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Integration Basics Guide - Nano Documentation
Integration Basics Guide
Block Lattice Design
Account, Key, Seed and Wallet IDs
Self-Signed Blocks
URI and QR Code Standards
Process a JSON blob block
Block Lattice Design¶
Nano's ledger is built on a data-structure called a "Block Lattice." Every account (private/public key pair) has their own blockchain (account-chain). Only the holder of the private key may sign and publish blocks to their own account-chain. Each block represents a transaction.
Send Send funds from users account to another account
Receive Receive funds from a given "Send" transaction
The system is akin to writing (send) and cashing (receive) a Cashier's Check. There are a few things to consider about transactions:
The receiving account does not have to be online during the Send transaction.
The transaction will stay as receivable indefinitely until a Receive transaction is created.
Once funds are sent, they cannot be revoked by the sender.
The Nano Network achieves consensus using the unique Open Representative Voting (ORV) model. In this setup, representatives (accounts where nano_node with the private keys are running 24/7) vote on transactions.
Below are some helpful things to remember about Nano's representatives and consensus:
A representative's voting power is directly proportional to the amount of funds delegated to that account by other users of the protocol.
An account's representative has no bearing on its transactions or nano_node operation.
Choosing a representative with good uptime that is also a unique entity (to prevent sybil attacks) helps maintain high Nano network security.
If an account's representative goes offline, the account's funds are no longer used to help secure the network; however, the account is unaffected.
Anyone that runs a full-time node may be a representative and be delegated voting weight from other users of the protocol.
An account can freely change its representative anytime within any transaction or explicitly by publishing a block which only changes the representative (sends no funds), which most wallets support.
Account, Key, Seed and Wallet IDs¶
When dealing with the various IDs in the node it is important to understand the function and implication of each one.
Similar IDs, Different Functions
There are several things that can have a similar form but may have very different functions, and mixing them up can result in loss of funds. Use caution when handling them.
Wallet ID¶
This is a series of 32 random bytes of data and is not the seed. It is used in several RPC actions and command line options for the node. It is a purely local UUID that is a reference to a block of data about a specific wallet (set of seed/private keys/info about them) in your node's local database file.
The reason this is necessary is because we want to store information about each account in a wallet: whether it's been used, what its account is so we don't have to generate it every time, its balance, etc. Also, so we can hold ad hoc accounts, which are accounts that are not derived from the seed. This identifier is only useful in conjunction with your node's database file and it will not recover funds if that database is lost or corrupted.
This is the value that you get back when using the wallet_create etc RPC commands, and what the node expects for RPC commands with a "wallet" field as input.
This is a series of 32 random bytes of data, usually represented as a 64 character, uppercase hexadecimal string (0-9A-F). This value is used to derive account private keys for accounts by combining it with an index and then putting that into the following hash function where || means concatenation and i is a 32-bit big-endian unsigned integer: PrivK[i] = blake2b(outLen = 32, input = seed || i)
Private keys are derived deterministically from the seed, which means that as long as you put the same seed and index into the derivation function, you will get the same resulting private key every time. Therefore, knowing just the seed allows you to be able to access all the derived private keys from index 0 to 2^{32} - 1
2^{32} - 1
(because the index value is a unsigned 32-bit integer).
Wallet implementations will commonly start from index 0 and increment it by 1 each time you create a new account so that recovering accounts is as easy as importing the seed and then repeating this account creation process.
It should be noted that Nano reference wallet is using described Blake2b private keys derivation path. However some implementations can use BIP44 deterministic wallets and mnemonic seed producing different private keys for given seed and indices. Additionally 24-word mnemonic can be derived from a Nano 64 length hex seed as entropy with clear notice for users that this is not BIP44 seed/entropy.
Python Bitcoinjs
Generates a deterministic key:
seed = b"\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x00\x01" # "0000000000000000000000000000000000000000000000000000000000000001"
index = 0x00000001.to_bytes(4, 'big') # 1
blake2b_state = hashlib.blake2b(digest_size=32)
blake2b_state.update(seed+index)
# where `+` means concatenation, not sum: https://docs.python.org/3/library/hashlib.html#hashlib.hash.update
# code line above is equal to `blake2b_state.update(seed); blake2b_state.update(index)`
PrivK = blake2b_state.digest()
print(blake2b_state.hexdigest().upper()) # "1495F2D49159CC2EAAAA97EBB42346418E1268AFF16D7FCA90E6BAD6D0965520"
Mnemonic words for Blake2b Nano seed using Bitcoinjs:
const mnemonic = bip39.entropyToMnemonic('0000000000000000000000000000000000000000000000000000000000000001')
// => abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon abandon diesel
// => '0000000000000000000000000000000000000000000000000000000000000001'
Account private key¶
This is also a 32 byte value, usually represented as a 64 character, uppercase hexadecimal string(0-9A-F). It can either be random (an ad-hoc key) or derived from a seed, as described above. This is what represents control of a specific account on the ledger. If you know or can know the private key of someone's account, you can transact as if you own that account.
Account public key¶
This is also a 32 byte value, usually represented as a 64 character, uppercase hexadecimal string (0-9A-F). It is derived from an account private key by using the ED25519 curve using Blake2b-512 as the hash function (instead of SHA-512). Usually account public keys will not be passed around in this form, rather the below address is used.
Account public address¶
This is what you think of as someone's Nano address: it's a string that starts with nano_ (previously xrb_), then has 52 characters which are the account public key but encoded with a specific base32 encoding algorithm to prevent human transcription errors by limiting ambiguity between different characters (no O and 0 for example). Then the final 8 characters are Blake2b-40 checksum of the account public key to aid in discovering typos, also encoded with the same base32 scheme (5 bytes).
So for address nano_1anrzcuwe64rwxzcco8dkhpyxpi8kd7zsjc1oeimpc3ppca4mrjtwnqposrs:
Encoded Account Public Key
nano_ 1anrzcuwe64rwxzcco8dkhpyxpi8kd7zsjc1oeimpc3ppca4mrjt wnqposrs
For basic address validation, the following regular expression can be used: ^(nano|xrb)_[13]{1}[13456789abcdefghijkmnopqrstuwxyz]{59}$. Validation of the checksum is also recommended, depending on the integration.
Prefixes: nano_ vs. xrb_
As of V19.0 the Nano node only returns nano_ addresses in all actions, but prior versions returned xrb_ addresses. These prefixes are interchangeable — everything after the _ remains the same. If you have an issue using one or the other prefix with any exchange or service, you can safely switch between nano_ and xrb_ prefixes as needed — they both represent the same account owned by your private key or seed.
Nano can be represented using more than one unit of measurement. While the most common unit is the nano
, the smallest unit is the raw
. Below is the formula for converting between raw
and nano
1 nano = 10^{30} raw
All RPC commands expect units to be represented as raw
Always keep units in integer raw
amounts to prevent any floating-point error or unit confusion.
Depending on your implementation language, you may require a big number library to perform arithmetic directly on raw
See Distribution and Units page for more details on units.
Because final balances are recorded rather than transaction amounts, API calls must be done carefully to avoid loss of funds. Incorrect arithmetic or use of fields may change an intended receive to a send to a non-existent address.
Blocks Specifications¶
Because final balances are recorded rather than transaction amounts, API calls must be done carefully to avoid sending erroneous amounts.
Block Format¶
Because each block contains the current state of the account, the "type" of the block is always "state". The following table presents the anatomy of a block, along with the format used within RPC calls for building blocks, and the serialized, binary representation:
Any transaction may also simultaneously change the representative. The above description of the "Change" action is for creating a block with an explicit representative change where no funds are transferred (balance is not changed).
In the completed, signed transaction json, the "link" field is always hexadecimal.
The first block on an account must be receiving funds (cannot be an explicit representative change). The first block is often referred to as "opening the account".
Self-Signed Blocks¶
If you choose to implement your own signing, the order of data (in bytes) to hash prior to signing is as follows.
All values are binary representations
No ASCII/UTF-8 strings.
Order of data:
block preamble (32-Bytes, value 0x6)
account (32-Bytes)
previous (32-Bytes)
representative (32-Bytes)
balance (16-Bytes)
link (32-Bytes)
The digital signing algorithm (which internally applies another Blake2b hashing) is applied on the resulting digest.
Private/public key usage
Make sure that your private key uses the correct partnering public key while signing as using an incorrect public key may leak information about your private key.
For details on how to create individual blocks for sending from, receiving to, opening or changing representatives for an account, please see the Creating Transactions section.
URI and QR Code Standards¶
Note: amount values should always be in RAW.
Note: Please use nano:// for deep links
Send to an address¶
nano:nano_<encoded address>[?][amount=<raw amount>][&][label=<label>][&][message=<message>]
Just the address
nano:nano_3wm37qz19zhei7nzscjcopbrbnnachs4p1gnwo5oroi3qonw6inwgoeuufdp
Address and an amount (as RAW)
nano:nano_3wm37qz19zhei7nzscjcopbrbnnachs4p1gnwo5oroi3qonw6inwgoeuufdp?amount=1000
Address and a label
nano:nano_3wm37qz19zhei7nzscjcopbrbnnachs4p1gnwo5oroi3qonw6inwgoeuufdp?label=Developers%20Fund%20Address
Send to an address with amount, label and message
nano:nano_3wm37qz19zhei7nzscjcopbrbnnachs4p1gnwo5oroi3qonw6inwgoeuufdp?amount=10&label=Developers%20Fund&message=Donate%20Now
Representative change¶
nanorep:nano_<encoded address>[?][label=<label>][&][message=<message>]
Change to representative with label and message
nanorep:nano_1stofnrxuz3cai7ze75o174bpm7scwj9jn3nxsn8ntzg784jf1gzn1jjdkou?label=Official%20Rep%202&message=Thank%20you%20for%20changing%20your%20representative%21
Private Key Import¶
nanokey:<encoded private key>[?][label=<label>][&][message=<message>]
Seed Import¶
nanoseed:<encoded seed>[?][label=<label>][&][message=<message>][&][lastindex=<index>]
Process a JSON blob block¶
(to be sent as the block argument to the RPC call process)
nanoblock:<blob>
Previous Get started integrating Nano |
indexByRange - Maple Help
Home : Support : Online Help : Connectivity : OpenMaple : Java Application Programming Interface : List Class : indexByRange
List.indexByRange
void indexByRange( int a, int b ) throws MapleException
The indexByRange function returns a List object corresponding to the sublist from positions a to b inclusive.
Valid values of a and b are
1..n
n
The input indices are one-based; that is, the index of the first element is 1 and the index of the last element equals the result of numElements. See subList for an equivalent which is zero-based.
List l2 = l.indexByRange( 2, 4 ); |
Component: cell_geometry
dd time V = -1.0 Cm i_Na + i_Ca + i_Ca_K + i_K + i_K1 + i_Kp + i_NaCa + i_NaK + i_ns_Ca + i_p_Ca + i_Ca_b + i_Na_b
i_Na = g_Na m 3.0 h j V - E_Na E_Na = R T F ln Nao Nai
alpha_m = 0.32 V + 47.13 1.0 -ⅇ -0.1 V + 47.13 beta_m = 0.08 ⅇ- V 11.0 dd time m = alpha_m 1.0 - m - beta_m m
alpha_h = 0.135 ⅇ 80.0 + V -6.8 if V < -40.0 0.0 otherwise beta_h = 3.56 ⅇ 0.079 V + 310000.0 ⅇ 0.35 V if V < -40.0 1.0 0.13 1.0 +ⅇ- V + 10.66 11.1 otherwisedd time h = alpha_h 1.0 - h - beta_h h
alpha_j = -127140.0 ⅇ 0.2444 V - 0.00003474 ⅇ -0.04391 V V + 37.78 1.0 +ⅇ 0.311 V + 79.23 if V < -40.0 0.0 otherwise beta_j = 0.1212 ⅇ -0.01052 V 1.0 +ⅇ -0.1378 V + 40.14 if V < -40.0 0.3 ⅇ -0.0000002535 V 1.0 +ⅇ -0.1 V + 32.0 otherwisedd time j = alpha_j 1.0 - j - beta_j j
g_K = 0.1128 Ko 5.4 E_K = R T F ln Ko + PNa_K Nao Ki + PNa_K Nai i_K = g_K Xi X 2.0 V - E_K
alpha_X = 0.0000719 V + 30.0 1.0 -ⅇ -0.148 V + 30.0 beta_X = 0.000131 V + 30.0 -1.0 +ⅇ 0.0687 V + 30.0 dd time X = alpha_X 1.0 - X - beta_X X
Xi = 1.0 1.0 +ⅇ V - 40.0 40.0
g_K1 = 0.75 Ko 5.4 E_K1 = R T F ln Ko Ki i_K1 = g_K1 K1_infinity V - E_K1
alpha_K1 = 1.02 1.0 +ⅇ 0.2385 V - E_K1 + 59.215 beta_K1 = 0.49124 ⅇ 0.08032 V - E_K1 + 5.476 +ⅇ 0.06175 V - E_K1 - 594.31 1.0 +ⅇ -0.5143 V - E_K1 + 4.753 K1_infinity = alpha_K1 alpha_K1 + beta_K1
E_Kp = E_K1 Kp = 1.0 1.0 +ⅇ 7.488 - V 5.98 i_Kp = g_Kp Kp V - E_Kp
i_NaCa = k_NaCa 1.0 K_mNa 3.0 + Nao 3.0 1.0 K_mCa + Cao 1.0 1.0 + k_sat ⅇ eta - 1.0 V F R T ⅇ eta V F R T Nai 3.0 Cao -ⅇ eta - 1.0 V F R T Nao 3.0 Cai
f_NaK = 1.0 1.0 + 0.1245 ⅇ -0.1 V F R T + 0.0365 sigma ⅇ- V F R T sigma = 1.0 7.0 ⅇ Nao 67.3 - 1.0 i_NaK = I_NaK f_NaK 1.0 1.0 + K_mNai Nai 1.5 Ko Ko + K_mKo
i_ns_Na = I_ns_Na 1.0 1.0 + K_m_ns_Ca Cai 3.0 i_ns_K = I_ns_K 1.0 1.0 + K_m_ns_Ca Cai 3.0 i_ns_Ca = i_ns_Na + i_ns_K I_ns_Na = Pns_Na V F 2.0 R T 0.75 Nai ⅇ V F R T - 0.75 Nao ⅇ V F R T - 1.0 I_ns_K = Pns_K V F 2.0 R T 0.75 Ki ⅇ V F R T - 0.75 Ko ⅇ V F R T - 1.0
i_p_Ca = I_pCa Cai K_mpCa + Cai
E_CaN = R T 2.0 F ln Cao Cai i_Ca_b = g_Cab V - E_CaN
E_NaN = E_Na i_Na_b = g_Nab V - E_NaN
i_Ca = i_Ca_max y O + O_Ca i_Ca_max = P_Ca 4.0 V F 2.0 R T 0.001 ⅇ 2.0 V F R T - 0.341 Cao ⅇ 2.0 V F R T - 1.0 i_Ca_K = p_k y O + O_Ca V F 2.0 R T Ki ⅇ V F R T - Ko ⅇ V F R T - 1.0 p_k = P_K 1.0 + i_Ca_max i_Ca_half alpha = 0.4 ⅇ V + 12.0 10.0 beta = 0.05 ⅇ- V + 12.0 13.0 alpha_a = alpha a beta_b = beta b gamma = 0.5625 Ca_SS dd time C0 = beta C1 + omega C_Ca0 - 4.0 alpha + gamma C0 dd time C1 = 4.0 alpha C0 + 2.0 beta C2 + omega b C_Ca1 - beta + 3.0 alpha + gamma a C1 dd time C2 = 3.0 alpha C1 + 3.0 beta C3 + omega b 2.0 C_Ca2 - beta 2.0 + 2.0 alpha + gamma a 2.0 C2 dd time C3 = 2.0 alpha C2 + 4.0 beta C4 + omega b 3.0 C_Ca3 - beta 3.0 + alpha + gamma a 3.0 C3 dd time C4 = alpha C3 + g O + omega b 4.0 C_Ca4 - beta 4.0 + f + gamma a 4.0 C4 dd time O = f C4 - g O dd time C_Ca0 = beta_b C_Ca1 + gamma C_Ca0 - 4.0 alpha_a + omega C_Ca0 dd time C_Ca1 = 4.0 alpha_a C_Ca0 + 2.0 beta_b C_Ca2 + gamma a C1 - beta_b + 3.0 alpha_a + omega b C_Ca1 dd time C_Ca2 = 3.0 alpha_a C_Ca1 + 3.0 beta_b C_Ca3 + gamma a 2.0 C2 - beta_b 2.0 + 2.0 alpha_a + omega b 2.0 C_Ca2 dd time C_Ca3 = 2.0 alpha_a C_Ca2 + 4.0 beta_b C_Ca4 + gamma a 3.0 C3 - beta_b 3.0 + alpha_a + omega b 3.0 C_Ca3 dd time C_Ca4 = alpha_a C_Ca3 + g_ O_Ca + gamma a 4.0 C4 - beta_b 4.0 + f_ + omega b 4.0 C_Ca4 dd time O_Ca = f_ C_Ca4 - g_ O_Ca
Component: L_type_Ca_channel_y_gate
dd time y = y_infinity - y tau_y y_infinity = 1.0 1.0 +ⅇ V + 55.0 7.5 + 0.1 1.0 +ⅇ- V + 21.0 6.0 tau_y = 20.0 + 600.0 1.0 +ⅇ V + 30.0 9.5
Component: RyR_channel_states
dd time P_C1 =- k_a_plus Ca_SS n P_C1 + k_a_minus P_O1 dd time P_O1 = k_a_plus Ca_SS n P_C1 - k_a_minus P_O1 + k_b_plus Ca_SS m P_O1 + k_c_plus P_O1 + k_b_minus P_O2 + k_c_minus P_C2 dd time P_O2 = k_b_plus Ca_SS m P_O1 - k_b_minus P_O2 dd time P_C2 = k_c_plus P_O1 - k_c_minus P_C2
Component: SERCA_pump
J_up = Vmax_f fb - Vmax_r rb 1.0 + fb + rb fb = Cai k_fb n_fb rb = Ca_NSR k_rb n_rb
Component: intracellular_Ca_fluxes
J_rel = v1 P_O1 + P_O2 Ca_JSR - Ca_SS J_tr = Ca_NSR - Ca_JSR tau_tr J_xfer = Ca_SS - Cai tau_xfer J_trpn = k_htrpn_plus Cai HTRPN_tot - HTRPNCa - k_htrpn_minus HTRPNCa + k_ltrpn_plus Cai LTRPN_tot - LTRPNCa - k_ltrpn_minus LTRPNCa
Component: intracellular_ionic_concentrations
betai = 1.0 1.0 + CMDN_tot K_mCMDN K_mCMDN + Cai 2.0 beta_SS = 1.0 1.0 + CMDN_tot K_mCMDN K_mCMDN + Ca_SS 2.0 beta_JSR = 1.0 1.0 + CSQN_tot K_mCSQN K_mCSQN + Ca_JSR 2.0 dd time Cai = betai - J_xfer + J_up + J_trpn + i_Ca_b +- 2.0 i_NaCa + i_p_Ca A_cap 2.0 V_myo F dd time Ca_SS = beta_SS J_rel V_JSR V_SS - J_xfer V_myo V_SS - i_Ca A_cap 2.0 V_SS F dd time Ca_JSR = beta_JSR J_tr - J_rel dd time Ca_NSR = J_up V_myo V_NSR - J_tr V_JSR V_NSR dd time Nai =- i_Na + i_Na_b + i_ns_Na + i_NaCa 3.0 + i_NaK 3.0 A_cap V_myo F dd time Ki =- i_Ca_K + i_K + i_K1 + i_Kp + i_ns_K +- i_NaK 2.0 A_cap V_myo F
Component: troponin
dd time HTRPNCa = k_htrpn_plus Cai HTRPN_tot - HTRPNCa - k_htrpn_minus HTRPNCa dd time LTRPNCa = k_ltrpn_plus Cai LTRPN_tot - LTRPNCa - k_ltrpn_minus LTRPNCa 0.333 + 0.667 1.0 - Force_norm
Component: tropomyosin_cross_bridges
f_01 = 3.0 f_XB f_12 = 10.0 f_XB f_23 = 7.0 f_XB g_01_SL = 1.0 g_XB_SL g_12_SL = 2.0 g_XB_SL g_23_SL = 3.0 g_XB_SL g_XB_SL = g_XB 1.0 + 1.0 - SL_norm 1.6 SL_norm = SL - 1.7 0.7 k_trop_np = k_trop_pn LTRPNCa LTRPN_tot K_trop_half N_trop N_trop = 3.5 + 2.5 SL_norm K_trop_half = 1.0 + K_trop_Ca 1.7 - 0.9 SL_norm -1.0 K_trop_Ca = k_ltrpn_minus k_ltrpn_plus dd time N0 = k_trop_pn P0 - k_trop_np N0 + g_01_SL N1 dd time P0 =- k_trop_pn + f_01 P0 + k_trop_np N0 + g_01_SL P1 dd time P1 =- k_trop_pn + f_12 + g_01_SL P1 + k_trop_np N1 + k_trop_np N1 + f_01 P0 + g_12_SL P2 dd time P2 =- f_23 + g_12_SL P2 + f_12 P1 + g_23_SL P3 dd time P3 =- g_23_SL P3 + f_23 P2
Component: force_computation
Force = zeta Force_norm Force_norm = phi_SL P1 + N1 + 2.0 P2 + 3.0 P3 Force_max Force_max = P1_max + 2.0 P2_max + 3.0 P3_max phi_SL = SL - 0.6 1.4 if SL < 2.0 ∧ SL > 1.7 1.0 if SL < 2.2 ∧ SL > 2.0 3.6 - SL 1.4 if SL < 2.3 ∧ SL > 2.2 P1_max = f_01 2.0 g_XB 3.0 g_XB g_XB 2.0 g_XB 3.0 g_XB + f_01 2.0 g_XB 3.0 g_XB + f_01 f_12 3.0 g_XB + f_01 f_12 f_23 P2_max = f_01 f_12 3.0 g_XB g_XB 2.0 g_XB 3.0 g_XB + f_01 2.0 g_XB 3.0 g_XB + f_01 f_12 3.0 g_XB + f_01 f_12 f_23 P3_max = f_01 f_12 f_23 g_XB 2.0 g_XB 3.0 g_XB + f_01 2.0 g_XB 3.0 g_XB + f_01 f_12 3.0 g_XB + f_01 f_12 f_23 |
Long-Run Equilibrium - Course Hero
Microeconomics/Perfect Competition/Long-Run Equilibrium
In the long run, in a perfectly competitive market, profits for all companies tend toward zero because supernormal profits lead to an influx of new competitors looking to claim some of that profit for themselves. Increasing supply in the market drives down prices to the point where the market is no longer attractive to new competitors, and prices will stabilize at that equilibrium point. However, companies can maximize their individual profits by calculating the quantity of output that will cause marginal cost and marginal revenue to be equal.
Over the long run (a span of time long enough for firms to freely enter or leave the market with no barriers or to change their output level), another assumption of perfect competition is that profits will tend toward zero. To understand why profits will tend toward zero under this system, it is first necessary to understand the difference between explicit costs and implicit costs and the type of profit with which each is associated.
An explicit cost is a cost involving a payment. Explicit costs are paid by a firm directly, such as costs paid to suppliers of materials, to its labor force, and to landlords, as well as taxes, tariffs, and other fees. Depreciation of assets, or a decrease in value of assets, is also counted as an explicit cost of doing business. The amount left over from total revenue once explicit cost is subtracted is known as accounting profit and is the profit reported by a firm when paying taxes or reporting to shareholders.
An implicit cost is a cost that does not require the buyer to pay cash, or that cannot easily be assigned a monetary value. Implicit costs are incurred by a firm through the loss of opportunity to generate revenue in other ways, such as using the firm's assets to do other kinds of business and management's opportunities to make money doing other kinds of work. These opportunity costs are included along with explicit costs in the calculations that give rise to economic profit, which is the type of profit considered in analyses of perfect competition.
The concepts of economic profit and opportunity cost demonstrate why, in a situation of perfect competition, profits must tend toward zero in the long run. If producers in an industry are earning greater economic profits than those in another industry, producers in the second industry are paying an opportunity cost by not producing goods in the first industry. Because perfect competition assumes there are no barriers to entry, producers in the second industry are likely to enter the market in the first industry. As producers enter the first industry, they will increase the supply in the first industry and reduce profits correspondingly. An increase in supply reduces the market price so that profits, being equal to
(\text{P}-\text{ATC}) \times \text{Q}
, will go down for each individual firm, where P is price, ATC is average total cost, and Q is quantity sold. This process will continue until there are zero economic profits, because with zero economic profits, there is no longer motivation for new producers to enter the first industry.
Meanwhile, in the second industry, supply will have dwindled as producers left for more fertile economic ground. Perfect competition assumes that demand is constant, so price (and therefore economic profits) will increase in the second industry until the point of zero economic losses is reached. With zero economic losses, producers will no longer be motivated to leave the second industry.
Exit and Entry in the Long Run
In perfect competition, all participants in a market know all the prices being charged for goods and services in that market as well as the profits being earned. With a lack of any barrier to entering or exiting markets, such as an Internet provider that can offer services easily in both Chicago and Philadelphia, firms are free to move to where they can maximize economic profit and minimize economic loss. Over time, this freedom of movement will act to reduce both economic profit and economic loss to zero. Exit and entry in the long run reflect the movement of companies into markets and the long-term equilibrium state.
For example, suppose that in a particular market, some producers are manufacturing candy bars and some are manufacturing frozen juice pops. In this fictional market, the producers of candy bars are making an economic profit and the producers of frozen juice pops are taking an economic loss. Suppose the candy bar firms are producing an output of 10,000 candy bars at a market price of $1 each and an average cost of $0.75 each, so that total profit equals $2,500
(\lbrack\$1\;\text{Price}-\$0.75\;\text{Average Cost}\rbrack \times 10\text{,}000\;\text{Units of Output})
. The juice pop makers are producing an output of 10,000 juice pops, with an average cost of $0.50 each, but their market price is only $0.45. Therefore, the juice pop makers are taking an overall economic loss of -$500
(\lbrack\$0.45\;\text{Price}-\$0.50\;\text{Average Cost}\rbrack \times 10\text{,}000\;\text{Units of Output})
Observing this situation, some clever juice pop makers decide to leave their industry and get into the candy bar business because candy bar firms seem to be doing pretty well by comparison. Assume firms that have been producing a total of 2,000 juice pops leave juice pop production for candy bar production, and with no economic barriers or additional costs, they are able to manufacture the same number of candy bars as they did juice pops.
These firms are now producing 2,000 candy bars, so there is now a total of 12,000 candy bars being produced and sold. This greater supply causes the price of candy bars to fall to $0.75. The average cost of candy bars is still $0.75, so zero economic profit is being made.
When the market price is $1.00 as determined by the market equilibrium (the place where supply S1 crosses demand D in the market), firms make a profit because this price (which is also marginal revenue, MR), is above their average total cost (ATC). Other firms see that profit is being made, which causes them to enter the market. This shifts supply right (S2), which moves the equilibrium point down the demand curve until it reaches $0.75, the new price. At this point, firms make normal profit, or zero profit, and no new firms will enter the market.
Meanwhile, over in the juice pop business, now only 8,000 juice pops are being made. This reduction in supply moves the market price up to $0.46 per juice pop. Losses are still being incurred in the juice pop business, but now at only $0.04 per juice pop instead of $0.05, a reduction in losses of a full 20%. The theory and conditions of perfect competition hold that this migration of producers from less profitable situations to more profitable ones will continue until profits on the one hand and losses on the other are both reduced to zero. The candy bar market further highlights the process of exit in the short-run. Some time in the future, the market price becomes $1.00, but the cost structure has changed, and average total cost is now $1.25 due to rising input costs. In this case, firms in the candy bar market are incurring short-term economic losses. Firms will begin to exit the market to reduce losses or search for profits elsewhere. Firms will exit, which reduces supply of candy bars in the market, and thus increases the market price. These exits will occur until the market reaches long-run equilibrium, where price equals average total cost and firms are making zero economic profit. In this new example, the equilibrium market price is $1.25 and average total cost is $1.25.
When market equilibrium (at the place where supply S1 crosses demand D in the market) sets the price at $1.00, firms suffer a loss because this price (which is also the firm's marginal revenue, MR), is below their average total cost (ATC). Some firms will leave the market to avoid taking losses. This shifts supply left (S2), which moves the equilibrium point up the demand curve until it reaches $1.25, the new price. At this point, firms make normal profit, or zero profit, and no more firms will leave the market.
<Calculation of Profit or Loss in the Short Run>Efficiency of Perfect Competition |
A technician has 10 resistors each of resistance
0.1 \Omega
. Find the largest and smallest resistance that he can obtain by combining these resistors.
1 \Omega \text{ and }0.1 \Omega
0.1 \Omega \text{ and }0.01 \Omega
1 \Omega \text{ and } 0.01 \Omega
10\Omega \text{ and }1 \Omega
V_A
in volts.
by JohnDonnie Celestre
Bill has two identical lamps each of resistance 5 ohms and two 9-volt-batteries. If he wants to light the two lamps for a duration as long as possible, how should he connect the lamps and batteries?
C D A B All four last equally long.
by Aditya Virani
When a charged capacitor is disconnected from its source it will eventually discharge. This is because a small amount of charge leaks through the dielectric between the plates of the capacitor. Suppose we fill a parallel plate capacitor with a ceramic of relative permittivity
\epsilon=2.1
and resistivity
\rho=1.4 \times 10^{13}~\Omega \cdot \mbox{m}
. The capacitor is charged by connecting it to a voltage source. How long will it take in seconds for the capacitor to lose half of the charge acquired after disconnecting it from the source?
A 120V source is connected across the plates of a parallel-plate capacitor. The capacitor is being submerged vertically at constant speed into a container filled with water. What current in Amps flows through the voltage source during this process if it takes
\tau=20 s
to totally submerge the capacitor? The capacitance (in air) of the capacitor is
C_{0}=12\mu F
and water's relative permittivity is
\epsilon=81.
Hint: If the capacitor is partially submerged at an instant of time try figuring out how to treat it as two separate capacitors at that instant. |
Lewin's equation - Wikipedia
Lewin's equation, B = f(P, E), is a heuristic formula proposed by psychologist Kurt Lewin as an explanation of what determines behavior.
2 Gestalt influence
3 Interaction of person and environment
3.1 Relative importance of P and E
3.2 Specific function linking P and E
4 Psychological reality
The formula states that behavior is a function of the person and his or her environment:[1]
{\displaystyle B=f(P,E)}
{\displaystyle B}
is behavior,
{\displaystyle P}
is person, and
{\displaystyle E}
This equation was first presented in Lewin's book, Principles of Topological Psychology, published in 1936.[2] The equation was proposed as an attempt to unify the different branches of psychology (e.g. child psychology, animal psychology, psychopathology) with a flexible theory applicable to all distinct branches of psychology.[3] This equation is directly related to Lewin's field theory. Field theory is centered around the idea that a person's life space determines their behavior.[2] Thus, the equation was also expressed as B = f(L), where L is the life space.[4] In Lewin's book, he first presents the equation as B = f(S), where behavior is a function of the whole situation (S).[5] He then extended this original equation by suggesting that the whole situation could be roughly split into two parts: the person (P) and the environment (E).[6] According to Lewin, social behavior, in particular, was the most psychologically interesting and relevant behavior.[7]
Lewin held that the variables in the equation (e.g. P and E) could be replaced with the specific, unique situational and personal characteristics of the individual. As a result, he also believed that his formula, while seemingly abstract and theoretical, had distinct concrete applications for psychology.[5]
Gestalt influence[edit]
Many scholars (and even Lewin himself[8]) have acknowledged the influence of Gestalt psychology on Lewin's work.[7] Lewin's field theory holds that a number of different and competing forces combine to result in the totality of the situation. A single person's behavior may be different in unique situations, as he or she is acting partly in response to these differential forces and factors (e.g. the environment, or E):
"A physically identical environment can be psychologically different even for the same man in different conditions."[9]
Similarly, two different individuals placed in exactly the same situation will not necessarily engage in the same behavior.
"Even when from the standpoint of the physicist the environment is identical or nearly identical for a child and or an adult, the psychological situation can be fundamentally different."[10]
For this reason, Lewin holds that the person (e.g. P) must be considered in conjunction with the environment. P consists of the entirety of a person (e.g. his or her past,[11] present, future,[12] personality,[10] motivations, desires). All elements within P are contained within the life space, and all elements within P interact with each other.
Lewin emphasizes that the desires and motivations within the person and the situation in its entirety, the sum of all these competing forces, combine to form something larger: the life space. This notion speaks directly to the gestalt idea that the "whole is greater than the sum of its parts."[7] The idea that the parts (e.g. P and E) of the whole (e.g. S) combine to form an interactive system has been called Lewin's 'dynamic approach,' a term that specifically refers to regarding "the elements of any situation...as parts of a system."[13]
Interaction of person and environment[edit]
Relative importance of P and E[edit]
Lewin explicitly stated that either the person or the environment may be more important in particular situations:
"Every psychological event depends upon the state of the person and at the same time on the environment, although their relative importance is different in different cases."[6]
Thus, Lewin believed he succeeded in creating an applicable theory that was also "flexible enough to do justice to the enormous differences between the various events and organisms."[14] In a sense, he held that it was inappropriate to pick a side on the classic psychological debate of nature versus nurture, as he held that "every scientific psychology must take into account whole situations, i.e., the state of both person and environment."[6] Further, Lewin stated that:
"The question whether heredity or environment plays the greater part also belongs to this kind of thinking. The transition of the Galilean thinking involved a recognition of the general validity of the thesis: An event is always the result of the interaction of several facts."[11]
Specific function linking P and E[edit]
Lewin defined an empirical law as "the functional relationship between various facts,"[15] where facts are the "different characteristics of an event or situation."[5] In Lewin's original proposal of his equation, he did not specify how exactly the person and the environment interact to produce behavior. Some scholars have noted that Lewin's use of the comma in his equation between the P and E represents Lewin's flexibility and receptiveness to multiple ways that these two may interact.[7] Lewin indeed held that the importance of the person or of the environment may vary on a case-by-case basis. The use of the comma may provide the flexibility to support this assertion.[7]
Psychological reality[edit]
Lewin differentiates between multiple realities. For example, the psychological reality encompasses everything that an individual perceives and believes to be true. Only what is contained within the psychological reality can affect behavior. In contrast, things that may be outside the psychological reality, such as bits of the physical reality or social reality, has no direct relation to behavior. Lewin states:
"The psychological reality...does not depend upon whether or not the content...exists in a physical or social sense....The existence or nonexistence...of a psychological fact are independent of the existence or nonexistence to which its content refers."[16]
As a result, the only reality that is contained within the life space is the psychological reality, as this is the reality that has direct consequences for behavior. For example, in Principles of Topological Psychology, Lewin continually reiterates the sentiment that "the physical reality of the object concerned is not decisive for the degree of psychological reality."[17] Lewin refers to the example of a "child living in a 'magic world.'"[17] Lewin asserts that, for this child, the realities of the 'magic world' are a psychological reality, and thus must be considered as an influence on their subsequent behavior, even though this 'magic world' does not exist within the physical reality. Likewise, scholars familiar with Lewin's work have emphasized that the psychological situation, as defined by Lewin, is strictly composed of those facts which the individual perceives or believes.[18]
Principle of contemporaneity[edit]
In Lewin's theoretical framework, the whole situation—or the life space, which contains both the person and the environment—is dynamic. In order to accurately determine behavior, Lewin's equation holds that one must consider and examine the life space at the exact moment when the behavior occurred. The life space, even moments after such behavior has occurred, is no longer exactly the same as it was when behavior occurred and thus may not accurately represent the whole situation that led to the behavior in the first place.[19] This focus on the present situation represented a departure from many other theories at the time. Most theories tended to focus on looking at an individual's past in order to explain their present behavior, such as Sigmund Freud's psychoanalysis.[2] Lewin's emphasis on the present state of the life space did not preclude the idea that an individual's past may impact the present state of the life space:
"[The] influence of the previous history is to be thought of as indirect in dynamic psychology: From the point of view of systematic causation, past events cannot influence present events. Past events can only have a position in the historical causal chains whose interweavings create the present situation."[20]
Lewin referred to this concept as the principle of contemporaneity.
^ The Sage Handbook of Methods in Social Psychology: Lewin's equation
^ a b c Christian Balkenius (1995). Natural Intelligence in Artificial Creatures. Lund University Cognitive Studies 37. Archived 2008-10-05 at the Wayback Machine (ISBN 91-628-1599-7): Chapter 4 – Reactive Behavior
^ Lewin, Kurt (1936). Principles of Topological Psychology. New York: McGraw-Hill. pp. 4–7.
^ Lewin, Kurt (1936). Principles of Topological Psychology. New York: McGraw-Hill. pp. 216.
^ a b c Lewin, Kurt (1936). Principles of Topological Psychology. New York: McGraw-Hill. pp. 11.
^ a b c d e Kihlstrom, John. "The Person-Situation Interaction". Retrieved November 5, 2015.
^ Lewin, Kurt (1936). Principles of Topological Psychology. New York: McGraw-Hill. pp. 24–25.
^ a b Lewin, Kurt (1936). Principles of Topological Psychology. New York: McGraw-Hill. pp. 24.
^ Gold, Martin (1992). "Metatheory and Field Theory in Social Psychology: Relevance or elegance?". Journal of Social Issues. 48 (2): 70. doi:10.1111/j.1540-4560.1992.tb00884.x.
^ Lewin, Kurt (1936). Principles of Topological Psychology. New York: McGraw-Hill. pp. 5.
^ a b Lewin, Kurt (1936). Principles of Topological Psychology. New York: McGraw-Hill. pp. 197.
^ Boring, Edwin (1950). A History of Experimental Psychology. New York: Appleton-Century-Crofts. p. 715.
Helbing, D. (2010). Quantitative Sociodynamics: Stochastic Methods and Models of Social Interaction Processes (2nd ed.). Springer.
Lewin, K. (1943). Defining the "Field at a Given Time." Psychological Review, 50, 292–310.
Lewin, K (1936). Principles of Topological Psychology. New York: McGraw-Hill.
Lewin, Sticky Minds
Retrieved from "https://en.wikipedia.org/w/index.php?title=Lewin%27s_equation&oldid=1029131027" |
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\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{7}
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p\left( 〈1,2;3,4〉 \right);
\textcolor[rgb]{0,0,1}{1}
\mathrm{Frac}\left(1.5\right);
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\mathrm{Frac}\left(\frac{3}{2}\right);
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p\left("a string"\right);
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p\left(\mathrm{expect}+\mathrm{an}+\mathrm{Error}\right); |
Compact multiclass model for support vector machines (SVMs) and other classifiers - MATLAB - MathWorks Australia
\begin{array}{cccc}& \text{Learner 1}& \text{Learner 2}& \text{Learner 3}\\ \text{Class 1}& 1& 1& 0\\ \text{Class 2}& -1& 0& 1\\ \text{Class 3}& 0& -1& -1\end{array}
\stackrel{^}{k}
\stackrel{^}{k}=\underset{k}{\text{argmin}}\frac{\sum _{l=1}^{B}|{m}_{kl}|g\left({m}_{kl},{s}_{l}\right)}{\sum _{l=1}^{B}|{m}_{kl}|}.
{L}_{d}\approx ⌈10{\mathrm{log}}_{2}K⌉
{L}_{s}\approx ⌈15{\mathrm{log}}_{2}K⌉
\Delta \left({k}_{1},{k}_{2}\right)=0.5\sum _{l=1}^{L}|{m}_{{k}_{1}l}||{m}_{{k}_{2}l}||{m}_{{k}_{1}l}-{m}_{{k}_{2}l}|, |
Solve the polynomial inequality x^{2}-2x+1>0 and graph the solution set on a
{x}^{2}-2x+1>0
The given polynomial inequality is
{x}^{2}-2x+1>0
To solve the inequality, find its critical points by equating it to zero.
{x}^{2}-2x+1=0
{x}^{2}-x-x+1=0
x\left(x-1\right)-1\left(x-1\right)=0
\left(x-1\right)\left(x-1\right)=0
x=1,\text{ }1
Now, make the sign analysis on Number line to determine about the inequality.
x<1
, then for the given polynomial the result is always positive and even if
x>1
, then also for the given polynomial the result is positive. Therefore, for all values of x except 1 ,the polynomial is positive.
Answer: Hence, the interval notation for the given polynomial inequality is
\left(-\mathrm{\infty },\text{ }1\right)\cup \left(1,\text{ }\mathrm{\infty }\right)
P\left(x\right)=-12{x}^{2}+2136x-41000
x=r\mathrm{cos}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y=r\mathrm{sin}
\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{{x}^{2}-{y}^{2}}{\sqrt{{x}^{2}+{y}^{2}}}
{C}_{1},{C}_{2},{C}_{3},{C}_{4}
Xi\left(x,y,{C}_{1},{C}_{2},{C}_{3},{C}_{4}\right)
Xi
x\in \left\{-\mathrm{\infty },\mathrm{\infty }\right\}
y\in \left\{-\mathrm{\infty },\mathrm{\infty }\right\}
max\left\{Xi\left(x,y,{C}_{1},{C}_{2},{C}_{3},{C}_{4}\right)\right\}
Write formulas for the isometries in terms of a complex variable
z=x+iy.
Evaluate the integral by making an appropriate change of variables.
Find the volume of the solid generated by rotating about the x-axis the region bounded by
y={4}^{x},\text{ }x=-3,\text{ }x=3
A real estate office handles a 60-unit apartment complex. When the rent is $530 per month, all units are occupied. For each $40 increase in rent, however, an average of one unit becomes vacant. Each occupied unit requires an average of $65 per month for service and repairs. What rent should be charged to obtain a maximum profit? |
q
q
p
such that FFT-based polynomial arithmetic can be used for this actual computation. The higher the degrees of f and rc are, the larger must be
{2}^{e}
p-1
\mathrm{with}\left(\mathrm{RegularChains}\right):
\mathrm{with}\left(\mathrm{FastArithmeticTools}\right):
\mathrm{with}\left(\mathrm{ChainTools}\right):
p≔962592769:
\mathrm{vars}≔[y,x]:
R≔\mathrm{PolynomialRing}\left(\mathrm{vars},p\right):
\mathrm{f1}≔x\left({y}^{2}+y+1\right)+2:
\mathrm{f2}≔\left(x+1\right)\left({y}^{2}+y+1\right)+{x}^{3}+x+1:
\mathrm{SCube}≔\mathrm{SubresultantChainSpecializationCube}\left(\mathrm{f1},\mathrm{f2},y,R,1\right)
\textcolor[rgb]{0,0,1}{\mathrm{SCube}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{subresultant_chain_specialization_cube}}
\mathrm{r2}≔\mathrm{ResultantBySpecializationCube}\left(\mathrm{f1},\mathrm{f2},x,\mathrm{SCube},R\right)
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If two quantities x and y vary inversely with each other, then which one of the following is true? from Class 12 TET Previous Year Board Papers | Mathematics 2018 Solved Board Papers
To fill a rectangular tank of area 700m2, 140 m3 of water is required. What will be the height of the water level in the tank?
Given the area of tank=700m3, Volume of tank = 140m2
Also Volume of tank = area of tank x height
\Rightarrow
140
=Height
\times 700
\Rightarrow
\frac{140}{700}
\frac{1}{5}
0.2
20
Which one of the following is most essential in learning mathematics at upper primary level?
Exploring different ways of solving a problem.
Memorising all formulas.
Copying correctly what teacher writes on the board.
Solving a problem many times.
If two quantities x and y vary inversely with each other, then which one of the following is true?
Product of their corresponding values remains constant
Summation of their corresponding values remains constant
Difference of their corresponding values remains constant
Ratio of their corresponding values remains constant
x\propto \frac{1}{y}
\Rightarrow
x=\frac{k}{y}
\because
k=constant)
\Rightarrow
k=xy
y\propto \frac{1}{x}
\Rightarrow
y=\frac{m}{x}
\Rightarrow
m=xy
From Eqs (1) and (2), we get k=m
Hence, the product of their corresponding values remains constant.
Which one of the following is the most suitable strategy to teach the skill of addition of money?
Doing lots of problems
Roleplay gives an opportunity to the learners to explore mathematics in the real-world context.
The strategy of questioning used in the mathematics class at upper primary level
makes the classroom noisy as the children would be taking too much
could create stress among children and may lead them to accept the teacher's authority
helps children to express their thoughts or understanding and think critically
should be discouraged as it demoralises the child who is unable to answer
After teaching the concept of multiplication to her class, a teacher asked her children to multiply 48 by 4. One of her students solved it orally as "To multiply 48 by 4, we first add 48 to 48, which makes 96 and then add another 96 to reach 192. So, the answer is 192". What can you say about his/her strategy of multiplication?
He/She has not understood the concept of multiplication.
The given problem is a multiplication problem and not addition problem.
He/She understood multiplication as repeated addition.
The child used a wrong method to multiply.He/She has to use the place value algorithm to multiply the numbers
Which one of the following methods is most suitable for teaching mathematics at the upper primary level?
Which one of the following is not the purpose of assessment?
A. Monitoring student's growth
B. Making instructional decision
C. Evaluating the effectiveness of curriculum
D. Ranking the children based on performance
Ranking the children based on performance is not the purpose of assessment. An assessment has an important place in the education system. The assessment shows us the capability of a student to achieve his aim.
Which one of the following should be taken up as initial activity in introducing the concept of 'time' to young learners?
Teaching children how to read time in clock
Teaching children how to calculate elapsed time
Conversion of time in different units
Discussing about the prior experiences with phrases related to time
Then, the value of y is
If y=4, then
https://www.zigya.com/share/VE1BRU5UVDEyMTkwNTY2 |
Vasicek Interest Rate Model Definition
The term Vasicek Interest Rate Model refers to a mathematical method of modeling the movement and evolution of interest rates. It is a single-factor short-rate model that is based on market risk. The Vasicek interest model is commonly used in economics to determine where interest rates will move in the future. Put simply, it estimates where interest rates will move in a given period of time and can be used to help analysts and investors figure out how the economy and investments will fare in the future.
The Vasicek Interest Rate Model is a single-factor short-rate model that predicts where interest rates will end up at the end of a given period of time.
It outlines an interest rate's evolution as a factor composed of market risk, time, and equilibrium value.
The model is often used in the valuation of interest rate futures and in solving for the price of various hard-to-value bonds.
The Vasicek Model values the instantaneous interest rate using a specific formula.
This model also accounts for negative interest rates.
Predicting how interest rates evolve can be difficult. Investors and analysts have many tools available to help them figure out how they'll change over time in order to make well-informed decisions about how their investments and the economy. The Vasicek Interest Rate Model is among the models that can be used to help estimate where interest rates will go.
As noted above, the Vasicek Interest Rate model, which is commonly referred to as the Vasicek model, is a mathematical model used in financial economics to estimate potential pathways for future interest rate changes. As such, it's considered a stochastic model, which is a form of modeling that helps make investment decisions.
It outlines the movement of an interest rate as a factor composed of market risk, time, and equilibrium value. The rate tends to revert toward the mean of these factors over time. The model shows where interest rates will end up at the end of a given period of time by considering current market volatility, the long-run mean interest rate value, and a given market risk factor.
\begin{aligned} &dr_t = a ( b - r^t ) dt + \sigma dW_t \\ &\textbf{where:} \\ &W = \text{Random market risk (represented by}\\ &\text{a Wiener process)} \\ &t = \text{Time period} \\ &a(b-r^t) = \text{Expected change in the interest rate} \\ &\text{at time } t \text{ (the drift factor)} \\ &a = \text{Speed of the reversion to the mean} \\ &b = \text{Long-term level of the mean} \\ &\sigma = \text{Volatility at time } t \\ \end{aligned}
drt=a(b−rt)dt+σdWtwhere:W=Random market risk (represented bya Wiener process)t=Time perioda(b−rt)=Expected change in the interest rateat time t (the drift factor)a=Speed of the reversion to the meanb=Long-term level of the meanσ=Volatility at time t
The model specifies that the instantaneous interest rate follows the stochastic differential equation, where d refers to the derivative of the variable following it. In the absence of market shocks (i.e., when dWt = 0) the interest rate remains constant (rt = b). When rt < b, the drift factor becomes positive, which indicates that the interest rate will increase toward equilibrium.
The Vasicek model is often used in the valuation of interest rate futures and may also be used in solving for the price of various hard-to-value bonds.
As mentioned earlier, the Vasicek model is a one- or single-factor short rate model. A single-factor model is one that only recognizes one factor that affects market returns by accounting for interest rates. In this case, market risk is what affects interest rate changes.
This model also accounts for negative interest rates. Rates that dip below zero can help central bank authorities during times of economic uncertainty. Although negative rates aren't commonplace, they have been proven to help central banks manage their economies. For instance, Denmark's central banks lowered interest rates below zero in 2012. European banks followed two years later followed by the Bank of Japan (BOJ), which pushed its interest rate into negative territory in 2016.
Vasicek Interest Rate Model vs. Other Models
The Vasicek Interest Rate Model isn't the only one-factor model that exists. The following are some of the other common models:
Merton's Model: This model helps determine the level of a company's credit risk. Analysts and investors can use the Merton Model to find out how positioned the company is to fulfill its financial obligations.
Cox-Ingersoll-Ross Model: This one-factor model also looks at how interest rates are expected to move in the future. The Cox-Ingersoll-Ross Model does so through current volatility, the mean rate, and spreads.
Hull-While Model: The Hull-While Model assumes that volatility will be low when short-term interest rates are near the zero-mark. This is used to price interest rate derivatives.
World Economic Forum. "Negative interest rates: absolutely everything you need to know." Accessed Dec. 28, 2021.
CFI. "Short Rate Model." Accessed Dec. 28, 2021. |
Glutamine—fructose-6-phosphate transaminase (isomerizing) - Wikipedia
Glutamine—fructose-6-phosphate transaminase 1, homodimer, Human
glucosamine-6-phosphate isomerase (glutamine-forming)
GlcN6P synthase
In enzymology, a glutamine-fructose-6-phosphate transaminase (isomerizing) (EC 2.6.1.16) is an enzyme that catalyzes the chemical reaction
L-glutamine + D-fructose 6-phosphate
{\displaystyle \rightleftharpoons }
L-glutamate + D-glucosamine 6-phosphate
Thus, the two substrates of this enzyme are L-glutamine and D-fructose 6-phosphate, whereas its two products are L-glutamate and D-glucosamine 6-phosphate.
This enzyme belongs to the family of transferases, specifically the transaminases, which transfer nitrogenous groups. The systematic name of this enzyme class is L-glutamine:D-fructose-6-phosphate isomerase (deaminating). This enzyme participates in glutamate metabolism and aminosugars metabolism.
