05v)+(.05e))/24)(x/(c/10000000))
You can estimate that with a 1000x multiplier and those simulated numbers you could earn ~$91 per hour, ~$2187 per day, or $798,386 per year from the play to earn pools. If you additionally factor in the 3% BUSD, you could earn an additional ~$1,248 per day or ~$455,520 per year from BUSD rewards.
You can estimate that with a 100x multiplier and those simulated numbers you could earn ~$9 per hour, ~$218 per day, or $79,570 per year from the play to earn pools.If you additionally factor in the 3% BUSD, you could earn an additional ~$120 per day or ~$43,800 per year from BUSD rewards.
You can estimate that with a 10x multiplier and those simulated numbers you could earn ~$1 per hour, ~$24 per day, or $8,760 per year from the play to earn pools.If you additionally factor in the 3% BUSD, you could earn an additional ~$12 per day or ~$4,380 per year from BUSD rewards.
Disclaimer: These are only estimates and it assumes we keep these numbers for the whole time it is estimated. Also, this does not including the amount you would be earning additionally if you are staking. |
Solutions of seepage equations in curvilinear coordinates | JVE Journals
K. A. Sidelnikov1 , A. M. Gubanov2 , V. E. Lyalin3 , M. A. Sharonov4
The article provides a method for solving equations of filtration in curvilinear coordinate systems through the application of the algorithm for constructing orthogonal difference grids on curved surfaces, allowing cost-effective solution of the quasi-three-dimensional non-stationary filtration problem with the possibility of application of these solvers to optimization problems.
Keywords: two-phase filtration, curvilinear coordinates, orthogonal grid, oil reservoir.
In the case of complex shapes study area is desirable to use of computational grids borders, adapted to current conditions. The use of grids with rectangular border approximation leads to considerable complication of calculation algorithms to implement the boundary conditions.
To build a grid on a curved surface, consider the geometry of a regular piece of the surface
S
defined by the equation in three-dimensional Euclidean space:
x=x\left(u,v\right),\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }y=y\left(u,v\right),\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }z=z\left(u,v\right),
u
v
some arbitrary coordinates. The orthogonal grid on the surface given by the equations
\xi =\xi \left(u,v\right)=const
\eta =\eta \left(u,v\right)=const
. For orthogonal lines on the surface of the relation:
\frac{\partial \eta }{\partial n}=\frac{\partial \xi }{\partial \tau } ,
n
\tau
– normal and tangent to the line.
\xi \left(u,v\right)
\eta \left(u,v\right)
must satisfy the system of equations [6]:
\frac{G{\eta }_{v}-F{\eta }_{u}}{W}={\xi }_{u},\mathrm{ }\mathrm{ }\mathrm{ }-\frac{E{\eta }_{u}-F{\eta }_{v}}{W}={\xi }_{v},
which is a generalization of the Cauchy-Riemann conditions.
To solve the system (3) differentiate the first equation by
v
, second by
u
, and add on them. As a result, we obtain a second differential operator:
{\nabla }^{2}\eta \equiv \frac{\partial }{\partial v}\left(\frac{G{\eta }_{v}-F{\eta }_{u}}{W}\right)+\frac{\partial }{\partial u}\left(\frac{E{\eta }_{u}-F{\eta }_{v}}{W}\right)=0,
E\left(u,v\right)={\left(\frac{\partial x}{\partial u}\right)}^{2}+{\left(\frac{\partial y}{\partial u}\right)}^{2}+{\left(\frac{\partial z}{\partial u}\right)}^{2}
G\left(u,v\right)={\left(\frac{\partial x}{\partial v}\right)}^{2}+{\left(\frac{\partial y}{\partial v}\right)}^{2}+{\left(\frac{\partial z}{\partial v}\right)}^{2}
F\left(u,v\right)=\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}+\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}+\frac{\partial z}{\partial u}\frac{\partial z}{\partial v}
– factors of fundamental quadratic form of the surface.
3. Algorithm and computational experiments
An algorithm for constructing orthogonal grid consists of consecutive solutions of the equation (4) and equation
{\nabla }^{2}\xi =0
with the appropriate boundary conditions. For variable
\eta
at one boundary set values
\eta =0
, on the other
\eta =1
, on the other borders
\frac{\partial \eta }{\partial n}=0
n
– normal to the boundary. From equation (2) are determined by the boundary conditions for the variable
\xi
{\nabla }^{2}\xi =0
is solved. Equations (1) and
\xi =\xi \left(u,v\right)=const
\eta =\eta \left(u,v\right)=const
define the position of the mesh nodes on the curved surface.
Equation (4) approximated the system of difference equations in the nine-pattern. The system of differential equations was solved by an iterative method of conjugate gradient regularization [1].
In the area shown in Fig. 1, a joint motion of the two phases: wetting and wetting liquids is considered. In this case, there are four unknown variables and hence four equations: two differential and two algebraic ones [2-4].
\nabla \left(h\left(\mathbf{r}\right){\mathrm{\lambda }}_{f}\left(\mathbf{r}\right)\left(\nabla {P}_{f}-{\gamma }_{f}\nabla Z\left(\mathbf{r}\right)\right)\right)=h\left(\mathbf{r}\right)\frac{\partial \left(\phi {S}_{f}/{B}_{f}\right)}{\partial t}-h\left(\mathbf{r}\right){Q}_{f}\left(t,\mathbf{r}\right),
{P}_{o}-{P}_{w}={P}_{c}, {S}_{w}+{S}_{o}=1,
f=\left\{w,o\right\}
{S}_{f}
f
{P}_{c}
– capillary pressure. It is believed that there is between
{S}_{f}
{P}_{c}
{\mathrm{\lambda }}_{f}\left(\mathbf{r}\right)={\mathbf{k}}_{f}\left(\mathbf{r}\right)/{\mu }_{f}
{\gamma }_{f}={\rho }_{f}g
\mathbf{r}\in \mathrm{\Omega }\subset {\stackrel{-}{\mathrm{\Omega }}}^{3}
{P}_{f}\left(\mathbf{r}\right)
\mathbf{r}\in \mathrm{\Omega }
\stackrel{-}{\mathrm{\Omega }}
\mathrm{\Gamma }
\mathbf{n}
{\mathbf{k}}_{f}
h
Z
– depth;
{\mu }_{f}
– liquid viscosity;
\rho {}_{f}{}^{\mathrm{ }}\mathrm{ }
– liquid density;
g
{Q}_{f}
\phi \left(\mathbf{r}\right)=\phi °\left(\mathbf{r}\right)\left(1+{c}_{\phi }\left(P-P°\right)\right)
\phi \left(\mathbf{r}\right)
{B}_{f}
- volumetric coefficient;
P
Fig. 1. Difference grid on curved surface
In curvilinear orthogonal coordinates
\left(\xi ,\eta \right)
, the divergence of the gradient operator is as follows:
\nabla \cdot \lambda \nabla P=\frac{1}{w}\left[\frac{\partial }{\partial \xi }\left(\frac{g\lambda {P}_{\xi }}{w}\right)+\frac{\partial }{\partial \eta }\left(\frac{e\lambda {P}_{\eta }}{w}\right)\right],
e={\left(\frac{\partial x}{\partial \eta }\right)}^{2}+{\left(\frac{\partial y}{\partial \eta }\right)}^{2}+{\left(\frac{\partial z}{\partial \eta }\right)}^{2}
g={\left(\frac{\partial x}{\partial \xi }\right)}^{2}+{\left(\frac{\partial y}{\partial \xi }\right)}^{2}+{\left(\frac{\partial z}{\partial \xi }\right)}^{2}
w=\sqrt{eg}
Then the difference approximation of two-dimensional filtering equation difference is a five-point pattern as in the Cartesian coordinates. The difference in rates is the difference in accounting factors
\frac{1}{w}
\frac{e}{w}
\frac{g}{w}
. Difference boundary conditions maintain their appearance. Therefore, the entire numerical algorithm, which is implemented into the Cartesian coordinates, is fully maintained.
Fig. 2. Isolines
S=const
In the following figures, the results of calculations for the reservoir, which has a curved shape, as in Figure 1 are shown. There is a problem of two-phase filtration. When
t=3.2
S=const
are shown (Figure 2), as well as the movement of water (Figure 3) and oil (Figure 4).
Fig. 3. Water flow
Fig. 6. Oil flow
The developed approach to the construction of the difference of the orthogonal curvilinear grid allows the calculation of filtration processes in formations with complex geometry. Accounting reservoir capacity with a relatively small thickness allows solving problems in a quasi-three-dimensional setting.
Golub J., Van Loan Charles Matrix Computations (3rd Edition). Johns Hopkins University Press, Baltimore, MD, USA, 1996, p. 728. [Search CrossRef]
Aziz H., Settarov E. Mathematical Modeling of Reservoir Systems. Institute of Computer Science, Moscow-Izhevsk, 2004, p. 416. Original Publication: “Nedra”, Moscow, 1982, (in Russian). [Search CrossRef]
Chanson, H. Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows. CRC Press, Taylor and Francis Group, Leiden, The Netherlands, 2009, p. 478. [Search CrossRef]
Economides Michael J., Nolte Kenneth G. Reservoir Stimulation. Wiley, USA, 2000, p. 856. [Search CrossRef] |
Support vector machine (SVM) for one-class and binary classification - MATLAB
f\left(x\right)=\left(x/s\right)\prime \beta +b.
\left\{\begin{array}{l}{\alpha }_{j}\left[{y}_{j}f\left({x}_{j}\right)-1+{\xi }_{j}\right]=0\\ {\xi }_{j}\left(C-{\alpha }_{j}\right)=0\end{array}
f\left({x}_{j}\right)=\varphi \left({x}_{j}\right)\prime \beta +b,
0.5\sum _{jk}{\alpha }_{j}{\alpha }_{k}G\left({x}_{j},{x}_{k}\right)
{\alpha }_{1},...,{\alpha }_{n}
\sum {\alpha }_{j}=n\nu
0\le {\alpha }_{j}\le 1
f\left(x\right)=x\prime \beta +b,
2/‖\beta ‖.
‖\beta ‖
0.5{‖\beta ‖}^{2}+C\sum {\xi }_{j}
{y}_{j}f\left({x}_{j}\right)\ge 1-{\xi }_{j}
{\xi }_{j}\ge 0
0.5\sum _{j=1}^{n}\sum _{k=1}^{n}{\alpha }_{j}{\alpha }_{k}{y}_{j}{y}_{k}{x}_{j}\prime {x}_{k}-\sum _{j=1}^{n}{\alpha }_{j}
\sum {\alpha }_{j}{y}_{j}=0
0\le {\alpha }_{j}\le C
\stackrel{^}{f}\left(x\right)=\sum _{j=1}^{n}{\stackrel{^}{\alpha }}_{j}{y}_{j}x\prime {x}_{j}+\stackrel{^}{b}.
\stackrel{^}{b}
{\stackrel{^}{\alpha }}_{j}
\stackrel{^}{\alpha }
\text{sign}\left(\stackrel{^}{f}\left(z\right)\right).
0.5\sum _{j=1}^{n}\sum _{k=1}^{n}{\alpha }_{j}{\alpha }_{k}{y}_{j}{y}_{k}G\left({x}_{j},{x}_{k}\right)-\sum _{j=1}^{n}{\alpha }_{j}
\sum {\alpha }_{j}{y}_{j}=0
0\le {\alpha }_{j}\le C
\stackrel{^}{f}\left(x\right)=\sum _{j=1}^{n}{\stackrel{^}{\alpha }}_{j}{y}_{j}G\left(x,{x}_{j}\right)+\stackrel{^}{b}.
{C}_{j}=n{C}_{0}{w}_{j}^{\ast },
{x}_{j}^{\ast }=\frac{{x}_{j}-{\mu }_{j}^{\ast }}{{\sigma }_{j}^{\ast }},
\begin{array}{c}{\mu }_{j}^{\ast }=\frac{1}{\sum _{k}{w}_{k}^{*}}\sum _{k}{w}_{k}^{*}{x}_{jk},\\ {\left({\sigma }_{j}^{\ast }\right)}^{2}=\frac{{v}_{1}}{{v}_{1}^{2}-{v}_{2}}\sum _{k}{w}_{k}^{*}{\left({x}_{jk}-{\mu }_{j}^{\ast }\right)}^{2},\\ {v}_{1}=\sum _{j}{w}_{j}^{*},\\ {v}_{2}=\sum _{j}{\left({w}_{j}^{*}\right)}^{2}.\end{array}
\sum _{j=1}^{n}{\alpha }_{j}=n\nu . |
Biotin carboxylase - Wikipedia
Biotin carboxylase homodimer, E.Coli
crystal structure of biotin carboxylase domain of acetyl-coenzyme a carboxylase from saccharomyces cerevisiae in complex with soraphen a
1dv1 / SCOPe / SUPFAM
In enzymology, a biotin carboxylase (EC 6.3.4.14) is an enzyme that catalyzes the chemical reaction
ATP + biotin-carboxyl-carrier protein + CO2
{\displaystyle \rightleftharpoons }
ADP + phosphate + carboxybiotin-carboxyl-carrier protein
The EC number for biotin carboxylase (6.3.4.14) indicates the enzymes class, subclass, sub-subclass, and serial number. A class number of 6 indicates the enzyme belongs to the ligases family of enzymes, as this enzyme is responsible for catalyzing a bond formation—namely between biotin and CO2. The subclass number for ligases are used to identify the type of bond being formed during catalysis; sub-subclass 3 is responsible for C-N bond-forming reactions. The sub-subclass integer value of 4 indicates that this particular ligase belongs to the 'other C-N ligases'--the other sub-subclass values identify the ligation product i.e. amide synthases, peptide synthases, cyclo-ligases, and ligases that use glutamine as the nitrogen donor. As the bond being formed in this reaction occurs between the carbon from CO2 and the nitrogen from the amide in biotin, it cannot be classified under any of the other sub-subclasses. The final integer, known as the serial number, is the number used to identify the specific enzyme in that sub-subclass, in this case the value being 14.[1]
The three substrates of this enzyme are ATP, biotin-carboxyl-carrier protein (BCCP), and CO2, whereas its three products are ADP, phosphate, and carboxybiotin-carboxyl-carrier protein.
The systematic name of this enzyme class is biotin-carboxyl-carrier-protein:carbon-dioxide ligase (ADP-forming). This enzyme is also called biotin carboxylase (component of acetyl CoA carboxylase). This enzyme participates in fatty acid biosynthesis. This enzyme participates in fatty acid biosynthesis by providing one of the catalytic functions of the Acetyl-CoA carboxylase complex. As previously mentioned, after the carboxybiotin product is formed, the carboxyltransferase unit of the complex will transfer the activated carboxy group from BCCP to Acetyl-CoA, forming a malonate analog known as malonyl-CoA. Malonyl-CoA serves as the primary carbon donor in fatty acid biosynthesis, followed by a series of reduction and dehydration reactions to remove the acyl group.[2]
Biotin carboxylases are a conserved enzyme present within biotin-dependent carboxylase complexes such as acetyl-CoA carboxylase. How biotin carboxylase functions is, within the relevant carboxylase complex, there is a biotin carboxyl-carrier protein which is covalently linked to biotin via a Lys-residue.[3] Both biotin carboxylase activity as well as the BCCP within the carboxylase complex are highly conserved among this enzyme class. The main source of variation for carboxylases arises from the carboxyltransferase component, as the molecule to which the carboxyl group is transferred (from biotin) dictates the necessary specificity component to catalyze this transfer.
The structure of biotin carboxylase heavily influences the reaction pathway the enzyme catalyzes, so discussion of this reaction pathway must also touch on how the substrates and intermediates are stabilized within the active site. Bicarbonate (HCO3−) is held within the active site of biotin carboxylase by hydrogen bonding with biotin as well as a bidentate ion pair interaction of the negatively charged oxygen's with Arg292 iminium ion.[3] It is hypothesized that the Glu296 residue of B.C. acts as a base, deprotonating bicarbonate molecule, thus facilitating nucleophilic attack of the carbonyl-oxygen on the terminal phosphate molecule of ATP. This initial reaction of the pathway can happen because the ATP is also held tightly within the active site pocket via non-covalent coordination of ATP with magnesium ions.
After this nucleophilic attack, the carbonate molecule is degraded to CO2 via electron pushing, producing a PO43- ion which then acts as a base and deprotonates the amide of the ureido ring within biotin. An enolate-like intermediate is formed, producing a negative charge on the oxygen, which is stabilized by the iminium ion of Arg338. The enolate then executes a nucleophilic attack on CO2 (which is being held in place through H-bonding with Glu296 residue), ultimately leading to the product of this enzymatic pathway: carboxybiotin.[3] After this reaction occurs, the carboxyltransferase enzyme present within the complex acts upon the carboxybiotin to transfer the carboxyl group to the target acceptor molecule i.e. acetyl Co-A, propionyl Co-A etc.
As of late 2007[update], 5 structures have been solved for this class of enzymes, with PDB accession codes 1BNC, 1DV1, 1DV2, 2GPS, and 2GPW.
The crystal structure has been determined for the biotin carboxylase (acetyl-CoA carboxylase) of Escherichia coli, but the eukaryotic B.C. is difficult to obtain info on as it is catalytically inactive in solution. E. coli biotin carboxylase is composed of two homogenous dimers made up of 3 domains: A, B, and C.[3] It is believed that the B domain of each monomer is essential to the function of this enzyme, as there is extreme flexibility of this domain seen in the crystal structure. Upon binding of the ATP substrate, a conformational change occurs where the B domain essentially closes over the active site. While this change is thought to bring ATP within close enough proximity for the reaction to occur, the active site was still solvent exposed. Because of this anomaly in the crystal structure, it is believed that the attachment of biotin to BCCP aids in this reaction pathway, essentially covering biotin within the active site, as evidence shows free biotin is not as great of a substrate for this enzyme when compared to biotin-BCCP.[4] A C-terminal conserved domain within this enzyme contains most of the active site residues.[5] The Glu296 and Arg338 are highly conserved residues among this subclass of enzymes, and work to stabilize the reaction intermediates and keep them within the active site pocket until the carboxylation is complete.[4]
This enzyme is vital to life and has maintained its function across a variety of organisms. While the structure itself may be divergent based on the biotin carboxylase function and which complex it is present in, the enzyme still works to serve the same function. Fatty acid synthesis provides sterols and other lipids essential to biochemical pathways, and the necessity for this enzyme function is confirmed by the highly conserved active site amino acid sequence.[3]
^ McDonald, Andrew (March 2019). "Class 6--Ligases" (PDF). Enzyme Database. Archived (PDF) from the original on 2009-01-06. Retrieved 6 October 2021.
^ Engelking, Larry R. (2015-01-01), Engelking, Larry R. (ed.), "Chapter 56 - Fatty Acid Biosynthesis", Textbook of Veterinary Physiological Chemistry (Third Edition), Boston: Academic Press, pp. 358–364, ISBN 978-0-12-391909-0, retrieved 2021-10-18
^ a b c d e Chou, Chi-Yuan; Yu, Linda P. C.; Tong, Liang (2009-04-24). "Crystal Structure of Biotin Carboxylase in Complex with Substrates and Implications for Its Catalytic Mechanism*". Journal of Biological Chemistry. 284 (17): 11690–11697. doi:10.1074/jbc.M805783200. ISSN 0021-9258. PMC 2670172.
^ a b Attwood, A (2002). "Chemical and catalytic mechanisms of carboxyl transfer reactions in biotin-dependent enzymes". Accounts of Chemical Research. 35: 113–120 – via Expasy.
^ Waldrop, G. L.; Rayment, I.; Holden, H. M. (1994). "Three-dimensional structure of the biotin carboxylase subunit of acetyl-CoA carboxylase". Biochemistry. 33 (34): 10249–10256. doi:10.1021/bi00200a004. PMID 7915138.
Dimroth P, Guchhait RB, Stoll E, Lane MD (1970). "Enzymatic carboxylation of biotin: molecular and catalytic properties of a component enzyme of acetyl CoA carboxylase". Proc. Natl. Acad. Sci. U.S.A. 67 (3): 1353–60. Bibcode:1970PNAS...67.1353D. doi:10.1073/pnas.67.3.1353. PMC 283359. PMID 4922289.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Biotin_carboxylase&oldid=1082972319" |
Assume that when adults with smartphones are randomly selected, 54%
Assume that when adults with smartphones are randomly selected, 54% use them in meetings or classes (based on data from an LG Smartphone survey).
If 12 adult smartphone users are randomly selected, find the probability that fewer than 3 of them use their smartphones in meetings or classes.
Probability of using smartphone in meetings or classes
=0.54
Number of selecting adult smartphone
=12
We have to find probability of selecting fewer than 3 of them use their smartphones in meetings or classes.
We can use binomial probability to find required probability.
P\left(X=x\right)=\frac{n!}{x!\left(n-x\right)!}×{p}^{x}×{\left(1-p\right)}^{n-x}
P\left(X<3\right)=P\left(X=2\right)+P\left(X=1\right)
P\left(X<3\right)=
P\left(X=2\right)=\frac{2!}{2!\left(12-2\right)!}\right\}×{0.54}^{2}×{\left(1-0.54\right)}^{12-2}
+P\left(X=1\right)=\frac{1!}{1!\left(12-1\right)!}×{0.54}^{1}×{\left(1-0.54\right)}^{12-1}
P\left(X<3\right)=0.0082+0.0013=0.0094
n=5
times. Use some form of technology to find the cumulative probability distribution given the probability
p=0.16
\begin{array}{|cc|}\hline k& P\left(X=k\right)\\ 0& \\ 1& \\ 2& \\ 3& \\ 4& \\ 5& \\ \hline\end{array}
What did you learn from binomial and Hyper- geometric distributions? Write a brief note of five lines on these distributions.
75% of all bald eagles survive their first year of life. Give your answers as decimals, not percents. If 30 bald eagles are randomly selected, find the probability that.
Between 20 and 26 (including 20 and 26) of them survive their first year of life. |
Revision as of 16:36, 17 January 2020 by Munich (talk | contribs) (→Horizontal and vertical profiles of the velocity components and Reynolds stresses)
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle \langle u\rangle }
{\displaystyle \langle w\rangle }
{\displaystyle \langle u'_{i}u'_{j}\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle ||{\vec {U}}||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {PIV} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {LES} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle -0.788}
{\displaystyle 0.03}
{\displaystyle -0.843}
{\displaystyle 0.037}
{\displaystyle -0.918}
{\displaystyle 0}
{\displaystyle -1.1}
{\displaystyle 0}
{\displaystyle -0.533}
{\displaystyle 0}
{\displaystyle -0.534}
{\displaystyle 0}
{\displaystyle -0.507}
{\displaystyle 0.036}
{\displaystyle -0.50}
{\displaystyle 0.04}
{\displaystyle -0.697}
{\displaystyle 0.051}
{\displaystyle -0.735}
{\displaystyle 0.06}
{\displaystyle -0.513}
{\displaystyle 0.017}
{\displaystyle -0.513}
{\displaystyle 0.02}
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }={\frac {x-x_{\mathrm {Cyl} }}{x_{\mathrm {Cyl} }-x_{\mathrm {V1} }}}}
{\displaystyle x_{\mathrm {Cyl} }=-0.5D}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {V1} }}
{\displaystyle \langle u(z)\rangle /u_{\mathrm {b} }}
{\displaystyle u(z)}
{\displaystyle x_{\mathrm {adj} }=-0.25}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle \langle u'u'(z)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle u'w'(z)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(z)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle u'w'\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle w'w'\rangle }
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u'w'\rangle }
{\displaystyle \langle w(x)\rangle /u_{\mathrm {b} }}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle w(x)\rangle }
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }\approx -0.1}
{\displaystyle x_{\mathrm {adj} }=-0.65}
{\displaystyle \langle u_{i}'u_{j}'(x)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(x)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u_{i}'u_{j}'\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.074u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.079u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.09u_{\mathrm {b} }^{2}}
{\displaystyle 0=P+\nabla T-\epsilon +C}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle C}
{\displaystyle v}
{\displaystyle P=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}}
{\displaystyle T=\underbrace {-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle } _{\text{turbulent fluctuations}}\underbrace {-{\frac {1}{\rho }}\langle u_{i}'p'\rangle } _{\text{pressure transport}}\underbrace {+2\nu \langle u_{j}'s_{ij}\rangle } _{\text{viscous diffusion}}}
{\displaystyle \epsilon =2\nu \langle s_{ij}s_{ij}\rangle }
{\displaystyle s_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}'}{\partial x_{j}}}+{\frac {\partial u_{j}'}{\partial x_{i}}}\right)}
{\displaystyle \epsilon _{\mathrm {total} }=\epsilon _{\mathrm {res} }+\epsilon _{\mathrm {SGS} }=2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle }
{\displaystyle C=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}}
{\displaystyle D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {PIV} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {LES} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle 0.3u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {LES} }\approx 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {PIV} }\approx 0.2u_{\mathrm {b} }^{3}/D}
{\displaystyle x=-0.7D}
{\displaystyle P}
{\displaystyle \nabla T_{\mathrm {turb,PIV} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb,LES} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x=-0.75D}
{\displaystyle 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb,LES} }\approx 0.35u_{\mathrm {b} }^{3}/D}
{\displaystyle \nabla T_{\mathrm {press,LES} }=-{\frac {1}{\rho }}{\frac {\partial \langle u_{i}'p'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {visc,LES} }=2\nu {\frac {\partial \langle u_{j}'s_{ij}\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb} }}
{\displaystyle \nabla T_{\mathrm {press} }}
{\displaystyle \langle w\rangle <0}
{\displaystyle w-}
{\displaystyle w'}
{\displaystyle p'<0}
{\displaystyle \nabla T_{\mathrm {visc} }}
{\displaystyle |0.05|u_{\mathrm {b} }^{3}/D}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle \epsilon _{\mathrm {PIV} }=2\nu \langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \epsilon _{\mathrm {LES,total} }=(2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle )\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P}
{\displaystyle \epsilon _{\mathrm {LES} }=0.066u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {max} }}
{\displaystyle \epsilon _{\mathrm {max} }}
{\displaystyle C_{\mathrm {PIV} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle C_{\mathrm {LES} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x\approx -0.63D}
{\displaystyle C}
{\displaystyle R_{\mathrm {PIV} }=P+\nabla T_{\mathrm {turb} }-\epsilon +C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle -\nabla T_{\mathrm {press,LES} }}
{\displaystyle R_{\mathrm {LES} }=P+\nabla T-\epsilon _{\mathrm {total} }+C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle <|0.01|u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb} }=-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }={\frac {\langle p\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }={\frac {\langle \tau _{\mathrm {w} }\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle z_{1}\approx 0.0036D\approx 10\mathrm {px} }
{\displaystyle z_{1}\approx 0.0005D}
{\displaystyle z-}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle |c_{\mathrm {f} }|=0.01}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle 50\times 171(n\times m)}
{\displaystyle 143\times 131(n\times m)}
{\displaystyle n\cdot m}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle -}
{\displaystyle \epsilon {\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle v\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {press} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {visc} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \epsilon _{\mathrm {total} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle c_{\mathrm {p} }} |
Which of the following are Arrhenius acids? (a) H_{2}O, (b)
\left(a\right){H}_{2}O,\left(b\right)Ca{\left(OH\right)}_{2},\left(c\right){H}_{3}P{O}_{4},\left(d\right)HI
a\right){H}_{2}O
{H}_{2}O⇒{H}^{+}+O{H}^{t}
{H}_{2}O
can be both an Arrhenius acid and base. It`s self-ionization both H^+ and OH^- ions.
b\right)Ca{\left(OH\right)}_{2}
Ca\left(OH{\right)}_{2}⇒C{a}^{2}+2O{H}^{t}
Ca{\left(OH\right)}_{2}
is not an Arrhenius acid. It is n Arrahenius bases since it produced
O{H}^{-}ion
c\right){H}_{3}P{O}_{4}
{H}_{3}P{O}_{4}⇒{H}_{3}P{O}_{4}^{-}+{H}^{+}
{H}_{3}P{O}_{4}
is an Arrahenius acid because it produced
{H}^{+}
ion.
d\right)HI
HI⇒{H}^{+}+{I}^{t}
HI
is an Arrhenius acid because it produced
{H}^{+}
We get result where
a\right){H}_{2}O,c\right){H}_{3}P{O}_{4},4\right)HI
are Arrhenius acids.
Water stands at a depth H in a large, open tank whose sidewalls are vertical. A hole is made in one of the walls at a depth h below the water surface.
a) at what distance R from the foot of the wall does the emerging stream strike the floor?
b) How far above the bottom of the tank could a second hole be cut so that the stream emerging from it could have the same range as for the first hole?
A point charge of magnitude qq is at the center of a cube with sides of length L. What is the electric flux
\mathrm{\Phi }
through each of the six faces of the cube? What would be the flux
{\mathrm{\Phi }}_{1}
through a face of the cube if its sides were of length
{L}_{1}
How many ways are there for a horse race with four horses to finish if ties are possible? [Note: Any number of the four horses may tie.)
The average thermal conductivity of the walls (includingwindows) and roof of a house in Figure P11.32 is 4.8
10-4kW/{m}^{\circ }C
,and their average thickness is 20.0cm. The house is heated with natural gas, with a heat of combustion(energy given off per cubic meter of gas burned) of 9300kcal/m3. How many cubic meters of gas must be burnedeach day to maintain an inside temperature of 28.0°C if the outside temperature is
{0.0}^{\circ }C
? Disregard radiation and loss by heat through theground.
\frac{{m}^{3}}{day} |
Model gain and phase uncertainty - MATLAB - MathWorks í•œêµ
Normalized uncertainty level, specified as a positive scalar. This value is the parameter É‘ that sets the size of the uncertainty disk (see Algorithms).
[0×1 string] (default) | string | cell array of character vector
Create a model of a SISO control loop, with gain uncertainty of ±6 dB and phase uncertainty of ±30°. Use open-loop transfer function
\mathit{L}=\frac{3.5}{{\mathit{s}}^{3}+2{\mathit{s}}^{2}+3\mathit{s}}
DGM = 1×2
Uncertain gain/phase "F" with relative gain change in [0.501,2] and phase change of ±36.8 degrees.
F represents the smallest uncertainty disk that can capture both the target gain and phase variation. The actual phase variation modeled by F is a little bigger than the target range of ±30°. To visualize the full range of gain and phase variations represented by F, including simultaneous gain and phase variations, use plot.
F: Uncertain gain/phase, gain × [0.501,2], phase ± 36.8 deg, 1 occurrences
Create a umargin block that models gain that can decrease by 10% but increase by 60% in the absence of phase variation, and a phase variation of ±15° in the absence of gain variation. To do so, use getDGM with the 'tight' option. This option finds the smallest disk that captures the gain and phase ranges you provide.
Uncertain gain/phase "F" with relative gain change in [0.86,1.6] and phase change of ±15 degrees.
