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Zero-phase response of digital filter - MATLAB zerophase - MathWorks América Latina
Zero-Phase Response of FIR Filter
Zero-Phase Response of Elliptic Filter
Zero-phase response of digital filter
[Hr,w] = zerophase(b,a)
[Hr,w] = zerophase(sos)
[Hr,w] = zerophase(d)
[Hr,w] = zerophase(___,nfft)
[Hr,w] = zerophase(___,nfft,"whole")
[Hr,f] = zerophase(___,fs)
[Hr,f] = zerophase(___,"whole",fs)
Hr = zerophase(___,w)
Hr = zerophase(___,f,fs)
[Hr,w,phi] = zerophase(___)
zerophase(___)
[Hr,w] = zerophase(b,a) returns the zero-phase response Hr and the angular frequencies w at which the zero-phase response is computed, given a filter defined by numerator coefficients b and denominator coefficients a. The function evaluates the zero-phase response at 512 equally spaced points on the upper half of the unit circle.
[Hr,w] = zerophase(sos) returns the zero-phase response for the second order sections matrix sos.
[Hr,w] = zerophase(d) returns the zero-phase response for the digital filter d. Use designfilt to generate d based on frequency-response specifications.
[Hr,w] = zerophase(___,nfft) uses nfft frequency points on the upper half of the unit circle to compute the zero-phase response. You can use an input combination from any of the previous syntaxes.
[Hr,w] = zerophase(___,nfft,"whole") uses nfft frequency points around the whole unit circle to compute the zero-phase response.
[Hr,f] = zerophase(___,fs) uses the sample rate fs to determine the frequencies f at which the zero-phase response is computed.
[Hr,f] = zerophase(___,"whole",fs) uses the sample rate fs to determine the frequencies f around the whole unit circle at which the zero-phase response is computed.
Hr = zerophase(___,w) returns the zero-phase response Hr evaluated at the angular frequencies specified in w.
Hr = zerophase(___,f,fs) returns the zero-phase response Hr evaluated at the frequencies specified in f.
[Hr,w,phi] = zerophase(___) returns the zero-phase response Hr, angular frequencies w, and continuous phase component phi.
zerophase(___) plots the zero-phase response versus frequency. If you input the filter coefficients or second order sections matrix, the function plots in the current figure window. If you input a digitalFilter, the function displays the step response in FVTool.
If the input to zerophase is single precision, the function calculates the zero-phase response using single-precision arithmetic. The output Hr is single precision.
Use designfilt to design a 54th-order FIR filter with a normalized cutoff frequency of 0.3π rad/sample, a passband ripple of 0.7 dB, and a stopband attenuation of 42 dB. Use the method of constrained least squares. Display the zero-phase response.
d = designfilt("lowpassfir",FilterOrder=Nf,CutoffFrequency=Fc, ...
PassbandRipple=Ap,StopbandAttenuation=As,DesignMethod="cls");
zerophase(d)
Design the same filter using fircls1, which uses linear units to measure the ripple and attenuation. Display the zero-phase response.
Design a 10th-order elliptic lowpass IIR filter with a normalized passband frequency of 0.4π rad/sample, a passband ripple of 0.5 dB, and a stopband attenuation of 20 dB. Display the zero-phase response of the filter on 512 frequency points around the whole unit circle.
d = designfilt("lowpassiir",FilterOrder=10,PassbandFrequency=0.4, ...
PassbandRipple=0.5,StopbandAttenuation=20,DesignMethod="ellip");
zerophase(d,512,"whole")
Create the same filter using ellip. Plot its zero-phase response.
zerophase(b,a,512,"whole")
Filter coefficients, specified as vectors. For FIR filters where a=1, you can omit the value a from the command.
Second order sections, specified as a k-by-6 matrix where the number of sections k must be greater than or equal to 2. If the number of sections is less than 2, the function considers the input to be the numerator vector b. Each row of sos corresponds to the coefficients of a second order (biquad) filter. The ith row of the sos matrix corresponds to [bi(1) bi(2) bi(3) ai(1) ai(2) ai(3)].
nfft — Number of frequency points
Number of frequency points used to compute the zero-phase response, specified as a positive integer scalar. For best results, set nfft to a value greater than the filter order.
Angular frequencies at which the zero-phase response is computed, specified as a vector and expressed in radians/sample. w must have at least two elements.
Frequencies at which the zero-phase response is computed, specified as a vector and expressed in hertz. f must have at least two elements.
Hr — Zero-phase response
Zero-phase response, returned as a vector.
The zero-phase response, Hr(ω), is related to the frequency response, H(ejω), by
H\left({e}^{j\omega }\right)={H}_{r}\left(\omega \right){e}^{j\phi \left(\omega \right)},
where φ(ω) is the continuous phase.
The zero-phase response is always real, but it is not the equivalent of the magnitude response. The former can be negative while the latter cannot be negative.
Angular frequencies at which the zero-phase response is computed, returned as a vector.
Frequencies at which the zero-phase response is computed, returned as a vector.
phi — Continuous phase component
Continuous phase component, returned as a vector. This quantity is not equivalent to the phase response of the filter when the zero-phase response is negative.
[1] Antoniou, Andreas. Digital Filters. New York: McGraw-Hill, Inc., 1993.
designfilt | digitalFilter | freqs | freqz | FVTool | grpdelay | invfreqz | phasedelay | phasez
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Home : Support : Online Help : Connectivity : MTM Package : rank
The function rank(A) computes the rank of a matrix by performing Gaussian elimination on the rows of the given matrix. The rank of the matrix A is the number of non-zero rows in the resulting matrix.
\mathrm{with}\left(\mathrm{MTM}\right):
A≔\mathrm{Matrix}\left([[1,2,1],[2,4,2],[2,8,1]]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{8}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{rank}\left(A\right)
\textcolor[rgb]{0,0,1}{2}
MTM[null]
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Generically split projective homogeneous varieties
15 March 2010 Generically split projective homogeneous varieties
Viktor Petrov, Nikita Semenov
Viktor Petrov,1 Nikita Semenov2
1Max-Planck-Institut für Mathematik
2Mathematisches Institut, Universität München
G
be an exceptional simple algebraic group over a field
k
X
G
-homogeneous variety such that
G
splits over
k\left(X\right)
. We classify such varieties
X
. This classification allows us to relate the Rost invariant of groups of type
{E}_{7}
and their isotropy and to give a two-line proof of the triviality of the kernel of the Rost invariant for such groups. Apart from this, it plays a crucial role in the solution of a problem posed by Serre for groups of type
{E}_{8}
Viktor Petrov. Nikita Semenov. "Generically split projective homogeneous varieties." Duke Math. J. 152 (1) 155 - 173, 15 March 2010. https://doi.org/10.1215/00127094-2010-010
Viktor Petrov, Nikita Semenov "Generically split projective homogeneous varieties," Duke Mathematical Journal, Duke Math. J. 152(1), 155-173, (15 March 2010)
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Same as [[#am|AM]], except that Arthur is exponential-time and can exchange exponentially long messages with Merlin.
Contains [[Complexity Zoo:M#maexp|MA<sub>EXP</sub>]], and is contained in [[Complexity Zoo:E#eh|EH]] and indeed [[Complexity Zoo:S#s2exppnp|S<sub>2</sub>-EXP•P<sup>NP</sup>]].
Contains [[Complexity Zoo:M#maexp|MA<sub>EXP</sub>]], and is contained in [[Complexity Zoo:E#eh|EH]] and indeed [[Complexity Zoo:S#s2exppnp|S<sub>2</sub>-EXP•P<sup>NP</sup>]].
If [[Complexity Zoo:C#conp|coNP]] is contained in [[#ampolylog|AM[polylog]]] then [[Complexity Zoo:E#eh|EH]] collapses to AM<sub>EXP</sub> [[zooref#pv04|[PV04]]].
===== <span id="amicoam" style="color:red">AM ∩ coAM</span> =====
The class of decision problems for which both "yes" and "no" answers can be verified by an [[#am|AM]] protocol.
{\displaystyle t(n)}
{\displaystyle t(n)}
{\displaystyle \Sigma _{0}=\Delta _{0}=\Pi _{0}}
{\displaystyle \times }
{\displaystyle \phi =\forall i<j\psi }
{\displaystyle \phi =\exists i<j\psi }
{\displaystyle j}
{\displaystyle \phi }
{\displaystyle \Sigma _{i+1}}
{\displaystyle \exists X_{1}\dots \exists X_{n},\psi }
{\displaystyle \psi \in \Delta _{i}}
{\displaystyle \Sigma _{i}}
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Estimate frequency response and spectrum using spectral analysis with frequency-dependent resolution - MATLAB spafdr - MathWorks Nordic
y\left(t\right)=G\left(q\right)u\left(t\right)+v\left(t\right)
G\left({e}^{i\omega }\right)
\frac{2\pi }{N{T}_{s}}
\frac{\pi }{{T}_{s}}
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Object for storing stereo camera system parameters - MATLAB - MathWorks Switzerland
tformOfCamera2
Intrinsic and extrinsic parameters of the two cameras
cameraParameters1
Geometric relationship between the two cameras
Accuracy of estimated parameters
Settings for camera parameter estimation
Object for storing stereo camera system parameters
The stereoParameters object stores the intrinsic and extrinsic parameters of two cameras and their geometric relationship.
You can create a stereoParameters object using the stereoParameters function described here. You can also create a stereoParameters object by using the estimateCameraParameters with an M-by-2-by-numImages-by-2 array of input image points, where M is the number of keypoint coordinates in each pattern.
stereoParams = stereoParameters(cameraParameters1,cameraParameters2,tformOfCamera2)
stereoParams = stereoParameters(cameraParameters1,cameraParameters2,rotationOfCamera2,translationOfCamera2)
stereoParams = stereoParameters(paramStruct)
stereoParams = stereoParameters(cameraParameters1,cameraParameters2,tformOfCamera2) returns a stereo camera system parameters object using the camera parameters from two cameras and a tformOfCamera2 object that specifies the transformation of camera 2 relative to camera 1. cameraParameters1 and cameraParameters2 are cameraParameters or cameraIntrinsics objects that contain the intrinsics of camera 1 and camera 2 respectively.
stereoParams = stereoParameters(cameraParameters1,cameraParameters2,rotationOfCamera2,translationOfCamera2) creates a stereoParameters object that contains the parameters of a stereo camera system, and sets the cameraParameters1, cameraParameters2, rotationOfCamera2, and translationOfCamera2 properties.
stereoParams = stereoParameters(paramStruct) creates an identical stereoParameters object from an existing stereoParameters object with parameters stored in paramStruct.
tformOfCamera2 — Transformation of camera 2
Transformation of camera 2 relative to camera 1, specified as a rigid3d.
Stereo parameters, specified as a stereo parameters struct. To get a paramStruct from an existing stereoParameters object, use the toStruct function.
cameraParameters1 — Parameters of camera 1
Parameters of camera 1, specified as a cameraParameters object. The object contains the intrinsic, extrinsic, and lens distortion parameters of a camera.
rotationOfCamera2 — Rotation of camera 2
Rotation of camera 2 relative to camera 1, specified as a 3-by-3 matrix.
The rotationOfCamera2 and the translationOfCamera2 represent the relative rotation and translation between camera 1 and camera 2, respectively. They convert camera 2 coordinates back to camera 1 coordinates using:
orientation1 = rotationOfCamera2 * orientation2
location1 = translationOfCamera2 * orienation2 + location2
where, orientation1 and location1 represent the absolute pose of camera 1, and orientation2 and location2 represent the absolute pose of camera 2.
translationOfCamera2 — Translation of camera 2
Translation of camera 2 relative to camera 1, specified as a 3-element vector.