As of late 2007, 12 structures have been solved for this class of enzymes, with PDB accession codes 1JXA, 1MOQ, 1MOR, 1MOS, 1XFF, 1XFG, 2BPL, 2J6H, 2POC, 2PUT, 2PUV, and 2PUW.
Ghosh S, Blumenthal HJ, Davidson E, Roseman S (1960). "Glucosamine metabolism. V. Enzymatic synthesis of glucosamine 6-phosphate". J. Biol. Chem. 235: 1265–73. PMID 13827775.
Gryder RM, Pogell BM (1960). "Further studies on glucosamine 6-phosphate synthesis by rat liver enzymes". J. Biol. Chem. 235: 558–62. PMID 13829889.
Leloir LF, Cardini CE (1953). "The biosynthesis of glucosamine". Biochim. Biophys. Acta. 12 (1–2): 15–22. doi:10.1016/0006-3002(53)90119-X. hdl:11336/140740. PMID 13115409.
Teplyakov A, Obmolova G, Badet-Denisot MA, Badet B (1999). "The mechanism of sugar phosphate isomerization by glucosamine 6-phosphate synthase". Protein Sci. 8 (3): 596–602. doi:10.1110/ps.8.3.596. PMC 2144271. PMID 10091662.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Glutamine—fructose-6-phosphate_transaminase_(isomerizing)&oldid=1046713841" |
The current in a 50 mH inductor is known to be i=120\
The current in a 50 mH inductor is known to be
i=120\text{ }mA,\text{ }t\le 0;\text{ }i={A}_{1}{e}^{-500t}+{A}_{2}{e}^{-2000t}A,t\ge 0
The voltage across the inductor (passive sign convention) is 3 V at t =0. a) Find the expression for the voltage across the inductor for t > 0. b) Find the time, greater than zero, when the power at the terminals of the inductor is zero.
p=vi
so we search for when the voltage of the current are zero:
i\left(t\right)=0.2{e}^{-500t}-0.08{e}^{-2000t}
i\left(t\right)=0⇒
0.2{e}^{-500t}=\frac{0}{08}{e}^{-2000t}
\frac{0.2}{0.08}=\frac{{e}^{-2000t}}{{e}^{-500t}}
\mathrm{ln}\left(2.5\right)=\mathrm{ln}\left({e}^{-1500t}
t=\frac{-\mathrm{ln}\left(2.5\right)}{1500}
t=-6.1\cdot {10}^{-4}\text{ }s
PSKWhich is not greater than zero. Lets
a) We know that
i\left(0\right)=120
mA so we can write
i\left(0\right)={A}_{1}{e}^{0}+{A}_{2}{e}^{0}={A}_{1}+{A}_{2}=0.12\text{ }A
We also kno that
v\left(0\right)=3\text{ }V
so we write
\frac{di}{dt}=-500{A}_{1}{e}^{-500t}-2000{A}_{2}{e}^{-2000t}\phantom{\rule{0ex}{0ex}}v=L\frac{di}{dt}=-25{A}_{1}{e}^{-500t}-100{A}_{2}{e}^{-2000t}\phantom{\rule{0ex}{0ex}}v\left(0\right)=-25{A}_{1}{e}^{0}-100{A}_{2}{e}^{0}=-25{A}_{1}-100{A}_{2}=3\text{ }V\phantom{\rule{0ex}{0ex}}v\left(0\right)=25{A}_{1}{e}^{0}-100{A}_{2}{e}^{0}=-25{A}_{1}-100{A}_{2}=3\text{ }V
Now we have two equations with two unknowns:
{A}_{1}+{A}_{2}=0.12\phantom{\rule{0ex}{0ex}}-25{A}_{1}-100{A}_{2}=3
We solve this and get
{A}_{1}=0.2\text{ }and\text{ }{A}_{2}=-0.08
Now we can write the expression for the voltage
v\left(t\right)=-25\left(0.2\right){e}^{-500t}-100\left(-0.08\right){e}^{-2000t}\phantom{\rule{0ex}{0ex}}v\left(t\right)=-5{e}^{-500t}+8{e}^{-2000t}\text{ }V
You help me, thanks!
4.50\cdot {10}^{4}
\theta
{13.0}^{\circ }
\theta
At constant pressure, which of these systems do work on the surroundings? Check all that apply.
2A\left(g\right)+3B\left(g\right)⇒4C\left(g\right)
2A\left(g\right)+3B\left(g\right)⇒4C\left(g\right)
A\left(g\right)+B\left(g\right)⇒3C\left(g\right)
A\left(s\right)+2B\left(g\right)⇒C\left(g\right)
{a}_{2}
, the magnitude of the centripetal acceleration of the star with mass
{m}_{2}
{a}_{2}=?
The fastest measured pitched baseball left the pitcher’s hand at a speed of 45.0 m/s. If the pitcher was in contact with the ball over a distance of 1.50 m and produced constant acceleration, (a) what acceleration did he give the ball, and (b) how much time did it take him to pitch it? |
Radius, Diameter, & Circumference of a Circle (Video + Examples)
Radius, Diameter, & Circumference of a Circle
Circumference Parts
In mathematics, a circle is the set of all coplanar points equidistant from a given point. That given point is the circle's center, and it does not lie on the circle. The circle is only the set of points (forming a curved line returning back on itself) that are in the same plane and the same distance from a center point, the center of a circle. The circle itself is not the inside space, the center point, or space outside of the circle. The circle is the line that circles back onto itself.
Without its identifying center point, the circle is nameless. The center of the circle is how the circle is named, so take the circle and place
Point I
in the middle, and now we have
Circle I
To get from the center point to the actual circle, we move in a straight line called a radius. Radius is the measure of that distance. The radius of a circle is one way to measure the size of the circle. Radius is always indicated by the small letter
. Here is a line segment,
IE
, with endpoint
I
at the circle's center and endpoint
E
on the circle itself:
Radius is
\frac{1}{2}
the diameter. Here is the radius formula:
r = \frac{1}{2}d
If we have two radii together, they can form central angles or a straight line across the circle. A straight line starting on the circle, passing through the center, and reaching to the circle again is a diameter. Here is a diameter for
Circle 2
, built by extending radius
IE
in the other direction, to
Point P
Instead of identifying that as two separate radii (the plural of radius), we can simply call
PE
a diameter of a circle. It is the distance across the entire circle. The diameter of the circle is always indicated by the lowercase letter
d
This means the diameter is
2
times the radius of the circle. Here is the diameter formula:
d = 2r
Circles show up everywhere, like pizza at dinner!
For polygons, the perimeter is the sum of the lengths of their sides. Circles have a perimeter, too, but we give it a special word: circumference (from Latin, to carry around). The circumference of a circle is the distance around the circle.
Let's take a look at the ratio of the circumference to diameter in these circles below.
Here is
Circle 1
with a diameter of
1 meter
Circle 2
2 meters
The distance all the way around
Circle 1
, the circumference of the circle, is 3.1415926 meters.
The circumference of
Circle 2
6.2831852 meters
Set up each circle's circumference to its diameter. These form ratios. See anything?
\frac{Circumference}{Diameter} : \frac{3.1415926}{1} = 3.1415926
\frac{6.2831852}{2} = 3.1415926
The ratio of the circumference
C
to diameter
d
of both circles simplify to the same value,
3.1415926
\frac{C}{d} = \textcolor[rgb]{0.8,0,0}{3.1415926}
The ratio of the circumference,
C
, of any circle to its diameter
d
is always the same value,
\textcolor[rgb]{0.8,0,0}{3.1415926}
, named using the Greek letter pi (as in apple pie), which looks like this:
\textcolor[rgb]{0.8,0,0}{\pi }
\frac{C}{d} = \textcolor[rgb]{0.8,0,0}{\pi }
When using pi, it is acceptable to round to two decimal places.
\pi
instead of that long number and show a formula of the circumference
C
d
. When we multiply both sides of the formula by
d
C = \pi d
Now we can find the circumference
C
of any circle as long as we know the diameter
d
. If you have the radius, you can still find the circumference of a circle, since the radius is equal to half the diameter:
C = 2\pi r
Let's try a practice problem and find the circumference of a circle that has a diameter of
20
Let's start with our formula, then plugin
20
for our diameter
d
C = \pi d
C = \pi · 20
\mathbf{C = }\mathbf{20}\mathbf{\pi }
You did it! We can leave our answer in terms of
\pi
, so the circumference of the circle is
20\pi
There are more parts to a circle left to cover. Imagine you sit down to a delicious, hot pizza, but it is not cut! You cut away a single slice, like this:
The portion of the crust in the cut piece is much smaller than the rest of the crust. That smaller portion is the minor arc of the circle. The larger part, the remaining circumference, is the major arc.
A minor arc is a portion of the circumference where the central angle measures less than
180°
. A major arc is a portion of the circumference where the central angle is greater than
180° |
Express each of the following number as a product of its prime factors. Use exponents to represent repeated multiplication, when applicable. An example is given below.
40 = 2 \cdot 20 = 2 \cdot 2 \cdot 10 = 2 \cdot 2 \cdot 2 \cdot 5 = 2 ^ { 3 } \cdot 5
30
Use the example above as a guide if you are having trouble. You can also refer to problem 3-64 for help with prime factorization.
2 · 3 · 5
300
This problem is very similar to part (a). It may also be helpful to think about how
300
relates to
30
. What prime factors could you multiply by
30
300
2^{2} · 3 · 5^{2}
17
17
down to prime factors? It is possible for a number to be the only prime factor of itself.
21
Follow the steps you did for parts (a) and (b). |
Riemann-Liouville and Higher Dimensional Hardy Operators for NonNegative Decreasing Function in Spaces
Muhammad Sarwar, Ghulam Murtaza, Irshaad Ahmed, "Riemann-Liouville and Higher Dimensional Hardy Operators for NonNegative Decreasing Function in Spaces", Abstract and Applied Analysis, vol. 2014, Article ID 621857, 5 pages, 2014. https://doi.org/10.1155/2014/621857
Muhammad Sarwar,1 Ghulam Murtaza,2 and Irshaad Ahmed 2
1Department of Mathematics, University of Malakand, Chakdara, Lower Dir 18800, Pakistan
2Department of Mathematics, GC University, Faisalabad, Faisalabad 38000, Pakistan
One-weight inequalities with general weights for Riemann-Liouville transform and -dimensional fractional integral operator in variable exponent Lebesgue spaces defined on are investigated. In particular, we derive necessary and sufficient conditions governing one-weight inequalities for these operators on the cone of nonnegative decreasing functions in spaces.
We derive necessary and sufficient conditions governing the one-weight inequality for the Riemann-Liouville operator and -dimensional fractional integral operator on the cone of nonnegative decreasing function in spaces.
In the last two decades a considerable interest of researchers was attracted to the investigation of the mapping properties of integral operators in so-called Nakano spaces (see, e.g., the monographs [1, 2] and references therein). Mathematical problems related to these spaces arise in applications to mechanics of the continuum medium. For example, Ružicka [3] studied the problems in the so-called rheological and electrorheological fluids, which lead to spaces with variable exponent.
Weighted estimates for the Hardy transform in spaces were derived in the papers [4] for power-type weights and in [5–9] for general weights. The Hardy inequality for nonnegative decreasing functions was studied in [10, 11]. Furthermore Hardy type inequality was studied in [12, 13] by Rafeiro and Samko in Lebesgue spaces with variable exponent.
Weighted problems for the Riemann-Liouville transform in spaces were explored in the papers [5, 14–16] (see also the monograph [17]).
Historically, one and two weight Hardy inequalities on the cone of nonnegative decreasing functions defined on in the classical Lebesgue spaces were characterized by Arino and Muckenhoupt [18] and Sawyer [19], respectively.
It should be emphasized that the operator is the weighted truncated potential. The trace inequity for this operator in the classical Lebesgue spaces was established by Sawyer [20] (see also the monograph [21], Ch.6 for related topics).
In general, the modular inequality for the Hardy operator is not valid (see [22], Corollary 2.3, for details). Namely, the following fact holds: if there exists a positive constant such that inequality is true for all , where ; ; ; and are nonnegative measurable functions, then there exists such that for almost every ; for almost every , and and take the same constant values a.e. for and .
To get the main result we use the following pointwise inequalities: for nonnegative decreasing functions, where , , , and are constants and are independent of , , and , and
In the sequel by the symbol we mean that there are positive constants and such that . Constants in inequalities will be mainly denoted by or ; the symbol means the interval .
We say that a radial function is decreasing if there is a decreasing function such that , . We will denote again by . Let be a measurable function, satisfying the conditions , .
Given such that and a nonnegative measurable function (weight) in , let us define the following local oscillation of : where is the ball with center 0 and radius .
We observe that is nondecreasing and positive function such that where and denote the essential infimum and supremum of on the support of , respectively.
By the similar manner (see [10]) the function is defined for an exponent and weight on :
Let be the class of nonnegative decreasing functions on and let be the class of all nonnegative radially decreasing functions on . Suppose that is measurable a.e. positive function (weight) on . We denote by the class of all nonnegative functions on for which
For essential properties of spaces we refer to the papers [23, 24] and the monographs [1, 2].
Under the symbol we mean the class of nonnegative decreasing functions on from .
Now we list the well-known results regarding one-weight inequality for the operator . For the following statement we refer to [18].
Theorem A. Let be constant such that . Then the inequity for a weight holds, if and only if there exists a positive constant such that for all
Condition (11) is called condition and was introduced in [18].
Theorem B (see [10]). Let be a weight on and such that , and assume that . The following facts are equivalent:(a)there exists a positive constant such that, for any , (b)for any , (c) a.e. and .
Proposition 1. For the operators , and , the following relations hold:(a)(b)
Proof. (a) Upper estimate: represent as follows: Observe that if , then . Hence where the positive constant does not depend on and . Using the fact that is decreasing we find that
Lower estimate follows immediately by using the fact that is nonnegative and the obvious estimate and .
(b) Upper estimate: let us represent the operator as follows: Since for we have that Taking into account the fact that is radially decreasing on we find that there is a decreasing function such that Let . Then we have It is easy to see that while using the fact that we find that Finally we conclude that Lower estimate follows immediately by using the fact that is nonnegative and the obvious estimate , where .
We will also need the following statement.
Lemma 2. Let be a constant such that . Then the inequality holds, if and only if there exists a positive constant C such that, for all ,
Proof. We will see that inequality (26) is equivalent to the inequality where , , and .
Indeed, using polar coordinates in we have
Conversely taking the test function , , in modular inequality (26), one can easily obtain inequality (27).
To formulate the main results we need to prove the following proposition.
Proposition 3. Let be a weight on and such that , and assume that . The following statements are equivalent:(a)there exists a positive constant such that, for any , (b)for any , (c) a.e. and .
Proof. We use the arguments of [10]. To show that (a) implies (b) it is enough to test the modular inequality (30) for the function , . Indeed, it can be checked that
Further, we find that Therefore To obtain (c) from (b) we are going to prove that condition (b) implies that is a constant function; namely, for all . This fact and the hypothesis on imply that , and hence, due to (7), Finally (31) means that . Let us suppose that is not constant. Then one of the following conditions holds:(i)there exists such that and, hence, there exists such that or(ii)there exists such that and then, for some , In case (i) we observe that condition (b), for , implies that Then using (36) we obtain, for , which is clearly a contradiction if we let . Similarly in case (ii) let us consider the same condition (b), for , and fix now . Taking into account (38) we find that which is a contradiction if we let .
Finally, the fact that condition (c) implies (a) follows from [18,Theorem 1.7].
Theorem 4. Let be a weight on and such that . Assume that . The following facts are equivalent:(i)there exists a positive constant such that, for any , (ii)condition (13) holds;(iii)condition of Theorem B is satisfied.
Proof. Proof follows by using Theorem B and Proposition 1(a).
Theorem 5. Let be a weight on and such that , and assume that . The following facts are equivalent:(i)there exists a positive constant such that, for any , (ii)condition (31) holds;(iii)condition (c) of Proposition 3 holds.
Proof. Proof follows by using Propositions 3 and 1(b).
The authors are grateful to Professor A. Meskhi for drawing their attention to the problem studied in this paper and helpful remarks. The authors are also grateful to the editor and anonymous reviewer for their careful review, valuable comments, and remarks to improve this paper.
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Lexical density - Wikipedia
Lexical density is a concept in computational linguistics that measures the structure and complexity of human communication in a language.[1] Lexical density estimates the linguistic complexity in a written or spoken composition from the functional words (grammatical units) and content words (lexical units, lexemes). One method to calculate the lexical density is to compute the ratio of lexical items to the total number of words. Another method is to compute the ratio of lexical items to the number of higher structural items in a composition, such as the total number of clauses in the sentences.[2][3]
The lexical density for an individual evolves with age, education, communication style, circumstances, unusual injuries or medical condition,[4] and his or her creativity. The inherent structure of a human language and one's first language may impact the lexical density of the individual's writing and speaking style. Further, human communication in the written form is generally more lexically dense than in the spoken form after the early childhood stage.[5][6] The lexical density impacts the readability of a composition and the ease with which the listener or reader can comprehend a communication.[7][8] The lexical density may also impact the memorability and retention of a sentence and the message.[9]
1.1 Ure Lexical density
1.2 Halliday Lexical density
The lexical density is the proportion of content words (lexical items) in a given discourse. It can be measured either as the ratio of lexical items to total number of words, or as the ratio of lexical items to the number of higher structural items in the sentences (for example, clauses).[2][3] A lexical item is typically the real content and it includes nouns, verbs, adjectives and adverbs. A grammatical item typically is the functional glue and thread that weaves the content and includes pronouns, conjunctions, prepositions, determiners, and certain classes of finite verbs and adverbs.[5]
Lexical density is one of the methods used in discourse analysis as a descriptive parameter which varies with register and genre. There are many proposed methods for computing the lexical density of any composition or corpus. Lexical density may be determined as:
{\displaystyle L_{d}=(N_{\mathrm {lex} }/N)\times 100}
{\displaystyle L_{d}}
= the analysed text's lexical density
{\displaystyle N_{\mathrm {lex} }}
= the number of lexical or grammatical tokens (nouns, adjectives, verbs, adverbs) in the analysed text
{\displaystyle N}
= the number of all tokens (total number of words) in the analysed text
Ure Lexical density[edit]
Ure proposed the following formula in 1971 to compute the lexical density of a sentence:
Ld = The number of lexical items/The total number of words * 100
Biber terms this ratio as "type-token ratio".[10]
Halliday Lexical density[edit]
In 1985, Halliday revised the denominator of the Ure formula and proposed the following to compute the lexical density of a sentence:[1]
Ld = The number of lexical items/The total number of clauses * 100
In some formulations, the Halliday proposed lexical density is computed as a simple ratio, without the "100" multiplier.[2][1]
Lexical density measurements may vary for the same composition depending on how a "lexical item" is defined and which items are classified as lexical or as a grammatical item. Any adopted methodology when consistently applied across various compositions provides the lexical density of those compositions. Typically, the lexical density of a written composition is higher than a spoken composition.[2][3] According to Ure, written forms of human communication in the English language typically have lexical densities above 40%, while spoken forms tend to have lexical densities below 40%.[2] In a survey of historical texts by Michael Stubbs, the typical lexical density of fictional literature ranged between 40% and 54%, while non-fiction ranged between 40% and 65%.[3][11][12]
The relation and intimacy between the participants of a particular communication impact the lexical density, states Ure, as do the circumstances prior to the start of communication for the same speaker or writer. The higher lexical density of written forms of communication, she proposed, is primarily because written forms of human communication involve greater preparation, reflection and revisions.[2] Human discussions and conversations involving or anticipating feedback tend to be sparser and have lower lexical density. In contrast, state Stubbs and Biber, instructions, law enforcement orders, news read from screen prompts within the allotted time, and literature that authors expect will be available to the reader for re-reading tend to maximize lexical density.[2][13][14] In surveys of lexical density of spoken and written materials across different European countries and age groups, Johansson and Strömqvist report that the lexical density of population groups were similar and depended on the morphological structure of the native language and within a country, the age groups sampled. The lexical density was highest for adults, while the variations estimated as lexical diversity, states Johansson, were higher for teenagers for the same age group (13-year-olds, 17-year-olds).[15][16]
^ a b c Michael Halliday (1985). Spoken and Written Language. Deakin University. pp. 61–64. ISBN 978-0-7300-0309-0.
^ a b c d e f g Erik Castello (2008). Text Complexity and Reading Comprehension Tests. Peter Lang. pp. 49–51. ISBN 978-3-03911-717-8.
^ a b c d Belinda Crawford Camiciottoli (2007). The Language of Business Studies Lectures: A Corpus-assisted Analysis. John Benjamins Publishing. p. 73. ISBN 978-90-272-5400-9.
^ Paul Yoder (2006). "Predicting Lexical Density Growth Rate in Young Children With Autism Spectrum Disorders". American Journal of Speech-Language Pathology. 15 (4): 362–373.
^ a b Michael Halliday (1985). Spoken and Written Language. Deakin University. pp. 61–75 (Chapter 5), 76-91 (Chapter 6). ISBN 978-0-7300-0309-0.
^ Victoria Johansson (2009). Developmental aspects of text production in writing and speech. Department of Linguistics and Phonetics, Centre for Languages and Literature, Lund University. pp. 1–16. ISBN 978-91-974116-7-7.
^ V To; S Fan; DP Thomas (2013). "Lexical density and Readability: A case study of English Textbooks". The International Journal of Language, Society and Culture. 37 (7): 61–71.
^ O'Loughlin, Kieran (1995). "Lexical density in candidate output on direct and semi-direct versions of an oral proficiency test". Language Testing. SAGE Publications. 12 (2): 217–237. doi:10.1177/026553229501200205. S2CID 145638000.
^ Perfetti, Charles A. (1969). "Lexical density and phrase structure depth as variables in sentence retention". Journal of Verbal Learning and Verbal Behavior. Elsevier BV. 8 (6): 719–724. doi:10.1016/s0022-5371(69)80035-6. ISSN 0022-5371.
^ Douglas Biber (2007). Discourse on the Move: Using Corpus Analysis to Describe Discourse Structure. John Benjamins Publishing. pp. 97–98 with footnote 7. ISBN 978-90-272-2302-9.
^ Mark Warschauer; Richard Kern (2000). Network-Based Language Teaching: Concepts and Practice. Cambridge University Press. pp. 107–108. ISBN 978-0-521-66742-5.
^ Michael Stubbs (1996). Text and Corpus Analysis: Computer Assisted Studies of Language and Culture. Wiley. pp. 71–73. ISBN 978-0-631-19512-2.
^ Nikola Dobrić; Eva-Maria Graf; Alexander Onysko (2016). Corpora in Applied Linguistics: Current Approaches. Cambridge Scholars Publishing. p. 57. ISBN 978-1-4438-9819-5.
^ Michael Stubbs (1986). "Lexical density: A technique and some findings". In Malcolm Coulthard (ed.). Talking about Text. University of Birmingham: English Language Research. pp. 27–42.
^ Victoria Johansson (2008). "Lexical diversity and lexical density in speech and writing: a developmental perspective". Linguistics and Phonetics Working Papers. Lund University. 53: 61–79.
^ Sven Strömqvist; Victoria Johansson; Sarah Kriz, H Ragnarsdottir, Ravid Aisenmann, Dorit Ravid (2002). "Toward a crosslinguistic comparison of lexical quanta in speech and writing". Written Language and Literacy. 5: 45–67. doi:10.1075/wll.5.1.03str. {{cite journal}}: CS1 maint: multiple names: authors list (link)
Ure, J (1971). Lexical density and register differentiation. In G. Perren and J.L.M. Trim (eds), Applications of Linguistics, London: Cambridge University Press. 443-452.
Lexical density 'Textalyser'
Retrieved from "https://en.wikipedia.org/w/index.php?title=Lexical_density&oldid=1012600523" |
hunterofdeath63 2021-12-10 Answered
In the given equation as follows , use a table of integrals with forms involving the trigonometric functions to find the indefinite integral:
\int \frac{1}{1+{e}^{2x}}dx
1+{e}^{2x}=t
. Differentiate both sides.
{e}^{2x}×2dx=dt
dx=\frac{dt}{2{e}^{2x}}
=\frac{dt}{2\left(t-1\right)}
Substitute the values into the integral. Also since the integral is indefinite; a constant of integration is to be added.
\int \frac{1}{1+{e}^{2x}}dx=\int \frac{1}{t}×\frac{dt}{2\left(t-1\right)}
=\frac{1}{2}\int \frac{dt}{t\left(t-1\right)}
=\frac{1}{2}\int \left[\frac{1}{t-1}-\frac{1}{t}\right]dt
=\frac{1}{2}×\left[\mathrm{ln}|t-1|-\mathrm{ln}|t|\right]+c
=\frac{1}{2}×\left[\mathrm{ln}|1+{e}^{2x}-1|-\mathrm{ln}|1+{e}^{2x}|\right]+c
=\frac{1}{2}×\left[\mathrm{ln}|{e}^{2x}|-\mathrm{ln}|1+{e}^{2x}|\right]+c
Hence the solution is obtained.
SlabydouluS62
\int \frac{1}{{e}^{2x}+1}dx
=-\frac{1}{2}\int \frac{1}{{e}^{-u}+1}du
=\int \frac{{e}^{u}}{{e}^{u}+1}du
=\int \frac{1}{v}dv
=\mathrm{ln}\left(v\right)
=\mathrm{ln}\left({e}^{u}+1\right)
-\frac{1}{2}\int \frac{1}{{e}^{-u}+1}du
=-\frac{\mathrm{ln}\left({e}^{u}+1\right)}{2}
=-\frac{\mathrm{ln}\left({e}^{u}+1\right)}{2}+C
Evaluating an Improper Integral :-, Determine whether the improper integral diverges or converges. Evaluate the integral if it converges :-
{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{4}{16+{x}^{2}}dx
\frac{{d}^{2}y}{{dx}^{2}}-8\frac{dy}{dx}=-16y
? Discuss what kind of differential equation is this, and when it may arise?
\left(\frac{dy}{dx}\right)-y-{e}^{3x}=0
\int \frac{{e}^{x}}{{e}^{2x}+2{e}^{x}+1}dx
\frac{dy}{dx}=4y
\int \left({x}^{\frac{1}{2}}+3{x}^{-\frac{2}{3}}\right)dx
\mathrm{tan}x{\mathrm{sec}}^{3}xdx |
Partial Fraction Decomposition - Vocabulary - Course Hero
College Algebra/Partial Fraction Decomposition/Vocabulary
degree of the term in a polynomial with the greatest degree
rational expression where the degree of the numerator is greater than or equal to the degree of the denominator
expression that cannot be written in a simpler form as a product of factors
factor of a polynomial in the form
(x + n)
n
for the rational expression
\frac{P}{Q}
, a rational expression with a denominator that has a degree less than the degree of
Q
, such that the sum of the partial fractions is equal to
\frac{P}{Q}
process of writing a rational expression as a sum of partial fractions, which are rational expressions with a denominator of lower degree
rational expression where the degree of the denominator is greater than the degree of the numerator
expression in the form
\frac{P}{Q}
P
Q
Q
is not zero
<Overview>Linear Factors in the Denominator |
A solution contains Cr^{3+}ions and Mg^{2+}Mg ions. The addition of
aramutselv 2022-01-10 Answered
A solution contains
C{r}^{3+}
ions and
M{g}^{2+}Mg
ions. The addition of 1.00 L of 1.51 M NaF solution causes the complete precipitati on of these ions as
Cr{F}_{3}\left(s\right)
Mg{F}_{2}\left(s\right)
. The total mass of the precipitate is 49.6g. Find the mass of
C{r}^{3+}
in the original solution.
C{r}^{3+}+3{F}^{-}⇒Cr{F}_{3}
M{g}^{2+}+2{F}^{-}⇒Mg{F}_{2}
\frac{x\text{ }mol}{1.0L}=1.51M
Solve for moles using Molarity
x=1.51mol
Multiply 1.51 by 1.0 to get 1.51 moles
49.6g\cdot \frac{1\text{ }mol}{109.0g}\cdot \frac{3\text{ }mol}{1\text{ }mol}=1.36mol
109.0 is obtained by the molar mass of CrF3
\left(52+3\left(19\right)\right)
49.6g\cdot \frac{1\text{ }mol}{62.3g}\cdot \frac{2\text{ }mol}{1\text{ }mol}=1.59\left(1-x\right)
\left(1.37x+1.59-1.59x\right)mol=1.51mol
x=0.36
0.36\cdot 49.6=18g
Multiply 0.36 by the total mass
18\cdot \frac{1\text{ }mol}{\left(52.0+3\left(19.0\right)\right)}\cdot \frac{52.0g}{1\text{ }mol}=8.6g\text{ }Cr3+
Convert 18 grams to moles then to moles of Cr3+
The molarity of the sodium fluoride (NaF) solution is 1.51 M.
The volume of the solution
=1.00L
Now, determine the number of moles of NaF present in the solution.
NaF=\text{Molarity of NaF}×\text{Volume of solution}
=1.51×1.00L
=1.51mol
1 mol of NaF gives 1 mole of
N{a}^{+}
ion and 1 mol of
{F}^{-}
ion. This indicates that there are 1.51 moles of F- ions present in the precipitate.
According to the given data, 1.55 moles of
{F}^{-}
ions are required to complete the precipitation of
Cr{F}_{3}\text{ }\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\text{ }Mg{F}_{2}
. Assume that there are x moles of
C{r}^{3+}
ions and y moles of
M{g}^{2+}
ions in the precipitate. Since
Cr{F}_{3}
consists of three fluoride ions for each
C{r}^{+3}
ion and
Mg{F}_{2}
consists of two fluoride ions for each
M{g}^{2+}
ion, the following equation can be written as:
3x+2y=1.51
The total mass of precipitate
=49.6g
Cr{F}_{3}=109.00\frac{g}{m}ol
Mg{F}_{2}=62.31\frac{g}{m}ol
Based on this data the following equation is:
109x+62.31y=49.6
Simplify and solve equations (1) and (2) to get x and y values:
Eq\left(1\right)×109⇒327x+218y=164.59
Eq\left(2\right)×3⇒327x+186.93y=148.8
31.07y=15.79
y=\frac{15.79}{31.07}=0.508mol
Substitute this y value in equation (1), to get
3x+2\left(0.508\right)=1.51
3x=0.494
X=0.1646mol
x=0.1646mol\text{ }\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\text{ }y=0.648mol
The molar mass of chromium ion
=52\frac{g}{m}ol
The number of moles of
C{r}^{3+}
ion is 0.1646 mol.
C{r}^{3+}
ion in the original solution
=moles×
=0.1646mol×52\frac{g}{m}ol
=8.5592g
Therefore, the mass of the
C{r}^{3+}
ion is 8.5592g.
A+10 nC charge is located at the origin. What is the strength of the electric field at the position (x,y)=(−5.0 cm,−5.0 cm)
A person accidentally leaves a car with the lights on. If eachof the two headlights uses 43 W andeach of the two taillights 7.0 W,for a total of 1.0E2 W, how longwill a fresh 12-V battery last if it is rated at 8.0E1 A·h?
Assume the full 12 V appears across each bulb.
Ropes 3 m and 5 m in length are fastened to a holiday decoration that is suspended over a town square. The decoration has a mass of 5 kg. The ropes, fastened at different heights, make angles of 52° and 40° with the horizontal. Find the tension in each wire and the magnitude of each tension. |
Simplify, please: \frac{12!}{8!4!}
\frac{12!}{8!4!}
\frac{12!}{8!4!}=
=\frac{12×11×10×9×8!}{8!4!}=
NKS
=\frac{12×11×10×9}{4!}=
=\frac{12×11×10×9}{4×3×2×1}=
=\frac{11×5×9}{1}=495
\frac{12!}{8!4!}=495
1. Use the appropriate table to solve this problem: 30% of hedgehog owners choose fresh food to feed their pets (the rest chooses dried cat food).
If 8 hedgehog owners are chosen at random, what is the probability that at most 2 of them feed their pets fresh food?
Fill in the following list:
n=p=
Then work the problem:
(give the answer to four decimal places)
According to sales information in the first quarter of 2016. 2.5% of new vehicles sold in the United States were hybrids. This is down from 2.8% for the same period a year earlier. An analysts
Solve the binomial probability when n=10, p=0.40, x=9
Find the mean and standard deviation of the number of success.
There is a binomial distribution with n = 6 trials, and a probability of success of p = 0.10. Find the probability that the number of successes x is exactly 2. P(x = 2) =…
Beth is taking an eleven-question multiple-choice test for which each question has three answer choices, only one of which is correct. Beth decides on answers by rolling a fair die and marking the first answer choice if the die shows 1 or 2, the second if the die shows 3 or 4, and the third if the die shows 5 or 6. Find the probability of the stated event. |
Sky High: Coordinating Muscles for Optimal Jump Performance - OpenSim Documentation - Global Site
In the study of human movement, experimental measurement is generally limited to the kinematics of the body segments, external reaction forces, and electromyographic (EMG) signals. While these data are essential for characterizing movement, important information is missing. For example, because the body is actuated by more muscles than it has degrees of freedom, we cannot uniquely solve for the muscle forces that give rise to an observed motion. Yet, knowledge of muscle force is essential for quantifying the stresses placed on bones and also for understanding the functional roles of muscles in normal and pathological movement. Using dynamic models of the musculoskeletal system to simulate movement provides not only a means of estimating muscle forces, but also a framework for investigating how the various components of the musculoskeletal system interact to produce movement.
The purpose of this lab is to introduce you to the components of a musculoskeletal model, illustrate how these components can be combined, and demonstrate the value of dynamic simulation. You will use the results of dynamic simulations for feedback as you manually edit the excitation signals applied to the muscles of the lower extremity, with the goal of making a musculoskeletal model jump as high as possible. Jumping was chosen as the activity for this lab because it has a well-defined objective (i.e., jump as high as possible) and, although still complex, its muscular coordination is relatively simple compared to that of walking.
In this example, you will use several features of the OpenSim GUI to search for a set of muscle excitations that maximizes the jump height. In the course of this example, you will:
investigate the functions of muscles when they are fired in isolation and in conjunction with other muscle groups;
quantify the magnitude of the articular contact forces in the hip;
compare the ground contact forces produced by your simulation to published literature in the field; and
compare the force in muscles to the maximum isometric force during jumping.
This lab was originally created by Jeff Reinbolt, B.J. Fregly, Clay Anderson, Allison Arnold, Silvia Blemker, Darryl Thelen, and Scott Delp using torque actuators. Daniel Jacobs contributed to improving and refining the example.
The Gait2392 and Gait2354 models are three-dimensional, 23-degree-of-freedom computer models of the human musculoskeletal system. The models were created by Darryl Thelen (University of Wisconsin-Madison) and Ajay Seth, Frank C. Anderson, and Scott L. Delp (Stanford University). The models feature lower extremity joint definitions adopted from Delp et al. (1990), low back joint and anthropometry adopted from Anderson and Pandy (1999), and a planar knee model adopted from Yamaguchi and Zajac (1989).
The Gait2392 model features 92 musculotendon actuators to represent 76 muscles in the lower extremities and torso. For the Gait2354 model, the number of muscles was reduced by Anderson to improve simulation speed for demonstrations and educational purposes. Seth removed the patella to avoid kinematic constraints; insertions of the quadriceps are handled with moving points in the tibia frame.
The default, unscaled version of these models represents a subject that is about 1.8 m tall and has a mass of 75.16 kg.
More details about how the models were constructed can be found on the Gait 2392 and 2354 Models page in the Musculoskeletal Models section.
Dynamic models of the musculoskeletal system typically consist of four important components:
The equations of motion for the body, or skeletal dynamics;
A model of musculoskeletal geometry;
A model of muscle–tendon mechanics; and
A model of activation dynamics.
Figure 1. Forward Dynamic Simulation
Figure 1 illustrates how these components are combined to execute a forward dynamic simulation. Based on a set of initial states, which include the muscle activations \vec{a}\left(t\right)
//<![CDATA[ \begin{array}{l}\vec{a}\left(t\right)\end{array} //]]>
, the muscle forces \vec{f}\left(t\right)
//<![CDATA[ \begin{array}{l}\vec{f}\left(t\right)\end{array} //]]>
, the generalized speeds \dot{\vec{q}}\left(t\right)
//<![CDATA[ \begin{array}{l}\dot{\vec{q}}\left(t\right)\end{array} //]]>
, and the generalized coordinates \vec{q}\left(t\right)
//<![CDATA[ \begin{array}{l}\vec{q}\left(t\right)\end{array} //]]>
, differential equations are used to compute the time rate of change of the states. At each time step of the simulation, the differential equations are numerically integrated to compute the states at the next time step. After each time step, the new states are used to calculate new derivatives and the forward dynamics process repeats, advancing the states in time until the final time of the simulation is reached. In the simulations you will conduct in this lab, a variable-step-size integrator is used.
The equations of motion for the body allow one to compute the accelerations of the body segments when forces and torques are applied to the body. The equations of motion can be expressed as follows:
(1) \ddot{\vec{q}} = I\left(\vec{q}\;\!\right)^{-1} \left( C\big(\vec{q}, \dot{\vec{q}}\,^2\big) + G\left(\vec{q}\;\!\right) + R\left(\vec{q}\;\!\right) \vec{f}_{M} + E\left(\vec{q}\;\!\right) \vec{f}_{E} \right)
Eq. (1) is simply an elaboration of Newton’s second law for a multi-link system, rearranged so that one can compute acceleration (i.e., \vec{a} = M\big(\vec{q}, \dot{\vec{q}}\;\!\big)^{-1} \vec{f}_M
//<![CDATA[ \begin{array}{l}\vec{a} = M\big(\vec{q}, \dot{\vec{q}}\;\!\big)^{-1} \vec{f}_M\end{array} //]]>
). The vector of generalized coordinates, \vec{q}
//<![CDATA[ \begin{array}{l}\vec{q}\end{array} //]]>
, is used to specify the position and orientation of the body segments. The time derivatives of \vec{q}
//<![CDATA[ \begin{array}{l}\vec{q}\end{array} //]]>
(i.e., \dot{\vec{q}}
//<![CDATA[ \begin{array}{l}\dot{\vec{q}}\end{array} //]]>
and \ddot{\vec{q}}
//<![CDATA[ \begin{array}{l}\ddot{\vec{q}}\end{array} //]]>
) represent the velocities and accelerations of the segments.
Depending on how one chooses to model the body, elements of \vec{q}
//<![CDATA[ \begin{array}{l}\vec{q}\end{array} //]]>
may be translations, orientations of segments with respect to the lab frame (segment angles), or orientations of segments with respect to other segments (joint angles). Implicit in one’s choice of generalized coordinates are one’s assumptions about how the joints of the body function. For example, one often models the hip joint as a three-degree-of-freedom ball-and-socket (spherical) joint, which requires three generalized coordinates: flexion-extension (q_1
//<![CDATA[ \begin{array}{l}q_1\end{array} //]]>
), abduction-adduction (q_2
//<![CDATA[ \begin{array}{l}q_2\end{array} //]]>
), and internal-external rotation (q_3
//<![CDATA[ \begin{array}{l}q_3\end{array} //]]>
The system mass matrix, I\left(\vec{q}\;\!\right)
//<![CDATA[ \begin{array}{l}I\left(\vec{q}\;\!\right)\end{array} //]]>
, characterizes the inertial properties of the body (i.e., masses and moments of inertia). The remaining terms in Eq. (1) express the generalized forces or torques that act on the body segments. C\big(\vec{q}, \dot{\vec{q}}\,^2\big)
//<![CDATA[ \begin{array}{l}C\big(\vec{q}, \dot{\vec{q}}\,^2\big)\end{array} //]]>
represents centripetal forces that arise from the angular velocities of the segments, G\left(\vec{q}\;\!\right)
//<![CDATA[ \begin{array}{l}G\left(\vec{q}\;\!\right)\end{array} //]]>
represents gravitational forces, R\left(\vec{q}\;\!\right) \vec{f}_{M}
//<![CDATA[ \begin{array}{l}R\left(\vec{q}\;\!\right) \vec{f}_{M}\end{array} //]]>
represents the moments applied at the joints by the muscles, and E\left(\vec{q}\;\!\right) \vec{f}_{E}
//<![CDATA[ \begin{array}{l}E\left(\vec{q}\;\!\right) \vec{f}_{E}\end{array} //]]>
represents external forces applied to the body, such as the ground reaction force. The matrix R\left(\vec{q}\;\!\right)
//<![CDATA[ \begin{array}{l}R\left(\vec{q}\;\!\right)\end{array} //]]>
is a matrix of moment arms that transform the muscle forces, \vec{f}_{M}
//<![CDATA[ \begin{array}{l}\vec{f}_{M}\end{array} //]]>
, into joint torques. The matrix E\left(\vec{q}\;\!\right)
//<![CDATA[ \begin{array}{l}E\left(\vec{q}\;\!\right)\end{array} //]]>
performs a similar function for the external forces, \vec{f}_{E}
//<![CDATA[ \begin{array}{l}\vec{f}_{E}\end{array} //]]>
For simple models, it is possible to derive the equations of motion by hand; however, for more complex models, this is generally not feasible (and it is highly error-prone!), and the equations of motion are generated on a computer. The jumping model used in this lab has 23 degrees of freedom (Anderson and Pandy, 1999), and the equations of motion for the jumping model were generated using OpenSim (Delp et al., 2007).
Accurately representing the path of a muscle from its origin to its insertion is one of the more challenging aspects of modeling the musculoskeletal system. Sometimes a muscle can be represented as a straight-line path between its origin and insertion. Other times, it is adequate to approximate the path as a series of straight line segments that pass through a series of via points (Delp et al., 2007). When modeling muscle paths in three dimensions, it is often necessary to simulate how muscles wrap over underlying bone or musculature. Cylinders, spheres, and ellipsoids have been used as wrapping surfaces (Van der Helm et al., 1992; Garner and Pandy, 2000; Arnold et al., 2000) (see figure at left).
Muscle–Tendon Mechanics
A muscle is not capable of generating force or relaxing 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 (see Guyton (1986) for review). 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, and so the time required for muscle force to fall can be considerably longer than the time for it to develop.
In the forward dynamic simulations you will conduct in this lab, activation dynamics is modeled using a first-order differential equation to relate the rate of change in activation (i.e., the concentration of calcium ions within the muscle) to excitation (i.e., the firing of motor units):
(2) \dot{a} = \frac{x^2-xa}{\tau_\textrm{rise}} + \frac{x-a}{\tau_\textrm{fall}}
//<![CDATA[ \begin{array}{l}a\end{array} //]]>
is the activation level of a muscle, x
//<![CDATA[ \begin{array}{l}x\end{array} //]]>
is the excitation level of a muscle, and \tau_\textrm{rise}
//<![CDATA[ \begin{array}{l}\tau_\textrm{rise}\end{array} //]]>
and \tau_\textrm{fall}
//<![CDATA[ \begin{array}{l}\tau_\textrm{fall}\end{array} //]]>
are the rise and fall time constants for activation, respectively. In the model, activation is allowed to vary continuously between zero (no contraction) and one (full contraction). In the body, the excitation level of a muscle is a function of both the number of motor units recruited and the firing frequency of the motor units. Some models for excitation–contraction coupling distinguish between these two control mechanisms (Hatze, 1976), but it is often not computationally feasible to use such models when conducting complex dynamic simulations.
Ground Contact Dynamics
The contact dynamics between the model bodies and the ground are represented with a continuous, nonlinear contact model published by Hunt and Crossley (1975). The contact force is modeled as
(3) f_\textrm{hc} = - \lambda\dot{x}x^n - kx^n
//<![CDATA[ \begin{array}{l}x\end{array} //]]>
and \dot{x}
//<![CDATA[ \begin{array}{l}\dot{x}\end{array} //]]>
are the interference and interference rate between the contacting surfaces, \lambda
//<![CDATA[ \begin{array}{l}\lambda\end{array} //]]>
is the damping constant, k
//<![CDATA[ \begin{array}{l}k\end{array} //]]>
is the spring constant, and n
//<![CDATA[ \begin{array}{l}n\end{array} //]]>
is the power constant. The contact surface between the foot and the ground is modeled as five spheres attached to the calcaneus and toe bodies on each foot.
A copy of these instructions can be saved as a .pdf or a Word document by selecting "Export to PDF" or "Export to Word" from the Tools menu in the top-right corner of this page. All the necessary files for this example are attached to the instructional page Coordinating Muscles for Optimal Jump Performance.
If you are completing this example as a laboratory exercise for a course on human movement, you will need to submit answers to the questions on the Questions: Optimal Jump Performance page.
Anderson, F.C., 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.
Arnold, A.S., Salinas, S., Asakawa, D.J., Delp, S.L. (2000). Accuracy of muscle moment arms estimated from MRI-based musculoskeletal models of the lower extremity. Computer Aided Surgery, 5(2):108–119.
Atkinson, L.V., Harley, P.J., Hudson, J.D. (1989). Numerical Methods with FORTRAN 77: A Practical Introduction. Addison–Wesley Publishing Company, Menlo Park.
Delp, S.L., Loan, J.P., Hoy, M.G., Zajac, F.E., Topp, E.L., Rosen, J.M. (1990). An interactive graphics-based model of the lower extremity to study orthopaedic surgical procedures. IEEE Transactions on Biomedical Engineering, 37(8):757–767.
Garner, B.A., Pandy, M.G. (2000). The obstacle-set method for representing muscle paths in musculoskeletal models. Computer Methods in Biomechanics and Biomedical Engineering, 3(1):1–30.
Guyton, A.C. (1986). Textbook of Medical Physiology, 7th ed. W. B. Saunders Company, Philadelphia.
McMahon, T.A. (1984). Muscles, Reflexes, and Locomotion. Princeton University Press, Princeton.
Symbolic Dynamics, Inc. (1996). SD/FAST User’s Manual, Version B.2. Mountain View.
Van der Helm, F.C.T., Veeger, H.E.J., Pronk, G.M., Van der Woude, L.H.V., Rozendal, R.H. (1992). Geometry parameters for musculoskeletal modeling of the shoulder system. Journal of Biomechanics, 2:129–144.
Zajac, F.E. (1989). Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control. CRC Critical Reviews in Biomedical Engineering (Edited by JR Bourne), 17(4):359–411.
Hunt, K.H., Crossley, F.R.E. (1975). Coefficient of restitution interpreted as damping in vibroimpact. Journal of Applied Mechanics, 42(2):440–445.
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Find the following limits or state that they do not
Find the following limits or state that they do not exist. \lim_{x\to1^+}
Find the following limits or state that they do not exist.
\underset{x\to {1}^{+}}{lim}\frac{x-1}{\sqrt{{x}^{2}-1}}
\underset{x\to {1}^{+}}{lim}\frac{x-1}{\sqrt{{x}^{2}-1}}
for evaluating given limit we substitute x=1+h, where
h\to 0
\underset{x\to {1}^{+}}{lim}\frac{x-1}{\sqrt{{x}^{2}-1}}=\underset{h\to {0}^{+}}{lim}\frac{1+h-1}{\sqrt{{\left(1+h\right)}^{2}-1}}
=\underset{h\to {0}^{+}}{lim}\frac{h}{\sqrt{{1}^{2}+2\left(1\right)\left(h\right)+{h}^{2}-1}}
=\underset{h\to {0}^{+}}{lim}\frac{h}{\sqrt{1+2h+{h}^{2}-1}}
=\underset{h\to {0}^{+}}{lim}\frac{h}{\sqrt{{h}^{2}+2h}}
=\underset{h\to {0}^{+}}{lim}\frac{h}{h\sqrt{1+\frac{2}{h}}}
=\underset{h\to {0}^{+}}{lim}\frac{1}{\sqrt{1+\frac{2}{h}}}
=\frac{1}{\sqrt{1+\mathrm{\infty }}}
=0
hence, given limit is equal to 0.
f\left(x\right)=x\sqrt{16-{x}^{2}}
\underset{x\to {0}^{+}}{lim}{x}^{{x}^{2}}
How can I calculate the following difficult limit:
\underset{x\to \mathrm{\infty }}{lim}\left(x+\mathrm{sin}x\right)\mathrm{sin}\frac{1}{x}
Limit of a function as it tends to zero
\underset{x\to 0}{lim}\frac{3{\left(\mathrm{tan}2x\right)}^{2}}{2{x}^{2}}
What method is best for solving this? The division by zero is causing me issues when trying to apply Sandwich theorem and/or Algebra of limits.
Find the limit, is the function continuous at the point being approached
\underset{x\to \frac{\pi }{6}}{lim}\sqrt{{\mathrm{csc}}^{2}x+5\sqrt{3}\mathrm{tan}x}
\underset{lim}{lim}its\left\{x\to 0\right\}\left\{\frac{\mathrm{tan}\left\{x\right\}-\mathrm{sin}\left\{x\right\}}{{x}^{3}}\right\}
So what I tried is:
\underset{x\to 0}{lim}\frac{\frac{\mathrm{sin}\left\{x\right\}}{\mathrm{cos}\left\{x\right\}}-\mathrm{sin}\left\{x\right\}}{{x}^{3}}=\underset{x\to 0}{lim}\frac{\left\{\mathrm{sin}\left\{x\right\}\right\}\left(\frac{1}{\mathrm{cos}x}-1\right)}{x\cdot {x}^{2}}
From here, using the rule
\underset{x\to 0}{lim}\frac{\mathrm{sin}\left\{x\right\}}{x}=1
it remains to evaluate
\underset{x\to 0}{lim}\frac{\frac{1}{\mathrm{cos}\left\{x\right\}}-1}{{x}^{2}}
\underset{n\to \infty }{\mathrm{lim}}\sqrt{n}\left(\sqrt[n]{n}-1\right)=0 |
Fuzzy logic - New World Encyclopedia
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Fuzzy logic, when construed in a wider sense, is the theory of fuzzy sets. The concept of fuzzy sets provides a convenient way to represent various notions with imprecision, vagueness, or fuzziness, for example young, tall, cold, and so forth, which we frequently employ in our everyday life. As such, fuzzy logic has the rationale of more closely resembling than traditional logic the way human beings actually think, where alternatives are not black and white but shades of gray. Fuzzy logic has had notable success in various engineering applications.
When construed in a narrower sense, fuzzy logic is an extension of ordinary two-valued logic in such a way that the points in interval units are allowed as truth-values. As the truth-values are generalized in such a way, usual truth-functional operations are generalized accordingly.