Uncertain gain/phase "u1" with relative gain change in [0.952,1.05] and phase change of ±2.79 degrees.
umargin converts the specified gain variation of ±5% to a disk-based uncertainty model, which also allows phase changes of about ±3°. Use plot to visualize the disk and the modeled range of gain and phase variation at each input and output.
u1: Uncertain gain/phase, gain × [0.952,1.05], phase ± 2.79 deg, 1 occurrences
y1: Uncertain gain/phase, gain × [0.952,1.05], phase ± 2.79 deg, 1 occurrences
Suppose that you also require the system to tolerate gain increase or decrease of up to 50% and phase variation of up to ±20° at the plant input. To check whether the system has these margins, create a umargin block that models these variations and insert it into the closed-loop model.
Consider a umargin block that captures a gain variation of a factor of 1.5 in either direction, and a phase variation of ±20°.
Uncertain gain/phase "F" with relative gain change in [0.667,1.5] and phase change of ±22.6 degrees.
When you create the umargin object, you provide the target disk-based gain variation DGM, which defines a disk of uncertainty. This value becomes the property F.GainChange, and umargin sets other properties based on this value. For instance, F.PhaseChange is automatically set to the phase range corresponding to the uncertainty disk defined by DGM. The properties F.DiskMargin and F.Skew are automatically set to the corresponding α and σ values of the disk (see Algorithms). Examine the values of these properties.
Uncertain gain/phase "F" with relative gain change in [0.818,1.22] and phase change of ±11.4 degrees.
If you manually change any of the properties that relate to the size of the modeled uncertainty, the block automatically updates the values of all of them. For instance, set F.GainChange to a new value that defines a different uncertainty disk. The properties of F are automatically updated to reflect the new corresponding phase range (F.PhaseChange) and the values of α and σ (F.DiskMargin and F.Skew) that describe the new disk.
Here, F.PhaseChange, F.DiskMargin, and F.Skew are all updated to describe the disk defined by the new value of F.GainChange. The new value of F.Skew is non-zero, because the new gain range is not symmetric (gmin ≠1/gmax).
F\left(s\right)=\frac{1+\mathrm{α}\left[\left(1â\mathrm{Ï}\right)/2\right]\mathrm{δ}\left(s\right)}{1â\mathrm{α}\left[\left(1+\mathrm{Ï}\right)/2\right]\mathrm{δ}\left(s\right)}.
δ(s) is a gain-bounded dynamic uncertainty, normalized so that it always varies within the unit disk (||δ||∞ < 1).
ɑ sets the amount of gain and phase variation modeled by F. For fixed σ, the parameter ɑ controls the size of the disk. For ɑ = 0, the multiplicative factor is 1, corresponding to the nominal L.
σ, called the skew, biases the modeled uncertainty toward gain increase or gain decrease.
The factor F takes values in a disk centered on the real axis and containing the nominal value F = 1. The disk is characterized by its intercept DGM = [gmin,gmax] with the real axis. gmin < 1 and gmin > 1 are the minimum and maximum relative changes in gain modeled by F, at nominal phase. The phase uncertainty modeled by F is the range DPM = [pmin,pmax] of phase values at the nominal gain (|F| = 1). For instance, in the following plot, the right side shows the disk F that intersects the real axis in the interval [0.71,1.4]. The left side shows that this disk models a gain variation of ±3 dB and a phase variation of ±19°. |
Consider the following polynomials in P_3:q(x)=-x^3+3x^2-x+5, r(x)=-4x^3+7x^2-x+10, u(x)=-5x^3+8x^2+10
{P}_{3}
q\left(x\right)=-{x}^{3}+3{x}^{2}-x+5
r\left(x\right)=-4{x}^{3}+7{x}^{2}-x+10
u\left(x\right)=-5{x}^{3}+8{x}^{2}+10
{p}_{1}\left(x\right)=-3{x}^{2}-3x+10
okomgcae
q\left(x\right)=-{x}^{3}+3{x}^{2}-x+5,r\left(x\right)=-4{x}^{3}+7{x}^{2}-x+10
u\left(x\right)=-5{x}^{3}+8{x}^{2}
Now, a polynomial p(x) is in span of {q,r,u} if there exist nonzero constants a,b and c such that
p\left(x\right)=aq+br+cu
{p}_{1}\left(x\right)=-3{x}^{2}-3x+10
{p}_{1}=aq+br+cu
-3{x}^{2}-3x+10=a\left(-{x}^{3}+3{x}^{2}-x+5\right)+b\left(-4{x}^{3}+7{x}^{2}-x+10\right)+c\left(-5{x}^{3}+8{x}^{2}+10\right)-3{x}^{2}-3x+10
=\left(a-4b-5c\right){x}^{3}+\left(3a+7b+8c\right){x}^{2}+\left(-a-b\right)x+5a+10b+10c
Equate the coefficients of both sides and find the values of a, b and c.
a-4b-5c=0
3a+7b+8c=-3
-a-b=-3
5a+10b+10c=10
a=-3+b
a=3-b
b=3-a
in equation (1) and equation (2) and (3)
-\left(3-b\right)-4b-5c=0
3\left(3-b\right)+7b+8c=-3
5\left(3-b\right)+10b+10c=10
-3+b-4b-5c=0
-3-3b-5c=0
-3b=5c+3
b=-\frac{5c+3}{3}
b=-\frac{5c+3}{3}
and in equation (5) and (6)
-9+4\left(-\frac{5c+3}{3}\right)+8c=-3
6{c}^{2}+2{c}^{4}-c
{x}^{4}–2{x}^{2}–3
2{a}^{2}-6a-4
-4{a}^{2}+6a+10
\frac{1}{{x}^{3}}+\frac{1}{{y}^{3}} |
Suppose that you are taking a multiple-choice exam with five
Suppose that you are taking a multiple-choice exam with five questions, each hav
Suppose that you are taking a multiple-choice exam with five questions, each have five choices, and one of them is correct. Because you have no more time left, you cannot read the question and you decide to select your choices at random for each question. Assuming this is a binomial experiment, calculate the binomial probability of obtaining exactly one correct answer.
Obtain the binomial probability of obtaining exactly one correct answer.
The binomial probability of obtaining exactly one correct answer is obtained below as follows:
Let X denotes the number of correct answers which follows binomial distribution with the probability of success 1/5 with the number of questions randomly selected is 5.
That is, \(\displaystyle{n}={5},{p}={\frac{{{1}}}{{{5}}}},{q}={\frac{{{4}}}{{{5}}}}\)
The probibility distribution is given by,
\(P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right) p^{x}(1-p)^{n-x}; here\ x=0,1,2, \cdots, n\ for\ 0 \leq p \leq 1\)
Where n is the number of trials and p is the probability of success for each trial.
Use Excel to obtain the probability value for x equals 1.
Follow the instruction to obtain the P-value:
2. Go to Formula bar.
3. In formula bar enter the function as“=BINOMDIST”
4. Enter the number of success as 1
5. Enter the Trails as 5
6. Enter the probability as 0.20
7. Enter the cumulative as False
From the Excel output, the P-value is 0.4096.
The binomial probability of obtaining exactly one correct answer is 0.4096.
P\left(x\right)
n=4,\text{ }x=1,\text{ }p=0.6
Assume that a procedure yields a binomial distribution with a trial repeated
n=5n=5
times. Use some form of technology to find the probability distribution given the probability
p=0.6p=0.6
(Report answers accurate to 2 decimal places.)
\text{Misplaced &}
A new experimental process for manufacturing a circuit for quantum computers is successful 80% of the time. Researchers want to calculate the probability that more than 15 of the next 20 attempts to make the circuit will be successful. Can they use the normal approximation to the binomial? Why or why not?
P\left(X=14\right),n=16,p=0.8
Please, convert the binomial probability to a normal distribution probability using continuity correction. P (x > 65).
A hand of 5 cards is dealt to each of three players from a standard deck of 52 cards.
(a)What is the probability that one of the players receives all four Aces?
(b)What is the probability that at least one player receives no hearts? |
Describe the transformations that must be applied to y=x^2 to create the graph of each of
Describe the transformations that must be applied to y=x^2 to create the graph of each of the following functions. a) y=1/4(x-3)^2+9b) y=((1/2)x)^2-7
y={x}^{2}
y=\frac{1}{4}{\left(x-3\right)}^{2}+9
y={\left(\left(\frac{1}{2}\right)x\right)}^{2}-7
To create the graph of the function a)
y=\frac{1}{4}{\left(x-3\right)}^{2}+9
We have (by the function)
1. vertical compression by
\frac{1}{4}
2. horizontal translation 3 units to the right
3. vertical translation 9 units up
y={\left(\left(\frac{1}{2}\right)x\right)}^{2}-7={\left(\left(\frac{1}{4}\right)x\right)}^{2}-7
\frac{1}{4}
2. vertical translation 7 units down
f\left(x\right)=1+\sqrt{x}
f\left(x\right)=\mathrm{ln}x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}g⟨x\right)=-\mathrm{ln}\left(2x\right) |
A teacher has taught measurement of area to class VIII children, but many of her students are confused between the usage of different units of area and volume. What could be the reason for such a confusion in children? from Class 12 TET Previous Year Board Papers | Mathematics 2018 Solved Board Papers
While teaching 'shapes', a teacher can plan a trip to historical places, as
A. it needs to provide leisure time as most of the syllabus has been completed in time.
B. it would be an opportunity to improve communication skill.
C. shapes are an integral part of every architecture and such trips encourage connections across disciplines.
D. Field trips are recommended by Education Board, so must be organised.
A student was asked to calculate the surface area of a cube. He calculated the volume.
The reason (s) of error in calculation is/are
A. the student finds the class boring as he does not like Mathematics class
B. the student is not fit to study in that class
C. the student is not able to understand the concept of surface area and volume
D. The student has understood the concept of surface area and volume
The student is not able to understand the concept of surface area and volume. So, the student finds volume besides of area.
In a meeting,"
\frac{4}{25}
of the members were female. What percent of the members was this?
Let the total number of members in meeting =
x
Then, the total number of females in meeting=
\frac{4x}{25}
\frac{Number\quad of\quad females}{Total\quad number\quad of\quad members\quad }\times 100
\Rightarrow \quad \quad \quad \quad \quad \frac{{\displaystyle \frac{4x}{25}}}{x}\times 100\%\quad =\quad 16
"Errors play a crucial role in learning of Mathematics." This statement is
True, because errors reflect the thinking of child
false, because mathematics is exact
true, because errors provide feedback about the marks they obtained
false, because errors occur due to carelessness
The errors are the windows of a child's thinking, the error they make represent the way they are thinking with respect to an idea.
Which one of the following can be the most appropriate aim of encouraging mathematical communication in classroom?
Children who have fear about mathematics should be able to interact in the class
To organise debates in the class regarding topics of mathematics.
Children should be able to recite theorems and formulas in mathematics class.
Children should be able to use a precise language while talking about mathematical statements and using them.
A shop reduced its prices by 10%. What is the new price of an item which was previously sold for
₹
₹
₹
₹
₹
₹
Given, the original price of an item=
₹500
and reduced percentage =
10%
\therefore
New price item=
original\quad price\times \frac{\left(100-reduced\quad percentage\right)}{100}
500\times \frac{(100-10}{100}
500\times \frac{90}{100}
₹450
Remedial teaching is helpful for
recapitulating the lesson
teaching in play-way method
removing learning difficulties of weak students
Which one of the following is not a mathematical process?
A teacher has taught measurement of area to class VIII children, but many of her students are confused between the usage of different units of area and volume. What could be the reason for such a confusion in children?
The children have not memorised different units
Different units have been introduced all together without relating them with daily life
The concept of measurement of area is a difficult topic for a class VIII learner
The children did not know the use of units for area
The purpose of a diagnostic test in Mathematics is
to give feedback to the parents
to fill the progress report
to plant the question paper for the end term examination
to know the gaps in children's understanding
The assessment is done in order to know about the gaps the learners have in their knowledge structure. Based on that assessment tests the teacher gets an understanding that how much further help the learners require.
https://www.zigya.com/share/VE1BRU5UVDEyMTkwNTc2 |
Vacuum airship - Wikipedia
Hypothetical airship concept
A vacuum airship, also known as a vacuum balloon, is a hypothetical airship that is evacuated rather than filled with a lighter-than-air gas such as hydrogen or helium. First proposed by Italian Jesuit priest Francesco Lana de Terzi in 1670,[1] the vacuum balloon would be the ultimate expression of lifting power per volume displaced. (Also called "FLanar", for F.Lana and Portuguese for wandering[2])
1.1 Double wall fallacy
3 Material constraints
4 Atmospheric constraints
From 1886 to 1900 Arthur De Bausset attempted in vain to raise funds to construct his "vacuum-tube" airship design, but despite early support in the United States Congress, the general public was skeptical. Illinois historian Howard Scamehorn reported that Octave Chanute and Albert Francis Zahm "publicly denounced and mathematically proved the fallacy of the vacuum principle", however the author does not give his source.[3] De Bausset published a book on his design[4] and offered $150,000 stock in the Transcontinental Aerial Navigation Company of Chicago.[5][6] His patent application was eventually denied on the basis that it was "wholly theoretical, everything being based upon calculation and nothing upon trial or demonstration."[7]
Double wall fallacy[edit]
In 1921, Lavanda Armstrong discloses a composite wall structure with a vacuum chamber "surrounded by a second envelop constructed so as to hold air under pressure, the walls of the envelope being spaced from one another and tied together", including a honeycomb-like cellular structure.[8]
In 1983, David Noel discussed the use of geodesic sphere covered with plastic film and "a double balloon containing pressurized air between the skins, and a vacuum in the centre".[9]
In 1982–1985 Emmanuel Bliamptis elaborated on energy sources and use of "inflatable strut rings".[10]
However, the double-wall design proposed by Armstrong, Noel, and Bliamptis would not have been buoyant. In order to avoid collapse, the air between the walls must have a minimum pressure (and therefore also a density) proportional to the fraction of the total volume occupied by the vacuum section, preventing the total density of the craft from being less than the surrounding air.
In 2004–2007, to address strength to weight ratio issues, Akhmeteli and Gavrilin addressed choice of four materials, specifically I220H beryllium (elemental 99%), boron carbide ceramic, diamond-like carbon, and 5056 Aluminum alloy (94.8% Al, 5% Mg, 0.12% Mn, 0.12%Cr) in a honeycomb double layer.[11] In 2021, they extended this research using a "finite element analysis was employed to demonstrate that buckling can be prevented" focusing on a "shell of outer radius R > 2.11 m containing two boron carbide face skins of thickness 4.23 x 10−5 R each that are reliably bonded to an aluminum honeycomb core of thickness 3.52 x 10−3 R".[12] At least two papers (in 2010 and 2016) have discussed the use of graphene as an outer membrane.[2][13]
An airship operates on the principle of buoyancy, according to Archimedes' principle. In an airship, air is the fluid in contrast to a traditional ship where water is the fluid.
The density of air at standard temperature and pressure is 1.28 g/l, so 1 liter of displaced air has sufficient buoyant force to lift 1.28 g. Airships use a bag to displace a large volume of air; the bag is usually filled with a lightweight gas such as helium or hydrogen. The total lift generated by an airship is equal to the weight of the air it displaces, minus the weight of the materials used in its construction including the gas used to fill the bag.
Vacuum airships would replace the helium gas with a near-vacuum environment. Having no mass, the density of this body would be near to 0.00 g/l, which would theoretically be able to provide the full lift potential of displaced air, so every liter of vacuum could lift 1.28 g. Using the molar volume, the mass of 1 liter of helium (at 1 atmospheres of pressure) is found to be 0.178 g. If helium is used instead of vacuum, the lifting power of every liter is reduced by 0.178 g, so the effective lift is reduced by 14%. A 1-liter volume of hydrogen has a mass of 0.090 g.
The main problem with the concept of vacuum airships is that, with a near-vacuum inside the airbag, the exterior atmospheric pressure is not balanced by any internal pressure. This enormous imbalance of forces would cause the airbag to collapse unless it were extremely strong (in an ordinary airship, the force is balanced by helium, making this unnecessary). Thus the difficulty is in constructing an airbag with the additional strength to resist this extreme net force, without weighing the structure down so much that the greater lifting power of the vacuum is negated.[2][11]
Material constraints[edit]
Compressive strength[edit]
From the analysis by Akhmeteli and Gavrilin:[11]
The total force on a hemi-spherical shell of radius
{\displaystyle R}
by an external pressure
{\displaystyle P}
{\displaystyle \pi R^{2}P}
. Since the force on each hemisphere has to balance along the equator, assuming
{\displaystyle h<<R}
{\displaystyle h}
is the shell thickness, the compressive stress (
{\displaystyle \sigma }
{\displaystyle \sigma =\pi R^{2}P/2\pi Rh=RP/2h}
Neutral buoyancy occurs when the shell has the same mass as the displaced air, which occurs when
{\displaystyle h/R=\rho _{a}/(3\rho _{s})}
{\displaystyle \rho _{a}}
{\displaystyle \rho _{s}}
is the shell density, assumed to be homogeneous. Combining with the stress equation gives
{\displaystyle \sigma =(3/2)(\rho _{s}/\rho _{a})P}
For aluminum and terrestrial conditions Akhmeteli and Gavrilin estimate the stress as
{\displaystyle 3.2\cdot 10^{8}}
Pa, of the same order of magnitude as the compressive strength of aluminum alloys.
Buckling[edit]
Akhmeteli and Gavrilin note, however, that the compressive strength calculation disregards buckling, and using R. Zoelli's formula for the critical buckling pressure of a sphere
{\displaystyle P_{cr}={\frac {2Eh^{2}}{\sqrt {3(1-\mu ^{2})}}}{\frac {1}{R^{2}}}}
{\displaystyle E}
is the modulus of elasticity and
{\displaystyle \mu }
is the Poisson ratio of the shell. Substituting the earlier expression gives a necessary condition for a feasible vacuum balloon shell:
{\displaystyle E/\rho _{s}^{2}={\frac {9P_{cr}{\sqrt {3(1-\mu ^{2})}}}{2\rho _{a}^{2}}}}
The requirement is about
{\displaystyle 4.5\cdot 10^{5}kg^{-1}m^{5}s^{-2}}
Akhmeteli and Gavrilin assert that this cannot even be achieved using diamond (
{\displaystyle E/\rho _{s}^{2}\approx 1\cdot 10^{5}}
), and propose that dropping the assumption that the shell is a homogeneous material may allow lighter and stiffer structures (e.g. a honeycomb structure).[11]
Atmospheric constraints[edit]
A vacuum airship should at least float (Archimedes law) and resist external pressure (strength law, depending on design, like the above R. Zoelli's formula for sphere). These two conditions may be rewritten as an inequality where a complex of several physical constants related to the material of the airship is to be lesser than a complex of atmospheric parameters. Thus, for a sphere (hollow sphere and, to a lesser extent, cylinder are practically the only designs for which a strength law is known) it is
{\displaystyle k_{\rm {L}}<{\sqrt {1-{\frac {P_{\rm {int}}}{P}}}}\cdot L_{\rm {a}}}
{\displaystyle P_{\rm {int}}}
is pressure within the sphere, while
{\displaystyle k_{\rm {L}}}
(«Lana coefficient») and
{\displaystyle L_{\rm {a}}}
(«Lana atmospheric ratio») are:[2]
{\displaystyle k_{\rm {L}}=2.79\cdot {\frac {\rho _{s}}{\rho _{\rm {atm}}}}\cdot {\sqrt {\frac {P_{\rm {atm}}}{E}}}\cdot (1-\mu ^{2})^{0.25}}
(or, when
{\displaystyle \mu }
is unknown,
{\displaystyle k_{\rm {L}}\approx 2.71\cdot {\frac {\rho _{s}}{\rho _{\rm {atm}}}}\cdot {\sqrt {\frac {P_{\rm {atm}}}{E}}}}
with an error of order of 3% or less);
{\displaystyle L_{\rm {a}}={\frac {\rho _{a}}{\rho _{\rm {atm}}}}\cdot {\sqrt {\frac {P_{\rm {atm}}}{P}}}}
{\displaystyle \rho _{a}}
{\displaystyle L_{\rm {a}}=10\cdot {\sqrt {\frac {P_{\rm {atm}}}{P}}}\cdot {\frac {M_{a}}{T_{a}}}}
{\displaystyle P_{\rm {atm}}=101325}
{\displaystyle Pa}
{\displaystyle \rho _{\rm {atm}}=1.22}
{\displaystyle kg/m^{3}}
are pressure and density of standard Earth atmosphere at sea level,
{\displaystyle M_{a}}
{\displaystyle T_{a}}
are molar mass (kg/kmol) and temperature (K) of atmosphere at floating area. Of all known planets and moons of the Sun system only the Venusian atmosphere has
{\displaystyle L_{\rm {a}}}
big enough to surpass
{\displaystyle k_{\rm {L}}}
for such materials as some composites (below altitude of ca. 15 km) and graphene (below altitude of ca. 40 km).[2] Both materials may survive in the Venusian atmosphere. The equation for
{\displaystyle L_{\rm {a}}}
shows that exoplanets with dense, cold and high-molecular (
{\displaystyle CO_{2}}
{\displaystyle O_{2}}
{\displaystyle N_{2}}
type) atmospheres may be suitable for vacuum airships, but it is a rare type of atmosphere.
In Edgar Rice Burroughs's novel Tarzan at the Earth's Core Tarzan travels to Pellucidar in a vacuum airship constructed of the fictional material Harbenite.
In Passarola Rising, novelist Azhar Abidi imagines what might have happened had Bartolomeu de Gusmão built and flown a vacuum airship.
Spherical vacuum body airships using the Magnus effect and made of carbyne or similar superhard carbon are glimpsed in Neal Stephenson's novel The Diamond Age.
In Maelstrom[14] and Behemoth:B-Max, author Peter Watts describes various flying devices, such as "botflies" and "lifters" that use "vacuum bladders" to keep them airborne.
In Feersum Endjinn by Iain M. Banks, a vacuum balloon is used by the narrative character Bascule in his quest to rescue Ergates. Vacuum dirigibles (airships) are also mentioned as a notable engineering feature of the space-faring utopian civilisation The Culture in Banks' novel Look to Windward, and the vast vacuum dirigible Equatorial 353 is a pivotal location in the final Culture novel, The Hydrogen Sonata.
^ "Francesco Lana-Terzi, S.J. (1631–1687); The Father of Aeronautics". Retrieved 13 November 2009.
^ a b c d e E. Shikhovtsev (2016). "Is FLanar Possible?". Retrieved 2016-06-19.
^ Scamehorn, Howard Lee (2000). Balloons to Jets: A Century of Aeronautics in Illinois, 1855–1955. SIU Press. pp. 13–14. ISBN 978-0-8093-2336-4.
^ De Bausset, Arthur (1887). Aerial Navigation. Chicago: Fergus Printing Co. Retrieved 2010-12-01.
^ "Aerial Navigation" (PDF). New York Times. February 14, 1887. Retrieved 2010-12-01.
^ "To Navigate the Air" (PDF). New York Times. February 19, 1887. Retrieved 2010-12-01.
^ Mitchell (Commissioner) (1891). Decisions of the Commissioner of Patents for the Year 1890. US Government Printing Office. p. 46. 50 O. G., 1766
^ US patent 1390745, Lavanda M Armstrong, "Aircraft of the lighter-than-air type", published Sep 13, 1921, assigned to Lavanda M Armstrong
^ David Noel (1983). "Lighter than Air Craft Using Vacuum" (PDF). Correspondence, Speculations in Science and Technology. 6 (3): 262–266.
^ US patent 4534525, Emmanuel Bliamptis, "Evacuated balloon for solar energy collection", published Aug 13, 1985, assigned to Emmanuel Bliamptis
^ a b c d US application 2007001053, AM Akhmeteli, AV Gavrilin, "US Patent Application 11/517915. Layered shell vacuum balloons", published Feb 23, 2006, assigned to Andrey M Akhmeteli and Andrey V Gavrilin
^ Akhmeteli, A.; Gavrilin, A.V. (2021). "Vacuum Balloon–A 350-Year-Old Dream". Eng. 2 (4): 480–491. doi:10.3390/eng2040030. {{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Zornes, David (2010). "Vacua Buoyancy Is Provided by a Vacuum Bag Comprising a Vacuum Membrane Film Wrapped Around a Three-Dimensional (3D) Frame to Displace Air, on Which 3D Graphene "Floats" a First Stack of Two-Dimensional Planar Sheets of Six-Member Carbon Atoms Within the Same 3D Space as a Second Stack of Graphene Oriented at a 90-Degree Angle". SAE International. SAE Technical Paper Series. 1. doi:10.4271/2010-01-1784.
^ Watts, Peter. "Maelstrom by Peter Watts". Rifters.com.
Alfred Hildebrandt (1908). Airships Past and Present: Together with Chapters on the Use of Balloons in Connection with Meteorology, Photography and the Carrier Pigeon. D. Van Nostrand Company. pp. 16–.
Collins, Paul (2009). "The rise and fall of the metal airship". New Scientist. 201 (2690): 44–45. Bibcode:2009NewSc.201...44C. doi:10.1016/S0262-4079(09)60106-8. ISSN 0262-4079.
Timothy Ferris (2000). Life Beyond Earth. Simon and Schuster. pp. 130–. ISBN 978-0-684-84937-9.
Le defi de Cyrano ; un ballon gonfle avec du vide : Fabrice David
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A-level Computing/AQA/Problem Solving, Programming, Operating Systems, Databases and Networking/Programming Concepts - Wikibooks, open books for an open world
A-level Computing/AQA/Problem Solving, Programming, Operating Systems, Databases and Networking/Programming Concepts
< A-level Computing | AQA | Problem Solving, Programming, Operating Systems, Databases and Networking
From the Specification : Programming Paradigms
Understand the need for and characteristics of a variety of programming paradigms.
Be familiar with the concept of an object class, an object, instantiation, encapsulation, inheritance.
Practical experience of programming using objects to model a simple problem.
Illustrate the use of recursive techniques in programming language
Abstract Data Types / Data Structures
Lists - Linear lists, Linked lists
Queues - linear, circular, priority
Be familiar with the concept of a list, a tree, a queue, a stack, a pointer and be familiar with the data structures and methods for representing these when a programming language does not support these structures as built-in types. Distinguish between static and dynamic structures and compare their uses. Use of free memory, the heap and pointers Be aware of a graph as a data structure to represent more complex relationships. Explain the terms graph, labelled graph, unlabelled graph, vertex, edge, digraph and arc. Know how an adjacency matrix and an adjacency list may be used to represent a graph.
Compare the use of each.
A tree is a connected undirected graph with no cycles. A rooted tree is a tree in which one vertex has been designed as the root and every edge is directed away from the root.
Be familiar with a simulation as a computer program or network of computers that attempts to simulate a model of a particular system.
Know that computer simulations can represent real and imaginary situations. Know that simulations allow users to study or try things that would be difficult or impossible to do in real life. Be familiar with simple simulation techniques involving queues.
1.3 Binary tree search
1.6 Stack, queue and list operations
1.7 Simple graph traversal algorithms
Standard Algorithms[edit | edit source]
You should be able to recognise and use trace tables on code that does the following:
Binary trees[edit | edit source]
A binary tree can be defined recursively as
an empty tree, containing no nodes or
a node, called the root, usually containing some data, with at most 2 subtrees, each of which is itself a binary tree
Node - a part of the tree holding some data
Branches - connections between nodes
Root - the node that has no parent, i.e., is at the top of the tree
Leaf - a node that has no subtrees, i.e., is at the bottom of the tree
Parent - a node that is connected to the root of a subtree
Level - all the nodes that are at the same depth in the tree, i.e., there are the same number of branches to get 'back to' the root, are at the same level. The root is at level 0
Child - the root of a subtree that is connected 'upwards' to its parent
Sibling - the next node at the same level
Binary tree search[edit | edit source]
Thinking of any particular binary tree, how many searches will it take to find a certain item?
Thinking of any particular binary tree, how many searches will it take to find a certain item (in the best and worst cases) if the tree has n nodes?
Does the number of searches (in both cases) differ depending on whether the item is in the tree or not?
Does the number of searches (in both cases) depend on the structure of the tree?
What is the worst case? What is the best case?
A binary tree has n items in it, and takes s searches (in the worst case) to find an item. If another n items are added to the tree how many searches will it need to find an item (in the worst case) if the tree is perfectly balanced, i.e., each node (apart from the leaf nodes) has two subtrees.
A perfectly balanced binary tree is one where every node, has 0 or 2 subtrees. The leaves have 0 subtrees, and the 'internal' nodes each have 2.
Stack, queue and list operations[edit | edit source]
Queue - FIFO Stack - LIFO
Simple graph traversal algorithms[edit | edit source]
(see: AQA supplementary materials, document called: 'Unit guide: COMP3 - Graph traversals')
A graph traversal algorithm aims to 'explore' every vertex.
If the algorithm is searching for a particular vertex (eg. the exit to a maze) then it considers whether each vertex is the one it is looking for.
After evaluation, if it has not found what it was looking for, the algorithm adds every vertex adjacent to the current one onto a 'list' of vertices to visit next.
The type of structure used to store this 'list' of vertices determines how the algorithm moves through the graph.
{\displaystyle \implies }
siblings are processed before their children
{\displaystyle \implies }
'Breadth-first traversal'
all vertices on the same level are processed before moving to the next level
{\displaystyle \implies }
children are processed before the siblings of the current vertex
{\displaystyle \implies }
'Depth-first traversal'
These algorithms also have to keep track of the state of each vertex:
The algorithm has yet to discover this vertex
A vertex is discovered once the algorithm has evaluated the vertex, and added its children to the list of vertices to visit. However not all of them have been evaluated yet
Completely explored/Processed
All adjacent vertices have been discovered
Simulations[edit | edit source]
Simulations model real and imaginary situations. They may be used to test scenarios that wouldn't be feasible in real life for cost or time reasons. Many computer games could be considered simulations.
You should know how to simulate simple queues.
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Sravanth C., Akshat Sharda, Skanda Prasad, and
Someguy Noname
Snell's law, also known as the law of refraction, is a law stating the relationship between the angles of incidence and refraction, when referring to light passing from one medium to another medium such as air to water, glass to air, etc.
Lateral Displacement and it's Calculation
Effects and Applications of Total Internal Reflection
Snell's Law - Problem Solving
Snell's Law states that the ratio of sine of angle of incidence and sine of angle of refraction is always constant for a given pair of media.
\dfrac{\sin i}{\sin r}=\text{constant}=n=\text{refractive index}
Let us consider that light enters from medium 1 to medium 2,
\therefore \dfrac{\sin i}{\sin r}=n_{21}=\dfrac{n_2}{n_1}=\dfrac{\color{#3D99F6}{v_1}}{\color{#3D99F6}{v_2}}=\dfrac{\color{#3D99F6}{\lambda_1}}{\color{#3D99F6}{\lambda_2}}
v_n
is the velocity of light in respective medium and
\lambda_n
is the wavelength of light in respective medium. You may be wondering how we obtained the expression in blue color, well if we define it in an easy way, the basic cause of refraction is due to the change in velocity of light by entering a medium of different refractive index. So, if a medium has less refractive index, then the velocity of light in that medium would be more but if a medium has more refractive index then the velocity of light in that medium would be comparatively less.