FundamentalMatrix — Fundamental matrix
Fundamental matrix, stored as a 3-by-3 matrix. The fundamental matrix relates the two stereo cameras, such that the following equation must be true:
\left[\begin{array}{cc}{P}_{2}& 1\end{array}\right]*FundamentalMatrix*{\left[\begin{array}{cc}{P}_{1}& 1\end{array}\right]}^{\text{'}}=0
P1, the point in image 1 in pixels, corresponds to the point, P2, in image 2.
EssentialMatrix — Essential matrix
Essential matrix, stored as a 3-by-3 matrix. The essential matrix relates the two stereo cameras, such that the following equation must be true:
\left[\begin{array}{cc}{P}_{2}& 1\end{array}\right]*EssentialMatrix*{\left[\begin{array}{cc}{P}_{1}& 1\end{array}\right]}^{\text{'}}=0
P1, the point in image 1, corresponds to P2, the point in image 2. Both points are expressed in normalized image coordinates, where the origin is at the camera’s optical center. The x and y pixel coordinates are normalized by the focal length fx and fy.
Average Euclidean distance between reprojected points and detected points over all image pairs, specified in pixels.
Number of calibration patterns that estimate the extrinsics of the two cameras, stored as an integer.
World coordinates of key points in the calibration pattern, specified as an M-by-2 array. M represents the number of key points in the pattern.
'mm' (default) | character vector
World points units, specified as a character vector. The character vector describes the units of measure.
toStruct Convert a stereo parameters object into a struct
[2] Heikkila, J, and O. Silven. “A Four-step Camera Calibration Procedure with Implicit Image Correction.” IEEE International Conference on Computer Vision and Pattern Recognition. 1997.
Use the toStruct method to pass a stereoParameters object into generated code. See the Code Generation for Depth Estimation From Stereo Video example.
stereoCalibrationErrors | intrinsicsEstimationErrors | extrinsicsEstimationErrors | cameraParameters | rigid3d | cameraIntrinsics
estimateCameraParameters | showReprojectionErrors | showExtrinsics | undistortImage | undistortPoints | detectCheckerboardPoints | generateCheckerboardPoints | reconstructScene | rectifyStereoImages | estimateFundamentalMatrix
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GU\left(n,q\right)
U\left(n,q\right)
n×n
{q}^{2}
q
GU\left(n,q\right)
n=2
q\le 20
n=3
q\le 5
n=4
q\le 4
n=5,6
q=2
\mathrm{with}\left(\mathrm{GroupTheory}\right):
\mathrm{GeneralUnitaryGroup}\left(2,2\right)
\textcolor[rgb]{0,0,1}{\mathbf{GU}}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\right)
\mathrm{GroupOrder}\left(\mathrm{GeneralUnitaryGroup}\left(2,4\right)\right)
\textcolor[rgb]{0,0,1}{300}
\mathrm{IdentifySmallGroup}\left(\mathrm{GeneralUnitaryGroup}\left(2,4\right)\right)
\textcolor[rgb]{0,0,1}{300}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{22}
\mathrm{GroupOrder}\left(\mathrm{GeneralUnitaryGroup}\left(4,q\right)\right)
\left(\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{q}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{q}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{q}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{q}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)
\mathrm{simplify}\left(\mathrm{GroupOrder}\left(\mathrm{GeneralUnitaryGroup}\left(2,{3}^{k}\right)\right)\right)
{\textcolor[rgb]{0,0,1}{81}}^{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{27}}^{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{9}}^{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{3}}^{\textcolor[rgb]{0,0,1}{k}}
\mathrm{simplify}\left(\mathrm{ClassNumber}\left(\mathrm{GeneralUnitaryGroup}\left(2,{3}^{k}\right)\right)\right)
{\textcolor[rgb]{0,0,1}{9}}^{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{3}}^{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
Here is a general formula for the order of the general unitary group of dimension
q
\mathrm{GroupOrder}\left(\mathrm{GeneralUnitaryGroup}\left(n,q\right)\right)
\left(\textcolor[rgb]{0,0,1}{q}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{q}}^{\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\prod }_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}}^{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{q}}^{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{-}{\left(\textcolor[rgb]{0,0,1}{-1}\right)}^{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\right)\right)
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Modulate using BPSK method - Simulink - MathWorks Deutschland
The BPSK Modulator Baseband block modulates a signal by using the binary phase shift keying (BPSK) method. The output is a baseband representation of the modulated signal. The input signal must be a discrete-time binary-valued signal. If the input bit is 0 or 1, then the modulated symbol is exp(jθ) or -exp(jθ), respectively. The Phase offset (rad) parameter specifies the value of θ in radians.
Input signal, specified as a scalar or vector with element values in the range [0,M – 1], where M is the modulation order. If you specify a binary vector, the number of elements must be an integer multiple of the number of bits per symbol. The number of bits per symbol is equal to log2(M).
Out — BPSK-modulated baseband signal
BPSK-modulated baseband signal, returned as a complex-valued scalar or vector.
Phase offset (rad) — Phase offset of zeroth point
Phase offset of the zeroth point of the constellation in radians, specified as a scalar.
Output data type, specified as one of these options.
Inherit via back propagation — The block matches the output data type and scaling to the next block in the model
<data type expression> — Enables parameters for which you specify additional details
Frame Synchronization Using Barker Code Preamble
Use a length 13 Barker code frame preamble for frame synchronization of data bits.
a ufix(ceil(log2(M))) only at the input for M-ary modulation.
{s}_{n}\left(t\right)=\sqrt{\frac{2{E}_{b}}{{T}_{b}}}\mathrm{cos}\left(2\pi {f}_{c}t+{\varphi }_{n}\right),
\left(n-1\right){T}_{b}\le t\le n{T}_{b},\text{ n}=\text{1},\text{2},\text{ 3},\dots
{s}_{n}\left(t\right)={e}^{-i{\varphi }_{n}}=\mathrm{cos}\left(\pi n\right).
{P}_{b}=Q\left(\sqrt{\frac{2{E}_{b}}{{N}_{0}}}\right),
BPSK Demodulator Baseband | M-PSK Modulator Baseband | QPSK Modulator Baseband | DBPSK Modulator Baseband
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Selecting random, distinct committees of validators is a big part of Ethereum 2.0; it is foundational for both its scalability and security. This selection is done by shuffling.
Shuffling a list of objects is a well understood problem in computer science. Notice, however, that this routine manages to shuffle a single index to a new location, knowing only the total length of the list. To use the technical term for this, it is oblivious. To shuffle the whole list, this routine needs to be called once per validator index in the list. By construction, each input index maps to a distinct output index. Thus, when applied to all indices in the list, it results in a permutation, also called a shuffling.
Why do this rather than a simpler, more efficient, conventional shuffle? It's all about light clients. Beacon nodes will generally need to know the whole shuffling, but light clients will often be interested only in a small number of committees. Using this technique allows the composition of a single committee to be calculated without having to shuffle the entire set: potentially a big saving on time and memory.
As stated in the code comments, this is an implementation of the "swap-or-not" shuffle, described in the cited paper. Vitalik kicked off a search for a shuffle with these properties in late 2018. With the help of Professor Dan Boneh of Stanford University, the swap-or-not was identified as a candidate a couple of months later, and adopted into the spec.
Use pivot to find another index in the list of validators, flip, which is pivot - index accounting for wrap-around in the list.
See the section on Shuffling for a more structured exposition and analysis of this algorithm (with diagrams!).
In practice, full beacon node implementations will run this once per epoch using an optimised version that shuffles the whole list, and cache the result of that for the epoch.
Used by compute_committee(), compute_proposer_index(), get_next_sync_committee_indices()
Uses bytes_to_uint64()
See also SHUFFLE_ROUND_COUNT
To account for the need to weight by effective balance, this function implements as a try-and-increment algorithm. A counter i starts at zero. This counter does double duty:
First i is used to uniformly select a candidate proposer with probability
1/N
N
is the number of active validators. This is done by using the compute_shuffled_index routine to shuffle index i to a new location, which is then the candidate_index.
Note that this dependence on the validators' effective balances, which are updated at the end of each epoch, means that proposer assignments are valid only in the current epoch. This is different from attestation committee assignments, which are valid with a one epoch look-ahead.
Used by get_beacon_proposer_index()
Uses compute_shuffled_index()
See also MAX_EFFECTIVE_BALANCE
compute_committee is used by get_beacon_committee() to find the specific members of one of the committees at a slot.
It may not be immediately obvious, but not all committees returned will be the same size (they can vary by one), and every validator in indices will be a member of exactly one committee. As we increment index from zero, clearly start for index == j + 1 is end for index == j, so there are no gaps. In addition, the highest index is count - 1, so every validator in indices finds its way into a committee.1
This method of selecting committees is light client friendly. Light clients can compute only the committees that they are interested in without needing to deal with the entire validator set. See the section on Shuffling for explanation of how this works.
Sync committees are assigned by a different process that is more akin to repeatedly performing compute_proposer_index().
Used by get_beacon_committee
This is trivial enough that I won't explain it. But note that it does rely on GENESIS_SLOT and GENESIS_EPOCH being zero. The more pernickety among us might prefer it to read,
return GENESIS_EPOCH + Epoch((slot - GENESIS_SLOT) // SLOTS_PER_EPOCH)
Maybe should read,
return GENESIS_SLOT + Slot((epoch - GENESIS_EPOCH) * SLOTS_PER_EPOCH))
Used by get_block_root()
See also SLOTS_PER_EPOCH, GENESIS_SLOT, GENESIS_EPOCH
Used by initiate_validator_exit(), process_registry_updates()
See also MAX_SEED_LOOKAHEAD
It is used by compute_fork_digest() and compute_domain().
Used by compute_fork_digest(), compute_domain()
Uses hash_tree_root()
See also ForkData
Extracts the first four bytes of the fork data root as a ForkDigest type. It is primarily used for domain separation on the peer-to-peer networking layer.
compute_fork_digest() is used extensively in the Ethereum 2.0 networking specification to distinguish between independent beacon chain networks or forks: it is important that activity on one chain does not interfere with other chains.
Uses compute_fork_data_root()
See also ForkDigest
Fun fact: this function looks pretty simple, but I found a subtle bug in the way tests were generated in a previous implementation.
Used by get_domain(), process_deposit()
See also Domain, DomainType GENESIS_FORK_VERSION
calculate the hash tree root of the object;
combine the hash tree root with the Domain inside a temporary SigningData object;
This is exactly equivalent to adding the domain to an object and taking the hash tree root of the whole thing. Indeed, this function used to be called compute_domain_wrapper_root().
Used by Many places
See also SigningData, Domain
Also not immediately obvious is that there is a subtle issue with committee sizes that was discovered by formal verification, although, given the max supply of ETH it will never be triggered.↩
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The stake in proof of stake provides three things: an anti-Sybil mechanism, an accountability mechanism, and an incentive alignment mechanism.
The 32 ETH stake size is a trade-off between network overhead, number of validators, and time to finality.
Combined with the Casper FFG rules, stakes provide economic finality: a quantifiable measure of the security of the chain.
A stake is the deposit that a full participant of the Ethereum 2 protocol must lock up. The stake is lodged permanently in the deposit contract on the Ethereum chain, and reflected in a balance in the validator's record on the beacon chain. The stake entitles a validator to propose blocks, to attest to blocks and checkpoints, and to participate in sync committees, all in return for rewards that accrue to its beacon chain balance.
In Ethereum 2 the stake has three key roles.
First, the stake is an anti-Sybil mechanism. Ethereum 2 is a permissionless system that anyone can participate in. Permissionless systems must find a way to to allocate influence among their participants. There must be some cost to creating an identity in the protocol, otherwise individuals could cheaply create vast numbers of duplicate identities and overwhelm the chain. In Proof of Work chains a participant's influence is proportional to its hash power, a limited resource1. In Proof of Stake chains participants must stake some of the chain's coin, which is again a limited resource. The influence of each staker in the protocol is proportional to the stake that they lock up.