2 Applications in Engineering
3 Formal Fuzzy Logics
3.1 Basic Fuzzy Propositional Logic
3.2 Versions of Fuzzy Propositional Logic
3.3 Basic Fuzzy Predicate Logic
Fuzzy logic studies fuzzy sets, which was first introduced by L. Zadeh in 1965. Zadeh maintains that the meanings of many words in natural language come with degrees. Twelve years old and 18 years old are clearly both young; however 12 years old is younger than 18 years old. To represent this, he introduces the concept of fuzzy subsets. A fuzzy subset of a given set U is a function from U into [1, 0]. The value that a given fuzzy set A assigns to an element x in U is called the degree of the membership of x in the fuzzy set A. Fuzzy subsets are usually referred to simply as fuzzy sets. Using this framework, the meaning of, say, the word “young” can be represented. Take the set of natural numbers and define some fuzzy set, as you like, that assigns values in the unit interval to natural numbers so that, say, 12 (years old) get some value (e.g. .95) higher than the value that 18 (e.g. .85). In that case, the value that each number gets assigned represents the degree of youth. The degree of the membership of 12 in the “youth” subset is higher than that of 18.
This concept of fuzzy sets generalizes the concept of sets in ordinary set theory. Given a set U, a subset S, in the ordinary sense, of U are determined by a function from U to [1, 0]. The elements of U that get 1 assigned represent the elements in S and the elements that get 0 assigned represent the elements that are not in S. The elements of U are all either in, or not in, the subset. However, fuzzy subsets are allowed to take any value in the unit interval other than just 1 and 0. In this sense, the sets in the ordinary sense are special cases of fuzzy sets.
In this image, cold, warm, and hot are functions mapping a temperature scale. A point on that scale has three "truth values"—one for each of the three functions. For the particular temperature shown, the three truth values could be interpreted as describing the temperature as, say, "fairly cold," "slightly warm," and "not hot."
A more sophisticated practical example is the use of fuzzy logic in high-performance error correction to improve information reception over a limited-bandwidth communication link affected by data-corrupting noise using turbo codes. The front-end of a decoder produces a likelihood measure for the value intended by the sender (0 or 1) for each bit in the data stream. The likelihood measures might use a scale of 256 values between extremes of "certainly 0" and "certainly 1." Two decoders may analyze the data in parallel, arriving at different likelihood results for the values intended by the sender. Each can then use as additional data the other's likelihood results, and repeats the process to improve the results until consensus is reached as to the most likely values.
Automobile and other vehicle subsystems, such as ABS and cruise control (e.g. Tokyo monorail)
Washing machines and other home appliances
Formal Fuzzy Logics
Fuzzy logic, when narrowly construed, is an extension of ordinary logics. The basic idea is that, in fuzzy extensions of logics, formulas can take any values in the unit interval, instead of just 1 or 0 as in ordinary logics.
Basic Fuzzy Propositional Logic
In basic fuzzy propositional logic, formulas are built, as in the language of ordinary propositional logic, from propositional variables, truth-functional connectives,
{\displaystyle \rightarrow }
{\displaystyle \wedge }
, and propositional constant 0. (
{\displaystyle \lnot \phi }
{\displaystyle \phi \rightarrow 0}
Interpretation functions on propositional variables are mappings from the set of propositional variables into [0, 1], and truth functional connectives are interpreted in terms of continuous t-norms. A t-norm
{\displaystyle \triangle }
is a binary operator on [0, 1] if
{\displaystyle \triangle }
{\displaystyle 1\triangle x=x}
{\displaystyle x\triangle y=y\triangle x}
{\displaystyle x\triangle (y\triangle z)=(x\triangle y)\triangle z}
{\displaystyle v\leq w}
{\displaystyle x\leq y}
{\displaystyle v\triangle x\leq w\triangle y}
A binary connective
{\displaystyle \triangle }
{\displaystyle \triangle }
{\displaystyle \epsilon >0}
{\displaystyle \delta >0}
such that wherever
{\displaystyle |x_{1}-x_{2}|<\delta }
{\displaystyle |y_{1}-y_{2}|<\delta }
{\displaystyle |(x_{1}\triangle y_{1})-(x_{2}\triangle y_{2})|<\epsilon }
Given a t-norm
{\displaystyle \triangle }
, the residuum
{\displaystyle \Rightarrow }
{\displaystyle x\Rightarrow y}
= max { z|
{\displaystyle x\triangle z\leq y}
A t-norm and its residuum interpret
{\displaystyle \wedge }
{\displaystyle \Rightarrow }
, and 0 in [0, 1] interprets the constant 0. Given an interpretation function e on propositional variables, a t-norm induces a valuation function
{\displaystyle e_{\triangle }}
on every formula. A formula
{\displaystyle \phi }
{\displaystyle t-tautology}
{\displaystyle e_{\triangle }=1}
There is a sound and complete axiomatization, i.e. the system in which a formula
{\displaystyle \phi }
is a t-tautology if and only if
{\displaystyle \phi }
Versions of Fuzzy Propositional Logic
_ukasiewicz fuzzy logic is a special case of basic fuzzy logic where conjunction is _ukasiewicz t-norm. It has axioms of basic logic plus additional axiom of double negation (so it is not intuitionistic logic), and its models correspond to MV-algebras.
Gödel fuzzy logic is a special case of basic fuzzy logic where conjunction is Gödel t-norm. It has axioms of basic logic plus additional axiom of idempotence of conjunction, and its models are called G-algebras.
Rational Pavelka logic is a generalization of multi-valued logic. It is an extension of _ukasziewicz fuzzy logic with additional constants.
Basic Fuzzy Predicate Logic
The language of basic fuzzy predicate logic consists of the same items as the first-order logic (variables, predicate symbols,
{\displaystyle \wedge }
{\displaystyle \rightarrow }
, 0, quantifiers). An interpretation consists of a nonempty domain and a function that maps an n-ary predicate symbol to an n-ary fuzzy relation (an n-ary fuzzy relation here is a mapping from n-ary tuples of objects in the domain to values in [0, 1]. An n-ary fuzzy relation that corresponds to a predicate symbol R represents the degrees in which n-ary tuples satisfy the formula
{\displaystyle Rx_{1}...x_{n}}
. Given a continuous t-norm, the connectives are interpreted as in the case of basic fuzzy propositional logic. The truth degree of a formula of the form
{\displaystyle \forall x\phi }
is defined as the infimum of the truth degrees of the instances of
{\displaystyle \phi }
, and that of a formula of the form
{\displaystyle \exists x\phi }
is defined as the supremum of the instances of
{\displaystyle \phi }
. The interpretations of basic fuzzy predicate logic generalize to so-called BL-algebra, and, based on the interpretation, a sound and complete axiomatization can be given (see Hájek 1998 for details).
Fuzzy logic is the same as "imprecise logic."
Fuzzy logic is not any less precise than any other form of logic: it is an organized and mathematical method of handling inherently imprecise concepts. The concept of "coldness" cannot be expressed in an equation, because although temperature is a quantity, "coldness" is not. However, people have an idea of what "cold" is, and agree that something cannot be "cold" at N degrees but "not cold" at N+1 degrees—a concept classical logic cannot easily handle due to the principle of bivalence.
In a widely circulated and highly controversial paper in 1993, Charles Elkan commented that "...there are few, if any, published reports of expert systems in real-world use that reason about uncertainty using fuzzy logic. It appears that the limitations of fuzzy logic have not been detrimental in control applications because current fuzzy controllers are far simpler than other knowledge-based systems. In future, the technical limitations of fuzzy logic can be expected to become important in practice, and work on fuzzy controllers will also encounter several problems of scale already known for other knowledge-based systems." Reactions to Elkan's paper are many and varied, from claims that he is simply mistaken, to others who accept that he has identified important limitations of fuzzy logic that need to be addressed by system designers. In fact, fuzzy logic wasn't largely used at that time, and today it is used to solve very complex problems in the AI area. Probably the scalability and complexity of the fuzzy system will depend more on its implementation than on the theory of fuzzy logic.
von Altrock, Constantin. 2002. Fuzzy Logic and NeuroFuzzy Applications Explained. ISBN 0133684652
Cox, Earl. 1994. The Fuzzy Systems Handbook. ISBN 0121942708
Elkan, Charles. The Paradoxical Success of Fuzzy Logic. November 1993. Available from Elkan's home page Retrieved September 18, 2008.
Hájek, Petr. “Metamathematics of fuzzy logic.” Kluwer, 1998.
Höppner, Frank, Frank Klawonn, Rudolf Kruse and Thomas Runkler. 1999. Fuzzy Cluster Analysis. ISBN 0471988642
Klir, George and Tina Folger. 1988. Fuzzy Sets, Uncertainty, and Information. ISBN 0133459845
Klir, George, UTE H. St. Clair and Bo Yuan. 1997. Fuzzy Set Theory Foundations and Applications.
Klir, George and Bo Yuan. 1995. Fuzzy Sets and Fuzzy Logic. ISBN 0131011715
Kosko, Bart. 1993. Fuzzy Thinking: The New Science of Fuzzy Logic. Hyperion. ISBN 078688021X
Nguyen, Hung T. 2006. A First Course in Fuzzy Logic, 3rd edition. Boca Raton: Chapman & Hall/CRC.
Passino, Kevin M. and Stephen Yurkovich. 1998. Fuzzy Control. Menlo Park, CA: Addison Wesley Longman.
Yager, Ronald and Dimitar Filev. 1994. Essentials of Fuzzy Modeling and Control. ISBN 0471017612
Zimmermann, Hans-Jürgen. 2001. Fuzzy Set Theory and its Applications. ISBN 0792374355
Web page about FSQL References and links about FSQL.
Fuzzy logic history
History of "Fuzzy logic"
Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Fuzzy_logic&oldid=1004788 |
ind = ltePUCCH1Indices(ue,chs) returns a matrix of resource element (RE) indices for the physical uplink control channel (PUCCH) format 1 transmission, given structures containing the UE-specific settings, and the channel transmission configuration.
[ind,info] = ltePUCCH1Indices(ue,chs) also returns a PUCCH information structure array.
Generate PUCCH format 1 RE indices for a 1.4 MHz bandwidth, PUCCH resource index 0. Use default values for all other parameters.
Initialize UE-specific and channel configuration structures (ue and chs). Generate PUCCH format 1 indices (ind).
Because there are three antennas, the indices are output as a three-column vector, and the info output structure contains three elements. View ind and the size of info to confirm this.
Generate the physical uplink control channel (PUCCH) format 1 indices for two transmit antenna paths and output in subscript indexing form.
Initialize UE-specific and channel configuration structures (ue and chs) and the indexing option parameter, opt. Generate PUCCH1 indices and information outputs (ind and info).
[ind,info] = ltePUCCH1Indices(ue,chs,{'sub'});
Because there are two antennas, the info output structure contains two elements. View the contents of the second info structure element.
0 (default) | 1 | optional
{n}_{PUCCH}^{\left(1\right)}
{N}_{RB}^{\left(2\right)}
0 (default) | 0,...,7 | integer | optional
Number of cyclic shifts used for format 1 in RBs with a mixture of Format 1 and Format 2 PUCCH, specified as an integer from 0 to 7. (
{N}_{cs}^{\left(1\right)}
When returned as a column integer vector, the resource allocation is the same in both slots of the subframe.
When returned as a two-column integer matrix, the resource allocations can vary for each slot in the subframe.
ltePUCCH1 | ltePUCCH1Decode | ltePUCCH1DRS | ltePUCCH1DRSIndices | ltePUCCH2Indices | ltePUCCH3Indices |
Urban Housing Prices, Labor Mobility and the Development of Urban High-Tech Industries—An Empirical Analysis Based on Panel Data in the Pearl River Delta Region
College of Economic, Jinan University, Guangzhou, China.
Niu, Z. (2019) Urban Housing Prices, Labor Mobility and the Development of Urban High-Tech Industries
—An Empirical Analysis Based on Panel Data in the Pearl River Delta Region. Modern Economy, 10, 1048-1061. doi: 10.4236/me.2019.103070.
{P}_{1m}{C}_{1m}+{P}_{1h}{C}_{1h}={W}_{1}
{C}_{1m}={\left({\int }_{0}^{n}{C}_{1i}^{1-1/\sigma }{d}_{i}\right)}^{1/\left(1-1/\sigma \right)}
{n}_{1}+{n}_{2}=n
\alpha =\mu /\left(\sigma -1\right)
{V}_{2}={\mu }^{\mu }{\left(1-\mu \right)}^{\left(1-\mu \right)}{W}_{2}/\left[{P}_{2h}^{1-\mu }{\left({S}_{n}{W}_{1}^{1-\sigma }+\left(1-{S}_{n}\right){W}_{2}^{1-\sigma }\right)}^{\alpha }\right]
{T}^{1-\sigma }=\Phi
{S}_{12}=\frac{{W}_{1}}{{W}_{2}}{\left(\frac{{P}_{1h}}{{P}_{2h}}\right)}^{\mu -1}{\Phi }^{-\alpha }\left[1-\frac{\alpha }{\Phi }\left(1-{\Phi }^{2}\right)\frac{{S}_{n}}{1-{S}_{n}}\right]
\mathrm{ln}{S}_{12}=\mathrm{ln}\frac{{W}_{1}}{{W}_{2}}+\left(\mu -1\right)\mathrm{ln}\frac{{P}_{1h}}{{P}_{2h}}-\alpha \mathrm{ln}\Phi +\mathrm{ln}\left(1-\frac{\alpha }{\Phi }\left(1-{\Phi }^{2}\right)\frac{{S}_{n}}{1-{S}_{n}}\right)
\frac{{S}_{n}}{1-{S}_{n}}=\frac{\Phi }{\alpha \left(1-{\Phi }^{2}\right)}\mathrm{ln}\frac{{W}_{1}}{{W}_{2}}+\frac{\Phi \left(\mu -1\right)}{\alpha \left(1-{\Phi }^{2}\right)}\mathrm{ln}\frac{{P}_{1h}}{{P}_{2h}}-\frac{\Phi }{1-{\Phi }^{2}}
\sigma >1
{T}^{1-\sigma }=\Phi
T>1
0<\Phi <1
1-{\Phi }^{2}>0
\alpha =\mu /\left(\sigma -1\right)
0<\mu <1
\sigma >1
\alpha >0
0<\mu <1
\mu -1<0
\Phi \ast \left(\mu -1\right)/\left(\alpha \ast \left(1-{\Phi }^{2}\right)\right)<0
{Y}_{it}={\alpha }_{0}+{Y}_{it-1}+{\alpha }_{1}R{W}_{it}^{2}+{\alpha }_{2}RH{P}_{it}+{X}_{it}+\mu
{Y}_{it}
{\alpha }_{0}
{\alpha }_{1}
{\alpha }_{2}
R{W}_{it}
RH{P}_{it}
\mu
{X}_{it}
RH{P}_{it}
R{W}_{it}^{2}
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[3] Hanson, G.H. and Spilimbergo, A. (1999) Illegal Immigration, Border Enforcement and Relative Wages: Evidence from Apprehensions at the US-Mexico Border. American Economic Review, 89, 1337-1357.
[4] Ji, Y. and Zhang, P. (2015) Urban Housing Price Level and Service Industry Agglomeration under Labor Flow—Based on the Threshold Regression Model Test of Panel Data of 284 Cities in China. Journal of Shijiazhuang University of Economics, No. 4, 12-18.
[5] Ding, X. (2018) Housing Prices, Employee Turnover and Industrial Upgrading. Market Weekly, No. 7, 19-20.
[6] Mark, N. (1998) Poor People on the Move: County-to-County Migration and the Spatial Concentration of Poverty. Journal of Regional Science, 38, 329-351.
[7] Saiz, A. (2007) Immigration and Housing Rents in American Cities. Journal of Urban Economics, 61, 345-371.
[8] Zhang, L., He, J., Ma, H. (2017) Escape from Beishangguang?—How Does Housing Prices Affect Labor Mobility. International Monetary Commentary, 11, 1070-1091.
[9] Dumais, G., Ellison, G. and Glaeser, E.L. (1997) Geographic Concentration as a Dynamic Process. NBER Working Paper No. 6270.
[10] Alexandre, S. (2006) Immigration, Firm Relocation and Welfare of Domestic Workers. 6th Annual Missouri Economics Conference Selected Papers, Columbia, 31 March-1 April 2006.
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[12] Wang, Y. and Yuan, H. (2010) Wage Difference, Labor Flow and Industrial Agglomeration—Based on the Explanation and Empirical Test of New Economic Geography. Journal of Finance and Economics, No. 3, 53-60.
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Every cubic polynomial can be categorised into one of four types: Type 1: Three real, dist
Every cubic polynomial can be categorised into one of four types: Type 1: Three real, distinct zeros: P(x)=a(x−α)(x−β)(x−γ),a≠0 Type 2: Two real zeros
P\left(x\right)=a\left(x-\alpha \right)\left(x-\beta \right)\left(x-\gamma \right),a\ne 0
P\left(x\right)=a\left(x-\alpha \right)2\left(x-\beta \right),a\ne 0
P\left(x\right)=a\left(x-\alpha \right)3,a\ne 0
P\left(x\right)=\left(x-\alpha \right)\left(ax2+bx+c\right),\mathrm{△}=b2-4ac<0,a\ne 0
\alpha \alpha
For the graph of the function
P\left(x\right)=\left(x-\alpha \right)\left(a{x}^{2}+bx+c\right),\mathrm{△}={b}^{2}-4ac<0,a=/0
There is only one x-intercept,
\left(\alpha ,0\right)
The other zeros are imaginary.
Copy and complete the anticipation guide in your notes. StatementThe quadratic formula can only be used when solving a quadratic equation.Cubic equations always have three real roots.The graph of a cubic function always passes through all four quadrants.The graphs of all polynomial functions must pass through at least two quadrants.The expression
x2>4
is only true if
x>2
.If you know the instantaneous rates of change for a function at
x=2
x=3
, you can predict fairly well what the function looks like in between.
x2>4
x>2
. If you know the instantaneous rates of change for a function at
x=2
x=3
x=-2
Graph the polynomial by transforming an appropriate graph of the form
y={x}^{n}
. Show clearly all x- and y-intercepts.
P\left(x\right)=-3{\left(x+2\right)}^{5}+96
\left(\frac{2}{7},-1\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}9+\frac{1}{3i}
Use your knowledge of the graphs of polynomial functions to make a rough sketch of the graph of
y=-2{x}^{3}+{x}^{2}-5x+2
f\left(x\right)=-{x}^{3}+4{x}^{2}-2x-1
The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Decide which, if any, are exponential, and give the exponential models for those that are.
\begin{array}{}x& -2& -1& 0& 1& 2\\ f\left(x\right)& 0.8& 0.2& 0.1& 0.005& 0.025\\ g\left(x\right)& 80& 40& 20& 10& 2\end{array} |
Laplace transforms A powerful tool in solving problems in engineering
Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t)
F\left(s\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-st}f\left(t\right)dt
where we assume s is a positive real number. For example, to find the Laplace transform of
f\left(t\right)={e}^{-t}
, the following improper integral is evaluated using integration by parts:
F\left(s\right)={\int }_{0}^{\mathrm{\infty }}{e}^{-st}{e}^{-t}dt={\int }_{0}^{\mathrm{\infty }}{e}^{-\left(s+1\right)t}dt=\frac{1}{s+1}
f\left(t\right)=\mathrm{cos}at\to F\left(s\right)=\frac{s}{{s}^{2}+{a}^{2}}
We have given the function.
f\left(t\right)=\mathrm{cos}at
To prove this we plug the values in the formula.
L\le ft\left\{\mathrm{cos}atright\right\}\left(s\right)={\int }_{0}^{\to +\mathrm{\infty }}{e}^{-st}\mathrm{cos}atdt
=\underset{L\to \mathrm{\infty }}{lim}{\int }_{0}^{L}{e}^{-st}\mathrm{cos}atdt
=\underset{L\to \mathrm{\infty }}{lim}{\left[\frac{{e}^{-st}\left(-s\mathrm{cos}at+a\mathrm{sin}at\right)}{{\left(-s\right)}^{2}+{a}^{2}}\right]}_{0}^{L}
=\underset{L\to \mathrm{\infty }}{lim}\left(\frac{{e}^{-sL}\left(-s\mathrm{cos}aL+a\mathrm{sin}aL\right)}{{s}^{2}+{a}^{2}}-\frac{{e}^{-s×0}\left(-s\mathrm{cos}\left(0×a\right)+a\mathrm{sin}\left(0×a\right)\right)}{{s}^{2}+{a}^{2}}\right)
=\underset{L\to \mathrm{\infty }}{lim}\left(\frac{s\mathrm{cos}\left(0×a\right)-a\mathrm{sin}\left(0×a\right)}{{s}^{2}+{a}^{2}}-\frac{{e}^{-sL}\left(-s\mathrm{cos}aL+a\mathrm{sin}aL\right)}{{s}^{2}+{a}^{2}}\right)
=\frac{s\mathrm{cos}\left(0×a\right)-a\mathrm{sin}\left(0×a\right)}{{s}^{2}+{a}^{2}}-0
=\frac{s\mathrm{cos}0-a\mathrm{sin}0}{{s}^{2}+{a}^{2}}
=\frac{s}{{s}^{2}+{a}^{2}}
\frac{dz}{dx}
\frac{dz}{dy}
z=\frac{xy}{{x}^{2}+{y}^{2}}
F\left(s\right)=\frac{s}{{R}^{2}{s}^{2}+16{\pi }^{2}}
R=70
-\left(p-q\right)
q-p
p+q
-p-q
p-q
g\left(t\right)=\left\{\begin{array}{ll}5\mathrm{sin}\left(3\left[t-\frac{\pi }{4}\right]\right)& t>\frac{\pi }{4}\\ 0& t<\frac{\pi }{4}\end{array}
2{y}^{\prime }-3y={e}^{2t},y\left(0\right)=1
y"+y=t,y\left(0\right)=0\text{ }and\text{ }{y}^{\prime }\left(0\right)=2
\left(2{y}^{2}-2xy+3x\right)dx+\left(y+4xy-{x}^{2}\right)dy=0 |
Is \( V_{\text{average}} = \dfrac{V_{\text{final}}+V_{\text{initial}}}{2} \) ? | Brilliant Math & Science Wiki
V_{\text{average}} = \dfrac{V_{\text{final}}+V_{\text{initial}}}{2}
The average velocity of a body over a course of time is the arithmetic mean of the initial and final velocities of the body.
Why some people say it's true: Average velocity means the average value of the initial and final velocities of the body.
Why some people say it's false: Average velocity is equal to the ratio of displacement to the time required for the displacement to occur, and not the average of the initial and final velocities.
\color{#D61F06}{\textbf{false}}
V_{\text{average}}
is not simply the average value of the initial and final velocities. Suppose that a body undergoes a change in displacement
\Delta \vec{r}
\Delta t
, then the average velocity of the body is defined as the constant velocity with which the body moves. It will undergo the same change in displacement in the same time interval:
\vec{V}_{\text{average}} = \dfrac{\Delta \vec{r} }{\Delta t}.
If the body has position vectors
\vec{r_{1}}
\vec{r_{2}}
at time instants
t_{1}
t_{2}
\vec{V}_{\text{average}} = \dfrac{\vec{r_{2}}-\vec{r_{1}}}{t_{2}-t_{1}}.
\vec{V}_{\text{average}}
\Delta \vec{r}
\left| \vec{V}_{\text{average}} \right| = \dfrac{1}{t_{2}-t_{1}} \cdot \left | \vec{r_{2}} - \vec{r_{1}} \right |.
Reply: (Explain why the argument is not vald)
(If relevant, add in an example problem that demonstates understanding of this misconception.)
V_{\text{average}} = \dfrac{V_{\text{final}}+V_{\text{initial}}}{2}
?. Brilliant.org. Retrieved from https://brilliant.org/wiki/is-v_average-dfracv_finalv_initial2/ |
Implement A-law compressor for source coding - Simulink - MathWorks France
A-Law Compressor
Implement A-law compressor for source coding
The A-Law Compressor block implements an A-law compressor for the input signal. The formula for the A-law compressor is
y=\left\{\begin{array}{ll}\frac{A|x|}{1+\mathrm{log}A}\mathrm{sgn}\left(x\right)\hfill & \text{for }0\le |x|\le \frac{V}{A}\hfill \\ \frac{V\left(1+\mathrm{log}\left(A|x|/V\right)\right)}{1+\mathrm{log}A}\mathrm{sgn}\left(x\right)\hfill & \text{for }\frac{V}{A}<|x|\le V\hfill \end{array}
where A is the A-law parameter of the compressor, V is the peak signal magnitude for x, log is the natural logarithm, and sgn is the sign function.
The most commonly used A value is 87.6.
The A-law parameter of the compressor.
The peak value of the input signal. This is also the peak value of the output signal.
A-Law Expander
[1] Sklar, Bernard. Digital Communications: Fundamentals and Applications. Englewood Cliffs, N.J., Prentice-Hall, 1988.
A-Law Expander | Mu-Law Compressor |
Evaluate the limit lim_{xrightarrowinfty}frac{4x^3-2}{3x^4+5x}
\underset{x\to \mathrm{\infty }}{lim}\frac{4{x}^{3}-2}{3{x}^{4}+5x}
\underset{x\to \mathrm{\infty }}{lim}\frac{4{x}^{3}-2}{3{x}^{4}+5x}
The limit.
\underset{x\to \mathrm{\infty }}{lim}\frac{4{x}^{3}-2}{3{x}^{4}+5x}
\underset{x\to \mathrm{\infty }}{lim}\frac{4{x}^{3}-2}{3{x}^{4}+5x}
On multiplying and dividing by
{x}^{4}
\underset{x\to \mathrm{\infty }}{lim}\frac{4{x}^{3}-2}{3{x}^{4}+5x}=\underset{x\to \mathrm{\infty }}{lim}\frac{{x}^{4}\left(\frac{4}{x}-\frac{2}{{x}^{4}}\right)}{{x}^{4}\left(3-\frac{5}{{x}^{3}}\right)}
On cancelling the common term,
\underset{x\to \mathrm{\infty }}{lim}\frac{4{x}^{3}-2}{3{x}^{4}+5x}=\underset{x\to \mathrm{\infty }}{lim}\frac{\left(\frac{4}{x}-\frac{2}{{x}^{4}}\right)}{\left(3-\frac{5}{{x}^{3}}\right)}
Substitute the variable with value,
\underset{x\to \mathrm{\infty }}{lim}\frac{4{x}^{3}-2}{3{x}^{4}+5x}=\underset{x\to \mathrm{\infty }}{lim}\frac{\left(4\left(0\right)-2\left(0\right)\right)}{\left(3-5\left(0\right)\right)}
\underset{x\to \mathrm{\infty }}{lim}\frac{4{x}^{3}-2}{3{x}^{4}+5x}=0
\underset{x\to \mathrm{\infty }}{lim}\left(3\cdot {2}^{1-x}+{x}^{2}\cdot {2}^{1-x}\right)
\underset{\left(x,y\right)\to \left(2,0\right)}{lim}\frac{1-\mathrm{cos}y}{x{y}^{2}}
\underset{x\to a}{lim}\frac{{x}^{2}-\sqrt{{a}^{3}x}}{\sqrt{ax}-a}=12
lim cos 2x
Use Taylor series to evaluate the following limits.
\underset{x\to 4}{lim}\frac{\mathrm{ln}\left(x-3\right)}{{x}^{2}-16}
The set of all points of discontinuity of the function
f\left(x\right)=\underset{n\to \mathrm{\infty }}{lim}{\mathrm{sin}}^{2n}\left(\pi \frac{x}{2}\right)
Show that the limit leads to an indeterminate form. Then carry out the two-step procedure: Transform the function algebraically and evaluate using continuity.
\underset{h\to 3}{lim}\frac{9-{h}^{2}}{h-3} |
Ground State Properties of Closed Shell 4He Nucleus under Compression
Department of Physics, Faculty of Science, Zarqa University, Al-Zarqa, Jordan
Received: July 3, 2017; Accepted: March 5, 2018; Published: March 8, 2018
\tau
\stackrel{^}{H}=\sum _{i=1}^{A}\text{ }\text{ }{\stackrel{^}{T}}_{i}+\sum _{i<j}^{A}\text{ }\text{ }{V}_{ij}
{\stackrel{^}{T}}_{i}
p
{\stackrel{^}{T}}_{i}={p}_{i}^{2}/2m
{{\stackrel{^}{H}}^{\prime }}_{eff}=\sum _{i=1}^{A}\text{ }{\stackrel{^}{T}}_{i}+\sum _{i<j}^{A}{\left({V}_{eff}\right)}_{ij}
{{\stackrel{^}{H}}^{\prime }}_{eff}
{\left({T}_{rel}\right)}_{ij}
{{\stackrel{^}{H}}^{\prime }}_{eff}={T}_{rel}+{V}_{eff}={T}_{rel}+{V}_{eff}^{NN}+{V}_{C}
{\left({T}_{rel}\right)}_{ij}
{\left({T}_{rel}\right)}_{ij}={\left({p}_{i}-{p}_{j}\right)}^{2}/2mA
{V}_{eff}
{V}_{eff}
\tau
{V}_{NN}
G\left(\omega \right)=V+VQ/\left(\omega -{H}_{0}\right)G\left(\omega \right)
\omega
{H}_{0}
{\stackrel{^}{{H}^{\prime }}}_{eff}
\hslash \omega
{r}_{rms}
{E}_{HF}
{\lambda }_{1}
{\lambda }_{2}
{E}_{HF}
{r}_{rms}
{\lambda }_{1},{\lambda }_{2}
\hslash {\omega }^{\prime }
{r}_{rms}
{r}_{rms}
{E}_{HF}
{r}_{rms}
{\rho }_{T}
{\rho }_{T}
{\rho }_{T}
{\rho }_{T}
{\rho }_{T}
{r}_{rms}=1.46\text{\hspace{0.17em}}\text{fm}
{\rho }_{T}
{r}_{rms}=\text{1}.\text{34}\text{\hspace{0.17em}}\text{fm}
{\rho }_{T}
{r}_{rms}=1.24\text{\hspace{0.17em}}\text{fm}
{r}_{rms}=1.24\text{\hspace{0.17em}}\text{fm}
{E}_{HF}
Abu-Sei’leek, M.H.E. (2018) Ground State Properties of Closed Shell 4He Nucleus under Compression. Journal of Applied Mathematics and Physics, 6, 458-467. https://doi.org/10.4236/jamp.2018.63042
1. Stoecker, H. and Sturm, C. (2011) The FAIR Start. Nuclear Physics A, 855, 506-509. https://doi.org/10.1016/j.nuclphysa.2011.02.117
2. Marshall, P., Stocker, W. and Chossy, T. (1999) Compressed Nuclei in Relativistic Thomas-Fermi Approximation. Physical Review C, 60, 064302. https://doi.org/10.1103/PhysRevC.60.064302
3. Ma, Z., Giai, N. and Toki, H. (1997) Compressibility of Nuclear Matter and Breathing Mode of Finite Nuclei in Relativistic Random Phase Approximation. Physical Review C, 55, 2385. https://doi.org/10.1103/PhysRevC.55.2385
4. Negele, J.W. (1970) Structure of Finite Nuclei in the Local-Density Approximation. Physical Review C, 1, 1260. https://doi.org/10.1103/PhysRevC.1.1260
5. Chossy, T.V. and Stocker, W. (2001) Compressed Nuclei in a Schematic Relativistic Mean-Field Description. Physics Letters B, 507, 109-114. https://doi.org/10.1016/S0370-2693(01)00461-0
6. Maessen, P.M., Rijken, T.A. and de Swart, J.J. (1989) Soft-Core Baryon-Baryon One-Boson-Exchange Models. II. Hyperon-Nucleon Potential. Physical Review C, 40, 2226. https://doi.org/10.1103/PhysRevC.40.2226
7. Reid, R.V. (1968) Local Phenomenological Nucleon-Nucleon Potentials. Annals of Physics, 50, 411-448. https://doi.org/10.1016/0003-4916(68)90126-7
8. Goodman, C.D., Austin, S.M., Bloom, S.D., Rapaort, J. and Satchler, G.R. (Eds.) (1980) The (p,n) Reaction and the Nucleon-Nucleon Force. Plenum Press, New York, p. 115.
9. Xing, Y.-Z., Zheng, Y.-M., Pornrad, S., Yan, Y.-P. and Chinorat, K. (2009) Differential Directed Flow of K+ Meson within Covariant Kaon Dynamics. Chinese Physics Letters, 26, 022501. https://doi.org/10.1088/0256-307X/26/2/022501
10. Frick, T., Müther, H., Polls, A. and Ramos, A. (2005) Correlations in Hot Asymmetric Nuclear Matter. Physical Review C, 71, 014313. https://doi.org/10.1103/PhysRevC.71.014313
11. Lenzi, S.M. (2009) Nuclear Structure. Journal of Physics: Conference Series, 168, 012009. https://doi.org/10.1088/1742-6596/168/1/012009
12. Stetcu, I., Barrett, B., Navrátil, P. and Johnson, C. (2004) Electromagnetic Transitions with Effective Operators. International Journal of Modern Physics E, 14, 95-103. https://doi.org/10.1142/S0218301305002813
13. Vary, J., Altramentov, O., Barrett, B., Hasan, M., et al. (2005) Ab Initio No-Core Shell Model—Recent Results and Future Prospects. The European Physical Journal A—Hadrons and Nuclei, 25, 475-480. https://doi.org/10.1140/epjad/i2005-06-214-x
14. Hasan, M., Vary, J. and Navrátil, P. (2004) Hartree-Fock Approximation for the ab initio No-Core Shell Model. Physical Review C, 69, 034332. https://doi.org/10.1103/PhysRevC.69.034332
15. Bozzolo, G. and Vary, J. (1984) Thermal Response of Light Nuclei with a Realistic Effective Hamiltonian. Physical Review Letters, 53, 903. https://doi.org/10.1103/PhysRevLett.53.903
16. Bozzolo, G. and Vary, J. (1985) Thermal Properties of 16O and 40Ca with a Realistic Effective Hamiltonian. Physical Review C, 31, 1909. https://doi.org/10.1103/PhysRevC.31.1909
17. Hasan, M.A., Kohler, S.H. and Vary, J.P. (1987) Excitation of the Δ(3,3) Resonance in Compressed Finite Nuclei from a Constrained Mean-Field Method. Physical Review C, 36, 2649. https://doi.org/10.1103/PhysRevC.36.2649
18. Abu-Sei’leek, M.H. (2011) Resonances-Excitation Calculation Studies Investigation of Δ(3, 3) in Ground State of 90Zr Cold Finite Heavy Nucleus at Equilibrium and Under Large Compression. Communications in Theoretical Physics, 55, 115. https://doi.org/10.1088/0253-6102/55/1/22
19. Abu-Sei’leek, M.H. and Hasan, M.A. (2010) Δ-Resonances in Ground State Properties of 2040Ca Spherical Cold Finite Nucleus at Equilibrium and under Compression. Comm. Communications in Theoretical Physics, 54, 339. https://doi.org/10.1088/0253-6102/54/2/25
20. Abu-Sei’leek, M.H. (2011) Hartree-Fock Calculation Studies Investigation of Δ(3,3) Resonances in the Ground State of Compressed Heavy Spherical Finite Nucleus 132Sn. International Journal of Pure and Applied Physics, 7, 73. https://www.ripublication.com/Volume/ijpapv7n1.htm
21. Abu-Sei’leek, M.H. (2011) Investigation of Δ(3,3) Resonances Effects on the Properties of Neutron-Rich Double Magic Spherical Finite Nucleus, 132Sn, in the Ground State and Under Compression. Pramana, 76, 573-589. https://doi.org/10.1007/s12043-011-0063-x
22. Abu-Sei’leek, M.H. (2011) Delta Excitation Calculation Studies in the Ground State of the Compressed Finite Heavy Doubly-Magic Nucleus 100Sn. Turkish Journal of Physics, 35, 273. http://journals.tubitak.gov.tr/physics/issues/fiz-11-35-3/fiz-35-3-5-1004-34.pdf
23. Abu-Sei’leek, M.H. (2010) Delta Excitation in Compressed Neutron-Rich Double Magic Spherical Finite Nucleus132Sn. Nuclear Physics Review, 27, 399-410. https://doi.org/10.11804/NuclPhysRev.27.04.399
24. Abu-Sei’leek, M.H. (2011) Delta Excitation Calculation Studies in Compressed Finite Spherical Nucleus 40Ca. Nuclear Physics Review, 28, 416-422. https://doi.org/10.11804/NuclPhysRev.28.04.416
25. Abu-Sei’leek, M.H. (2011) Doubly-Magic 100Sn Nucleus with Delta Excitation under Compression. Journal of the Physical Society of Japan, 80, 104201. https://doi.org/10.1143/JPSJ.80.104201
26. Abu-Sei’leek, M.H. (2014) Neutron-Rich 208Pb Nucleus with Delta Excitation under Compression. Turkish Journal of Physics, 38, 253-260. https://doi.org/10.3906/fiz-1402-4
27. Abu-Sei’leek, M.H. (2016) Delta Excitation in the Compressed Finite Nucleus 90Zr. Journal of Applied Mathematics and Physics, 4, 586-593. https://doi.org/10.4236/jamp.2016.43064
28. Tilley, D.R., Walle, H.R. and Hale, G.M. (1992) Energy Levels of Light Nuclei A = 4. Nuclear Physics A, 541, 1-104. https://doi.org/10.1016/0375-9474(92)90635-W |
Algebraic_code-excited_linear_prediction Knowpia
Algebraic code-excited linear prediction (ACELP) is a patented[1] speech coding algorithm by VoiceAge Corporation in which a limited set of pulses is distributed as excitation to a linear prediction filter. It is a linear predictive coding (LPC) algorithm that is based on the code-excited linear prediction (CELP) method and has an algebraic structure.
The ACELP method is widely employed in current speech coding standards such as AMR, EFR, AMR-WB (G.722.2), VMR-WB, EVRC, EVRC-B, SMV, TETRA, PCS 1900, MPEG-4 CELP and ITU-T G-series standards G.729, G.729.1 (first coding stage) and G.723.1.[2][3][4][5] The ACELP algorithm is also used in the proprietary ACELP.net codec.[6]
ACELP is a patented technology and registered trademark of VoiceAge Corporation[7] in Canada and/or other countries and was developed in 1989 by the researchers at the Université de Sherbrooke in Canada.[8]{{{{C}}}}
The main advantage of ACELP is that the algebraic codebook it uses can be made very large (> 50 bits) without running into storage (RAM/ROM) or complexity (CPU time) problems.
The ACELP algorithm is based on that used in code-excited linear prediction (CELP), but ACELP codebooks have a specific algebraic structure imposed upon them.
A 16-bit algebraic codebook shall be used in the innovative codebook search, the aim of which is to find the best innovation and gain parameters. The innovation vector contains, at most, four non-zero pulses.
In ACELP, a block of N speech samples is synthesized by filtering an appropriate innovation sequence from a codebook, scaled by a gain factor g c, through two time-varying filters.
The long-term (pitch) synthesis filter is given by:
{\displaystyle {\frac {1}{B(z)}}={\frac {1}{1-g_{p}z^{-T}}}}
The short-term synthesis filter is given by:
{\displaystyle {\frac {1}{A(z)}}={\frac {1}{1+\sum _{i=1}^{P}a_{i}z^{-i}}}}
Voiceage have kept very tight control of the product. Audible Inc. use a modified version for their speaking books. It is also licensed conference-calling software, speech compression toys and has become one of the 3GPP formats. With the patent ending on 9 February 2018, designers of narrow-band speech (such as emergency services) have the option of ACELP which the customer can optionally pay for now or for standard usage after the patent expires.
^ US patent 5717825, "Algebraic code-excited linear prediction speech coding method", issued 10 February 1998
^ ACELP map, VoiceAge Corporation, Archive.org
^ VoiceAge Corporation - related standards
^ VoiceAge Corporation (13 October 2007). "Codec Technologies". Archived from the original on 13 October 2007. Retrieved 20 September 2009.
^ VoiceAge Corporation. "Codec Technologies". VoiceAge Corporation. Archived from the original on 18 October 2009. Retrieved 20 September 2009.
^ VoiceAge Corporation. "ACELP.net — Beyond the Standards". Archived from the original on 14 October 2007. Retrieved 3 January 2010.
^ Trademarks
^ Transfer of technology |
Evaluating the Average Power Delivered by a Wind Turbine - MATLAB & Simulink Example
{\mathit{P}}_{\mathit{w}}=\frac{\rho {\text{\hspace{0.17em}}\mathit{A}\text{\hspace{0.17em}}\mathit{u}}^{3}}{2}
{\mathit{m}}^{2}
\mathrm{kg}/{m}^{3}
\mathit{m}/\mathit{s}
{\mathit{P}}_{\mathit{e}}=\frac{{\mathit{C}}_{\mathrm{tot}}\text{\hspace{0.17em}\hspace{0.17em}}\rho {\text{\hspace{0.17em}}\mathrm{Au}}^{3}}{2}
{\mathit{C}}_{\mathrm{tot}}=\mathrm{overall}\text{\hspace{0.17em}}\mathrm{efficiency}={\mathit{C}}_{\mathit{p}}{\mathit{C}}_{\mathit{t}}{\mathit{C}}_{\mathit{g}}
{\mathit{P}}_{\mathrm{er}}
{\mathit{C}}_{\mathrm{totR}}
{\mathit{P}}_{\mathrm{er}}=\frac{{\mathit{C}}_{\mathrm{totR}}\text{\hspace{0.17em}\hspace{0.17em}}\rho {\text{\hspace{0.17em}}\mathrm{Au}}^{3}}{2}
{\mathit{u}}_{\mathit{r}}
{\mathit{u}}_{\mathit{c}}
{\mathit{u}}_{\mathit{f}}
{u}_{c}
{u}_{r}
{\mathit{u}}_{\mathit{r}}
{\mathit{u}}_{\mathit{f}}
\left\{\begin{array}{cl}0& \text{ if }u<{u}_{c}\\ {C}_{1}+{C}_{2} {u}^{k}& \text{ if }{u}_{c}\le u\wedge u\le {u}_{r}\\ \mathrm{Per}& \text{ if }u\le {u}_{f}\wedge {u}_{r}\le u\\ 0& \text{ if }{u}_{f}<u\end{array}
{\mathit{C}}_{1}
{\mathit{C}}_{2}
\frac{\mathrm{Per} {{u}_{c}}^{k}}{{{u}_{c}}^{k}-{{u}_{r}}^{k}}
-\frac{\mathrm{Per}}{{{u}_{c}}^{k}-{{u}_{r}}^{k}}
f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)=\frac{\left(\frac{b}{a}\right)\phantom{\rule{0.16666666666666666em}{0ex}}{\left(\frac{u}{a}\right)}^{b-1}}{{\mathrm{e}}^{{\left(\frac{u}{a}\right)}^{b}}}
P{e}_{average}={\int }_{0}^{\infty }Pe\left(u\right)\phantom{\rule{0.16666666666666666em}{0ex}}f\left(u\right)du
{u}_{c}
{\mathit{u}}_{\mathit{f}}
P{e}_{average}={C}_{1}\phantom{\rule{0.16666666666666666em}{0ex}}\left({\int }_{{u}_{c}}^{{u}_{r}}f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)\phantom{\rule{0.16666666666666666em}{0ex}}du\right)+{C}_{2}\phantom{\rule{0.16666666666666666em}{0ex}}\left({\int }_{{u}_{c}}^{{u}_{r}}{u}^{b}f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)du\right)+\mathrm{P}\mathrm{e}\mathrm{r}\phantom{\rule{0.16666666666666666em}{0ex}}\left({\int }_{{u}_{r}}^{{u}_{f}}f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)\phantom{\rule{0.16666666666666666em}{0ex}}du\right)
x={\left(\frac{u}{a}\right)}^{b}
\mathrm{d}\mathrm{x}=\left(\frac{b}{a}\right)\phantom{\rule{0.16666666666666666em}{0ex}}{\left(\frac{u}{a}\right)}^{b-1}\phantom{\rule{0.16666666666666666em}{0ex}}\mathrm{d}\mathrm{u}
\int f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)\phantom{\rule{0.16666666666666666em}{0ex}}du=\int \frac{1}{{\mathrm{e}}^{x}}\phantom{\rule{0.16666666666666666em}{0ex}}dx
\int {u}^{b}\phantom{\rule{0.16666666666666666em}{0ex}}f\phantom{\rule{-0.16666666666666666em}{0ex}}\left(u\right)\phantom{\rule{0.16666666666666666em}{0ex}}du={a}^{b}\phantom{\rule{0.16666666666666666em}{0ex}}\left(\int \frac{x}{{\mathrm{e}}^{x}}\phantom{\rule{0.16666666666666666em}{0ex}}dx\right)
{\left(\frac{u}{a}\right)}^{b}
-{\mathrm{e}}^{-{\left(\frac{u}{a}\right)}^{b}}
-{a}^{b} {\mathrm{e}}^{-{\left(\frac{u}{a}\right)}^{b}} \left({\left(\frac{u}{a}\right)}^{b}+1\right)
\begin{array}{l}\mathrm{Per} {\sigma }_{2}-\mathrm{Per} {\mathrm{e}}^{-{\left(\frac{{u}_{f}}{a}\right)}^{b}}+\frac{\mathrm{Per} {{u}_{c}}^{k} {\mathrm{e}}^{-{\left(\frac{{u}_{c}}{a}\right)}^{b}}}{{\sigma }_{1}}-\frac{\mathrm{Per} {{u}_{c}}^{k} {\sigma }_{2}}{{\sigma }_{1}}-\frac{\mathrm{Per} {a}^{b} {\mathrm{e}}^{-{\left(\frac{{u}_{c}}{a}\right)}^{b}} \left({\left(\frac{{u}_{c}}{a}\right)}^{b}+1\right)}{{\sigma }_{1}}+\frac{\mathrm{Per} {a}^{b} {\sigma }_{2} \left({\left(\frac{{u}_{r}}{a}\right)}^{b}+1\right)}{{\sigma }_{1}}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}={{u}_{c}}^{k}-{{u}_{r}}^{k}\\ \\ \mathrm{ }{\sigma }_{2}={\mathrm{e}}^{-{\left(\frac{{u}_{r}}{a}\right)}^{b}}\end{array} |
Raiments of the Eye - OSRS Wiki
A player wearing the Raiments of the Eye.
Raiments of the Eye are a set of robes that grants 10% more runes when runecrafting per piece, with a 20% set bonus for a total of 60% when the full outfit is worn. The hat can also be attuned to an imbued tiara to allow it to function as one. Only one tiara can be attuned at a time, and attuning does not consume the tiara.
Pieces of the outfit are purchased from the Guardians of the Rift reward shop for a total of 1350 Abyssal pearls. This equates to roughly 180 games of Guardians of the Rift and 630 searches.[1] The hat, robe top, and robe bottoms can also be recoloured with various Abyssal dyes that are obtained as a rare reward from the Rewards Guardian.
The entire set, and its coloured variants, can be stored in a magic wardrobe in a player-owned house costume room.
1 Extra rune mechanics
1.1 Impact on Achievement Diary tasks
Extra rune mechanics[edit | edit source]
With the full outfit, the player is guaranteed at least 1.6x the runes they would normally craft with the essence they have. If that quantity does not divide into 6 easily, there is a chance to get a bonus 6 runes depending on the remainder.[2] For example, if the player could normally craft 28 runes, 60% additional runes would be a total of 44.8 runes. The player would receive at least 45 runes, with an unknown chance to receive 50 runes.
Impact on Achievement Diary tasks[edit | edit source]
With this information, a formula can be developed to determine what rune multiplier is required for certain Achievement Diary tasks while wearing the full outfit. Let X represent the number of runes necessary for the task, and Y the effective number of runes you need to make:[3]
{\displaystyle Y={\bigg \lceil }{\frac {X}{1.6}}{\bigg \rceil }}
{\displaystyle Rune_{Mult}={\bigg \lceil }{\frac {Y}{28}}{\bigg \rceil }}
This implies that tasks to craft double the quantity of runes (e.g. Cosmic, Nature, Astral) cannot be bypassed using the outfit. The effective number of runes would be 35, and the rune multiplier would be 1.25, rounding up to 2. The tasks which do benefit are listed below:
Level normally
With raiments
Hard Falador Craft 140 mind runes. 56 42
Elite Lumbridge & Draynor Craft 140 water runes. 76 57
Elite Varrock Craft 100 earth runes. 78 52
Elite Falador Craft 252 air runes. 88 55
The outfit does not reduce the Runecraft level required for the diary cape, which is 91 for the Elite Karamja Diary.
Hat of the eye 400
Robe top of the eye 350
Robe bottoms of the eye 350
Boots of the eye 250 N/A
Abyssal dye N/A N/A
A player wearing the Raiments' default colors.
A player wearing red Raiments.
A player wearing green Raiments.
A player wearing blue Raiments.
Concept art for the Raiments of the Eye, by Mod Jerv.
Concept art for coloured versions of the Raiments, by Mod Jerv.
^ With Abyssal pearls dropping an average of 15 at a time at a 1/7 drop rate, it would take an average of 630 searches of the Rewards Guardian to obtain enough for one complete set.
^ Jagex. Mod Husky's Twitter account. 23 March 2022. (Archived from the original on 23 March 2022.) Mod Husky: "That's not how it works either, It'll always be at least 1.6x runes and if that doesn't divide into 6 easily you have a chance to get bonus 6 depending on the remainder"
^ Jagex. Mod Husky's Twitter account. 23 March 2022. (Archived from the original on 23 March 2022.) Mod Husky: "That would be correct yes"
Retrieved from ‘https://oldschool.runescape.wiki/w/Raiments_of_the_Eye?oldid=14289143’ |
Consider the following sets of rational functions. af(x)=\frac{a}{x} for a={−2,−1,−0.5,0.5,2,4}
Consider the following sets of rational functions.
af\left(x\right)=\frac{a}{x}
for a={−2,−1,−0.5,0.5,2,4},
h\left(x-c\right)=\frac{1}{x-c}
for c=[−4,−2,−0.5,0.5,2,4], h(x−c)=1x−c for c=[−4,−2,−0.5,0.5,2,4], g(bx)=1bx for b={−2,−1,−0.5,0.5,2,4} for b={−2,−1,−0.5,0.5,2,4},
k\left(x\right)+d=\frac{1}{x}+d
for d={−4,−2,−0.5,0.5,2,4} for d={−4,−2,−0.5,0.5,2,4}.
c. Choose two functions from any set. Find the slope between consecutive points on the graphs.
For the first set slopes are 0.5 and 1, for the second set slopes are -0.5 and 0.5 for the third set slopes are
\frac{1}{3}
and -\frac{1}{3} and for the fourth set slopes are 1 and -1.
In general, how does one determine if a rational function is regular? I have the particular problem of determining in which points of the circle
V\left({x}^{2}+{y}^{2}-1\right)\subseteq {A}^{2}
is the rational function
\alpha =\frac{y-1}{x}
regular?