\therefore v \propto \dfrac{1}{n} \Rightarrow \dfrac{v_1}{v_2}=\dfrac{n_2}{n_1}=n_{21}
Question: A ray of light travelling in air is incident on the plane surface of a transparent medium. The angle of incidence is found to be
45^{\circ}
30^{\circ}
. Find the refractive index of the medium.
Solution: We know that
\hat i=45^{\circ}
\hat r=30^{\circ}
Therefore refractive index,
\begin{aligned} n=\dfrac{\sin i}{\sin r} &= \dfrac{\sin 45^{\circ}}{\sin 30^{\circ}}\\ &= \dfrac{1/\sqrt{2}}{1/2}= \sqrt{2} \end{aligned}
60^\circ
45^\circ
Absolute Refractive Index:
When we compare the speed of light in a medium to that of the speed of the light in vacuum, then we would be dealing with something called absolute refractive index. We generally refer to the absolute refractive index of a medium when we say that a certain object's refractive index is
x
The expression for the absolute refractive index of a medium would thus be:
\text{absolute refractive index}=\dfrac{\text{speed of light in vacuum}}{\text{speed of light in the given medium}} = \dfrac{c}{v}
Note: As the speed of light is at its maximum in vacuum, the absolute refractive index always greater than
1
. Also note that the refractive index is a relative quantity and thus it had no units.
Question: The absolute refractive index of a glass window is
1.5
. What is the speed of light when it is traveling through the glass window? Assume that the speed of light in vacuum
=3\times 10^8m/s
Solution: According to the question, we have:
\dfrac{\text{speed of light in vacuum}}{\text{speed of light in the given medium}}=1.5\\ \implies \dfrac{3\times 10^8}{\text{speed of light in the given medium}}=1.5\\ \implies \text{speed of light in the given medium}=\dfrac{3\times 10^8}{1.5}=\boxed{2\times 10^8 m/s}
Question: The absolute refractive index of diamond is
2.42
. What is the speed of light in diamond? (Take speed of light in vacuum=
3 \times 10^8 m/s
Solution: Absolute refractive index of diamond is
=\dfrac{\text{speed of light in vacuum}}{\text{speed of light in diamond}}\quad\therefore\dfrac{c}{v}=2.42\\ \implies v=\dfrac{c}{2.42} \implies v=\dfrac{3 \times 10^8}{2.42} \\\boxed{v=1.24 \times 10^8 m/s}
Refraction of a ray of light in a glass slab
In this case, we will try to prove
\angle i_1=\angle r_2
or the incident ray is parallel to the emergent ray,
Applying Snell's Law when the light is incident on the glass slab's surface,
\dfrac{\sin i_1}{\sin r_1}=n=\text{refractive index of glass}
Now, applying Snell's Law when the light ray is leaving the glass slab through another surface,
\dfrac{\sin i_2}{\sin r_2}=\dfrac{1}{n}\Rightarrow \dfrac{\sin r_2}{\sin i_2}=n=\text{refractive index of glass} \\ \therefore \dfrac{\sin i_1}{\sin r_1}=\dfrac{\sin r_2}{\sin i_2}
\angle r_1=\angle i_2
as they are alternate angles, thus,
\sin r_1=\sin i_2
\therefore \sin i_1=\sin r_2\Rightarrow \angle i_1=\angle r_2
So, the incident ray is parallel to the emergent ray but it is laterally displaced from it.
Question: A ray of light travelling in air falls on the surface of a transparent glass slab. The ray makes and angle of
45^{\circ}
with the normal to the surface. Find the angle made by the refracted ray with the normal within the slab. Given that refractive index of the glass slab is
\sqrt{2}
n=\dfrac{\sin i}{\sin r} = \dfrac{\sin 45^{\circ}}{\sin r}
, here the refractive index is
\sqrt{2}
\begin{aligned} \dfrac{\sin 45^{\circ}}{\sin r}&=\sqrt{2}\\ \implies\sin r &= \dfrac{1}{\sqrt{2}}\times \sin 45^{\circ}\\ =\dfrac{1}{\sqrt{2}}\times \dfrac{1}{\sqrt{2}} &=\dfrac{1}{2} \end{aligned}
\sin r
\dfrac{1}{2}
, the angle of refraction would be
r=\sin^{-1}\left(\dfrac 12\right)=30^\circ
As discussed earlier, the emergent ray is parallel to the incident ray but appears slightly shifted, and this shift in the position of the emergent ray as compared to the incident ray is called Lateral displacement.
The perpendicular distance between the incident ray and the emergent ray is defined as lateral shift. This shift depends upon the angle of incidence, the angle of refraction and the thickness of the medium. It is given by the following expression:
S_{\text{Lateral}}=\dfrac{t}{\cos r}\sin{(i-r)}
We shall now try to derive the above stated formula for a Glass slab. In the figure given below,
AB
is the incident ray,
BC
is the refracted ray and
CD
is the emergent ray. The ray is striking the slab at an angle of
i_1
and it is emerging from the slab at an angle of
r_2
Refraction of a ray of light in a glass slab with it's corresponding angles
\triangle BCK
\sin (i_1-r_1)=\dfrac{CK}{BC} \Rightarrow CK=BC \sin (i_1-r_1)
\triangle BCN'
\cos r_1=\dfrac{BN'}{BC}=\dfrac{t}{BC} \Rightarrow BC=\dfrac{t}{\cos r_1}
t
is the thickness of slab.
BC
in the first equation,
S_L=\text{Lateral Displacement }(CK)=t\dfrac{\sin(i_1-r_1)}{\cos r_1}
Question: The thickness of a glass slab is
0.25m
, it has a refractive index of
1.5
. A ray of light is incident on the surface of the slab at an angle of
60^\circ
Find the lateral displacement of the light ray when it emerges from the other side of the mirror. You may assume that the speed of light is
3\times 10^8 m/s
Solution: From the previous topics, we know:
\text{refractive index}=\dfrac{\sin i}{\sin r}=1.5\text{ (in this case)}\\\sin r=\dfrac{1.5}{\sin 60}\approx 0.57735\\\implies r = \sin^{-1}(0.57735)\approx 35.25^\circ
Now, applying the values in the formula for lateral displacement we get:
S_L=\dfrac{0.25}{\cos(35.25)}\times\sin(60-35.25)\approx 0.1281 m =\boxed{12.81cm}
Many a time you might have seen the floor of the swimming pool raised/ the letters appearing to be raised under a glass slab, ever wondered why this happens? If you observe clearly, you'll find that refraction explains it. Let's see the definition.
The vertical distance by which an object appears to be shifted when an object placed in one medium is observed from another medium of different refractive indices, is called Normal shift. It is given by the formula:
S_{\text{Normal}}=t\left(1-\dfrac{1}{_{\text r}n_{\text d}}\right)\quad\text{where}\quad _{\text r}n_{\text d}=\mu=\dfrac{\text{real depth}}{\text{apparent depth}}
\text{refractive index}(\mu)=\dfrac{\sin i}{\sin r}=\dfrac{\text{real depth}}{\text{apparent depth}}
The thickness of a glass slab is
0.2m
, and it is placed over a flat book, the refractive index of the glass slab is
1.5
. A student looks through it and finds that the normal shift is
x
x
Solution: We know that:
\begin{aligned} S_N&=t\left(1-\dfrac{1}{\mu}\right)\\ &=0.2\left(1-\dfrac{1}{1.5}\right)\\ &=0.2\times \dfrac 13= 0.066m \end{aligned}
When light travels from a denser to rarer medium with an angle greater than the critical angle, the ray of light does not deviate in its path or does not refract, but it undergoes a reflection known as total internal reflection. The angle beyond which light in a given medium undergoes total internal reflection is called the critical angle.
The critical angle differs from medium to medium. If the refractive index of a given medium is
\mu
, then it's critical angle is given by the formula: [1]
\mu =\dfrac { 1 }{ \sin{ \theta }_{ c } }\quad\\\theta_c=\sin^{-1}\left(\dfrac 1\mu\right)
Sparkle of the diamond
Whenever your mom wears it you notice it, yes the sparkling beauty of the diamond never misses our eye. But have you ever wondered why the diamond sparkles? Well it is due to the phenomenon we've been discussing now, total internal reflection. Sparking beauty of the hope diamond [2]
This very old illusion ,which had fooled many people, is due to the magic of Total Internal relfection! Mirage is an optical illusion caused by refraction and total internal reflection. We know that the temperature of air varies with height, and also refractive index depends on the temperature of the medium.
Mirage Formation on a road [3]
During hot summers, the Surface of the Earth gets hotter, and the layers of air with decreasing temperature are formed. But the hot air has a refractive index lower than the cold air, that is hot air is optically rarer than cold air, and we know if a ray of light passes through a rarer medium from a denser medium, then the light rays bend away from the normal. So, at some points the light rays get totally reflected internally and reach the eyes of an observer, creating the reflection of an object on the surface of the Earth.
Very similar to mirage formation,thus phenomenon makes the objects appear to be levitating in the sky. This is mostly seen in the polar regions (as opposed to mirages, which are generally frowned in hot deserts). In these places the surface of the Earth is very cold and as we go up, layers of air with increasing temperatures are formed. As a result, the layers of atmosphere near the Earth have a higher refractive index than the layers above them, this layer is called as an inversion layer.
The objects appear to be floating due to the phenomemnon of looming [4]
When the light from any object (normally ships) reaches an observer, it undergoes a series of refractions which makes the light rays bend away from the normal, and at a point, they reach a stage where the angle incidence is greater than the critical angle and thus the rays undergo total internal reflection and reach the eye of an observer and creates and optical illusion that the object is really floating in the sky!
Optical fibres are the devices used to transfer light signals over large distances with negligible loss of energy. It is a revolutionary idea in terms of communication. But it's working is based on this simple phenomenon of total internal reflection. If you take a close look at an optical fibre you will observe that it consists of a thin transparent material, this is know as the core. This core is coated with something known as cladding and has a higher refractive index than the surrounding medium[5], it prevents the absorption of light by any means.
The internal structure and the transfer of light signal in a single optical fibre [6]
When the light rays enter the acceptance cone, some rays which are incident at an angle greater than the critical angle gets reflected internally and then it undergoes a series of Total Internal reflections until it reaches the other end of the firbe. But we should note that not all of the rays get reflected internally because they may not have struck the surface at the required angle (as seen in the figure above).
60^\circ
45^\circ
60.0^\circ
30.0^\circ
45.0^\circ
15.0^\circ
1.5 \times 10^8\text{ m/s}
3.0 \times 10^8\text{ m/s}.
n_{\text{air}} = 1.00
n_{\text{glass}} = 1.50,
n_{\text{coating}} = 1.46
[1] Total Internal reflection, rp-encyclopedia.com. Retrieved 16:45, March 15, 2016, from https://www.rpphotonics.com/totalinternalreflection.html.
[3] Image credit http://epod.usra.edu/blog/2010/03/highway-mirage.html: Universities Space Research Association
[4] Image from https://en.m.wikipedia.org/wiki/File:Illustrationofloomingrefrationphenomenon.jpg under the creative Commons license for reuse and modification.
[5] Optical Fibres, rp-encyclopedia.com. Retrieved 08:56, March 17, 2016, from https://www.rpphotonics.com/fibers.html.
[6] Image credit http://www.pacificcable.com/Fiber-Optic-Tutorial.html.
Cite as: Snell's Law. Brilliant.org. Retrieved from https://brilliant.org/wiki/snells-law/ |
Evaluate the line integral, where C is the given curve \int_C
Evaluate the line integral, where C is the given curve \int_C (\frac{x}{y})ds , C:x
{\int }_{C}\left(\frac{x}{y}\right)ds,C:x={t}^{3},y={t}^{4},1\le t\le 4
\left({x}^{2}+2xy-4{y}^{2}\right)dx-\left({x}^{2}-8xy-4{y}^{2}\right)dy=0
x={t}^{2},y=2t,0\le t\le 5
{\left(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}+\frac{{z}^{2}}{{c}^{2}}\right)}^{2}=\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-\frac{{z}^{2}}{{c}^{2}}.
Calculate this triple integral over D:
\int \int \int \sqrt{10+{y}^{2}-{x}^{2}}dxdydz
over the region bounded D:
D=\left\{\left(x,y,z\right)\mid {x}^{2}+{z}^{2}\le {y}^{2}+10,\text{ }{x}^{2}+{y}^{2}\le 9\right\}
\int 6{x}^{2}{\left({x}^{3}+2\right)}^{99}dx
{\int }_{0}^{1}{\int }_{3y}^{3}{e}^{{x}^{2}}dxdy
\int \frac{{e}^{t}}{\sqrt{{e}^{2t}+4}}dt |
Chromatography - New World Encyclopedia
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This gas chromatography system records the concentration of a chemical (acrylonitrile) in the air in different parts of a chemical laboratory.
Chromatography (from Greek χρώμα chroma, meaning "color") is the collective term for a family of laboratory techniques for the separation of mixtures. Basically, its a group of different methods used to separate or analyze mixtures. It involves passing a mixture through a stationary phase, which separates the analyte to be measured from other molecules in the mixture and allows it to be isolated.
5 Capillary-action chromatography
6.2 Fast performance liquid chromatography
8 Countercurrent chromatography
The main use of chromatography is to purify a certain material from a mixture. It is so precise that it can be used either to separate proteins that only vary by one amino acid as well as to purify volatile or soluble material.
Different types of Chormatography and some uses are:
Liquid Chromatography—used to analyze water samples to look for pollution, metal ions, organic compounds. Liquid is used to mesh hydrophilic and insoluble molecules
Paper Chromatography—most common method which uses paper and through capillary action, solvents are pulled up and separated
Gas Chromatography—used in forensics in analyzing fibers, blood. Helium is used to move a gas mixture through a column
Thin Layer Chromatography—checks purity of compounds, such as pestisides or substances in food. TLC uses absorbent material on flat glass or plastic plates
Chromatography means "to show with colors." It was the Russian botanist Mikhail Semyonovich Tsvet (1872-1919) who invented the first chromatography technique in 1900, during his research on chlorophyll. He used a liquid-adsorption column containing calcium carbonate to separate plant pigments. The method was described on December 30, 1901, at the 11th Congress of Naturalists and Doctors (XI съезд естествоиспытателей и врачей) in St. Petersburg. The first printed description was in 1903, in the Proceedings of the Warsaw Society of Naturalists, section of biology. He first used the term "chromatography" in print in 1906, in his two papers about chlorophyll in the German botanical journal, Berichte der Deutschen Botanischen Gesellschaft. In 1907, he demonstrated his chromatograph for the German Botanical Society. Interestingly, Mikhail's surname "Tsvet" means "color" in Russian, so some have suggested that he named the procedure chromatography (literally "color writing") to ensure that he, a commoner in Tsarist Russia, would be remembered for his work.
In 1952, Archer John Porter Martin and Richard Laurence Millington Synge were awarded the Nobel Prize in Chemistry for their invention of partition chromatography.[1] Since then, the technology has advanced rapidly. Researchers found that the principles underlying Tsvet's chromatography could be applied in many different ways, giving rise to the different varieties of chromatography described below. Simultaneously, advances continually improved the technical performance of chromatography, allowing the separation of increasingly similar molecules.
The analyte is the substance which is to be purified or isolated during chromatography
Analytical chromatography is used to determine the identity and concentration of molecules in a mixture
A chromatogram is the visual output of the chromatograph. Different peaks or patterns on the chromatogram correspond to different components of the separated mixture
Plotted on the x-axis is the retention time and plotted on the y-axis a signal (for example obtained by UV spectroscopy) corresponding to the amount of analyte exiting the system.
A chromatograph takes a chemical mixture carried by liquid or gas and separates it into its component parts as a result of differential distributions of the solutes as they flow around or over the stationary phase
The mobile phase is the analyte and solvent mixture which travels through the stationary phase
Preparative chromatography is used to nondestructively purify sufficient quantities of a substance for further use, rather than analysis.
The retention time is the characteristic time it takes for a particular molecule to pass through the system under set conditions.
The stationary phase is the substance which is fixed in place for the chromatography procedure and is the phase to which solvents and the analyte travels through or binds to. Examples include the silica layer in thin layer chromatography.
The bonded phase is the phase which is covalently bonded to the support particles or to the inside wall of the column tubing.
The Column Length is proportional to the number of theoretical plates. Shorter columns show higher resolutions.
Chromatography is a separation method that defines the differences in partitioning behavior between a mobile phase and a stationary phase to separate the components in a mixture. Components of a mixture may be interacting with the stationary phase based on charge, relative solubility or adsorption. There are two theories of chromatography, the plate and rate theories.
The retention is a measure of the speed at which a substance moves in a chromatographic system. The retention volume of a solute is that volume of mobile phase that passes through the column between the injection point and the peak maximum.In continuous development systems like HPLC or GC, where the compounds are eluted with the eluent, the retention is usually measured as the retention time Rt or tR, the time between injection and detection. In interrupted development systems like TLC the retention is measured as the retention factor Rf, the run length of the compound divided by the run length of the eluent front:
{\displaystyle R_{f}={\frac {distance\ moved\ by\ compound}{distance\ moved\ by\ solvent}}}
During the chromatographic process the analyte experiences zone broadening as a result of diffusion. Two analytes with different retention times yet with large broadening do not resolve and this is why in any chromatographic system broadening needs to be minimized. This is done by selecting the proper stationary and mobile phase, the eluent velocity, the track length and temperature. The Van Deemter's equation gives an ideal eluent velocity taking into account several physical parameters.
The plate theory of chromatography was developed by Archer John Porter Martin and Richard Laurence Millington Synge. They stated that each plate should be a broken into a specific length, and the solute spends a finite (limited) amount of time in each place. The size of the cell is there equilibriate between the two phases. The smaller the plate, the faster the equilibrium and the more plate in the column. This is further related to column efficiency.
The plate theory describes the chromatography system, the mobile and stationary phases, as being in equilibrium. The partition coefficient K is based on this equilibrium, and is defined by the following equation:
{\displaystyle K={\frac {Concentration\ of\ solute\ in\ stationary\ phase}{Concentration\ of\ solute\ in\ mobile\ phase}}}
K is assumed to be independent of concentration, and can change if experimental conditions are changed, for example temperature is increased or decreased. As K increases, it takes longer for solutes to separate. For a column of fixed length and flow, the retention time
{\displaystyle (t_{R})}
and retention volume
{\displaystyle (V_{r})}
can be measured and used to calculate K.
Capillary-action chromatography
This old technique is used to analyze complex mixtures, such as ink, by separating the different chemicals it was made from. The method involves placing a small spot of sample solution onto a strip of chromatography paper. The paper is placed into a jar containing a shallow layer of solvent and sealed. As the solvent rises through the paper it meets the sample mixture which starts to travel up the paper with the solvent. Different compounds in the sample mixture travel different distances according to how strongly they interact with the paper. Mixtures of different characteristics (size and solubility) travel at different speeds. This allows the calculation of an Rf value and can be compared to standard compounds to aid in the identification of an unknown substance.
Column chromatography encompasses a number of techniques based around utilizing a vertical glass column filled with some form of solid adsorbent, with the sample to be separated placed on top of this support. The top is then filled with a liquid, which flows down through the column. Similarly to other forms of chromatography, differences in rates of movement through the solid medium are translated to different exit times from the bottom of the column for the various elements of the original sample. If the solvent moves down by gravity, it is gravity column chromatography. If the solvent moves down by air pressure, it is known as flash chromatography.
In 1978, W.C. Still introduced a modified version of column chromatography called flash column chromatography (flash).[2] The technique is very similar to the traditional column chromatography, except that the solvent is driven through the column by applying positive pressure. This method is used solely in organic teaching labs because of its ease and environmentally friendly nature. This allowed most separations to be performed in less than 20 minutes, with improved separations compared to the old method. Modern flash chromatography systems are sold as pre-packed plastic cartridges, and the solvent is pumped through the cartridge. Systems may also be linked with detectors and fraction collectors providing automation. The introduction of gradient pumps resulted in quicker separations and less solvent usage.
Fast performance liquid chromatography (FPLC) is a term applied to several chromatography techniques which are used to purify proteins. Many of these techniques are identical to those carried out under high performance liquid chromatography.
FPLC involves using a pump and column which withstand high pressure so separations are quickly induced.
High performance liquid chromatography (HPLC) is a form of column chromatography used frequently in biochemistry and analytical chemistry. The analyte is forced through a column (stationary phase) by a liquid (mobile phase) at high pressure, which decreases the time the separated components remain on the stationary phase and thus the time they have to diffuse within the column.
Specific techniques which come under this broad heading are listed below. It should also be noted that the following techniques can also be considered fast protein liquid chromatography if no pressure is used to drive the mobile phase through the stationary phase. It is not dissimilar from Aqueous Normal Phase Chromatography.
Ion exchange chromatography (IEC) is a column chromatography based on charge. It is used to separate charged compounds including amino acids, peptides, and proteins. The stationary phase is usually an ion exchange resin that carries charged functional groups which interact with oppositely charged groups of the compound to be retained. Ion exchange chromatography is commonly used to purify proteins using FPLC.
Size exclusion chromatography (SEC) is also known as gel permeation chromatography (GPC) or gel filtration chromatography and separates particles on the basis of size by using porous particles. Smaller molecules enter a porous media and take longer to exit the column, whereas larger particles leave the column faster. It is generally a low resolution chromatography and thus it is often reserved for the final, "polishing" step of a purification. It is also useful for determining the tertiary structure and quaternary structure of purified proteins, especially since it can be carried out under native solution conditions.
Affinity chromatography is based on selective non-covalent interaction between an analyte and specific molecules. Its set to separate all molecules of a specificity from the whole-lot of molecules of a mixture. It is often used in biochemistry in the purification of proteins bound to tags. These fusion proteins are labeled with compounds such as His-tags, biotin, or antigens, which bind to the stationary phase specifically. After purification, some of these tags are usually removed and the pure protein is obtained.
Gas chromatography (GC) is based on a partition equilibrium of analyte between a solid stationary phase and a mobile gas. It involves a vaporised sample being injected onto the chromatography column head. The stationary phase is adhered to the inside of a small-diameter glass tube (a capillary column) or a solid matrix inside a larger metal tube (a packed column). It is widely used in analytical chemistry; though the high temperatures used in GC make it unsuitable for high molecular weight biopolymers, frequently encountered in biochemistry, it is well suited for use in the petrochemical, environmental monitoring, and industrial chemical fields. It is also used extensively in chemistry research.
Countercurrent chromatography (CCC) is a type of liquid-liquid chromatography, where both the stationary and liquid phases are liquids. It involves mixing a solution of liquids, allowing them to settle into layers, and then separating the layers. The liquid is passed through coil columns synchronized by coil planet centrifuge. The coiled force keeps the stationary phase against the continuous mobile phase.
↑ Nobel Prize, The Nobel Prize in Chemistry 1952. Retrieved April 7, 2008.
↑ W.C. Still, M. Kahn, and A. Mitra. J. Org. Chem. 1978, 43(14): 2923-2925.
Jones, Loretta and Peter Atkins. 2000. Chemistry: Molecules, Matter and Change. Gordonsville, VA: W. H. Freeman. ISBN 0716735954
Kazakevich, Yuri and Rosario LoBrutto, eds. 2007. HPLC for Pharmaceutical Scientists. Hoboken, NJ: John Wiley & Sons. ISBN 0471681628
Miller, James M. 2005. Chromatography: Concepts and Contrasts. Hoboken, NJ: Wiley-Interscience. ISBN 0471472077
Poole, Colin F. 2003. The Essence of Chromatography. Amsterdam: Elsevier Science. ISBN 0444501991
Chromatography Introductory Theory.
Chromatography Online online books about chromatography.
Learning by Simulations Overlapping peaks and their quantification.
MIT OpenCourseWare Digital Lab Techniques Manual has videos of both Thin-layer and Column Chromatography.
Chromatography Equations Calculator.
History of "Chromatography"
Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Chromatography&oldid=1033036 |
2,6 abscisa -1 ordinate is 4 Write down the coordinates of points - Maths - Introduction to Graphs - 10799105 | Meritnation.com
2,6 abscisa -1 ordinate is 4. Write down the coordinates of points.
\mathrm{We} \mathrm{know} \mathrm{that} :\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{abscissa} = \mathrm{x}-\mathrm{coordinate}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{ordinate} = \mathrm{y}-\mathrm{coordinate}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{We} \mathrm{have} \mathrm{abscissa} = -1 \mathrm{and} \mathrm{ordinate} = 4\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{So}, \mathrm{coordinate} \mathrm{of} \mathrm{the} \mathrm{point} \mathrm{is} \left(-1, 4\right)
Subhasmita Patra answered this
points (-1,4) |
\dots \mathrm{f1}{!}^{i}\mathrm{f2}{!}^{j}\mathrm{f3}{!}^{k}\dots
i,j,k
\mathrm{f1}
\mathrm{f2}
\mathrm{f3}
\sqrt{\mathrm{\pi }}=\mathrm{\Gamma }\left(\frac{1}{2}\right)
0<i
j,k<0
\mathrm{f1}-\mathrm{f2}-\mathrm{f3}=n
\frac{\left(\genfrac{}{}{0}{}{\mathrm{f1}}{\mathrm{f2}}\right)c\mathrm{f2}!\mathrm{f3}!}{\mathrm{f1}!}
c
is a correction factor depending on
\mathrm{f3}
i,0<j
k<0
\mathrm{f3}-\mathrm{f1}-\mathrm{f2}=n
\frac{c\mathrm{f3}!}{\mathrm{f1}!\mathrm{f2}!\left(\genfrac{}{}{0}{}{\mathrm{f2}}{\mathrm{f1}}\right)}
\dots \mathrm{f1}{!}^{i}\mathrm{f2}{!}^{j}\dots
i,j
\frac{\mathrm{f1}}{\mathrm{f2}}
r
1<|r|
\frac{\mathrm{f2}!\left(\genfrac{}{}{0}{}{\mathrm{f1}}{\mathrm{f2}}\right)\left(\mathrm{f1}-\mathrm{f2}\right)!}{\mathrm{f1}!}
|r|<1
\frac{\mathrm{f2}!}{\mathrm{f1}!\left(\genfrac{}{}{0}{}{\mathrm{f2}}{\mathrm{f1}}\right)\left(\mathrm{f2}-\mathrm{f1}\right)!}
a≔\frac{n!}{k!\left(n-k\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{!}}{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{!}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{n}}{\textcolor[rgb]{0,0,1}{k}}\right)
a≔\frac{n\left({n}^{2}+m-k+2\right)\left({n}^{2}+m\right)!}{k!\left({n}^{2}+m-k+2\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\right)\textcolor[rgb]{0,0,1}{!}}{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{!}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{k}}\right)}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
a≔\frac{{m!}^{3}}{\left(3m\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{!}}^{\textcolor[rgb]{0,0,1}{3}}}{\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{m}}\right)\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{m}}\right)}
a≔\frac{\mathrm{\Gamma }\left(m+\frac{3}{2}\right)}{\mathrm{sqrt}\left(\mathrm{\pi }\right)\mathrm{\Gamma }\left(m\right)}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\right)}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}}\right) |
Fundamentals of Transportation/Traffic Flow/Solution - Wikibooks, open books for an open world
Fundamentals of Transportation/Traffic Flow/Solution
{\displaystyle q=N\left({\frac {3600}{t_{measured}}}\right)=4\left({\frac {3600}{15}}\right)=960veh/hr\,\!}
{\displaystyle k={\frac {N}{L}}={\frac {4*1000}{280}}=14.2veh/km\,\!}
{\displaystyle {\overline {v_{t}}}={\frac {1}{N}}\sum \limits _{n=1}^{N}{v_{n}}={\frac {1}{4}}\left({72+90+80+88}\right)=82.5km/hr\,\!}
{\displaystyle {\begin{array}{l}{\overline {v_{s}}}={\frac {N}{\sum \limits _{n=1}^{N}{\frac {1}{v_{i}}}}}={\frac {4}{{\frac {1}{72}}+{\frac {1}{90}}+{\frac {1}{80}}+{\frac {1}{88}}}}=81.86\\t_{i}=L/v_{i}\\t_{A}=L/v_{A}=0.28/88=0.00318hr\\t_{B}=L/v_{B}=0.28/80=0.00350hr\\t_{C}=L/v_{C}=0.28/90=0.00311hr\\t_{D}=L/v_{D}=0.28/72=0.00389hr\\{\overline {v_{s}}}={\frac {NL}{\sum \limits _{n=1}^{N}{t_{n}}}}={\frac {4*0.28}{\left({0.00318+0.00350+0.00311+0.00389}\right)}}=81.87km/hr\\\end{array}}\,\!}
Retrieved from "https://en.wikibooks.org/w/index.php?title=Fundamentals_of_Transportation/Traffic_Flow/Solution&oldid=3291077" |
Solubility Equilibrium | Brilliant Math & Science Wiki
Divyanshi Verma and Jimin Khim contributed
The solubility of any compound is given by the enthalpy change in the solution. The solubility of any ionic compounds in water is given by the ionization constant at equilibrium; here it refers to the non-changement in the ionization constant of water. The ionization constant for acids and bases in water is especially sought out by finding the equilibrium constant for a particular temperature which is the concentration of the products to the reactants at equilibrium when the substance is not completely dissolved, likely in the case of non-polar solvents. In case of complete dissolution, it is more obvious that the ionization constant is the product of the reactants, i.e. the conjugate acids and bases as the compound is no more in the pure form and it's dissociated into ions. But in all these processes, the ionization constant of water (Kw) always remains constant. This is in accordance with the very famous Le Chatelier's principle. As the change in the equilibrium of the system (here water as neutral) by adding of a solute will shift the reaction to such a direction as to counter effect the changes caused by the added substance, although the Kw does not change, the concentration of H+ and OH- do change as there is a shift in the direction of reaction as well as more H+ and OH- ions from other compounds:
In solubility of any compound, the solution enthalpy, as previously mentioned, depends upon the enthalpy change, which again depends upon the enthalpy or the energy required to break one mole of any ionic compound into its gaseous state ions, more familiar lattice enthalpy, but not only the lattice enthalpy can give information about solubility but also how much energy it is then required to dissolve the ions in the solvent (if water) and then the hydration enthalpy. The hydration enthalpy is negative, which means it is exothermic in nature. On adding these enthalpies, the enthalpy of solution is figured out:
This also tells if the enthalpy of solvation (hydration enthalpy) is less than the lattice enthalpy, then the compound will not dissolve sufficiently. Therefore, it becomes essential for any polar or non-polar compound that its hydration enthalpy must be big enough to compensate for the lattice enthalpy which is endothermic.