Second, the stake provides accountability. There is a direct cost to acting in a harmful way in Ethereum 2. Specific types of harmful behaviour can be uniquely attributed to the stakers that performed them, and their stakes can be reduced or taken away entirely in a process called slashing. This allows us to quantify the economic security of the protocol in terms of what it would cost an attacker to do something harmful.
Third, the stake aligns incentives. Stakers necessarily own some of what they are guarding, and are incentivised to guard it well.
The size of the stake in Ethereum 2 is 32 ETH per validator.
This value is a compromise. It tries to be as small as possible to allow wide participation, while remaining large enough that we don't end up with too many validators. In short, if we reduced the stake, we would potentially be forcing stakers to run more expensive hardware on higher bandwidth networks, thus increasing the forces of centralisation.
The main practical constraint on the number of validators in a monolithic2 L1 blockchain is the messaging overhead required to achieve finality. Like other PBFT-style consensus algorithms, Casper FFG requires two rounds of all-to-all communication to achieve finality. That is, for all nodes to agree on a block that will never be reverted.
Following Vitalik's notation, if we can tolerate a network overhead of
\omega
messages per second, and we want a time to finality of
, then we can have participation from at most
n
validators, where
n \le \frac{\omega f}{2}
We would like to keep
\omega
small to allow the broadest possible participation by validators, including those on slower networks. And we would like
f
to be as short as possible since a shorter time to finality is much more useful than a longer time3. Taken together, these requirements imply a cap on
, the total number of validators.
This is a classic scalability trilemma. Personally, I don't find these pictures of triangles very intuitive, but they have become the canonical way to represent the trade-offs.
A version of the scalability trilemma: pick any two.
Our ideal might be to have high participation (large
n
) with low overhead (low
\omega
) – lots of stakers on low-spec machines –, but finality would take a long time since message exchange would be slow.
We could have very fast finality and high participation, but would need to mandate that stakers run high spec machines on high bandwidth networks in order to participate.
Or we could have fast finality on reasonably modest machines by severely limiting the number of participants.
It's not clear exactly how to place Ethereum 2 on such a diagram, but we definitely favour participation over time to finality: maybe "x" marks the spot. One complexity is that participation and overhead are not entirely independent: we could decrease the stake to encourage participation, but that would increase the hardware and networking requirements (the overhead), which will tend to reduce the number of people able or willing to participate.4
To put this in concrete terms, the hard limit on the number of validators is the total Ether supply divided by the stake size. With a 32 ETH stake, that's about 3.6 million validators today, which is consistent with a time to finality of 768 seconds (two epochs), and a message overhead of 9375 messages per second5. That's a substantial number of messages per second to handle. However, we don't ever expect all Ether to be staked, perhaps around 10-20%. In addition, due to the use of BLS aggregate signatures, messages are highly compressed to an asymptotic 1-bit per validator.
Given the capacity of current p2p networks, 32 ETH per stake is about as low as we can go while delivering finality in two epochs. Anecdotally, my staking node continually consumes about 3.5mb/s in up and down bandwidth. That's about 30% of my upstream bandwidth on residential ADSL. If the protocol were more any chatty it would rule out home staking for many.
An alternative approach might be to cap the number of validators active at any one time to put an upper bound on the number of messages exchanged. With something like that in place, we could explore reducing the stake below 32 ETH, allowing many more validators to participate, but each participating only on a part-time basis.
Note that this analysis overlooks the distinction between nodes (which actually have to handle the messages) and validators (a large number of which can be hosted by a single node). A design goal of the Ethereum 2 protocol is to minimise any economies of scale, putting the solo-staker on as equal as possible footing with staking pools. Thus we ought to be careful to apply our analyses to the most distributed case, that of one-validator per node.
Fun fact: the original hybrid Casper FFG PoS proposal (EIP-1011) called for a minimum deposit size of 1500 ETH as the system design could handle up to around 900 active validators. While 32 ETH now represents a great deal of money for most people, decentralised staking pools that can take less than 32 ETH are now becoming available.
The requirement for validators to lock up stakes, and the introduction of slashing conditions allows us to quantify the security of the beacon chain in some sense.
The main attack we wish to prevent is one that rewrites the history of the chain. The cost of such an attack parameterises the security of the chain. In proof of work, this is the cost of acquiring an overwhelming (51%) of hash power for a period of time. Interestingly, a successful 51% attack in proof of work costs essentially nothing, since the attacker claims all of the block rewards on the rewritten chain.
In Ethereum's proof of stake protocol we can measure security in terms of economic finality. That is, if an attacker wished to revert a finalised block on the chain, what would be the cost?
This turns out to be easy to quantify. To quote Vitalik's Parametrizing Casper,
H_1
is economically finalized if enough validators sign a message attesting to
H_1
, with the property that if both
H_1
and a conflicting
H_2
are finalized, then there is evidence that can be used to prove that at least
\frac{1}{3}
of validators were malicious and therefore destroy their entire deposits.
Ethereum's proof of stake protocol has this property. In order to finalise a checkpoint (
H_1
), two-thirds of the validators must have attested to it. To finalise a conflicting checkpoint (
H_2
) requires two-thirds of validators to attest to that as well. Thus, at least one-third of validators must have attested to both checkpoints. Since individual validators sign their attestations, this is both detectable and attributable: it's easy to submit the evidence on-chain that those validators contradicted themselves, and they can be punished by the protocol.
In the Altair implementation, if one-third of validators were to be slashed simultaneously, they would have around two-thirds of their stake burned (20 ETH each). The plan is to increase this later to their full stake (32 ETH each). At that point with, say, nine million Ether staked, the cost of reverting a finalised block would be three million of the attackers' Ether being permanently burned and the attackers being expelled from the network.
It is obligatory at this point to quote (or paraphrase) Vlad Zamfir: comparing proof of stake to proof of work, "it’s as though your ASIC farm burned down if you participated in a 51% attack".
For more on the mechanics of economic finality, see below under Slashing, and for more on the rationale and justification, see the section on Casper FFG. [TODO: link to Casper FFG when written.]
Parametrizing Casper: the decentralization/finality time/overhead tradeoff presents some early reasoning about the trade-offs for different stake sizes. Things have moved on somewhat since then, most notably with the advent of BLS aggregate signatures.
Why 32 ETH validator sizes? from Vitalik's Serenity Design Rationale.
Vitalik's discussion document around achieving single slot finality looks at the participation/overhead/finality trade-off space from a different perspective.
In the Bitcoin white paper, Satoshi wrote that, "Proof-of-work is essentially one-CPU-one-vote", although ASICs and mining farms have long subverted this. Proof of Stake is one-stake-one-vote.↩
A monolithic blockchain is one in which all nodes process all information, be it transactions or consensus-related. Pretty much all blockchains to date, including Ethereum, have been monolithic. One way to escape the scalability trilemma is to go "modular".
More on the general scalability trilemma: Why sharding is great by Vitalik.
More on modularity: The lay of the modular blockchain land by Polynya.
In an unfinished paper Vitalik attempts to quantify the "protocol utility" for different times to finality.
...a blockchain with some finality time
f
has utility roughly
-\log(f)
, or in other words increasing the finality time of a blockchain by a constant factor causes a constant loss of utility. The utility difference between 1 minute and 2 minute finality is the same as the utility difference between 1 hour and 2 hour finality.
He goes on to make a justification for this (p.10).↩
Exercise for the reader: try placing some of the other monolithic L1 blockchains within the trade-off space.↩
Vitalik's estimate of 5461 is too low since he omits the factor of two in the calculation.↩
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Inverse limit - Wikiversity
1 Wiki2Reveal Slide for Inverse Limit
5.1 Family of Groups
5.2 Family of Group Homomorphisms
5.3 Inverse System and Inverse Limit
5.5.2 Existence and Uniqueness of Inverse Systems
5.5.3 Category C and partially ordered sets
6.1 Example 1 - p-adic Integers
6.2 Example 2 - p-adic Solenoids
6.3 Example 3 - Factor Groups
6.4 Example 4 - Formal Power Series
6.5 Example 5 - Pro-finite Groups
6.6 Example 6 - Natural Projections
6.7 Example 7 - Inverse system and inverse limits - Category Sets
6.8 Example 8 - Topological Spaces
6.9 Example 9 - Set of Strings in Computer Science
7 Derived functors of the inverse limit
7.1 Ordered and Countable
7.2 Mittag-Leffler condition
7.3 Origine of the Name
7.4 Examples for Mittag-Leffler condition
8 Related concepts and generalizations
Wiki2Reveal Slide for Inverse Limit[edit | edit source]
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In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category, and they are a special case of the concept of a limit in category theory.
This learning resource in Wikiversity has the objective to provide a Wiki2Reveal environment for lectures and seminars about Algebra. The presentation can be used in lectures to explain the topic and the lecturer can annotate the slides with a stylus in the browsers or students can use the slides in seminars to explain the topic of inverse limits.
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The formal definition consists of the following steps:
Definition of the inverse system that creates mappings between groups
the mappings are homomorphisms between the groups
Family of Groups[edit | edit source]
We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (
{\displaystyle I}
, ≤) be a directed poset (not all authors require I to be directed). Let (Ai)i∈I be a family of groups.
Family of Group Homomorphisms[edit | edit source]
Suppose we have a family of homomorphisms
{\displaystyle f_{ij}:A_{j}\to A_{i}}
{\displaystyle i\leq j}
(note the order) with the following properties:
{\displaystyle f_{ii}}
{\displaystyle A_{i}}
{\displaystyle f_{ik}=f_{ij}\circ f_{jk}\quad {\text{ for all }}\quad i\leq j\leq k.}
{\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})}
is called an inverse system of groups and morphisms over
{\displaystyle I}
, and the morphisms
{\displaystyle f_{ij}}
are called the transition morphisms of the system.
Inverse System and Inverse Limit[edit | edit source]
We define the inverse limit of the inverse system
{\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})}
as a particular subgroup of the direct product of the
{\displaystyle A_{i}}
's:
{\displaystyle A=\varprojlim _{i\in I}{A_{i}}=\left\{\left.{\vec {a}}\in \prod _{i\in I}A_{i}\;\right|\;a_{i}=f_{ij}(a_{j}){\text{ for all }}i\leq j{\text{ in }}I\right\}.}
The inverse limit
{\displaystyle A}
comes equipped with natural projections
{\displaystyle \pi \setminus \!\!\!\setminus \pi _{i}}
>:
{\displaystyle A}
{\displaystyle A_{i}}
which pick out the ith component of the direct product for each
{\displaystyle i}n
{\displaystyle I}
. The inverse limit and the natural projections satisfy a universal property described in the next section.
This same construction may be carried out if the
{\displaystyle A_{i}}
's are sets,[1] semigroups,[1] topological spaces,[1] rings, modules (over a fixed ring), algebras (over a fixed ring), etc., and the homomorphisms are morphisms in the corresponding category. The inverse limit will also belong to that category.
General definition[edit | edit source]
The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let
{\textstyle (X_{i},f_{ij})}
be an inverse system of objects and morphisms in a category C (same definition as above). The inverse limit of this system is an object X in C together with morphisms
{\displaystyle \pi }
i: X → Xi (called projections) satisfying
{\displaystyle \pi }
{\displaystyle f_{ij}}
∘
{\displaystyle \pi }
j for all i ≤ j.
The pair (X,
{\displaystyle \pi }
i) must be universal in the sense that for any other such pair (Y, ψi) there exists a unique morphism u: Y → X such that the diagram
commutes for all i ≤ j. The inverse limit is often denoted
{\displaystyle X=\varprojlim X_{i}}
with the inverse system
{\textstyle (X_{i},f_{ij})}
being understood.
Existence and Uniqueness of Inverse Systems[edit | edit source]
In some categories, the inverse limit of certain inverse systems does not exist. If it does, however, it is unique in a strong sense: given any two inverse limits
{\displaystyle X}
{\displaystyle X'}
of an inverse system, there exists a unique isomorphism
{\displaystyle X':\to }
commuting with the projection maps.