"A rational function is defined as the quotient of polynomials in which the denominator has a degree of at least 1"
If we are talking merely about x, then I get the concept. A rational function f(x) could be written as "
\frac{p\left(x\right)}{q\left(x\right)}
q\left(x\right)\ne 0
The issue that I'm having is that of talking about rational functions of n variables. For instance, what would be the meaning of 'f(x,y) is a rational function of
x\text{ }\mathrm{&}\text{ }{y}^{\prime }
H\left(x\right)=\frac{B\left(x\right)}{A\left(x\right)}
{H}^{\prime }\left(x\right)=\frac{\left({B}^{\prime }\left(x\right)\star A\left(x\right)-B\left(x\right)\star {A}^{\prime }\left(x\right)\right)}{\left(A\left(x\right){\right)}^{2}}
k
B\left(x\right)
A\left(x\right)
r
r\ge 2
{H}^{\prime }\left(x\right)
{B}^{\prime }\left(x\right)\star A\left(x\right)-B\left(x\right)\star {A}^{\prime }\left(x\right)
r-1
Do all rational functions have vertical asymptotes? Why or why not? If not, give an example of a rational function that does not have a vertical asymptote.
Find all the points of discontinuity of the function
\frac{{y}^{2}-18y+80}{\left(y-4\right)\left(y-1\right)}
\int \frac{1-\sqrt{x}}{1+\sqrt{x}}dx |
mrpandey's Math Blog ·
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mrpandey's Math Blog
SPOJ Problem: Adjacent Bit Counts
30 Apr 2018 ♦ combinatorics, number-theory
I was trying to solve this problem on SPOJ. It’s a dynamic-programming problem. I tried to find some recursive relation. I spent hours. But no breakthrough.
I even tried looking for hints in comments and I came to know it’s a 3D dynamic-programming problem. Some of the users were even able to reduce it to a 2D technique.
Then I realized I need to rethink the entire approach. So somehow, I started treating it like a combinatorics problem and I landed on a solution — a formula that guarantees linear time execution. I love moments like these when you solve an interesting problem with an uncommon approach. And that’s why I am writing this post.
Beggar's Method
Find the number of integer solutions i.e. ordered pairs
\left(x_1, x_2, \ldots , x_r \right) |
Functional derivative (variational derivative) - MATLAB functionalDerivative - MathWorks France
\frac{\delta S}{\delta y}\left(x\right)
S\left[y\right]={\int }_{a}^{b}f\left[x,y\left(x\right),y\text{'}\left(x\right),...\right]\text{\hspace{0.17em}}dx
S\left[y\right]={\int }_{b}^{a}y\left(x\right)\mathrm{sin}\left(y\left(x\right)\right)\phantom{\rule{0.2222222222222222em}{0ex}}dx
y
f\left[y\left(x\right)\right]=y\left(x\right)\phantom{\rule{0.16666666666666666em}{0ex}}\mathrm{sin}\left(y\left(x\right)\right)
S
\mathrm{sin}\left(y\left(x\right)\right)+\mathrm{cos}\left(y\left(x\right)\right) y\left(x\right)
S\left[u,v\right]={\int }_{b}^{a}\left({u}^{2}\left(x\right)\frac{dv\left(x\right)}{dx}+v\left(x\right)\frac{{d}^{2}u\left(x\right)}{d{x}^{2}}\right)\phantom{\rule{0.2222222222222222em}{0ex}}dx
u
v
f\left[u\left(x\right),v\left(x\right),{u}^{\prime \prime }\left(x\right),{v}^{\prime }\left(x\right)\right]={u}^{2}\frac{dv}{dx}+v\frac{{d}^{2}u}{d{x}^{2}}
S
\left(\begin{array}{c}\frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }v\left(x\right)+2 u\left(x\right) \frac{\partial }{\partial x}\mathrm{ }v\left(x\right)\\ \frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }u\left(x\right)-2 u\left(x\right) \frac{\partial }{\partial x}\mathrm{ }u\left(x\right)\end{array}\right)
\frac{m {\left(\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)\right)}^{2}}{2}-\frac{k {x\left(t\right)}^{2}}{2}
S\left[x\right]={\int }_{{t}_{1}}^{{t}_{2}}L\left[t,x\left(t\right),\underset{}{\overset{˙}{x}}\left(t\right)\right]\phantom{\rule{0.16666666666666666em}{0ex}}dt
S\left[x\left(t\right)\right]
-m \frac{{\partial }^{2}}{\partial {t}^{2}}\mathrm{ }x\left(t\right)-k x\left(t\right)=0
x\left(0\right)=10
\underset{}{\overset{˙}{x}}\left(0\right)=0
10 \mathrm{cos}\left(\frac{\sqrt{k} t}{\sqrt{m}}\right)
y\left(x\right)
a
b
g
t={\int }_{a}^{b}\sqrt{\frac{1+{{y}^{\prime }}^{2}}{2gy}}\phantom{\rule{0.16666666666666666em}{0ex}}dx.
t
y
\frac{\delta t}{\delta y}\left(x\right)=0
2 y\left(x\right) \frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }y\left(x\right)+{\left(\frac{\partial }{\partial x}\mathrm{ }y\left(x\right)\right)}^{2}=-1
F\left(y\left(x\right)\right)=g\left(x\right)
\begin{array}{l}\left(\begin{array}{c}y\left(x\right)={C}_{2}-x \mathrm{i}\\ y\left(x\right)={C}_{3}+x \mathrm{i}\\ {\sigma }_{1}={C}_{4}+x\\ {\sigma }_{1}={C}_{5}-x\\ \frac{{C}_{1}+y\left(x\right)}{y\left(x\right)}=0\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}={C}_{1} \mathrm{atan}\left(\sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}\right)-y\left(x\right) \sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}\end{array}
y\left(x\right)
{C}_{1} \mathrm{atan}\left(\sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}\right)-y\left(x\right) \sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}={C}_{4}+x
{C}_{1} \mathrm{atan}\left(\sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}\right)-y\left(x\right) \sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}={C}_{5}-x
y
{C}_{1}+y\left(x\right)=0
y\left(0\right)=5
y\left(4\right)=1
{C}_{1}
{C}_{5}
-6.4199192418473511250705556729108 \mathrm{atan}\left(\sqrt{\frac{6.4199192418473511250705556729108}{y\left(x\right)}-1}\right)-y\left(x\right) \sqrt{\frac{6.4199192418473511250705556729108}{y\left(x\right)}-1}=-x-5.8078336827583088482183433150164
x
y\left(x\right)
x
y
y\left(x\right)
y
y\left(x\right)
x
0<x<4
1<y<5
u\left(x,y\right)
F\left[u\right]={\int }_{{y}_{1}}^{{y}_{2}}{\int }_{{x}_{1}}^{{x}_{2}}f\left[x,y\left(x\right),u\left(x,y\right),{u}_{x},{u}_{y}\right]\phantom{\rule{0.2777777777777778em}{0ex}}dx\phantom{\rule{0.2777777777777778em}{0ex}}dy={\int }_{{y}_{1}}^{{y}_{2}}{\int }_{{x}_{1}}^{{x}_{2}}\sqrt{1+{u}_{x}^{2}+{u}_{y}^{2}}\phantom{\rule{0.2777777777777778em}{0ex}}dx\phantom{\rule{0.2777777777777778em}{0ex}}dy
{u}_{x}
{u}_{y}
u
x
y
\begin{array}{l}-\frac{{\left(\frac{\partial }{\partial y}\mathrm{ }u\left(x,y\right)\right)}^{2} \frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }u\left(x,y\right)+\frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }u\left(x,y\right)+{{\sigma }_{1}}^{2} \frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ }u\left(x,y\right)-2 \frac{\partial }{\partial y}\mathrm{ }{\sigma }_{1} \frac{\partial }{\partial y}\mathrm{ }u\left(x,y\right) {\sigma }_{1}+\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ }u\left(x,y\right)}{{\left({{\sigma }_{1}}^{2}+{\left(\frac{\partial }{\partial y}\mathrm{ }u\left(x,y\right)\right)}^{2}+1\right)}^{3/2}}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}=\frac{\partial }{\partial x}\mathrm{ }u\left(x,y\right)\end{array}
S\left[y\right]={\int }_{a}^{b}f\left[x,y\left(x\right),y\text{'}\left(x\right),...,{y}^{\left(n\right)}\left(x\right)\right]\text{\hspace{0.17em}}dx,
\delta y\left(x\right)=\epsilon \varphi \left(x\right)
DS\left[y\right]=\underset{\epsilon \to 0}{\mathrm{lim}}\frac{S\left[y+\epsilon \varphi \right]-S\left[y\right]}{\epsilon }={\int }_{a}^{b}\frac{\delta S}{\delta y}\left(x\right)\varphi \left(x\right)dx\text{.}
\frac{\delta S}{\delta y}\left(x\right)
\begin{array}{lll}\frac{\delta S}{\delta y}\left(x\right)\hfill & =\hfill & \frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial {y}^{\text{'}}}+\frac{{d}^{2}}{d{x}^{2}}\frac{\partial f}{\partial {y}^{\text{'}\text{'}}}-...+{\left(-1\right)}^{n}\frac{{d}^{n}}{d{x}^{n}}\left(\frac{\partial f}{\partial {y}^{\left(n\right)}}\right)\hfill \\ \hfill & =\hfill & \sum _{i=0}^{n}{\left(-1\right)}^{i}\frac{{d}^{i}}{d{x}^{i}}\left(\frac{\partial f}{\partial {y}^{\left(i\right)}}\right).\hfill \end{array} |
Pulse_repetition_frequency Knowpia
The pulse repetition frequency (PRF) is the number of pulses of a repeating signal in a specific time unit, normally measured in pulses per second. The term is used within a number of technical disciplines, notably radar.
In radar, a radio signal of a particular carrier frequency is turned on and off; the term "frequency" refers to the carrier, while the PRF refers to the number of switches. Both are measured in terms of cycle per second, or hertz. The PRF is normally much lower than the frequency. For instance, a typical World War II radar like the Type 7 GCI radar had a basic carrier frequency of 209 MHz (209 million cycles per second) and a PRF of 300 or 500 pulses per second. A related measure is the pulse width, the amount of time the transmitter is turned on during each pulse.
The PRF is one of the defining characteristics of a radar system, which normally consists of a powerful transmitter and sensitive receiver connected to the same antenna. After producing a brief pulse of radio signal, the transmitter is turned off in order for the receiver units to hear the reflections of that signal off distant targets. Since the radio signal has to travel out to the target and back again, the required inter-pulse quiet period is a function of the radar's desired range. Longer periods are required for longer range signals, requiring lower PRFs. Conversely, higher PRFs produce shorter maximum ranges, but broadcast more pulses, and thus radio energy, in a given time. This creates stronger reflections that make detection easier. Radar systems must balance these two competing requirements.
Using older electronics, PRFs were generally fixed to a specific value, or might be switched among a limited set of possible values. This gives each radar system a characteristic PRF, which can be used in electronic warfare to identify the type or class of a particular platform such as a ship or aircraft, or in some cases, a particular unit. Radar warning receivers in aircraft include a library of common PRFs which can identify not only the type of radar, but in some cases the mode of operation. This allowed pilots to be warned when an SA-2 SAM battery had "locked on", for instance. Modern radar systems are generally able to smoothly change their PRF, pulse width and carrier frequency, making identification much more difficult.
Sonar and lidar systems also have PRFs, as does any pulsed system. In the case of sonar, the term pulse repetition rate (PRR) is more common, although it refers to the same concept.
Electromagnetic (e.g. radio or light) waves are conceptually pure single frequency phenomena while pulses may be mathematically thought of as composed of a number of pure frequencies that sum and nullify in interactions that create a pulse train of the specific amplitudes, PRRs, base frequencies, phase characteristics, et cetera (See Fourier Analysis). The first term (PRF) is more common in device technical literature (Electrical Engineering and some sciences), and the latter (PRR) more commonly used in military-aerospace terminology (especially United States armed forces terminologies) and equipment specifications such as training and technical manuals for radar and sonar systems.
The reciprocal of PRF (or PRR) is called the pulse repetition time (PRT), pulse repetition interval (PRI), or inter-pulse period (IPP), which is the elapsed time from the beginning of one pulse to the beginning of the next pulse. The IPP term is normally used when referring to the quantity of PRT periods to be processed digitally. Each PRT having a fixed number of range gates, but not all of them being used. For example, the APY-1 radar used 128 IPP's with a fixed 50 range gates, producing 128 Doppler filters using an FFT. The different number of range gates on each of the five PRF's all being less than 50.
Within radar technology PRF is important since it determines the maximum target range (Rmax) and maximum Doppler velocity (Vmax) that can be accurately determined by the radar.[1] Conversely, a high PRR/PRF can enhance target discrimination of nearer objects, such as a periscope or fast moving missile. This leads to use of low PRRs for search radar, and very high PRFs for fire control radars. Many dual-purpose and navigation radars—especially naval designs with variable PRRs—allow a skilled operator to adjust PRR to enhance and clarify the radar picture—for example in bad sea states where wave action generates false returns, and in general for less clutter, or perhaps a better return signal off a prominent landscape feature (e.g., a cliff).
Pulse repetition frequency (PRF) is the number of times a pulsed activity occurs every second.
This is similar to cycle per second used to describe other types of waveforms.
PRF is inversely proportional to time period
{\displaystyle \mathrm {T} }
which is the property of a pulsed wave.
{\displaystyle \mathrm {T} ={\frac {1}{\text{PRF}}}}
PRF is usually associated with pulse spacing, which is the distance that the pulse travels before the next pulse occurs.
{\displaystyle {\text{Pulse Spacing}}={\frac {\text{Propagation Speed}}{\text{PRF}}}}
PRF is crucial to perform measurements for certain physics phenomenon.
For example, a tachometer may use a strobe light with an adjustable PRF to measure rotational velocity. The PRF for the strobe light is adjusted upward from a low value until the rotating object appears to stand still. The PRF of the tachometer would then match the speed of the rotating object.
Other types of measurements involve distance using the delay time for reflected echo pulses from light, microwaves, and sound transmissions.
PRF is crucial for systems and devices that measure distance.
Different PRF allow systems to perform very different functions.
A radar system uses a radio frequency electromagnetic signal reflected from a target to determine information about that target.
PRF is required for radar operation. This is the rate at which transmitter pulses are sent into air or space.
Range ambiguityEdit
A real target in 100 km or a second-sweep echo in a distance of 400 km
A radar system determines range through the time delay between pulse transmission and reception by the relation:
{\displaystyle {\text{Range}}={\frac {c\tau }{2}}}
For accurate range determination a pulse must be transmitted and reflected before the next pulse is transmitted. This gives rise to the maximum unambiguous range limit:
{\displaystyle {\text{Max Range}}={\frac {c\tau _{\text{PRT}}}{2}}={\frac {c}{2\,{\text{PRF}}}}\qquad {\begin{cases}\tau _{\text{PRT}}={\frac {1}{\text{PRF}}}\end{cases}}}
The maximum range also defines a range ambiguity for all detected targets. Because of the periodic nature of pulsed radar systems, it is impossible for some radar system to determine the difference between targets separated by integer multiples of the maximum range using a single PRF. More sophisticated radar systems avoid this problem through the use of multiple PRFs either simultaneously on different frequencies or on a single frequency with a changing PRT.
The range ambiguity resolution process is used to identify true range when PRF is above this limit.
Low PRFEdit
Systems using PRF below 3 kHz are considered low PRF because direct range can be measured to a distance of at least 50 km. Radar systems using low PRF typically produce unambiguous range.
Unambiguous Doppler processing becomes an increasing challenge due to coherency limitations as PRF falls below 3 kHz.
For example, an L-Band radar with 500 Hz pulse rate produces ambiguous velocity above 75 m/s (170 mile/hour), while detecting true range up to 300 km. This combination is appropriate for civilian aircraft radar and weather radar.
{\displaystyle {\text{300 km range}}={\frac {C}{2\times 500}}}
{\displaystyle {\text{75 m/s velocity}}={\frac {500\times C}{2\times 10^{9}}}}
Low PRF radar have reduced sensitivity in the presence of low-velocity clutter that interfere with aircraft detection near terrain. Moving target indicator is generally required for acceptable performance near terrain, but this introduces radar scalloping issues that complicate the receiver. Low PRF radar intended for aircraft and spacecraft detection are heavily degraded by weather phenomenon, which cannot be compensated using moving target indicator.
Medium PRFEdit
Range and velocity can both be identified using medium PRF, but neither one can be identified directly. Medium PRF is from 3 kHz to 30 kHz, which corresponds with radar range from 5 km to 50 km. This is the ambiguous range, which is much smaller than the maximum range. Range ambiguity resolution is used to determine true range in medium PRF radar.
Medium PRF is used with Pulse-Doppler radar, which is required for look-down/shoot-down capability in military systems. Doppler radar return is generally not ambiguous until velocity exceeds the speed of sound.
A technique called ambiguity resolution is required to identify true range and speed. Doppler signals fall between 1.5 kHz, and 15 kHz, which is audible, so audio signals from medium-PRF radar systems can be used for passive target classification.
For example, an L band radar system using a PRF of 10 kHz with a duty cycle of 3.3% can identify true range to a distance of 450 km (30 * C / 10,000 km/s). This is the instrumented range. Unambiguous velocity is 1,500 m/s (3,300 mile/hour).
{\displaystyle {\text{450 km}}={\frac {C}{0.033\times 2\times 10,000}}}
{\displaystyle {\text{1,500 m/s}}={\frac {10,000\times C}{2\times 10^{9}}}}
The unambiguous velocity of an L-Band radar using a PRF of 10 kHz would be 1,500 m/s (3,300 mile/hour) (10,000 x C / (2 x 10^9)). True velocity can be found for objects moving under 45,000 m/s if the band pass filter admits the signal (1,500/0.033).
Medium PRF has unique radar scalloping issues that require redundant detection schemes.
High PRFEdit
Systems using PRF above 30 kHz function better known as interrupted continuous-wave (ICW) radar because direct velocity can be measured up to 4.5 km/s at L band, but range resolution becomes more difficult.
High PRF is limited to systems that require close-in performance, like proximity fuses and law enforcement radar.
For example, if 30 samples are taken during the quiescent phase between transmit pulses using a 30 kHz PRF, then true range can be determined to a maximum of 150 km using 1 microsecond samples (30 x C / 30,000 km/s). Reflectors beyond this range might be detectable, but the true range cannot be identified.
{\displaystyle {\text{150 km}}={\frac {30\times C}{2\times 30,000}}}
{\displaystyle {\text{4,500 m/s}}={\frac {30,000\times C}{2\times 10^{9}}}}
It becomes increasingly difficult to take multiple samples between transmit pulses at these pulse frequencies, so range measurements are limited to short distances.[2]
SonarEdit
Sonar systems operate much like radar, except that the medium is liquid or air, and the frequency of the signal is either audio or ultra-sonic. Like radar, lower frequencies propagate relatively higher energies longer distances with less resolving ability. Higher frequencies, which damp out faster, provide increased resolution of nearby objects.
Signals propagate at the speed of sound in the medium (almost always water), and maximum PRF depends upon the size of the object being examined. For example, the speed of sound in water is 1,497 m/s, and the human body is about 0.5 m thick, so the PRF for ultrasound images of the human body should be less than about 2 kHz (1,497/0.5).
As another example, ocean depth is approximately 2 km, so sound takes over a second to return from the sea floor. Sonar is a very slow technology with very low PRF for this reason.
Light waves can be used as radar frequencies, in which case the system is known as lidar. This is short for "LIght Detection And Ranging," similar to the original meaning of the initialism "RADAR," which was RAdio Detection And Ranging. Both have since become commonly-used english words, and are therefore acronyms rather than initialisms.
Laser range or other light signal frequency range finders operate just like radar at much higher frequencies. Non-laser light detection is utilized extensively in automated machine control systems (e.g. electric eyes controlling a garage door, conveyor sorting gates, etc.), and those that use pulse rate detection and ranging are at heart, the same type of system as a radar—without the bells and whistles of the human interface.
Unlike lower radio signal frequencies, light does not bend around the curve of the earth or reflect off the ionosphere like C-band search radar signals, and so lidar is useful only in line of sight applications like higher frequency radar systems.
^ "Pulse Repetition Frequency". Radartutorial.
^ "Continuous Wave Radar". Retrieved January 29, 2011. [permanent dead link] |
A wildlife biologist examines frogs for a genetic trait he
A wildlife biologist examines frogs for a genetic trait he suspects may be linked to sensitivity to industrial toxins in the environment. Previous research had established that this trait is usually found in 1 of every 8 frogs. He collects and examines a dozen frogs.
A wildlife biologist examines frogs for a genetic trait he suspects may be linked to sensitivity to industrial toxins in the environment.
Previous research had established that this trait is usually found in 1 of every 8 frogs. He collects and examines a dozen frogs.
If the frequency of the trait has not changed, what’s the probability he finds the trait in
a) none of the 12 frogs?
b) at least 2 frogs?
c) 3 or 4 frogs?
d) no more than 4 frogs?
1. binompdf(12, .125, 0) = .201
2. binomcdf(12, .875, 10) = .453
3. binompdf(12, .125, 3) + binompdf(12, . 125, 4) = .171
4. binomcdf(12, .125, 4) = .989 Result:A. .201
If X is a binomial random variable, for what value of
\theta
is the probability b(x; n,
\theta
) a maximum?
In a 10-question true/false test, what is the probability of guessing correctly on questions 1 through 4 exactly 2 times?
Compute the probability of X successes, using Table B in appendix A.
n=12,p=0.90,x=2
I believe they are using the the binomial probability formula and im not sure how to start if you can give a step by step anwser explanation that would be helpful.
n=20
p=0.2
P\left(X\le 12\right)
In Exercises, X denotes a binomial random variable with parameters n and p. For each exercise, indicate which area under the appropriate normal curve would be determined to approximate the specified binomial probability.
P\left(4<X<8\right) |
Find the volume of the cone shown in problem 10-142.
From problem 10-142, you learned that the volume of a cone is one-third the volume of a cylinder with the same base area and height.
The equation for the volume of a cylinder is:
\text{Volume = (area of base)(height)}
So the volume of a cone is:
\text{Volume } =\; \frac{1}{3}(\text{area of base})(\text{height})
The base is a circle with a radius of 6 inches. So using the equation for area of a circle:
\text{Area of base} = (6^2)π= 36π
The height of the cylinder is
8
inches. Substitute the known values into the volume equation for a cone:
\text{Volume}=\frac{1}{3}(36\pi )(8)
Now simplify to get the answer. |
Income Elasticity of Demand - Course Hero
Microeconomics/Elasticity/Income Elasticity of Demand
Demand is susceptible to shifts based on many occurrences. Demand can increase or decrease based on things that happen to consumers. One factor that can shift the demand curve (a graph showing growth or lessening of demand) is income. Therefore, economists are interested in how consumers respond to changes in income. The income elasticity of demand is the responsiveness of the quantity demanded to changes in a consumer's income, as measured by the percentage change in the quantity demanded divided by the percentage change in consumers' income. It tells how much consumers respond to a change in income and in what direction. If a consumer receives a large pay raise, they are likely to spend more money. The income elasticity of demand (
E_y
) is calculated by dividing the percentage change in quantity demanded (
\%\Delta \text{Q}_{\text{D}}
) by the percentage change in income (
\% \Delta Y
\text{E}_{y}=\frac{\%\Delta \text{Q}_{\text{D}}}{\%\Delta Y}
The sign of the income elasticity depends on the type of good or service in question. The absolute value is not taken when calculating this type of elasticity, so calculations must be made to determine whether it is positive or negative. A normal good is a good for which demand rises when income rises; most goods are normal goods. Therefore, the numerator (the percentage change in demand) and the denominator (the percentage change in income) will have the same sign, and the income elasticity of demand will be positive. For example, suppose that the income elasticity of demand for hot tubs is 2. This means that a 10% increase in income (the denominator) will lead to a 20% increase in the quantity of hot tubs demanded (the numerator). Because both quantity demanded and income are increasing, the numerator and denominator are both positive, so the ratio between them will be positive as well.
To use another example, suppose a person's salary is $100,000 and this person goes to (consumes) 10 movies per year. Now suppose this person gets a pay raise and their income increases to $110,000 (a change of
+10\%
), so they begin to go to 15 movies a year (an increase of 50%). This would mean an income elasticity of demand of
50\%/10\%=5
, which would make movies a luxury good. When this person's income went up, they consumed disproportionately more of this particular good relative to their income increase.
The relative size of the income elasticity for normal goods can help determine if the good is a necessity or a luxury (both necessities and luxuries are normal goods). Goods that are necessities have an income elasticity between zero and 1. This occurs because a change in income doesn't have a large impact on the amount demanded for necessities. However, the income elasticity for luxuries is greater than 1. A fall in income will lead to a greater decline in the demand for luxuries, making the income elasticity value greater. For example, if the moviegoer discussed above suffered a pay cut of 10% but their movie consumption dropped by 20%, that would mean the movies were a luxury good for that person, because the income elasticity of their demand for movies would be
20\%/10\%=2
An inferior good is a good for which demand falls when income rises. Therefore, the numerator (the percentage change in demand) and the denominator (the percentage change in income) will have different signs, and the income elasticity of demand will be negative. Suppose that the income elasticity of demand for ramen noodles is –0.7. A 10% increase in income leads to a 7% decline in the quantity of ramen noodles demanded. Because the denominator is positive and the numerator is negative, the ratio will be negative. Therefore, the income elasticity for ramen noodles, an inferior good, is negative.
<Price Elasticity of Demand and Total Revenue>Cross-Price Elasticity of Demand |
A rocket starts from rest and moves upward from the
A rocket starts from rest and moves upward from the surface of the earth. For th
A rocket starts from rest and moves upward from the surface of the earth. For the first 10.0 s of its motion, the vertical acceleration of the rocket is given by
{a}_{y}=12.80\frac{m}{{s}^{3}}
)t, where the + y-direction is upward. (a) What is the height of the rocket above the surface of the earth at t = 10.0 s? (b) What is the speed of the rocket when it is 325 m above the surface of the earth?
a) Since the acceleration is constant constant, integrate the first kinematics equation:
{v}_{y}={\int }_{0}^{t}{a}_{y}dt={\int }_{0}^{10}2.8tdt=1.4{t}^{2}+{v}_{0}
y={\int }_{0}^{t}vdt
{v}_{0}=0
{y}_{0}=0
y={\int }_{0}^{10}1.4{t}^{2}dt=0.467{t}^{3}{\mid }_{0}^{10}=467
b) Substitute y=325 m in the equation:
y=\int vdt=1.4\int {t}^{2}dt=0.467{t}^{3}
325=0.476{t}^{3}
\therefore t=8.86\text{ }s
then substitute this time in the equation:
{v}_{y}={\int }_{0}^{t}{a}_{y}dt={\int }_{0}^{8.86}2.8dt=1.4{t}^{2}\mid {0}^{8.86}=110
Result: a) y=467 m
{v}_{y}=110
\stackrel{\to }{{R}_{A}}
\stackrel{\to }{{R}_{A}}
{360}^{m}
{40}^{\circ }
{123}^{\circ }
{5.0}^{s}
\stackrel{\to }{{R}_{B}}
\stackrel{\to }{{R}_{B}}
{880}^{m}
{\stackrel{\to }{R}}_{BA}={\stackrel{\to }{R}}_{B}-{\stackrel{\to }{R}}_{A}
{\stackrel{\to }{R}}_{BA}
{\stackrel{\to }{R}}_{BA}
A hollow sphere of inner radius 8.0cm and outer radius 9.0cmfloats half -submerged in a liquid of density 800
k\frac{g}{{m}^{3}}
(a) What is the mass of the sphere? (b) calculate the density of the material of which the sphere is made.
Prove the general power rule of derivatives using the inverse property
\left({x}^{n}={e}^{n\mathrm{ln}x}\right)
The following is an 8051 instruction: CJNE A, # Q ,AHEAD
a) what is the opcode for this instruction?
b) how many bytes long is this instruction?
c) explain the purpose of each byte of this instruction.
d) how many machine cycles are required to execute this instruction?
e) If an 8051 is operating from a 10 MHz crystal, how longdoes this instruction take to execute? |
Prove the general power rule of derivatives using the inverse
\left({x}^{n}={e}^{n\mathrm{ln}x}\right)
The general power rule:
\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}
\frac{d}{dx}\left({x}^{n}\right)=\frac{d}{dx}\left({e}^{n\mathrm{ln}x}\right)\text{ }\text{ }\text{ }\left[\because {x}^{n}={e}^{n\mathrm{ln}x}\right]
={e}^{n\mathrm{ln}x}\frac{d}{dx}\left(n\mathrm{ln}x\right)
={e}^{n\mathrm{ln}x}\frac{n}{x}
={x}^{n}×\frac{n}{x}\text{ }\text{ }\text{ }\left[\because {x}^{n}={e}^{n\mathrm{ln}x}\right]
=n{x}^{n-1}
Thus, the general power rule proved.
{30}^{\circ }
a,b,c:F=I\cdot l\cdot B=7.06×{10}^{-3}
Without using the fact that the area of a triangle is
\frac{1}{2}bh
but instead only using the area of a rectangle
l×w
, explain why the area of the following triangles is equal to
\frac{1}{2}bh
. Be sure that your expression mathes your moving and additivity story.
Using the two equations E=hv and
c=\lambda v
derive an equation expressing E in terms of h, c, and
\lambda
{\int }_{0}^{3}{\int }_{0}^{\pi /3}{\int }_{0}^{4}r×\mathrm{cos}\theta drd\theta dz
{\int }_{0}^{\pi /2}{\int }_{0}^{3}{\int }_{0}^{4-z}zdrdzd\theta
{\int }_{0}^{pi/2}{\int }_{0}^{\pi /2}{\int }_{0}^{2}{\rho }^{2}d\rho d\theta d\varphi
{\int }_{0}^{\pi /4}{\int }_{0}^{\pi /4}{\int }_{0}^{\mathrm{cos}\left(\varphi \right)}\mathrm{cos}\theta d\rho d\varphi d\theta |
Energy Solution to the Chern-Simons-Schrödinger Equations
Hyungjin Huh, "Energy Solution to the Chern-Simons-Schrödinger Equations", Abstract and Applied Analysis, vol. 2013, Article ID 590653, 7 pages, 2013. https://doi.org/10.1155/2013/590653
Hyungjin Huh 1
1Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea
We prove that the Chern-Simons-Schrödinger system, under the condition of a Coulomb gauge, has a unique local-in-time solution in the energy space . The Coulomb gauge provides elliptic features for gauge fields . The Koch- and Tzvetkov-type Strichartz estimate is applied with Hardy-Littlewood-Sobolev and Wente's inequalities.
We study herein the initial value problem of the Chern-Simons-Schrödinger (CSS) equations where denotes the imaginary unit; , , and for ; is the complex scalar field; is the gauge field; is the covariant derivative for , and is a coupling constant representing the strength of interaction potential. The summation convention used involves summing over repeated indices and Latin indices are used to denote .
The CSS system of equations was proposed in [1, 2] to deal with the electromagnetic phenomena in planar domains, such as the fractional quantum Hall effect or high-temperature superconductivity. We refer the reader to [3, 4] for more information on the physical nature of these phenomena.
The CSS system exhibits conservation of mass and the conservation of total energy Note that the terms are missing in (3) when compared to the Maxwell-Schrödinger equations studied in [5].
To figure out the optimal regularity for the CSS system, we observe that the CSS system is invariant under scaling: Therefore, the scaled critical Sobolev exponent is for . In view of (2) we may say that the initial value problem of the CSS system is mass critical.
The CSS system is invariant under the following gauge transformations: where is a smooth function. Therefore, a solution to the CSS system is formed by a class of gauge equivalent pairs . In this work, we fix the gauge by imposing the Coulomb gauge condition of , under which the Cauchy problem of the CSS system may be reformulated as follows: where the initial data . For the formulation of (6)–(8) we refer the reader to Section 3.
The initial value problem of the CSS system was investigated in [6, 7]. It was shown in [6] that the Cauchy problem is locally well posed in , and that there exists at least one global solution, , provided that the initial data are made sufficiently small in by finding regularized equations. They also showed, by deriving a virial identity, that solutions blow up in finite time under certain conditions. Explicit blow-up solutions were constructed in [8] through the use of a pseudo-conformal transformation. The existence of a standing wave solution to the CSS system has also been proved in [9, 10].
The adiabatic approximation of the Chern-Simons-Schrödinger system with a topological boundary condition was studied in [11], which provides a rigorous description of slow vortex dynamics in the near self-dual limit.
Taking the conservation of energy (3) into account, it seems natural to consider the Cauchy problem of the CSS system with initial data . Our purpose here is to supplement the original result of [6] by showing that there is a unique local- in-time solution in the energy space . We follow a rather direct means of constructing the solution and prove the uniqueness. We adapt the idea discussed in [12, 13] where a low regularity solution of the modified Schrödinger map (MSM) was studied. In fact, the CSS and MSM systems have several similarities except for the defining equation for . In the MSM, can be written roughly as , where denotes the Riesz transform. The local existence of a solution to the MSM was proved in [12] for the initial data in with , and similarly, the uniqueness was proved in [14] for with . To show the existence and uniqueness of the solution to the CSS system, the estimate of the gauge field, , is important for situations in which special structures of nonlinear terms in the defining equation for are used. The following describes are our main results.
Theorem 1. Let initial data belong to . Then, there exists a local-in-time solution, , to (6)–(8) that satisfies where , , and .
Theorem 2. Let and be solutions to (6)–(8) on in the distribution sense with the same initial data to that outlined vide supra. Moreover, one assumes that for some constant . One then has for .
We present some preliminaries in Section 2. Theorems 1 and 2 are proved in Sections 3 and 4, respectively. We conclude the current section by providing a few notations. We denote space time derivatives by and is used for spacial derivatives. We use the standard Sobolev spaces , with the norm and with the norm , where and . The space denotes . We define the space time norm as . We use to denote various constants. Because we are interested in local solutions, we may assume that . Thus, we replace the smooth function of with . We also use the convention of writing as shorthand for .
We collect here a few lemmas used for the proof of Theorems 1 and 2. The following lemma is reminiscent of Wente's inequality (see [15, 16]).
Lemma 3. Let and be two functions in and let be the solution of where is small at infinity. Then, and
The following energy estimate in [17, 18] is used for estimating a solution to the magnetic Schrödinger equation.
Lemma 4. Let u be a solution of where and are real-valued functions. Then, for there exists an absolute constant such that wherein one means the homogeneous Sobolev space when and simply when .
The following type of Strichartz estimate was used in [19, 20] for the study of the Benjamin-Ono equation. We refer to [12] for the counterpart to the Schrödinger equation.
Lemma 5. Let and be a solution to the equation Then, for and , one has where and .
We use the following Gagliardo-Nirenberg inequality with the specific constant [21], especially for the proof of Theorem 2.
Theorem 1 is proved in this section. Because the local well-posedness for smooth data is already known in [6], we simply present an a priori estimate for the solution to (6)–(8). Let us first explain (8). To derive it, note the following identities: where and . Note that the second-order terms are cancelled out. Combined with the above algebra, the equation for comes from the second and third equations in (1): We then have the formulation (6)–(8) in which is the only dynamical variable and , , and are determined through (7) and (8).
The constraint equation and the Coulomb gauge condition provide an elliptic feature of ; that is, the components can be determined from by solving the elliptic equations Taking into account that the Coulomb gauge condition in Maxwell dynamics deduces a wave equation, the previous observation was used in [6]. Using (20), we have the following representation of :
3.1. Estimates for and
We are now ready to estimate several quantities of . Making use of (20) and the representation (21), we obtain the following estimates for .
Proposition 7. Let and . One also assumes that if or if . Then, one has
Proof. The above can be checked by applying Calderon-Zygmund and Hardy-Littlewood-Sobolev inequalities. We refer to [2, Section 2] for the details.
To estimate , the special algebraic structure and divergence form of the nonlinear terms in (19) are used.
Proposition 8. Let be the solution of (19). Then, one has
Proof. Decompose as follows: We first estimate the quantity . Applying Lemma 3 to (24), we deduce that To estimate we use the Gagliardo-Nirenberg inequality with small : Applying Hardy-Littlewood-Sobolev's inequality to (25) we deduce where Proposition 7 and Lemma 6 are used. We can also derive the following from (25): The first term can be estimated as follows: where is used. The second term can be estimated as follows: where is used. Therefore, we obtain with , that is, , Therefore, we conclude that On the other hand, Lemma 3 shows that We also have from (25) that Therefore, we have
3.2. The Energy Solution to (CSS)
We now prove Theorem 1. Let us define where , , and . We derive the following estimate: from which Theorem 1 is proved by standard argument; see [2, Section 3].
To control , we apply Lemma 4 to the solution of (6)–(8).
Proposition 9. Let be a solution to (6)–(8). Then, one has where and .
Proof. From the conservation of mass, we derive the first estimate. We apply Lemma 4 to (6) with and . Combined with Proposition 7, we have where . We are then left to estimate . By Proposition 8, we obtain Combining (40) and (41), we obtain where and .
To estimate , we apply Lemma 5 to the solution of (6)–(8).
Proposition 10. Let be a solution to (6)–(8). Then, one has where , and .
Proof. Applying Lemma 5 with and , we obtain where , and . Considering Proposition 8, we obtain The other terms can be treated, as mentioned in Section 1, by similar arguments to those in [2, Section 3]. Applying Proposition 7, we have Plugging estimates (45)–(48) into (44) with , we obtain
We finally obtain the estimate (38) by combining Propositions 9 and 10, which proves Theorem 1.
In this section, we prove the uniqueness of the solution to (6). The basic rationale is borrowed from [12, 22].
Let and be solutions of (6)–(8) with the same initial data. If we set , then the equation for is We will derive where is a constant in Theorem 2 and . Then we have Considering and , we obtain Letting , for the time interval satisfying , we conclude that for , which thus proves Theorem 2.
In the remainder of this section, we derive inequality (51). Multiplying to both sides of (50) and integrating the imaginary part of , we have The integrals (II)–(V), that is, those not containing , can be controlled by applying similar arguments to those described in [2, Section 4]. Integral (II) can be estimated, considering , by for which we omit the proof.
We simply present how to control integral (I), for which we have where , . Applying Lemma 6, we obtain To control , we consider the equation for Decomposing and as (24) and (25), we have Taking into account we can rewrite the equation for as follows: where should be noted. Using the Hardy-Littlewood-Sobolev inequality, we have where and , from which we deduce . Then, we have
The term can be bounded as follows: Since , we have Since , we may check Then, we have Combining estimates (57) and (69), and denoting , we obtain where . We then obtain (51) by combining (55) and (70).
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2011-0015866), and was also partially supported by the TJ Park Junior Faculty Fellowship.
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Copyright © 2013 Hyungjin Huh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
Matrices C and D are shown below C=begin{bmatrix}2&1&0 0&3&40&2&1 end{bmatrix},
Matrices C and D are shown belowC=begin{bmatrix}2&1&0 0&3&40&2&1 end{bmatrix},D=begin{bmatrix}a & b&-0.4 0&-0.2&0.80&0.4&-0.6 end{bmatrix}What values of a and b will make the equation CD=I true?a)a=0.5 , b=0.1b)a=0.1 , b=0.5c)a=-0.5 , b=-0.1
Matrices C and D are shown below
C=\left[\begin{array}{ccc}2& 1& 0\\ 0& 3& 4\\ 0& 2& 1\end{array}\right],D=\left[\begin{array}{ccc}a& b& -0.4\\ 0& -0.2& 0.8\\ 0& 0.4& -0.6\end{array}\right]
What values of a and b will make the equation CD=I true?
a)a=0.5 , b=0.1
b)a=0.1 , b=0.5
c)a=-0.5 , b=-0.1
The given matrices are,
C=\left[\begin{array}{ccc}2& 1& 0\\ 0& 3& 4\\ 0& 2& 1\end{array}\right]\text{ and }D=\left[\begin{array}{ccc}a& b& -0.4\\ 0& -0.2& 0.8\\ 0& 0.4& -0.6\end{array}\right]
Now multiply the matrices C and D as shown below.
CD=\left[\begin{array}{ccc}2& 1& 0\\ 0& 3& 4\\ 0& 2& 1\end{array}\right]\left[\begin{array}{ccc}a& b& -0.4\\ 0& -0.2& 0.8\\ 0& 0.4& -0.6\end{array}\right]
=\left[\begin{array}{ccc}2a+0+0& 2b-0.2+0& -0.8+0.8+0\\ 0+0+0& 0-0.6+1.6& 0+2.4-2.4\\ 0+0+0& 0-0.4+0.4& 0+1.6-0.6\end{array}\right]
=\left[\begin{array}{ccc}2a& 2b-0.2& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]
Now equate the matrix CD to the identity matrix I and obtain the values of a and b as follows.
\left[\begin{array}{ccc}2a& 2b-0.2& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]
Two matrices are equal , only if the corresponding elements are equal.
2a=1
⇒a=0.5
2b-0.2=0
⇒b=0.1
Therefore, the CD = I is true for a = 0.5 and b =0.1.
why does the canvas top of a convertible bulge out when thecar is travelling at high speed? [HINT: the windshielddeflects air upward, pushing streamlines closer together.]
A student presses a book between his hands, as the drawing indicates. The forces that he exerts on the front and back covers of the book are perpendicular to the book and are hortizontal. The book weighs 31 N. The coefficients of static friction between hishands and the book is .40. To keep the book from falling, what is the magnitude of the minimum pressing force that each hand must exert. |
Next, look up the probability in the binomial probability distribution
Next, look up the probability in the binomial probability distribution table. (b) Find the probability of getting exactly two heads.
(b) Find the probability of getting exactly two heads.
(c) Find the probability of getting two or more heads.
A binomial probability is given. Write the probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability:
P\left(x>73\right)
The probability of getting _ 73 successes.
Which of the following is the normal probability statement that corresponds to the binomial probability statement?
P\left(72.5<x<73.5\right)
P\left(x<72.5\right)
P\left(x>72.5\right)
P\left(x<73.5\right)
P\left(x>73.5\right)
A binomial probability is given P(x>73)
Which probability statement that corresponds to the binomial probability statement?
They gave a test of 22 multiple-choice questions. Each question has 4 possible answers, only one of which is correct. What is the probability that you will guess at most three correctly?
A binomial random variable has mean 1.8 and variance 1.44. Determine complete binomial probability distribution.
n=20\text{ }\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\text{ }p=.70
P\left(x\ge 16\right)
How do you find the probability of at least one success when n independent Bernoulli trials are carried out with probability of success p? |
Settings Files and XML Tag Definitions - OpenSim Documentation - Global Site
The settings files are XML files whose tags specify properties to be used by OpenSim for performing the residual reduction. The tags used for each type of settings file are defined in the following sections:
RRA Setup File
A setup file provides the high-level information required for the first pass of the residual reduction algorithm. It references three other files: an actuators file, the constraints file, and a tasks file. All of these files are explained in detail below. An example of the setup file is given in Example 1 below.
In the setup file, the property settings for RRA are enclosed in <RRATool>. The types of properties listed in the XML setup files for RRA include model files, actuator and control information, integration parameters, kinematics and ground reaction data files, tracking information, optimization parameters, and output information.
The <model_file> property specifies the name of the .osim file to load. In RRA, the value of this property in the above example is subject01_simbody.osim, the .osim file representing the dynamic subject-specific model. The output of RRA <output_model_file> in Example 1 is subject01_simbody_adjusted.osim.
The <maximum_number_of_integrator_steps> property indicates the maximum number of steps RRA during an entire run before termination. During each integration, the maximum number of seconds that may elapse is specified by <maximum_integrator_step_size>. Increasing the <integrator_error_tolerance> will decrease the integrator step size, while decreasing the <integrator_fine_tolerance> will increase the integrator step size. Other simulation parameters like <initial_time>, <final_time>, and <cmc_time_window> are described earlier in the Simulation section for RRA (How RRA Works).
The coordinates that should be followed by the model during RRA are specified within a file, indicated by the <task_set_file> and </task_set_file> tags. In Example 1, that file is gait2354_RRA_Tasks.xml. See RRA Tasks File for information about the property tags used within a task file.
Actuator and Control Information
A model has some set of actuators that can apply forces to its skeleton. For example, the dynamic subject-specific model in Example 1 has 54 muscles as its default set of actuators. These actuators can be modified using the tag <replace_force_set>. If the value of the property <replace_force_set> is true, then the actuators specified in any file listed under the property <force_set_files> will replace the corresponding model's actuators. If the value of <replace_force_set> is false, then the actuators specified in the files listed under <force_set_files> will be added to the model's existing actuator set. Details about the actuators file are given in RRA Actuators File below.
The <constraints_file> property specifies the name of an XML file containing minimum and maximum values for the control values for the model's actuators. An actuator's actual range of values is equal to the range from the minimum to maximum control values times its optimal force. The maximum and minimum control values, as well as the optimal force, are specified in the constraints file. See Example 3 below for more information about the constraints file and the best method for altering an actuator's range.
Kinematics and Ground Reaction Data Files
RRA will attempt to make the model follow the kinematics (generalized coordinates as functions of time) specified in the <desired_kinematics_file>, which should be a motion (.mot) or a storage (.sto) file.
Prior to simulation, the kinematics to be tracked by RRA are low-pass filtered at a frequency specified by the tag <lowpass_cutoff_frequency>. The value of this frequency is assumed to be in Hertz (Hz). A negative value for this property leads to no filtering. The default value is –1.0, i.e., no filtering.
The target consists of an objective function (J) that is a weighted (w) sum of squared actuator controls (x) plus the sum of desired acceleration (\ddot{q}_j\,^*
//<![CDATA[ \begin{array}{l}\ddot{q}_j\,^*\end{array} //]]>
) errors:
J = \sum_{i=1}^{nx} x_i^2 + \sum_{j=1}^{nq} w_j \left( \ddot{q}_j\,^* - \ddot{q}_j \right)^2
The first summation minimizes and distributes loads across actuators and the second drives the model accelerations (\ddot{q}_j
//<![CDATA[ \begin{array}{l}\ddot{q}_j\end{array} //]]>
) toward the desired accelerations (\ddot{q}_j\,^*
//<![CDATA[ \begin{array}{l}\ddot{q}_j\,^*\end{array} //]]>
The behavior of the optimizer is controlled using three tags: <optimizer_derivative_dx>, <optimizer_convergence_criterion>, <optimizer_max_iterations>.
The <optimizer_derivative_dx> tag determines the perturbation size used by the optimizer to compute numerical derivatives. Valid values for <optimizer_derivative_dx> range from 1.0e-4 to 1.0e-8.
The <optimizer_convergence_criterion> tag specifies the depth of optimization required for convergence. The smaller the value of this property, the better the solution is. But decreasing this value also can increase computation time.
The <optimizer_max_iterations> property limits the number of iterations the optimizer can take at each time step in searching for an optimal solution.
The optimizer can be set to print out details of what it is doing by using the <optimizer_print_level> tag. Valid values for this property are 0, 1, 2, or 3, where a value of 0 means no printing, a value of 3 means detailed printing, and 1 and 2 represent levels in-between.
Results printed by RRA will be output into the directory specified by the tag <results_directory>. The name of the file that is created is determined by the <output_model_file> tag. By default, the name of the <output_model_file> is adjusted_model.osim. The precision of RRA output is specified in the property <output_precision>, which is 8 by default.
If the <adjust_com_to_reduce_residuals> property is true, the file that is output contains data where the mass center of the body is specified by the tag <adjusted_com_body>. The <adjusted_com_body> should normally be the heaviest segment in the model. For the gait model, torso is usually the best choice. The body name must correspond to the body name in the navigator (or model file). The average residuals computed during RRA will be printed out to a file in the <results_directory>.
RRA Actuators File
The RRA actuators file uses the <ForceSet> and </ForceSet>tags to describe actuators that replace the model's actuators. Example 2 below shows an actuator file for RRA (gait2354_RRA_Actuators.xml). Forces are specified in newtons and torques in newton-meters.
RRA Tasks File
Each generalized coordinate task contains the following properties: <on>, <wrt_body>, <express_body>, <active>, <weight>, <kp>, <kv>, <ka>, <r0>, <r1>, <r2>, <coordinate>, and <limit>.
<on> indicates whether or not the coordinate(s) specified in the task should be followed or not. The body to which the task is applied is specified using the tags <wrt_body> and </wrt_body>. The reference frame which is used to specify the coordinates to follow is determined by the property <express_body>, which refers to a specific body. For example, if a point on body 2 is to be followed, and the point's coordinates are expressed in the frame of body 1, then <wrt_body> would be 2 and <express_body> would be 1.
The <active> property is an array of three flags, each flag indicating whether a component of a task is active. For example, the trajectory of a point in space could have three components (x, y, z). This allows the tracking of each coordinate of the point to be made active (true) or inactive (false). The definition of a flag depends on the type of coordinate: <CMC_Joint> or <CMC_Point>. For a task that tracks a joint coordinate (like all tasks in Example 34 below), only the first of the three flags is valid.
The tags <kp>, <kv>, < ka> are parameters in the proportional-derivative (PD) control law used to compute desired accelerations for tracking the experimental kinematics computed by the inverse kinematics (IK) solver (see Inverse Kinematics). This control law contains a position error feedback gain (stiffness) and a velocity error feedback gain (damping). The stiffness for each tracked coordinate is specified within the <kp> property, while the damping is specified within the <kv> property. An acceleration feed-forward gain is also allowed, but in the above example, this gain is set to 1, i.e., there is no acceleration gain. The acceleration gain is specified within the <ka> property.
The <r0>, <r1>, and <r2> properties indicate direction vectors for the three components of a task, respectively. These properties are not used for tasks representing tracking of a single joint coordinate, such as the tasks in Example 4 below.
The name of the coordinate to be tracked is specified within the property <coordinate>. The error limit on the tracking accuracy for a coordinate is specified within the <limit> property. If the tracking errors approach this limit during simulation, the weighting for this coordinate is increased.
RRA Example Files
Example 1: XML file for the setup file for RRA
subject01_Setup_RRA.xml
<RRATool name="subject01_walk1_RRA">
<!--Replace the model's force set with sets specified in <force_set_files>? If false, the force set is appended to.-->
<replace_force_set> true </replace_force_set>
<force_set_files> gait2354_RRA_Actuators.xml </force_set_files>
<results_directory> ResultsRRA/ </results_directory>
<!--Output precision. It is 20 by default.-->
<output_precision> 8 </output_precision>
<!--Flag indicating whether or not to compute equilibrium values for states other than the coordinates or speeds.