In the equilibrium process of solubility of a compound, the rate of forward reaction (the ions formed) must be equal to the rate of formation of the solid/compound back. This is generally a case with the saturated solution when no more solute dissolves as the system attains equilibrium. The solute ions will form, collide, and react again to form the solute back, a dynamic equilibrium state is attained, so the overall concentration of the pure solute does not change. And the equilibrium constant at this equilibrium condition is called solubility product constant, which is the product of the concentration of the products and equal to the molar solubility of the solute/compound.
The values of Ksp constant are
1.1\cdot 10^{-10}
2\cdot 10^{-15}
for BaSO4 and AgCN, respectively. What will be the concentration of the ions dissociated and which one is sparingly soluble?
Cite as: Solubility Equilibrium. Brilliant.org. Retrieved from https://brilliant.org/wiki/solubility-equilibrium/ |
In the given equation as follows , use a table
In the given equation as follows , use a table of integrals to find the indefini
In the given equation as follows , use a table of integrals to find the indefinite integral:
\int xar\mathcal{s}c\left({x}^{2}+1\right)dx
In the given equation as follows , use a table of integrals to find the indefinite integral:-
\int xar\mathcal{s}c\left({x}^{2}+1\right)dx
\int x{\mathrm{sec}}^{-1}\left({x}^{2}+1\right)dx
Let us substitute,
{x}^{2}+1=z
2xdx=dz
\int \frac{1}{2}{\mathrm{sec}}^{-1}zdz
=\frac{1}{2}\left[{\mathrm{sec}}^{-1}z\int dz-\int \left[\frac{d}{dz}\left({\mathrm{sec}}^{-1}z\right)\int dz\right]\right]
=\frac{1}{2}\left[z{\mathrm{sec}}^{-1}z-\frac{1}{\sqrt{{z}^{2}-1}}z\right]
=\frac{1}{2}\left[\left({x}^{2}+1\right){\mathrm{sec}}^{-1}\left({x}^{2}+1\right)-\frac{{x}^{2}+1}{\sqrt{{\left({x}^{2}+1\right)}^{2}-1}}\right]+C
x={t}^{2},y=2t,0\le t\le 5
\left({x}^{2}+2xy-4{y}^{2}\right)dx-\left({x}^{2}-8xy-4{y}^{2}\right)dy=0
\int 2{x}^{3}\mathrm{cos}\left({x}^{2}\right)dx
{\int }_{0}^{6}|2x-4|dx
{\int }_{0}^{\frac{\pi }{4}}\left[\mathrm{sec}ti+{\mathrm{tan}}^{2}tj-t\mathrm{sin}tk\right]dt
\left({x}^{2}+{y}^{2}\right)dV
{x}^{2}+{y}^{2}+{z}^{2}=4
{x}^{2}+{y}^{2}+{z}^{2}=9
\int \mathrm{sin}2xdx |
Graphical_timeline_of_the_Big_Bang Knowpia
This timeline of the Big Bang shows a sequence of events as currently theorized by scientists.
It is a logarithmic scale that shows
{\displaystyle 10\cdot \log _{10}}
second instead of second. For example, one microsecond is
{\displaystyle 10\cdot \log _{10}0.000001=10\cdot (-6)=-60}
. To convert −30 read on the scale to second calculate
{\displaystyle 10^{-{\frac {30}{10}}}=10^{-3}=0.001}
second = one millisecond. On a logarithmic time scale a step lasts ten times longer than the previous step.
"Timeline of the Big Bang - The Big Bang and the Big Crunch - The Physics of the Universe". www.physicsoftheuniverse.com. Retrieved 2018-05-17. |
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle \langle u\rangle }
{\displaystyle \langle w\rangle }
{\displaystyle \langle u'_{i}u'_{j}\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle ||{\vec {U}}||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
. The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.
{\displaystyle ||{\vec {U}}_{\mathrm {PIV} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {LES} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle -0.788}
{\displaystyle 0.03}
{\displaystyle -0.843}
{\displaystyle 0.037}
{\displaystyle -0.918}
{\displaystyle 0}
{\displaystyle -1.1}
{\displaystyle 0}
{\displaystyle -0.533}
{\displaystyle 0}
{\displaystyle -0.534}
{\displaystyle 0}
{\displaystyle -0.507}
{\displaystyle 0.036}
{\displaystyle -0.50}
{\displaystyle 0.04}
{\displaystyle -0.697}
{\displaystyle 0.051}
{\displaystyle -0.735}
{\displaystyle 0.06}
{\displaystyle -0.513}
{\displaystyle 0.017}
{\displaystyle -0.513}
{\displaystyle 0.02}
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }={\frac {x-x_{\mathrm {Cyl} }}{x_{\mathrm {Cyl} }-x_{\mathrm {V1} }}}}
{\displaystyle x_{\mathrm {Cyl} }=-0.5D}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {V1} }}
{\displaystyle \langle u(z)\rangle /u_{\mathrm {b} }}
{\displaystyle u(z)}
{\displaystyle x_{\mathrm {adj} }=-0.25}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle \langle u_{i}'u_{j}'(z)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(z)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle x_{\mathrm {adj} }=-1.5}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle w'w'\rangle }
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u'w'\rangle }
{\displaystyle \langle w(x)\rangle /u_{\mathrm {b} }}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle w(x)\rangle }
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }\approx -0.1}
{\displaystyle x_{\mathrm {adj} }=-0.65}
{\displaystyle \langle u_{i}'u_{j}'(x)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(x)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u_{i}'u_{j}'\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.074u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.079u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.09u_{\mathrm {b} }^{2}}
{\displaystyle 0=P+\nabla T-\epsilon +C}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle C}
{\displaystyle v}
{\displaystyle P=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}}
{\displaystyle T=\underbrace {-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle } _{\text{turbulent fluctuations}}\underbrace {-{\frac {1}{\rho }}\langle u_{i}'p'\rangle } _{\text{pressure transport}}\underbrace {+2\nu \langle u_{j}'s_{ij}\rangle } _{\text{viscous diffusion}}}
{\displaystyle \epsilon =2\nu \langle s_{ij}s_{ij}\rangle }
{\displaystyle s_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}'}{\partial x_{j}}}+{\frac {\partial u_{j}'}{\partial x_{i}}}\right)}
{\displaystyle \epsilon _{\mathrm {total} }=\epsilon _{\mathrm {res} }+\epsilon _{\mathrm {SGS} }=2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle }
{\displaystyle C=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}}
{\displaystyle D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {PIV} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {LES} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle 0.3u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {LES} }\approx 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {PIV} }\approx 0.2u_{\mathrm {b} }^{3}/D}
{\displaystyle x=-0.7D}
{\displaystyle P}
{\displaystyle \nabla T_{\mathrm {turb,PIV} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb,LES} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x=-0.75D}
{\displaystyle 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb,LES} }\approx 0.35u_{\mathrm {b} }^{3}/D}
{\displaystyle \nabla T_{\mathrm {press,LES} }=-{\frac {1}{\rho }}{\frac {\partial \langle u_{i}'p'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {visc,LES} }=2\nu {\frac {\partial \langle u_{j}'s_{ij}\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb} }}
{\displaystyle \nabla T_{\mathrm {press} }}
{\displaystyle \langle w\rangle <0}
{\displaystyle w-}
{\displaystyle w'}
{\displaystyle p'<0}
{\displaystyle \nabla T_{\mathrm {visc} }}
{\displaystyle |0.05|u_{\mathrm {b} }^{3}/D}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle \epsilon _{\mathrm {PIV} }=2\nu \langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \epsilon _{\mathrm {LES,total} }=(2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle )\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P}
{\displaystyle \epsilon _{\mathrm {LES} }=0.066u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {max} }}
{\displaystyle \epsilon _{\mathrm {max} }}
{\displaystyle C_{\mathrm {PIV} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle C_{\mathrm {LES} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x\approx -0.63D}
{\displaystyle C}
{\displaystyle R_{\mathrm {PIV} }=P+\nabla T_{\mathrm {turb} }-\epsilon +C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle -\nabla T_{\mathrm {press,LES} }}
{\displaystyle R_{\mathrm {LES} }=P+\nabla T-\epsilon _{\mathrm {total} }+C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle <|0.01|u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb} }=-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }={\frac {\langle p\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }={\frac {\langle \tau _{\mathrm {w} }\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle z_{1}\approx 0.0036D\approx 10\mathrm {px} }
{\displaystyle z_{1}\approx 0.0005D}
{\displaystyle z-}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle |c_{\mathrm {f} }|=0.01}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle 50\times 171(n\times m)}
{\displaystyle 143\times 131(n\times m)}
{\displaystyle n\cdot m}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle -}
{\displaystyle \epsilon {\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle v\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {press} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {visc} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \epsilon _{\mathrm {total} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle c_{\mathrm {p} }} |
Category electric permittivity +> CalculatePlus
Category electric permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ε (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. In electrostatics, the permittivity plays an important role in determining the capacitance of a capacitor.
In the simplest case, the electric displacement field D resulting from an applied electric field E is
{\displaystyle \mathbf {D} =\varepsilon \mathbf {E} .}
More generally, the permittivity is a thermodynamic function of state. It can depend on the frequency, magnitude, and direction of the applied field. The SI unit for permittivity is farad per meter (F/m).
The permittivity is often represented by the relative permittivity εr which is the ratio of the absolute permittivity ε and the vacuum permittivity ε0
{\displaystyle \kappa =\varepsilon _{\mathrm {r} }={\frac {\varepsilon }{\varepsilon _{0}}}}
This dimensionless quantity is also often and ambiguously referred to as the permittivity. Another common term encountered for both absolute and relative permittivity is the dielectric constant which has been deprecated in physics and engineering as well as in chemistry.
Relative permittivity is directly related to electric susceptibility (χ) by
{\displaystyle \chi =\kappa -1}
Latest from category electric permittivity
100 eps to deci vacuum permittivity (vacuum permittivities to deps)
1 μeps to zetta vacuum permittivities (micro vacuum permittivity to Zeps)
1 nano vacuum permittivity to eps (neps to vacuum permittivity)
2 ceps to Meps (centi vacuum permittivities to mega vacuum permittivities)
1,000 meps to aeps (milli vacuum permittivities to atto vacuum permittivities)
100 μeps to zetta vacuum permittivities (micro vacuum permittivities to Zeps)
5 pico vacuum permittivities to ueps (peps to micro vacuum permittivities)
9 peta vacuum permittivities to mega vacuum permittivities (Peps to Meps)
9 eps to giga vacuum permittivities (vacuum permittivities to Geps)
6 eps to giga vacuum permittivity (vacuum permittivities to Geps)
1 ceps to eps (centi vacuum permittivity to vacuum permittivity)
7 deps to Eeps (deci vacuum permittivities to exa vacuum permittivities)
8 peta vacuum permittivities to ceps (Peps to centi vacuum permittivities)
100 peta vacuum permittivities to neps (Peps to nano vacuum permittivities)
1,000 ceps to eps (centi vacuum permittivities to vacuum permittivities)
1,000 ceps to yocto vacuum permittivity (centi vacuum permittivities to yeps)
2 zepto vacuum permittivities to nano vacuum permittivities (zeps to neps)
100 ceps to pico vacuum permittivity (centi vacuum permittivities to peps)
100 kilo vacuum permittivities to pico vacuum permittivities (keps to peps)
6 hecto vacuum permittivities to vacuum permittivity (heps to eps)
9 pico vacuum permittivities to milli vacuum permittivities (peps to meps)
7 pico vacuum permittivities to zepto vacuum permittivity (peps to zeps)
100 deci vacuum permittivities to centi vacuum permittivities (deps to ceps)
10 pico vacuum permittivities to aeps (peps to atto vacuum permittivities)
Meps to ceps (mega vacuum permittivity to centi vacuum permittivity)
7 meps to micro vacuum permittivities (milli vacuum permittivities to μeps)
9 centi vacuum permittivities to exa vacuum permittivity (ceps to Eeps)
4 Meps to vacuum permittivity (mega vacuum permittivities to eps)
4 micro vacuum permittivities to femto vacuum permittivity (μeps to feps)
8 centi vacuum permittivities to mega vacuum permittivity (ceps to Meps)
2 zetta vacuum permittivities to centi vacuum permittivities (Zeps to ceps)
10 zepto vacuum permittivities to hecto vacuum permittivity (zeps to heps)
aeps to meps (atto vacuum permittivity to milli vacuum permittivity)
1 centi vacuum permittivity to myria vacuum permittivities (ceps to myeps)
2 Meps to kilo vacuum permittivities (mega vacuum permittivities to keps)
micro vacuum permittivity to zepto vacuum permittivities (μeps to zeps)
Peps to myeps (peta vacuum permittivity to myria vacuum permittivity)
Meps to nano vacuum permittivity (mega vacuum permittivity to neps)
1 peta vacuum permittivity to myria vacuum permittivity (Peps to myeps)
8 peta vacuum permittivities to zepto vacuum permittivities (Peps to zeps)
9 Eeps to myria vacuum permittivity (exa vacuum permittivities to myeps)
8 Meps to milli vacuum permittivity (mega vacuum permittivities to meps)
4 pico vacuum permittivities to vacuum permittivity (peps to eps)
4 femto vacuum permittivities to Teps (feps to tera vacuum permittivities)
5 Meps to micro vacuum permittivities (mega vacuum permittivities to μeps)
10 meps to nano vacuum permittivity (milli vacuum permittivities to neps)
4 meps to deca vacuum permittivity (milli vacuum permittivities to daeps)
2 peta vacuum permittivities to peps (Peps to pico vacuum permittivities)
6 centi vacuum permittivities to meps (ceps to milli vacuum permittivities)
2 peta vacuum permittivities to centi vacuum permittivities (Peps to ceps)
deca vacuum permittivities to kilo vacuum permittivity (daeps to keps)
μeps to zetta vacuum permittivities (micro vacuum permittivity to Zeps)
10 hecto vacuum permittivities to zetta vacuum permittivity (heps to Zeps)
4 meps to nano vacuum permittivity (milli vacuum permittivities to neps)
9 micro vacuum permittivities to myria vacuum permittivity (μeps to myeps)
5 Meps to ceps (mega vacuum permittivities to centi vacuum permittivities)
2 myria vacuum permittivities to yeps (myeps to yocto vacuum permittivities)
myeps to Peps (myria vacuum permittivity to peta vacuum permittivity)
5 Meps to yotta vacuum permittivities (mega vacuum permittivities to Yeps)
5 Meps to tera vacuum permittivities (mega vacuum permittivities to Teps)
5 Meps to deci vacuum permittivity (mega vacuum permittivities to deps)
5 Meps to meps (mega vacuum permittivities to milli vacuum permittivities)
5 Meps to nano vacuum permittivity (mega vacuum permittivities to neps)
5 Meps to zeps (mega vacuum permittivities to zepto vacuum permittivities)
5 Meps to yocto vacuum permittivities (mega vacuum permittivities to yeps)
1,000 Meps to exa vacuum permittivities (mega vacuum permittivities to Eeps)
100 Meps to exa vacuum permittivities (mega vacuum permittivities to Eeps)
10 Meps to exa vacuum permittivities (mega vacuum permittivities to Eeps)
9 Meps to exa vacuum permittivities (mega vacuum permittivities to Eeps)
1 Meps to exa vacuum permittivities (mega vacuum permittivity to Eeps)
5 exa vacuum permittivities to Meps (Eeps to mega vacuum permittivities)
Meps to exa vacuum permittivities (mega vacuum permittivity to Eeps)
4 atto vacuum permittivities to yotta vacuum permittivity (aeps to Yeps)
4 atto vacuum permittivities to exa vacuum permittivities (aeps to Eeps)
4 atto vacuum permittivities to Peps (aeps to peta vacuum permittivities)
4 atto vacuum permittivities to giga vacuum permittivities (aeps to Geps)
4 atto vacuum permittivities to hecto vacuum permittivity (aeps to heps)
4 atto vacuum permittivities to vacuum permittivity (aeps to eps)
4 atto vacuum permittivities to deci vacuum permittivity (aeps to deps)
4 atto vacuum permittivities to centi vacuum permittivity (aeps to ceps)
4 atto vacuum permittivities to micro vacuum permittivities (aeps to μeps)
4 peps to atto vacuum permittivities (pico vacuum permittivities to aeps)
eps to exa vacuum permittivities (vacuum permittivity to Eeps)
eps to giga vacuum permittivity (vacuum permittivity to Geps)
eps to myria vacuum permittivity (vacuum permittivity to myeps)
eps to kilo vacuum permittivities (vacuum permittivity to keps)
eps to deca vacuum permittivity (vacuum permittivity to daeps)
eps to centi vacuum permittivities (vacuum permittivity to ceps)
eps to nano vacuum permittivity (vacuum permittivity to neps)
eps to atto vacuum permittivities (vacuum permittivity to aeps)
eps to yocto vacuum permittivity (vacuum permittivity to yeps) |
Construct Linear Time Invariant Models - MATLAB & Simulink
Zero/Pole/Gain Models
LTI Object Properties
Additional LTI Input and Output Properties
Input and Output Names
LTI Model Characteristics
Model Predictive Control Toolbox™ software supports the same LTI model formats as does Control System Toolbox™ software. You can use whichever is most convenient for your application and convert from one format to another. For more details, see Basic Models.
A transfer function (TF) relates a particular input/output pair of (possibly vector) signals. For example, if u(t) is a plant input and y(t) is an output, the transfer function relating them might be:
\frac{Y\left(s\right)}{U\left(s\right)}=G\left(s\right)=\frac{s+2}{{s}^{2}+s+10}{e}^{-1.5s}
This TF consists of a numerator polynomial, s+2, a denominator polynomial, s2+s+10, and a delay, which is 1.5 time units here. You can define G using Control System Toolbox tf function:
Gtf1 = tf([1 2], [1 1 10],'OutputDelay',1.5)
exp(-1.5*s) * ------------
s^2 + s + 10
Like the TF format, the zero/pole/gain (ZPK) format relates an input/output pair of (possibly vector) signals. The difference is that the ZPK numerator and denominator polynomials are factored, as in
G\left(s\right)=2.5\frac{s+0.45}{\left(s+0.3\right)\left(s+0.1+0.7i\right)\left(s+0.1-0.7i\right)}
(zeros and/or poles are complex numbers in general).
You define the ZPK model by specifying the zero(s), pole(s), and gain as in
poles = [-0.3, -0.1+0.7*i, -0.1-0.7*i];
Gzpk1 = zpk(-0.45,poles,2.5);
The state-space format is convenient if your model is a set of LTI differential and algebraic equations.
The linearized model of a Continuously Stirred Tank Reactor (CSTR) is shown in CSTR Model. In the model, the first two state variables are the concentration of reagent (here referred to as CA and measured in kmol/m3) and the temperature of the reactor (here referred to as T, measured in K), while the first two inputs are the coolant temperature (Tc, measured in K, used to control the plant), and the inflow feed reagent concentration CAf, measured in kmol/m3, (often considered as unmeasured disturbance).
A state-space model can be defined as follows:
This defines a continuous-time state-space model stored in the variable CSTR. The model is continuous time because no sampling time was specified, and therefore a default sampling value of zero (which means that the model is continuous time) is assumed. You can also specify discrete-time state-space models. You can specify delays in both continuous-time and discrete-time models.
The ss function in the last line of the above code creates a state-space model, CSTR, which is an LTI object. The tf and zpk commands described in Transfer Function Models and Zero/Pole/Gain Models also create LTI objects. Such objects contain the model parameters as well as optional properties.
The following code sets some optional input and outputs names and properties for the CSTR state-space object:
CSTR=setmpcsignals(CSTR,'MV',1,'UD',2,'MO',1,'UO',2)
The first three lines specify labels for the input, output and state variables. The next four specify the signal type for each input and output. The designations MV, UD, MO, and UO mean manipulated variable, unmeasured disturbance, measured output, and unmeasured output. (See MPC Signal Types for definitions.) For example, the code specifies that input 2 of model CSTR is an unmeasured disturbance. The last line causes the LTI object to be displayed, generating the following lines in the MATLAB® Command Window:
The optional InputName and OutputName properties affect the model displays, as in the above example. The software also uses the InputName and OutputName properties to label plots and tables. In that context, the underscore character causes the next character to be displayed as a subscript.
As mentioned in MPC Signal Types, Model Predictive Control Toolbox software supports three input types and two output types. In a Model Predictive Control Toolbox design, designation of the input and output types determines the controller dimensions and has other important consequences.
For example, suppose your plant structure were as follows:
Two manipulated variables (MVs)
Three measured outputs (MOs)
One measured disturbance (MD)
Two unmeasured outputs (UOs)
Two unmeasured disturbances (UDs)
The resulting controller has four inputs (the three MOs and the MD) and two outputs (the MVs). It includes feedforward compensation for the measured disturbance, and assumes that you wanted to include the unmeasured disturbances and outputs as part of the regulator design.
If you didn't want a particular signal to be treated as one of the above types, you could do one of the following:
Eliminate the signal before using the model in controller design.
For an output, designate it as unmeasured, then set its weight to zero.
For an input, designate it as an unmeasured disturbance, then define a custom state estimator that ignores the input.
By default, the software assumes that unspecified plant inputs are manipulated variables, and unspecified outputs are measured. Thus, if you didn't specify signal types in the above example, the controller would have four inputs (assuming all plant outputs were measured) and five outputs (assuming all plant inputs were manipulated variables).
Since the D matrix is zero, the output does not instantly respond to change in the input. The Model Predictive Control Toolbox software prohibits direct (instantaneous) feedthrough from a manipulated variable to an output. For example, the CSTR state-space model could include direct feedthrough from the unmeasured disturbance, CAf, to either CA or T but direct feedthrough from Tc to either (measured or not) output would violate this restriction. When the model has a direct feedthrough from Tc, you can add a small delay at this input to circumvent the problem.
For CSTR, the default Model Predictive Control Toolbox assumptions are incorrect. You must set its InputGroup and OutputGroup properties, as illustrated in the above code, or modify the default settings when you load the model into MPC Designer.
Use setmpcsignals to make type definition. For example:
CSTR = setmpcsignals(CSTR,'UD',2,'UO',2);
sets InputGroup and OutputGroup to the same values as in the previous example. The CSTR display would then include the following lines:
Notice that setmpcsignals sets unspecified inputs to Manipulated and unspecified outputs to Measured.
Control System Toolbox software provides functions for analyzing LTI models. Some of the more commonly used are listed below. Type the example code at the MATLAB prompt to see how they work for the CSTR example.
damp(CSTR) Displays the damping ratio, natural frequency, and time constant of the poles of CSTR.
pzmap(CSTR)
Plots the poles and zeros of CSTR.
pole(CSTR)
Calculates the poles of CSTR (to check stability, etc.).
tzero(CSTR)
Calculates the transmission zeros of CSTR.
dcgain(CSTR)
Calculates the steady state gain matrix of CSTR.
Plots unit-step responses of CSTR.
stepinfo(CSTR) Calculates rise time, settling time, and other step-response characteristics of CSTR.
impulse(CSTR)
Plots the unit-impulse responses of CSTR.
sigma(CSTR)
Plots the singular values of the frequency response of CSTR.
bode(CSTR)
Plots the Bode frequency responses of CSTR.
nyquist(CSTR)
Plots the Nyquist frequency responses of CSTR.
nichols(CSTR)
Plots the Nichols frequency responses of CSTR.
linearSystemAnalyzer(CSTR)
Opens the Linear System Analyzer with the CSTR model loaded. You can then display model characteristics by making menu selections.
tf | zpk | ss | setmpcsignals |
Equation in slope intercept form. line f has an equation
Equation in slope intercept form. line f has an equation of y=-9/10x-7 line g in
Equation in slope intercept form. line f has an equation of y=-9/10x-7 line g includes the point (-3,-5)and is perpendicular to line f. What is the equation of line g?
Given, line f has an equation of
y=-\frac{9}{10}x-7
\therefore
Slope of line f,
{m}_{1}=-\left(\frac{9}{10}\right)
Line g is perpendicular to line f
\therefore
Slope of line g,
{m}_{2}=-\left(-\frac{1}{\frac{9}{10}}\right)=\frac{10}{9}
line g includes the point (-3,-5)
\therefore
equation of line g:
\left(y-\left(-5\right)\right)=\left(\frac{10}{9}\right)\left(x-\left(-3\right)\right)
⇒\left(y+5\right)=\left(\frac{10}{9}\right)\left(x+3\right)
A=\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right],b\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]
\stackrel{―}{x}=
Find the values of c and d that make the equation true.
c+3ci=4+di
Solve for s in the equation 6s=1.
(a)Use the substitution method to justify that the given system of equations has no solution.
(b)What do you know about the two lines in this system of equations?
What is the solution of a system of linear equations?
Solve the system of equations x+y=-1 and 5x-7y=79 by combining the equations. |
an object dropped from cliff falls with a constant acceleration of 10m/s2 find its speed after 2s after it was - Maths - Linear Inequalities - 7342619 | Meritnation.com
acceleration. a = 10 m/s2
v\quad =\quad u\quad +\quad at\phantom{\rule{0ex}{0ex}}=\quad 0\quad +\quad \left(10\right)\quad \times \quad 2\phantom{\rule{0ex}{0ex}}=\quad 20\quad \mathrm{m}/\mathrm{s}
So, after 2 sec, its velocity is 20 m/s. |
Department of Mathematics, Petroleum Institute, Abu Dhabi, United Arab Emirates.
A familiar and natural decomposition of square matrices leads to the construction of a Pfaffian with the same value as the determinant of the square matrix.
Determinant, Pfaffian
Miller, G. (2017) A Note on a Natural Correspondence of a Determinant and Pfaffian. Open Journal of Discrete Mathematics, 7, 1-2. doi: 10.4236/ojdm.2016.71001.
A square matrix is the sum of its symmetric and skew symmetric parts, S and C respectively:
M=S+C
For a 2 by 2 matrix it happens that the determinant is
|M|=|S|+|C|
But this is not true in general. However, in block matrix form, of any size, we do have
|\begin{array}{cc}C& S\\ -S& -C\end{array}|={|M|}^{2}
This follows from the properties of determinants according to the sequence of equalities:
|\begin{array}{cc}C& S\\ -S& -C\end{array}|=|\begin{array}{cc}S+C& S\\ -S-C& -C\end{array}|=|\begin{array}{cc}S+C& S\\ 0& S-C\end{array}|={|M|}^{2}
|S-C|=|{\left(S-C\right)}^{\text{T}}|=|{S}^{\text{T}}-{C}^{\text{T}}|=|S+C|=|M|
B=SJ
. The matrix J has zero entries except for all ones on the secondary diagonal (NE to SW). The matrix B is just S with the order of columns reversed.
Form the skew matrix in block form
\left(\begin{array}{cc}C& B\\ -{B}^{\text{T}}& {}^{\text{T}}C\end{array}\right)
We have indicated the transpose across the secondary diagonal by
{}^{\text{T}}C
. (In general,
{}^{\text{T}}A=J{A}^{\text{T}}J
Then, we claim that the Pfaffian formed by the triangular array above the main diagonal in this matrix has the same value as the determinant of the original matrix M.
\left(\begin{array}{cc}3& -3\\ 7& 4\end{array}\right)=\left(\begin{array}{cc}3& 2\\ 2& 4\end{array}\right)+\left(\begin{array}{cc}0& -5\\ 5& 0\end{array}\right)
and the determinant is 33.
We calculate the equivalent Pfaffian as
‖\begin{array}{ccc}-5& 2& 3\\ & 4& 2\\ & & -5\end{array}‖=25-4+12=33
using the “cofactor” expansion of a Pfaffian. For the cofactor expansion see [1] .
|\begin{array}{cc}C& B\\ -{B}^{\text{T}}& {}^{\text{T}}C\end{array}|={|M|}^{2}
The calculation follows.
|\begin{array}{cc}C& B\\ -{B}^{\text{T}}& {}^{\text{T}}C\end{array}|=|\begin{array}{cc}C& SJ\\ -JS& -JCJ\end{array}|=|\begin{array}{cc}I& 0\\ 0& J\end{array}||\begin{array}{cc}C& S\\ -S& -C\end{array}||\begin{array}{cc}I& 0\\ 0& J\end{array}|=|\begin{array}{cc}C& S\\ -S& -C\end{array}|
The last equation follows by taking the determinant of the three factors. And as mentioned above,
|\begin{array}{cc}C& S\\ -S& -C\end{array}|={|M|}^{2}.
So the construction of the Pfaffian array delivers the value of the original determinant, up to sign.
It is easy enough to verify by calculation that the correct sign is given in the case of a two by two matrix rewritten as a Pfaffian array. In general the expansion of the determinant will contain a term which is a product of the diagonal elements of the matrix regardless of the other entries. In the same way, the expansion of the Pfaffian will contain this same term, independent of all other entries. So the sign of the Pfaffian is correct.
The author thanks the Petroleum Institute for continued financial support.
[1] Ishikawa, M. and Wakayama, M. (2006) Applications of the Minor Summation Formula III: Plucker Relations, Lattice Paths and Pfaffians. Journal of Combinatorial Theory, Series A, 113, 113-155. |
Quantile predictions for out-of-bag observations from bag of regression trees - MATLAB - MathWorks América Latina
L={Q}_{1}-1.5*IQR
U={Q}_{3}+1.5*IQR,
{w}_{tj}\left(x\right)=\frac{I\left\{{X}_{j}\in {S}_{t}\left(x\right)\right\}}{\sum _{k=1}^{{n}_{\text{train}}}I\left\{{X}_{k}\in {S}_{t}\left(x\right)\right\}},
{w}_{j}^{\ast }\left(x\right)=\frac{1}{T}\sum _{t=1}^{T}{w}_{tj}\left(x\right). |
In this problem set, we will use angle brackets to denote points, e.g. \(\point{1\\0},\) and square brackets to denote vectors, e.g.
\vector{3\\2}.
This is not standard notation. Typically both are denoted by square brackets (or parentheses), and distinguished by context (if at all).
Compute the following expressions if they make sense, or answer "meaningless" if they do not make sense.
\(\point{1\\2} - \point{2\\1}\)
\(\point{-1\\\tfrac12} + \vector{1\\\tfrac32}\)
\(\point{1\\0} + \point{0\\1}\)
\vector{-3\\1} + \vector{2\\2}
\vector{1\\1\\1} + \vector{2\\1}
2\vector{1\\\tfrac12}
\(2\point{1\\\tfrac12}\)
2\vector{1\\\tfrac32\\-1} - \vector{0\\\tfrac12\\-1}
\vector{-1\\1}
\(\point{0\\2}\)
meaningless: can't add a point to a point
\vector{-1\\3}
meaningless: can't add vectors of different sizes (more abstractly, can't add vectors belonging to different vector spaces)
\vector{2\\1}
meaningless: can't multiply a point by a scalar
\vector{2\\\tfrac52\\-1} |
Demographic Models - Course Hero
Introduction to Biology/Populations/Demographic Models
Learn all about demographic models in just a few minutes! Jessica Pamment, professional lecturer at DePaul University, explains two types of demographic models that determine growth patterns: exponential growth models and logistic growth models.