Category C and partially ordered sets[edit | edit source]
Inverse systems and inverse limits in a category C admit an alternative description in terms of functors. Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j if and only if i ≤ j. An inverse system is then just a contravariant functor I → C. Let
{\displaystyle C^{I^{\mathrm {op} }}}
be the category of these functors (with natural transformations as morphisms). An object X of C can be considered a trivial inverse system, where all objects are equal to X and all arrow are the identity of X. This defines a "trivial functor" from C to
{\displaystyle C^{I^{\mathrm {op} }}.}
The direct limit, if it exists, is defined as a right adjoint of this trivial functor.
The following list provides examples of inverse limits
Example 1 - p-adic Integers[edit | edit source]
The ring of p-adic integers is the inverse limit of the rings
{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }
(see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers
{\displaystyle (n_{1},n_{2},\dots )}
such that each element of the sequence "projects" down to the previous ones, namely, that
{\displaystyle n_{i}\equiv n_{j}{\mbox{ mod }}p^{i}}
{\displaystyle i<j.}
The natural topology on the p-adic integers is the one implied here, namely the product topology with cylinder sets as the open sets.
Example 2 - p-adic Solenoids[edit | edit source]
The ring of p-adic solenoids is the inverse limit of the rings
{\displaystyle \mathbb {R} /p^{n}\mathbb {Z} }
(see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of real numbers
{\displaystyle (n_{1},n_{2},\dots )}
{\displaystyle n_{i}\equiv n_{j}{\mbox{ mod }}p^{i}}
{\displaystyle i<j.}
Example 3 - Factor Groups[edit | edit source]
Let p be a prime number. Consider the direct system composed of the factor groups
{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }
and the homomorphisms
{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} \rightarrow \mathbb {Z} /p^{n+1}\mathbb {Z} }
induced by multiplication by
{\displaystyle p}
. The inverse limit of this system is the circle group
{\displaystyle \mathbb {T} =\mathbb {R} /\mathbb {Z} }
, expressed in a positional number system of base
{\displaystyle p}
. This is similar to the construction of the real numbers by Cauchy sequences, using elements of the Prüfer group instead of the rational numbers or base
{\displaystyle p}
decimal fractions.
Example 4 - Formal Power Series[edit | edit source]
{\displaystyle R[t]}
of formal power series over a commutative ring R can be thought of as the inverse limit of the rings
{\displaystyle \textstyle R[t]/t^{n}R[t]}
, indexed by the natural numbers as usually ordered, with the morphisms from
{\displaystyle \textstyle R[t]/t^{n+j}R[t]}
{\displaystyle \textstyle R[t]/t^{n}R[t]}
given by the natural projection.
Example 5 - Pro-finite Groups[edit | edit source]
Pro-finite groups are defined as inverse limits of (discrete) finite groups.
Example 6 - Natural Projections[edit | edit source]
Let the index set I of an inverse system (Xi,
{\displaystyle f_{ij}}
) have a greatest element m. Then the natural projection
{\displaystyle \pi }
m: X → Xm is an isomorphism.
Example 7 - Inverse system and inverse limits - Category Sets[edit | edit source]
In the category of sets, every inverse system has an inverse limit, which can be constructed in an elementary manner as a subset of the product of the sets forming the inverse system. The inverse limit of any inverse system of non-empty finite sets is non-empty. This is a generalization of Kőnig's lemma in graph theory and may be proved with Tychonoff's theorem, viewing the finite sets as compact discrete spaces, and then applying the finite intersection property characterization of compactness.
Example 8 - Topological Spaces[edit | edit source]
In the category of topological spaces, every inverse system has an inverse limit. It is constructed by placing the initial topology on the underlying set-theoretic inverse limit. This is known as the limit topology.
Example 9 - Set of Strings in Computer Science[edit | edit source]
The set of infinite strings is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are discrete, the limit space is totally disconnected. This is one way of realizing the p-adic numbers and the Cantor set (as infinite strings).
Derived functors of the inverse limit[edit | edit source]
For an abelian category C, the inverse limit functor
{\displaystyle \varprojlim :C^{I}\rightarrow C}
is left exact.
Ordered and Countable[edit | edit source]
If I is ordered (not simply partially ordered) and countable, and C is the category Ab of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms fij that ensures the exactness of
{\displaystyle \varprojlim }
. Specifically, Eilenberg constructed a functor
{\displaystyle \varprojlim {}^{1}:\operatorname {Ab} ^{I}\rightarrow \operatorname {Ab} }
(pronounced "lim one") such that if (Ai, fij), (Bi, gij), and (Ci, hij) are three inverse systems of abelian groups, and
{\displaystyle 0\rightarrow A_{i}\rightarrow B_{i}\rightarrow C_{i}\rightarrow 0}
is a short exact sequence of inverse systems, then
{\displaystyle 0\rightarrow \varprojlim A_{i}\rightarrow \varprojlim B_{i}\rightarrow \varprojlim C_{i}\rightarrow \varprojlim {}^{1}A_{i}}
is an exact sequence in Ab.
Mittag-Leffler condition[edit | edit source]
If the ranges of the morphisms of an inverse system of abelian groups (Ai, fij) are stationary, that is, for every k there exists j ≥ k such that for all i ≥ j :
{\displaystyle f_{kj}(A_{j})=f_{ki}(A_{i})}
one says that the system satisfies the Mittag-Leffler condition.
Origine of the Name[edit | edit source]
The name "Mittag-Leffler" for this condition was given by Bourbaki in their chapter on uniform structures for a similar result about inverse limits of complete Hausdorff uniform spaces. Mittag-Leffler used a similar argument in the proof of Mittag-Leffler's theorem.
Examples for Mittag-Leffler condition[edit | edit source]
The following situations are examples where the Mittag-Leffler condition is satisfied:
a system in which the morphisms fij are surjective
a system of finite-dimensional vector spaces or finite abelian groups or modules of finite length or Artinian modules.
An example where
{\displaystyle \varprojlim {}^{1}}
is non-zero is obtained by taking I to be the non-negative integers, letting Ai = piZ, Bi = Z, and Ci = Bi / Ai = Z/piZ. Then
{\displaystyle \varprojlim {}^{1}A_{i}=\mathbf {Z} _{p}/\mathbf {Z} }
where Zp denotes the p-adic integers.
Further results[edit | edit source]
More generally, if C is an arbitrary abelian category that has enough injectives, then so does CI, and the right derived functors of the inverse limit functor can thus be defined. The nth right derived functor is denoted
{\displaystyle R^{n}\varprojlim :C^{I}\rightarrow C.}
In the case where C satisfies Grothendieck's axiom (AB4*), Jan-Erik Roos generalized the functor lim1 on AbI to series of functors limn such that
{\displaystyle \varprojlim {}^{n}\cong R^{n}\varprojlim .}
It was thought for almost 40 years that Roos had proved (in Sur les foncteurs dérivés de lim. Applications. ) that lim1 Ai = 0 for (Ai, fij) an inverse system with surjective transition morphisms and I the set of non-negative integers (such inverse systems are often called "Mittag-Leffler sequences"). However, in 2002, Amnon Neeman and Pierre Deligne constructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim1 Ai ≠ 0. Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct if C has a set of generators (in addition to satisfying (AB3) and (AB4*)).
Barry Mitchell has shown (in "The cohomological dimension of a directed set") that if I has cardinality
{\displaystyle \aleph _{d}}
(the dth infinite cardinal), then Rnlim is zero for all n ≥ d + 2. This applies to the I-indexed diagrams in the category of R-modules, with R a commutative ring; it is not necessarily true in an arbitrary abelian category (see Roos' "Derived functors of inverse limits revisited" for examples of abelian categories in which limn, on diagrams indexed by a countable set, is nonzero for n > 1).
Related concepts and generalizations[edit | edit source]
The categorical dual of an inverse limit is a direct limit (or inductive limit). More general concepts are the limits and colimits of category theory. The terminology is somewhat confusing: inverse limits are a class of limits, while direct limits are a class of colimits.
Explain how the ordered sets in
{\displaystyle I}
is linked to properties of the group homomorphisms.
Direct, or inductive limit
↑ 1.0 1.1 1.2 John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.
Bourbaki, Nicolas (1989), Algebra I, Springer, ISBN 978-3-540-64243-5, OCLC 40551484
Bourbaki, Nicolas (1989), General topology: Chapters 1-4, Springer, ISBN 978-3-540-64241-1, OCLC 40551485
Mac Lane, Saunders (September 1998), Categories for the Working Mathematician (2nd ed.), Springer, ISBN 0-387-98403-8
Mitchell, Barry (1972), "Rings with several objects", Advances in Mathematics - MR 0294454, 8: 1–161, doi:10.1016/0001-8708(72)90002-3
Neeman, Amnon (2002), "A counterexample to a 1961 "theorem" in homological algebra (with appendix by Pierre Deligne)", Inventiones Mathematicae MR: 1906154, 148 (2): 397–420, doi:10.1007/s002220100197
Roos, Jan-Erik (1961), "Sur les foncteurs dérivés de lim. Applications", C. R. Acad. Sci. Paris MR: 0132091, 252: 3702–3704
Roos, Jan-Erik (2006), "Derived functors of inverse limits revisited", J. London Math. Soc., Series 2 - MR 2197371, 73 (1): 65–83, doi:10.1112/S0024610705022416
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Source: Wikiversity https://en.wikiversity.org/wiki/Inverse%20limit
Wikipedia2Wikiversity[edit | edit source]
This page was based on the following Wikipedia source page:
Inverse limit https://en.wikipedia.org/wiki/Inverse%20limit
Date: 7/16/2021 - Source History Wikipedia
Retrieved from "https://en.wikiversity.org/w/index.php?title=Inverse_limit&oldid=2301170"
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Ice cream - Wackypedia - The nonsensical encyclopedia anyone can mess up
Mice Cream was invented by Tac on the eighth day
A young Gordon Ramsay keeps a straight face as he attempts to submit his Invisible Sherbet recipe to Ben and Jerry.
Ice cream is often called many things such as love potion, Wednesday juice and chicken bucks, though really it's just a substance stoopid people think gives them brain power and wonder sex.
1 ICECREAM - THE OFFICIAL WINNER OF CONNECT FOUR
2 What The Fuck is Icecream?!
3 Icecream TODAY
4 Underground Icecream
5 How to eat icecream: A step by step Guide
6 Icecream in the future
ICECREAM - THE OFFICIAL WINNER OF CONNECT FOUR[edit]
ONLY SPECIAL PEOPLE MAY READ THIS SPECIAL ARTICLE ABOUT ICECREAM!
What The Fuck is Icecream?![edit]
Icecream is a creamy iced substance that helps the world go round. Much like Superman, it gives people extreme powers and fat. It is eaten all over the world by douchebags and people too lazy to eat a substantial meal. It is the most famous dessert in the history of man EVUH and has one up to 78 Emmy awards and 5 Grammys. It has released 34 albums and 7 behind the scenes DVDs.
It is a well known fact that icecream tastes good with EVERYTHING, no matter what it is.
It's also a lethal poison when prepared in a certain way, but that's an article for another day.
{\displaystyle icecream+lovejuice=wickedcoolinsectrepelent}
Now, Icecream was invented in 16dickedy9 when a bunch of monks found a freezer in their non-existant bathtub(which was clearly from the 80s) and decided to put a cow in it. Six months and 36 excruciating hours later, the first tub of icecream was born. They called it Willy the Blood bucket because really it was just a bucket with some frozen milk and a feotus in it. After eating this mysterious substance, the monks became addicted and took all of the cows in the country and put them in the freezer too. This is also how cows became extinct.
Icecream TODAY[edit]
Icecream today is only eaten by lonely, desperate housewives with nothing better to do with their lives than drink wine and eat away thier depressing lives.