For example, equilibrium muscle fiber lengths or muscle forces.-->
<solve_for_equilibrium_for_auxiliary_states> false </solve_for_equilibrium_for_auxiliary_states>
<!--Integrator error tolerance. When the error is greater, the integrator step size is decreased.-->
<!--XML file (.xml) containing the external loads applied to the model as a set of PrescribedForce(s).-->
<!--Motion (.mot) or storage (.sto) file containing the desired point trajectories.-->
<!--Motion (.mot) or storage (.sto) file containing the desired kinematic trajectories.-->
<desired_kinematics_file> subject01_walk1_ik.mot </desired_kinematics_file>
<!--File containing the tracking tasks. Which coordinates are tracked and with what weights are specified here.-->
<task_set_file> gait2354_RRA_Tasks.xml </task_set_file>
<constraints_file> gait2354_RRA_ControlConstraints.xml </constraints_file>
<!--File containing the controls output by RRA. These can be used to place constraints on the residuals during CMC.-->
<!--Low-pass cut-off frequency for filtering the desired kinematics. A negative value results in no filtering.
The default value is -1.0, so no filtering.-->
<lowpass_cutoff_frequency> -1.00000000 </lowpass_cutoff_frequency>
<!--Preferred optimizer algorithm (currently support "ipopt" or "cfsqp", the latter requiring the osimFSQP library.-->
<!--Perturbation size used by the optimizer to compute numerical derivatives. A value between 1.0e-4 and 1.0e-8
is usually appropriate.-->
<!--Convergence criterion for the optimizer. The smaller this value, the deeper the convergence. Decreasing this number
can improve a solution, but will also likely increase computation time.-->
<!--Flag (true or false) indicating whether or not to make an adjustment in the center of mass of a body to reduced DC offsets
in MX and MZ. If true, a new model is writen out that has altered anthropometry.-->
<adjust_com_to_reduce_residuals> true </adjust_com_to_reduce_residuals>
<!--Initial time used when computing average residuals in order to adjust the body's center of mass. If both initial and final
time are set to -1 (their default value) then the main initial and final time settings will be used.-->
<initial_time_for_com_adjustment> -1.00000000 </initial_time_for_com_adjustment>
<!--Final time used when computing average residuals in order to adjust the body's center of mass.-->
<final_time_for_com_adjustment> -1.00000000 </final_time_for_com_adjustment>
<!--Name of the body whose center of mass is adjusted. The heaviest segment in the model should normally be chosen. For a gait model, the
torso segment is usually the best choice.-->
<adjusted_com_body> torso </adjusted_com_body>
<!--Name of the output model file (.osim) containing adjustments to anthropometry made to reduce average residuals. This file is written
if the property adjust_com_to_reduce_residuals is set to true. If a name is not specified, the model is written out to a
file called adjusted_model.osim.-->
<output_model_file> subject01_RRA_adjusted.osim </output_model_file>
<!--True-false flag indicating whether or not to turn on verbose printing for cmc.-->
</RRATool>
Example 2: XML file for the actuator set file for RRA
<ForceSet name="gait2354_RRA">
<PointActuator name="default">
<max_force> 10000.000 </max_force>
<min_force> -10000.000 </min_force>
<optimal_force> 1000.00000000 </optimal_force>
<body> </body_A>
<point> 0.000 0.000 0.000 </point>
<direction> 1.000 0.000 0.000 </direction>
</PointActuator >
<TorqueActuator name="default">
<max_force> 1000.000 </max_force>
<min_force> -1000.000 </min_force>
<optimal_force> 300.00000000 </optimal_force>
<body_A> </body_A>
<axis> 1.000 0.000 0.000 </axis>
<body_B> </body_B>
</TorqueActuator>
<CoordinateActuator name="default">
<coordinate> </coordinate>
</CoordinateActuatorCoordinateActuator>
<!-- Residuals -->
<PointActuator name="FX">
<optimal_force> 4.00000000 </optimal_force>
<point> -0.0724376 0.00000000 0.00000000 </point>
<direction> 1 0 0 </direction>
<PointActuator name="FY">
<PointActuator name="FZ">
<TorqueActuator name="MX">
<body_A> pelvis </body_A>
<body_B> ground </body_B>
<TorqueActuator name="MY">
<TorqueActuator name="MZ">
<CoordinateActuator name="hip_flexion_r">
<!-..additional <CoordinateActuator> tags cut for brevity..->
Example 3: XML file for the control constraints file for RRA
<ControlSet name="gait2354_RRA">
<default_min> -20.0 </default_min>
<default_max> 20.0 </default_max>
<ControlLinear name="FY.excitation">
<ControlLinear name="FZ.excitation">
<ControlLinear name="MX.excitation">
<ControlLinear name="MY.excitation">
<ControlLinear name="MZ.excitation">
<ControlLinear name="hip_flexion_r.excitation" />
<ControlLinear name="hip_adduction_r.excitation" />
<ControlLinear name="hip_rotation_r.excitation" />
<ControlLinear name="knee_angle_r.excitation" />
<ControlLinear name="ankle_angle_r.excitation" />
<ControlLinear name="hip_flexion_l.excitation" />
<ControlLinear name="hip_adduction_l.excitation" />
<ControlLinear name="hip_rotation_l.excitation" />
<ControlLinear name="knee_angle_l.excitation" />
<ControlLinear name="ankle_angle_l.excitation" />
<ControlLinear name="lumbar_extension.excitation" />
<ControlLinear name="lumbar_bending.excitation" />
<ControlLinear name="lumbar_rotation.excitation" />
Example 4: XML file for the tasks file for RRA
<CMC_TaskSet name="gait2354_RRA">
<CMC_Joint name="default">
<on> false </on>
<wrt_body> -1 </wrt_body>
<express_body> -1 </express_body>
<active> false false false </active>
<weight> 1 1 1 </weight>
<kp> 1 1 1 </kp>
<kv> 1 1 1 </kv>
<ka> 1 1 1 </ka>
<limit> 0 </limit>
<CMC_Joint name="pelvis_tx">
<coordinate> pelvis_tx </coordinate>
<CMC_Joint name="pelvis_ty">
<coordinate> pelvis_ty </coordinate>
<!-- . . additional <CMC_Joint> tags cut for brevity . . -->
Next: Computed Muscle Control
Previous: How to Use the RRA Tool |
School of Biochemistry, Devi Ahilya University, Takshashila Campus, Indore, India.
Waghmare, R. and Gadre, R. (2018) Impact of Essential Micronutrient, Zn, on Growth and Chlorophyll Biosynthesis in Young Zea mays Seedlings. American Journal of Plant Sciences, 9, 1855-1867. doi: 10.4236/ajps.2018.99135.
\text{Chla}\left({\text{μgml}}^{-\text{1}}\right)=\text{12}.\text{21}\left({\text{A}}_{\text{663}}\right)-\text{2}.\text{81}\left({\text{A}}_{\text{646}}\right)
\text{Chlb}\left({\text{μgml}}^{-\text{1}}\right)=\text{2}0.\text{13}\left({\text{A}}_{\text{646}}\right)-\text{5}.0\text{3}\left({\text{A}}_{\text{663}}\right)
\text{TotalChlorophyll}=\text{Chla}+\text{Chlb}
\text{Caratenoids}\left({\text{μgml}}^{-\text{1}}\right)=\text{1}000\left({\text{A}}_{\text{47}0}\right)-\text{3}.\text{27}\left(\text{Chla}\right)-\text{1}0\text{4}\left(\text{Chlb}\right)/\text{229}.
[1] Jelakovic, S., Kopriva, S., Suss, K.H. and Schulz, G.E. (2003) Structure and Catalytic Mechanism of the Cytosolic D-ribulose-5-phosphate 3-epimerase from Rice. Journal of Molecular Biology, 326, 127-135.
[2] Marschner, H. (1995) Mineral Nutrition of Higher Plants. Academic Press, London, Vol. 2, 889.
[3] Sagardoy, R., Vázquez, S., Florez-Sarasa, I.D., Albacete, A., Ribas-Carbó, M.J., Abadía, F.J. and Morales, F. (2010) Stomatal and Mesophyll Conductances to CO2 Are the Main Limitations to Photosynthesis in Sugar Beet (Beta vulgaris) Plants Grown with Excess Zinc. New Phytologist, 187, 145-158.
[4] Cui, Y. and Zhao, N. (2011) Oxidative Stress and Change in Plant Metabolism of Maize (Zea mays L.) Growing in Contaminated Soil with Elemental Sulfur and Toxic Effect of Zinc. Plant Soil and Environment, 57, 34-39.
[5] Bonnet, M., Camares, O. and Veisseire, P. (2000) Effect of Zinc and Influence of Acremonium lolli on Growth Parameters, Chlorophyll a Fluorescence and Antioxidant Enzyme Activity of Ryegrass (Lolium perenne L. cv Apollo). Journal of Experimental Botany, 51, 945-953.
[6] Manivasagaperumal, R., Balamurugan, S., Thiyagarajan, G. and Sekar, J. (2011) Effect of Zinc on Germination, Seedling Growth and Biochemical Content of Cluster Bean (Cyamopsis tetragonoloba (L.) Taub). Current Botany, 2, 11-15.
[7] Sagardoy, R., Morales, F., Lopez-Millan, A.F., Abadia, A. and Abadia, J. (2009) Effects of Zinc Toxicity on Sugar Beet (Beta vulgaris L.) Plants Grown in Hydroponics. Plant Biology, 11, 339-350.
[8] Mirshekali, H., Hadi, H., Amirnia, R. and Verdiloo, H.K. (2012) Effect of Zinc Toxicity on Plant Productivity, Chlorophyll and Zn Contents of Sorghum (Sorghum Bicolor) and Common Lambsquarter (Chenopodium album). International Journal of Agriculture Research and Review, 2, 247-254.
[9] Ozdener, Y. and Aydin, B.K. (2010) The Effect of Zinc on the Growth and Physiological and Biochemical Parameters in Seedlings of Eruca sativa (L.) (Rocket). Acta Physiologiae Plantarum, 32, 469-476.
[10] Beale, S.I. (1999) Enzymes of Chlorophyll Biosynthesis. Photosynthesis Research, 60, 43-73.
[11] Jaffe, E.K. (2000) The Porphobilinogen Synthase Family of Metalloenzymes. Acta Crystallographica, 56, 115-128.
[12] Lowry, O.H., Rosebrough, N.J., Farr, A.C. and Randall, R.J. (1951) Protein Measurement with the Folin-Phenol Reagent. The Journal of Biological Chemistry, 193, 265-275.
[13] Webb, J.M. and Levy, H.B. (1958) New Developments in Chemical Determination of Nucleic Acid. In: Glick, D., Ed., Methods of Biochemical Analysis, Inter Science Publication, New York City, Vol. 6, 1-30.
[14] Bates, L.S., Walderren, R.P. and Teare, I.D. (1973) Rapid Determination of Free Proline for Water-Studies. Plant Soil, 39, 205-207.
[15] Lichtenthaler, H.K. and Welburn, A.R. (1983) Determination of Total Carotenoids and Chlorophylls a and b of Extracts in Different Solvents. Biochemical Society Transactions, 11, 591-592.
[16] Tewari, A.K. and Tripathy, B.C. (1998) Temperature-Stress-Induced Impairment of Chlorophyll Biosynthetic Reactions in Cucmber and Wheat. Plant Physiology, 177, 851-858.
[17] Mauzerall, D. and Granik, S. (1956) The Occurrence and Determination of δ-Aminoleveulinic Acid Dehydratase and Porophobilinogen in Urine. The Journal of Biological Chemistry, 219, 435-446.
[18] Jain, M. and Gadre, R. (2004) Inhibition of 5-Amino Levulinic Acid Dehydratase Activity by Arsenic in Excised Etiolated Maize Leaf Segments during Greening. Journal of Plant Physiology, 161, 251-255.
[19] Samreen, T., Humaria, S.H.U., Ullah, S. and Javid, M. (2017) Zn Effect on Growth Rate, Chlorophyll, Protein and Mineral Contents of Hydroponically Grown Mungbeans Plant (Vigna radiata). Arabian Journal of Chemistry, 10, S1802-S1807.
[20] Hasan, M.K., Cheng, Y., Kanwar, M.K., Chu, X.Y., Ahammed, G.J. and Qi, Z.Y. (2017) Responses of Plant Proteins to Heavy Metal Stress—A Review. Frontiers in Plant Science, 8, 1492.
[21] Mishra, S. and Agrawal, S.B. (2006) Interactive Effects between Supplemental UV-B Radiation and Heavy Metals on Growth and Biochemical Characterstics of Spinacea oleracea L. Brazilian Journal of Plant Physiology, 18, 1-8.
[22] Lalelou, F.S., Kolvanagh, J.S. and Fateh, M. (2013) Effect of Various Concentrations of Zinc on Chlorophyll, Starch, Soluble Sugars and Proline in Naked Pumpkin (Cucurbita pepo). International Journal of Farming and Allied Sciences, 2, 1198-1202.
[23] Tucker, R.M. (1999) Essential Plant Nutrients: Their Presence in North Carolina Soils and Role in Plant Nutrition. 1-9.
[24] Tchuinmogne, S.T., Hault, C., Auoules, A. and Balange, A.P. (1989) Inhibitory Effect of Gabaculine on 5-Aminolevulinnate Dehydratase in Raddish Seedlings. Plant Physiology, 90, 1293-1297. |
Constrained Electrostatic Nonlinear Optimization, Problem-Based - MATLAB & Simulink - MathWorks í•œêµ
Consider the electrostatics problem of placing 20 electrons in a conducting body. The electrons will arrange themselves in a way that minimizes their total potential energy, subject to the constraint of lying inside the body. All the electrons are on the boundary of the body at a minimum. The electrons are indistinguishable, so the problem has no unique minimum (permuting the electrons in one solution gives another valid solution). This example was inspired by Dolan, Moré, and Munson [1].
\left(x,y,z\right)
\begin{array}{l}zâ¤-|x|-|y|\\ {x}^{2}+{y}^{2}+\left(z+1{\right)}^{2}â¤1.\end{array}
x
y
z
energy=\underset{i<j}{â}\frac{1}{âelectron\left(i\right)-electron\left(j\right)â}.
[1] Dolan, Elizabeth D., Jorge J. Moré, and Todd S. Munson. “Benchmarking Optimization Software with COPS 3.0.†Argonne National Laboratory Technical Report ANL/MCS-TM-273, February 2004. |
How do you solve \cos(x)=\frac{1}{2}
\mathrm{cos}\left(x\right)=\frac{1}{2}
\mathrm{cos}\left(x\right)>0
And x will be in the first/fourth quadrants
\mathrm{cos}\left(x\right)=\frac{1}{2}
x={\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{3}
is an angle in the first quadrant.
x=\left(2\pi -\frac{\pi }{3}\right)=\frac{5\pi }{3}
is an angle in the fourth quadrant.
General solutions for
\mathrm{cos}\left(x\right)=\frac{1}{2}
x=\frac{\pi }{3}+2\pi n
x=\frac{5\pi }{3}+2\pi n
\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)
Can anyone see a way to simplify one of these expressions?
{\mathrm{cos}}^{2}\theta \mathrm{sin}\varphi +{\mathrm{sin}}^{2}\theta \mathrm{cos}\varphi
{\mathrm{sin}}^{2}\theta \mathrm{sin}\varphi -{\mathrm{cos}}^{2}\theta \mathrm{cos}\varphi
\mathrm{sin}x+\sqrt{3}\mathrm{cos}x=\sqrt{2}
\mathrm{sin}x+\sqrt{3}\mathrm{cos}x=\sqrt{2}
\mathrm{cos}\left(x-\frac{\pi }{6}\right)=\mathrm{cos}\left(\frac{\pi }{4}\right)
x=2n\pi ±\frac{\pi }{4}+\frac{\pi }{6}
x=2n\pi +\frac{5\pi }{12}
x=2n\pi -\frac{\pi }{12}
Prove the identity whether is true or false
\mathrm{cos}2\theta =\frac{1-{\mathrm{tan}}^{2}\theta }{1+{\mathrm{tan}}^{2}\theta }
\frac{1}{2\mathrm{csc}2x}={\mathrm{cos}}^{2}x\mathrm{tan}x
Choose the sequence of steps below that verifies the identity
{\mathrm{cos}}^{2}x\mathrm{tan}x={\mathrm{cos}}^{2}x\frac{\mathrm{sin}x}{\mathrm{cos}x}=\mathrm{cos}x\mathrm{sin}x=\frac{\mathrm{sin}2x}{2}=\frac{1}{2\mathrm{csc}2x}
{\mathrm{cos}}^{2}x\mathrm{tan}x={\mathrm{cos}}^{2}x\frac{\mathrm{cos}x}{\mathrm{sin}x}=\mathrm{cos}x\mathrm{sin}x=\frac{\mathrm{sin}2x}{2}=\frac{1}{2\mathrm{csc}2x}
{\mathrm{cos}}^{2}x\mathrm{tan}x={\mathrm{cos}}^{2}x\frac{\mathrm{cos}x}{\mathrm{sin}x}=\mathrm{cos}x\mathrm{sin}x=2\mathrm{sin}2x=\frac{1}{2\mathrm{csc}2x}
{\mathrm{cos}}^{2}x\mathrm{tan}x={\mathrm{cos}}^{2}x\frac{\mathrm{sin}x}{\mathrm{cos}x}=\mathrm{cos}x\mathrm{sin}x=2\mathrm{sin}2x=\frac{1}{2\mathrm{csc}2x}
\frac{2\mathrm{sin}x+\mathrm{sin}2x}{2\mathrm{sin}x-\mathrm{sin}2x}={\mathrm{csc}}^{2}x+2\mathrm{csc}x\mathrm{cot}x+{\mathrm{cot}}^{2}x
{\mathrm{csc}}^{2}x+2\mathrm{csc}x\mathrm{cot}x+{\mathrm{cot}}^{2}x=\frac{1}{{\mathrm{sin}}^{2}x}+\frac{2\mathrm{cos}x}{{\mathrm{sin}}^{2}x}+\frac{{\mathrm{cos}}^{2}x}{{\mathrm{sin}}^{2}x}
=\frac{{\mathrm{cos}}^{2}x+2\mathrm{cos}x+1}{{\mathrm{sin}}^{2}x}
=\frac{1-{\mathrm{sin}}^{2}x+1+\frac{\mathrm{sin}2x}{\mathrm{sin}x}}{{\mathrm{sin}}^{2}x}
=\frac{\frac{2\mathrm{sin}x-{\mathrm{sin}}^{3}x+\mathrm{sin}2x}{\mathrm{sin}x}}{{\mathrm{sin}}^{2}x}
=\frac{2\mathrm{sin}x-{\mathrm{sin}}^{3}x+\mathrm{sin}2x}{{\mathrm{sin}}^{3}x} |
Generalized Growth of Special Monogenic Functions
Susheel Kumar, "Generalized Growth of Special Monogenic Functions", Journal of Complex Analysis, vol. 2014, Article ID 510232, 5 pages, 2014. https://doi.org/10.1155/2014/510232
Susheel Kumar1
1Department of Mathematics, Jaypee University of Information Technology, Samirpur 177601 (H.P.), India
We study the generalized growth of special monogenic functions. The characterizations of generalized order, generalized lower order, generalized type, and generalized lower type of special monogenic functions have been obtained in terms of their Taylor’s series coefficients.
Clifford analysis offers the possibility of generalizing complex function theory to higher dimensions. It considers Clifford algebra valued functions that are defined in open subsets of for arbitrary finite and that are solutions of higher-dimensional Cauchy-Riemann systems. These are often called Clifford holomorphic or monogenic functions.
In order to make calculations more concise, we use the following notations, where is -dimensional multi-index and : Following Almeida and Kraußhar [1] and Constales et al. [2, 3], we give some definitions and associated properties.
By we denote the canonical basis of the Euclidean vector space . The associated real Clifford algebra is the free algebra generated by modulo , where is the neutral element with respect to multiplication of the Clifford algebra . In the Clifford algebra , the following multiplication rule holds: where is Kronecker symbol. A basis for Clifford algebra is given by the set with , where , . Each can be written in the form with . The conjugation in Clifford algebra is defined by , where and for , . The linear subspace is the so-called space of paravectors which we simply identify with . Here, is scalar part and is vector part of paravector . The Clifford norm of an arbitrary is given by Also, for , we have . Each paravector has an inverse element in which can be represented in the form . In order to make calculations more concise, we use the following notation: The generalized Cauchy-Riemann operator in is given by If is an open set, then a function is called left (right) monogenic at a point if (). The functions which are left (right) monogenic in the whole space are called left (right) entire monogenic functions.
Following Abul-Ez and Constales [4], we consider the class of monogenic polynomials of degree , defined as Let be -dimensional surface area of -dimensional unit ball and let be -dimensional sphere. Then, the class of monogenic polynomials described in (6) satisfies (see [5], pp. 1259) Also following Abul-Ez and De Almeida [5], we have
Now following Abul-Ez and De Almeida [5], we give some definitions which will be used in the next section.
Definition 1. Let be a connected open subset of containing the origin and let be monogenic in . Then, is called special monogenic in , if and only if its Taylor’s series near zero has the form (see [5], pp. 1259)
Definition 2. Let be a special monogenic function defined on a neighborhood of the closed ball . Then, where is the maximum modulus of (see [5], pp. 1260).
Definition 3. Let be a special monogenic function whose Taylor’s series representation is given by (9). Then, for the maximum term of this special monogenic function is given by (see [5], pp. 1260) Also the index with maximal length for which maximum term is achieved is called the central index and is denoted by (see [5], pp. 1260)
Definition 4. Let be a special monogenic function whose Taylor’s series representation is given by (9). Then, the order and lower order of are defined as (see [5], pp. 1263)
Definition 5. Let be a special monogenic function whose Taylor’s series representation is given by (9). Then, the type and lower type of special monogenic function having nonzero finite generalized order are defined as (see [5], pp. 1270) For generalization of the classical characterizations of growth of entire functions, Seremeta [6] introduced the concept of the generalized order and generalized type with the help of general growth functions as follows.
Let denote the class of functions satisfying the following conditions:(i) is defined on and is positive, strictly increasing, and differentiable, and it tends to as (ii), for every function such that as .
Let denote the class of functions satisfying conditions (i) and (i), (ii)for every ; that is, is slowly increasing.
Following Srivastava and Kumar [7] and Kumar and Bala ([8–10]), here we give definitions of generalized order, generalized lower order, generalized type, and generalized lower type of special monogenic functions. For special monogenic function and functions , we define the generalized order and generalized lower order of as If in above equation we put and , then we get definitions of order and lower order as defined by Abul-Ez and De Almeida (see [5], pp. 1263). Hence, their definitions of order and lower order are special cases of our definitions.
Further, for , we define the generalized type and generalized lower type of special monogenic function having nonzero finite generalized order as If in above equation we put , , and , then we get definitions of type and lower type as defined by Abul-Ez and De Almeida (see [5], pp. 1270). Hence, their definitions of type and lower type are special cases of our definitions.
Abul-Ez and De Almeida [5] have obtained the characterizations of order, lower order, type, and lower type of special monogenic functions in terms of their Taylor’s series coefficients. In the present paper we have obtained the characterizations of generalized order, generalized lower order, generalized type and generalized lower type of special monogenic functions in terms of their Taylor’s series coefficients. The results obtained by Abul-Ez and De Almeida [5] are special cases of our results.
Theorem 6. Let be a special monogenic function whose Taylor’s series representation is given by (9). If and , then the generalized order of is given as
Proof. Write Now, first we prove that . The coefficients of a monogenic Taylor’s series satisfy Cauchy’s inequality; that is, Also from (15), for arbitrary and all , we have Now, from inequality (19), we get Since , (see [11], pp. 148) so the above inequality reduces to Putting in the above inequality, we get, for all large values of , or or or Since , . Hence, proceeding to limits as , we get Since is arbitrarily small, so finally we get Now, we will prove that . If , then there is nothing to prove. So let us assume that . Therefore, for a given there exists such that, for all multi-indices with , we have or Now, from the property of maximum modulus (see [11], pp. 148), we have or On the lines of the proof of the theorem given by Srivastava and Kumar (see [7], Theorem 2.1, pp. 666), we get Combining this with inequality (28), we get (17). Hence, Theorem 6 is proved.
Next, we prove the following.
Theorem 7. Let be a special monogenic function whose Taylor’s series representation is given by (9). Also let and ; then the generalized type of is given as
Proof. Write Now, first we prove that . From (16), for arbitrary and all , we have where . Now, using (19), we get Now, as in the proof of Theorem 6, here this inequality reduces to Putting , we get or or Now, proceeding to limits as , we get Since is arbitrarily small, so finally we get Now, we will prove that . If , then there is nothing to prove. So let us assume that . Therefore, for a given there exists such that, for all multi-indices with , we have or Now, from the property of maximum modulus (see [11], pp. 148), we have or On the lines of the proof of the theorem given by Srivastava and Kumar (see [7], Theorem 2.2, pp. 670), we get Combining this with (43), we get (34). Hence, Theorem 7 is proved.
Theorem 8. Let be a special monogenic function whose Taylor’s series representation is given by (9). If and then the generalized lower order of satisfies Further, if is a nondecreasing function of , then equality holds in (49).
Proof. The proof of the above theorem follows on the lines of the proof of Theorem 6 and [7] Theorem 2.4 (pp. 674). Hence, we omit the proof.
Theorem 9. Let be a special monogenic function whose Taylor’s series representation is given by (9). Also let and ; then the generalized lower type of satisfies Further, if is a nondecreasing function of , then equality holds in (51).
The author is very thankful to the referee for the valuable comments and observations which helped in improving the paper.
R. de Almeida and R. S. Kraußhar, “On the asymptotic growth of entire monogenic functions,” Zeitschrift für Analysis und ihre Anwendungen, vol. 24, no. 4, pp. 791–813, 2005. View at: Publisher Site | Google Scholar | MathSciNet
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{ℝ}^{n+1}
M. A. Abul-Ez and D. Constales, “Basic sets of polynomials in Clifford analysis,” Complex Variables: Theory and Application, vol. 14, no. 1–4, pp. 177–185, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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Copyright © 2014 Susheel Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
Discrete Math Question Consider the relation R on Z defined by
Discrete Math Question Consider the relation R on Z defined by the rule that (a,b)
\left(a,b\right)\in R
a+2b
a=b=1
a+2b=1+2\left(1\right)=1+2=3
, which is not even, so
\left(1,1\right)\mathrm{¬}\left\{\in \right\}R
a\in Z,\left(a,a\right)\mathrm{¬}\left\{\in \right\}R
a=2,b=1
a+2b=2+2\left(1\right)=4
, which is even, so
\left(2,1\right)\in R
Now, take
a=1,b=2
1+2b=1+2\left(2\right)=5
\left(1,2\right)\mathrm{¬}\left\{\in \right\}R
a,b\in Z,\left(a,b\right)\in R
\left(b,a\right)\mathrm{¬}\left\{\in \right\}R
\left(a,b\right)\in R
\left(b,c\right)\in R
then a and b must be even.
a+2c
will clearly be even.
\left(A-B\right)-C=\left(A-C\right)-\left(B-C\right)
\left(a,b\right)\in R
U=\left\{x:x\in \mathbb{N},x>10\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}x<40\right\},\phantom{\rule{4pt}{0ex}}A=5,10,20,40
{A}^{c}=\left\{\text{All natural numbers between greater than 10 and less than 40 except for 20}\right\}
\mathrm{\forall }n\ge 1,\text{ }{1}^{3}+{2}^{3}+{3}^{3}+\cdots +{n}^{3}=\frac{{n}^{2}{\left(n+1\right)}^{2}}{4} |
Micro-optical spatial and spectral elements
1 November 2009 Micro-optical spatial and spectral elements
Pradeep Srinivasan, Yigit Ozan Yilmaz, Raymond C. Rumpf, Eric G. Johnson
Pradeep Srinivasan,1 Yigit Ozan Yilmaz,1 Raymond C. Rumpf,2 Eric G. Johnson1
1The Univ. of North Carolina at Charlotte (United States)
2Prime Research, LC (United States)
Optical Engineering, 48(11), 110501 (2009). https://doi.org/10.1117/1.3258651
Optical interference filters have been used to achieve a transmission notch with excellent sidelobe suppression within the stop band of a photonic crystal structure.1 A one-dimensional (1-D) photonic crystal or distributed Bragg reflector consists of alternating quarter-wave-thick layers of high- and low-index materials. A transmission notch is generated when a defect is introduced into a central layer within the structure. The spectral location within the stop band is a function of the optical thickness of the defect layer. This concept has been applied to realize transmission filters for direct integration onto image sensors.
Filters with space-variant spectral transmission have been demonstrated by patterning and etching a subwavelength array of holes with spatially varying fill fraction through the volume of the multilayer structure.2, 3 Excellent transmission tuning was achieved for a range of fill fractions. The approach has the advantage that growth and patterning/etching steps are decoupled, and it results in a simplified fabrication process. However, for small fill fractions, the effective index of the layers and their contrast reduces. The spectral linewidth of transmission becomes broader as a result. The fabrication of such filters for visible wavelengths is challenging due to the large aspect ratio of the holes that are required.4 Transmission filters for direct integration onto an image sensor have been demonstrated at visible wavelengths by spatially modulating the physical thickness of the defect layer.5 The device performs well for transmission tuning across the entire stop band since the mirrors’ reflectivity remains constant. In this approach, the defect layer was patterned and etched using a
{2}^{N}
patterning and etching approach that requires multiple alignment, lithography, and etching steps. This ultimately limits the number of discrete transmission wavelengths that can be obtained. Theoretically, the transmission can be tuned continuously across the stop band.
In this letter, we show that by incorporating diffractive phase functions in the defect layer, both discrete and pseudo-continuous spectral and spatial transmission tuning can be achieved. The spectral transmission notch results when the accumulated phase for the wavelength propagating through the structure equals zero (or
2m\pi
). The spatial transmission profile at that spectral location corresponds to the spatial contours of equal optical thickness across the defect layer. The schematic diagram of an 8-level diffractive element with a spiral phase that was incorporated on the defect layer is shown in Fig. 1. The spectral transmission under broadband illumination will consist of eight discrete wavelengths corresponding to each level of the element. The spatial transmission resulting due to the element designed to have a topological charge of value
m=2
will consist of triangular wedges separated by
180\phantom{\rule{0.3em}{0ex}}\mathrm{deg}
with wavelength-dependent orientation at each wavelength, as shown in Fig. 1 (Ref. 6). These devices represent a novel application of interference filters, since they multiplex spatial and spectral optical functionalities and realize wavelength tunable spatial filtering.
(a) Schematic of a charge 2, 8-level diffractive vortex element to be incorporated in the defect layer. The optical thickness of the diffractive changes by
2\pi
over an angular span of
180\phantom{\rule{0.3em}{0ex}}\mathrm{deg}
. (b) The micro-optical spatial and spectral element (MOSSE) with the vortex lens patterned and etched on the defect layer of a photonic crystal filter. The spatial transmission profile is expected to consist of triangular areas with wavelength-dependent angular orientation. (Color online only.)
A target structure composed of alternating layers of silicon oxide
\left(\mathrm{Si}{\mathrm{O}}_{\mathrm{x}}\right)
and nitride
\left({\mathrm{Si}}_{\mathrm{x}}{\mathrm{N}}_{\mathrm{y}}\right)
layers was analyzed by numerical modeling using rigorous coupled wave analysis (RCWA).7 Eight pairs of quarter-wave-thick oxide and nitride layers formed the distributed Bragg reflector (DBR) mirrors on either side of an oxide defect layer grown on a silicon substrate. In the simulation, the thickness of the defect layer was varied and the location and spectral width of the transmission notch were analyzed. Continuous transmission tuning was achieved across the photonic crystal stop band, as shown in Fig. 2. While the location of the stop band changes with defect thickness, its width was
\sim 10\phantom{\rule{0.3em}{0ex}}\mathrm{nm}
throughout the tuning range. High transmission
\left(>90%\right)
and excellent sidelobe suppression
\left(>15\phantom{\rule{0.3em}{0ex}}\mathrm{dB}\right)
were achieved across the tuning range. The linewidth can be made narrower and the suppression ratio can be improved by increasing the number of layers in the dielectric DBR or by using materials with higher index contrast.
Continuous tuning of the transmission as obtained across the stop band was achieved. High transmission
\left(>90%\right)
and excellent sidelobe suppression was achieved throughout the tuning range. Color bar on the right indicates percent transmission.
In order to generate an accurate design, the refractive index of the component films was characterized across the wavelength range of interest by separately depositing the thin films of oxide and nitride using plasma-enhanced chemical vapor deposition (PECVD) on a
500\text{-}\mu \mathrm{m}
-thick silicon substrate. Refractive index values of 1.4608 and 1.948 were extracted from the parameters measured on a ellipsometer at
1.55\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}
for the oxide and nitride films, respectively. As per design, eight pairs of
265\text{-}\mathrm{nm}
-thick silicon oxide
\left(\mathrm{Si}{\mathrm{O}}_{\mathrm{x}}\right)
199\text{-}\mathrm{nm}
-thick silicon nitride
\left({\mathrm{Si}}_{\mathrm{x}}{\mathrm{N}}_{\mathrm{y}}\right)
followed by a
450\text{-}\mathrm{nm}
-thick defect layer of
\mathrm{Si}{\mathrm{O}}_{\mathrm{x}}
were deposited using PECVD on a silicon substrate, and the wafer was removed from the growth chamber. Shipley 1813 positive photoresist was spin-coated to a thickness of
1.1\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}
. The fabrication of diffractives on the defect layer was accomplished by the use of additive lithography on a GCA 6300 g-line stepper tool, which is a simple one-step process for fabricating multilevel optical elements.8
The developed patterns were transferred into the defect layer by dry etching in
\mathrm{C}\mathrm{H}{\mathrm{F}}_{3}∕{\mathrm{O}}_{2}
plasma chemistry. The selectivity of etching was defined as the ratio of etched oxide to the etched photoresist. The low sag diffractives (on the order of
500\phantom{\rule{0.3em}{0ex}}\mathrm{nm}
or less) were transferred using a low-selectivity process. The selectivity was controlled by changing the amount of oxygen in the plasma chemistry. Increasing amounts of oxygen in the chemistry increased the photoresist etch rate linearly but did not modify the oxide etch rate significantly. A cross-sectional scanning electron micrograph (SEM) of one of the levels of the fabricated filter is shown in Fig. 3. Heights of the levels were measured using a DekTak profilometer. The top DBR was then grown, and the device fabrication was completed.
Cross-sectional scanning electron micrograph (SEM) of the fabricated multilayer stack obtained by cleaving through one of the diffractive levels. The layered structure consisting of nitride (lighter regions) and oxide (darker regions) layers with an oxide defect layer is clearly seen in the micrograph.
The profilometer measured levels corresponding to each of the eight levels in the diffractive are summarized in Fig. 4. The heights of the diffractive levels deviated from the target heights by 10%. While this change would impact the performance as a diffractive optical element, the filter performance for the intended application would not be impacted, since the transmission at these thicknesses are spaced farther than the spectral FWHM. The devices were interrogated with an unpolarized laser source that had a tuning range from
1520\phantom{\rule{0.3em}{0ex}}\mathrm{nm}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}1630\phantom{\rule{0.3em}{0ex}}\mathrm{nm}
coupled to a single-mode fiber with pigtailed collimator producing a
320\text{-}\mu \mathrm{m}
beam. Four of the vortex levels had defect layers with thicknesses appropriate for transmission in the wavelength range of the tunable laser. The results from simulation and experiment are compared in Fig. 4. The transmission peaks from the four diffractive levels were located at
1532\phantom{\rule{0.3em}{0ex}}\mathrm{nm}
1552\phantom{\rule{0.3em}{0ex}}\mathrm{nm}
1575\phantom{\rule{0.3em}{0ex}}\mathrm{nm}
1610\phantom{\rule{0.3em}{0ex}}\mathrm{nm}
. The transmission line had a full width at half maximum (FWHM) of
10\phantom{\rule{0.3em}{0ex}}\mathrm{nm}
in simulation and experiment. The spatial transmission through the space-variant filter element was imaged onto a CCD camera. The beam from the tunable laser was amplified using an erbium-doped fiber amplifier (EDFA), and the beam diameter was expanded to
3\phantom{\rule{0.3em}{0ex}}\mathrm{mm}
. This was done to ensure that large sections of the vortex element were illuminated by the beam. As expected, the transmission through the element was composed in triangular wedges separated by
180\phantom{\rule{0.3em}{0ex}}\mathrm{deg}
in angular space with wavelength-dependent orientation, as shown in Fig. 4. Wavelength-tunable spatial filters were demonstrated by integrating diffractive optical elements in the defect layer of an interference filter.
(a) Optical microscope image of the charge 2 vortex elements and schematic. (b) Simulated (dotted lines) and experimental transmission spectra. (c) Images of the spatial transmission through the vortex element at
1532\phantom{\rule{0.3em}{0ex}}\mathrm{nm}
1553\phantom{\rule{0.3em}{0ex}}\mathrm{nm}
. The vortex element was illuminated using a beam of diameter
3\phantom{\rule{0.3em}{0ex}}\mathrm{mm}
A novel implementation of interference filters was proposed and demonstrated in this paper. Multiplexed spatial and spectral transmission profiles were obtained by patterning diffractive optical elements in the defect layer of a photonic crystal transmission filter. The transmission characteristics were studied by simulation, and the experimental results match closely with the expected transmission profiles. Space-variant spectral transmission was achieved, and the peak location and linewidth matched the model simulations. Triangular wedges with wavelength-dependent orientation were transmitted spatially. Other transmission profiles can be obtained by incorporating appropriate phase functions on the defect layer. The devices couple spatial and spectral filtering and can be used as pupil filters for advanced multispectral imaging systems. The demonstrated devices have direct applications in other pupil filtering applications, hyperspectral imaging, and engineered illumination.
This work was funded in part through a National Science Foundation CAREER Grant (ECS0348280).
P. H. Lissberger, “Properties of all-dielectric interference filters. i. a new method of calculation,” J. Opt. Soc. Am., 49 (2), 121 –122 (1959). https://doi.org/10.1364/JOSA.49.000121 0030-3941 Google Scholar
P. Filloux and N. Paraire, “Use of multilayer structures, periodically etched, to implement compact diffractive optical devices,” J. Opt. A, Pure Appl. Opt., 4 (5), 175 –181 (2002). https://doi.org/10.1088/1464-4258/4/5/367 1464-4258 Google Scholar
A. Mehta, R. C. Rumpf, Z. Roth, and E. G. Johnson, “Nanofabrication of a space-variant optical transmission filter,” Opt. Lett., 31 (19), 2903 –2905 (2006). https://doi.org/10.1364/OL.31.002903 0146-9592 Google Scholar
G. Shambat, R. Athale, G. Euliss, M. Mirotznik, E. Johnson, and V. Smolski, “Reconfigurable photonic crystal filters for multi-band optical filtering on a monolithic substrate,” Proc. SPIE, 7041 70410P-12 (2008). https://doi.org/10.1117/12.794239 0277-786X Google Scholar
Y. Inaba, M. Kasano, K. Tanaka, and T. Yamaguchi, “Degradation-free MOS image sensor with photonic crystal color filter,” IEEE Electron Device Lett., 27 (6), 457 –459 (2006). https://doi.org/10.1109/LED.2006.874126 0741-3106 Google Scholar
J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander Jr, “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett., 97 (5), 053901 (2006). https://doi.org/10.1103/PhysRevLett.97.053901 0031-9007 Google Scholar
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Pradeep Srinivasan, Yigit Ozan Yilmaz, Raymond C. Rumpf, and Eric G. Johnson "Micro-optical spatial and spectral elements," Optical Engineering 48(11), 110501 (1 November 2009). https://doi.org/10.1117/1.3258651
Fabrication of ultra thin Si nanopillar arrays for polarization independent...
Fabrication issues in micromachined tunable optical filters
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Fabrication and optical characterization of Si3N4 2D photonic crystals for...
Pradeep Srinivasan, Yigit Ozan Yilmaz, Raymond C. Rumpf, Eric G. Johnson, "Micro-optical spatial and spectral elements," Opt. Eng. 48(11) 110501 (1 November 2009) https://doi.org/10.1117/1.3258651 |
Perpendicular Lines Practice Problems Online | Brilliant
If the vertical line is perpendicular to both of the horizontal lines, which of the following statements about the blue and purple angles is false?
They are equal They are complementary They are supplementary They are right angles
If the blue and green lines are perpendicular, which of the following statements is true about the yellow and purple angles?
They are right angles They are equal They are complementary They are supplementary
\angle a = 53^\circ
\angle b = 37^\circ
, which of the following statements are true:
l
m
l
n
III) None of the lines in this diagram are perpendicular.
Only statement III is true. Only statement II is true. Statements I and II are both true. Only statement I is true.
l \perp m
n \perp p
\angle a = 31^\circ
\angle b
13 ^ \circ
31 ^ \circ
59 ^ \circ
90 ^ \circ
What is the angle measure (in degrees) between 2 perpendicular lines?
30 ^ \circ
45 ^ \circ
90 ^ \circ
180 ^ \circ |
Moving average - Simulink - MathWorks í•œêµ
Data over which the block computes the moving average. The block accepts real-valued or complex-valued multichannel inputs, that is, m-by-n size inputs, where m ≥ 1 and n ≥ 1. The block also accepts variable-size inputs. During simulation, you can change the size of each input channel. However, the number of channels cannot change.
\begin{array}{l}{w}_{N,\mathrm{λ}}=\mathrm{λ}{w}_{Nâ1,\mathrm{λ}}+1\\ {\stackrel{¯}{x}}_{N,\mathrm{λ}}=\left(1â\frac{1}{{w}_{N,\mathrm{λ}}}\right){\stackrel{¯}{x}}_{Nâ1,\mathrm{λ}}+\left(\frac{1}{{w}_{N,\mathrm{λ}}}\right){x}_{N}\end{array}
{\stackrel{¯}{x}}_{N,\mathrm{λ}}
{x}_{N}
{\stackrel{¯}{x}}_{Nâ1,\mathrm{λ}}
{w}_{N,\mathrm{λ}}
\left(1â\frac{1}{{w}_{N,\mathrm{λ}}}\right){\stackrel{¯}{x}}_{Nâ1,\mathrm{λ}}
{w}_{N,\mathrm{λ}} |
Use the method of Laplace transformation to solve initial value
Use the method of Laplace transformation to solve initial value problem. \frac{dx}{dt}=x-2y , x(0)=-1, y(0)=2 , \frac{dy}{dt}=5x-y
\frac{dx}{dt}=x-2y,x\left(0\right)=-1,y\left(0\right)=2
\frac{dy}{dt}=5x-y
We know in Laplace transformation
L\left\{{y}^{n}\left(t\right)\right\}={s}^{n}\overline{y}\left(s\right)-{s}^{n-1}y\left(0\right)-{s}^{n-2}{y}^{\prime }\left(0\right)\dots {y}^{n-1}\left(0\right)
L\left\{{t}^{n}\right\}=\frac{n!}{{s}^{n+1}},L\left\{\mathrm{cos}at\right\}=\frac{s}{{s}^{2}+{a}^{2}},L\left\{\mathrm{sin}\left(at\right)\right\}=\frac{a}{{s}^{2}+{a}^{2}}
Now Given IVP is
\frac{dx}{dt}=x-2y\left(1\right)x\left(0\right)=-1,y\left(0\right)=2
\frac{dy}{dt}=5x-y\left(2\right)
⇒{x}^{\prime }\left(t\right)-x+2y=0
{y}^{\prime }\left(t\right)-5x+y=0
Taking laplace transformation on both the equation and both the side
L\left\{x\left(t\right)\right\}-L\left\{x\left(t\right)\right\}+2L\left\{y\left(t\right)\right\}=0
⇒sx\left(s\right)-x\left(0\right)-x\left(s\right)+2y\left(s\right)=0
⇒\left(s-1\right)x\left(s\right)+2y\left(s\right)=-1\left(3\right)
L\left\{{y}^{\prime }\left(t\right)\right\}-5L\left\{x\left(t\right)\right\}+L\left\{y\left(t\right)\right\}=0
⇒sy\left(s\right)-y\left(0\right)-5x\left(s\right)+y\left(s\right)=0
⇒\left(s+1\right)y\left(s\right)-5x\left(s\right)=2\left(4\right)
⇒5x\left(s-1\right)x\left(s\right)+2y\left(s\right)=-1
\left(s-1\right)×\left(s+1\right)y\left(s\right)-5x\left(s\right)=2
\left(\left({s}^{2}-1\right)+10\right)y\left(s\right)=2\left(s-1\right)-5
⇒y\left(s\right)=\frac{2\left(s-1\right)-5}{{s}^{2}+9}=2\frac{s}{{s}^{2}+9}-\frac{7}{3}\frac{3}{{s}^{2}+9}
Taking inverse laplace transformation both the side . We get
f\left(t\right)=36t+5{\int }_{0}^{t}f\left(t-u\right)\mathrm{sin}\left(5u\right)du
Find the Laplace transforms of the following time functions.
Solve problem 1(a) and 1 (b) using the Laplace transform definition i.e. integration. For problem 1(c) and 1(d) you can use the Laplace Transform Tables.
f\left(t\right)=1+2t
f\left(t\right)=\mathrm{sin}\omega t\text{Hint: Use Euler’s relationship, }\mathrm{sin}\omega t=\frac{{e}^{\left(}j\omega t\right)-{e}^{\left(}-j\omega t\right)}{2j}
f\left(t\right)=\mathrm{sin}\left(2t\right)+2\mathrm{cos}\left(2t\right)+{e}^{-t}\mathrm{sin}\left(2t\right)
L\left\{{\int }_{0}^{t}\frac{1-{e}^{u}}{u}du\right\}
Find the Laplace transform by the method of definition.
f\left(t\right)={e}^{\frac{t}{5}}
2\cdot \left(\frac{dy}{dx}\right)+2y=0
Solve the Homogenous Differential Equations.
\left(x-y\mathrm{ln}y+y\mathrm{ln}x\right)dx+x\left(\mathrm{ln}y-\mathrm{ln}x\right)dy=0
{\int }_{0}^{\mathrm{\infty }}\left\{\mathrm{cos}\left(xt\right)over1+{t}^{2}\right\}dt |
Null vector - Wikipedia
Vector on which a quadratic form is zero
This article is about zeros of a quadratic form. For the zero element in a vector space, see Zero vector. For null vectors in Minkowski space, see Minkowski space § Causal structure.
A null cone where
{\displaystyle q(x,y,z)=x^{2}+y^{2}-z^{2}.}
In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0.
In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.
A quadratic space (X, q) which has a null vector is called a pseudo-Euclidean space.
A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, X = A + B, where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres:
{\displaystyle \bigcup _{r\geq 0}\{x=a+b:q(a)=-q(b)=r,a\in A,b\in B\}.}
The null cone is also the union of the isotropic lines through the origin.
The light-like vectors of Minkowski space are null vectors.
The four linearly independent biquaternions l = 1 + hi, n = 1 + hj, m = 1 + hk, and m∗ = 1 – hk are null vectors and { l, n, m, m∗ } can serve as a basis for the subspace used to represent spacetime. Null vectors are also used in the Newman–Penrose formalism approach to spacetime manifolds.[1]
A composition algebra splits when it has a null vector; otherwise it is a division algebra.
In the Verma module of a Lie algebra there are null vectors.
^ Patrick Dolan (1968) A Singularity-free solution of the Maxwell-Einstein Equations, Communications in Mathematical Physics 9(2):161–8, especially 166, link from Project Euclid
Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. (1984). Modern Geometry: Methods and Applications. Translated by Burns, Robert G. Springer. p. 50. ISBN 0-387-90872-2.
Shaw, Ronald (1982). Linear Algebra and Group Representations. Vol. 1. Academic Press. p. 151. ISBN 0-12-639201-3.
Neville, E. H. (Eric Harold) (1922). Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions. Cambridge University Press. p. 204.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Null_vector&oldid=1079965044" |
Active networking - Wikipedia
Telecommunications routing system
This article is about the network architecture. For the technology company, see ACTIVE Network, LLC (company).
Active network architecture is composed of execution environments (similar to a unix shell that can execute active packets), a node operating system capable of supporting one or more execution environments. It also consists of active hardware, capable of routing or switching as well as executing code within active packets. This differs from the traditional network architecture which seeks robustness and stability by attempting to remove complexity and the ability to change its fundamental operation from underlying network components. Network processors are one means of implementing active networking concepts. Active networks have also been implemented as overlay networks.
2 How it relates to other networking paradigms
2.1 Active networking and software-defined networking
4 Nanoscale active networks
What does it offer?[edit]
Active networking allows the possibility of highly tailored and rapid "real-time" changes to the underlying network operation. This enables such ideas as sending code along with packets of information allowing the data to change its form (code) to match the channel characteristics. The smallest program that can generate a sequence of data can be found in the definition of Kolmogorov complexity. The use of real-time genetic algorithms within the network to compose network services is also enabled by active networking.
How it relates to other networking paradigms[edit]
Active networking relates to other networking paradigms primarily based upon how computing and communication are partitioned in the architecture.
Active networking and software-defined networking[edit]
Active networking is an approach to network architecture with in-network programmability. The name derives from a comparison with network approaches advocating minimization of in-network processing, based on design advice such as the "end-to-end argument". Two major approaches were conceived: programmable network elements ("switches") and capsules, a programmability approach that places computation within packets traveling through the network. Treating packets as programs later became known as "active packets". Software-defined networking decouples the system that makes decisions about where traffic is sent (the control plane) from the underlying systems that forward traffic to the selected destination (the data plane). The concept of a programmable control plane originated at the University of Cambridge in the Systems Research Group, where (using virtual circuit identifiers available in Asynchronous Transfer Mode switches) multiple virtual control planes were made available on a single physical switch. Control Plane Technologies (CPT) was founded to commercialize this concept.
Fundamental challenges[edit]
Active network research addresses the nature of how best to incorporate extremely dynamic capability within networks.[1]
In order to do this, active network research must address the problem of optimally allocating computation versus communication within communication networks.[2] A similar problem related to the compression of code as a measure of complexity is addressed via algorithmic information theory.
One of the challenges of active networking has been the inability of information theory to mathematically model the active network paradigm and enable active network engineering. This is due to the active nature of the network in which communication packets contain code that dynamically change the operation of the network. Fundamental advances in information theory are required in order to understand such networks.[3]
An active network channel uses executable code in the packet to impact the channel controlling the relationship between the transmitted sequence
{\displaystyle X}
and the received sequence
{\displaystyle Y}
{\displaystyle X}
is composed of a data portion
{\displaystyle X^{data}}
and a code portion
{\displaystyle X^{code}}
. Upon incorporation of
{\displaystyle X^{code}}
, the channel medium may change its operational state and capabilities.[4]
Nanoscale active networks[edit]
As the limit in reduction of transistor size is reached with current technology, active networking concepts are being explored as a more efficient means accomplishing computation and communication.[5][6] More on this can be found in nanoscale networking.
Nanoscale networking
^ Bush, S. F. (2005). "A Simple Metric for Ad Hoc Network Adaptation" (PDF). IEEE Journal on Selected Areas in Communications Journal. 23 (23): 2272–2287. doi:10.1109/JSAC.2005.857204. S2CID 17916856. Archived from the original (PDF) on 2011-07-11.