A demographic describes the life statistics of a population, such as births and deaths. It also includes life statistics of populations of animal and plant species in a given area. Each of the following studies uses demographics, the statistical data of populations: An environmental group is taking a census of elephants by flying over a massive preserve and counting elephants from the sky. Conservationists, concerned about invasive plant species encroaching on a wild grassland, do an inventory of noxious cheatgrass, highway ice plant, and red broom to determine how much more territory these species now occupy. A town performs a census to determine its population growth by age to determine future needs for day care, elementary schools, and senior citizen facilities. Scientists use these data to plan growth and use of land, service needs, and facility requirements. Determining growth patterns requires the analysis of demographic models, and the two most commonly used are exponential models and logistic models.
Exponential growth of a population is the rate of population growth in situations where food and resources are unlimited. Exponential growth assumes ideal conditions, which are impossible in nature; conditions might be ideal for a period of time (and exponential growth can occur), but they will not be ideal indefinitely. For example, a female house mouse (mouse A) and her mate establish a home in an old granary. Food is consistently available, conditions are excellent for reproduction, and there is no risk of predation. Mouse A produces a litter of four pups, two females and two males. Ten weeks later, mouse A and her two daughters (B and C) all deliver litters of healthy pups. Of the 15 pups, 8 are female. In an additional 10-week period, all females—through mouse K—produce new litters. Now, the mouse population has exploded from 2 to 77 mice in five months.
Scientists can create ideal conditions by controlling the breeding of individuals within a controlled population, ensuring adequate food, providing medical attention to deal with disease, and preventing predation. This is the method by which zoos around the world monitor controlled breeding of endangered species. The American bison is a case study in exponential growth patterns. At one point, the bison population on the Great Plains ranged between 15 million and 100 million individuals, but human interference reduced this massive population to about 1,000 bison by 1888. Conservation efforts began in the early 1900s. Through assisted breeding and protection, the bison population has expanded exponentially to nearly 450,000 individuals in 2010.
At one point, the bison population on the Great Plains ranged between 15 and 100 million individuals, but human interference reduced the massive population to about 1,000 bison by 1888. Conservation efforts began in the early 1900s. Through assisted breeding and protection, the bison population has expanded exponentially (the rate of population growth in situations where food and resources are unlimited) to nearly 450,000 individuals in 2010.
The calculation for an exponential growth model uses the equation
\frac{\Delta N}{\Delta t}=rN
where ∆N is the change in the population divided by ∆t, the change in time, and rN is the rate of increase. This equals the rate of increase in a population. In the case of the mice, the change in number (∆N) would be 75 mice (77 offspring minus 2 original parents) divided by the change in time (∆t), 20 weeks, which equals a population rate increase of 3.75 mice per week. With the American bison, the 1888 population of 1,300 increased to 450,000 over a period of 122 years, or 448,700 bison divided by 122 years, which equals a rate of 3,678 bison per year.
A logistic growth model indicates how a population grows more slowly when it reaches the carrying capacity of its environment. The carrying capacity of an ecosystem varies as the food supply varies. In the Arctic, snowy owls, Arctic foxes, and stoats (a type of weasel) feed on lemmings. When lemming populations rise, so do the populations of owls, foxes, and stoats. When food is scarce, lemming populations decline, and predator populations decline accordingly because their food source is less available. In temperate forests, wildfires can destroy grasses, forbs, vines, and wildflowers and reduce tree populations. Ash from the fires nourishes the soil, and exponential growth follows the disruption of the original growth pattern.
Human populations have also encountered limiting factors to population growth. Specifically, epidemic diseases have dramatically changed worldwide populations in the past. The Black Plague wiped out 30–50% of the human population between 1347 and 1351. World War I caused 18 million deaths and was quickly followed by the Spanish flu pandemic, which killed 20–50 million people.
Adjustments to the exponential growth equation provide a mathematical solution to a logistic equation. Begin with the equation
\frac{\Delta N}{\Delta t}=rN
\frac{\Delta N}{\Delta t}
is the rate of change in the population. The term
\left(\frac{(K-N)}K\right)
represents the carrying capacity of the ecosystem, and the term rN represents the population size. The equation for logistic growth is
\frac{\Delta N}{\Delta t}=rN\;\left(\frac{(K-N)}K\right)
. The equation adjusts for the limited resources that may affect reproduction or growth rates.
Carrying capacity is the maximum number of individuals of a population that a habitat can support. As a population reaches the carrying capacity of its environment, changes in population growth level out. Logistic growth models adjust for the carrying capacity of a population and illustrate how a population grows more slowly when it reaches the carrying capacity of its environment.
As an example, consider a population of seals. A small group of seals colonize a new island. The seal population grows exponentially, following the exponential growth equation. Over time, the number of seals nears carrying capacity of the island: seals must compete with each other for limited food and places to raise their young, and not all of them survive this competition. The population growth plateaus, following the logistic growth equation.
<Population Limiting Factors>Human Population Traits |
rootbound - Maple Help
Home : Support : Online Help : Mathematics : Factorization and Solving Equations : Roots : rootbound
compute bound on complex roots of a polynomial
rootbound(p, x)
polynomial in x with numeric coefficients
Returns a positive integer N such that
|r|<N
for all complex roots r of p. In general, this bound is better than Cauchy's bound of
⌈1+\frac{\mathrm{maxnorm}\left(p\right)}{|\mathrm{lcoeff}\left(p\right)|}⌉
p≔x↦{x}^{4}-10\cdot {x}^{2}+1:
\mathrm{rootbound}\left(p\left(x\right),x\right)
\textcolor[rgb]{0,0,1}{4}
\mathrm{ceil}\left(1+\frac{\mathrm{maxnorm}\left(p\left(x\right)\right)}{\mathrm{abs}\left(\mathrm{lcoeff}\left(p\left(x\right)\right)\right)}\right)
\textcolor[rgb]{0,0,1}{11}
\mathrm{fsolve}\left(p\left(x\right)=0,x\right)
\textcolor[rgb]{0,0,1}{-3.146264370}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-0.3178372452}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.3178372452}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3.146264370}
q≔x↦\left(x-3.2\right)\cdot \left(x-2.5\right)\cdot \left(x-0.3\right)\cdot \left(x+1.3\right)\cdot \left(x+2.5\right)\cdot \left(x+3.6\right):
\mathrm{rootbound}\left(q\left(x\right),x\right)
\textcolor[rgb]{0,0,1}{6}
\mathrm{ceil}\left(1+\frac{\mathrm{maxnorm}\left(\mathrm{expand}\left(q\left(x\right)\right)\right)}{\mathrm{abs}\left(\mathrm{lcoeff}\left(q\left(x\right)\right)\right)}\right)
\textcolor[rgb]{0,0,1}{78}
Monagan, M.B. "A Heuristic Irreducibility Test for Univariate Polynomials." J. of Symbolic Comp. Vol. 13 No. 1. Academic Press, (1992): 47-57. |
So far you have been able to extend the rules for real numbers to add, subtract, and multiply complex numbers, but what about dividing? Can you use what you know about real numbers to divide one complex number by another? In other words, if a problem looks like this:
\cfrac { 3 + 2 i } { - 4 + 7 i }
What needs to be done to get an answer in the form of a single complex number,
a + bi
Natalio had an idea. He said, “I'll bet we can use the conjugate!”
“How?” asked Ricki.
“Well, it's a fraction. Can't we multiply the numerator and denominator by the same number?” Natalio replied.
Natalio was not very clear in his explanation. Show Ricki what he meant they should do.
\frac{(3+2i)(-4-7i)}{(-4+7i)(-4-7i)}=\frac{2-29i}{65}
Using Natalio's ideas you probably still came up with a fraction in part (a), but the denominator should be a whole number. To write a complex number such as
\frac { c + d i } { m }
a + bi
, just use the distributive property to rewrite the result as
\frac { c } { m } + \frac { d } { m } i
. Rewrite your result for part (a) in this form.
\frac{2}{65}-\frac{29}{65}i |
Existence Results for a px-Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition
2013 Existence Results for a
p\left(x\right)
-Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition
Libo Wang, Minghe Pei
We consider the existence and multiplicity of solutions for the
p\left(x\right)
-Kirchhoff-type equations without Ambrosetti-Rabinowitz condition. Using the Mountain Pass Lemma, the Fountain Theorem, and its dual, the existence of solutions and infinitely many solutions were obtained, respectively.
Libo Wang. Minghe Pei. "Existence Results for a
p\left(x\right)
-Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition." J. Appl. Math. 2013 1 - 8, 2013. https://doi.org/10.1155/2013/914210
Libo Wang, Minghe Pei "Existence Results for a
p\left(x\right)
-Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-8, (2013) |
Polluted bootstrap percolation in three dimensions
February 2021 Polluted bootstrap percolation in three dimensions
Janko Gravner, Alexander E. Holroyd, David Sivakoff
Janko Gravner,1 Alexander E. Holroyd,2 David Sivakoff3
1Mathematics Department, University of California, Davis
3Departments of Statistics and Mathematics, The Ohio State University
In the polluted bootstrap percolation model, vertices of the cubic lattice
{\mathbb{Z}}^{3}
are independently declared initially occupied with probability p or closed with probability q, where
\mathit{p}+\mathit{q}\le 1
. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least 3 occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as
\mathit{p},\mathit{q}\to 0
. We show that this density converges to 1 if
\mathit{q}\ll {\mathit{p}}^{3}{\left(log{\mathit{p}}^{-1}\right)}^{-3}
for both standard and modified rules. Our principal result is a complementary bound with a matching power for the modified model: there exists C such that the final density converges to 0 if
\mathit{q}>\mathit{C}{\mathit{p}}^{3}
. For the standard model, we establish convergence to 0 under the stronger condition
\mathit{q}>\mathit{C}{\mathit{p}}^{2}
Janko Gravner. Alexander E. Holroyd. David Sivakoff. "Polluted bootstrap percolation in three dimensions." Ann. Appl. Probab. 31 (1) 218 - 246, February 2021. https://doi.org/10.1214/20-AAP1588
Received: 1 June 2019; Published: February 2021
Keywords: Bootstrap percolation , cellular automaton , critical scaling
Janko Gravner, Alexander E. Holroyd, David Sivakoff "Polluted bootstrap percolation in three dimensions," The Annals of Applied Probability, Ann. Appl. Probab. 31(1), 218-246, (February 2021) |
GDP Deflator - Course Hero
Macroeconomics/GDP/GDP Deflator
The GDP deflator is a number that represents the current prices of various goods and services versus their past prices of a given year. It used to measure the level of price changes over time relative to a base year. This allows economists to measure and track inflation or deflation. If current prices are used to measure GDP, true economic output can be over- or understated. The GDP deflator compensates for changing price levels over time (inflation or deflation). This measure is an implicit index of the price level that allows economists to measure real changes in economic output. Economists also use it to convert nominal GDP in any given year into real GDP.
Economists analyze changes in real GDP to determine the growth of output in an economy. Real GDP measures the total value produced using constant prices, isolating the effect of price changes. As a result, real GDP is a better gauge of changes in the output level of an economy.
Note the GDP price deflator in the U.S. over a period of 25 years. It is compared to the Consumer Price Index for all urban consumers (CPI-U) in the same period. GDP Deflator is a measure of the prices of all goods and services while the CPI-U is a measure of only goods bought by urban consumers. Economists monitor both of these economic indicators.
\text{GDP}\;\text{Deflator}=\frac{\text{Nominal}\;\text{GDP}}{\text{Real}\;\text{GDP}}\times100
The GDP deflator is a tool used to measure the level of price changes over time so that current prices can be accurately compared to historical prices. If the nominal GDP and the GDP deflator are both known quantities but real GDP is not, the following formula can be used to solve for real GDP:
\text{Real}\;\text{GDP}=\frac{\text{Nominal}\;\text{GDP}}{\text{GDP}\;\text{Deflator}}\times100
<Calculating GDP>Measuring and Analyzing Changes in GDP |
To John Scott 11 December [1862]1
Decr. 11th.—
I have read your paper with much interest.2 You ask for remarks on matter, which is alone really important, shall you think me impertinent (I am sure I do not mean to be so) if I hazard a remark on style, which is of more importance than some think? In my opinion (whether or no worth much) your paper would have been much better if written more simply & less elaborated,—more like your letters. It is a golden rule always to use, if possible, a short old Saxon word. Such a sentence as “so purely dependent is the incipient plant on the specific morphological tendency”3—does not sound to my ears like good mother English—it wants translating.—
Here & there you might, I think, have condensed some sentences: I go on plan of thinking every single word which can be omitted without actual loss of sense as a decided gain.— Now perhaps you will think me a muddling intruder; Anyhow it is the advice of an old hackneyed writer who sincerely wishes you well.—
Your remarks on the two sexes counteracting variability in product of the one is new to me.4 But I cannot avoid thinking that there is something unknown & deeper in seminal generation. Reflect on the long succession of embryological changes in every animal. Does a bud ever produce cotyledons or embryonic leaves? I have been much interested by your remarks on inheritance at corresponding ages; I hope you will, as you say, continue to attend to this.5 Is it true that female plants always produce female by parthenogenesis?6 If you can answer this I shd. be glad; it bears on my Primula work: I thought on subject but gave up investigating what had been observed, because female Bee by parthenogenesis produces males alone. Your paper has told me much that in my ignorance was quite new to me.—
Thanks about P. Scotica. If any important criticisms are made on Primula to Bot. Soc. I shd. be glad to hear them.7 If you think fit, you may state that I repeated the crossing experiments on P. Sinensis & Cowslip with the same result this Spring as last year—indeed with rather more marked difference in fertility of the two crosses.8 In fact had I then proved Linum case I would not have wasted time in repetition.— I am determined I will at once publish on Linum.—9
We seem predestined to work on same subjects: for 3 or 4 summers I have worked hard at Drosera & Dionæa & have almost a volume of materials, which I suppose some day I shall publish; ie if ever I have time to work my materials into shape.10 Irritability of plants has been a hobby-horse to me. I suspect it will turn out that irritability of same nature, only intensified in Dionæa &c. is very common with plants.—11 If your paper on Drosera is published, I shd. be grateful for a copy.—12 Two German papers have been written on subject, but I now forget where; & there is good French paper on Structure, which I daresay you know13 I was right to be cautious in supposing you in error about Siphocampylus (no flowers were enclosed):14 I hope that you will make out whether the pistil presents two definite lengths; I shall be astounded if it does.—
I do not fully understand your objections to N. Selection;15 if I do, I presume they would apply with full force to, for instance, Birds. Reflect on modification of Arab-Turk Horse with our English Race-Horse.—16
I have had satisfaction to tell my publisher to send my Journal & Origin to your address.—17
I suspect with your fertile mind, you would find it far better to experiment on your own choice; but if on reflection you would like to try some which interest me, I shd. be truly delighted, & in this case would write in some detail.18 If you have means, to repeat Gärtners experiments on vars. of Verbascum or on Maize (see Origin) such experiments would be preeminently important.—19 I could never get vars. of Verbascum.20 I could suggest experiments on Potatoes analogous with case of Passiflora; even this case of Passiflora,, often as it has been repeated, might be with advantage repeated.21
I have worked like a slave (having counted about 9000 seeds) on Melastomas on meaning of the two sets of very different stamens, & as yet have been shamefully beaten, & I now cry for aid.—22
I could suggest what I believe very good scheme (at least Dr. Hooker thought so) for systematic degeneration of culinary plants & so find out their origin; but this would be laborious & work of years.—23
I am tired, so pray believe me, yours very faithfully | Ch. Darwin
P.S. I have been thinking that if you do not complete your beginning on the non-dimorphic Primula, I should like extremely to do so & would of course fully acknowledge your work.24 What I shd. do would be to fertilise a dozen or score of flowers with own pollen & another score of flowers with pollen of others, the most different, & count the seed of each capsule.—
Now could you aid me (unless you resolve to do the work yourself) & procure me from any Edinburgh nursery-garden
\frac{1}{2}
a dozen plants (or fewer, if so many could not be got) &
\frac{1}{2}
dozen of any of the other non-dimorphic species. They could be sent in pots in box addressed. “C. Darwin care of Down Postman, Bromley Kent” & I could pay by P.O.— They give or lend me all plants at Kew; but they are very weak in Primulas. I am sick of ordering plants at London nurseries; I so often get wrong thing. Could you aid me in this?—
The year is provided by the relationship to the letter from John Scott, 6 December [1862].
Scott 1862a. See letter from John Scott, 6 December [1862]. CD’s annotated copy of this paper is in the Darwin Pamphlet Collection–CUL.
In his paper, Scott aimed, by analysing the nature of fern spores, to shed new light on their ‘apparently anomalous properties’, and particularly on ‘that peculiar facility afforded by spores for the reproduction and perpetuation of any accidental variation of the parts upon which they originate’ (Scott 1862a, p. 210). Scott contrasted the reproductive organs in ferns with those in higher plants. Since in the latter the embryo was ‘the modified resultant of two originally distinct organs’ (p. 214), there would, he claimed, ‘necessarily be a greater tendency to efface any individual peculiarities of these than would have been the case, had the embryo been the product of a single organ.’ See also letter from John Scott, 6 December [1862] and n. 15.
Scott 1862a, pp. 217–8. Scott prefaced his remarks by quoting CD’s statement (Origin, p. 13): ‘That at whatever period a peculiarity first appears, it tends to appear in the offspring at a corresponding age, though sometimes earlier.’
CD refers to the Botanical Society of Edinburgh (see letters from John Scott, 6 December [1862] and 17 December [1862]).
Having discovered in 1861 that certain species of Linum were dimorphic, CD carried out crossing experiments in the summer of 1862 on L. perenne and L. grandiflorum. According to his ‘Journal’ (Correspondence vol. 10, Appendix II), CD wrote his paper, ‘Two forms in species of Linum’, between 11 and 21 December 1862.
See letter from John Scott, 6 December [1862]. In 1860, CD began to experiment on the sensitivity to various substances of the insectivorous plants, Drosera rotundifolia and Dionaea muscipula (see Correspondence vol. 8). He had hoped to continue and complete the experiments in the summer of 1861, but subsequently decided to postpone them (see Correspondence vol. 9, letter to J. D. Hooker, 4 February [1861], and letter to Daniel Oliver, 11 September [1861]). He carried out further experiments in May and September 1862 (see DAR 54: 29–49, 74–5). However, he did not again work extensively on this subject until 1872 (LL 3: 322); his findings were published in 1875 as Insectivorous plants.
See also letter to J. D. Hooker, 26 September [1862].
Scott read his paper on Drosera and Dionaea (Scott 1862b) before the Botanical Society of Edinburgh on 11 December 1862. CD cited Scott’s paper in Insectivorous plants, pp. 1 n.–2 n.
In a brief overview of the literature on insectivorous plants in Insectivorous plants, p. 1 n., CD referred to five papers on Drosera rotundifolia published by German and French writers before 1862: Milde 1852 and Nitschke 1861a and 1861b (in Botanische Zeitung), and Grönland 1855 and Trécul 1855 (in Annales des Sciences Naturelles (Botanique)).
See letter to John Scott, 3 December [1862], and letter from John Scott, 6 December [1862]. See also letter from John Scott, 17 December [1862].
CD apparently refers to the fact that an objection to natural selection based on the effect of blending inheritance on incipient varieties, would be particularly telling in cases where individuals paired for life (as with birds). CD had cited the gradual improvement of ‘the whole body of English racehorses’ by selection and training in Origin p. 35, noting that they had come to surpass the parent Arab stock. See also Correspondence vol. 10, Appendix VI.
John Murray. See letter to John Scott, 3 December [1862], and letter from John Scott, 6 December [1862].
The references are to Origin, pp. 269–71, and Gärtner 1844 and 1849; CD’s extensively annotated copies of the latter works are in the Darwin Library–CUL (see Marginalia 1: 248–98). CD considered the experiments particularly important because of their bearing on an objection raised against natural selection by Thomas Henry Huxley (see letter to T. H. Huxley, 18 December [1862] and nn. 8–12, and Correspondence vol. 10, Appendix VI).
See, for example, Correspondence vol. 9, letters to J. D. Hooker, 28 September [1861], 18 October [1861], 23 October [1861], and 1 November [1861], and this volume, letter to C. C. Babington, 20 January [1862], and letter to W. E. Darwin, 4 [July 1862] and n. 4.
The reference is to the fact that in Passiflora it had been found that plants could be ‘much more easily fertilised by the pollen of a distinct species, than with its own’ (Natural selection, p. 400; see also Origin, p. 251). Scott reported the results from a series of experiments on sterility and hybridisation in Passiflora, begun in 1863, in Scott 1864b.
See letter to J. D. Hooker, 14 [October 1862], n. 7.
Hooker discussed this project with CD during a visit to Down House in April 1860 (see Correspondence vol. 8, letter to Asa Gray, 25 April [1860]). See also letter from C. W. Crocker, 31 October 1862 and n. 12.
Scott had apparently sent CD an account of his experiments on Primula in the missing portion of his letter of [20 November – 2 December 1862]). After further research, Scott wrote a paper on the subject (Scott 1864c), which CD later cited in Forms of flowers.
Grönland, Johannes. 1855. Note sur les organes glanduleux du genre Drosera. Annales des sciences naturelles (botanique) 4th ser. 3: 297–303.
Milde, Karl August Julius. 1852. Ueber die Reizbarkeit der Blätter von Drosera rotundifolia. Botanische Zeitung 10: 540. [Vols. 8,10]
Trécul, Auguste. 1855. Organisation des glandes pédicellées des feuilles du Drosera rotundifolia. Annales des sciences naturelles (botanique) 4th ser. 3: 303–11.
DAR 93: B37, B49–52 |
Convert baseband signal to RF signal - Simulink - MathWorks India
\left(Available\text{ }\text{\hspace{0.17em}}power\text{\hspace{0.17em}}gain+I/Q\text{\hspace{0.17em}}gain\text{\hspace{0.17em}}mismatch\right)
\frac{{R}_{\text{load}}}{{R}_{\text{source}}}>{R}_{\text{ratio}}
\frac{{R}_{\text{load}}}{{R}_{\text{source}}}<\frac{1}{{R}_{\text{ratio}}}
{R}_{\text{ratio}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\sqrt{1+{\epsilon }^{2}}+\epsilon }{\sqrt{1+{\epsilon }^{2}}-\epsilon }
\epsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{{10}^{\left(0.1{R}_{\text{p}}\right)}-1}
\frac{{R}_{\text{load}}}{{R}_{\text{source}}}>{R}_{\text{ratio}}
\frac{{R}_{\text{load}}}{{R}_{\text{source}}}<\frac{1}{{R}_{\text{ratio}}}
{R}_{\text{ratio}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\sqrt{1+{\epsilon }^{2}}+\epsilon }{\sqrt{1+{\epsilon }^{2}}-\epsilon }
\epsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{{10}^{\left(0.1{R}_{\text{p}}\right)}-1} |
Sound Synthesis Theory/Sound in the Digital Domain - Wikibooks, open books for an open world
1 Sound in the Digital Domain
1.2 Digital <-> Analogue Conversion
1.3.1 Nyquist-Shannon sampling theorem
1.3.2 Sampling accuracy and bit depth
Sound in the Digital Domain[edit | edit source]
Digital systems (e.g. computers) and formats (e.g. CD) are clearly the most popular and commonplace methods of storing and manipulating audio. Since the introduction of the compact disc in the early 1980s, the digital format has provided increasingly greater storage capacity and the ability to store audio information at an acceptable quality. Although analogue formats still exist (vinyl, tape), they typically serve a niche audience. Digital systems are ubiquitous in modern music technology. It must be stressed that there is no argument as to whether one domain, be it analogue or digital is superior, but the following provides some desirable features of working with audio in the digital domain.
Storage. The amount of digital audio data capable of being stored on a modern hard drive is far greater than a tape system. Furthermore, we can choose the quality of the captured audio data, which relates directly to file size and other factors.
Control. By storing audio information in digital, we can perform powerful and complex operations on the data that would be extremely difficult to realise otherwise.
Durability. Digital audio can be copied across devices without any loss of information. Furthermore, many systems employ error correction codes to compensate for wear and tear on a physical digital format such as a compact disc.
Digital <-> Analogue Conversion[edit | edit source]
Acoustic information (sound waves) are treated as signals. As demonstrated in the previous chapter, we traditionally view these signals as varying amplitude over time. In analogue systems, this generally means that the amplitude is represented by a continuous voltage; but inside a digital system, the signal must be stored as a stream of discrete values.
Figure 2.1. An overview of the digital <-> analogue conversion process.
Digital data stored in this way has no real physical meaning; one could describe a song on a computer as just an array of numbers; these numbers are meaningless unless there exists within the system a process that can interpret each number in sequence appropriately. Fig. 2.1 shows an overview of the process of capturing analogue sound and converting it into a digital stream of numbers for storage and manipulation in such a system. The steps are as follows:
An input such as a microphone converts acoustic air pressure variations (sound waves) into variations in voltage.
An analogue to digital converter (ADC) converts the varying voltage into a stream of digital values by taking a 'snapshot' of the voltage at a point in time and assigning it a value depending on its amplitude. It typically takes these 'snapshots' thousands of times a second, the rate at which is known as the sample rate.
The numerical data is stored on the digital system and then subsequently manipulated or analysed by the user.
The numerical data is re-read and streamed out of the digital system.
A digital to analogue converter (DAC) converts the stream of digital values back to a varying voltage.
A loudspeaker converts the voltage to variations in air pressure (sound).
Although the signal at each stage comes in a different form (sound energy, digital values etc.), the information is analogous. However, due to the nature of the conversion process, this data may become manipulated and distorted. For instance, low values for sample rates or other factors at the ADC might mean that the continuous analogue signal is not represented with enough detail and subsequently the information will be distorted. There are also imperfections in physical devices such as microphones which further "colour" the signal in some way. It is for this reason that musicians and engineers aim to use the most high-quality equipment and processes in order to preserve the integrity of the original sound throughout the process. Musicians and engineers must consider what other processes their music will go through before consumption, too (radio transmission etc.).
Sound waves in their natural acoustic form can be considered continuous; that is, their time-domain graphs are smooth lines on all zoom factors without any breaks or jumps. We cannot have these breaks, or discontinuities because sound cannot switch instantaneously between two values. An example of this may be an idealised waveform like a square wave - on paper, it switches between 1 and -1 amplitude at a point instantaneously; however a loudspeaker cannot, by the laws of physics, jump between two points in no time at all, the cone has to travel through a continuous path from one point to the next.
Figure 2.2. Discrete samples (red) of a continuous waveform (grey).
Sampling is the process of taking a continuous, acoustic waveform and converting it into a digital stream of discrete numbers. An ADC measures the amplitude of the input at a regular rate creating a stream of values which represent the waveform in digital. The output is then created by passing these values to the DAC, which drives a loudspeaker appropriately. By measuring the amplitude many thousands of times a second, we create a "picture" of the sound which is of sufficient quality to human ears. The more and more we increase this sample rate, the more accurately a waveform is represented and reproduced.
Nyquist-Shannon sampling theorem[edit | edit source]
The frequency of a signal has implications for its representation, especially at very high frequencies. As discussed in the previous chapter, the frequency of a sine wave is the number of cycles per second. If we have a sample rate of 20000 samples per second (20 kHz), it is clear that a high frequency sinusoid such as 9000 Hz is going to have fewer "snapshots" than a sinusoid at 150 Hz. Eventually there reaches a point where there are not enough sample points to be able to record the cycle of a waveform, which leads us to the following important requirement:
The sample rate must be greater than twice the maximum frequency represented.
Why is this? The minimum number of sample points required to represent a sine wave is two, but we need at least slightly more than this so that we're not dependent phase (samples at exactly twice the sine wave frequency, the samples may fall on the peaks of the sine wave, or on the zero crossings). It may seem apparent at this time that using just two points to represent a continuous curve such as a sinusoid would result in a crude approximation - a square wave. And, inside the digital system, this is true. However, both ADCs and DACs have low-pass filters set at half the sample rate (the highest representable frequency). What this means for input and output is that any frequency above the cut-off point is removed and it follows from this that the crude sine representation - a square wave in theory - becomes filtered down to a single frequency (i.e. a sine wave). From this, we have two mathematical results:
{\displaystyle F_{s}>2f_{max}}
{\displaystyle F_{N}={\frac {F_{s}}{2}}}
{\displaystyle F_{s}}
is the sample rate,
{\displaystyle f_{max}}
is the highest frequency in the signal.
{\displaystyle F_{N}}
is the Nyquist frequency. Frequencies over the Nyquist frequency are normally blocked by filters before conversion to the digital domain when recording; without such processes there would be frequency component foldover, otherwise known as aliasing.
Sampling accuracy and bit depth[edit | edit source]
It has been established that the higher the sample rate, the more accurate the representation of a waveform in a digital system. However, although there are many reasons and arguments for higher sample rates, there are two general standards: 44100 samples per second and 48000 samples per second, with the former being most commonplace. The main consideration for this is the fact that the human hearing range extends, at maximum, to an approximate limit (that varies from person to person) of 20000 Hz. Frequencies above this are inaudible. Considering the example of 44.1 kHz, we find that the Nyquist frequency evaluates to 22050 Hz, which is more than the human hearing system is capable of perceiving. There are other reasons for this particular sample rate, but that is beyond the scope of this book.
Figure 2.3. Effects of increased sample rate and bit depth on representing a continuous analogue signal.
There is one more important factor to consider when considering the sampling process: bit depth. Bit depth represents the precision with which the amplitude is measured. In the same way that there are a limited amount of samples per second in a conversion process, there are also a limited amount of amplitude values for a sample point, and the greater the number, the greater the accuracy. A common bit resolution found in most standard digital audio systems (Hi-Fi, Compact Disc) is 16 binary bits which allows for a range of 65536 (
{\displaystyle 2^{16}}
) individual amplitude values at a point in time. Lower bit values result in a greater distortion of the sound - a two bit system (
{\displaystyle 2^{2}}
) only allows for four different amplitudes, which results in a massively inaccurate approximation of the input signal.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Sound_Synthesis_Theory/Sound_in_the_Digital_Domain&oldid=3267668" |
Glycerol dehydrogenase (NADP+) - Wikipedia
In enzymology, a glycerol dehydrogenase (NADP+) (EC 1.1.1.72) is an enzyme that catalyzes the chemical reaction
glycerol + NADP+
{\displaystyle \rightleftharpoons }
D-glyceraldehyde + NADPH + H+
Thus, the two substrates of this enzyme are glycerol and NADP+, whereas its 3 products are D-glyceraldehyde, NADPH, 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 glycerol:NADP+ oxidoreductase. This enzyme is also called glycerol dehydrogenase (NADP+). This enzyme participates in glycerolipid metabolism.