Oh God*shnockleshnockle*! My son has aids and soccer practice this afternoon and I still haven't finished cleaning the house, washing our goat and doing the shopping and the rest of my 18 children feel like their not getting enough out of the washing machine these days. The worst part of it all is that my husband hasn't been home for the last 6 years and the police said that they found his skeleton 3 years ago but I know better! He's just hiding from me! OH GOD I NEED ICECREAM! - irritated mother of 5
Underground Icecream[edit]
Underground icecream is the underground sport of eating icecream in waffle cones. Only the most courageous of people do it - i.e.. [[those good for nothing teenagers]] and tweens who think their game enough for it. Adults cannot play because the sport is simple too hardcore and dangerous.
The underground icecream was created in 1978 when an angsy teen individual tried to commit a tasty suicide by filling a waffle cone with vanilla icecream and stuffing it into his mouth. Though this suicide attempt didn't turn out, the teen quickly realised that waffle cones were really delicious with icecream and thus spread the news of his discovery.
Underground has various difficulty flavours that one can move up, much like the belts in karatay. In order to move up to another flavour, an individual must complete a set of 5 tasks to prove that he or she is mentally and physically ready for a more extreme level.
The flavours go like this-
Preparation stage- Vanilla
Easiest- Rainbow coloured Vanilla
Slightly harder than Easiest- Strawberry/Vanilla wave
Harder than Slightly harder than easiest but still pretty easy- Full Strawberry
Easier than medium but getting harder- Banana/Vanilla wave
Medium- Full Banana
Slightly harder than Medium- Chocolate/Vanilla wave
Harder still than medium- Full Chocolate
Difficult- Cookies and Cream
Getting pretty damn hard- Coconut
Really Hard- Pistachio
Hardest you can get before going into expert levels- Mango Sorbet
The following levels are the extra special expert levels that one can only reach by completing all other flavours and proving themselves as a true underground icecream eater. Most people never reach these levels in a lifetime.
Expert- Apple Sorbet
Good Expert- Orange sorbet
Great Expert- Watermelon sorbet
Super Expert- Pineapple sorbet
Super Special Expert- Double Choc Orange
Super Special Sexy Expert- Double scoop Vanilla and Chocolate
Awesome Special Expert- Double scoop Banana and Coconut
Wicked Awesome Special Expert- Double Scoop Cookies and Cream and Pistachio
Wicked Cool like Awesome Expert- Triple scoop Coconut, Mango Sorbet and Orange Sorbet
Extra Wicked Cool like Awesome Expert- Triple Scoop Watermelon, Pineapple and Double Choc Orange
Super Special Sexy Wicked Cool like Awesome Expert Expert- Quadruple scoop Mango, Apple, Pineapple and Double Choc Orange
Only one person in history has ever reached the Super Special Sexy Wicked Cool like Awesome Expert Expert stage. This person is known as Hugabert Quenchlimon Chinkysmut Jr though this name is rumoured to be fake. Hugabert Quenchlimon Chinkysmut Jr reached this level only 5 minutes before turning 20 and being deemed a full adult. It took him a full 11 years to read this level, having been pushed into the cruel, dangerous world of underground icecream eating at the tender age of 9. He is also the youngest known person to enter underground icecream, most kids entering at age 9 and 3/4.
How to eat icecream: A step by step Guide[edit]
Buy some icecream or make some the way the prehistoric monks of 16dickety9 did
Make a spoon from pipecleaners and washing detergent
Break your Microwave with a chainsaw
Check the date. If it's Tuesday, remember to feed the prisoners you're holding in your basement
Get a proper spoon and a bowl/icecream cone
Place your icecream container on a flat, hard surface
Take off the icecream lid
Hold your spoon by the handle and lower it to the icecream
Dig the spoon into the icecream
Pull the spoon through the icecream until you have a reasonably sized scoop
Carefully lift the spoon
Move the spoon so it's over your bowl
Put the icecream into the bowl. Make sure the spoon isn't still attached to the icecream
Kill some people with a toothpick and steal the Saturday. This is an essential step so be sure not to get it wrong
Repeat steps 9-15 until desired amount of icecream is in the bowl
Take your bowl of icecream and spoon to a place where you can sit down and eat
Take your spoon and use it to scoop up a little bit of icecream
Place icecream in your mouth
Move jaws up and down
Swallow icecream
Repeat steps 19-22 untl icecream is gone
Scream 'I SCREAM YOU SCREAM WE ALL SCREAM FOR ICE CREAM' 3 times
Clean bowl and place icecream back into the freezer
Well done! If you just followed all those steps, you've just finished your very first bowl of icecream! Congratulations. Give yourself a pat on the back and a round of applause, you should be proud of yourself.
Icecream in the future[edit]
Many leading scientists believe than in the future, icecream will remain exactly the same except with more daring flavours and different ways of serving it. One theory is that instead of eating it out of a cone or a bowl, people will eat it out of a box made out of rat poison. Another theory is that people will have in injected directly into their blood streams, though it's more likely that its be injected directly into the bowel so people can eat it straight from the toilet bowl. Many people believe that that theory is a load of edible crap.
Retrieved from "https://wackypedia.risteq.net/w/index.php?title=Ice_cream&oldid=165951"
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How Solutions Form - Course Hero
General Chemistry/Solutions and Colloids/How Solutions Form
Solutions are homogeneous mixtures of two or more substances that have an enthalpy change (heat absorbed or released) associated with their formation.
A solution is a homogeneous mixture of two or more substances. The substance that dissolves a material to form a solution is called the solvent. The dissolved material in a solution is a solute. Thus the solvent dissolves the solute, and together they form a solution. A solution is perhaps most commonly thought of as the liquid that results when a solid solute dissolves into a liquid solvent, as in the case of sugar dissolved in water. In fact solids, liquids, and gases can participate as a solution. A mixture of two or more gases, such as air, is a solution. Gas can also be dissolved in a liquid or in a solid. Carbonated beverages contain dissolved CO2 gas, and H2 gas can be dissolved in solid palladium (Pd). Any mixture of two liquids, such as ethanol and water, is also a solution. Two or more solids can also be mixed to form a new solid, as in the case of many alloys. An alloy is a mixture of two or more metallic elements.
In a solution, the solute particles are evenly distributed throughout the mixture with the solvent, and a solution is therefore homogeneous by definition. When a solution forms, three things must happen. First, the solute-solute molecules move apart from one another as the solute dissolves. The solvent-solvent molecules must also move apart from one another to make room for the dissolved solute molecules. Because the like molecules are attracted to one another, both of these processes are endothermic. Lastly, as the molecules mix, the solute and solvent molecules are drawn to one another by attractive intermolecular forces, an exothermic process. Energy is absorbed by an endothermic reaction and released by an exothermic reaction. Enthalpy (H) is the internal energy of a system plus the work needed to displace the environment to produce the components of the system. The enthalpy change (
{\Delta H}
) of endothermic processes is always positive, and the enthalpy change of exothermic processes is always negative. The sum of the
{\Delta H}
associated with the three steps in the formation of a solution, called the enthalpy of solution (
{\Delta H}_{\rm{soln}}
), is the net enthalpy change of the process.
This three-step process can be summarized as:
1. Solute molecules spread apart:
\Delta H_1>0
2. Solvent molecules spread apart:
\Delta H_2>0
3. Solvent and solute molecules move together:
\Delta H_3<0
\Delta H_{\rm{soln}}=\Delta H_1+\Delta H_2+\Delta H_3
Pure components start at a certain enthalpy (energy of the particles plus work needed to produce components of the system). The solvent and solute molecules use energy to spread out, so the spread-out molecules exist at a higher enthalpy than at the start. When the molecules mix together, they release energy because of the attractive forces between them. If the energy released,
\Delta H_3
, is less than the sum of
\Delta H_1
\Delta H_2
\Delta H_{\rm{soln}}>0
, and the process is endothermic. If
\Delta H_3>\Delta H_1+\Delta H_2
, the process is exothermic.
When a solute molecule is completely surrounded by solvent molecules, it is said to be solvated. A solute is hydrated if it is solvated by water molecules. Hydration enthalpy is the energy released when one mole of solute becomes completely hydrated. This energy explains why water is able to dissolve ionic compounds, which have strong attractive forces between the positive and negative ions. The hydration enthalpy for ionic compounds is approximately equal to the energy needed to separate the ionic compound into its ion components.
<Vocabulary>Will a Solution Form?
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What is rest? from Physics Motion in Straight Line Class 11 CBSE
When an object changes it's position with time, with respect to it's surroundings, the object is said to be in motion.
A particle moves in a circular orbit. What is the displacement undergone by the particle in one complete revolution?
A drunkard walking in a narrow lane takes 7 steps forward and 4 steps backward followed by same strokes. Each step is 0.5m long and requires 1sec. Find the net displacement of drunkard in three such strokes and average velocity of the drunkard.
Given, the drunkard walks 7 steps forward and 4 steps backward.
So, the net displacement undertaken by the drunkard = 3 steps in one stroke
Also given, length of 1 step = 0.5 m
Time taken to take 1 step = 1 sec
∴ Displacement in 1 stroke = 3 x 0.5=1.5m
Displacement in 3 strokes = 3 x 1.5 =4.5m
Time taken to walk one step is 1 sec.
Since to complete one stroke, drunkard walks 7 steps forward and 4 steps backward i.e. total of 11 steps, therefore drunkard takes 11s to complete one stroke.
Time taken to complete three strokes, t = 11 x 3 = 33s
Now, average velocity of drunkard is given by,
\phantom{\rule{0ex}{0ex}} \overline{)\mathrm{v}}=\frac{4.5}{33}=\frac{15}{110}\mathrm{m}/\mathrm{s}
When an object does not change its position with time, with respect to it's surroundings. In such a case the object is said to be at rest.
What does a speedometer record-average speed or instantaneous speed?
https://www.zigya.com/share/UEhFTjExMDM3Nzg0
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Rank features for classification using minimum redundancy maximum relevance (MRMR) algorithm - MATLAB fscmrmr - MathWorks América Latina
fscmrmr
Select Features and Compare Accuracies of Two Classification Models
Minimum Redundancy Maximum Relevance (MRMR) Algorithm
Specify 'UseMissing',true to use missing values in predictors for ranking
Rank features for classification using minimum redundancy maximum relevance (MRMR) algorithm
idx = fscmrmr(Tbl,ResponseVarName)
idx = fscmrmr(Tbl,formula)
idx = fscmrmr(Tbl,Y)
idx = fscmrmr(X,Y)
idx = fscmrmr(___,Name,Value)
[idx,scores] = fscmrmr(___)
idx = fscmrmr(Tbl,ResponseVarName) ranks features (predictors) using the MRMR algorithm. The table Tbl contains predictor variables and a response variable, and ResponseVarName is the name of the response variable in Tbl. The function returns idx, which contains the indices of predictors ordered by predictor importance. You can use idx to select important predictors for classification problems.
idx = fscmrmr(Tbl,formula) specifies a response variable and predictor variables to consider among the variables in Tbl by using formula.
idx = fscmrmr(Tbl,Y) ranks predictors in Tbl using the response variable Y.
idx = fscmrmr(X,Y) ranks predictors in X using the response variable Y.
idx = fscmrmr(___,Name,Value) specifies additional options using one or more name-value pair arguments in addition to any of the input argument combinations in the previous syntaxes. For example, you can specify prior probabilities and observation weights.
[idx,scores] = fscmrmr(___) also returns the predictor scores scores. A large score value indicates that the corresponding predictor is important.
Rank the predictors based on importance.
[idx,scores] = fscmrmr(X,Y);
The drop in score between the first and second most important predictors is large, while the drops after the sixth predictor are relatively small. A drop in the importance score represents the confidence of feature selection. Therefore, the large drop implies that the software is confident of selecting the most important predictor. The small drops indicate that the difference in predictor importance are not significant.