^ Bush, S. F. (2002). "Active Virtual Network Management Prediction: Complexity as a Framework for Prediction, Optimization, and Assurance" (PDF). Proceedings of the 2002 DARPA Active Networks Conference and Exposition (DANCE 2002). IEEE Computer Society Press: 534–553. arXiv:cs/0203014. Bibcode:2002cs........3014B. doi:10.1109/DANCE.2002.1003518. ISBN 0-7695-1564-9. S2CID 1202234. Archived from the original (PDF) on 2011-07-11.
^ Bush, Stephen F. (2011). "Toward in vivo nanoscale communication networks: utilizing an active network architecture". Front. Comput. Sci. 5 (3): 316–326. doi:10.1007/s11704-011-0116-9. S2CID 3436762.
^ ``NANA: A Nanoscale Active Network Architecture by Patwardhan, J. P.; Dwyer, C. L.; Lebeck, A. R. & Sorin, D. J., ACM Journal on Emerging Technologies in Computing Systems (JETC), ACM Journal on Emerging Technologies in Computing Systems) Vol. 2, No. 1, Pages 1–30, January 2006, 3, 1–31.
^ Nanoscale Communication Networks, Bush, S. F., ISBN 978-1-60807-003-9, Artech House, 2010 https://www.amazon.com/Nanoscale-Communication-Networks-Stephen-Bush/dp/1608070034
Programmable Networks for IP Service Deployment" by Galis, A., Denazis, S., Brou, C., Klein, C.- Artech House Books, London, June 20;, 450 pp. ISBN 1-58053-745-6.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Active_networking&oldid=1068121255" |
\Let f(x) =x^{5} + 5x^{4} + 6x +3. a. Conduct a
\Let f(x) =x^{5} + 5x^{4} + 6x +3. a. Conduct a sign analysi
Letf\left(x\right)={x}^{5}+5{x}^{4}+6x+3.
F\left(x\right)={x}^{5}+5{x}^{4}+6x+3
differentioting the above ef wet (x) we heve
{F}^{1}\left(x\right)=5{x}^{4}+20{x}^{3}+6
again differentiat wet (x),we have
{F}^{11}\left(x\right)=20{x}^{3}+60{x}^{2}
(a.) for beundasies (endpoints) of he interval we apply.
{F}^{11}\left(x\right)=0
20{x}^{3}+60{x}^{2}=0
ov,
20{x}^{2}\left(x+3\right)=0
The roots of the abore equation are 0,0 8-3
b) for concave up,
{F}^{11}\left(x\right)>0
So,for
x\succ 3,
F(x) is concave up.
(c.) Infection points are those points at which
Function changes its concarity that is concave up to concave down ov vice-versa .
In the secHon(a) ,we see that
{F}^{11}\left(x\right)
cnange its sign once at -3 where concarity changes from concave down to concave up.
So, the point of Inflection is -3.
P\left(x\right)=-12{x}^{2}+2136x-41000
x=r\mathrm{cos}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y=r\mathrm{sin}
\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{{x}^{2}-{y}^{2}}{\sqrt{{x}^{2}+{y}^{2}}}
{a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)
{a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)
{x}^{6}-1=0
In each of the following problems , use the information given to determine
\left(f+g\right)\left(-1\right)
\left(f-g\right)\left(-1\right)
\left(fg\right)\left(-1\right)
\left(\frac{f}{g}\right)\left(-1\right)
f=\left\{\left(5,2\right),\left(0,-1\right),\left(-1,3\right),\left(-2,4\right)\right\}\text{ and }g=\left\{\left(-1,3\right),\left(0,5\right)\right\}
f=\left\{\left(3,15\right),\left(2,-1\right),\left(-1,1\right)\right\}\text{ and }g\left(x\right)=-2
\left({P}_{2}=0.6\right)
\left({P}_{3}=0.8\right)
\left({P}_{2}=0.6\right)
\left(1-{P}_{3}=0.2\right)
\left(1-{P}_{2}=0.4\right)
\left({P}_{4}=0.6\right)
Find the indefinite integral by making a change of variables.
\int {x}^{2}\sqrt{1-x}dx |
Protocol Design - Distribution and Units - Nano Documentation
Protocol Design - Distribution and Units
Distribution and Units Distribution and Units Table of contents
Protocol Design - Distribution and Units¶
Page may be migrating
This page may be migrated into another page or section - TBD.
Divisibility¶
There are three important aspects of divisibility of the supply which are satisfied by the final distributed amount:
The supply needs to be able to be divided up amongst a large number of users with users possibly wanting several accounts.
Each account needs to be able to represent an adequate dynamic range of value.
The supply should be able to deal with deflation over time as accounts are abandoned.
The distribution of Nano (formerly RaiBlocks) was performed through solving manual captchas starting in late 2015 and ending in October 2017. Distribution stopped after ~39% of the Genesis amount was distributed and the rest of the supply was burnt.1
Genesis: nano_3t6k35gi95xu6tergt6p69ck76ogmitsa8mnijtpxm9fkcm736xtoncuohr3
Landing: nano_13ezf4od79h1tgj9aiu4djzcmmguendtjfuhwfukhuucboua8cpoihmh8byo
Faucet: nano_35jjmmmh81kydepzeuf9oec8hzkay7msr6yxagzxpcht7thwa5bus5tomgz9
Burn: nano_1111111111111111111111111111111111111111111111111111hifc8npp
During distribution the Genesis seed was kept in cold storage and funds were moved to the Landing account once per week to minimize the number of live, undistributed blocks. These were subsequently moved into the Faucet account for distribution until the faucet was closed and remaining funds sent to the Burn account.
With 2^{128} - 1
2^{128} - 1
raw (i.e. FFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFF HEX raw) in the original Genesis account, upon closing of the faucet and burning of the remaining funds, the total supply which is 100% in circulation ended at ~133,248,297 nano (or more precisely 133248297920938463463374607431768211455 raw). Since then, additional funds have been sent to the known burn address slightly lowering the amount in circulation as a result. This amount can be found using the available_supply RPC.
Unit Dividers¶
A 128 bit integer is used to represent account balances. The reference wallet uses nano as a divider.
nano (NANO/Nano) 1000000000000000000000000000000 10^{30}
10^{30}
Mnano
raw 1 10^{0}
10^{0}
NOTE: 1 raw is the smallest possible division and is used in QR codes as amount, while nano is the current standard division used for human readable elements in most wallets, on exchanges, etc.
A set of SI prefixes2 from the base nano has been previously used to make the numbers more accessible and avoid confusion in certain scenarios, but this approach is not common (e.g., micronano or μnano for 10^{24}
10^{24}
https://medium.com/nanocurrency/the-nano-faucet-c99e18ae1202 ↩
The SI prefixes are metric prefixes that were standardized for use in the International System of Units (SI) by the International Bureau of Weights and Measures (BIPM). https://www.bipm.org/en/measurement-units/si-prefixes ↩
Previous Resource Usage
Next Signing, Hashing and Key Derivation |
Schrödinger's Cat | Brilliant Math & Science Wiki
Adam Strandberg and Eli Ross contributed
Two worlds "splitting off"- one where the cat is alive and one where the cat is dead. Taken from [MWI].
Schrödinger's cat is a thought experiment designed to show how certain interpretations of quantum mechanics lead to counterintuitive results.
In the experiment, a cat is placed inside a box with a vial of poisonous gas. A mallet is set up such that it breaks the vial of gas if a particular radioactive atom decays, killing the cat. Since the radioactive decay is a quantum system, whether the cat lives or dies is determined by quantum mechanical behavior. This leads to the conclusion that before the box is opened, the cat is simultaneously alive and dead.
Erwin Schrödinger originally proposed the idea as an absurd example showing that the Copenhagen Interpretation of quantum mechanics - the most popular philosophical interpretation at the time - could not possibly be true [1]. However, it has lived on as a thought experiment fueling both physical theories and the popular imagination.
Quantum States and Superposition
The two main ideas behind quantum theory are the idea of quantized, or discrete, states, and the idea of superposition.
A physical system is defined by the set of possible states in which it can be observed. For instance, an electron can be observed either in a spin up state or in a spin down state, but never a combination of the two. Similarly, a particle emitted in radioactive decay is observed to be either emitted or not emitted, never partway emitted. However, particles prepared in identical ways will not always be observed to have the same state. Identical uranium atoms will decay or not decay at random times, though the more time has passed, the more likely the atom is to decay. The state of the system will change such that it becomes more likely to decay.
To determine how states change over time, the idea of a superposition of states is required. A superposition is a vector addition of two states. Quantum states can be in any superposition of the observable states in the system. For instance, an electron can be in a state that is 50% spin up and 50% spin down. As time passes, the electron may change states, perhaps smoothly oscillating between 100% spin up and 100% spin down, passing through the 50/50 state at each time. The Schrödinger equation, the analog of Newton's laws for quantum mechanics, describes exactly how the electron will transition between these different superpositions of states.
But this seems to be at odds with the previous statement. How can a system be an a superposition of states if it can only ever be observed in one state? The Copenhagen interpretation of quantum mechanics holds that when an observer observes a system, it "collapses" from a superposition of multiple states down to a single state in a probabilistic way. This is distinct from the smooth changes in superposition that happen due to the Schrödinger equation.
When Schrödinger's cat is observed, it is either alive or dead. But when it isn't, it is in a superposition of alive and dead.
Technical note: the space of valid states is the set of complex vectors with an eigenbasis given by the observable states and magnitude 1. For linear combinations of eigenvectors, the probability of observing each component of the state vector is given by the magnitude of the coefficient of that component. Linear algebra is important for a deep understanding of quantum mechanics: most results come directly from the properties of operators on complex vector spaces.
The collapse interpretation has an issue, though. Instead of just you observing a cat in a box, imagine putting yourself and the cat in a room and having your friend wait outside the room. You run the experiment, open the box, record your observations, and only then have your friend open the room and see what your observations were. From your perspective, the cat is in a superposition of alive and dead until you open the box, at which point a collapse occurs. You are then in the definite state of having seen the cat, a state which persists until your friend opens the door. But from your friend's perspective, up until they open the door, you are still in a superposition of having seen the cat dead and having seen it alive.
Even worse, this setup can be repeated again and again, such that every new observer is placed in a larger room. Observer
n
always thinks that the state has collapsed before observer
n + 1
. The idea that different observers will disagree on the state of reality in this experiment is problematic.
The many-worlds interpretation of quantum mechanics solves this problem by rejecting the idea of collapse entirely. It instead claims that there is always a superposition of two "world-branches," one where the cat is dead and one where the cat is alive. When you open the box, there is now a superposition of two worlds with two versions of you. In one world, the cat is alive and you see the cat is alive. In the other world, the cat is dead and you see the cat is dead.
[1] Schrödinger, Erwin; Translated by Trimmer, John. The Present Situation in Quantum Mechanics. Proceedings of the American Philosophical Society. Retrieved on 7 Mar 2016 from http://www.tuhh.de/rzt/rzt/it/QM/cat.html
[flickr] Flickr user chwalker01. Retrieved on 7 Mar 2016 from https://www.flickr.com/photos/31690139@N02/2965956885
Cite as: Schrödinger's Cat. Brilliant.org. Retrieved from https://brilliant.org/wiki/schrodingers-cat/ |
Discrete-time or continuous-time low-pass filter - Simulink - MathWorks í•œêµ
Low-Pass Filter (Discrete or Continuous)
Discrete-time or continuous-time low-pass filter
The Low-Pass Filter (Discrete or Continuous) block implements a low-pass filter in conformance with IEEE 421.5-2016[1]. In the standard, the filter is referred to as a Simple Time Constant.
To configure the filter for continuous time, set the Sample time property to 0. This representation is equivalent to the continuous transfer function:
G\left(s\right)=\frac{K}{Ts+1},
K is the filter gain.
T is the filter time constant.
From the preceeding transfer function, the filter defining equations are:
\left\{\begin{array}{c}\stackrel{Ë}{x}\left(t\right)=\frac{1}{T}\left(Ku\left(t\right)âx\left(t\right)\right)\\ y\left(t\right)=x\left(t\right)\end{array}\text{â}\text{â}\text{â}y\left(0\right)=x\left(0\right)=K{u}_{0},
u is filter input.
x is filter state.
y is filter output.
To configure the filter for discrete time, set the Sample time property to a positive, nonzero value, or to -1 to inherit the sample time from an upstream block. The discrete representation is equivalent to the transfer function:
G\left(z\right)=K\frac{\left({T}_{s}/T\right){z}^{â1}}{1+\left({T}_{s}/Tâ1\right){z}^{â1}},
Ts is the filter sample time.
From the discrete transfer function, the filter equations are defined using the forward Euler method:
\left\{\begin{array}{c}x\left(n+1\right)=\left(1â\frac{{T}_{s}}{T}\right)x\left(n\right)+K\left(\frac{{T}_{s}}{T}\right)u\left(n\right)\\ y\left(n\right)=x\left(n\right)\end{array}\text{â}\text{â}\text{â}y\left(0\right)=x\left(0\right)=K{u}_{0},
u is the filter input.
x is the filter state.
y is the filter output.
The anti-windup method limits the integrator state between the lower saturation limit A and upper saturation limit B:
A<=x<=B\text{â}.
Because the state is limited, the output can respond immediately to a reversal of the input sign when the integral is saturated. This block diagram depicts the implementation of the anti-windup saturation method in the filter.
Set the time constant to a value smaller than or equal to the sample time to ignore the dynamics of the filter. When bypassed, the block feeds the gain-scaled input directly to the output:
Tâ¤{T}_{s}ây=Ku
Low-pass filter input signal. The block uses the input initial value to determine the state initial value.
Low-pass filter output.
Gain — Filter gain
Low-pass filter gain.
Time constant — Filter time constant
Low-pass filter time constant. In the discrete implementation, set this value to less than the Sample time to bypass the dynamics of the filter.
Low-pass filter upper state limit. Set this to inf for an unsaturated upper limit, or to a finite value to prevent upper windup of the filter's integrator.
Low-pass filter lower state limit. Set this to -inf for an unsaturated lower limit, or to a finite value to prevent lower windup of the filter's integrator.
Filtered Derivative (Discrete or Continuous) | Lead-Lag (Discrete or Continuous) | Washout (Discrete or Continuous) | Integrator (Discrete or Continuous) | Integrator with Wrapped State (Discrete or Continuous) |
While running, a 70-kg student generates thermal energy at a rate of 1
While running, a 70-kg student generates thermal energy at a rate of 1200 W. For the runner to maintain a constant body temperature of
{37}^{\circ }C
, this energy must be removed by perspiration or other mechanisms. If these mechanisms failed and the energy could not flow out of the student’s body, for what amount of time could a student run before irreversible body damage occurred? (Protein structures in the body are irreversibly damaged if body temperature rises to
{44}^{\circ }C
or higher. The specific heat of a typical human body is
3480\frac{J}{k}g\cdot K
, slightly less than that of water. The difference is due to the presence of protein, fat, and minerals, which have lower specific heats.)
The time taht could the student runs before body damage occured
{m}_{student}=70kgPower=1200\frac{J}{s}
{T}_{1}=37{C}^{0}{T}_{2}=44{C}^{0}C=3480\frac{J}{k}g.K
Calculate the quantity heat of the body
{Q}_{body}={m}_{student}×C×\delta T
=70kg×3480\frac{J}{k}g.K×\left(44-37\right){C}^{0}
=1.7×{10}^{6}J
From the Power (rate) we would find the time where
Time=\frac{Q}{P}
=\frac{1.27×{10}^{6}J}{1200\frac{J}{s}}
=1400\mathrm{sec}
=23minute
So the time before damage occurs to the body is
1421 seconds or 23.67 minutes
Q=Heat=1200W
c=\text{Specific heat of human body}=3480\frac{J}{k}gK
\mathrm{△}T=\text{Change in temperature}={\left(44-37\right)}^{\circ }C
t=\text{Time taken}
Q=\frac{mc\mathrm{△}T}{t}
⇒t=\frac{mc\mathrm{△}T}{Q}
⇒t=\frac{70×3480×\left(44-37\right)}{1200}
⇒t=1421s
The time student jogs before irreversible body damage occurs is 1421 seconds or 23.67 minutes.
q=mass×\text{specific heat}×deltaT
q=70kg×3480J/kg\cdot K×7=??
q×\left(\frac{1sec}{1200J}\right)=
time in sec.
Check my thinking. 1421 seconds (about 24 minutes)
ra\frac{d}{{s}^{2}}
-1.59\cdot {10}^{4}
ra\frac{d}{{s}^{2}}
\stackrel{\to }{F}
A buffer contains significant amounts of acetic acid and sodium acetate. Write equations showing how this buffer neutralizes added acid and added base.
It is given in the table that the midyear population of India (in millions) for the last half of the 20 th century. The year are given in alternate 10 years with respect to their population data given.
Draw a scatter plot, semiloge plot, and log-log plot for the data and find the type of model.
Finde an exponential model for the population.
Calculate the population in 2010 by comparing itwith the population of 1173 million and show it’s conclusion. |
15x less than 73, where (i)x belongs to N and (ii) Z solve the inequation and represent the solution - Maths - Linear Inequalities - 7543797 | Meritnation.com
15\mathrm{x}<73\phantom{\rule{0ex}{0ex}}\mathrm{Now}\quad \mathrm{considering}\quad \mathrm{first}\quad \mathrm{case}\quad \mathrm{when},\quad \mathrm{x}\in \mathrm{N}\quad ,\quad \phantom{\rule{0ex}{0ex}}15\mathrm{x}<73\phantom{\rule{0ex}{0ex}}\mathrm{or}\quad \quad \mathrm{x}<\frac{73}{15}\quad \mathrm{or}\quad \mathrm{x}<4.8\quad \phantom{\rule{0ex}{0ex}}\mathrm{But}\quad \mathrm{x}\quad \mathrm{is}\quad \mathrm{natural}\quad \mathrm{number}\quad \mathrm{so}\quad \mathrm{x}=1,2,3,4\quad \mathrm{are}\quad \mathrm{the}\quad \mathrm{required}\quad \mathrm{value}\quad \mathrm{of}\quad \mathrm{x}.\quad \phantom{\rule{0ex}{0ex}}
on the number line these values are shown as A,B,C and D.
Case-2-
when\quad x\in Z\quad then,\quad \phantom{\rule{0ex}{0ex}}x<4.8\quad can\quad be\quad shown\quad as,\quad \phantom{\rule{0ex}{0ex}}x\in (-\infty ,\quad 4.8)\quad \phantom{\rule{0ex}{0ex}}and\quad on\quad number\quad line\quad it\quad can\quad be\quad shown\quad as,\quad \phantom{\rule{0ex}{0ex}} |
Fundamentals of Neuroscience/Electrical Currents - Wikiversity
1 Basic Properties of Electrical Currents
1.2.2 Atomic Charge
1.2.3 Ionic Charge
1.2.4 Electric Transfer from Ion to Ion
1.2.5 Ion Currents
1.2.6 Electrical Transfer from High to Low Potential
1.2.7 Charge Separation across a Dielectric
1.2.8 The Cellular Membrane as a Capacitor
Basic Properties of Electrical CurrentsEdit
To Introduce the electron
To Introduce the idea of Electron Shells
To Introduce the idea of charge of an Atom
To Introduce the idea of Ions as charged Atoms
To Introduce the idea of Amperage as Current
To introduce the idea of Ions moving in a particular direction as an Ion Current
To introduce the idea of Voltage as EMF
To introduce the idea of Capacitance
Although Electricity in its many forms is an interesting subject, in its own right, the idea of this course is to give you enough of an understanding of how Electricity works in the Neuron to be able to make sense out of more advanced training opportunities. As such we aren't going to get into the advanced uses of electricity, or electronics, instead we will cover the basics, and then veer off from the normal approach to electricity to introduce concepts needed to understand how electricity works in neurons.
The ElectronEdit
At the heart of electricity and electronics, and even neural potentials, lies at least theoretically a sub-atomic particle called an electron. The electron is negatively charged, and orbits the nucleus of the atom at about the speed of light. It is held in place by the electro-weak force, which is in turn caused by charge attraction between the light electron and the heavier protons that with neutrons make up the nucleus of the atom. It is thought that electricity happens when electrons break free of the atom they are normally part of, and wander among other atoms in the same cluster.
Atomic ChargeEdit
Since electrons can wander away from their original atoms, and get picked up by other atoms, the atoms that have more electrons than normal are considered to be negatively charged, and the atoms that have fewer electrons than normal are considered to be positively charged. Since electricity flows in the direction of lowest potential, Atoms that are negatively charged, will discharge their electrons to atoms that are positively charged, if the intervening atoms allow the transfer of electrons. We say that the electrons flow from the Negative charge to the positive charge, possibly since the original theory was that electricity was some sort of fluid.
Ionic ChargeEdit
We call an atom that is carrying a charge, whether it is negative or positive an Ion. In chemical systems based on water, many ions are created by the nature of water which levers apart many common chemicals into their component ions. If the external shell of the atom, called the Valence Shell, has an odd number of electrons, the ion usually has a negative charge, and if it has an even number of electrons the ion usually has a positive charge. It is possible in some cases to strip off two valence electrons, in which case you end up with a double positively charged ion.
Electric Transfer from Ion to IonEdit
The Ability to allow a charge of electricity to travel from one location to another depends on the resistance of the intervening atoms to having their electrons stolen, and replaced. The looser the electro-weak forces holding onto the electrons, the less resistance to flow is found, and thus the greater the potential flow given a particular pool of electrons on one side of the system, and the pool of positively charged ions on the other. Electrical Charge is measured in Coulombs, where [1]
1 Coulomb = 6.25 * 1018 electrons.
The charge of a single electron is 1.6 * 10-19 Coulombs.
1 AMP is the amount of current flowing when 1 Coulomb per second passes a particular point in one second.
Ion CurrentsEdit
Since Current is measured in Coulombs per second passing a point, it doesn't really matter whether the electrons are moving or the ions are moving. In fact it is the nature of electronics that we can postulate a flow counter to the flow of electrons that indicates the electrical force pushing the ions. We use this in welding by forcing metal ions to jump from a rod that consists of an electrical conductor surrounded by flux, to another piece of metal thus joining or building up the metal where it lands. In Neurons we often have to measure ionic currents where a specific ion is being transferred across the membrane, and thus affecting the charge stored in the neuron.
Electrical Transfer from High to Low PotentialEdit
In order for an electron to move from one atom to another, it requires an electro-motive force, to push it against the resistance of the intervening atoms. This force, measured in Volts, is the amount of energy symbolized by E needed to move an amount of charge symbolized by Q.
{\displaystyle V=E/Q}
One Volt is defined as the amount of potential difference between two points when one joule of energy is used to move one Coulomb of charge from the one point to the other. A joule is the amount of energy needed to move an object one meter against an opposing force of one newton (0.225 lb.)
Given either Amperage and Voltage, voltage and resistance, or Amperage and resistance, you can calculate the third of these three factors, using the formula
{\displaystyle I=V/R}
which is called Ohms law.
Charge Separation across a DielectricEdit
When resistance is high enough, electrons cannot flow across the resistant material, however they are still repulsed by each other by the electro-weak force, so when a resistance that is large enough to stop electron flow is found, the electrons tend to gather on one side of the material and the positive ions tend to gather on the other, until the voltage generated across the material exceeds it's dielectric constant, and a current begins to flow. We call this tendency for high resistance to result in a charge separation, capacitance. We call the highly resistive material a Dielectric, and the charge capacity of the capacitor is directly related to the plate size on each side of the dielectric.
Capacitors are measured in Farads (F) and
{\displaystyle C=Q/V}
1F = 1 Coulomb / 1 Volt
The Cellular Membrane as a CapacitorEdit
By transferring ions across the Cellular membrane which is made of highly resistive materials, the net effect is that of turning the membrane into the dielectric of a capacitor. As ionic currents add to or subtract from the charge building up inside the neuron, the ions line up along the cell membrane attracted to the opposite charged ions on the other side of the membrane. If an ion is moving away from its own type of charge, the capacitance of the membrane acts to speed it on its way, if it can find a pore, or ion channel to flow through. However to move towards it's own type of charge, energy must be spent in the form of voltage to pass the current across the membrane. Thus ion channels that pump ions against the charge gradient of the membrane capacitance must burn energy usually in the form of ATP.
As the charge builds up the Potential between the inside of the membrane and the outside of the membrane increases until it reaches the dielectric constant of the membrane, at which point the membrane depolarizes. Usually resulting in the firing of the cell. Thus understanding capacitance is important to understanding the nature of the cell membrane and how it impacts the firing of Neurons.
1 An electron is:
a small sub-atomic particle
a Heavy sub-atomic particle
a large sub-atomic particle
the lightest sub-atomic particle
similar to a positron
2 An electron travels:
as slow as a cucumber in January
always from right to left
at or near the speed of light
at or near the speed of sound
3 An electron has:
a positive electric charge
A charge account at Macy's
4 When EMF is applied:
Only the Electron is affected
Both the electron and the atom it is attached to are accelerated
Both the electron and the atom it is attached to are accelerated in the same direction
Only the Atom is affected
5 Amperage is:
A way of telling how much power a vacuum has
A measure of how much charge stays in one place
equivalent to a Coulomb
A measure of moving charge
6 Voltage is:
How much power a portable drill has
A measure of the difference in Potential between two points
A measure of how much charge moves how quickly
The opposite of EMF
7 Capacitance is:
something that only happens when there is high resistance
A measure of how much charge moves past a certain point
A mechanism for cleaning toothbrushes
A way to separate charge accounts
↑ Principles of Electric Circuits 2nd Edition, Thomas L. Floyd, (1985,1981) Charles E. Merrill Pub. (Bell and Howell) Columbus Ohio, ISBN 0-675-20402-X
Retrieved from "https://en.wikiversity.org/w/index.php?title=Fundamentals_of_Neuroscience/Electrical_Currents&oldid=2292554" |
Equations Of A Circle | Standard & General Form (Video & Examples)
Equations Of A Circle | Standard & General Form
Circles are everywhere, and every circle can be described mathematically by either of two formulas. The first takes advantage of the Pythagorean Theorem. The second formula applies either the standard form or general form for the equation. We will look at both formulas.
Equation Of A Circle Examples
When you consider a circle on a coordinate graph is the set of all points equidistant from a center point, you can see that those points can be described as an
\left(x, y\right)
value on the graph. Move right or left so many boxes (that's the x value), and then move up or down to the
y
With the circle's center point also an
\left(x, y\right)
value, you can create a right triangle with the two sides
x
boxes left or right from that center point, and
y
boxes up or down from that same center point.
r
of the circle -- the distance from the center point to the circle itself -- now becomes the hypotenuse for every possible right triangle for every possible point.
A circle has infinite points, since points are dimensionless positions in space. So, at least in the pure science of mathematics, an infinite number of right triangles exist that satisfy
{a}^{2} + {b}^{2} = {c}^{2}
and every one of them has a vertex (of the hypotenuse and one side of the triangle) lying on the circle.
The Pythagorean Theorem shows a relationship between the two sides of a right triangle and its hypotenuse:
{a}^{2} + {b}^{2} = {c}^{2}
Here is a circular orbit of a satellite around Mars:
[insert drawing of Mars with satellite on a circular orbit]
We never want the satellite to hit Mars, but we do want to be close enough to "see" Martian features with radar, cameras, magnetometers, and lasers.
To stay in one spot above a hypothetically spherical Mars (geostationary orbit), you need an orbital radius of
20,428
If the center of Mars -- the core of the planet -- is
\left(0, 0\right)
on a graph, our
x
values extend outward from the core, while our
y
values extend at right angles to those
x
values. Our hypotenuse of every right triangle must be
20,428 km
, and our circular orbit can be calculated as x2 + y2 = r2.
{x}^{2} + {y}^{2} = {r}^{2}
{x}^{2} + {y}^{2} = {20,428}^{2}
{x}^{2} + {y}^{2} = 417,303,184 km
You can isolate either the
x
y
value to find the other. Plugging in any particular value for
x
will return a value for
y
\left(x, y\right)
values will fall on the satellite's orbital path.
Say you are an orbital mechanics engineer. You know your
c
value, your hypotenuse, is
20,428 km
. You want to see the longer leg value (the
y
value) for a short-leg value (the
x
value) of
10,000 km
y = \sqrt{{c}^{2} - {x}^{2}}
y = \sqrt{{20,428}^{2} - {10,000}^{2}}
\mathbf{y} \mathbf{\approx } \mathbf{17,813.006} \mathbf{km}
Orbits are one thing; circles that are not centered at
\left(0, 0\right)
are another. What do we do if, say, the center point of a circle on a graph is at
\left(4, 7\right)
\left(0, 0\right)
[insert drawing with a circle with radius r {no actual value is needed for this} and a center point at
\left(4, 7\right)
We need to compensate for this circle that "slipped away" from
\left(0, 0\right)
, so we subtract the x-value and y-value from our original formula:
{\left(x - 4\right)}^{2} + {\left(y - 7\right)}^{2} = {r}^{2}
This will work even when the
\left(x, y\right)
coordinates are negative:
[insert drawing with a circle with radius r and center point (-3, -5)]
{\left(x - -3\right)}^{2} + {\left(y - -5\right)}^{2} = {r}^{2}
{\left(x + 3\right)}^{2} + {\left(y + 5\right)}^{2} = {r}^{2}
The Standard Form of a circle is that expression we just derived from the Pythagorean Theorem! We cannot use
\left(x, y\right)
for all the graph points, so we use other letters to identify the coordinates of the center of the circle, in this case
\left(a, b\right)
{\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {r}^{2}
From the Standard Form you have the
\left(a, b\right)
value to find the center point. We have our radius
r
. We can graph the circle.
We can also use algebra to rearrange the equation to the General Form of a circle. This is not intuitive, so let's plug in some
\left(a, b\right)
r
[insert drawing of circle on graph with center point at (2, 3) and a radius r of 5]
{\left(x - a\right)}^{2} + {\left(y - b\right)}^{2} = {r}^{2}
{\left(x - 2\right)}^{2} + {\left(y - 3\right)}^{2} = {5}^{2}
Let's expand that so you can more easily see how it turns into the General Form:
{x}^{2} - 4x + 4 + {y}^{2} - 6y + 9 = 25
Pull like terms together, set the equation equal to 0, and we have this:
{x}^{2} + {y}^{2} - 4x - 6y + 4 + 9 - 25 = 0
{x}^{2} + {y}^{2} - 4x - 6y - 12 = 0
This is the General Form of a circle. You can recognize it because the two leading terms will always be
{x}^{2}
{y}^{2}
. The generic General Form Equation looks like this:
{x}^{2} + {y}^{2} + Ax + By + C = 0
Choose between Standard Form and General Form based on the information you have in the problem. Having two ways to solve the equation of a circle -- using Pythagoras, or the Standard or General Forms -- gives you power.
Here is some information about a circle. Which method will you choose?
[insert drawing of graph with circle center point identified at
\left(-4, 5\right)
and r identified as 6; this creates a circle reaching all four quadrants]
We know the center
\left(-4, 5\right)
and the radius,
r = 6
. Start with the Standard Form, which is really just a derivation of the Pythagorean Theorem (
{a}^{2} + {b}^{2} = {c}^{2}
{\left(x - -4\right)}^{2} + {\left(y - 5\right)}^{2} = {6}^{2}
{\left(x + 4\right)}^{2} + {\left(y - 5\right)}^{2} = 36
Expand and set it to equal 0:
{x}^{2} + 8x + 16 + {y}^{2} + 10y + 25 - 36 = 0
{x}^{2} + {y}^{2} + 8x + 10y + 16 + 25 - 36 = 0
{x}^{2} + {y}^{2} + 8x + 10y + 5 = 0
That is the General Form of the equation, derived from the Standard Form, which derives from the Pythagorean Theorem! |
Difference between revisions of "Publications/boutry.21.dgmm.1" - LRDE
Difference between revisions of "Publications/boutry.21.dgmm.1"
| abstract = In Mathematical Morphology (MM), dynamics are used to compute markers to proceed for example to watershed-based image decomposition. At the same time, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) and represents the stability of the extrema of a Morse function. Since these concepts are similar on Morse functions, we studied their relationship and we found, and proved, that they are equal on 1D Morse functions. Here, we propose to extend this proof to <math>n</math>-D<math>n \geq 2</math>, showing that this equality can be applied to <math>n</math>-D images and not only to 1D functions. This is a step further to show how much MM and MT are related.
| abstract = In Mathematical Morphology (MM), dynamics are used to compute markers to proceed for example to watershed-based image decomposition. At the same time, persistence is a concept coming from Persistent Homology (PH) and Morse Theory (MT) and represents the stability of the extrema of a Morse function. Since these concepts are similar on Morse functions, we studied their relationship and we found, and proved, that they are equal on 1D Morse functions. Here, we propose to extend this proof to <math>n</math>-D, <math>n \geq 2</math>, showing that this equality can be applied to <math>n</math>-D images and not only to 1D functions. This is a step further to show how much MM and MT are related.
| lrdepaper = http://www.lrde.epita.fr/dload/papers/boutry.21.dgmm.1.pdf
month = <nowiki>{</nowiki>May<nowiki>}</nowiki>,
month = may,
address = <nowiki>{</nowiki>Uppsala, Sweden<nowiki>}</nowiki>,
series = <nowiki>{</nowiki>Lecture Notes in Computer Science<nowiki>}</nowiki>,
{\displaystyle n}
{\displaystyle n\geq 2}
{\displaystyle n} |
Radioactive substances decay exponentially. For example, a sample of Carbon-14(^{14}C)will
Radioactive substances decay exponentially. For example, a sample of Carbon-14(^{14}C)will lose half of its mass every 5730 years. (In other words, th
Radioactive substances decay exponentially. For example, a sample of Carbon
-14{\left(}^{14}C\right)
will lose half of its mass every 5730 years. (In other words, the half-life of
{14}_{C}
is 5730 years.) Let A be the initial mass of the sample. Model the decay of
{}^{14}C
using a discrete-time model... (a) using
\delta t=5730
years. (b) using Δt=1year.
A initial mass
{}^{\left\{4\right\}}C
is 5730 years
\delta t=5730
Let m(t) represent the mass of the Carbon-14 after ¢ periods of 5730 years
Initinlly, the mass is equal to A.
(0) =A
After 1 period of 5730 years, the mass m(t-1) of 5730 years ago is divided by half as the halflife of Carbon-14 is 5730 years.
m\left(t\right)=\frac{m\left(t-1\right)}{2}
for t>0
Combining these two expressions, we than obtain:
m\left(t\right)=\left\{\begin{array}{ll}A& \text{ if }t=0\\ \frac{m\left(t-1\right)}{2}& \text{ if }t=0\end{array}
\delta t=1
Let m(t) represent the mass of the Carbon-14 after t years.
m(0) =A
After 5730 years, the mass A is divided by half as the halfife of Carbou-14 is 5730 years.
m\left(5730\right)=\frac{A}{2}
Combining these two expressions, we then obtain:
m\left(t\right)=\left\{\begin{array}{ll}A& \text{ if }t=0\\ \frac{m\left(t-5730\right)}{2}& \text{ if }t=5730x\text{ for some positive integer }x\end{array}
Or you could also use the formula
m\left(t\right)=A{\left(\frac{1}{2}\right)}^{\frac{t}{5730}}
instead (formula for the halfiine), but the book claims that we need to use recurrence relations in this case (which makes is very hard to properly defined it in this case).
Determine whether these statements are true or false
\varnothing \in \left\{\varnothing \right\}
\varnothing \in \left\{\varnothing ,\left\{\varnothing \right\}\right\}
\left\{\varnothing \right\}\in \left\{\varnothing \right\}
\left\{\varnothing \right\}\in \left\{\left\{\varnothing \right\}\right\}
\left\{\varnothing \right\}\subset \left\{\varnothing ,\left\{\varnothing \right\}\right\}
\left\{\left\{\varnothing \right\}\right\}\subset \left\{\varnothing ,\left\{\varnothing \right\}\right\}
\left\{\left\{\varnothing \right\}\right\}\subset \left\{\left\{\varnothing \right\},\left\{\varnothing \right\}\right\}
f\left(x\right)=\frac{5}{3}\left(\frac{4}{5}{\right)}^{x}
y=5.2\cdot {3}^{x}
Determine the type of graph Increasing Linear, Decreasing Linear, Positive Quadratic, Negative Quadratic, Exponential Growth, or Exponential decay and give proof.
Is that exponential growth or exponential decay?
f\left(t\right)=70{\left(0.4\right)}^{t}
You learned about the row reduction matrix method for solving a system of linear equations. For example, consider the following system: |
Ramsey Theory | Brilliant Math & Science Wiki
Ramsey theory is the study of questions of the following type: given a combinatorial structure (e.g. a graph or a subset of the integers), how large does the structure have to be to guarantee the existence of some substructure (e.g. subgraph, subset) with a given property? The theory has applications in the design of communications networks and other purely graph-theoretical contexts, as well as interesting problems in elementary number theory.
Ramsey's theorem and Ramsey numbers
Proof of Ramsey's theorem
Bounds on Ramsey numbers
The most well-known example of Ramsey theory is furnished by Ramsey's theorem, which generalizes the following brainteaser.
Show that any party with at least
6
people will contain a group of three mutual friends or a group of three mutual non-friends.
Solution: Call the people A, B, C, D, E, F. Either A has three friends or three non-friends. Without loss of generality, suppose that B,C,D are all friends with A. Then if any pair of them are friends with each other, that pair plus A forms a group of three mutual friends. If no two of them are friends, then they are a group of three mutual non-friends.
\square
N = 6
is the minimal party size that guarantees this property. Consider the following graph: There are
5
vertices, each of which represents a person. Friends are connected by red edges, and non-friends are connected by blue edges. A group of three mutual friends would be represented by a red triangle, and a group of three mutual non-friends would be represented by a blue triangle. But neither of these is present anywhere in the graph, so
N=5
is not sufficient.
The genesis of Ramsey theory is in a theorem which generalizes the above example, due to the British mathematician Frank Ramsey.
Fix positive integers
m,n
. Every sufficiently large party will contain a group of
mutual friends or a group of
n
mutual non-friends.
It is convenient to restate this theorem in the language of graph theory, which will make it easier to generalize. This requires some definitions:
A complete graph
K_n
is a graph on
vertices where every pair of vertices is connected by an edge.
A clique inside a graph is a set of vertices which are pairwise connected to each other; in other words, a clique of size
n
in a graph is a copy of
K_n
inside the graph.
So Ramsey's theorem, restated, is: Fix positive integers
m,n
. Every complete graph on sufficiently many vertices, with every edge colored blue or red, will contain a red clique of
m
vertices or a blue clique of
n
vertices. (Here a "red clique" means that every edge connecting two vertices in the clique is red. The vertices are not colored; the edges are.)
The Ramsey number
R(m,n)
is the smallest party size that guarantees a group of
mutual friends or a group of
n
mutual non-friends. Alternatively, it is the minimum number of vertices a complete graph must have so that if every edge is colored blue or red, there is a red clique of
m
n
The theorem generalizes to an arbitrary (finite) number of colors; there is a Ramsey number
R(n_1,n_2,\ldots,n_k)
that guarantees a monochromatic clique of
n_k
vertices with color
k
on a sufficiently large graph.
R(m,1) = R(1,m) = 1
trivially. (There is no coloring requirement of a clique with 1 vertex.)
R(m,2) = R(2,m) = m
. This is because in
K_m
, either all the edges are colored red, in which case there is a red clique on
m
vertices; or there is a blue edge somewhere, in which case the two vertices it connects are a blue clique on
2
vertices. In other words, either everyone at the party is friends with everyone else, or there are two people who are not friends.
R(3,3) = 6
. This is a restatement of the example in the introduction.
The goal is to show that
R(m,n)
exists. Induct on
m+n
m
n
1
, we are done. For the inductive hypothesis, we show that a complete graph on
R(m-1,n) + R(m,n-1)
vertices satisfies the condition of the problem.
To see this, take a vertex from the graph. Consider the subsets
V_r
V_b
of vertices connected to this vertex by red and blue edges, respectively. Then
|V_r| + |V_b| = R(m-1,n)+R(m,n-1)-1
|V_r| \ge R(m-1,n)
|V_b| \ge R(m,n-1)
V_r
contains either a blue clique on
vertices, in which case we are done, or a red clique on
m-1
vertices; but putting that clique together with the original vertex produces a red clique on
m
vertices. The latter case is similar. So the proof is complete by induction.
\square
So the proof gives an inequality
R(m,n) \le R(m-1,n) + R(m,n-1).
Note that this immediately shows
R(3,3) \le R(3,2) + R(2,3) = 3+3 = 6
. The proof of the theorem is in fact a generalization of the proof that
R(3,3) = 6
R(m,n)
is a very difficult problem in general, even for small
m
n
. The inequality for
R(m,n)
looks a bit like Pascal's identity, and in fact an easy induction using Pascal's identity shows that
R(m,n) \le \binom{m+n-2}{m-1}.
Exact values of
R(m,n)
3 \le m \le n
are only known for
(m,n) = (3,3),(3,4),\ldots,(3,9),(4,4),(4,5)
. Current research shows only that
43 \le R(5,5) \le 49
Better upper bounds are hard to come by, since in the absence of an elegant logical argument they require an enumeration of all possible colorings of
K_n
and a demonstration that every coloring gives a monochromatic clique of the right size. Lower bounds only require one specific coloring that does not admit such a clique (e.g. the one for
R(3,3) \ge 6
given in the introduction).
Cite as: Ramsey Theory. Brilliant.org. Retrieved from https://brilliant.org/wiki/ramsey-theory/ |
In recent decades there has been increased coverage of animal welfare issues, the health risks of high consumption of animal products and the contribution of farming animals to climate change. Multiple high-profile organisations have called for reduced animal consumption through reducetarian, vegetarian and vegan diets [1, 2].
Each of these three issues has been studied extensively and solving these issues has gained broad support. A balanced vegan diet has the opportunity to reduce the negative impacts of all three issues but campaigning for veganism has not been very successful and veganism still has relatively low prevalence of under 10%. For example, in the United Kingdom, where the first vegan society was founded, prevalence of veganism is estimated at 1.16% by the most recent survey [3].
An alternative to veganism called flexitarianism or reducetarianism has been gaining ground. The idea of these diets is to reduce consumption of animal products by partially replacing them with plant-based foods. If one were to adopt a flexitarian diet it would be confusing to decide which species to avoid since consumption of different species have different levels of associated harm on the three scales. For example, poultry has lower health risks and contributes less to climate change than beef but is considered a poor choice from an animal welfare perspective due to the poor conditions of the animals and the high number of animals used.
The goal of this tool is to allow the user to specify their relative concern on two issues: animal welfare and climate change. The tool ranks animal species according to the harm induced by their consumption while taking into accord the user’s values. This ranking could be used to decide which animal products should be replaced with plant-based products.
Rankings based on welfare have been developed previously by various individuals and groups such as Peter Hurford, Brian Tomasik, Charity Entrepreneurship and Dominik Peters. This tool is a minor extension of the work of Dominik Peters that also considers emissions in addition to welfare. I want to thank Dominik for kindly providing the data and methodology that he used.
Animal suffering subscale
To estimate the negative impact on animal welfare, the tool calculates the number of hours animals have spent in a farm in order to produce an amount of food which provides 2000 kcal of energy. For example, Dominik retrieved data on production yields and slaughter age from providers of breed chickens such as Amgen and Lohmann Tierzucht.
I calculated the amount of produce required for 2000 kcal of energy using data from nutritiondata.self.com. In the cases of “dairy cow”, “caged hen” and “cage-free hen” it is meant consuming dairy or eggs. Of course different preparations from the same animal can have varying nutritional value but for the sake of simplicity one value is used per species. The same energy value was used for caged hen and cage-free hen eggs. Likewise with broiler and slow-growing broiler meat.
suffering_s
designate the suffering subscale score of species
lifespan_s,\ production_s,\ refweight_s
designate the average lifespan of an animal in hours, weight of produce per animal and weight of produce required for 2000 kcal of energy. The basic score is the number of hours suffered on a farm to produce 2000 kcal of produce:
suffering_s = \frac{lifespan_s}{production_s} \times refweight_s
There are some issues with this basic approach. For example, our confidence in different species being sentient varies. If we just account for hours lived to produce 2000 kcal then the harm of smaller species will dominate. But the user of the tool might have low confidence in shrimp being sentient and we want to account for that.
There are arguments in favour of and against brain weighting [4] but if we were to believe that capacity for welfare is linked to brain mass or neuron count we could use either one to scale the suffering scores. Dominik used Carl Shulman’s data on brain mass and neuron count [5]. The user can also choose to apply different functions to the neuron count data. The tool supports linear, logarithmic, square-root and square transformation of neuron counts. Some believe that cognitive abilities increase sublinearly in regards to neuron count so the default transform is square-root but the user can choose other transforms or disable brain weighting. When brain weighting is enabled, the hours spent in a farm are scaled by the ratio of the brain mass (or neuron count) of the species and that of a chicken. If
n_s
f
designate the neuron count of species
and user chosen neuron scaling function then the suffering scale can be adjusted by multiplying with the scalar
f(n_s) / f(n_{broiler})
It is also likely that different species of farm animals do not suffer equally due to the different conditions they are raised in. We allow the user to specify their belief in the relative suffering of different species. The default values are from Brian Tomasik [6]. The suffering of a beef cow is set at 1 and the user can specify a species’ relative suffering in relation to that of a beef cow.
Different animal products might have different price elasticities. Price elasticity of supply shows the responsiveness of production to change in price. Elasticity of demand shows the responsiveness of demand to change in price. Cumulative elasticity is the net effect on supply. If someone spares 10 chickens a year by not eating chickens the actual change could be less than 10. Decreased chicken meat price due to lower demand might motivate someone else to eat more chicken. The user can choose to factor in cumulative elasticity in order to account for this effect. Two sources are provided: the book “Compassion, by the Pound” [7] and the work of the organisation Animal Charity Evaluators [8].
The user can also choose to factor in sleeping time if they assume that animals do not suffer while sleeping and liveability which also factors in animals who die before slaughter.
Climate change subscale
The climate change subscale measures the CO2 equivalent greenhouse gases produced per kilogram of animal produce. This value is scaled according to the amount of produce required for 2000 kcal. CO2 equivalent emissions data has been collected from lifecycle analyses [9, 10]. Let
emissions_s
designate the CO2 equivalent gases produced per kilogram of produce of species
. The climate subscale score of species
is thus:
climate_s = emissions_s \times refweight_s
The elasticity parameters apply both to the suffering and climate subscales. That means if elasticity is enabled then
climate_s
is multiplied by the cumulative elasticity factor.
Note that only the impact on climate change is considered. There are other negative environmental impacts. Saltwater fishing causes marine pollution and fish farms cause eutrophication. But since the risk of climate change outweighs other environmental risks related to the consumption of animal produce I consider these omissions acceptable.
It is sometimes argued that buying local food is more important than reducing meat consumption. In general the climate impact of food is dominated by production [11] so this tool does not make a distinction between where animals were farmed. Imported plant-based food tends to have lower emissions than local animal produce. Life cycle analysis of animal products already includes transportation and this is considered sufficient.
The tool also does not consider which plant-based foods are substituted for animal products. Plant-based food production in general causes significantly lower emissions [12, 13] but from the perspective of the environment it might make sense to prefer whole foods because these do not require additional energy-intensive processing.
An issue with using CO2 equivalent greenhouse gas emissions as a measure of warming is that farming different species puts different types of greenhouse gases in the environment. The high impact of ruminants is caused by their methane emissions. While methane warms the atmosphere more than CO2 it is also removed from the atmosphere significantly faster. Climate scientists sometimes use a metric called CO2-equivalent with Global Warming Potential 100 which considers methane to cause 25x as much warming as an equivalent amount of CO2 over a century. Some physicists disagree with this approach [14]. If one is concerned about the effects of warming over thousands of years as opposed to a hundred years this approach understates the impact of CO2 compared to methane.
A weighted product model is used to combine the subscales. Weighted product models are dimensionless and are used for ranking options when making decisions. Because the scores are dimensionless they are normalised to the range
[0, 100]
w_{suffering}
w_{climate}
designate the suffering and climate weights. The combined score of species
is calculated using:
harm_s = suffering_s^{w_{suffering}} \cdot climate_s^{w_{climate}}
A product model ensures that the subscales affect the combined score equivalently. A 1% increase in CO2 emissions changes the combined score by the same amount that a 1% increase in the animal suffering subscale would. Adding weights to the model allows us to change the relative contribution of each subscale to the combined score.
Why is there no health impacts subscale?
I considered designing a health subscale using data from the Global Burden of Disease (GBD) study but eventually opted against it.
Understanding nutrition is notoriously difficult. It is impossible to conduct trials that assess long-term health impacts of diets due to costs and ethical concerns with assigning people to diets with unknown health effects. Due to this dietary decisions must be made based mostly on observational data which is not as reliable as randomised controlled trials. Even well studied questions such as the impact of saturated fat consumption have not been fully resolved [15, 16].
GBD data has been aggregated from a large number of sources by a large team. It might be close to the consensus of nutritional science but even if we have reasons to trust the data it would be difficult to use due to non-linear effects. If two people decided to eat one less chicken this year their climate and animal welfare impact would be similar even if one of them previously ate 10 chickens per year and the other ate a single chicken each year. Positive health effects of reduced meat consumption on the other hand have diminishing returns and the reduction of their health risks would depend on the current composition of their diets.
Adequately planned reducetarian, vegetarian and vegan diets are believed to be healthy based on existing evidence [17] but positive effects over conventional diets are not considered in this tool due the uncertainty of the effects and difficulties of modeling.
Multiple issues with the method used to estimate suffering are outlined in [18]. The tool could be helpful to make decisions in the face of uncertainty but is not a true measure of harm.
The model does not consider the suffering of wild animals which could significantly exceed that of farm animals [19]. There is also no consideration of which plant-based products are substituted for animal products. Farming of some plants causes less emissions [12] or wild animal suffering [20].
I consider the greatest limitation of the tool the fact that setting subscale priorities based on intuitions can be misleading. It would make sense to compare emissions and harm based on the underlying values which cause us to be concerned about the issues in the first place. If for example I am motivated by increased welfare, it would be helpful to estimate the welfare impacts of climate change and factory farming on a common scale.
Lifespan Dominik Peters
Production Dominik Peters
Sleeping time Dominik Peters
Pre-slaughter mortality Dominik Peters
Neuron count Dominik Peters
Brain mass Dominik Peters
Elasticity factors Animal Charity Evaluators [8]; Compassion, by the pound [7]
Emissions Dominik Peters; Cao et al. [10]
Caged hen Boiled egg
Cage-free hen Boiled egg
Broiler Cooked breast
Slow-growth broiler Cooked breast
Pig Cooked ground pork
Turkey Cooked meat
Beef cow Cooked ground beef
Dairy cow Whole milk
Lamb Cooked ground lamb
Duck Cooked duck
Shrimp Cooked crustaceans, shrimp
[1] “Climate Change and Land — IPCC.” https://www.ipcc.ch/report/srccl/.