Kormann AW, Hurst RO, Flynn TG (1972). "Purification and properties of an NADP+-dependent glycerol dehydrogenase from rabbit skeletal muscle". Biochim. Biophys. Acta. 258 (1): 40–55. doi:10.1016/0005-2744(72)90965-5. PMID 4400494.
Toews CJ (1967). "The kinetics and reaction mechanism of the nicotinamide-adinine dinucleotide phosphate-specific glycerol dehydrogenase of rat skeletal muscle". Biochem. J. 105 (3): 1067–1073. PMC 1198427. PMID 16742532.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Glycerol_dehydrogenase_(NADP%2B)&oldid=917404940" |
Spud has done it again. He's lost another polynomial function. This one was a cubic, written in standard form. He knows that there were two complex zeros,
-2 ± 5i
and one real zero,
-1
. What could his original function have been?
Use the three zeros to write the polynomial in factored form.
p(x) = (x - (- 1))(x - (-2 + 5i))(x - (-2 - 5i))
Multiply the two complex polynomials.
(x - (-2 + 5i))(x - (-2 - 5i))
x^2 + 4x + 29
Multiply the result by
(x + 1)
(x + 1)(x^2 + 4x + 29)
p(x) = x^3 + 5x^2 + 33x + 29 |
From Charles and Emma Darwin to J. D. Hooker [10 July 1865]1
I do not know whether you are at Kew. When you can find time I shd. much like to hear how Teesdale suited you & Mrs. H.2 I have had a very bad time, with incessant vomiting, but have been better for last 4 days— The ice to spine did nothing.3 I fear I am much too weak to risk seeing you, even if gooseberries could tempt you.4
We are reading aloud on your recommendation Tylor & are deeply interested by it: what a clever man— do you know who & what he is?—5
Have you read last nor of H. Spencers Biology:6 I shd very much like to hear what you think of it, especially about the umbellifers.7 When writing the Origin I consulted you & you did not think that the compactness of the head accounted for the difference of the outer flowers; but then I do not think we took into account the difference between a globular & flat corymb.8 I am not satisfied by his views on irregular flowers; the peloric Gloxinia being upright seems a good argument; but then the peloric Snap dragon I find forms the same angle with the stem as does the very irregular common flower.9
The last number of the Nat. Hist review seems to me very good. That on H. Spencer I suppose is by Masters.10 Let me have a short note from you whenever you can spare time.
All scientific work has been stopped with me for the last 2
\frac{1}{2}
months.11
Remind Oliver to have the kindness to let me hear if there are any articles in German periodicals concerning my subjects.12 what is Oliver doing? Dear Dr Hooker13
I do hope Charles is making a little progress in spite of frequent returns of the sickness but there is a degree of vigour about him on the well days which makes me hope that his constitution is making a struggle. If he conquers this sickness I do hope you will be able to come & see him before long & I am sure there is nobody in the world he cares so much to see. He had one terribly bad week. We liked Dr Chapman so very much we were quite sorry the ice failed for his sake as well as ours.14
Will you give my love to Mrs Hooker. I trust she is stronger15
The date is established by the relationship between this letter and the letter from J. D. Hooker, 13 July 1865. The Monday before 13 July 1865 was 10 July.
Joseph Dalton and Frances Harriet Hooker and George and Sarah Bentham travelled to York on 26 or 27 June 1865, and then to Middleton-in-Teesdale, County Durham (see letter from J. D. Hooker, [15 June 1865], and Jackson 1906, p. 202).
CD had been following John Chapman’s ice treatment since 20 May (see letter to J. D. Hooker, [17 June 1865] and n. 11). Emma Darwin recorded in her diary (DAR 242) that CD suffered several bouts of vomiting and flatulence throughout May and June, but indicated that his health was good on 6, 9, and 10 July 1865 and good for part of 7 and 8 July, though with some sickness.
See letter from J. D. Hooker, [after 17 June 1865] and n. 6.
Hooker first recommended Tylor 1865 to CD in May (see letter from J. D. Hooker, [26 May 1865] and n. 13) but ill health had prevented CD from reading it earlier. CD made several references to Tylor 1865 in Descent and cited the second edition, Tylor 1870, three times in Expression. There are annotated copies of Tylor 1865 and 1870 in the Darwin Library–CUL (see Marginalia 1: 810–11). For more on the career of Edward Burnett Tylor and the relation of his work in anthropology to Darwinism, see Leopold 1980 and Stocking 1987.
The most recent instalment of Herbert Spencer’s Principles of biology, part 14, was published in June 1865 (see Spencer 1864–7, 1: Preface). Spencer’s list of subscribers included CD and Hooker (Spencer 1904, 2: 484). CD’s instalments of Spencer 1864–7 are in the Darwin Library–CUL as a bound volume (see also Marginalia 1: 769–73).
The inflorescences of the Umbelliferae (also referred to as the Apiaciae in Lindley 1853) are discussed in Spencer 1864–7, 2: 156–7.
No correspondence between CD and Hooker on this subject has been found; however, in Origin, p. 145, CD wrote, ‘in the case of the corolla of the Umbelliferae, it is by no means, as Dr. Hooker informs me, in species with the densest heads that the inner and outer flowers most frequently differ’. He prefaced this by stating (p. 144): I know of no case better adapted to show the importance of the laws of correlation in modifying important structures, independently of utility and, therefore, of natural selection, than that of the difference between the outer and inner flowers in some Compositous and Umbelliferous plants. Spencer 1864–7, 2: 156–7, gives several examples of Umbelliferae species, some with flat inflorescences and differently shaped inner and outer flowers (when the flowers are crowded together), and others with globular inflorescences that do not have differently shaped flowers. Moreover, he argues that in species where the umbellules themselves are clustered, the tendency towards greater irregularity of outer flowers is increased as one moves to the periphery of the cluster. Spencer thus attempts to show that a causal relationship exists between the shape of the inflorescence and the shape of individual flowers within it and concludes (ibid., p. 157): Considering how obviously these various forms are related to the various conditions, we should be scarcely able, even in the absence of all other facts, to resist the conclusion that the differences in the conditions are the causes of the differences in the forms. The above passage was scored by CD in his copy of part 14. He also wrote on the back cover, ‘Hooker disagrees about Umbellifers’ (see Marginalia 1: 770). CD’s statements in Origin were not substantially changed in later editions.
Spencer suggested that the shape of the flower of Gloxinia erecta was related to the angle it formed relative to the stem. The peloric regular flower, which was radially symmetrical, was associated with an upright attitude, while the normal irregular flower, which was bilaterally symmetrical, inclined at an acute angle from the stem (Spencer 1864–7, 2: 151–2). CD had kept Hooker informed about his recent crossing experiments on peloric and common Antirrhinum majus as well as sending him flower specimens (see letter to J. D. Hooker, 1 June [1865]).
CD refers to Maxwell Tylden Masters. The article on Spencer’s Principles of biology in the issue of Natural History Review for July 1865, pp. 373–85, was unsigned. The review focused on aspects of plant teratology, a subject that Masters had studied extensively (see Masters 1869, and letter from M. T. Masters, 7 February 1865). CD’s unbound copy of the July 1865 issue of Natural History Review is in the Darwin Library–CUL. The article is lightly annotated with an additional note written by CD on a separate sheet at the back of the issue.
CD recorded in his Journal that he had been ill since 22 April 1865 (see Correspondence vol.13, Appendix II; see also n. 3, above). By ‘scientific work’ CD usually meant writing (see letter to T. H. Huxley, 12 July [1865]); he was still making botanical experiments (see, for example, letter to J. D. Hooker, 1 June [1865], where he mentions his work on snapdragons).
Daniel Oliver was an editor of phanerogamic botany for the Natural History Review. He often referred German and French language publications on botanical topics to CD (see, for example, Correspondence vol. 12, letter from Daniel Oliver, 14 June 1864).
Emma Darwin added her own note to Hooker on the back page of CD’s letter.
See n. 3, above. For CD’s comments on the effects of the ice treatment, see the letter to John Chapman, 7 June 1865, and the letter to J. D. Hooker, [17 June 1865].
Frances Harriet Hooker had suffered a miscarriage (see letter from J. D. Hooker, [2 June 1865]).
Leopold, Joan. 1980. Culture in comparative and evolutionary perspective: E. B. Tylor and the making of Primitive culture. Berlin: Dietrich Reimer Verlag.
Spencer, Herbert. 1904. An autobiography. 2 vols. London: Williams and Norgate.
Tylor, Edward Burnett. 1870. Researches into the early history of mankind and the development of civilization. 2d edition. London: John Murray.
AL(S) 4pp |
Read File - Maple Help
Home : Support : Online Help : Connectivity : Web Features : Worksheet Package : Read File
parse a worksheet into an XML data structure
ReadFile( filename, format=output_format )
string; name of a worksheet file
The ReadFile command parses a worksheet, returning an XML data structure that represents the worksheet. It requires a single argument filename that must be a Maple string and is assumed to name a worksheet file. The file is opened, and its contents read and parsed. The parsed worksheet 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.
You can save a worksheet in XML format by using the procedure WriteFile in the XMLTools package.
Note: Maple worksheets (.mw) files are saved in an XML-based format. You can display the structure of a Maple worksheet in XML applications.
\mathrm{with}\left(\mathrm{Worksheet}\right):
\mathrm{dir}≔\mathrm{kernelopts}\left(\mathrm{mapledir}\right):
\mathrm{mws}≔\mathrm{ReadFile}\left(\mathrm{cat}\left(\mathrm{dir},"/examplesclassic/obj.mws"\right)\right):
\mathrm{type}\left(\mathrm{mws},'\mathrm{Worksheet}:-\mathrm{worksheet}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{XMLTools}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{ElementStatistics}\left(\mathrm{mws}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}
[\textcolor[rgb]{0,0,1}{"Worksheet"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"View-Properties"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Label-Scheme"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Version"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Task"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Styles"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Hyperlink"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Section"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Title"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{10}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Layout"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{33}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Input"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{104}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Group"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{104}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Font"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{110}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Text-field"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{146}]
Worksheet[FromString]
XMLTools[WriteFile] |
What Determine the International Competitiveness of Chinese Publishing Industry?
Y=\left\{Y\left(k\right)|k=1,2,\cdots ,n\right\}
\left\{{X}_{1}\left(k\right),{X}_{2}\left(k\right),\cdots ,{X}_{i}\left(k\right)\right\}=\left\{\begin{array}{c}{X}_{11},{X}_{12},\cdots ,{X}_{1n}\\ {X}_{21},{X}_{22},\cdots ,{X}_{2n}\\ {X}_{m1},{X}_{m2},\cdots ,{X}_{mn}\end{array}\right\}
{x}_{i}\left(k\right)=\frac{{X}_{i}\left(k\right)}{{X}_{i}\left(l\right)}
{x}_{i}\left(l\right)
\Delta \left(\mathrm{min}\right)=\underset{i}{\mathrm{min}}\left(\underset{i}{\mathrm{min}}|{X}_{0}\left(k\right)-{X}_{i}\left(k\right)|\right)
\Delta \left(\text{max}\right)=\underset{i}{\text{max}}\left(\underset{i}{\text{max}}|{X}_{0}\left(k\right)-{X}_{i}\left(k\right)|\right)
{\xi }_{i}\left(k\right)=\frac{\underset{i}{\mathrm{min}}\underset{k}{\mathrm{min}}|{X}_{0}\left(k\right)-{X}_{i}\left(k\right)|+\rho \underset{i}{\mathrm{max}}\underset{k}{\mathrm{max}}|{X}_{0}\left(k\right)-{X}_{i}\left(k\right)|}{|{X}_{0}\left(k\right)-{X}_{i}\left(k\right)|+\rho \underset{i}{\mathrm{max}}\underset{k}{\mathrm{max}}|{X}_{0}\left(k\right)-{X}_{i}\left(k\right)|}
{\xi }_{i}\left(k\right)
{X}_{i}
{X}_{0}
\rho \in \left(0,1\right)
{r}_{i}=\sum _{i=1}^{N}{\xi }_{i}\left(k\right)
{r}_{i}
{X}_{i}
{X}_{0}
Wei, L. and Yang, H. (2019) What Determine the International Competitiveness of Chinese Publishing Industry? American Journal of Industrial and Business Management, 9, 789-798. https://doi.org/10.4236/ajibm.2019.94052
1. Zhang, B. and Du, X.Y. (2012) Analysis on International Competitiveness and Influencing Factors of America’s Cultural Industries. International Business: Journal of International Business and Economics, No. 4, 81-91.
2. Tang, X.H. and Li, B. (2012) The International Competiveness of the Japanese Cultural Industry and the Prospects for Development. Contemporary Economy of Japan, No. 4, 47-55.
3. Lan, Q.W., Zheng, P. and Han, J. (2012) International Competitiveness Comparison and Promotion Strategy of China’s Cultural Industry-Based on 2011 Cross-Section Data. Finance & Trade Economics, No. 8, 80-87.
4. Huang, Y.J. (2013) Empirical Research on the International Competitiveness of American Cultural Industry. Liaoning University, Shenyang.
5. Fu, L.P., Song, J.S., Deng, J., et al. (2010) A Summary of the Research on the Competitiveness and Evaluation of Cultural Industry. Academic Forum, 33, 168-171.
6. Qu, G.M. (2012) Comparison of International Competitiveness of Creative Industry between China and USA: Based on RCA, TC and “Diamond” Model. Journal of International Trade, No. 3, 79-89.
7. Shang, T. and Tao, Y.F. (2011) On International Division of Labor in China’s Creative Industry: An Analysis Based on the Comparison between Developed and Developing Countries. World Economy Study, No. 2, 40-47.
8. Nie, L. (2013) A Study of the International Competitiveness of the Trade in Creative Products of the BRICs. Journal of International Trade, No. 2, 111-122.
9. Wang, G.A. and Zhao, X.Q. (2013) A Comparative Study on International Competitiveness of Sino-US Movie and Television Industry: Based on the Perspective of Global Value Chain. Journal of International Trade, No. 1, 58-67.
10. Gao, H. and Yan, L.T. (2017) A Comparative Analysis of International Competitiveness of Creative Industries between China and Japan: New Measurement Based on the Change of Creative Products and Service Trade. Contemporary Economy of Japan, No. 1, 66-80.
11. Huang, X.R. and Tian, C.Q. (2014) Research on the International Competitiveness of China’s Publishing Industry. China Publishing Journal, No. 3, 8-12.
12. Rosenblatt, G. (2015) The Automation of the Publishing Industry.
13. Chen, B.B. (2015) Grey Relational Analysis of Factors Influencing the International Competitiveness of China’s Transport Services Trade. Reform & Opening, No. 23, 10-12. |
Metastability in loss networks with dynamic alternative routing
April 2022 Metastability in loss networks with dynamic alternative routing
Sam Olesker-Taylor1
Consider N stations interconnected with links, each of capacity K, forming a complete graph. Calls arrive to each link at rate λ and depart at rate 1. If a call arrives to a link
\mathit{x}\mathit{y}
, connecting stations x and y, which is at capacity, then a third station z is chosen uniformly at random and the call is attempted to be routed via z: if both links
\mathit{x}\mathit{z}
\mathit{z}\mathit{y}
have spare capacity, then the call is held simultaneously on these two; otherwise the call is lost.
We analyse an approximation of this model. We show rigorously that there are three phases according to the traffic intensity
\mathit{\alpha }:=\mathit{\lambda }/\mathit{K}
: for
\mathit{\alpha }\in \left(0,{\mathit{\alpha }}_{\mathit{c}}\right)\cup \left(1,\infty \right)
, the system has mixing time logarithmic in the number of links
\mathit{n}:=\left(\genfrac{}{}{0.0pt}{}{\mathit{N}}{2}\right)
\mathit{\alpha }\in \left({\mathit{\alpha }}_{\mathit{c}},1\right)
the system has mixing time exponential in n, the number of links. Here
{\mathit{\alpha }}_{\mathit{c}}:=\frac{1}{3}\left(5\sqrt{10}-13\right)\approx 0.937
is an explicit critical threshold with a simple interpretation. We also consider allowing multiple rerouting attempts. This has little effect on the overall behaviour; it does not remove the metastability phase.
Finally, we add trunk reservation: in this, some number σ of circuits are reserved; a rerouting attempt is only accepted if at least
\mathit{\sigma }+1
circuits are available. We show that if σ is chosen sufficiently large, depending only on α, not K or n, then the metastability phase is removed.
The author was supported by EPSRC Doctoral Training Grant #1885554.
The question of studying mixing times for this model was originally raised by Nathanaël Berestycki. I would like to thank Perla Sousi, my Ph.D. supervisor, for reading this paper and giving lots of constructive feedback. I would also like to thank Frank Kelly, for numerous helpful discussions on this work and related stochastic networks discussions. He introduced me to the topic through his Cambridge Part III lecture course and his book [12] with Elena Yudovina; I have become thoroughly interested in the topic as a result.
I also thank the anonymous referee for helpful comments which improved the clarity and presentation of the paper. They also alerted me to the analogous metastable behaviour exhibited by the Chayes–Machta dynamics in the random cluster model and to the references [1, 3].
Sam Olesker-Taylor. "Metastability in loss networks with dynamic alternative routing." Ann. Appl. Probab. 32 (2) 1362 - 1399, April 2022. https://doi.org/10.1214/21-AAP1711
Received: 1 August 2020; Revised: 1 April 2021; Published: April 2022
Primary: 60K20 , 60K25 , 60K30 , 90B15 , 90B18 , 90B22
Keywords: dynamic alternative routing , Loss network , metastability , Mixing times
Sam Olesker-Taylor "Metastability in loss networks with dynamic alternative routing," The Annals of Applied Probability, Ann. Appl. Probab. 32(2), 1362-1399, (April 2022) |
Lactaldehyde reductase - Wikipedia
In enzymology, a lactaldehyde reductase (EC 1.1.1.77) is an enzyme that catalyzes the chemical reaction
(R)[or (S)]-propane-1,2-diol + NAD+
{\displaystyle \rightleftharpoons }
(R)[or (S)]-lactaldehyde + NADH + H+
The 3 substrates of this enzyme are (R)-propane-1,2-diol, (S)-propane-1,2-diol, and NAD+, whereas its 4 products are (R)-lactaldehyde, (S)-lactaldehyde, 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)[or (S)]-propane-1,2-diol:NAD+ oxidoreductase. Other names in common use include propanediol:nicotinamide adenine dinucleotide (NAD+) oxidoreductase, and L-lactaldehyde:propanediol oxidoreductase. This enzyme participates in pyruvate metabolism and glyoxylate and dicarboxylate metabolism.
As of late 2007, 3 structures have been solved for this class of enzymes, with PDB accession codes 1RRM, 2BI4, and 2BL4.
Ting SM, Sellinger OZ, Miller ON (1964). "The metabolism of lactaldehyde. VI. The reduction of D- and L-lactaldehyde in rat liver". Biochim. Biophys. Acta. 89: 217–225. doi:10.1016/0926-6569(64)90210-x. PMID 14203169.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Lactaldehyde_reductase&oldid=917454919" |
ROBOImplant II: Development of a Noninvasive Controller/Actuator for Wireless Correction of Orthopedic Structural Deformities | J. Med. Devices | ASME Digital Collection
Jonathan A. Liu,
UCB/UCSF Joint Graduate Group in Bioengineering
Mozziyar Etemadi,
James A. Heller,
Department of Bioengineering, and Therapeutic Sciences,
Emeritus of Surgery, Pediatrics Obstetrics, Gynecology and Reproductive Sciences, Director Emeritus, Fetal Treatment Center, Division of Pediatric Surgery, Department of Surgery, University of California
Associate Professor of Bioengineering and Therapeutic Sciences, Biomedical Medical Devices Laboratory, Department of Bioengineering and Therapeutic Sciences,
e-mail: [email protected]
Jonathan A. Liu Engineer
Mozziyar Etemadi MD/PhD Candidate
James A. Heller Engineer
Dillon Kwiat Engineer
Richard Fechter Clinical Engineer
Michael R. Harrison Professor
Liu, J. A., Etemadi, M., Heller, J. A., Kwiat, D., Fechter, R., Harrison, M. R., and Roy, S. (August 14, 2012). "ROBOImplant II: Development of a Noninvasive Controller/Actuator for Wireless Correction of Orthopedic Structural Deformities." ASME. J. Med. Devices. September 2012; 6(3): 031006. https://doi.org/10.1115/1.4007183
An implantable actuator and its accompanying driver circuit are presented for the purpose of lengthening bones and correcting skeletal deformities without requiring physical contact between the operator and the implanted device. This system utilizes magnetic coupling to form a magnetic gear, allowing an external motor to drive an implantable telescoping rod. The accompanying electronics are able to monitor the progress, in the form of turns delivered, as well as detect procedural errors, such as magnet decoupling. The force applied by the implanted telescoping rod can be accurately measured by monitoring the current necessary for the external controller to drive the extension of the implanted rod. After characterization, the system was shown to reliably deliver extension distances within 34
μm
and maintain coupling out to 70 mm. The system is also able to measure torques as low as 0.12 mN m. System variability and accuracy of external monitoring are addressed.
Actuators, Control equipment, Engines, Magnets, Motors, Torque, Orthopedics, Errors, Gears, Electronics
Pearls and Pitfalls of Deformity Correction and Limb Lengthening via Monolateral External Fixation
.http://www.uiortho.com/images/stories/ioj/1996iowaorthojournal.pdfhttp://www.uiortho.com/images/stories/ioj/1996iowaorthojournal.pdf
Complications of Bone Lengthening
H.-C. C.
Improved Comfort in Lower Limb Lengthening With the Intramedullary Skeletal Kinetic Distractor: Principles and Preliminary Clinical Experiences
A Fully Implantable Motorized Intramedullary Nail for Limb Lengthening and Bone Transport
Meswania
Design and Characterization of a Novel Permanent Magnet Synchronous Motor Used in a Growing Prosthesis for Young Patients With Bone Cancer
Apparatus for Transmitting Power
Non-Contact Magnetic Gear for Micro Transmission Mechanism
Micro Electro Mechanical Systems, MEMS ’91, Proceedings: An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots
, Jan. 30–Feb. 2, pp.
Unalloyed Titanium For Implants In Bone Surgery
Analysis and Design of Magnetic Torque Couplers and Magnetic Gears
The 4th International Power Electronics and Motion Control Conference
, IPEMC 2004, Xi’an, China, Aug. 14–16, pp.
Theoretical Computations of the Magnetic Coupling Between Magnetic Gears
“Magnetic Coupling Studies Between Radial Magnetic Gears
Simulation Study of the Magnetic Coupling Between Radial Magnetic Gears |
Properties of Sobolev-type metrics in the space of curves | EMS Press
Properties of Sobolev-type metrics in the space of curves
A.C.G. Mennucci
A. Yezzi
G. Sundaramoorthi
M
where objects
c\in M
are curves, which we parameterize as
c:S^1\to \real^n
n\ge 2
S^1
is the circle). We study geometries on the manifold of curves, provided by Sobolev--type Riemannian metrics
H^j
. These metrics have been shown to regularize gradient flows used in Computer Vision applications, see \cite{ganesh
_sobol_activ_contour08} and references therein. We provide some basic results of
H^j
metrics; and, for the cases
j=1,2
, we characterize the completion of the space of smooth curves. We call these completions \emph{``
H^1
H^2
Sobolev--type Riemannian Manifolds of Curves''}. \gsinsert{This result is fundamental since it is a first step in proving the existence of geodesics with respect to these metrics.} As a byproduct, we prove that the Fr\'echet distance of curves (see \cite{Michor-Mumford}) coincides with the distance induced by the ``Finsler
L^\infinity
metric'' defined in \S2.2 in \cite{YM
A.C.G. Mennucci, A. Yezzi, G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves. Interfaces Free Bound. 10 (2008), no. 4, pp. 423–445 |
2016 Density by Moduli and Lacunary Statistical Convergence
Vinod K. Bhardwaj, Shweta Dhawan
We have introduced and studied a new concept of
-lacunary statistical convergence, where
f
is an unbounded modulus. It is shown that, under certain conditions on a modulus
f
, the concepts of lacunary strong convergence with respect to a modulus
f
f
-lacunary statistical convergence are equivalent on bounded sequences. We further characterize those
\theta
{S}_{\theta }^{f}={S}^{f}
{S}_{\theta }^{f}
{S}^{f}
denote the sets of all
f
-lacunary statistically convergent sequences and
f
-statistically convergent sequences, respectively. A general description of inclusion between two arbitrary lacunary methods of
-statistical convergence is given. Finally, we give an
{S}_{\theta }^{f}
-analog of the Cauchy criterion for convergence and a Tauberian theorem for
{S}_{\theta }^{f}
-convergence is also proved.
Vinod K. Bhardwaj. Shweta Dhawan. "Density by Moduli and Lacunary Statistical Convergence." Abstr. Appl. Anal. 2016 1 - 11, 2016. https://doi.org/10.1155/2016/9365037
Received: 15 November 2015; Accepted: 5 January 2016; Published: 2016
Vinod K. Bhardwaj, Shweta Dhawan "Density by Moduli and Lacunary Statistical Convergence," Abstract and Applied Analysis, Abstr. Appl. Anal. 2016(none), 1-11, (2016) |
2017 Approximation Properties of
q
-Bernoulli Polynomials
M. Momenzadeh, I. Y. Kakangi
q
-analogue of Euler-Maclaurin formula and by introducing a new
q
-operator we drive to this form. Moreover, approximation properties of
q
-Bernoulli polynomials are discussed. We estimate the suitable functions as a combination of truncated series of
q
-Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches: first we solve the differential equations by estimating functions and second we may apply these techniques for operator theory.
M. Momenzadeh. I. Y. Kakangi. "Approximation Properties of
q
-Bernoulli Polynomials." Abstr. Appl. Anal. 2017 1 - 6, 2017. https://doi.org/10.1155/2017/9828065
Received: 8 August 2017; Accepted: 29 October 2017; Published: 2017
M. Momenzadeh, I. Y. Kakangi "Approximation Properties of
q
-Bernoulli Polynomials," Abstract and Applied Analysis, Abstr. Appl. Anal. 2017(none), 1-6, (2017) |
On the dynamics of certain actions of free groups on closed real analytic manifolds | EMS Press
On the dynamics of certain actions of free groups on closed real analytic manifolds
Let M be a closed connected real analytic manifold; let
\Gamma
be a free group on two generators. The set of analytic actions of
\Gamma
on M endowed with Taken‘s topology contains a nonempty open subset whose corresponding actions share three properties: (a) they have every orbit dense, (b) they leave invariant no geometric structure on M, (c) any homeomorphism conjugating two of them is analytic.
Michel Belliart, On the dynamics of certain actions of free groups on closed real analytic manifolds. Comment. Math. Helv. 77 (2002), no. 3, pp. 524–548 |
Effects of Ultrasounds on the Heat Transfer Enhancement From a Circular Cylinder to Distilled Water in Subcooled Boiling | J. Thermal Sci. Eng. Appl. | ASME Digital Collection
Federica Baffigi,
Department of Energetics “L.Poggi,”
, Via Diotisalvi 2, Pisa 56126, Italy
e-mail: [email protected]
Carlo Bartoli Professor
Baffigi, F., and Bartoli, C. (March 1, 2011). "Effects of Ultrasounds on the Heat Transfer Enhancement From a Circular Cylinder to Distilled Water in Subcooled Boiling." ASME. J. Thermal Sci. Eng. Appl. March 2011; 3(1): 011001. https://doi.org/10.1115/1.4003510
The main aim of this work is to investigate experimentally the influence of ultrasonic waves, on the heat transfer enhancement, from a stainless steel circular cylinder to distilled water, in subcooled boiling conditions. This study has carried on for a few years at the Department of Energetics “L.Poggi.” The effect was observed since the 1960s: Different authors had investigated the cooling effect due to the ultrasonic waves at different heat transfer regimes, especially from a thin platinum wire to water. They had found out that the highest heat transfer coefficient enhancement was in subcooled boiling conditions. So this paper has the purpose to clarify the physical phenomenon and optimize a large range of variables involved in the mechanism. It reports the experimental results obtained with ultrasound at the frequency of 38 kHz, at two different subcooling degrees,
ΔTsub=25°C
35°C
. The heat fluxes applied on the cylinder, the ultrasonic generator power
Pgen
, and also the placement of the heater inside the ultrasonic generator tank were varied. The ultrasonic waves seem to be very useful for a practical application in the last generation electronic components’ cooling: They need dissipating huge heat fluxes and avoiding high temperatures
(≈150°C)
, after that they could damage themselves.
boiling, cooling, distillation, shapes (structures), stainless steel, ultrasonic effects, ultrasonic waves, water
Boiling, Circular cylinders, Cylinders, Heat transfer, Heat transfer coefficients, Subcooling, Ultrasonic waves, Water, Temperature, Generators, Cooling, Heat flux, Ultrasound, Stainless steel
The Influence of Acoustical Vibrations on Heat Transfer by Natural Convection From a Horizontal Cylinder to Water
Ultrasonic Enhancement of Saturated and Subcooled Pool Boiling
Effects of Ultrasonic Waves on Natural Convection, Nucleate Boiling, and Film Boiling Heat Transfer From a Wire to a Saturated Liquid
Enhancement of Natural Convection and Pool Boiling Heat Transfer via Ultrasonic Vibrations
Effects of Ultrasonic Vibration on Transient Boiling Heat Transfer During Rapid Quenching of a Thin Wire in Water
3D CMOS SOI for High Performance Computing
,” ISLPED 98, Monterey, CA, Aug. 10–12, pp.
Microcooling Systems for High Density Packaging
Thermal Analysis of Three-Dimensional (3-D) Integrated Circuits (ICs)
Opportunities for Reduced Power Dissipation Using Three-Dimensional Integration
,” Proceedings of the IEEE 2002 International Interconnect Technology Conference, pp.
System Design Issue for 3D System-in-Package (SiP)
Numerical Prediction of Electronic Component Operational Temperature: A Perspective
3D Processing Technology and Its Impact on iA32 Microprocessors
Proceedings of the IEEE International Conference on Computer Design (ICCD)
Heat Transfer Enhancement From a Circular Cylinder to Distilled Water by Ultrasonic Waves: Preliminary Remarks
Proceedings of the ExHFT-7, Seventh World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics
, Krakow, Poland, Jun. 28–Jul. 3.