Find important predictors by using fscmrmr. Then compare the accuracies of the full classification model (which uses all the predictors) and a reduced model that uses the five most important predictors by using testckfold.
The output arguments of fscmrmr include only the variables ranked by the function. Before passing a table to the function, move the variables that you do not want to rank, including the response variable and weight, to the end of the table so that the order of the output arguments is consistent with the order of the table.
Rank the predictors in adultdata. Specify the column salary as the response variable.
[idx,scores] = fscmrmr(adultdata,'salary','Weights','fnlwgt');
The five most important predictors are relationship, capital_loss, capital_gain, education, and hours_per_week.
Compare the accuracy of a classification tree trained with all predictors to the accuracy of one trained with the five most important predictors.
Create a classification tree template using the default options.
C = templateTree;
Define the table tbl1 to contain all predictors and the table tbl2 to contain the five most important predictors.
tbl1 = adultdata(:,adultdata.Properties.VariableNames(idx(1:13)));
tbl2 = adultdata(:,adultdata.Properties.VariableNames(idx(1:5)));
Pass the classification tree template and the two tables to the testckfold function. The function compares the accuracies of the two models by repeated cross-validation. Specify 'Alternative','greater' to test the null hypothesis that the model with all predictors is, at most, as accurate as the model with the five predictors. The 'greater' option is available when 'Test' is '5x2t' (5-by-2 paired t test) or '10x10t' (10-by-10 repeated cross-validation t test).
[h,p] = testckfold(C,C,tbl1,tbl2,adultdata.salary,'Weights',adultdata.fnlwgt,'Alternative','greater','Test','5x2t')
h equals 0 and the p-value is almost 1, indicating failure to reject the null hypothesis. Using the model with the five predictors does not result in loss of accuracy compared to the model with all the predictors.
Now train a classification tree using the selected predictors.
mdl = fitctree(adultdata,'salary ~ relationship + capital_loss + capital_gain + education + hours_per_week', ...
'Weights',adultdata.fnlwgt)
If fscmrmr uses a subset of variables in Tbl as predictors, then the function indexes the predictors using only the subset. The values in the 'CategoricalPredictors' name-value pair argument and the output argument idx do not count the predictors that the function does not rank.
fscmrmr considers NaN, '' (empty character vector), "" (empty string), <missing>, and <undefined> values in Tbl for a response variable to be missing values. fscmrmr does not use observations with missing values for a response variable.
To specify a subset of variables in Tbl as predictors, use a formula. If you specify a formula, then fscmrmr does not rank any variables in Tbl that do not appear in formula.
fscmrmr considers NaN, '' (empty character vector), "" (empty string), <missing>, and <undefined> values in Y to be missing values. fscmrmr does not use observations with missing values for Y.
Example: 'CategoricalPredictors',[1 2],'Verbose',2 specifies the first two predictor variables as categorical variables and specifies the verbosity level as 2.
If fscmrmr uses a subset of input variables as predictors, then the function indexes the predictors using only the subset. The CategoricalPredictors values do not count the response variable, observation weight variable, or any other variables that the function does not use.
By default, if the predictor data is in a table (Tbl), fscmrmr assumes that a variable is categorical if it is a logical vector, unordered categorical vector, character array, string array, or cell array of character vectors. If the predictor data is a matrix (X), fscmrmr assumes that all predictors are continuous. To identify any other predictors as categorical predictors, specify them by using the CategoricalPredictors name-value argument.
'empirical' determines class probabilities from class frequencies in the response variable in Y or Tbl. If you pass observation weights, fscmrmr uses the weights to compute the class probabilities.
fscmrmr normalizes the weights in each class ('Weights') to add up to the value of the prior probability of the respective class.
UseMissing — Indicator for whether to use missing values in predictors
Indicator for whether to use missing values in predictors, specified as either true to use the values for ranking, or false to discard the values.
fscmrmr considers NaN, '' (empty character vector), "" (empty string), <missing>, and <undefined> values to be missing values.
If you specify UseMissing as true, then fscmrmr uses missing values for ranking. For a categorical variable, fscmrmr treats missing values as an extra category. For a continuous variable, fscmrmr places NaN values in a separate bin for binning.
If you specify UseMissing as false, then fscmrmr does not use missing values for ranking. Because fscmrmr computes mutual information for each pair of variables, the function does not discard an entire row when values in the row are partially missing. fscmrmr uses all pair values that do not include missing values.
Example: "UseMissing",true
Example: UseMissing=true
Verbosity level, specified as the comma-separated pair consisting of 'Verbose' and a nonnegative integer. The value of Verbose controls the amount of diagnostic information that the software displays in the Command Window.
0 — fscmrmr does not display any diagnostic information.
1 — fscmrmr displays the elapsed times for computing Mutual Information and ranking predictors.
≥ 2 — fscmrmr displays the elapsed times and more messages related to computing mutual information. The amount of information increases as you increase the 'Verbose' value.
fscmrmr normalizes the weights in each class to add up to the value of the prior probability of the respective class.
If fscmrmr uses a subset of variables in Tbl as predictors, then the function indexes the predictors using only the subset. For example, suppose Tbl includes 10 columns and you specify the last five columns of Tbl as the predictor variables by using formula. If idx(3) is 5, then the third most important predictor is the 10th column in Tbl, which is the fifth predictor in the subset.
A large score value indicates that the corresponding predictor is important. Also, a drop in the feature importance score represents the confidence of feature selection. For example, if the software is confident of selecting a feature x, then the score value of the next most important feature is much smaller than the score value of x.
The mutual information between two variables measures how much uncertainty of one variable can be reduced by knowing the other variable.
The mutual information I of the discrete random variables X and Z is defined as
I\left(X,Z\right)={\sum }_{i,j}P\left(X={x}_{i},Z={z}_{j}\right)\mathrm{log}\frac{P\left(X={x}_{i},Z={z}_{j}\right)}{P\left(X={x}_{i}\right)P\left(Z={z}_{j}\right)}.
If X and Z are independent, then I equals 0. If X and Z are the same random variable, then I equals the entropy of X.
The fscmrmr function uses this definition to compute the mutual information values for both categorical (discrete) and continuous variables. fscmrmr discretizes a continuous variable into 256 bins or the number of unique values in the variable if it is less than 256. The function finds optimal bivariate bins for each pair of variables using the adaptive algorithm [2].
The MRMR algorithm [1] finds an optimal set of features that is mutually and maximally dissimilar and can represent the response variable effectively. The algorithm minimizes the redundancy of a feature set and maximizes the relevance of a feature set to the response variable. The algorithm quantifies the redundancy and relevance using the mutual information of variables—pairwise mutual information of features and mutual information of a feature and the response. You can use this algorithm for classification problems.
The goal of the MRMR algorithm is to find an optimal set S of features that maximizes VS, the relevance of S with respect to a response variable y, and minimizes WS, the redundancy of S, where VS and WS are defined with mutual information I:
{V}_{S}=\frac{1}{|S|}{\sum }_{x\in S}I\left(x,y\right),
{W}_{S}=\frac{1}{{|S|}^{2}}{\sum }_{x,z\in S}I\left(x,z\right).
|S| is the number of features in S.
Finding an optimal set S requires considering all 2|Ω| combinations, where Ω is the entire feature set. Instead, the MRMR algorithm ranks features through the forward addition scheme, which requires O(|Ω|·|S|) computations, by using the mutual information quotient (MIQ) value.
{\text{MIQ}}_{x}=\frac{{V}_{x}}{{W}_{x}},
where Vx and Wx are the relevance and redundancy of a feature, respectively:
{V}_{x}=I\left(x,y\right),
{W}_{x}=\frac{1}{|S|}{\sum }_{z\in S}I\left(x,z\right).
The fscmrmr function ranks all features in Ω and returns idx (the indices of features ordered by feature importance) using the MRMR algorithm. Therefore, the computation cost becomes O(|Ω|2). The function quantifies the importance of a feature using a heuristic algorithm and returns a score (scores). A large score value indicates that the corresponding predictor is important. Also, a drop in the feature importance score represents the confidence of feature selection. For example, if the software is confident of selecting a feature x, then the score value of the next most important feature is much smaller than the score value of x. You can use the outputs to find an optimal set S for a given number of features.
fscmrmr ranks features as follows:
Select the feature with the largest relevance,
\underset{x\in \Omega }{\mathrm{max}}{V}_{x}
. Add the selected feature to an empty set S.
Find the features with nonzero relevance and zero redundancy in the complement of S, Sc.
If Sc does not include a feature with nonzero relevance and zero redundancy, go to step 4.
Otherwise, select the feature with the largest relevance,
\underset{x\in {S}^{c},\text{\hspace{0.17em}}{W}_{x}=0}{\mathrm{max}}{V}_{x}
. Add the selected feature to the set S.
Repeat Step 2 until the redundancy is not zero for all features in Sc.
Select the feature that has the largest MIQ value with nonzero relevance and nonzero redundancy in Sc, and add the selected feature to the set S.
\underset{x\in {S}^{c}}{\mathrm{max}}{\text{MIQ}}_{x}=\underset{x\in {S}^{c}}{\mathrm{max}}\frac{I\left(x,y\right)}{\frac{1}{|S|}{\sum }_{z\in S}I\left(x,z\right)}.
Repeat Step 4 until the relevance is zero for all features in Sc.
Add the features with zero relevance to S in random order.
The software can skip any step if it cannot find a feature that satisfies the conditions described in the step.
[1] Ding, C., and H. Peng. "Minimum redundancy feature selection from microarray gene expression data." Journal of Bioinformatics and Computational Biology. Vol. 3, Number 2, 2005, pp. 185–205.
[2] Darbellay, G. A., and I. Vajda. "Estimation of the information by an adaptive partitioning of the observation space." IEEE Transactions on Information Theory. Vol. 45, Number 4, 1999, pp. 1315–1321.
R2020a: Specify 'UseMissing',true to use missing values in predictors for ranking
Starting in R2020a, you can specify whether to use or discard missing values in predictors for ranking by using the 'UseMissing' name-value pair argument. The default value of 'UseMissing' is false because most classification training functions in Statistics and Machine Learning Toolbox™ do not use missing values for training.
In R2019b, fscmrmr used missing values in predictors by default. To update your code, specify 'UseMissing',true.
relieff | sequentialfs | fsulaplacian | fscnca
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Analyzing and Computing Cash Flows - MATLAB & Simulink - MathWorks Deutschland
Interest Rates/Rates of Return
Present or Future Values
Financial Toolbox™ cash-flow functions compute interest rates and rates of return, present or future values, depreciation streams, and annuities.
Some examples in this section use this income stream: an initial investment of $20,000 followed by three annual return payments, a second investment of $5,000, then four more returns. Investments are negative cash flows, return payments are positive cash flows.
Stream = [-20000, 2000, 2500, 3500, -5000, 6500,...
9500, 9500, 9500];
Several functions calculate interest rates involved with cash flows. To compute the internal rate of return of the cash stream, execute the toolbox function irr
ROR = irr(Stream)
ROR =
The rate of return is 11.72%.
The internal rate of return of a cash flow may not have a unique value. Every time the sign changes in a cash flow, the equation defining irr can give up to two additional answers. An irr computation requires solving a polynomial equation, and the number of real roots of such an equation can depend on the number of sign changes in the coefficients. The equation for internal rate of return is
\frac{c{f}_{1}}{\left(1+r\right)}+\frac{c{f}_{2}}{{\left(1+r\right)}^{2}}+\dots +\frac{c{f}_{n}}{{\left(1+r\right)}^{n}}+Investment=0,
where Investment is a (negative) initial cash outlay at time 0, cfn is the cash flow in the nth period, and n is the number of periods. irr finds the rate r such that the present value of the cash flow equals the initial investment. If all the cfns are positive there is only one solution. Every time there is a change of sign between coefficients, up to two additional real roots are possible.