[2] W. Willett, J. Rockström, B. Loken, M. Springmann, T. Lang, S. Vermeulen, T. Garnett, D. Tilman, F. DeClerck, A. Wood, and others, “Food in the anthropocene: The eat–lancet commission on healthy diets from sustainable food systems,” The Lancet, vol. 393, no. 10170, pp. 447–492, 2019.
[3] “Statistics | The Vegan Society.” https://www.vegansociety.com/news/media/statistics.
[4] B. Tomasik, “Is Brain Size Morally Relevant?” https://reducing-suffering.org/is-brain-size-morally-relevant/.
[5] C. Shulman, “How are brain mass (and neurons) distributed among humans and the major farmed land animals?” https://reflectivedisequilibrium.blogspot.com/2013/09/how-is-brain-mass-distributed-among.html.
[6] B. Tomasik, “How Much Direct Suffering Is Caused by Various Animal Foods.” https://reducing-suffering.org/how-much-direct-suffering-is-caused-by-various-animal-foods/.
[7] F. B. Norwood and J. Lusk, Compassion, by the pound: The economics of farm animal welfare. New York: Oxford University Press, 2011.
[8] Animal Charity Evaluators, “Online Ads Impact.” https://docs.google.com/spreadsheets/d/1iNDQIt9MRD4r1ws5M_2hQ-MNjMY-bcUra0fpOmF4Am0/edit#gid=0.
[9] K. Hamerschlag and K. Venkat, Meat eater’s guide to climate change+ health: Lifecycle assessments: Methodology and results 2011. Environmental Working Group, 2011.
[10] L. Cao, J. S. Diana, G. A. Keoleian, and Q. Lai, “Life cycle assessment of chinese shrimp farming systems targeted for export and domestic sales,” Environmental science & technology, vol. 45, no. 15, pp. 6531–6538, 2011.
[11] C. L. Weber and H. S. Matthews, “Food-miles and the relative climate impacts of food choices in the united states.” ACS Publications, 2008.
[12] J. Poore and T. Nemecek, “Reducing food’s environmental impacts through producers and consumers,” Science, vol. 360, no. 6392, pp. 987–992, 2018.
[13] M. Springmann, H. C. J. Godfray, M. Rayner, and P. Scarborough, “Analysis and valuation of the health and climate change cobenefits of dietary change,” Proceedings of the National Academy of Sciences, vol. 113, no. 15, pp. 4146–4151, 2016.
[14] M. Allen, “Short-lived promise? The science and policy of cumulative and short-lived climate pollutants,” University of Oxford, 2015.
[15] Y. Zhu, Y. Bo, and Y. Liu, “Dietary total fat, fatty acids intake, and risk of cardiovascular disease: A dose-response meta-analysis of cohort studies,” Lipids in health and disease, vol. 18, no. 1, p. 91, 2019.
[16] L. Hooper, N. Martin, O. F. Jimoh, C. Kirk, E. Foster, and A. S. Abdelhamid, “Reduction in saturated fat intake for cardiovascular disease,” Cochrane Database of Systematic Reviews, no. 5, 2020.
[17] W. J. Craig, A. R. Mangels, and others, “Position of the american dietetic association: Vegetarian diets.” Journal of the American Dietetic Association, vol. 109, no. 7, pp. 1266–1282, 2009.
[18] H. Browning, “If I Could Talk to the Animals: Measuring Subjective Animal Welfare,” PhD thesis, College of Arts; the Social Sciences, The Australian National University, 2020.
[19] B. Tomasik, “The Importance of Wild-Animal Suffering,” Foundational Research Institute. Apr-2015.
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By Ville SokkSource code |
Arc Measure Formula | How to Find Angle Measure of an Arc (Video)
Arc Measure Formula, Definition, & How To Find
Arc Measure Definition
An arc is a segment of a circle around the circumference. An arc measure is an angle the arc makes at the center of a circle, whereas the arc length is the span along the arc. This angle measure can be in radians or degrees, and we can easily convert between each with the formula
\pi radians = 180°
You can also measure the circumference, or distance around, a circle. If you take less than the full length around a circle, bounded by two radii, you have an arc. That curved piece of the circle and the interior space is called a sector, like a slice of pizza. When you cut up a circular pizza, the crust gets divided into arcs.
Arc Measure vs. Arc Length
Arc Measure Formula
How To Find The Measure of an Arc
Identifying Arc Angle Indicated
If we cut across a delicious, fresh pizza, we have two halves, and each half is an arc measuring
180°
. If we make three additional cuts in one side only (so we cut the half first into two quarters and then each quarter into two eighths), we have one side of the pizza with one big,
180°
arc and the other side of the pizza with four,
45°
arcs like this:
The half of the pizza that is one giant slice is a major arc since it measures
180°
(or more). The other side of the pizza has four minor arcs since they each measure less than
180°
The arc is the fraction of the circle's circumference that lies between the two points on the circle. An arc has two measurements:
The arc's length is a distance along the circumference, measured in the same units as the radius, diameter or entire circumference of the circle; these units will be linear measures, like inches, cm, m, yards, and so on
The arc's angle measurement, taken at the center of the circle the arc is part of, is measured in degrees (or radians)
Do not confuse either arc measurement (length or angle) with the straight-line distance of a chord connecting the two points of the arc on the circle. The chord's length will always be shorter than the arc's length.
To be able to calculate an arc measure, you need to understand angle measurements in both degrees and radians. An angle is measured in either degrees or radians. A circle measures 360 degrees, or
2\pi radians
, whereas one radian equals 180 degrees. So degrees and radians are related by the following equations:
360° = 2\pi radians
180° = \pi radians
The relationship between radians and degrees allows us to convert to one another with simple formulas. To convert degrees to radians, we take the degree measure multiplied by pi divided by 180.
Let's convert 90 degrees into radians for example:
90° × \left(\frac{\pi }{180°}\right)
\frac{90\pi }{180}
\frac{\mathbf{\pi }}{\mathbf{2}} \mathbf{radians}
Now let's convert
\frac{\pi }{3} radians
to degrees:
\frac{\pi }{3} × \frac{180}{\pi }
\frac{180\pi }{3\pi }
\frac{\mathbf{180}}{\mathbf{3}} \mathbf{=} \mathbf{60}\mathbf{°}
Once you got the hang of radians, we can use the arc measure formula which requires the arc length,
s
, and the radius of the circle,
r
, to calculate.
arc measure = \frac{arc length}{radius} = \frac{s}{r}
Let's try an example where our arc length is 3 cm, and our radius is 4 cm as seen in our illustration:
Start with our formula, and plug in everything we know:
arc measure = \frac{s}{r}
arc measure = \frac{3}{4}
Now we can convert
\frac{3}{4} radians
into degrees by multiplying by 180 dividing by
\pi
\left(\frac{3}{4}\right)\left(\frac{180}{\pi }\right)
\mathbf{42.9718} \mathbf{\approx } \mathbf{43}\mathbf{°}
You need to know the measurement of the central angle that created the arc (the angle of the two radii) to calculate arc length. The arc length is the fractional amount of the circumference of the circle. The circumference of any circle is found with
2\pi r
r = radius
. If you have the diameter, you can also use
\pi d
d = diameter
The formula for finding arc length is:
Arc length = \left(\frac{arc angle}{360°}\right) \left(2\pi r\right)
Let's try an example with this pizza:
Our pie has a diameter of 16 inches, giving a radius of 8 inches. We know the slice is
60°
. So the formula for this particular pizza slice is:
= \frac{60°}{360°} · 2·\pi ·8
= \frac{1}{6} · 16\pi
\approx 8.3775"
An arc angle's measurement is shown as
m\stackrel{⌢}{AB}
A
B
are the two points on the circle creating the arc. The
means measurement, and the short curved line over the
\stackrel{⌢}{AB}
indicates we are referring to the arc. The two points derived from the central angle (the angle of the two radii emerging from the center point).
One important distinction between arc length and arc angle is that, for two circles of different diameters, same-angle sectors from each circle will not have the same arc length. Arc length changes with the radius or diameter of the circle (or pizza).
Now that you have eaten your way through this lesson, you can identify and define an arc and distinguish between major arcs and minor arcs. You are also able to measure an arc in linear units and degrees and use the correct symbol,
m\stackrel{⌢}{AB}
(where A and B are the two points on the circle), to show arc length.
After working your way through this lesson and video, you will learn to:
Identify and define an arc
Distinguish between major arcs and minor arcs
Measure an arc in linear units and degrees
Use correct symbols to show arc length |
Abstract: Introduction: Breast cancer cases, mastectomy and following reconstruction procedures are growing in numbers. Despite being lifesaving, mastectomies have a destructive psychological impact on the patients. On the other hand, breast reconstruction improves psychological damages within the same population. Various techniques for nipple reconstruction were described in literature. Trillium flap is an innovative technique to reconstruct neo-nipple with several advantages that make it superior to other popular flaps. Objectives: To come up with an innovative design for reconstructing a neo-nipple post mastectomy, that is superior to other popular flaps. Results: The Trillium flap design has less visible and easily camouflaged scars, is geometry-based, specific, well-detailed and flexible to produce a tailored nipple with any desired height and diameter and ensures the flaps good vascularity and the neo-nipple projection sustainability. Conclusion: Trillium flap is an innovative technique to reconstruct neo-nipple with several advantages that make it superior to other popular flaps. The results shown in the study are for experimental procedures done on human tissue samples of excised flaps from abdominoplasties and brachioplasties. Further application on actual cases is needed with monitoring of neo-nipple projection sustainability on the long term.
Keywords: Trillium, Flap, Breast, Nipple, NAC, Reconstruction, Neo-Nipple, Mastectomy
It is estimated that there are more than 3.8 million women living in the United States with a history of invasive breast cancer, and 268,600 women will be newly diagnosed in 2019. More than 150,000 breast cancer survivors are living with metastatic disease, three-quarters of whom were originally diagnosed with stage I through III cancer [1].
Mastectomy is done for 34% of patients with early-stage (stage I or II) breast cancer, more than two-thirds (68%) of patients with stage III disease and only 12% of patients with stage IV [2] (Figure 1).
Being diagnosed with breast cancer and undergoing mastectomy lead to serious psychological issues regarding self-esteem, self-consciousness and sexual intimacy suggesting the need for cognitive interventions [3] [4] [5] [6] [7]. On the other hand, breast reconstruction was found to have a benefit for improving the psychological damages in patients with breast cancer [8] [9].
There are various reports of nipple areola complex (NAC) reconstruction with flaps in the medical literature [10]. Without significant differences, all techniques have nearly an equal rate of complications e.g. flap vascular compromise on the short term and loss of projection on the long term [11].
2. Ideal Nipple Areola Complex (NAC)
· Height and diameter:
In a morphologic study of nipple-areola complex in 600 breasts, Sanuki et al. analyzed the results statistically to come up with the findings shown in Table 1. They also found that the mean diameter of the areola in women who gave birth was 0.5 cm larger than that of those who did not. They classified the sample according to the relation between nipple height and diameter as shown in Table 2 [12].
Figure 1. Female Breast Cancer Treatment Patterns (%) by Stage, 2016. *A small number of these patients received chemotherapy. †A small number of these patients received radiation therapy (RT). BCS indicates breast-conserving surgery; chemo, chemotherapy (includes targeted therapy and immunotherapy) [2].
Table 1. Diameter of the nipple-areola complex and height of the nipple found by Sanuki et al. [12].
Table 2. Classification of nipple shape according to the relation between nipple height and diameter found by Sanuki et al. [12].
The optimal NAC proportions were found by Hauben et al. to be with the proportion of the upper to the lower pole at a ratio of 45:55. The angulation of the nipple was upward at a mean angle of 20˚ from the nipple meridian. The areola-breast and nipple-areola proportions were 1:3.4 and 1:3, respectively [13].
Schiffman, with the patient standing or sitting, utilizes a line from the midclavicular point (MC) to the mid-nipple (N). At the same time, a line is marked in the center of the chest wall from the center of the sternum superiorly to the mid-xiphoid process. The inframammary fold is palpated from underneath the breast inferiorly and the tip of the finger palpated superficially and marked on the MC to N line [10].
Another study by Lewin et al. determined the preferences for the nipple-areola complex on the female breast in our study population. The NAC placement preferred by both genders had a ratio of 40:60x and 50:50y (Figure 2), which means that it was best situated in the middle of the breast gland vertically and slightly lateral to the midpoint horizontally [14].
3. Geometric Basics
Circle circumference can be calculated as follows:
C: Circumference, π: Pi (22/7), r: Radius, ø: Diameter.
C=2\text{π}r=\pi ø=\frac{22ø}{7}\approx 3ø
Circle/cylinder circumference nearly equals three times its diameter.
Supposing that a nipple is a cylinder, and my design depends on building that cylinder out of three equal vertical flaps, the width of one flap equals 1/3 of its circumference equals the cylinder diameter. To make that possible, the length sides of each one of the three equal flaps should be parallel tangential lines to the circle that forms the cylinder roof (Figure 3 and Figure 4).
Figure 2. The coordinate system of the breast [14].
Figure 3. The width of one flap equals 1/3 of its circumference equals the cylinder diameter.
Figure 4. The length sides (B, B) of each one of the three equal flaps should be parallel tangential lines to the circle that forms the cylinder roof. The width of one flap equals the cylinder diameter.
1) Visually, scars on the areola are three short scars in three different directions and less noticeable than one long scar in the same direction like most of other popular nipple reconstruction flaps.
2) Visually, scars on the areola are three short scars that will never exceed the areola region. After camouflage tattooing, scars will be well-hidden.
3) The design is flexible to be rotated to partially include any previous scars, i.e. mastectomy scar, in the excision diamond shaped area (X).
4) The design is geometry-based, specific, well-detailed (Figure 5 and Figure 6) and flexible to produce a tailored nipple with any desired height and diameter according to guidelines illustrated in ideal nipple areola complex section above.
5) Building the nipple by three random flaps ensures the flaps good vascularity more than most of the popular nipple reconstruction flaps that makes the vascularity of the neo-nipple components depend on one base.
6) Choosing the flaps to be three not more or less is a balance point that has both advantages of preventing flaps vascular compromise, by being not more than three flaps, and defying/redistributing the retraction forces that will oppose the neo-nipple projection, by being not less than three flaps.
7) Projection is held in place by four Polydioxanone sutures that will dissolve after nearly 180 days, the thing that ensures the neo-nipple projection sustainability.
8) The flap design looks like Trillium Grandiflorum flower for having a significant center (pistil) and reciprocating three petals and three sepals (Figure 7).
Figure 5. 1. The general flap design. 2: Details regarding the equal sides (A)s or (B)s. A: Neo-nipple diameter, B: Neo-nipple height. 3: The three diamond-shaped skin excision zones (X) (Excision direction to be vertical on sides between (X) and (U) zones and bevelled outwards, from (X), on sides between (X) and (E) zones to ensure that the neo-nipple will look cylidrical not prismatic), The three flap elevations zones (E), The three undermining zones for easy flap mobilization (U). 4: Subcutaneous 3/0 Polydioxanone approximation purse string suture that holds the three flaps bases together (S1) (Approximation is preferred to be done while tracting the tip of each flap vertically upwards), Three 3/0 Polydioxanone sutures that hold base corners of each two adjacent flaps (S2).
Three designs found in the literature can be mixed up with Trillium flap. Differences are shown in Table 3 and Figure 8 below:
Figure 6. A guide for the main steps of Trilllium flap. 8: The neo-nipple projection.
Figure 7. The flap (A) looks like Trillium Grandiflorum flower (B) [15] for having a significant center (pistil) and reciprocating three petals and three sepals.
Figure 8. Trillium Flap (D) and its lookalikes (A [10], B [18], C [20] ). Essential differences are still there in technique and indications of use.
Table 3. Trillium Flap and its lookalikes. Essential differences are still there in technique and indications of use.
Trillium flap is an innovative technique to reconstruct neo-nipple with several advantages that make it superior to other popular flaps. The photos of results included in Figure 6 are for experimental procedures done on excised flaps from abdominoplasties and brachioplasties.
Further application on actual cases is needed with monitoring of neo-nipple projection sustainability on the long term.
Cite this paper: Alsayed, A. (2020) Trillium Flap for Postmastectomy Neo-Nipple Reconstruction (A Novel Technique). Modern Plastic Surgery, 10, 9-16. doi: 10.4236/mps.2020.101002.
[1] Mariotto, A.B., Etzioni, R., Hurlbert, M., Penberthy, L. and Mayer, M. (2017) Estimation of the Number of Women Living with Metastatic Breast Cancer in the United States. Cancer Epidemiology, Biomarkers & Prevention, 26, 809-815.
[2] American College of Surgeons Commission on Cancer (2019) National Cancer Database, 2016 Data Submission. American College of Surgeons Commission on Cancer, Chicago, IL.
[3] Zhao, R., Qiao, Q., Yue, Y., et al. (2003) The Psychological Impact of Mastectomy on Women with Breast Cancer. Chinese Journal of Plastic Surgery, 19, 294-296.
[4] Nozawa, K., Ichimura, M., Oshima, A., et al. (2015) The Present State and Perception of Young Women with Breast Cancer towards Breast Reconstructive Surgery. International Journal of Clinical Oncology, 20, 324-331.
[5] Gopie, J.P., Mureau, M.A., Seynaeve, C., et al. (2013) Body Image Issues after Bilateral Prophylactic Mastectomy with Breast Reconstruction in Healthy Women at Risk for Hereditary Breast Cancer. Familial Cancer, 12, 479-487.
[6] Fallbjörk, U., Rasmussen, B.H., Karlsson, S. and Salander, P. (2013) Aspects of Body Image after Mastectomy Due to Breast Cancer—A Two-Year Follow-up Study. European Journal of Oncology Nursing, 17, 340-345.
[7] Schover, L.R. (1994) Sexuality and Body Image in Younger Women with Breast Cancer. Journal of the National Cancer Institute Monographs, 16, 177-182.
[8] Chen, W., Lv, X., Xu, X., Gao, X. and Wang, B. (2018) Meta-Analysis for Psychological Impact of Breast Reconstruction in Patients with Breast Cancer. Breast Cancer, 25, 464-469.
[9] Rowland, J.H., Holland, J.C., Chaglassian, T. and Kinne, D. (1993) Psychological Response to Breast Reconstruction. Expectations for and Impact on Postmastectomy Functioning. Psychosomatics, 34, 241-250.
[10] Shiffman, M.A. (2018) History of Nipple-Areolar Complex Reconstruction. In: Nipple-Areolar Complex Reconstruction, Springer, Cham, 10-11.
[11] Davis, G.B., Miller, T. and Lee, G. (2018) Nipple Reconstruction: Risk Factors and Complications. In: Shiffman, M., Ed., Nipple-Areolar Complex Reconstruction, Springer, Cham, 624-625.
[12] Sanuki, J., Fukuma, E. and Uchida, Y. (2009) Morphologic Study of Nipple-Areola Complex in 600 Breasts. Aesthetic Plastic Surgery, 33, 295-297.
[13] Hauben, D.J., Adler, N., Silfen, R. and Regev, D. (2003) Breast-Areola-Nipple Proportion. Aesthetic Plastic Surgery, 50, 510-513.
https://doi.org/10.1097/01.SAP.0000044145.34573.F0
[14] Lewin, R., Amoroso, M., Plate, N., Clara, C. and Selvaggi, G. (2016) The Aesthetically Ideal Position of the Nipple-Areola Complexon the Breast. Aesthetic Plastic Surgery, 40, 724-732.
[15] Ramsey, D. (2007) Trillium grandiflorum. Self-Photographed. Photo Taken at the Mt. Cuba Center Where It Was Identified.
https://commons.wikimedia.org/wiki/File:White_Trillium_Trillium_grandiflorum_Flower_2613px.jpg
[16] Huang, W. (2003) A New Method for Correction of Inverted Nipple with Three Periductal Dermofibrous Flaps. Aesthetic Plastic Surgery, 27, 301-304.
[17] Hsiao, S., Huang, W., Yu, C., Tsai, Y. and Tung, K. (2008) Refinement of Three Periductal Dermofibrous Flaps Method for Correction of Inverted Nipples.
[18] Kim, J.T. and Singh, J. (2018) Correction of Inverted Nipples with Twisting and Locking Principle. In: Shiffman, M., Ed., Nipple-Areolar Complex Reconstruction, Springer, Cham, 315-329.
[19] Berson, M.I. (1946) Construction of Pseudoareola. Surgery, 20, 808.
[20] Sisti, A., Tassinari, J., Cuomo, R., Brandi, C., Nisi, G., Grimaldi, L. and D’Aniello, C. (2018) Nipple-Areola Complex Reconstruction. In: Shiffman, M., Ed., Nipple-Areolar Complex Reconstruction, Springer, Cham, 359-368. |
Subexpressions or terms of symbolic expression - MATLAB children - MathWorks Benelux
Find Child Subexpressions of Symbolic Expression
Find Child Subexpressions of Equation
Find Child Subexpressions of Integral
Plot Taylor Approximation of Expression
Find Child Subexpressions of Elements of Matrix
children returns cell arrays
Subexpressions or terms of symbolic expression
Starting in R2020b, the syntax subexpr = children(expr) for a scalar input expr returns subexpr as a nonnested cell array instead of a vector. You can use subexpr = children(expr,ind) to index into the returned cell array of subexpressions. For more information, see Compatibility Considerations.
subexpr = children(expr)
subexpr = children(A)
subexpr = children(___,ind)
subexpr = children(expr) returns a nonnested cell array containing the child subexpressions of the symbolic expression expr. For example, the child subexpressions of a sum are its terms.
subexpr = children(A) returns a nested cell array containing the child subexpressions of each expression in the symbolic matrix A.
subexpr = children(___,ind) returns the child subexpressions of a symbolic expression expr or a symbolic matrix A as a cell array indexed by ind.
Find the child subexpressions of the symbolic expression
{x}^{2}+xy+{y}^{2}
. The subexpressions are returned in a nonnested cell array. children uses internal sorting rules when returning the subexpressions. You can index into each element of the cell array by using subexpr{i}, where i is the cell index. The child subexpressions of a sum are its terms.
subexpr = children(x^2 + x*y + y^2)
subexpr=1×3 cell array
{[x*y]} {[x^2]} {[y^2]}
s1 = subexpr{1}
x y
{x}^{2}
{y}^{2}
You can also index into each element of the subexpressions by specifying the index ind in the children function.
s1 = children(x^2 + x*y + y^2,1)
x y
{x}^{2}
{y}^{2}
To convert the cell array of subexpressions into a vector, you can use the command [subexpr{:}].
V = [subexpr{:}]
\left(\begin{array}{ccc}x y& {x}^{2}& {y}^{2}\end{array}\right)
Find the child subexpressions of the equation
{x}^{2}+xy={y}^{2}+1
. The child subexpressions of the equation are returned in a 1-by-2 cell array. Index into all elements of the cell array. The subexpressions of an equation are the left and right sides of that equation.
subexpr = children(x^2 + x*y == y^2 + 1)
{[x^2 + y*x]} {[y^2 + 1]}
subexpr{:}
{x}^{2}+y x
{y}^{2}+1
Next, find the child subexpressions of the inequality
\mathrm{sin}\left(x\right)<\mathrm{cos}\left(x\right)
. Index into all elements of the returned cell array. The child subexpressions of an inequality are the left and right sides of that inequality.
subexpr = children(sin(x) < cos(x))
{[sin(x)]} {[cos(x)]}
\mathrm{sin}\left(x\right)
\mathrm{cos}\left(x\right)
Find the child subexpressions of an integral
{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2777777777777778em}{0ex}}dx
. The child subexpressions are returned as a cell array of symbolic expressions.
syms f(x) a b
subexpr = children(int(f(x),a,b))
{[f(x)]} {[x]} {[a]} {[b]}
\left(\begin{array}{cccc}f\left(x\right)& x& a& b\end{array}\right)
Find the Taylor approximation of the
\mathrm{cos}\left(x\right)
function near
x=2
t = taylor(cos(x),x,2)
\mathrm{cos}\left(2\right)+\frac{\mathrm{sin}\left(2\right) {\left(x-2\right)}^{3}}{6}-\frac{\mathrm{sin}\left(2\right) {\left(x-2\right)}^{5}}{120}-\mathrm{sin}\left(2\right) \left(x-2\right)-\frac{\mathrm{cos}\left(2\right) {\left(x-2\right)}^{2}}{2}+\frac{\mathrm{cos}\left(2\right) {\left(x-2\right)}^{4}}{24}
The Taylor expansion has six terms that are separated by
+
–
\mathrm{cos}\left(x\right)
function. Use children to separate out the terms of the expansion. Show that the Taylor expansion approximates the function more closely as more terms are included.
fplot(cos(x),[0 4])
s = s + children(t,i);
fplot(s,[0 4],'--')
legend({'cos(x)','1 term','2 terms','3 terms','4 terms','5 terms','6 terms'})
Call the children function to find the child subexpressions of the following symbolic matrix input. The result is a 2-by-2 nested cell array containing the child subexpressions of each element of the matrix.
symM = [x + y, sin(x)*cos(y); x^3 - y^3, exp(x*y^2) + 3]
\left(\begin{array}{cc}x+y& \mathrm{cos}\left(y\right) \mathrm{sin}\left(x\right)\\ {x}^{3}-{y}^{3}& {\mathrm{e}}^{x {y}^{2}}+3\end{array}\right)
s = children(symM)
s=2×2 cell array
To unnest or access the elements of the nested cell array s, use braces. For example, the {1,1}-element of s is a 1-by-2 cell array of symbolic expressions.
s11 = s{1,1}
s11=1×2 cell array
{[x]} {[y]}
Unnest each element of s using braces. Convert the nonnested cell arrays to vectors using square brackets.
s11vec = [s{1,1}{:}]
s11vec =
\left(\begin{array}{cc}x& y\end{array}\right)
\left(\begin{array}{cc}{x}^{3}& -{y}^{3}\end{array}\right)
\left(\begin{array}{cc}\mathrm{cos}\left(y\right)& \mathrm{sin}\left(x\right)\end{array}\right)
\left(\begin{array}{cc}{\mathrm{e}}^{x {y}^{2}}& 3\end{array}\right)
If each element of the nested cell array s contains a nonnested cell array of the same size, then you can also use the ind input argument to access the elements of the nested cell array. The index ind allows children to access each column of subexpressions of the symbolic matrix input symM.
scol1 = children(symM,1)
scol1=2×2 cell array
{[x ]} {[cos(y) ]}
{[x^3]} {[exp(x*y^2)]}
[scol1{:}].'
\left(\begin{array}{c}x\\ {x}^{3}\\ \mathrm{cos}\left(y\right)\\ {\mathrm{e}}^{x {y}^{2}}\end{array}\right)
{[y ]} {[sin(x)]}
{[-y^3]} {[3 ]}
\left(\begin{array}{c}y\\ -{y}^{3}\\ \mathrm{sin}\left(x\right)\\ 3\end{array}\right)
symbolic number | symbolic variable | symbolic function | symbolic expression
Input expression, specified as a symbolic number, variable, function, or expression.
Input matrix, specified as a symbolic matrix.
ind — Index of child subexpressions to return
Index of child subexpressions to return, specified as a positive integer.
If children(expr) returns a nonnested cell array of child subexpressions, then indexing with children(expr,ind) returns the ind-th element of the cell array.
If children(A) returns a nested cell array of child subexpressions, where each cell element has the same size, then indexing with children(A,ind) returns the ind-th column of the nonnested cell array.
R2020b: children returns cell arrays
In versions before R2020b, the syntax subexpr = children(expr) returns a vector subexpr that contains the child subexpressions of the scalar symbolic expression expr. The syntax subexpr = children(A) returns a nonnested cell array subexpr that contains the child subexpressions of the symbolic matrix A.
Starting in R2020b, the syntax subexpr = children(expr) returns subexpr as a cell array instead of a vector, and the syntax subexpr = children(A) returns subexpr as a nested cell array instead of a nonnested cell array. You can use subexpr = children(expr,ind) to index into the returned cell arrays of subexpressions. For example, see Plot Taylor Approximation of Expression. You can also unnest and access the elements of a cell array by indexing into the cell array using curly braces. To convert subexpr from a nonnested cell array to a vector, you can use the command [subexpr{:}].
coeffs | lhs | numden | rhs | subs |
Quantifying the Quantum Threat
Software Engineering Security Engineering Applied Cryptography Performance Optimization
2022/04/09 Nicolas Portmann pqc
Quantum computers are no longer an abstract idea as they were when Peter Shor developed his efficient quantum algorithms for discrete logarithms and factoring [S94] in 1994. And while there is still a lot of progress to be made until the quantum computers developed by IBM, Google, and the likes are practically useful. We should, nevertheless, consider the implications of large-scale quantum computers while we still have time to react. Like Peter Shor developed his quantum algorithms before physicists at Oxford University built the first 2-qubit computer in 1998, we should have our strategy for coping with large-scale quantum computers before they pose an actual threat.
One of the most frequently asked questions about quantum computers is "How big is the threat to classical cryptography posed by quantum computers?". Unless an unknown player secretly developed a quantum computer far ahead of the R&D departments of billion-dollar companies such as Google, the answer is unsurprising that there is currently no threat at all. However, if we look into the future, things aren't as clear-cut.
x
: time that products and data must remain secure
y
: time it takes to migrate to post-quantum cryptography
z
: time it takes until cryptographically-relevant quantum computers will be available
In his theorem [M15] illustrated above, Dr. Michele Mosca pointed out that if
x + y > z
, you should be worried. As cryptographers, we can do nothing to impact
z
; cryptographically-relevant quantum computers will become available, and we can do nothing about it.
x
is also out of our control for the most part. Some information just has to remain secure for a relatively long time. Even if you decide to skip reading the following more technical remainder of this post, know that the only thing we can affect is
y
. Acting now to make our organizations and systems crypto-agile is the best thing to focus on to prepare our systems for quantum adversaries.
crypto agility is not as much about using protocols that support various cryptographic algorithms and key lengths. It is about being able to make and execute the decision to swap protocols or primitives in a live system. This is just as much an organizational challenge as it is an engineering one.
To quantify the threat posed by quantum algorithms, we need metrics to measure and compare their efficiency. Its requirements in space and time typically define the performance of an algorithm. On traditional hardware, we usually compare the memory consumption and the runtime of algorithms. Terminology aside, not a lot changes when analyzing quantum algorithms. The metrics we use to compare the efficiency and space-time tradeoffs of quantum algorithms are the number of logical qubits (space complexity, also referred to as the width of a quantum circuit) and the gate depth (time complexity, best measured in the number of sequential Toffoli gates) [KHJ18] required to execute the algorithm.
Logical vs. Physical Qubits
Manufacturers of quantum computers like to advertise with big numbers. The latest IBM quantum computer packs 127 qubits [IBM21]. They are, however, prone to error and not directly useful in the business of breaking cryptographic algorithms. To satisfy quantum algorithms, we need clean, error-corrected "logical qubits". Many (~13 [ED+21]) physical qubits can be arranged to encode one logical qubit. A quantum algorithm's requirements in space are measured in logical qubits, while quantum computer marketing material usually refers to physical qubits.
Quantum gates are the building blocks of quantum algorithms. Taking the total number of gates as the measurement for the time complexity would not be accurate, as each kind of gate introduces a different overhead to the total runtime. The runtime is therefore often expressed by the number of Toffoli gates in the circuit because [KHJ18]
all quantum mechanically allowed computations can be implemented by Toffoli (and single) gates.
circuits based on Clifford gates don't provide an advantage over classical computing. Toffoli gates are non-Clifford gates and are essential in delivering a quantum benefit.
logical Toffoli gates are expected to be the primary source of time bottlenecks in real applications.
The Quantum Impact on Classical Cryptography
The quantum algorithms we know today impact different areas of classical (pre-quantum) cryptography differently. As we will see later, asymmetric cryptography based on the factorization of integers (e.g., RSA) or the discrete logarithm problem (e.g., ECDH) is impacted most. Symmetric cryptography (e.g., AES) is also impacted but to a far lesser degree.
Shor's algorithm for factoring integers [S94] is the primary cause of concern for asymmetric cryptography schemes such as RSA [RSA78]. The security of RSA is based on the fact that it is infeasible to infer the private key from the public key. The public key contains the modulus
n
e
. Factoring the prime factors (
p
q
n
- which is not computationally feasible on classical hardware - allows for a full recreation of the private key
d
. The following table lists the space and time requirements to break different key lengths (security levels) of RSA.
Note that the T-Depth in the table below is merely an estimate as no exact numbers are provided in the paper quoted below. The estimate is provided based on the given order of the total gate count
O(n^3 log n)
# qubits
T-Depth (roughly!)
112
4096
8.58 * 10^9
128
6144
6.87 * 10^{10}
RSA-15360
256
30720
3.62 * 10^{12}
Source: [HRS17]
An elliptic curve private key is a random number in the range of
1 .. n-1
n
is the order of the group of point
G
. The public key
Q
is a point defined as
Q = G*k
. The elliptic curve discrete logarithm problem (ECDLP [HL15]) is the computationally hard problem of determining
k
Q
G
. Using Shor's discrete logarithm quantum algorithm [S94], the ECDLP becomes computationally feasible under the requirements listed in the table below.
Note that the requirements for breaking RSA keys of equivalent security strength to ECC keys are roughly equal in time but more complex in space.
ECC-256
128
2330
1.16 * 10^{11}
192
3484
4.15 * 10^{11}
256
4719
1.05 * 10^{12}
Source: [RN+17]
Grover's algorithm [G97] can be used in known-plaintext attacks [JN+19] that take a reversible quantum implementation of the AES [NIST2001] cipher as well as some plaintext/ciphertext pairs and will search for the corresponding key. Multiple (2 or 3) plaintext/ciphertext pairs are required to ensure Grover finds the correct key. All requirements to break AES at different security levels are listed below.
# PT/CT pairs
AES-128
3329
1.74 * 10^{21}
2
AES-192
3969
7.45 * 10^{30}
2
AES-256
6913
3.37 * 10^{40}
3
Source: [JN+19]
Various proposals have been made to improve the security of symmetric algorithms against quantum adversaries [GT12], [BUK19], but none of them have been considered for standardization. It appears as if Grover's algorithm cannot be improved upon or parallelized to achieve significantly better results than presented above [Z08]. There may, however, be entirely different attack vectors against symmetric cryptography outside of Grover's algorithm [BSS21], [HHL09].
A more subjective quantification of the quantum threat can be found in the 2021 Quantum Threat Timeline Report from the Global Risk Institute [MP22]. 46 Experts estimated the likelihood that quantum computers will be able to break RSA-2048 in 24h in different time frames.
Source: [MP22]
The majority of experts questioned estimate a likelihood of 50%, or above that in 15 years from now, a quantum computer will be able to break RSA-2048 in 24 hours.
On the one hand, it will probably take us years until we see the first quantum computer with a cryptographically relevant number of clean, logical qubits (> 4000). Even once this milestone is reached, it may still be infeasible at first to run quantum algorithms with a depth of trillions of Toffoli gates as required to break RSA or ECDH in a reasonable amount of time.
On the other hand, we have to expect the capabilities of quantum computers to grow at an exponential rate, following a quantum equivalent to Moore's law [DS13]. Additionally, we must assume that our most sensitive information communicated today will be stored now and decrypted later - once quantum attacks on current classical cryptography become feasible. We should also consider that it takes certain industries years to develop and adopt new standards. This leaves us with an immediate need for action to protect our data with long-term security requirements [BSI20], [ANSSI22].
Quantum computers will break asymmetric cryptography based on factorization or the discrete logarithm problem in polynomial time.
Symmetric cryptography is relatively safe against quantum computers. Doubling the key length (e.g., from AES-128 to AES-256) for long-term keys is currently sufficient.
Delaying the adoption of quantum-secure cryptography is risky due to "harvest now, decrypt later" attacks.
Standardization of post-quantum cryptography is still ongoing, and the maturity level of the candidates for standardization should not be overestimated [ANSSI22].
© Nicolas Portmann |
5+4+2(-3)+7+(-5)
Draw the expression using
+
-
The first three terms have been drawn for you.
+ \quad + \quad + \qquad \quad + \quad + \qquad \quad -\ -\ -\\ \quad + \quad + \qquad \quad \ \ + \quad + \qquad \quad -\ -\ -
Simplify the expression to find the number it represents.
The last two terms are drawn below. You can move the drawings from this part and the previous parts to help you make zeroes and determine the answer.
+\ +\ +\ +\ +\ +\ + \qquad -\ -\ -\ -\ -
Remove the last number from the expression
(-5)
and find the sum again. Show how this would change your drawing. How much larger or smaller is the answer? Explain how your answer makes sense when compared to the answer in part (b).
You can use the drawings from the parts above to help you with this problem.
Use the following eTool to answer this problem. |
differintiate e2x Sin3x Cos4x [ Hint given in the solution is 2sin3xcos4x = sin7x - sinx by std trignometry formulae - Maths - Limits and Derivatives - 8787507 | Meritnation.com
differintiate e2x Sin3x Cos4x
[ Hint given in the solution is 2sin3xcos4x = sin7x - sinx by std trignometry formulae
As i understand Sin2A = 2SinACosA . Therefore 2sin3xcos3x = sin6x
So as given in hint 2sin3xcos4x should it be corrected to2sin3xcos3x (cos3x in place of cos4x)
Correct me if i am wrong.]
Let y = {e}^{2x} \mathrm{cos} 4x . \mathrm{sin} 3x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒y = \frac{1}{2}{e}^{2x}\left(2 \mathrm{cos} 4x . \mathrm{sin} 3x\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒y = \frac{1}{2}{e}^{2x}\left[\mathrm{sin}\left(4x+3x\right) - \mathrm{sin}\left(4x-3x\right)\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒y = \frac{1}{2}{e}^{2x}\left(\mathrm{sin} 7x - \mathrm{sin} x\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒y = \frac{1}{2}\left[{e}^{2x} . \mathrm{sin} 7x - {e}^{2x} . \mathrm{sin} x\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dx} = \frac{1}{2}\left[{e}^{2x} . 7 \mathrm{cos} 7x + \mathrm{sin} 7x . 2 {e}^{2x} - {e}^{2x} . \mathrm{cos} x - \mathrm{sin} x . 2 {e}^{2x} \right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dx} = \frac{{e}^{2x}}{2}\left(7 \mathrm{cos} 7x + 2 \mathrm{sin} 7x - \mathrm{cos} x - 2 \mathrm{sin} x\right) |
Use symbols to write the logical form of the following
Use symbols to write the logical form of the following arguments. If valid, identify the rule of inference that guarantees its validity. If you study hard for your discrete math final you will get an A.
Use symbols to write the logical form of the following arguments. If valid, identify the rule of inference that guarantees its validity. Qtherwise, state whether the converse or the inverse error has been made.
a) If you study hard for your discrete math final you will get an A.
b) Jane got an A on her discrete math final.
c) Therefore, Jane must have studied hard.
Let P be the statement where
P:\text{Studying hard for discrete math final}
Let Q be the statement, where
Q:\text{Getting A in discrete math final}
Definition: "If A is true then B is true." The logical form of this statement is:
A\to \text{ }B
"If you study hard for your discrete math final you will get A"
Note that P:
P:\text{Studying hard for discrete math final}
Q:\text{Getting A in discrete math final.}
The logical form of the given statement is:
P\to \text{ }Q
The given statements is:
"Jane got an A on her discrete math final. therefore Jane must have studied hard."
"Jane got an A on her discrete math final." Hence the statement Q is true for Jane. This statement implies that "Jane must have studied hard." Hence P is true. If Q is true then P is true.
The logical form of the statement is:
Q\to \text{ }P
P\to \text{ }Q
From the second and third statements,
Q\to \text{ }P
For example, If the fruit is banana then it is yellow in color. if the fruit is yellow in color one can not assure whether the fruit is a banana.
Hence, converse or inverse error has been made.
\left(A-B\right)-C=\left(A-C\right)-\left(B-C\right)
\cap \cup
\left(a,b\right)\in R
The following problem is solved by using factors and multiples and features the strategies of guessing and checking and making an organized list.
A factory uses machines to sort cards into piles. On one occasion a machine operator obtained the following curious result.
When a box of cards was sorted into 7 equal groups, there were 6 cards left over, when the box of cards was sorted into 5 equal groups, there were 4 left over, and when it was sorted into 3 equal groups, there were 2 left.
If the machine cannot sort more than 200 cards at a time, how many cards were in the box?
1620-sq-ft
Transcribed Image Text 3) Let U = {ne N:ns 200}, find the number of elements of U that are divisible by 2, 3, or 5. |
Ohm's law (microscopic interpretation) Practice Problems Online | Brilliant
Suppose that the number of conduction electrons per unit volume in a certain metal is
n=1.23 \times 10^{29} \text{ m}^{-3}
and the mean free time between collisions for the conduction electrons in that metal is
\tau=3.42 \times 10^{ -15} \text{ s}.
What is the resistivity
\rho
of that metal?
e=1.60 \times 10^{-19}\text{ C}
and the mass of electron is
m=9.11 \times 10^{-31}\text{ kg}.
8.46 \times 10^{-8} \,\Omega\cdot\text{m}
7.53 \times 10^{-9} \,\Omega\cdot\text{m}
5.67 \times 10^{-8} \,\Omega\cdot\text{m}
4.75 \times 10^{-9} \,\Omega\cdot\text{m}
If the number of conduction electrons per unit volume in copper is
8.47 \times 10^{28}\text{ m}^{-3}
and the resistivity of copper is
1.61 \times 10^{-8}\,\Omega\cdot\text{m},
what is the mean free time
\tau
between collisions for the conduction electron in copper?
e=1.60 \times 10^{-19}\text{ C}
m=9.10 \times 10^{-31}\text{ kg}.
3.18 \times 10^{-17} \text{ s}
2.32 \times 10^{-15} \text{ s}
2.03 \times 10^{-16} \text{ s}
2.61 \times 10^{-14} \text{ s}
If the mean free time between collisions for the conduction electrons in copper is
\tau=2.5 \times 10^{-14}\text{ s}
and their effective speed
v_{\text{eff}}
1.5 \times 10^6\text{ m/s},
what is the mean free path
\lambda
for the conduction electron in copper?
37.5 \text{ nm}
18.7 \text{ nm}
12.5 \text{ nm}
56.2 \text{ nm}
The measured resistivity of aluminium at
25\,^\circ\text{C}
2.72 \times 10^{-8}\,\Omega\cdot\text{m}.
The valency, density, and the atomic mass of aluminium are
3,
2.68\text{g/cm}^3,
27,
respectively. Assuming that each aluminium atom contributes three free conduction electrons to the metal, what is the mean free time between collisions for the conduction electrons in aluminium at a temperature of
25\,^\circ\text{C}?
e=1.602 \times 10^{-19}\text{ C}.
The mass of electron is
m=9.109 \times 10^{-31}\text{ kg}.
The Avogadro constant is
N_A=6.022 \times 10^{23}\text{ mol}^{-1}.
7.28 \times 10^{-15} \text{ s}
2.18 \times 10^{-13} \text{ s}
2.18 \times 10^{-15} \text{ s}
7.28 \times 10^{-13} \text{ s}
The measured electron drift mobility in silver is
57\text{ cm}^2\text{V}^{-1}\text{s}^{-1}
27\,^\circ\text{C}.
The atomic mass and density of silver are
107.87\text{ g/mol}
10.70\text{ g/cm}^3,
respectively. Assuming that each silver atom contributes one conduction electron, what is the resistivity of Ag at
27\,^\circ\text{C}?
e=1.602 \times 10^{-19}\text{ C}.
m=9.109 \times 10^{-31}\text{ kg}.
N_A=6.022 \times 10^{23}\text{ mol}^{-1}.
1.83 \text{ n}\Omega\cdot\text{m}
12.83 \text{ n}\Omega\cdot\text{m}
18.34 \text{ n}\Omega\cdot\text{m}
5.50 \text{ n}\Omega\cdot\text{m} |
Suppose unpolarized light of intensity 150 W/m^{2} falls on the polarizer at an
Suppose unpolarized light of intensity 150 W/m^{2} falls on the polarizer at an angle 30. What is the light intensity reaching the photocell?
150\frac{W}{{m}^{2}}
Intensity of unpolarised light
I=150\frac{W}{{m}^{2}}
Intensity reaching the photocell
{I}^{\prime }=\frac{I}{2}=75\frac{W}{{m}^{2}}
\left(1-{\frac{t}{40}}^{2}\right)5000=V0\le t\le 40
A compound consisting of C, H and O only, has a molar mass of 331.5g/mol. Combustion of 0.1000g of this compound caused a 0.2921g increase in the mass of the CO2 absorber and a 0.0951 g increasein the mass of the H2O absorber. What is the empirical formula ofthe compound?
What is ment by absorber ? product?
A major information system has 1140 modules.There are 96 modules that perform control and coordinationfunctions and 490 modules who function depends on prior processing.The system processes approximately 220 data objects that each havean average of three attributes. There are 140 unique data baseitems and 90 different database segments. Finally, 600 modules havesingle entry and exit points. Compute the DSQI for this system
A baton twirler throws a spinning baton directly upward.As it goes up and returns to the twirlers |
IPv4 - CS Notes IPv4 | CS Notes
IP (the Internet Protocol) is a routing and addressing protocol. When it was first proposed, IP was intended as a way to connect multiple LANs, but now you can imagine IP as a global virtual LAN [1, P. 185].
Although IP version 6 was formalized in 1998, the Internet still mostly uses version 4 of the protocol (IPv4).
Public IP addresses are administratively assigned. Originally they were assigned by IANA, but IANA now delegates the task to other organizations [1, P. 24].
IP addresses are 32 bits (4 bytes) long. Commonly, IPv4 addresses are represented in dotted decimal notation, where each byte is written in decimal, separated by a period. For example, 172.16.254.1.
Figure: IPv4 address
IP addresses are hierarchical. They are split into two parts: a network part and a host part. The network part has the same value for all hosts on a LAN [2, P. 443].
This addressing strategy is known as CIDR (Classless Interdomain Routing), pronounced cider.
The length of the network part determines how many host addresses are available on a network. For example, if the network part is 16 bits, then 16 bits are free for the host part, which means a maximum of 65534 addressed hosts on the network (
2^{16} - 2 \text{ reserved IP addresses}
Prefixes are written in the form A.B.C.D/P, where P is the number of bits used for the network part. For example, an IP address with a 16-bit network part would be denoted as 172.16.0.0/16.
A network part can’t be inferred from the IP address, so routing protocols must provide the length of the network part when sharing route information [2, P. 443].
The length of the network part can be used to create a subnet mask, which produces the network part when logically ANDed with an IP address.
Figure: Calculating network part using subnet mask
The advantage of splitting IP addresses into parts is that routers can forward packets based solely on their network part. The host part is only used when a packet has arrived in the network specified with the network part [1, P. 193].
Subnets enable a network to be split into multiple smaller networks, rather than requiring networks to be entirely LAN switched [1, P. 195].
Subnets work by assigning a subnet mask to each of the internal networks. In order to route packets internally, a main router must know the subnet mask for each of its subnets. The main router can determine which network to forward a packet to by bitwise ANDing the destination IP address with each of the subnet masks in turn, until a match is found.
The IPv4 header contains the following information:
Destination and source address.
Indication of IPv4 (vs IPv6).
A Time To Live (TTL) value, to prevent routing loops.
A field indicating what comes next in the packet (e.g. TCP or UDP).
The header format is as follows:
|Version| IHL | DS Field | * | Total Length |
* ECN
Version is for the IPv4 version (0100).
IHL represents the total length of the header length in 32-bit words. Since IHL is 4 bits, the max header size is 15 32-bit words [1, P. 186].
The DS (Differentiated services) Field is used to specify preferential handling for certain packets, e.g. those involved in VoIP.
Total Length is the length of the datagram in octets.
Time to Live is used to stop routing loops. It’s decremented by 1 at each router. If it reaches 0, the packet is dropped.
Protocol indicates the protocol used in the packet’s data. Common values are 6 (TCP), and 17 (UDP).
Header Checksum is used to verify that the header wasn’t corrupted during transmission.
The Source Address and Destination Address fields contain the IPv4 addresses of the sender and the recipient.
IP addresses are given to interfaces, rather than nodes or hosts [1, P. 188].
One example is the loopback interface. For most machines, localhost resolves to the IPv4 loopback address 127.0.0.1. “Delivering packets to the loopback interface is simply a form of interprocess communication” [1, P. 188].
There are also other special interfaces. For example, when VPN connections are created, “each end of the logical connection typically terminates at a virtual interface” [1, P. 188].
“When a computer hosts a virtual machine, there is almost always a virtual network to connect the host and virtual systems. The host will have a virtual interface to connect to the virtual network. The host may act as a NAT router for the virtual machine, “hiding” that virtual machine behind its own IP address, or it may act as an Ethernet switch, in which case the virtual machine will need an additional public IP address” [1, P. 188].
“Routers always have at least two interfaces on two separate IP networks”. Normally the router would have a separate IP address for each interface, although some point-to-point interfaces can be used without IP addresses [1, P. 189].
A multihomed host is a non-router host with multiple non-loopback network interfaces. For example, many laptops have an Ethernet interface and a Wi-Fi interface. These interfaces can be used simultaneously if they both have a different IP address [1, P. 189].
It’s also possible to assign multiple different IP addresses to a single interface. Sometimes this is done to enable two different IP networks to share the same LAN [1, P. 189].
IPv4 has a few assigned special addresses.
Loopback addresses. The default loopback address is 127.0.0.1, however, any IPv4 address beginning with 127 can serve as a loopback address [1, P. 190].
Private addresses. Private addresses are IP addressses that are intended for internal use only. There are three standard private-address blocks:
Broadcast addresses are IPv4 addresses intended to be used with LAN broadcasting. The common forms are 255.255.255.255 to broadcast to the network the device is on. Historically 0.0.0.0 was also used as a broadcast address. You can also broadcast to a different network by filling the host part of an IP address with all 1-bits. This is why all host ranges have
2^n - 2
n
is the number of host bits [1, P. 190].
Multicast Addresses: Multicasting means sending packets to a specified set of addresses. Multicast addresses have the first byte beginning 1110 [1, P. 190].
IPv4 supports fragmentation to break up large packets into smaller chunks. This means large packets can be sent over networks that cannot support the full size of the packet. The fragments are reassembled once they have been received by the destination host [1, P. 191].
IP follows a path fragmentation and reassembly process where reassembly is done at the far end of the path, rather than by intermediate routers [1, P. 191].