Heat Transfer Enhancement From a Circular Cylinder to Distilled Water by Ultrasonic Waves in Subcooled Boiling Conditions
,” ITP09, Interdisciplinary Transport Phenomena VI: Fluid, Thermal; Biological, Materials and Space Sciences, Volterra, Italy, Oct. 4–9. |
2-oxoglutarate synthase - Wikipedia
In enzymology, a 2-oxoglutarate synthase (EC 1.2.7.3) is an enzyme that catalyzes the chemical reaction
2-oxoglutarate + CoA + 2 oxidized ferredoxin
{\displaystyle \rightleftharpoons }
succinyl-CoA + CO2 + 2 reduced ferredoxin
The 3 substrates of this enzyme are 2-oxoglutarate, CoA, and oxidized ferredoxin, whereas its 3 products are succinyl-CoA, CO2, and reduced ferredoxin.
This enzyme belongs to the family of oxidoreductases, specifically those acting on the aldehyde or oxo group of donor with an iron-sulfur protein as acceptor. The systematic name of this enzyme class is 2-oxoglutarate:ferredoxin oxidoreductase (decarboxylating). Other names in common use include 2-ketoglutarate ferredoxin oxidoreductase, 2-oxoglutarate:ferredoxin oxidoreductase, KGOR, 2-oxoglutarate ferredoxin oxidoreductase, and 2-oxoglutarate:ferredoxin 2-oxidoreductase (CoA-succinylating). This enzyme participates in the Citric acid cycle. Some forms catalyze the reverse reaction within the Reverse Krebs cycle, as a means of carbon fixation.
Buchanan BB, Evans MC (1965). "The synthesis of alpha-ketoglutarate from succinate and carbon dioxide by a subcellular preparation of a photosynthetic bacterium". Proc. Natl. Acad. Sci. U.S.A. 54 (4): 1212–8. Bibcode:1965PNAS...54.1212B. doi:10.1073/pnas.54.4.1212. PMC 219840. PMID 4286833.
Gehring U, Arnon DI (1972). "Purification and properties of -ketoglutarate synthase from a photosynthetic bacterium". J. Biol. Chem. 247 (21): 6963–9. doi:10.1016/S0021-9258(19)44680-2. PMID 4628267.
Dorner E, Boll M (2002). "Properties of 2-Oxoglutarate:Ferredoxin Oxidoreductase from Thauera aromatica and Its Role in Enzymatic Reduction of the Aromatic Ring". J. Bacteriol. 184 (14): 3975–83. doi:10.1128/JB.184.14.3975-3983.2002. PMC 135165. PMID 12081970.
Mai X, Adams MW (1996). "Characterization of a fourth type of 2-keto acid-oxidizing enzyme from a hyperthermophilic archaeon: 2-ketoglutarate ferredoxin oxidoreductase from Thermococcus litoralis". J. Bacteriol. 178 (20): 5890–6. doi:10.1128/jb.178.20.5890-5896.1996. PMC 178443. PMID 8830683.
Schut GJ, Menon AL, Adams MW (2001). "2-keto acid oxidoreductases from Pyrococcus furiosus and Thermococcus litoralis". Methods Enzymol. Methods in Enzymology. 331: 144–58. doi:10.1016/S0076-6879(01)31053-4. ISBN 978-0-12-182232-3. PMID 11265457.
Retrieved from "https://en.wikipedia.org/w/index.php?title=2-oxoglutarate_synthase&oldid=1040189508" |
L-alanine + H2O + NAD+
{\displaystyle \rightleftharpoons }
pyruvate + NH3 + NADH + H+
The 2 substrates of this enzyme are L-alanine, water, and nicotinamide adenine dinucleotide+ because water is 55M and does not change, whereas its 4 products are pyruvate, ammonia, NADH, and hydrogen ion.
This enzyme participates in taurine and hypotaurine metabolism and reductive carboxylate cycle (CO2 fixation).
This enzyme belongs to the family of oxidoreductases, specifically those acting on the CH-NH2 group of donors with NAD+ or NADP+ as acceptor. The systematic name of this enzyme class is L-alanine:NAD+ oxidoreductase (deaminating). Other names in common use include AlaDH, L-alanine dehydrogenase, NAD+-linked alanine dehydrogenase, alpha-alanine dehydrogenase, NAD+-dependent alanine dehydrogenase, alanine oxidoreductase, and NADH-dependent alanine dehydrogenase. T
Alanine dehydrogenase contains both a N-terminus[1] and C-terminus domains.[2][3]
^ Pfam PF05222
^ Tripathi SM, Ramachandran R (2008). "Crystal structures of the Mycobacterium tuberculosis secretory antigen alanine dehydrogenase (Rv2780) in apo and ternary complex forms captures "open" and "closed" enzyme conformations". Proteins. 72 (3): 1089–95. doi:10.1002/prot.22101. PMID 18491387. S2CID 23999004.
O'Connor RJ, Halvorson H (March 1961). "The substrate specificity of L-alanine dehydrogenase". Biochimica et Biophysica Acta. 48 (1): 47–55. doi:10.1016/0006-3002(61)90513-3. PMID 13730044.
Pierard A; Wiame JM (1960). "Proprietes de la L(+)-alanine-deshydrogenase". Biochim. Biophys. Acta. 37 (3): 490–502. doi:10.1016/0006-3002(60)90506-0. PMID 14432812.
Yoshida A, Freese E (February 1965). "Enzymic properties of alanine dehydrogenase of Bacillus subtilis". Biochimica et Biophysica Acta (BBA) - Enzymology and Biological Oxidation. 96 (2): 248–62. doi:10.1016/0926-6593(65)90009-3. PMID 14298830.
Tripathi SM, Ramachandran R (August 2008). "Crystal structures of the Mycobacterium tuberculosis secretory antigen alanine dehydrogenase (Rv2780) in apo and ternary complex forms captures "open" and "closed" enzyme conformations". Proteins. 72 (3): 1089–95. doi:10.1002/prot.22101. PMID 18491387. S2CID 23999004.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Alanine_dehydrogenase&oldid=988271342" |
Planetary gear train with stepped planet gear set - MATLAB - MathWorks í•œêµ
Compound Planetary Gear
Ring (R) to planet (P) teeth ratio (NR/NP)
Planetary gear train with stepped planet gear set
The Compound Planetary Gear block represents a planetary gear train with composite planet gears. Each composite planet gear is a pair of rigidly connected and longitudinally arranged gears of different radii. One of the two gears engages the centrally located sun gear while the other engages the outer ring gear.
The block models the compound planetary gear as a structural component based on the Simscape™ Driveline™ Sun-Planet and Ring-Planet blocks. The figure demonstrates the equivalent block diagram for the Compound planetary gear block.
To increase the fidelity of the gear model, specify properties such as gear inertia, meshing losses, and viscous losses. By default, gear inertia and viscous losses are assumed to be negligible. The block enables you to specify the inertias of the internal planet gears. To model the inertias of the carrier, sun, and ring gears, connect Simscape Inertia blocks to ports C, S, and R.
The Compound Planetary Gear block imposes two kinematic and two geometric constraints.
{r}_{C}{\mathrm{Ï}}_{C}={r}_{S}{\mathrm{Ï}}_{S}+{r}_{P1}{\mathrm{Ï}}_{P},
{r}_{R}{\mathrm{Ï}}_{R}={r}_{C}{\mathrm{Ï}}_{C}+{r}_{P2}{\mathrm{Ï}}_{P},
{r}_{C}={r}_{S}+{r}_{P1},
{r}_{R}={r}_{C}+{r}_{P2},
rP1 is the radius of planet gear 1.
The ring-planet and planet-sun gear ratios are:
{g}_{RP}={r}_{R}/{r}_{P2}={N}_{R}/{N}_{P2}
{g}_{PS}={r}_{P1}/{r}_{S}={N}_{P1}/{N}_{S},
gRP is the ring-planet gear ratio.
NR is the number of teeth on the ring gear.
NP2 is the number of teeth on planet gear 2.
gPS is the planet-sun gear ratio.
NS is the number of teeth on the sun gear.
In terms of the gear ratios, the key kinematic constraint is:
\left(\text{1 }+ {g}_{RP}{g}_{PS}\right){\mathrm{Ï}}_{C} = {\mathrm{Ï}}_{S} + {g}_{RP}{g}_{PS}{\mathrm{Ï}}_{R}.
The four degrees of freedom reduce to two independent degrees of freedom. The gear pairs are (1, 2) = (P2, R) and (S, P1).
The gear ratio gRP must be strictly greater than one.
The torque transfers are:
{g}_{RP}{\mathrm{Ï}}_{P2} + {\mathrm{Ï}}_{R}– {\mathrm{Ï}}_{loss}\left(P2,R\right)\text{ }=\text{ }0
{g}_{PS}{\mathrm{Ï}}_{S} + {\mathrm{Ï}}_{P1} – {\mathrm{Ï}}_{loss}\left(S,P1\right)=0,
Ï„P2 is torque transfer for planet gear 2.
Ring (R) to planet (P) teeth ratio (NR/NP) — Ring to planet gear rotation ratio
Fixed ratio, gRP, of the ring gear to the planet gear rotations as defined by the number of ring gear teeth divided by the number of planet gear teeth. This gear ratio must be strictly greater than 1.
Fixed ratio, gPS, of the planet gear to the sun gear rotations as defined by the number of planet gear teeth divided by the number of sun gear teeth. This gear ratio must be strictly greater than 0.
Sun-planet and ring-planet ordinary efficiencies — Torque transfer efficiencies
[.96 .98] (default) | vector
Vector of torque transfer efficiencies, [ηSP ηRP], for sun-planet and ring-carrier gear wheel pair meshings, respectively. The vector element must be in the interval (0,1].
Sun-planet efficiency — Sun-planet efficiency
[.95, .9, .85] (default) | vector | rot
Ring-planet efficiency — Ring-planet efficiency
[.95 .9 .85] (default) | vector
Vector of component efficiencies, ηRP—the ratio of output power to input power, that the block uses to construct a 1-D temperature-efficiency lookup table.
Sun-carrier and planet-carrier power thresholds — Minimum efficiency power threshold for the sun-carrier and planet carrier gear couplings
Vector of power thresholds above which the full efficiency factors apply. Enter the thresholds in the order sun-carrier, planet-carrier. Below these values, a hyperbolic tangent function smooths the efficiency factor.
[0 0] N*m/(rad/s) (default) | vector
Vector of viscous friction coefficients, [μS μP], for the sun-carrier and planet-carrier gear motions, respectively.
Planetary Gear | Ravigneaux Gear | Ring-Planet | Sun-Planet |
Field (mathematics)/Related Articles - Citizendium
Field (mathematics)/Related Articles
< Field (mathematics)
A list of Citizendium articles, and planned articles, about Field (mathematics).
See also changes related to Field (mathematics), or pages that link to Field (mathematics) or to this page or whose text contains "Field (mathematics)".
Auto-populated based on Special:WhatLinksHere/Field (mathematics). Needs checking by a human.
Bra-ket notation [r]: The notation ⟨ψ|φ⟩ for the inner product of ψ and φ, and related notations. [e]
{\displaystyle i^{2}=-1}
Dual space [r]: The space formed by all functionals defined on a given space. [e]
Elliptic curve [r]: An algebraic curve of genus one with a group structure; a one-dimensional abelian variety. [e]
Linear map [r]: Function between two vector spaces that preserves the operations of vector addition and scalar multiplication. [e]
Matrix inverse [r]: The equivalent of the reciprocal defined for certain matrices. [e]
Normal extension [r]: A field extension which contains all the roots of an irreducible polynomial if it contains one such root. [e]
Ordered field [r]: A field with a total order which is compatible with the algebraic operations. [e]
Rational number [r]: Add brief definition or description
Trace (mathematics) [r]: Add brief definition or description
Vector field [r]: Add brief definition or description
Retrieved from "https://citizendium.org/wiki/index.php?title=Field_(mathematics)/Related_Articles&oldid=643218" |
Write File - Maple Help
Home : Support : Online Help : Connectivity : Web Features : Worksheet Package : Write File
write a Maple XML tree data structure representing a Maple worksheet to a file as a MWS document
WriteFile(fileName, xmlTree, format=output_format)
Maple XML tree; worksheet
The WriteFile command writes the specified XML document xmlTree to the file fileName in the specified format.
It is assumed that the XML document that is written to the file represents a valid worksheet. (Maple performs only a surface check.)
\mathrm{with}\left(\mathrm{Worksheet}\right):
\mathrm{WriteFile}\left("temp.mws",\mathrm{xmlTree},\mathrm{format}="mws"\right) |
Heat - Citizendium
Energy of the hot gas flame flows into the kettle and the liquid water in it.
Heat is a form of energy that is transferred between two bodies that are in thermal contact and have different temperatures. For instance, the bodies may be two compartments of a vessel separated by a heat-conducting wall and containing fluids of different temperatures on either side of the wall. Or one body may consist of hot radiating gas and the other may be a kettle with cold water, as shown in the picture. Heat flows spontaneously from the higher-temperature to the lower-temperature body. The effect of this transfer of energy usually, but not always, is an increase in the temperature of the colder body and a decrease in the temperature of the hotter body.
1 Change of aggregation state
3 Equivalence of heat and work
5 Forms of heat
7 Semantic caveats
Change of aggregation state
A vessel containing a fluid may lose or gain energy without a change in temperature when the fluid changes from one aggregation state to another. For instance, a gas condensing to a liquid does this at a certain fixed temperature (the boiling point of the liquid) and releases condensation energy. When a vessel, containing a condensing gas, loses heat to a colder body, then, as long as there is still vapor left in it, its temperature remains constant at the boiling point of the liquid, even while it is losing heat to the colder body. In a similar way, when the colder body is a vessel containing a melting solid, its temperature will remain constant while it is receiving heat from a hotter body, as long as not all solid has been molten. Only after all of the solid has been molten and the heat transport continues, the temperature of the colder body (then containing only liquid) will rise.
For example, the temperature of the tap water in the kettle shown in the figure will rise quickly to the boiling point of water (100 °C). Then, when the flame is not switched off, the temperature inside the kettle remains constant at 100 °C for quite some time, even though heat keeps on flowing from flame to kettle. When all liquid water has evaporated—when the kettle has boiled dry—the temperature of the kettle will quickly rise again until it obtains the temperature of the burning gas, then the heat flow will finally stop. (Most likely, though, the handle and maybe the metal of the kettle, too, will have melted before that).
At present the unit for the amount of heat is the same as for any form of energy. Before the equivalence of mechanical work and heat was clearly recognized, two units were used. The calorie was the amount of heat necessary to raise the temperature of one gram of water from 14.5 to 15.5 °C and the unit of mechanical work was basically defined by force times path length (in the old cgs system of units this is erg). Now there is one unit for all forms of energy, including heat. In the International System of Units (SI) it is the joule, but the British Thermal Unit and calorie are still occasionally used. The unit for the rate of heat transfer is the watt (J/s).
Although heat and work are forms of energy that both obey the law of conservation of energy, they are not completely equivalent. Work can be completely converted into heat, but the converse is not true. When converting heat into work, part of the heat is not—and cannot be—converted to work, but flows to the body of lower temperature that is out of necessity present to generate a heat flow.
The important distinction between heat and temperature (heat being a form of energy and temperature a measure of the amount of that energy present in a body) was clarified by Count Rumford, James Prescott Joule, Julius Robert Mayer, Rudolf Clausius, and others during the late 18th and 19th centuries. Also it became clear by the work of these men that heat is not an invisible and weightless fluid, named caloric, as was thought by many 18th century scientists, but a form of motion. The molecules of the hotter body are (on the average) in more rapid motion than those of the colder body. The first law of thermodynamics, discovered around the middle of the 19th century, states that the (flow of) heat is a transfer of part of the internal energy of the bodies. In the case of ideal gases, internal energy consists only of kinetic energy and it is indeed only this motional energy that is transferred when heat is exchanged between two containers with ideal gases. In the case of non-ideal gases, liquids and solids, internal energy also contains the averaged inter-particle potential energy (attraction and repulsion between molecules), which depends on temperature. So, for non-ideal gases, liquids and solids, also potential energy is transferred when heat transfer occurs.
The actual transport of heat may proceed by electromagnetic radiation (as an example one may think of an electric heater where usually heat is transferred to its surroundings by infrared radiation, or of a microwave oven where heat is given off to food by microwaves), conduction (for instance through a metal wall; metals conduct heat by the aid of their almost free electrons), and convection (for instance by air flow or water circulation).
If two systems, 1 (cold) and 2 (hot), are isolated from the rest of the universe (i.e., no other heat flows than from 2 to 1 and no work is performed on the two systems) then the entropy Stot = S1 + S2 of the total system 1 + 2 increases upon the spontaneous flow of heat. This is in accordance with the second law of thermodynamics that states that spontaneous thermodynamic processes are associated with entropy increase. In general, the entropy S of a system at absolute temperature T increases with
{\displaystyle \Delta S={\frac {Q}{T}}}
when it receives an amount of heat Q > 0. Entropy is an additive (size-extensive) property.
The hotter system 2 loses an amount of heat to the colder system 1. In absolute value the exchanged amounts of heat are the same by the law of conservation of energy (no energy escapes to the rest of the universe), hence
{\displaystyle \Delta S_{\mathrm {tot} }=\Delta S_{1}+\Delta S_{2}=Q\left({\frac {1}{T_{1}}}-{\frac {1}{T_{2}}}\right)>0\quad {\hbox{because}}\quad T_{2}>T_{1}.}
Here it is assumed that the amount of heat Q is so small that the temperatures of the two systems are constant. One can achieve this by considering a small time interval of heat exchange and/or very large systems.
Remark: the expression ΔS = Q/T is only strictly valid for a reversible (also known as quasistatic) flow of energy. It is possible[1] to define:
{\displaystyle \Delta S\equiv \Delta S_{\mathrm {ext} }+\Delta S_{\mathrm {int} }\quad {\hbox{with}}\quad \Delta S_{\mathrm {ext} }\equiv {\frac {Q}{T}}\quad {\hbox{and}}\quad \Delta S_{\mathrm {int} }\geq 0}
It is assumed that ΔSint is much smaller than ΔSext, so that it can be neglected.
Semantic caveats
It is strictly speaking not correct to say that a hot object "possesses much heat"—it is correct to say, however, that it possesses high internal energy. The word "heat" is reserved to describe the process of transfer of energy from a high temperature object to a lower temperature one (in short called "heating of the cold object"). The reason that the word "heat" is to be avoided for the internal energy of an object is that the latter can have been acquired either by heating or by work done on it (or by both). When we measure internal energy, there is no way of deciding how the object acquired it—by work or by heat. In the same way as one does not say that a hot object "possesses much work", one does not say that it "possesses much heat". Yet, terms as "heat reservoir" (a system of temperature higher than its environment that for all practical purposes is infinite) and "heat content" (a synonym for enthalpy) are commonly used and are incorrect by the same reasoning.
The molecules of a hot body are in agitated motion and, as said, it cannot be measured how they became agitated, by work or by heat. Often, especially outside physics, the random molecular motion is referred to as "thermal energy". In classical (phenomenological) thermodynamics this is an intuitive, but undefined, concept. In statistical thermodynamics, thermal energy could be defined (but rarely ever is) as the average kinetic energy of the molecules constituting the body. Kinetic and potential energy of molecules are concepts that are foreign to classical thermodynamics, which predates the general acceptance of the existence of molecules.
As a result Carathéodory was able to obtain the laws of thermodynamics without recourse to fictitious machines or objectionable concepts as the flow of heat.[2]
↑ E. A. Guggenheim, Thermodynamics, 5th edition, North Holland (1967). p. 17
↑ H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry, 2nd edition, Van Nostrand Company, New York (1956) p. 29
Retrieved from "https://citizendium.org/wiki/index.php?title=Heat&oldid=625422" |
Use your knowledge of the unit circle to explain why the graphs of
y = \sinθ
y = \cos(θ -\frac { \pi } { 2 })
\frac{\pi}{2}
90°
Sketch a unit circle. Choose any angle. Find the sine and cosine of the angle. Now subtract
90°
from your angle and find the sine and cosine again. What happened? If you draw an accurate reference triangle, it will help you see what is going on here. |
Matrices - Course Hero
College Algebra/Matrices
Matrices are arrays of numbers or organized data. Operations with matrices include addition, subtraction, and multiplication. Matrices may also have inverses. The methods for finding the inverse of a matrix include using a formula, technology, or matrix operations. Matrices can represent systems of linear equations. A variety of techniques can be used to solve systems of equations represented as matrices.
m\times n
n\times p
m\times p
The inverse of a matrix can be used to solve a matrix equation of the form
AX=B
Gaussian elimination is a method of solving systems of linear equations by using an augmented matrix and row operations.
Row operations can be used to determine whether a linear system of equations is consistent and independent, inconsistent, or dependent. |
Architectures - EnergyFlow
eval_filters
Energy Flow Networks (EFNs) and Particle Flow Networks (PFNs) are model architectures designed for learning from collider events as unordered, variable-length sets of particles. Both EFNs and PFNs are parameterized by a learnable per-particle function \Phi
\Phi
and latent space function F
An EFN takes the following form:
\text{EFN}=F\left(\sum_{i=1}^M z_i \Phi(\hat p_i)\right)
where z_i
is a measure of the energy of particle i
, such as z_i=p_{T,i}
z_i=p_{T,i}
, and \hat p_i
\hat p_i
is a measure of the angular information of particle i
, such as \hat p_i = (y_i,\phi_i)
\hat p_i = (y_i,\phi_i)
. Any infrared- and collinear-safe observable can be parameterized in this form.
A PFN takes the following form:
\text{PFN}=F\left(\sum_{i=1}^M \Phi(p_i)\right)
where p_i
is the information of particle i
, such as its four-momentum, charge, or flavor. Any observable can be parameterized in this form. See the Deep Sets framework for additional discussion.
Since these architectures are not used by the core EnergyFlow code, and require the external TensorFlow and scikit-learn libraries, they are not imported by default but must be explicitly imported, e.g. from energyflow.archs import *. EnergyFlow also contains several additional model architectures for ease of using common models that frequently appear in the intersection of particle physics and machine learning.
Base class for all architectures contained in EnergyFlow. The mechanism of specifying hyperparameters for all architectures is described here. Methods common to all architectures are documented here. Note that this class cannot be instantiated directly as it is an abstract base class.
energyflow.archs.archbase.ArchBase(*args, **kwargs)
Accepts arbitrary arguments. Positional arguments (if present) are dictionaries of hyperparameters, keyword arguments (if present) are hyperparameters directly. Keyword hyperparameters take precedence over positional hyperparameter dictionaries.
*args : arbitrary positional arguments
Each argument is a dictionary containing hyperparameter (name, value) pairs.
*kwargs : arbitrary keyword arguments
Hyperparameters as keyword arguments. Takes precedence over the positional arguments.
Default NN Hyperparameters
Common hyperparameters that apply to all architectures except for LinearClassifier.
loss='categorical_crossentropy' : str
The loss function to use for the model. See the Keras loss function docs for available loss functions.
optimizer='adam' : Keras optimizer or str
A Keras optimizer instance or a string referring to one (in which case the default arguments are used).
metrics=['accuracy'] : list of str
The Keras metrics to apply to the model.
compile_opts={} : dict
Dictionary of keyword arguments to be passed on to the compile method of the model. loss, optimizer, and metrics (see above) are included in this dictionary. All other values are the Keras defaults.
output_dim=2 : int
The output dimension of the model.
output_act='softmax' : str or Keras activation
Activation function to apply to the output.
filepath=None : str
The file path for where to save the model. If None then the model will not be saved.
save_while_training=True : bool
Whether the model is saved during training (using the ModelCheckpoint callback) or only once training terminates. Only relevant if filepath is set.
save_weights_only=False : bool
Whether only the weights of the model or the full model are saved. Only relevant if filepath is set.
modelcheck_opts={'save_best_only':True, 'verbose':1} : dict
Dictionary of keyword arguments to be passed on to the ModelCheckpoint callback, if it is present. save_weights_only (see above) is included in this dictionary. All other arguments are the Keras defaults.
patience=None : int
The number of epochs with no improvement after which the training is stopped (using the EarlyStopping callback). If None then no early stopping is used.
earlystop_opts={'restore_best_weights':True, 'verbose':1} : dict
Dictionary of keyword arguments to be passed on to the EarlyStopping callback, if it is present. patience (see above) is included in this dictionary. All other arguments are the Keras defaults.
name_layers=True : bool
Whether to give the layers of the model explicit names or let them be named automatically. One reason to set this to False would be in order to use parts of this model in another model (all Keras layers in a model are required to have unique names).
compile=True : bool
Whether the model should be compiled or not.
summary=True : bool
Whether a summary should be printed or not.
Train the model by fitting the provided training dataset and labels. Transparently calls the .fit() method of the underlying model.
*args : numpy.ndarray or tensorflow.data.Dataset
Either the X_train and Y_train NumPy arrays or a TensorFlow dataset.
Keyword arguments passed on to the .fit() method of the underlying model. Most relevant for neural network models, where the TensorFlow/Keras model docs contain detailed information on the possible arguments.
The return value of the the underlying model's .fit() method.
predict(X_test, **kwargs)
Evaluate the model on a dataset. Note that for the LinearClassifier this corresponds to the predict_proba method of the underlying scikit-learn model.
X_test : numpy.ndarray
The dataset to evaluate the model on.
Keyword arguments passed on to the underlying model when predicting on a dataset.
The value of the model on the input dataset.
The underlying model held by this architecture. Note that accessing an attribute that the architecture does not have will resulting in attempting to retrieve the attribute from this model. This allows for interrogation of the EnergyFlow architecture in the same manner as the underlying model.
For neural network models:
model.layers will return a list of the layers, where model is any EnergFlow neural network.
model.coef_ will return the coefficients, where model is any EnergyFlow LinearClassifier instance.
Energy Flow Network (EFN) architecture.
energyflow.archs.EFN(*args, **kwargs)
See ArchBase for how to pass in hyperparameters as well as defaults common to all EnergyFlow neural network models.
Required EFN Hyperparameters
The number of features for each particle.
Phi_sizes (formerly ppm_sizes) : {tuple, list} of int
The sizes of the dense layers in the per-particle frontend module \Phi
\Phi
. The last element will be the number of latent observables that the model defines.
F_sizes (formerly dense_sizes) : {tuple, list} of int
The sizes of the dense layers in the backend module F
Default EFN Hyperparameters
Phi_acts='relu' (formerly ppm_acts) : {tuple, list} of str or Keras activation
Activation functions(s) for the dense layers in the per-particle frontend module \Phi
\Phi
. A single string or activation layer will apply the same activation to all layers. Keras advanced activation layers are also accepted, either as strings (which use the default arguments) or as Keras Layer instances. If passing a single Layer instance, be aware that this layer will be used for all activations and may introduce weight sharing (such as with PReLU); it is recommended in this case to pass as many activations as there are layers in the model. See the Keras activations docs for more detail.
F_acts='relu' (formerly dense_acts) : {tuple, list} of str or Keras activation
Activation functions(s) for the dense layers in the backend module F
. A single string or activation layer will apply the same activation to all layers.
Phi_k_inits='he_uniform' (formerly ppm_k_inits) : {tuple, list} of str or Keras initializer
Kernel initializers for the dense layers in the per-particle frontend module \Phi
\Phi
. A single string will apply the same initializer to all layers. See the Keras initializer docs for more detail.
F_k_inits='he_uniform' (formerly dense_k_inits) : {tuple, list} of str or Keras initializer
Kernel initializers for the dense layers in the backend module F
. A single string will apply the same initializer to all layers.
latent_dropout=0 : float
Dropout rates for the summation layer that defines the value of the latent observables on the inputs. See the Keras Dropout layer for more detail.
F_dropouts=0 (formerly dense_dropouts) : {tuple, list} of float
Dropout rates for the dense layers in the backend module F
. A single float will apply the same dropout rate to all dense layers.
Phi_l2_regs=0 : {tuple, list} of float
L_2
-regulatization strength for both the weights and biases of the layers in the \Phi
\Phi
network. A single float will apply the same L_2
-regulatization to all layers.
F_l2_regs=0 : {tuple, list} of float
L_2
-regulatization strength for both the weights and biases of the layers in the F
network. A single float will apply the same L_2
mask_val=0 : float
The value for which particles with all features set equal to this value will be ignored. The Keras Masking layer appears to have issues masking the biases of a network, so this has been implemented in a custom (and correct) manner since version 0.12.0.
num_global_features=None : int
Number of additional features to be concatenated with the latent space observables to form the input to F. If not None, then the features are to be provided at the end of the list of inputs.
eval_filters(patch, n=100, prune=True)
Evaluates the latent space filters of this model on a patch of the two-dimensional geometric input space.
patch : {tuple, list} of float
Specifies the patch of the geometric input space to be evaluated. A list of length 4 is interpretted as [xmin, ymin, xmax, ymax]. Passing a single float R is equivalent to [-R,-R,R,R].
n : {tuple, list} of int
The number of grid points on which to evaluate the filters. A list of length 2 is interpretted as [nx, ny] where nx is the number of points along the x (or first) dimension and ny is the number of points along the y (or second) dimension.
prune : bool
Whether to remove filters that are all zero (which happens sometimes due to dying ReLUs).
(numpy.ndarray, numpy.ndarray, numpy.ndarray)
Returns three arrays, (X, Y, Z), where X and Y have shape (nx, ny) and are arrays of the values of the geometric inputs in the specified patch. Z has shape (num_filters, nx, ny) and is the value of the different filters at each point.
List of input tensors to the model. EFNs have two input tensors: inputs[0] corresponds to the zs input and inputs[1] corresponds to the phats input.
Weight tensor for the model. This is the zs input where entries equal to mask_val have been set to zero.
List of tensors corresponding to the layers in the \Phi
\Phi
network.
List of tensors corresponding to the summation layer in the network, including any dropout layer if present.
List of tensors corresponding to the layers in the F
Output tensor for the model.
Particle Flow Network (PFN) architecture. Accepts the same hyperparameters as the EFN.
energyflow.archs.PFN(*args, **kwargs)
List of input tensors to the model. PFNs have one input tensor corresponding to the ps input.
Weight tensor for the model. A weight of 0 is assigned to any particle which has all features equal to mask_val, and 1 is assigned otherwise.
\Phi
List of tensors corresponding to the layers in the F
energyflow.archs.CNN(*args, **kwargs)
Required CNN Hyperparameters
input_shape : {tuple, list} of int
The shape of a single jet image. Assuming that data_format is set to channels_first, this is (nb_chan,npix,npix).
filter_sizes : {tuple, list} of int
The size of the filters, which are taken to be square, in each convolutional layer of the network. The length of the list will be the number of convolutional layers in the network.
num_filters : {tuple, list} of int
The number of filters in each convolutional layer. The length of num_filters must match that of filter_sizes.