Another toolbox rate function, effrr, calculates the effective rate of return given an annual interest rate (also known as nominal rate or annual percentage rate, APR) and number of compounding periods per year. To find the effective rate of a 9% APR compounded monthly, enter
Rate = effrr(0.09, 12)
The Rate is 9.38%.
A companion function nomrr computes the nominal rate of return given the effective annual rate and the number of compounding periods.
The toolbox includes functions to compute the present or future value of cash flows at regular or irregular time intervals with equal or unequal payments: fvfix, fvvar, pvfix, and pvvar. The -fix functions assume equal cash flows at regular intervals, while the -var functions allow irregular cash flows at irregular periods.
Now compute the net present value of the sample income stream for which you computed the internal rate of return. This exercise also serves as a check on that calculation because the net present value of a cash stream at its internal rate of return should be zero. Enter
NPV = pvvar(Stream, ROR)
The NPV is very close to zero. The answer usually is not exactly zero due to rounding errors and the computational precision of the computer.
Other toolbox functions behave similarly. The functions that compute a bond's yield, for example, often must solve a nonlinear equation. If you then use that yield to compute the net present value of the bond's income stream, it usually does not exactly equal the purchase price, but the difference is negligible for practical applications.
The toolbox includes functions to compute standard depreciation schedules: straight line, general declining-balance, fixed declining-balance, and sum of years' digits. Functions also compute a complete amortization schedule for an asset, and return the remaining depreciable value after a depreciation schedule has been applied.
This example depreciates an automobile worth $15,000 over five years with a salvage value of $1,500. It computes the general declining balance using two different depreciation rates: 50% (or 1.5), and 100% (or 2.0, also known as double declining balance). Enter
Decline1 = depgendb(15000, 1500, 5, 1.5)
Decline1 =
4500.00 3150.00 2205.00 1543.50 2101.50
6000.00 3600.00 2160.00 1296.00 444.00
These functions return the actual depreciation amount for the first four years and the remaining depreciable value as the entry for the fifth year.
Several toolbox functions deal with annuities. This first example shows how to compute the interest rate associated with a series of loan payments when only the payment amounts and principal are known. For a loan whose original value was $5000.00 and which was paid back monthly over four years at $130.00/month:
Rate = annurate(4*12, 130, 5000, 0, 0)
The function returns a rate of 0.0094 monthly, or about 11.28% annually.
The next example uses a present-value function to show how to compute the initial principal when the payment and rate are known. For a loan paid at $300.00/month over four years at 11% annual interest
Principal = pvfix(0.11/12, 4*12, 300, 0, 0)
The function returns the original principal value of $11,607.43.
The final example computes an amortization schedule for a loan or annuity. The original value was $5000.00 and was paid back over 12 months at an annual rate of 9%.
[Prpmt, Intpmt, Balance, Payment] = ...
amortize(0.09/12, 12, 5000, 0, 0);
This function returns vectors containing the amount of principal paid,
Prpmt = [399.76 402.76 405.78 408.82 411.89 414.97
418.09 421.22 424.38 427.56 430.77 434.00]
the amount of interest paid,
Intpmt = [37.50 34.50 31.48 28.44 25.37 22.28
19.17 16.03 12.88 9.69 6.49 3.26]
the remaining balance for each period of the loan,
Balance = [4600.24 4197.49 3791.71 3382.89 2971.01
2556.03 2137.94 1716.72 1292.34 864.77
434.00 0.00]
and a scalar for the monthly payment.
Payment = 437.26
irr | effrr | nomrr | fvfix | fvvar | pvfix | pvvar
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Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : LHPDO : Overview
Overview of the LHPDO Object
LHPDO Object Methods
The LHPDO object is designed and created to represent linear homogeneous partial differential operators (LHPDOs).
There is a collection of methods that are available for a LHPDO object, including (i) method allowing the LHPDO object to act as an operator / function (ii) methods for exploring properties of LHPDO (e.g. specification of domain and codomain). Some existing Maple builtins are extended for allowing LHPDO object.
All methods of the LHPDO object become available only once a valid LHPDO object is constructed successfully. To construct a LHPDO object, see LieAlgebrasOfVectorFields[LHPDO].
The LHPDO object is exported by the LieAlgebrasOfVectorFields package. See Overview of the LieAlgebrasOfVectorFields package for more detail.
A LHPDO Delta acts as an operator on an
m
\left(m\ge 1\right)
of scalar expressions, mapping it to an
s
\left(s\ge 0\right)
of scalars. Thus the input to Delta is a list of
m
s
elements. Each component of Delta takes linear homogeneous combinations of derivatives of the inputs.
The "independent variables" (with respect to which the inputs may be differentiated) may be accessed via the GetIndependents method. The integers
m
s
respectively are accessed via the GetDependentsCount and GetSystemCount methods. The inputs may be functions of some subset of the independent variables; the dependencies allowed for the inputs may be accessed via the GetDependencies method.
After a LHPDO object Delta is successfully constructed, each method of Delta can be accessed by either the short form method(Delta, arguments) or the long form Delta:-method(Delta, arguments).
The most important method of an LHPDO object is that it can act as a differential operator. See LHPDO Object as Operator for more detail.
After a LHPDO object is constructed, the following methods are available to extract its properties:
The following Maple builtins functions are extended so that they work for a LHPDO object: type, expand, has, hastype, indets, normal, simplify. See LHPDO Object Overloaded Builtins for more detail.
\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):
Construct a LHPDO object from some differential expressions, linear homogeneous with respect to
\left[u\left(x,y\right),v\left(x,y\right)\right]
\mathrm{\Delta }≔\mathrm{LHPDO}\left([\mathrm{diff}\left(u\left(x,y\right),x\right)-\mathrm{diff}\left(v\left(x,y\right),y\right),\mathrm{diff}\left(u\left(x,y\right),y\right)+\mathrm{diff}\left(v\left(x,y\right),x\right)]\right)
\textcolor[rgb]{0,0,1}{\mathrm{\Delta }}\textcolor[rgb]{0,0,1}{≔}\left(\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\right)\textcolor[rgb]{0,0,1}{→}\left[\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{-}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{v}\right)\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{v}\right]
The operator Delta operates on an ordered pair of inputs (u,v) and returns an ordered pair of expressions:
\mathrm{\Delta }\left([2xy,{y}^{2}-{x}^{2}]\right)
[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]
\mathrm{\Delta }\left([f\left(x,y\right),g\left(x,y\right)]\right)
[\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{∂}}{\textcolor[rgb]{0,0,1}{∂}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)]
\mathrm{GetIndependents}\left(\mathrm{\Delta }\right)
[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]
\mathrm{GetDependentsCount}\left(\mathrm{\Delta }\right)
\textcolor[rgb]{0,0,1}{2}
\mathrm{GetSystemCount}\left(\mathrm{\Delta }\right)
\textcolor[rgb]{0,0,1}{2}
\mathrm{GetDependencies}\left(\mathrm{\Delta }\right)
[[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}]]
Build another operator U...
U≔\mathrm{LHPDO}\left([x\left({\mathrm{cos}\left(a\right)}^{2}+{\mathrm{sin}\left(a\right)}^{2}\right)\mathrm{diff}\left(u\left(x,t\right),t,t\right)+x\left(x-1\right)\mathrm{diff}\left(u\left(x,t\right),x,x\right)-{x}^{2}\mathrm{diff}\left(u\left(x,t\right),x,x\right)],\mathrm{indep}=[x,t],\mathrm{dep}=[u]\right)
\textcolor[rgb]{0,0,1}{U}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{→}\left[\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right)\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\right)}^{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right)\right)\right]
Apply various Maple builtins to operator U, these have been extended to understand the LHPDO data type.
\mathrm{type}\left(U,\mathrm{LHPDO}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{expand}\left(U\right)
\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{→}\left[{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right)\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right)\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right)\right)\right]
\mathrm{simplify}\left(U\right)
\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{→}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right)\right)\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∂}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right)\right)\right]
\mathrm{indets}\left(U,\mathrm{name}\right)
{\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}}
\mathrm{hastype}\left(U,\mathrm{trig}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{has}\left(U,[u,v]\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
LHPDO Object overloaded builtins
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Inductor - New World Encyclopedia
Previous (Induction (philosophy))
Next (Indulgences)
An inductor is a passive electrical component that can store energy in a magnetic field created by passing an electric current through it. A simple inductor is a coil of wire. When an electric current is passed through the coil, a magnetic field is formed around it. This magnetic field causes the inductor to resist changes in the amount of current passing through it.
An inductor's ability to store magnetic energy is measured by its inductance, in units of henries. The inductance of a coil is directly proportional to the number of turns in the coil. Inductance also varies with the coil's radius and the material (or "core") around which the coil is wound.
1.1 Hydraulic analogy
4 Calculations for electric circuits
6 Inductance formulae
Inductors are widely used in analog circuits and signal processing. Large inductors, in combination with capacitors, are useful as chokes in power supplies, to remove fluctuations from direct current output. Small inductor/capacitor combinations are useful in making tuned circuits for radio reception and broadcasting. In addition, inductors are employed in transformers for power grids, and as energy storage devices in some switched-mode power supplies.
When an electric current first begins to pass through an inductor (coil of wire), the inductor resists the flow of current, as a magnetic field is building up around the inductor.[1] After this field has been built up, the inductor allows current to flow through it normally. When the electricity supply is turned off, the magnetic field around the coil supports the flow of current for a brief interval, before the field disappears.
Inductance (L) (measured in henries) is an effect resulting from the magnetic field that forms around a current-carrying conductor that tends to resist changes in the current. Electric current through the conductor creates a magnetic flux proportional to the current. A change in this current creates a change in magnetic flux that, in turn, by Faraday's law generates an electromotive force (EMF) that acts to oppose this change in current. Inductance is a measure of the amount of EMF generated for a unit change in current. For example, an inductor with an inductance of 1 henry produces an EMF of 1 volt when the current through the inductor changes at the rate of 1 ampere per second.
The number of loops, the size of each loop, and the material it is wrapped around all affect the inductance. For example, the magnetic flux linking these turns can be increased by coiling the conductor around a material with a high permeability such as iron. This can increase the inductance by 2000 times, although less so at high frequencies.
An "ideal inductor" has inductance but no resistance or capacitance, and it does not dissipate energy. A real inductor has not only inductance but also some resistance (due to resistivity of the wire) and some capacitance. At some frequency, usually much higher than the working frequency, a real inductor behaves as a resonant circuit (due to its self capacitance). In addition to dissipating energy in the resistance of the wire, magnetic core inductors may dissipate energy in the core due to hysteresis, and at high currents may show other departures from ideal behavior due to nonlinearity.
The behavior of an inductor can be described by using a hydraulic analogy.[1] An inductor can be modeled by the flywheel effect of a heavy turbine rotated by the flow. When water (current) first starts to flow, the stationary turbine will cause an obstruction in the flow and high pressure (voltage) opposing the flow until it gets turning. Once it is turning, if there is a sudden interruption of water flow, the turbine will continue to turn by inertia, generating a high pressure to keep the flow moving. (Magnetic interactions in transformers are not modeled hydraulically.)
A choke with two 47mH windings, as may be found in a power supply.
Inductors are used extensively in analog circuits and signal processing. Along with capacitors and other components, inductors are used to form tuned circuits that can emphasize or filter out specific signal frequencies. This can range from the use of large inductors as chokes in power supplies, which in conjunction with filter capacitors remove residual hum or other fluctuations from the direct current output, to such small inductances as generated by a ferrite bead or torus around a cable to prevent radio frequency interference from being transmitted down the wire. Smaller inductor/capacitor combinations provide tuned circuits used in radio reception and broadcasting, for instance.