The Identification field in the IP header is used to group fragmented IP packets. Its value should be different for each packet. Fragments of a packet keep the same Identification value as their original packet, so it’s possible to identify fragments of a packet by comparing their Identification value [1, P. 191].
The Fragment Offset field marks the start position of the data portion of a fragment within the data portion of the original packet. This is used to reassemble the packet [1, P. 191].
TCP normally uses Path MTU Discovery to discover the maximum transmission size that is supported over the network. It will then keep packets under this size in order to avoid IP fragmentation. However, it’s not uncommon for fragmentation to occur over UDP, as in the NFS protocol [1, P. 192].
It’s worth noting that IPv6 doesn’t support fragmentation [1, P. 185].
NAT (Network Address Translation) is an approach to use a single IP address for a network of IP-connected devices.
Instead of assigning an IP address to each host in an internal network, a public IP address is assigned only to a gateway router. The gateway router, known as a NAT router, connects the internal network to the Internet.
All hosts in the internal network are assigned private IP addresses. When an internal host makes a request, the NAT router will translate the source private IP address into its own public IP address, and keep the translation in a special NAT forwarding table. When the NAT gateway receives a response from the remote machine, it will check its NAT forwarding table, see that the request is for the internal host, replace the destination IP address with the private source IP address, and forward the packet to the internal host [1, Pp. 200-1].
Figure: NAT router [1, P. 201]
The NAT forwarding table includes port numbers, so that it can distinguish between two different internal hosts attempting to connect to the same external host. If two internal hosts attempt to reach the same host from the same port, then the NAT router will need to rewrite one of the source port numbers to be able to distinguish between packets destined for each router. |
Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations | J. Comput. Nonlinear Dynam. | ASME Digital Collection
Olivier A. Bauchau,
e-mail: [email protected]
Alexander Epple,
e-mail: [email protected]
e-mail: [email protected]
Bauchau, O. A., Epple, A., and Bottasso, C. L. (March 9, 2009). "Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations." ASME. J. Comput. Nonlinear Dynam. April 2009; 4(2): 021007. https://doi.org/10.1115/1.3079826
This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size,
h
; hence, the condition number of the Jacobian and error propagation are both
O(h0)
: the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.
differential algebraic equations, integration, Jacobian matrices, joining processes, linearisation techniques, many-body problems, mechanical products
Equations of motion, Jacobian matrices, Multibody systems, Simulation, Multibody dynamics, Differential algebraic equations, Algebra, Errors
Laulusa
A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems. Part I
A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems. Part II
Computer Aided Analysis and Optimization of Mechanical Systems Dynamics
ODE Methods for the Solution of Differential/Algebraic Systems
Lötstedt
Numerical Solution of Nonlinear Differential Equations With Algebraic Constraints. II: Practical Implications
Time Integration of the Equations of Motion in Mechanism Analysis
An Integrated Approach to Mechanism Analysis
,” Ph.D. thesis, Université de Liège, Belgium.
Analysis of Transient Algorithms With Particular Reference to Stability Behavior
Engineering Optimization. Methods and Applications
Numerical Optimization Techniques for Engineering: With Applications
Numerical Integration of Second Order Differential-Algebraic Systems in Flexible Mechanism Dynamics
A Modified Lagrangian Formulation for the Dynamic Analysis of Constrained Mechanical Systems
Modeling Rotorcraft Dynamics With Finite Element Multibody Procedures
Sequential Quadratic Programming Methods For Nonlinear Programming
Computer-Aided Analysis and Optimization of Mechanical System Dynamics
Computational Schemes for Flexible, Nonlinear Multi-Body Systems |
Lowest Common Multiple Practice Problems Online | Brilliant
2
3
A bus stop has two bus lines: the red line and the blue line. The red line stops every
18
minutes, and the blue line stops every
27
minutes. If the buses of both lines have just simultaneously stopped at the bus stop, how many minutes later will the buses of the two lines meet at this bus stop again?
3^a 5^b
3^4
3^5
3^a 5^b ? |
For the following exercises, enter the data from each table into a graphing calculator and
For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the da
\begin{array}{|ccccccccccc|}\hline x& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13\\ f\left(x\right)& 9.429& 9.972& 10.415& 10.79& 11.115& 11.401& 11.657& 11.889& 12.101& 12.295\\ \hline\end{array}
List all of the elements of
\left\{I,J,K\right\}×\left\{Q,R\right\}.
An engineering company has four openings and the applicant pool consists of 6 database administrators and eight network engineers. All are equally qualified so the hiring will be done randomly. The hiring committee consists of 4 women and 4 men.
a. If one person on the hiring committee is chosen at random to draw the names out of a hat, what is the probability that the person drawing the names is a woman.
b. How many ways can the group who are to be hired be formed if they are no restrictions on composition?
c. How many ways can 3 database administrators be chosen?
d. How many ways can 1 network engineers be chosen?
e. What is the probability that the random selection of the four persons to be hired will result in 3 database administrators and 1 network engineer?
Ali Baba's Car Wash Service Centre is open 6 days a week, but its busiest day is always on Sunday. From the previous data, Ali Baba estimates that dirty cars arrive at the rate of one every two minutes, One car at a time is cleaned in this example of a single-channel waiting line. Assuming Poisson arrivals and exponential service times, find the following:
Calculate the value of probability that there are no car in the system.
A random sample of 25 of the 400 member of the Bigtime Theater Company is surveyed about how may plays each has acted in. Which statement below would be an appropriate inference about the data?
3,5,5,3,4,4,1,3,6,10,1,3,4,5,1,2,4,2,3,2,5,5,5,5,6
1) Most members have acted in 2 plays.
2) Most members have acted in more than 6 plays.
3) Most members have acted in between 1-4 plays.
4) The number of members who have acted in 10 plays is greater than the number of members who have acted in 1 play.
A.C. Neilsen reported that children between the ages of 2 and 5 watch an average of 25 hours of television per week. Assume the amount watched by a randomly selected child is normally distributed and the standard deviation is 3 hours. If 20 children between the ages of 2 and 5 are randomly selected, find the standard error of the average number of hours watched by them in the sample.
Some non-linear regressions can also be estimated using a linear regression model (using "linearization"). Assume that the data below show the selling prices y (in dollars) of a certain equipment against its age x (in years). We'd like to fit a non-linear regression im the form
\stackrel{^}{y}=c{d}^{X}
to estimate parameters c and d from the data by linearizing the model through
\mathrm{ln}\stackrel{^}{y}=\mathrm{ln}c+\left(\mathrm{ln}d\right)x={b}_{0}+{b}_{1}x
\begin{array}{|cccc|}\hline x& y& x& y\\ 1& 6312& 3& 5387\\ 2& 5697& 5& 4973\\ 2& 5734& 5& 4892\\ \hline\end{array}
Using Excel ot other software, the non-linear regression model
\stackrel{^}{y}=c{d}^{X}
\stackrel{^}{y}=?{\left(?\right)}^{X} |
MediaWiki file: WikiDiff3.php
Location: includes/diff/
Classes: WikiDiff3 • RangeDifference
New version of the difference engine. This diff implementation is mainly lifted from the LCS algorithm of the Eclipse project which in turn is based on Myers' "An O(ND) difference algorithm and its variations" citeseer.ist.psu.edu with range compression (see Wu et al.'s "An O(NP) Sequence Comparison Algorithm".
This implementation supports an upper bound on the execution time. Complexity:
{\displaystyle O((M+N)D)}
{\displaystyle O(M+N+D^{2})}
{\displaystyle O(M+N)}
Retrieved from "https://www.mediawiki.org/w/index.php?title=Manual:WikiDiff3.php&oldid=4244524" |
Frequency offset estimation using cyclic prefix - MATLAB lteFrequencyOffset - MathWorks í•œêµ
{N}_{\text{RB}}^{\text{DL}}
{N}_{\text{RB}}^{\text{UL}}
Average frequency offset estimate, returned as a scalar value expressed in Hertz. This function can only accurately estimate frequency offsets of up to ±7.5 kHz (a range of 15 kHz, the subcarrier spacing). |
If we randomly select 10 people, using Binomial probability distribution,
If we randomly select 10 people, using Binomial probability distribution, find the probability that a) at least 8 people like the drink.
An advertising agency is conducting a survey to after introducing a fruit drink in the market.
As per the survey about 60 % of the people like the drink.
If we randomly select 10 people,
using Binomial probability distribution, find the probability that
a) at least 8 people like the drink.
b) at most 3 people like the drink
In a poll men and women were asked"When someone yelled or snapped at you at work, how did you want to respond Twenty percent of the women in the survey said that they felt like crying (TimeApril 4, 2011) Suppose that this result is true for the current population of women employees. Arandom sample of 24 women employees is selectedUse the binomial probabilities table or technology to find the probability that the number of women employees in this sample of 24 who will hold the above opinion in response to the said question is
Find a basis for the set of vectors in ℝ3 in the plane
x+2y+z=0
A binomial probability is given:
P\left(x\le 124\right)
Find the answer that corresponds to the binomial probability statement.
P\left(x>123.5\right)
P\left(x<123.5\right)
P\left(x<124.5\right)
P\left(x>124.5\right)
P\left(123.5<x<124.5\right)
If x is a binomial random variable, what is the probability of x for
n=4,\text{ }x=1,\text{ }p=0.3
P\left(x\right)
n=16,\text{ }x=3,\text{ }p-\frac{1}{5}
Drone Deliveries Based on a Pitney Bowes survey, assume that 42% of consumers are comfortable having drones deliver their purchases. Suppose we want to find the probability that when five consumers are randomly selected, exactly two of them are comfortable with the drones. What is wrong with using the multiplication rule to find the probability of getting two consumers comfortable with drones followed by three consumers not comfortable, as in this calculation: (0.42) (0.42) (0.58) (0.58) (0.58) = 0.0344? |
A person stands on a scale inside an elevator at rest. The Scale reads 800N. a)
A person stands on a scale inside an elevator at rest. The Scale reads 800N. a) What is the person's mass? (b) The elevator accelerates upward momentarily scale read then? (c) The elevator then moves with a steady speed of 5m/s. What Does the scale read?
mass m=
\frac{W}{g}
= weight/acceleration due togravity
=\frac{800}{9.8}=81.6326\text{ }kg
Key idea is the scale reading is equal to the magnitude of normal force N on the passenger from the scale.
N-{F}_{g}=ma
N={F}_{g}+ma
This tells us that the scale reading , which is equal to N depends on the vertical acceleration a of the cab. Substituting mg for Fg gives us N = mg +ma = m (g+a)
For any choice of acceleration. a. if it accelerates up the weight will be more because a +ve
if it accelerates downward then weight will decrease because a is - ve then.
If moving with constant speed a =0 so the weight will remain the same as 800N
When suddenly accelates up let us say acceleration = a
Then scale reads m(g + a) = 81.83( 9.8 +a)
When it is moving with constant speed means not moving with any additional acceleration . a = 0
Scale reading = mg = 800N
\frac{F}{16}
\frac{F}{8}
\frac{F}{4}
\frac{F}{2}
\underset{\theta \to \frac{\pi }{4}}{lim}\frac{\mathrm{cos}\theta -\mathrm{sin}\theta }{\theta -\frac{\pi }{4}}
This limit takes
\frac{0}{0}
form when
\theta =\frac{\pi }{4}
\frac{0}{0}
form is an indeterminate form. So how do I make it determinate?
F1=7.50N
F2=5.30N
are applied tangentially to a wheel with radius 0.330m. What is the net trque on the wheel due tothese two forces for an axis perpendicular to the wheel and passingthrough its center.
Discuss the continuity of the function below at x=-3. Justify you work with limits.
h\left(x\right)=\left\{\begin{array}{ll}\frac{{x}^{2}+2x-5}{x+7},& x\ne -3\\ 8,& x=-3\end{array} |
Plot simulated time response of dynamic system to arbitrary inputs; simulated response data - MATLAB lsim - MathWorks India
\begin{array}{cc}sys\left(s\right)=\frac{{\omega }^{2}}{{s}^{2}+2s+{\omega }^{2}},& \omega =62.83\end{array}.
sys\left({z}^{-1}\right)=\frac{{a}_{0}+{a}_{1}{z}^{-1}+\dots +{a}_{n}{z}^{-n}}{1+{b}_{1}{z}^{-1}+\dots +{b}_{n}{z}^{-n}},
y\left[k\right]={a}_{0}u\left[k\right]+\dots +{a}_{n}u\left[k-n\right]-{b}_{1}y\left[k-1\right]-\dots -{b}_{n}\left[k-n\right].
\begin{array}{c}x\left[n+1\right]=Ax\left[n\right]+Bu\left[n\right],\\ y\left[n\right]=Cx\left[n\right]+Du\left[n\right].\end{array} |
A person jumps from a fourth-story window 15.0 m above a firefighter's safety ne
A person jumps from a fourth-story window 15.0 m above a firefighter's safety net. The survivor stretches the net 1.0 m before coming to rest. (a) Wha
A person jumps from a fourth-story window 15.0 m above a firefighter's safety net. The survivor stretches the net 1.0 m before coming to rest. (a) What was the average deceleration experienced by the survivor when slowed to rest by the net? (b)What would you do to make it "safer" (that is, generate a smaller deceleration): would you stiffen or loosen the net? Explain.
First, we need to know how fast the person was falling when he hit the net. We know how far he fell, so it is a simple task of determining the his velocity:
v={v}_{0}+at
which means we need to know how long he was falling:
d={v}_{0}+\frac{1}{2}a{t}^{2}
I will assume is initial velocity was zero. We can then solve the above equation for t: (keep in mind d is the distance he fell,and a is gravity, both of these are negative quantities so the negatives cancel)
t=\sqrt{\frac{2d}{a}}\approx 1.75\text{ }s
plugging t into the velocity equation:
v=\left(-9.8\right)\left(1.75\right)\approx 17.15\text{ }\frac{m}{s}
Now the average deceleration can be found by taking the change in velocity over the change in time. But alas, we do not have tright off hand, so we need something else What we do know is that it took 1m to stop the person. Lets
Lets calculate speed just before hitting net
{v}_{0}=\sqrt{2gh}=\sqrt{2\left(9.8\text{ }m/{s}^{2}\right)\left(15m\right)}
{v}_{0}=17.15\text{ }m/s
a) Person comes to rest in 1m
{v}_{f}^{2}-{v}_{r}^{2}=2ad
{0}^{2}-\left(17.15{\right)}^{2}=2\left(-a\right)\left(1m\right)
a=147\text{ }m/{s}^{2}
a=\frac{{v}_{0}^{2}}{2d}
for smaller a ,d have to be larger
so, loosen the net
1950\text{ }N\cdot m
{y}_{1}=\left(3.0cm\right)\mathrm{cos}\left(4.0x-1.6t\right)
{y}_{2}=\left(4.0cm\right)\mathrm{sin}\left(5.0x-2.0t\right)
where y and x are in centimeters and t is inseconds. Find the superposition of the waves
{y}_{1}+{y}_{2}
x= 0.500, t=0.
For the cellar of a new house a hole is dug in the ground, withvertical sides going down 2.40m. A concrete foundation wallis built all the way across the 9.6m width of the excavation. This foundation wall is 0.183m away from the front of the cellarhole. During a rainstorm, drainage from the streetfills up the space in front of the concrete wall, but not thecellar behind the wall. The water does not soak into the clays oil. Find the force the water causes on the foundation wall. For comparison, the weight of the water is given by:
2.40m\cdot 9.60m\cdot 0.183m\cdot 1000k\frac{g}{{m}^{3}}\cdot 9.8\frac{m}{{s}^{2}}=41.3kN
A=\left[\begin{array}{ccc}1& 0& -2\\ 3& 1& 0\\ 1& 0& -3\end{array}\right],\text{ }B=\left[\begin{array}{ccc}1& 0& -3\\ 3& 1& 0\\ 1& 0& -2\end{array}\right]
Find an elementary matrix E such that EA=B. |
1. 2M(s)+6HCl(aq) \Rightarrow 2MCl3(aq)+3H2(g)2M(s)+6HCl(aq) \Rightarrow 2MCl3(aq)+3H2(g) \triangle H_{1}=-819.0kJ 2. HCl(g) \Rightarrow HCl(aq)HCl(g)
2M\left(s\right)+6HCl\left(aq\right)⇒2MCl3\left(aq\right)+3H2\left(g\right)2M\left(s\right)+6HCl\left(aq\right)⇒2MCl3\left(aq\right)+3H2\left(g\right)
\mathrm{△}{H}_{1}=-819.0kJ
HCl\left(g\right)⇒HCl\left(aq\right)HCl\left(g\right)⇒HCl\left(aq\right)
\mathrm{△}{H}_{2}=-74.8kJ
H2\left(g\right)+Cl2\left(g\right)⇒2HCl\left(g\right)H2\left(g\right)+Cl2\left(g\right)⇒2HCl\left(g\right)
\mathrm{△}{H}_{3}=-1845.0kJ
MCl3\left(s\right)⇒MCl3\left(aq\right)
\mathrm{△}{H}_{4}=-128.0kJ
Use the given information to determine the enthalpy of the reaction
2M\left(s\right)+3C{l}_{2}\left(g\right)⇒2MC{l}_{3}\left(s\right)
\mathrm{△}H=?
In the above reactions we only need 2 reactions to calculate the enthalpy of the following reaction;
2M\left(s\right)+3Cl2\left(g\right)⇒MCl3\left(g\right)
We have 4 chemical equations from which 2 are important:
2M\left(s\right)+6HCl\left(aq\right)⇒2MC{l}_{3}\left(aq\right)+3{H}_{2}\left(g\right)
\mathrm{△}{H}_{1}=819KJ
{H}_{2}\left(g\right)+C{l}_{2}\left(g\right)⇒2HCl\left(g\right)
\mathrm{△}{H}_{3}=-1845KJ
Multiplying equation 1st by 3
3{H}_{2}\left(g\right)+3C{l}_{2}\left(g\right)⇒6HCl\left(g\right)
\mathrm{△}{H}_{3}=3×\left(-1845KJ\right)=-5535KJ
Adding equation 1st and 3rd
2M\left(s\right)+3C{l}_{2}\left(g\right)⇒2MC{l}_{3}\left(aq\right)
\mathrm{△}{H}_{reaction}
=\left(\mathrm{△}{H}_{1}+\mathrm{△}{H}_{3}\right)=-4716KJ
Therefore the enthalpy of formation
=-4716KJ
Chemical reaction 1:
2M\left(s\right)+6HCl\left(aq\right)⇒2MCl3\left(aq\right)+3H2\left(g\right);\mathrm{△}H1=-556.0kJ
. Chemical reaction 2:
HCl\left(g\right)⇒HCl\left(aq\right);\mathrm{△}H2=-74.8kJ
H2\left(g\right)+Cl2\left(g\right)⇒2HCl\left(g\right);\mathrm{△}H3=-1845.0kJ
MCl3\left(s\right)⇒MCl3\left(aq\right);\mathrm{△}H4=-342.0kJ
2M\left(s\right)+3Cl2\left(g\right)⇒2MCl3\left(s\right);\mathrm{△}H5=?.\mathrm{△}H5=\mathrm{△}H1+6·\mathrm{△}H2+3\cdot \mathrm{△}H3-2·\mathrm{△}H4
\mathrm{△}H5=-550kJ+6·\left(-74,8kJ\right)+3\cdot \left(-1845kJ\right)-2\cdot \left(-342kJ\right)
\mathrm{△}H5=-550kJ-448,8kJ-5535kJ+684kJ
\mathrm{△}H5=-5849,8kJ
\phantom{\rule{0ex}{0ex}}2M\left(s\right)+3C{l}_{2}\left(g\right)⇒2MC{l}_{3}\left(s\right)\phantom{\rule{0ex}{0ex}}\mathrm{△}H=-944kJ+6×\left(-74.8kJ\right)+3×\left(-1845kJ\right)+\left(2\right)×\left(234kJ\right)\phantom{\rule{0ex}{0ex}}\mathrm{△}H=-944kJ-448.8-5535kJ+468kJ\phantom{\rule{0ex}{0ex}}\mathrm{△}H=-6459.8kJ
{25.0}^{\circ }
{30}^{\circ }
{P}_{n}\left(F\right)
is generated by {1,x,....,show pn(f) is generated by
1,x,....,{x}_{n}
Show transcribed image text Show
{P}_{n}\left(F\right)
1,x,....,{x}_{n}
Gold, which has a density of
19.32\frac{g}{c}{m}^{3}
, is the most ductile metal and can be pressed into a thin leaf or drawn out into a long fiber.
(a) If a sample of gold, with a mass of 27.63 g, is pressed into a leaf of
1.000\mu m
thickness, what is the area of the leaf?
b) If, instead, the gold is drawn out into a cylindrical fiber of radius
2.500\mu m
, what is the length of the fiber?
Which of the following are Arrhenius acids?
\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 |
Requires Uploaded Supporting Analysis Find \frac{dy}{dx} by implicit differentiation. y^2-2x=4y
Requires Uploaded Supporting Analysis Find
\frac{dy}{dx}
{y}^{2}-2x=4y
{y}^{2}-2x=4y
\frac{d}{dx}\left({y}^{2}-2x\right)=\frac{d}{dx}\left(4y\right)
\frac{d}{dx}\left({y}^{2}\right)-\frac{d}{dx}\left(2x\right)=4\frac{dy}{dx}
2y\frac{dy}{dx}-2=4\frac{dy}{dx}
\frac{dy}{dx}\left(2y-4\right)=2
\frac{dy}{dx}=\frac{2}{2y-4}
\frac{dy}{dx}=\frac{1}{y-2}
P\left(x\right)=-12{x}^{2}+2136x-41000
x=r\mathrm{cos}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}y=r\mathrm{sin}
\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{{x}^{2}-{y}^{2}}{\sqrt{{x}^{2}+{y}^{2}}}
p=126-0.5x
C=50x+39.75
Which interval is wider: (a) the 95% confidence interval for the conditional mean of the response variable at a particular set of values of the predictor variables or (b) the 95% prediction interval for the response variable at the same set of values of the predictor variables?
y=140t+\frac{1}{2}{t}^{2}-{t}^{3},\text{ }0\le t\le 8
a) After how many hours will the hourly number of units be maximized? hr
\frac{units}{hr}
{x}^{4}+2{x}^{3}+22{x}^{2}+50x-75
The complex zeros of f are ?
Use the complex zeros to factor f.
Lynbrook West, an apartment complex, has 100 two-bedroom units. The monthly profit (in dollars) realized from renting x apartments is represented by the following function.
P\left(x\right)=-11{x}^{2}+1830x-34000
(a) What is the actual profit realized from renting the 41st unit, assuming that 40 units have already been rented? |
Area Of A Circle | Formula For Radius, Diameter, & Circumference
Area Of A Circle Using Diameter
Area and Circumference Formula
How To Find Area With Circumference
A circle is not a square, but a circle's area (the amount of interior space enclosed by the circle) is measured in square units. Finding the area of a square is easy: length times width.
A circle, though, has only a diameter, or distance across. It has no clearly visible length and width, since a circle (by definition) is the set of all points equidistant from a given point at the center.
Yet, with just the diameter, or half the diameter (the radius), or even only the circumference (the distance around), you can calculate the area of any circle.
Recall that the relationship between the circumference of a circle and its diameter is always the same ratio,
3.14159265
, pi, or
\pi
. That number,
\pi
, times the square of the circle's radius gives you the area of the inside of the circle, in square units.
If you know the radius,
r
, in whatever measurement units (mm, cm, m, inches, feet, and so on), use the formula π r2 to find area,
A
A = \pi {r}^{2}
The answer will be square units of the linear units, such as
m{m}^{2}
c{m}^{2}
{m}^{2}
, square inches, square feet, and so on.
Here is a circle with a radius of 7 meters. What is its area?
[insert drawing of 14-m-wide circle, with radius labeled 7 m]
A = \pi ·{r}^{2}
A = \pi × {7}^{2}
A = \pi × 49
\mathbf{A} \mathbf{=} \mathbf{153.9380} {\mathbf{m}}^{\mathbf{2}}
If you know the diameter,
d
, in whatever measurement units, take half the diameter to get the radius,
r
, in the same units.
Here is the real estate development of Sun City, Arizona, a circular town with a diameter of
1.07
kilometers. What is the area of Sun City?
First, find half the diameter, given, to get the radius:
\frac{1.07}{2} = 0.535 km = \mathbf{535} \mathbf{m}
Plug in the radius into our formula:
A = \pi ·{r}^{2}
A = \pi × {535}^{2}
A = \pi × 286,225
A = 899,202.3572 {m}^{2}
To convert square meters,
{m}^{2}
, to square kilometers,
k{m}^{2}
1,000,000
\mathbf{A} \mathbf{=} \mathbf{0.8992} {\mathbf{km}}^{\mathbf{2}}
Sun City's westernmost circular housing development has an area of nearly 1 square kilometer!
Try these area calculations for four different circles. Be careful; some give the radius,
r
, and some give the diameter,
d
Remember to take half the diameter to find the radius before squaring the radius and multiplying by
\pi
A 406-mm bicycle wheel
The London Eye Ferris wheel with a radius of 60 meters
A 26-inch bicycle wheel
The world's largest pizza had a radius of 61 feet, 4 inches (736 inches)
Do not peek at the answers until you do your calculations!
A 406-mm bicycle wheel has a radius,
r
, of 203 mm:
A = \pi {r}^{2}
A = \pi × 203 m{m}^{2}
A = 637.7433 m{m}^{2}
The London Eye Ferris wheel's 60-meter radius:
A = \pi {r}^{2}
A = \pi × 60 {m}^{2}
A = 188.4955 {m}^{2}
A 26-inch bicycle wheel has a radius,
r
, of 13 inches:
A = \pi {r}^{2}
A = \pi × 13 i{n}^{2}
A = 530.9291 i{n}^{2}
The world's largest pizza with its 736-inch radius:
A = \pi {r}^{2}
A = \pi × 736 i{n}^{2}
A = 1,701,788.17 i{n}^{2}
11,817.97 f{t}^{2}
of pizza! Yum! Anyway, how did you do on the four problems?
If you have no idea what the radius or diameter is, but you know the circumference of the circle,
C
, you can still find the area.
Circumference (the distance around the circle) is found with this formula:
C = 2\pi r
That means we can take the circumference formula and "solve for
," which gives us:
r = \frac{C}{2\pi }
r
in our original formula with that new expression:
A = \pi {\left(\frac{C}{2\pi }\right)}^{2}
That expression simplifies to this:
A = \frac{{C}^{2}}{4\pi }
That formula works every time!
How To Find The Area With Circumference
Here is a beautiful, reasonable-sized pizza you and three friends can share. You happen to know the circumference of your pizza is
50.2655
inches, but you do not know its total area. You want to know how many square inches of pizza you will each enjoy.
[insert cartoon drawing of typical 16-inch pizza but do not label diameter]
50.2655
inches for
C
A = \frac{{50.2655}^{2}}{4\pi }
A = \frac{2,526.6204}{4\pi }
\mathbf{A} \mathbf{=} \mathbf{201.0620} {\mathbf{in}}^{\mathbf{2}}
Equally divide that total area for a full-sized pizza among four friends, and you each get
50.2655 i{n}^{2}
of pizza! That's about a third of a square foot for each of you! Yum, yum! |
Semantic analysis - CS Notes Semantic analysis | CS Notes
Semantic analysis is a compiler process which validates that source code is semantically consistent with the language definition. An example is type checking. It also often includes gathering additional information for future phases (e.g. type information) [1, P. 8].
Examples of validations made during semantic analysis:
All identifiers are declared before usage
Inheritance relationships are valid
Methods in a class are defined only once
Semantic analysis is the last phase of the compiler frontend.
Most semantic analyses can be implemented as a recursive descent of an AST.
The scope of a name binding is the part of a program where the name binding is valid (where the name can be used to refer to the entity). The same name may refer to different entities in different parts of the program.
Name bindings can have restricted scope, e.g. in C, where block scope restricts scope to a subset of a function.
Lexical scope (aka static scope) is where the scope only depends on the position of the identifier in the source text—the scope isn’t based on run-time behavior. Most programming languages use static scope.
Dynamic scope is where the scope of an identifier depends on the execution of a program (e.g. the closest binding in the execution of the language). Lisp used to be dynamically scoped.
The “most closely nested” rule is where an identifier refers to the definition in the closest enclosing scope, such that the declaration precedes the use. C++ uses the “most closely nested rule”.
A symbol table is a data structure that tracks the current bindings of identifiers.
When performing semantic analysis on a portion of the AST, the defined identifiers must be known.
A type is an attribute of data that defines the operations that can be performed on the data, the values that the data can take, and the way the data is stored.
The three main benefits of types in a compiler:
Statically-typed languages are typechecked during compilation (e.g. C, Java). Dynamically-typed languages are typechecked at run-time (e.g. JS, Python).
Most statically-typed languages have escape mechanisms to circumvent the type system, like unsafe casts in C and Java.
Implicit type conversion is where a value of type
T
is coerced into an expected type
E
T
is an invalid type for the operation being performed on it. A strongly-typed language typically doesn’t perform implicit type conversions, whereas a weakly-typed language does perform implicit type conversions. e.g. '1' + 1 throws an error in strongly-typed Python, and evaluates to '11' in weakly-typed JS.
A type signature defines the types of the parameters and the return value of a function or method.
Type inference is where the compiler automatically detects the type of an expression. For example, a variable could be declared without a type annotation and the compiler could infer the type at compile-time (e.g. var in C#).
A sound type system has the property that if a variable is declared with a particular type, then it will have that type at run-time. A sound type system has the ability to catch every possible bug that might happen at run-time.
A complete type system has the property that it will only ever catch bugs that will happen at run-time. This comes at the cost of sometimes missing errors that will happen at run-time.
Type checking is the process of verifying and enforcing type constraints. Static type checking is done at compile-time as part of semantic analysis.
Type checking can be implemented as a post-order tree walk, where each leaf node has a known type and each non-leaf node’s type can be inferred from the types of its children.
Pseudo-code for typechecking an expression:
def type_check(environment, node):
if type(node) is AddExpressionNode:
return type_check_add_expr(environment, node.e1, node.e2)
## .. case for each node
def typecheck_add_expr(environment, e1, e2):
t1 = type_check(environment, e1)
if not type(t1) == TInt:
raise TypeCheckError('expected int')
A type rule is an inference rule that describes how a type system assigns a type to a syntactic construct. Type rules can be applied by a type system to verify that a program is well-typed and to determine the type of each expression.
e
\tau
e\!:\!\tau
. The type environment is written as
\Gamma
The notation for inference is the same as for inference rules. In general:
The sequents above the line are premises that must be fulfilled in order for the rule to be applied, yielding the conclusion (the bottom sequents below the line). The turnstyle (
\vdash
) is read as “it is provable that …”.
A type environment is a function that maps identifiers to types, giving types for free variables in an expression.
When type checking, the environment is usually passed down the AST from the root towards the leaves.
Let environment
\Gamma
be a function mapping identifiers to types.
\Gamma \vdash e : T
, is read “under the assumption that free variables have the type given by
\Gamma
, it is provable that the expression
e
T
\Gamma[T/x]
is a function that applied to
x
T
In some languages method names and identifiers exist in different namespaces, therefore you can have both a method and a variable foo. This is implemented by using different environments (e.g. one for identifiers, and one for method names).
Subtyping is a form of type polymorphism where a subtype is related to another datatype (the supertype) by some notion of substitutability.
Figure: Subtyping hierarchy
Y
X
, the subtyping relation is written
Y \le X
In OO, subclasses can only add methods or override methods with the same type signature.
Variance refers to how type constructs (like function return types) use subtyping relations. An example is covariance, which is commonly used for function return type. Covariance of a return type
X
would allow any subtype
S
S \le X
) to be used in place of type
X
Supertype allowed
Subtype allowed
Invariance No No
Covariance No Yes
Contravariance Yes No
Bivariance Yes Yes
In pure OO languages, the Least upper bound (LUB) of two types
S
T
is their lowest common ancestor in the hierarchy tree.
In language where conditional expressions evaluate to a value, the type of an expression would be
LUB(T_1, ..., T_N)
T_1, ..., T_N
are the types corresponding to each consequent expression.
[1] A. V. Aho, M. S. Lam, R. Sethi, and J. D. Ullman, Compilers: Principles, Techniques, and Tools (2nd Edition). USA: Addison-Wesley Longman Publishing Co., Inc., 2006.
[2] K. D. Cooper and T. Linda, Engineering a Compiler, 2nd ed. Morgan Kaufmann Publishers, 2012. |
Revision as of 09:47, 19 November 2012 by Dave.Ellacott (talk | contribs) (→CFD Methods)
{\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}}
The key physical features of the UFR (Description section) present significant difficulties for all the existing approaches to turbulence representation, whether from the standpoint of solution fidelity (for the conventional (U)RANS models) or in terms of computational expense for full LES (especially if the turbulent boundary layers are to be resolved). For this reason, most of the computational studies of multi-body flows, in general, and the TC configuration, in particular, are currently relying upon hybrid RANS-LES approaches. This is true also with regard to simulations carried out in the course of the BANC-I and II Workshops and in the framework of the ATAAC project, where different hybrid RANS-LES models of the DES type were used (see Table 3) [1]. |
Concurrency - CS Notes Concurrency | CS Notes
Effective use of concurrency can dramatically speed up a program, so it pays to learn how to use it well.
There are two common models in concurrent programming:
Shared memory—concurrent units interact by reading and writing shared objects in memory. e.g. two processors on a machine sharing the same physical memory, two threads in a program sharing the same objects.
Message passing—concurrent units interact by sending messages to each other via a communication channel. e.g. two processes interacting via stdio streams.
A process is an executing instance of a program that is isolated from other processes on the same machine, e.g. it has its own address space.
A thread (thread of execution) is a context that usually runs within a process and will normally have shared address space with other threads.
A fiber is like a thread, except that it uses cooperative multitasking (where the fiber explicitly yields control to other fibers), as opposed to preemptive multitasking—where a scheduler decides when to stop running one thread and start running another.
A critical region (or critical section) is a section of code where shared resources are read and written.
Critical regions need to be protected, e.g. by locking, to stop unexpected behavior.
Consider a shared resource with a single critical region: a global integer with an operation that increments it:
This might translate into the following assembly:
movl i(%rip), %eax # move current value of i to register eax
addl $1, %eax # add 1 to i in register eax
movl %eax, i(%rip) # write back the new value of i
Assume there are two threads of execution that both enter the critical region, and the initial value of i is 7. The desired outcome would be:
| Thread 1 | Thread 2 |
| movl i(%rip) | - |
| addl \$1, %eax | - |
| movl %eax, i(%rip) | - |
| - | movl i(%rip) |
| - | addl \$1, %eax |
| - | movl %eax, i(%rip) |
This would result in i being set to 9. However, it’s possible that the instructions will execute in a different order:
This would result in i being set to 8, rather than 9.
This is known as a race condition. A race condition is where multiple threads execute simultaneously in a critical region, possibly causing behavior that varies between executions.
The solution is for the set of increment instructions to be performed atomically as a single instruction. Most processors provide an instruction to atomically read, increment, and write-back. But if the critical region contains multiple instructions that don’t have an atomic equivalent, then locks can be used instead.
Atomic operations are instruction sequences that execute as one unit. Atomic operations are the foundation of synchronization methods [1, P. 175].
A lock is a way to prevent multiple threads of execution from entering a critical region at the same time.
A lock works like a lock on a door. When a thread enters a critical region, it locks the region. The thread is then free to execute instructions without being interrupted. When the thread leaves the critical region, it unlocks the region so that other processes can enter it [1, P. 165].
Cooperative locks are advisory and voluntary. They are a programming construct that must be followed by all threads in order for them to provide atomic access to critical regions [1, P. 166]. Most locks are cooperative.
Mandatory locks will throw exceptions if a thread attempts to access a locked resource.
Locks can be implemented with atomic instructions that can test the value of an integer and set it to a new value only if it’s zero. For example, locks are implemented on x86 with an instruction called copy and exchange [1, P. 166].
Spin locks work by looping continuously until a lock becomes available (busy waiting):
while (__sync_lock_test_and_set(&exclusion, 1)) {}
__sync_lock_release(&exclusion);
Note: __sync_lock_test_and_set is an atomic exchange operator and an acquire fence.__sync_lock_release() writes 0 to exclusion atomically and forces a release fence. See memory fences for more details.
Spin locks are efficient if threads are blocked for a short time, because they avoid OS process rescheduling.
A futex (fast userspace mutex) is a 32-bit value whose address is supplied to the futex() system call, which is used to implement basic locking.
The two basic operations of a futex() system call:
FUTEX_WAIT(addr, val)—if the value stored at addr is val, then the thread sleeps waiting for a FUTEX_WAIT opeartion on the futex word.
FUTEX_WAKE(addr, num)—wake at most num of the waiters that are waiting on the futex word at addr.
A deadlock is a condition in a group of threads where no thread can proceed because they are each waiting for another member to take an action, such as releasing a lock.
A self-deadlock is where a thread attempts to acquire a lock that it already holds—causing the thread to wait forever because the thread is waiting and unable to release it [1, P. 169].
Another example is a case with multiple locks. Consider
n
threads and
n
locks. If each thread is waiting for a lock held by another thread, none of the threads will be able to progress. A common case is where
n
is 2, known as the deadly embrace or ABBA deadlock [1, P. 170].
You can prevent deadlocks by following simple rules:
Implement lock ordering. If two or more locks are acquired at the same time, they must be acquired in the same order.
Avoid starvation. “Ask yourself, does this code finish? If foo does not occur, will bar wait forever?”
Don’t double acquire the same lock.
A memory fence (memory barrier) is a type of CPU instruction that causes the CPU (or compiler) to enforce an ordering constraint on memory operations issued before and/or after the barrier.
Memory fences are required because:
Most modern CPUs employ performance optimizations that can result in out-of-order execution.
Compilers can reorder memory instructions.
A full memory fence ensures that no memory operations will move before or after the barrier.
A release memory fence ensures that no memory operation which appears before the memory barrier can be reordered to after the barrier.
An acquire memory fence ensures that no memory operation which appears after the memory barrier can be reordered to appear before the memory barrier. |
Research:Mapping Citation Quality - Meta
Duration: 2019-January – 2019-?
2.2 Article Characteristics
2.3 Lists of Articles Missing Citations
4 Results: Quantitative
4.1 Citation Quality VS Article Quality (and popularity)
4.2 Breakdown of Citation Quality by Topic
4.3 Citation Quality of Articles Marked as "Missing Sources"
5 Results: Qualitative Examples
5.1 Low Citation Quality Examples
5.2 High Citation Quality Examples
This project is a follow-up "Citation Needed" research project, where we developed multilingual models to automatically detect sentences needing citations in Wikipedia articles. Here, we apply these models at scale on hundreds of thousands of articles from English, French, and Italian Wikipedias, and quantify the proportion of unsourced content in these spaces, and its distribution over different topics.
We want to check for unsourced content across different types of articles.
We sample 5% of articles in English, French and Italian Wikipedia, and then randomly sample sentences across these subsets. Below a summary of the data used for these experiments.
Average Sentences / Article
English 4120432 421101 9.8
Italian 483460 72862 6.6
French 738477 109310 6.7
Article CharacteristicsEdit
Using mw:ORES ORES and the Pageviews API, we tag articles with 3 dimensions, when possible:
Topic: using ORES' topic model, we label all English Wikipedia articles with one or more predicted topics. For non-English articles, we propagate the topic assigned to their corresponding article in English Wikipedia (when existing), found through Wikidata
Quality: for English and French Wikipedias, we retain the article quality scores assigned by ORES
Popularity: for English Wikipedia, we also use the Page views API to get the number of page views up to May 2019.
Lists of Articles Missing CitationsEdit
To compare the results of our algorithms with real annotated data, we download the lists of articles that have been marked by editors as "Missing Sources". For example, for english Wikipedia, we use Quarry to get the list of all articles in the category All_articles_needing_additional_references (353070), for Italian, all articles in the Categoria:Senza Fonti (167863), and for French, the Catégorie:Article_manquant_de_références (84911).
After tagging each article according to topic, quality and popularity, we compute how "well sourced" an article is. To do so, for each article, we calculate the citation quality, namely the proportion of "well-sourced" sentences among the sample in our data. To label each sentence according to their citation need, we run the citation need models, and annotate each sentence with a binary "citation need" label
{\displaystyle y}
according to the model output:
{\displaystyle y=[{\hat {y}}]}
{\displaystyle [\cdot ]}
is the rounding function and \hat{y} is the predicted continuous label.
Next, if we consider:
{\displaystyle p}
as the number of sentences predicted as "needing citations"
{\displaystyle c}
as the real "citation label" for a sentence
{\displaystyle c=0}
if the sentence doesn't have an inline citation in the original text
{\displaystyle c=1}
if the sentence has an inline citation in the original text
{\displaystyle P}
{\displaystyle p}
sentences needing citations according to the algorithm, namely the ones for which
{\displaystyle y=1}
The citation quality
{\displaystyle Q}
for an article is then calculated as:
{\displaystyle Q={\frac {1}{p}}\sum _{i\in P}c}
{\displaystyle Q=0}
the quality is very low, none of the sentences classified by the model as needing citations are actually recognized as
We consider articles for which
{\displaystyle n\geq 5}
Results: QuantitativeEdit
We present here some results correlating overall article quality and article citation quality
Citation Quality VS Article Quality (and popularity)Edit
We correlate, for each article, the citation quality score
{\displaystyle Q}
with the article quality score as output by ORES
{\displaystyle AQ}
, using the Pearson correlation coefficients. For the two languages where ORES' quality scores are available, we observe a strong correlation (statistically significant) between these 2 quantities. While this is somehow expected, these results provide a "sanity check" for the statistical soundness and accuracy of our article citation quality score.
{\displaystyle \rho (Q,AQ)}
For English, we also computed the correlation between citation quality and article popularity. We found here a correlation of
{\displaystyle \rho =0.09}
, a significant value, though weaker than the correlation between citation quality and article quality. This correlation is probably due to the fact that very popular articles tend also to be of high quality (there is a significant correlation of
{\displaystyle \rho =0.14}
between article quality and popularity).
Breakdown of Citation Quality by TopicEdit
We cross the topic information we have about each article with the overall citation quality. We find that, across languages, the most well sourced articles are the Medicine and Biology articles. "Language and Literature", the topic category hosting most biographies, also rank among the top well-sourced topics. We find that articles in Mathematics and Physics tend to be marked as poorly sourced. This is probably due to the fact that these articles don't report many inline citations, as the proof of the scientific claims is in the formulas/equations that follow, and these articles tend to have a few references cited in the general. We will see more insights about these corner cases in the qualitative analysis section.
Citation Quality of Articles Marked as "Missing Sources"Edit
To get an aggregated view of the citation quality scores for articles marked by the community as "missing sources", and compare it with the articles not marked as "missing sources" we compute the average scores assigned by our models on the two groups of articles. We see that, in average, articles marked as "missing sources" receive a much lower citation quality score.
{\displaystyle Q}
for Marked Articles
{\displaystyle Q}
for non-marked Articles
{\displaystyle Q}
To get a closer look at the behavior of individual articles, we plot below the citation quality scores of articles marked as "missing sources" for English Wikipedia. Each vertical line in the plot below represents one article. The color indicates whether the article is marked as "missing sources" (magenta) or not (green). The height of the line is the citation quality score assigned to the article. As we can see, most articles marked as "missing sources" have low citation quality scores. However, we see some cases where articles with low-quality scores are not marked as "missing sources", as well as articles with high-quality scores are marked as "missing sources". We will give an in-depth view of these cases in the Qualitative Analysis section. We get similar results for French and Italian Wikipedias.
Results: Qualitative ExamplesEdit
We show here some specific examples of articles with high/low citation quality score. We limit this qualitative analysis to English Wikipedia, for ease of understanding.
Low Citation Quality ExamplesEdit
Some articles with Low Citation Quality scores have been already marked as "missing sources" by the Wikimedia community, for example:
Places en:Riverview School District (Pennsylvania)
People en:Kenges Rakishev
In other cases, articles detected as "low citation quality" by our models have not been recognized as missing citations. Some of them are also biographies:
Literature: en:Norwegian literature
Biographies: en:Bihari brothers
Some articles about scientific topics are detected as low citation quality. Should we consider those articles as missing sources?
Chemistry: en:Stoichiometry has a 0 citation quality score due to unsourced sentences like
"A stoichiometric reactant is a reactant that is consumed in a reaction, as opposed to a catalytic reactant, which is not consumed in the overall reaction because it reacts in one step and is regenerated in another step."
Computing: en:Analogical modeling has entire paragraphs left unsourced, and the model detects as missing citations sentences like
"In bitwise logical operations (e.g., logical AND, logical OR), the operand fragments may be processed in any arbitrary order because each partial depends only on the corresponding operand fragments (the stored carry bit from the previous ALU operation is ignored)."
In some cases the model makes mistakes, one of the most common is about lists of fictional characters
Books: en:List_of_Wild_Cards_characters
TV Series: en:List_of_Marvel_Comics_characters
High Citation Quality ExamplesEdit
Some articles detected by our model as "high citation quality" are clear examples of very well sourced pages:
For example, Stephen Hawking's article is a formerly featured article
Important articles for knowledge dissemination, such as the one Vaccine controversies, are examples of very well sourced articles.
The model recognizes that sentences in the "plot" section of an article about movies/books shouldn't be cited, and therefore considers b-level quality articles like The Mummy, Tomb of the Dragon Emperor as high citation quality articles.
When Physics articles contain well sourced sections about historical facts, as well as technical sections, the model detects them as "high citation quality", see for example the article on Catenary.
The model samples sentences from the article. In some cases, the unsampled sentences are marked as citation needed, and therefore they fall in the category "missing sources".
For example, generally well sourced articles, such as the [[en:Cavaquinho | Cavaquinho] article, have a few sentences marked as citation needed, but the model outputs a citation quality score of 1.0.
Similarly, in Joe Dolan's biography, there is one sentence marked as "citation needed". However, this is a generally well sourced articles, and thus our model gives a high citation quality score.
This article about the 1974 Greek Referendum has a section marked as "missing references". However, the model is not sampling from that section, thus giving a very high citation quality score to that article.
Retrieved from "https://meta.wikimedia.org/w/index.php?title=Research:Mapping_Citation_Quality&oldid=22294956" |
Find two positive numbers that satisfy the given requirements. -The
Find two positive numbers that satisfy the given requirements. -The sum is S and
Find two positive numbers that satisfy the given requirements. -The sum is S and the product is a maximum.
f\left(x\right)=x\left(s-x\right)=sx-{x}^{2}
A CI is desired for the true average stray-load loss A (watts) for a certain type of induction motor when the line current is heldat 10 amps for a speed of 1500 rpm. Assume that stray-load loss isnormally distributed with A = 3.0.
In this problem part (a) wants you to compute a 95% CI for A when n =25 and the sample mean = 58.3.
A well-insulated rigid tank contains 3 kg of saturated liquid-vapor mixture of water at 200 kPa. Initially, three-quarters of the mass is in the liquid phase. An electric resistance heater placed in the tank is now turned on and kept on until all the liquid in the tank is vaporized. Determine the entropy change of the steam during this process. Answer: 11.1 kJ/K |
To calculate: The equation 100y^{2} + 4x=x^{2} + 104 in one of standard forms of
To calculate: The equation 100y^{2} + 4x=x^{2} + 104 in one of standard forms of the conic sections and identify the conic section.
To calculate: The equation
100{y}^{2}\text{ }+\text{ }4x={x}^{2}\text{ }+\text{ }104
in one of standard forms of the conic sections and identify the conic section.
Step 1Formula: The general equation of the hyperbola is
\frac{{x}^{2}}{{a}^{2}}\text{ }-\text{ }\frac{{y}^{2}}{{b}^{2}}=1
where coordinate of the focus
\left(±\text{ }c,\text{ }0\right)\text{ }\text{and}\text{ }{c}^{2}={a}^{2}\text{ }+\text{ }{b}^{2}
Step 2Calculation:Consider the equation of the conic sections
100{y}^{2}\text{ }+\text{ }4x={x}^{2}\text{ }+\text{ }104
100{y}^{2}\text{ }-\text{ }{x}^{2}\text{ }+\text{ }4x=104
100{y}^{2}\text{ }-\text{ }\left({x}^{2}\text{ }-\text{ }4x\right)=104
100{y}^{2}\text{ }-\text{ }\left({x}^{2}\text{ }-\text{ }4x\text{ }+\text{ }4\right)=104\text{ }-\text{ }4
100{y}^{2}\text{ }-\text{ }{\left(x\text{ }-\text{ }2\right)}^{2}=100
\frac{{y}^{2}}{1}\text{ }-\text{ }\frac{{\left(x\text{ }-\text{ }2\right)}^{2}}{{10}^{2}}=1
Therfore the standard form of the conic section is
\frac{{y}^{2}}{1}\text{ }-\text{ }\frac{{\left(x\text{ }-\text{ }2\right)}^{2}}{{10}^{2}}=1
And since, general equation of the hyperbola is
\frac{{x}^{2}}{{a}^{2}}\text{ }-\text{ }\frac{{y}^{2}}{{b}^{2}}=1
, hence, it's a hyperbola.
\frac{8}{\sqrt{15}-\sqrt{11}}
The polar equation of the conic with the given eccentricity and directrix and focus at origin:
r=41\text{ }+\text{ }\mathrm{cos}\theta
\left(a\right)4{x}^{2}-9{y}^{2}=12\left(b\right)-4x+9{y}^{2}=0
\left(c\right)4{y}^{2}+9{x}^{2}=12\left(d\right)4{x}^{3}+9{y}^{3}=12
r=\frac{1}{\left(1+\mathrm{cos}\theta \right)}
Is there a parametrization of a hyperbola
{x}^{2}-{y}^{2}=1
by functions x(t) and y(t) both birational?
Consider the hyperbola
{x}^{2}-{y}^{2}=1
. I am aware of some parametrizations like:
\left(x\left(t\right),y\left(t\right)\right)=\left(\frac{{t}^{2}+1}{2t},\frac{{t}^{2}-1}{2t}\right)
\left(x\left(t\right),y\left(t\right)\right)=\left(\frac{{t}^{2}+1}{{t}^{2}-1},\frac{2t}{{t}^{2}-1}\right)
\left(x\left(t\right),y\left(t\right)\right)=\left(\text{cosh}t,\text{sinh}t\right)
\left(x\left(t\right),y\left(t\right)\right)=\left(\mathrm{sec}\left(t\right),\mathrm{tan}\left(t\right)\right)
The first and the second are by rational functions x(t) and y(t). But the functions are not birational(i.e. with rational inverse of each).
Is there a parametrization where:
- x(t) is rational with inverse also rational, and
- y(t) is rational with inverse also rational?
Is possible, to find a parametrization where both are rational and at least one of the has inverse rational? |
Subsets and Splits