Default CNN Hyperparameters
dense_sizes=None : {tuple, list} of int
The sizes of the dense layer backend. A value of None is equivalent to an empty list.
pool_sizes=0 : {tuple, list} of int
Size of maxpooling filter, taken to be a square. A value of 0 will not use maxpooling.
conv_acts='relu' : {tuple, list} of str or Keras activation
Activation function(s) for the conv layers. A single string or activation layer will apply the same activation to all conv layers. Keras advanced activation layers are also accepted, either as strings (which use the default arguments) or as Keras Layer instances. If passing a single Layer instance, be aware that this layer will be used for all activations and may introduce weight sharing (such as with PReLU); it is recommended in this case to pass as many activations as there are layers in the model.See the Keras activations docs for more detail.
dense_acts='relu' : {tuple, list} of str or Keras activation
Activation functions(s) for the dense layers. A single string or activation layer will apply the same activation to all dense layers.
conv_k_inits='he_uniform' : {tuple, list} of str or Keras initializer
Kernel initializers for the convolutional layers. A single string will apply the same initializer to all layers. See the Keras initializer docs for more detail.
dense_k_inits='he_uniform' : {tuple, list} of str or Keras initializer
Kernel initializers for the dense layers. A single string will apply the same initializer to all layers.
conv_dropouts=0 : {tuple, list} of float
Dropout rates for the convolutional layers. A single float will apply the same dropout rate to all conv layers. See the Keras Dropout layer for more detail.
num_spatial2d_dropout=0 : int
The number of convolutional layers, starting from the beginning of the model, for which to apply SpatialDropout2D instead of Dropout.
dense_dropouts=0 : {tuple, list} of float
Dropout rates for the dense layers. A single float will apply the same dropout rate to all dense layers.
paddings='valid' : {tuple, list} of str
Controls how the filters are convoled with the inputs. See the Keras Conv2D layer for more detail.
data_format='channels_last' : {'channels_first', 'channels_last'}
Sets which axis is expected to contain the different channels. 'channels_first' appears to have issues with newer versions of tensorflow, so prefer 'channels_last'.
Dense Neural Network architecture.
energyflow.archs.DNN(*args, **kwargs)
Required DNN Hyperparameters
dense_sizes : {tuple, list} of int
The number of nodes in the dense layers of the model.
Default DNN Hyperparameters
acts='relu' : {tuple, list} of str or Keras activation
Activation functions(s) for the dense layers. A single string or activation layer will apply the same activation to all dense layers. Keras advanced activation layers are also accepted, either as strings (which use the default arguments) or as Keras Layer instances. If passing a single Layer instance, be aware that this layer will be used for all activations and may introduce weight sharing (such as with PReLU); it is recommended in this case to pass as many activations as there are layers in the model.See the Keras activations docs for more detail.
k_inits='he_uniform' : {tuple, list} of str or Keras initializer
Kernel initializers for the dense layers. A single string will apply the same initializer to all layers. See the Keras initializer docs for more detail.
dropouts=0 : {tuple, list} of float
Dropout rates for the dense layers. A single float will apply the same dropout rate to all layers. See the Keras Dropout layer for more detail.
l2_regs=0 : {tuple, list} of float
L_2
-regulatization strength for both the weights and biases of the dense layers. A single float will apply the same L_2
Linear classifier that can be either Fisher's linear discriminant or logistic regression. Relies on the scikit-learn implementations of these classifiers.
energyflow.archs.LinearClassifier(*args, **kwargs)
See ArchBase for how to pass in hyperparameters.
linclass_type='lda' : {'lda', 'lr'}
Controls which type of linear classifier is used. 'lda' corresponds to LinearDisciminantAnalysis and 'lr' to Logistic Regression. If using 'lr' all arguments are passed on directly to the scikit-learn class.
Linear Discriminant Analysis Hyperparameters
solver='svd' : {'svd', 'lsqr', 'eigen'}
Which LDA solver to use.
tol=1e-12 : float
Threshold used for rank estimation. Notably not a convergence parameter.
LR_hps={} : dict
Dictionary of keyword arguments to pass on to the underlying LogisticRegression model. |
Plot response pattern of microphone - MATLAB - MathWorks América Latina
Azimuth Response and Directivity of Cardioid Microphone
Response of Cardioid Microphone in U/V Space
3-D Response of Cardioid Microphone over Restricted Range of Angles
Plot response pattern of microphone
plotResponse(H,FREQ)
plotResponse(H,FREQ,Name,Value)
plotResponse(H,FREQ) plots the element response pattern along the azimuth cut, where the elevation angle is 0. The operating frequency is specified in FREQ.
plotResponse(H,FREQ,Name,Value) plots the element response with additional options specified by one or more Name,Value pair arguments.
Element System object™
Operating frequency in Hertz specified as a scalar or 1–by-K row vector. FREQ must lie within the range specified by the FrequencyVector property of H. If you set the 'RespCut' property of H to '3D', FREQ must be a scalar. When FREQ is a row vector, plotResponse draws multiple frequency responses on the same axes.
Cut angle specified as a scalar. This argument is applicable only when RespCut is 'Az' or 'El'. If RespCut is 'Az', CutAngle must be between –90 and 90. If RespCut is 'El', CutAngle must be between –180 and 180.
Specify the polarization options for plotting the antenna response pattern. The allowable values are |'None' | 'Combined' | 'H' | 'V' | where
For antennas that do not support polarization, the only allowed value is 'None'. This parameter is not applicable when you set the Unit parameter value to 'dbi'.
Azimuth angles for plotting element response, specified as a row vector. The AzimuthAngles parameter sets the display range and resolution of azimuth angles for visualizing the radiation pattern. This parameter is allowed only when the RespCut parameter is set to 'Az' or '3D' and the Format parameter is set to 'Line' or 'Polar'. The values of azimuth angles should lie between –180° and 180° and must be in nondecreasing order. When you set the RespCut parameter to '3D', you can set the AzimuthAngles and ElevationAngles parameters simultaneously.
Elevation angles for plotting element response, specified as a row vector. The ElevationAngles parameter sets the display range and resolution of elevation angles for visualizing the radiation pattern. This parameter is allowed only when the RespCut parameter is set to 'El' or '3D' and the Format parameter is set to 'Line' or 'Polar'. The values of elevation angles should lie between –90° and 90° and must be in nondecreasing order. When you set the RespCut parameter to '3D', you can set the ElevationAngles and AzimuthAngles parameters simultaneously.
U coordinate values for plotting element response, specified as a row vector. The UGrid parameter sets the display range and resolution of the U coordinates for visualizing the radiation pattern in U/V space. This parameter is allowed only when the Format parameter is set to 'UV' and the RespCut parameter is set to 'U' or '3D'. The values of UGrid should be between –1 and 1 and should be specified in nondecreasing order. You can set the UGrid and VGrid parameters simultaneously.
V coordinate values for plotting element response, specified as a row vector. The VGrid parameter sets the display range and resolution of the V coordinates for visualizing the radiation pattern in U/V space. This parameter is allowed only when the Format parameter is set to 'UV' and the RespCut parameter is set to '3D'. The values of VGrid should be between –1 and 1 and should be specified in nondecreasing order. You can set the VGrid and UGrid parameters simultaneously.
h = phased.CustomMicrophoneElement;
h.PolarPatternFrequencies = [500 1000];
h.PolarPattern = mag2db([...
0.5+0.5*cosd(h.PolarPatternAngles);...
0.6+0.4*cosd(h.PolarPatternAngles)]);
plotResponse(h,[fc 2*fc],'RespCut','Az','Format','Polar');
plotResponse(h,[fc 2*fc],'RespCut','Az','Format','Line','Unit','dbi');
u
-cut of the response of a custom cardioid microphone that is designed to operate in the frequency range 500-1000 Hz.
plotResponse(h,fc,'Format','UV');
Plot the 3-D response of a custom cardioid microphone in space but with both the azimuth and elevation angles restricted to the range -40 to 40 degrees in 0.1 degree increments.
Plot the 3-D response.
plotResponse(h,fc,'Format','polar','RespCut','3D',...
'Unit','mag','AzimuthAngles',[-40:0.1:40],...
'ElevationAngles',[-40:0.1:40]); |
statplots(deprecated)/boxplot - Maple Help
Home : Support : Online Help : statplots(deprecated)/boxplot
stats[statplots, boxplot]
stats[statplots, boxplot](data,
\mathrm{arg}=\mathrm{value}
statplots[boxplot(data,
\mathrm{arg}=\mathrm{value}
boxplot(data,
\mathrm{arg}=\mathrm{value}
shift=x
(optional, default=0) plot center shifted to x
(optional, default=1) width of the box plot
format=notched
optionally format as a notched-box plot
additional plot options
The function boxplot of the subpackage stats[statplots, ...] gives a box plot summarizing the data in data.
A box plot comprises these elements:
1) A box with
- a central line showing the median,
- a lower line showing the first quartile,
- an upper line showing the third quartile;
2) Two lines extending from the central box of maximal length 3/2 the interquartile range but not extending past the range of the data;
3) Outliers, points that lie outside the extent of the previous elements.
Using the parameter format=notched will create a box plot with one additional feature. The sides of the box are indented, or notched, at the median line. This notch has a maximal width of 3.14 times the interquartile range divided by the square root of the total weight of the data. The notch is constrained inside the first and third quartiles.
When there is more than one statistical list in data, a box plot will be produced for each data set. Each box will be plotted on the same axes, side by side.
Box plots are quite useful to compare similar data sets. For example, one can quickly compare the monthly mid-day temperature over the period of one year by producing one boxplot per month, and displaying them side by side.
Notched-box plots can be used to determine if two random samples were drawn from the same population. If the notches of each plot overlap, it is acceptable to say that both data sets have the same distribution. However, overlapping notches do not guarantee that the data sets are identically distributed.
Another use of box-plots is as a one-dimensional summary that is inserted in the edge of a scatter plot.
Classes are assumed to be represented by the class mark, for example
10..12
11
. Missing data are ignored.
\mathrm{with}\left(\mathrm{stats}\right):
\mathrm{data1}≔[2.93,2.58,2.85,4.26,2.94,4.33,1.71,4.42,3.59,4.35,2.07,1.16,2.36,1.16,4.72]:
\mathrm{data2}≔[2.46,4.34,0.182,3.22,5.37,10.5,3.11,-1.99,-0.865,2.56,10.6,10.9,6.56,7.22,4.84]:
p≔\mathrm{statplots}[\mathrm{boxplot}]\left(\mathrm{data1},\mathrm{axes}=\mathrm{FRAMED}\right):
p
change axes to frame:
\mathrm{plots}[\mathrm{display}]\left(p,\mathrm{axes}=\mathrm{FRAME}\right)
Usually, there are more than one box plot in a statistical plot. Two sets of data can be compared in this manner:
\mathrm{with}\left(\mathrm{stats}[\mathrm{statplots}]\right):
\mathrm{boxplot}\left(\mathrm{data1},\mathrm{data2},\mathrm{shift}=1\right)
Plotting data1 versus data2 with box plots in the margin of a scatter plot:
\mathrm{P1}≔\mathrm{scatterplot}\left(\mathrm{data2},\mathrm{data1}\right):
One of the box plots above x=14
\mathrm{P2}≔\mathrm{boxplot}\left(\mathrm{data1},\mathrm{width}=\frac{1}{2},\mathrm{shift}=14\right):
The other box plots above x=6
\mathrm{P3}≔\mathrm{boxplot}\left(\mathrm{data2},\mathrm{width}=\frac{1}{2},\mathrm{shift}=6\right):
Flip that one so that it is now on its side, centered at y=6
\mathrm{P3b}≔\mathrm{xyexchange}\left(\mathrm{P3}\right):
Display the composite plot.
\mathrm{plots}[\mathrm{display}]\left({\mathrm{P1},\mathrm{P2},\mathrm{P3b}},\mathrm{view}=[-3..15,0..6.5],\mathrm{axes}=\mathrm{FRAME}\right)
Examples of notched box plots (use the abbreviated command):
\mathrm{with}\left(\mathrm{stats}[\mathrm{statplots}]\right):
\mathrm{boxplot}\left(\mathrm{data1},\mathrm{format}=\mathrm{notched},\mathrm{axes}=\mathrm{FRAME}\right)
In the following plot, one could conclude that data3 and data4 are samples from the same population, and that data5 was drawn from a different population. This is exactly the situation, since data3 and data4 come from a random sample of a chi-square distribution with three degrees of freedom, whereas data5 comes from a Student's t distribution with three degrees of freedom.
\mathrm{data3}≔[2.44,1.90,3.54,0.270,4.05,1.43,7.90,8.48,4.56,0.907,3.22,1.22,1.55,6.55,1.72,0.507,0.792,2.62,3.02,0.718]:
\mathrm{data4}≔[6.20,1.55,7.02,4.06,2.58,0.917,1.87,0.731,0.883,0.548,5.14,1.28,1.07,0.398,0.665,0.230,4.88,5.02,6.65,1.92]:
\mathrm{data5}≔[0.0637,-1.15,-0.410,0.993,0.624,0.848,-0.257,-0.385,-0.180,0.282,-0.176,-0.410,1.09,1.94,0.618,0.520,2.61,-1.07,-0.719,0.717]:
\mathrm{boxplot}\left(\mathrm{data3},\mathrm{data4},\mathrm{data5},\mathrm{format}=\mathrm{notched},\mathrm{width}=\frac{1}{2},\mathrm{axes}=\mathrm{FRAMED}\right) |
Lens grinding formula - zxc.wiki
Lens grinding formula
The lens grinder formula , also known as the lens maker equation , indicates how the refractive power of a thin spherical lens is related to its shape. The shape of the lens is described by the radii of the spheres that form the surfaces of the lenses. Further parameters that have an influence on the refractive power are the thickness of the lens, the refractive index of its material and the refractive index of the surrounding medium.
{\ displaystyle R_ {1}, R_ {2}}
the spherical radii (here it should be noted that the two radii have the same sign if the center points are on the same side of the lens [convex-concave lens], but different signs if the lens is biconvex or biconcave; see also: spherical Lenses ),
{\ displaystyle d}
the thickness of the lens (measured at the height of the optical axis ),
{\ displaystyle n_ {0}}
the refractive index of the medium outside the lens,
{\ displaystyle n}
the refractive index of the lens material,
{\ displaystyle f}
the focal length of the lens and
{\ displaystyle D}
the refractive power.
For optical systems with the same media in object space (1) and image space (2) ( ), the following generally applies:
{\ displaystyle n_ {1} = n_ {2} = n_ {0}}
{\ displaystyle D = {\ frac {1} {f}} = {\ frac {n-n_ {0}} {n_ {0}}} \ left ({\ frac {1} {R_ {1}}} - {\ frac {1} {R_ {2}}} + {\ frac {(n-n_ {0}) d} {nR_ {1} R_ {2}}} \ right)}
If the external medium is air under the same conditions, the following applies approximately:
{\ displaystyle \ Rightarrow n_ {1} = n_ {2} = n_ {0} \ approx 1}
{\ displaystyle D = {\ frac {1} {f}} = \ left (n-1 \ right) \ left ({\ frac {1} {R_ {1}}} - {\ frac {1} {R_ {2}}} + {\ frac {(n-1) d} {nR_ {1} R_ {2}}} \ right)}
For thin lenses , the thickness of which is much smaller than the spherical radii, the equation for the so-called lens grinder formula is simplified:
{\ displaystyle D = {\ frac {1} {f}} = \ left (n-1 \ right) \ left ({\ frac {1} {R_ {1}}} - {\ frac {1} {R_ {2}}} \ right) \ ;.}
^ Wilhelm Raith, Clemens Schaefer : Electromagnetism (= textbook of experimental physics . Volume 2 ). 8th, completely revised edition. Walter de Gruyter, Berlin a. a. 1999, ISBN 3-11-016097-8 , pp. 386-387 .
↑ Eugene Hecht: Optics . 4th, revised edition. Oldenbourg, Munich a. a. 2005, ISBN 3-486-27359-0 , pp. 267 .
↑ Wolfgang Zinth , Ursula Zinth: Optics. Rays of light - waves - photons . 3rd, improved edition. Oldenbourg, 2011, ISBN 978-3-486-70534-8 , pp. 93 .
This page is based on the copyrighted Wikipedia article "Linsenschleiferformel" (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. |
Bisphosphoglycerate phosphatase - Wikipedia
Bisphosphoglycerate phosphatase
Bisphosphoglycerate phosphatase (bifunctional) homodimer, Human
In enzymology, a bisphosphoglycerate phosphatase (EC 3.1.3.13) is an enzyme that catalyzes the chemical reaction
2,3-bisphospho-D-glycerate + H2O
{\displaystyle \rightleftharpoons }
3-phospho-D-glycerate + phosphate
Thus, the two substrates of this enzyme are 2,3-bisphospho-D-glycerate and H2O, whereas its two products are 3-phospho-D-glycerate 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 2,3-bisphospho-D-glycerate 2-phosphohydrolase. Other names in common use include 2,3-diphosphoglycerate phosphatase, diphosphoglycerate phosphatase, 2,3-diphosphoglyceric acid phosphatase, 2,3-bisphosphoglycerate phosphatase, and glycerate-2,3-diphosphate phosphatase. This enzyme participates in glycolysis/gluconeogenesis.
As of late 2007, 7 structures have been solved for this class of enzymes, with PDB accession codes 1YFK, 1YJX, 2F90, 2H4X, 2H4Z, 2H52, and 2HHJ.
JOYCE BK, GRISOLIA S (1958). "Studies on glycerate 2,3-diphosphatase". J. Biol. Chem. 233 (2): 350–4. PMID 13563500.
RAPOPORT S, LUEBERING J (1951). "Glycerate-2,3-diphosphatase". J. Biol. Chem. 189 (2): 683–94. PMID 14832286.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bisphosphoglycerate_phosphatase&oldid=1050854475" |
A Treatise on Electricity and Magnetism/Part IV/Chapter VII - Wikisource, the free online library
A Treatise on Electricity and Magnetism/Part IV/Chapter VII
Theory of Electric Circuits
Exploration of the Field by Means of the Secondary Circuit
109318A Treatise on Electricity and Magnetism — Theory of Electric CircuitsJames Clerk Maxwell
THEORY OF ELECTRIC CIRCUITS.
578.1 We may now confine our attention to that part of the kinetic energy of the system which depends on squares and products of the strengths of the electric currents. We may call this the Electrokinetic Energy of the system. The part depending on the motion of the conductors belongs to ordinary dynamics, and we have shewn that the part depending on products of velocities and currents does not exist.
{\displaystyle A_{1}}
{\displaystyle A_{2}}
, &c. denote the different conducting circuits. Let their form and relative position be expressed in terms of the variables
{\displaystyle x_{1}}
{\displaystyle x_{2}}
, &c., the number of which is equal to the number of degrees of freedom of the mechanical system. We shall call these the Geometrical Variables.
{\displaystyle y_{1}}
, denote the quantity of electricity which has crossed a given section of the conductor
{\displaystyle A_{1}}
, since the beginning of the time t. The strength of the current will be denoted by
{\displaystyle {\dot {y}}_{1}}
, the fluxion of this quantity.
We shall call
{\displaystyle {\dot {y}}_{1}}
the actual current, and
{\displaystyle y_{1}}
the integral current. There is one variable of this kind for each circuit in the system.
{\displaystyle T}
denote the electrokinetic energy of the system. It is a homogeneous function of the second degree with respect to the strengths of the currents, and is of the form
{\displaystyle T={\frac {1}{2}}L_{1}{\dot {y}}_{1}^{2}+{\frac {1}{2}}L_{2}{\dot {y}}_{2}^{2}+\And c.+M_{12}{\dot {y}}_{1}{\dot {y}}_{2}+\And c.,}
{\displaystyle L}
{\displaystyle M}
, &c. are functions of the geometrical variables
{\displaystyle x_{1}}
{\displaystyle x_{2}}
, &c. The electrical variables
{\displaystyle y_{1}}
{\displaystyle y_{2}}
, &c., do not enter into the expression.
{\displaystyle L_{1}}
{\displaystyle L_{2}}
, &c., the electric moments of inertia of the circuits
{\displaystyle A_{1}}
{\displaystyle A_{2}}
{\displaystyle M_{12}}
the electric product of inertia of the two circuits
{\displaystyle A_{1}}
{\displaystyle A_{2}}
. When we wish to avoid the language of the dynamical theory, we shall call
{\displaystyle L_{1}}
the coefficient of self-induction of the circuit
{\displaystyle A_{1}}
{\displaystyle M_{12}}
the coefficient of mutual induction of the circuits
{\displaystyle A_{1}}
{\displaystyle A_{2}}
{\displaystyle M_{12}}
is also called the potential of the circuit
{\displaystyle A_{1}}
{\displaystyle A_{2}}
. These quantities depend only on the form and relative position of the circuits. We shall find that in the electromagnetic system of measurement they are quantities of the dimension of a line. See Art. 627.
{\displaystyle T}
{\displaystyle {\dot {y}}_{1}}
we obtain the quantity
{\displaystyle p_{1}}
which, in the dynamical theory, may be called the momentum corresponding to
{\displaystyle y_{1}}
. In the electric theory we shall call
{\displaystyle p_{1}}
the electrokinetic momentum of the circuit
{\displaystyle A_{1}}
. Its value is
{\displaystyle p_{1}=L_{1}{\dot {y}}_{1}+M_{12}{\dot {y}}_{2}+\And c.}
{\displaystyle A_{1}}
is therefore made up of the product of its own current into its coefficient of self-induction, together with the sum of the products of the currents in the other circuits, each into the coefficient of mutual induction of
{\displaystyle A_{1}}
and that other circuit.
On Electromotive Force.
{\displaystyle E}
be the impressed electromotive force in the circuit
{\displaystyle A}
, arising from some cause, such as a voltaic or thermoelectric battery, which would produce a current independently of magneto-electric induction.
{\displaystyle R}
be the resistance of the circuit, then, by Ohm's law, an electromotive force
{\displaystyle R{\dot {y}}}
is required to overcome the resistance, leaving an electromotive force
{\displaystyle E-R{\dot {y}}}
available for changing the momentum of the circuit. Calling this force
{\displaystyle Y'}
, we have, by the general equations,
{\displaystyle Y'={\frac {dp}{dt}}-{\frac {dT}{dy}}}
{\displaystyle T}
does not involve
{\displaystyle y}
, the last term disappears.
Hence, the equation of electromotive force is
{\displaystyle E-R{\dot {y}}=Y'={\frac {dp}{dt}},}
{\displaystyle E=R{\dot {y}}+{\frac {dp}{dt}}.}
The impressed electromotive force
{\displaystyle E}
is therefore the sum of two parts. The first,
{\displaystyle R{\dot {y}}}
, is required to maintain the current
{\displaystyle {\dot {y}}}
against the resistance
{\displaystyle R}
. The second part is required to increase the electromagnetic momentum
{\displaystyle p}
. This is the electromotive force which must be supplied from sources independent of magneto-electric induction. The electromotive force arising from magneto-electric induction alone is evidently
{\displaystyle -{\frac {dp}{dt}}}
, or the rate of decrease of the electrokinetic momentum of the circuit.
580.] Let X' be the impressed mechanical force arising from external causes, and tending to increase the variable x. By the general equations
{\displaystyle X'={\frac {d}{dt}}{\frac {dT}{d{\dot {x}}}}-{\frac {dT}{dx}}.}
Since the expression for the electrokinetic energy does not contain the velocity
{\displaystyle ({\dot {x}})}
, the first term of the second member disappears, and we find
{\displaystyle X'=-{\frac {dT}{dx}}.}
Here X' is the external force required to balance the forces arising from electrical causes. It is usual to consider this force as the reaction against the electromagnetic force, which we shall call X, and which is equal and opposite to X'.
{\displaystyle X={\frac {dT}{dx}}.}
or, the electromagnetic force tending to increase any variable is equal to the rate of increase of the electrokinetic energy per unit increase of that variable, the currents being maintained constant.
If the currents are maintained constant by a battery during a displacement in which a quantity, W, of work is done by electromotive force, the electrokinetic energy of the system will be at the same time increased by W. Hence the battery will be drawn upon for a double quantity of energy, or 2W, in addition to that which is spent in generating heat in the circuit. This was first pointed out by Sir W. Thomson[1]. Compare this result with the electrostatic property in Art. 93.
581.] Let A1 be called the Primary Circuit, and A2 the Secondary Circuit. The electrokinetic energy of the system may be written
{\displaystyle T={\frac {1}{2}}L{\dot {y}}_{1}^{2}+M{\dot {y}}_{1}{\dot {y}}_{2}+{\frac {1}{2}}N{\dot {y}}_{2}^{2}}
where L and N are the coefficients of self-induction of the primary and secondary circuits respectively, and M is the coefficient of their mutual induction.
Let us suppose that no electromotive force acts on the secondary circuit except that due to the induction of the primary current. We have then
{\displaystyle E_{2}=R_{2}{\dot {y}}_{2}+{\frac {d}{dt}}(M{\dot {y}}_{1}+N{\dot {y}}_{2})=0.}
Integrating this equation with respect to t, we have
{\displaystyle R_{2}y_{2}+M{\dot {y}}_{1}+N{\dot {y}}_{2}=C,{\text{ a constant,}}}
where y2 is the integral current in the secondary circuit.
The method of measuring an integral current of short duration will be described in Art. 748, and it is easy in most cases to ensure that the duration of the secondary current shall be very short.
Let the values of the variable quantities in the equation at the end of the time t be accented, then, if y2 is the integral current, or the whole quantity of electricity which flows through a section of the secondary circuit during the time t,
{\displaystyle R_{2}y_{2}=M{\dot {y}}_{1}+N{\dot {y}}_{2}-(M'{\dot {y}}_{1}'+N'{\dot {y}}_{2}').}
If the secondary current arises entirely from induction, its initial value
{\displaystyle {\dot {y}}_{2}}
must be zero if the primary current is constant, and the conductors at rest before the beginning of the time t.
If the time t is sufficient to allow the secondary current to die away,
{\displaystyle {\dot {y}}_{2}'}
, its final value, is also zero, so that the equation becomes
{\displaystyle R_{2}y_{2}=M{\dot {y}}_{1}-M'{\dot {y}}_{1}'.}
The integral current of the secondary circuit depends in this case on the initial and final values of
{\displaystyle M{\dot {y}}_{1}}
Induced Currents.
582.] Let us begin by supposing the primary circuit broken, or
{\displaystyle {\dot {y}}_{1}=0}
, and let a current
{\displaystyle {\dot {y}}_{1}'}
be established in it when contact is made.
The equation which determines the secondary integral current is
{\displaystyle R_{2}y_{2}=-M{\dot {y}}_{1}'.\,}
When the circuits are placed side by side, and in the same direction, M is a positive quantity. Hence, when contact is made in the primary circuit, a negative current is induced in the secondary circuit.
When the contact is broken in the primary circuit, the primary current ceases, and the induced current is y2 where
{\displaystyle R_{2}y_{2}=M{\dot {y}}_{1}.\,}
The secondary current is in this case positive.
If the primary current is maintained constant, and the form or relative position of the circuits altered so that M becomes 'M', the integral secondary current is y2, where
{\displaystyle R_{2}y_{2}=(M-M'){\dot {y}}_{1}.}
In the case of two circuits placed side by side and in the same direction M diminishes as the distance between the circuits in creases. Hence, the induced current is positive when this distance is increased and negative when it is diminished.
These are the elementary cases of induced currents described in Art. 530.
On Mechanical Action between the Two Circuits.
583.] Let x be any one of the geometrical variables on which the form and relative position of the circuits depend, the electromagnetic force tending to increase x is
{\displaystyle X={\frac {1}{2}}{\dot {y}}_{1}^{2}{\frac {dL}{dx}}+{\dot {y}}_{1}{\dot {y}}_{2}{\frac {dM}{dx}}+{\frac {1}{2}}{\dot {y}}_{2}^{2}{\frac {dN}{dx}}.}
If the motion of the system corresponding to the variation of x is such that each circuit moves as a rigid body, L and N will be independent of x, and the equation will be reduced to the form
{\displaystyle X={\dot {y}}_{1}{\dot {y}}_{2}{\frac {dM}{dx}}.}
Hence, if the primary and secondary currents are of the same sign, the force X, which acts between the circuits, will tend to move them so as to increase M.
If the circuits are placed side by side, and the currents flow in the same direction, M will be increased by their being brought nearer together. Hence the force X is in this case an attraction.
584.] The whole of the phenomena of the mutual action of two circuits, whether the induction of currents or the mechanical force between them, depend on the quantity M, which we have called the coefficient of mutual induction. The method of calculating this quantity from the geometrical relations of the circuits is given in Art. 524, but in the investigations of the next chapter we shall not assume a knowledge of the mathematical form of this quantity. We shall consider it as deduced from experiments on induction, as, for instance, by observing the integral current when the secondary circuit is suddenly moved from a given position to an infinite distance, or to any position in which we know that M = 0.
↑ Nichol's Cyclopaedia of Physical Science, ed. 1860, Article, 'Magnetism, Dynamical Relations of'.
Retrieved from "https://en.wikisource.org/w/index.php?title=A_Treatise_on_Electricity_and_Magnetism/Part_IV/Chapter_VII&oldid=1823453" |
Z-test - Wikipedia
(Redirected from Z test)
"Z test" redirects here. For the "Z-test" procedure in the graphics pipeline, see Z-buffering.
Third, calculate the standard score:
{\displaystyle Z={\frac {({\bar {X}}-\mu _{0})}{\sigma }},}
Use in location testing[edit]
The term "Z-test" is often used to refer specifically to the one-sample location test comparing the mean of a set of measurements to a given constant when the sample variance is known. For example, if the observed data X1, ..., Xn are (i) independent, (ii) have a common mean μ, and (iii) have a common variance σ2, then the sample average X has mean μ and variance
{\displaystyle {\frac {\sigma ^{2}}{n}}}
To calculate the standardized statistic
{\displaystyle Z={\frac {({\bar {X}}-\mu _{0})}{s}}}
, we need to either know or have an approximate value for σ2, from which we can calculate
{\displaystyle s^{2}={\frac {\sigma ^{2}}{n}}}
. In some applications, σ2 is known, but this is uncommon.
{\displaystyle \mathrm {SE} ={\frac {\sigma }{\sqrt {n}}}={\frac {12}{\sqrt {55}}}={\frac {12}{7.42}}=1.62}
{\displaystyle {\sigma }}
is the population standard deviation.
{\displaystyle z={\frac {M-\mu }{\mathrm {SE} }}={\frac {96-100}{1.62}}=-2.47}
Z-tests other than location tests[edit]
Location tests are the most familiar Z-tests. Another class of Z-tests arises in maximum likelihood estimation of the parameters in a parametric statistical model. Maximum likelihood estimates are approximately normal under certain conditions, and their asymptotic variance can be calculated in terms of the Fisher information. The maximum likelihood estimate divided by its standard error can be used as a test statistic for the null hypothesis that the population value of the parameter equals zero. More generally, if
{\displaystyle {\hat {\theta }}}
is the maximum likelihood estimate of a parameter θ, and θ0 is the value of θ under the null hypothesis,
{\displaystyle {\frac {{\hat {\theta }}-\theta _{0}}{{\rm {SE}}({\hat {\theta }})}}}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Z-test&oldid=1083448333" |
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