Two (or more) inductors that have coupled magnetic flux form a transformer, which is a fundamental component of every electric utility power grid. The efficiency of a transformer may decrease as the frequency increases due to eddy currents in the core material and skin effect on the windings. Size of the core can be decreased at higher frequencies and, for this reason, aircraft use 400 hertz alternating current rather than the usual 50 or 60 hertz, greatly saving on weight through the use of smaller transformers.[2]
An inductor is used as the energy storage device in some switched-mode power supplies. The inductor is energized for a specific fraction of the regulator's switching frequency, and de-energized for the remainder of the cycle. This energy transfer ratio determines the input-voltage to output-voltage ratio. This XL is used in complement with an active semiconductor device to maintain very accurate voltage control.
Inductors are also employed in electrical transmission systems, where they are used to depress voltages from lightning strikes and to limit switching currents and fault current. In this field, they are more commonly referred to as reactors.
As inductors tend to be larger and heavier than other components, their use has been reduced in modern equipment; solid-state switching power supplies eliminate large transformers, for instance, and circuits are designed to use only small inductors, if any; larger values are simulated by use of gyrator circuits.
Inductors. (Major scale shown in centimeters.)
An inductor is usually constructed as a coil of conducting material, typically copper wire, wrapped around a core either of air or of ferromagnetic material. Core materials with a higher permeability than air increase the magnetic field and confine it closely to the inductor, thereby increasing the inductance. Low frequency inductors are constructed like transformers, with cores of electrical steel laminated to prevent eddy currents. "Soft" ferrites are widely used for cores above audio frequencies, since they don't cause the large energy losses at high frequencies that ordinary iron alloys do. This is because of their narrow hysteresis curves, and their high resistivity prevents eddy currents. Inductors come in many shapes. Most are constructed as enamel coated wire wrapped around a ferrite bobbin with wire exposed on the outside, while some enclose the wire completely in ferrite and are called "shielded." Some inductors have an adjustable core, which enables changing of the inductance. Inductors used to block very high frequencies are sometimes made by stringing a ferrite cylinder or bead on a wire.
Small value inductors can also be built on integrated circuits using the same processes that are used to make transistors. Aluminum interconnect is typically used, laid out in a spiral coil pattern. However, the small dimensions limit the inductance, and it is far more common to use a circuit called a "gyrator" which uses a capacitor and active components to behave similarly to an inductor.
Calculations for electric circuits
An inductor opposes changes in current. An ideal inductor would offer no resistance to a constant direct current; however, only superconducting inductors have truly zero electrical resistance.
In general, the relationship between the time-varying voltage v(t) across an inductor with inductance L and the time-varying current i'(t) passing through it is described by the differential equation:
{\displaystyle v(t)=L{\frac {di(t)}{dt}}}
When there is a sinusoidal alternating current (AC) through an inductor, a sinusoidal voltage is induced. The amplitude of the voltage is proportional to the product of the amplitude (
{\displaystyle I_{P}}
) of the current and the frequency (f) of the current.
{\displaystyle i(t)=I_{P}\sin(2\pi ft)\,}
{\displaystyle {\frac {di(t)}{dt}}=2\pi fI_{P}\cos(2\pi ft)}
{\displaystyle v(t)=2\pi fLI_{P}\cos(2\pi ft)\,}
In this situation, the phase of the current lags that of the voltage by 90 degrees. #
If an inductor is connected to a DC current source, with value I via a resistance, R, and then the current source short circuited, the differential relationship above shows that the current through the inductor will discharge with an exponential decay:
{\displaystyle \ i(t)=I(e^{\frac {-tR}{L}})}
When using the Laplace transform in circuit analysis, the transfer impedance of an ideal inductor with no initial current is represented in the s domain by:
{\displaystyle Z(s)=Ls\,}
L is the inductance, and
s is the complex frequency
{\displaystyle LI_{0}\,}
(Note that the source should have a polarity that opposes the initial current)
{\displaystyle {\frac {I_{0}}{s}}}
{\displaystyle I_{0}}
{\displaystyle {\frac {1}{L_{\mathrm {eq} }}}={\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}+\cdots +{\frac {1}{L_{n}}}}
{\displaystyle L_{\mathrm {eq} }=L_{1}+L_{2}+\cdots +L_{n}\,\!}
The energy (measured in joules, in SI) stored by an inductor is equal to the amount of work required to establish the current through the inductor, and therefore the magnetic field. This is given by:
{\displaystyle E_{\mathrm {stored} }={1 \over 2}LI^{2}}
where L is inductance and I is the current through the inductor(****).
An ideal inductor will be lossless, irrespective of the amount of current through the winding. However, typically inductors have winding resistance from the metal wire forming the coils. Since the winding resistance appears as a resistance in series with the inductor, it is often called the series resistance. The inductor's series resistance converts electrical current through the coils into heat, thus causing a loss of inductive quality. The quality factor (or Q) of an inductor is the ratio of its inductive reactance to its resistance at a given frequency, and is a measure of its efficiency. The higher the Q factor of the inductor, the closer it approaches the behavior of an ideal, loss less, inductor.
The Q factor of an inductor can be found through the following formula, where R is its internal electrical resistance and
{\displaystyle \omega {}L}
is capacitive or inductive reactance at resonance:
{\displaystyle Q={\frac {\omega {}L}{R}}}
By using a ferromagnetic core, the inductance is greatly increased for the same amount of copper, multiplying up the Q. Cores however also introduce losses that increase with frequency. A grade of core material is chosen for best results for the frequency band. At VHF or higher frequencies an air core is likely to be used.
An almost ideal inductor (Q approaching infinity) can be created by immersing a coil made from a superconducting alloy in liquid helium or liquid nitrogen. This supercools the wire, causing its winding resistance to disappear. Because a superconducting inductor is virtually lossless, it can store a large amount of electrical energy within the surrounding magnetic field (see superconducting magnetic energy storage).
Inductance formulae
The table below lists some common formulae for calculating the theoretical inductance of several inductor constructions.
Cylindrical coil[3]
{\displaystyle L={\frac {\mu _{0}KN^{2}A}{l}}}
{\displaystyle \pi }
Straight wire conductor
{\displaystyle L=l\left(\ln {\frac {4l}{d}}-1\right)\cdot 200\times 10^{-9}}
l = length of conductor (m)
d = diameter of conductor (m)
{\displaystyle L=5.08\cdot l\left(\ln {\frac {4l}{d}}-1\right)}
L = inductance (nH)
l = length of conductor (in)
d = diameter of conductor (in)
Short air-core cylindrical coil
{\displaystyle L={\frac {r^{2}N^{2}}{9r+10l}}}
Multilayer air-core coil
{\displaystyle L={\frac {0.8r^{2}N^{2}}{6r+9l+10d}}}
Flat spiral air-core coil
{\displaystyle L={\frac {r^{2}N^{2}}{(2r+2.8d)\times 10^{5}}}}
r = mean radius of coil (m)
d = depth of coil (outer radius minus inner radius) (m)
{\displaystyle L={\frac {r^{2}N^{2}}{8r+11d}}}
Toroidal core (circular cross-section)
{\displaystyle L=\mu _{0}\mu _{r}{\frac {N^{2}r^{2}}{D}}}
{\displaystyle \pi }
↑ 1.0 1.1 How Stuff Works, How Inductors Work. Retrieved February 23, 2009.
↑ Aspi Wadia, Aircraft electrical systems and why they operate at 400 Hz frequency. Retrieved February 23, 2009.
↑ 3.0 3.1 Hantaro Nagaoka, The Inductance Coefficients of Solenoids, Journal of the College of Science, Imperial University, Tokyo, Japan. 27:18. Retrieved February 23, 2009.
Giancoli, Douglas. 2007. Physics for Scientists and Engineers, with Modern Physics, 4th ed. Mastering Physics Series. Upper Saddle River, NJ: Prentice Hall. ISBN 978-0136139263.
Gibilisco, Stan. 2005. Electricity Demystified. New York: McGraw-Hill. ISBN 0071439250.
Hughes, Edward, et al. 2002. Electrical & Electronic Technology, 8th ed. Harlow: Prentice Hall. ISBN 058240519X.
Tipler, Paul Allen, and Gene Mosca. 2004. Physics for Scientists and Engineers, Volume 2: Electricity and Magnetism, Light, Modern Physics, 5th ed. New York: W.H. Freeman. ISBN 0716708108
Young, Hugh D., and Roger A. Freedman. 2003. Physics for Scientists and Engineers, 11th edition. San Francisco: Pearson. ISBN 080538684X.
How Inductors Work. How Stuff Works.
Chapter 25. Capacitance and Inductance. Electricity and Magnetism. (A chapter from an online textbook.)
Coil Inductance Calculator. 66pacific.com.
Inductor history
History of "Inductor"
Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Inductor&oldid=1009440
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Extended Kohler’s Rule of Magnetoresistance
Jing Xu, Fei Han, Ting-Ting Wang, Laxman R. Thoutam, Samuel E. Pate, Mingda Li, Xufeng Zhang, Yong-Lei Wang, Roxanna Fotovat, Ulrich Welp, Xiuquan Zhou, Wai-Kwong Kwok, Duck Young Chung, Mercouri G. Kanatzidis, Zhi-Li Xiao
A notable phenomenon in topological semimetals is the violation of Kohler’s rule, which dictates that the magnetoresistance MR obeys a scaling behavior of
\mathrm{MR}=f\left(H/{\rho }_{0}\right)
\mathrm{MR}=\left[\rho \left(H\right)-{\rho }_{0}\right]/{\rho }_{0}
H
is the magnetic field, with
\rho \left(H\right)
{\rho }_{0}
being the resistivity at
H
and zero field, respectively. Here, we report a violation originating from thermally induced change in the carrier density. We find that the magnetoresistance of the Weyl semimetal TaP follows an extended Kohler’s rule
\mathrm{MR}=f\left[H/\left({n}_{T}{\rho }_{0}\right)\right]
{n}_{T}
describing the temperature dependence of the carrier density. We show that
{n}_{T}
is associated with the Fermi level and the dispersion relation of the semimetal, providing a new way to reveal information on the electronic band structure. We offer a fundamental understanding of the violation and validity of Kohler’s rule in terms of different temperature responses of
{n}_{T}
. We apply our extended Kohler’s rule to
{\mathrm{BaFe}}_{2}{\left({\mathrm{As}}_{1-x}{\mathrm{P}}_{x}\right)}_{2}
to settle a long-standing debate on the scaling behavior of the normal-state magnetoresistance of a superconductor, namely,
\mathrm{MR}\sim {\mathrm{tan}}^{2}{\theta }_{H}
{\theta }_{H}
is the Hall angle. We further validate the extended Kohler’s rule and demonstrate its generality in a semiconductor, InSb, where the temperature-dependent carrier density can be reliably determined both theoretically and experimentally.
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Improving the Gini Coefficient Formula's Accuracy
§ Improving the Gini Coefficient Formula's Accuracy
Just a few days ago, to my big surprise, Vitalik released a blog post about a topic critical to the functioning of Rug Pull Index. A blog post titled: "Against overuse of the Gini coefficient."
While I'm super interested to learn more about V's thoughts - I haven't had any time to read it yet. Still, a friend of mine, and also ex-BigchainDBer, Ryan Henderson, first made me aware of an inaccuracy in RPI's Gini coefficient calculation for small and/or extreme case populations as e.g. a population of two with incomes of 0 and 1. He, too, had recently used the Gini coefficient in a paper on neural networks.
Originally using Wikipedia's formula, for a maximally inequal populalation e.g. of 0 and 1, the Gini coefficient calculation ended up being
G=\frac{1}{2}
, where as we'd expect it to be
G = 1
However, after having some more discussions with Ryan and my friend Jost Arndt, we settled on a more intuitive derivation of the Gini coefficient for Rug Pull Index that should also produce reasonable results for small or extreme populations. For anyone interested, I updated the /specification page with the latest formula.
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Forecast conditional variances from conditional variance models - MATLAB forecast - MathWorks í•œêµ
For example, to generate forecasts Y from a GARCH(0,2) model, forecast requires presample responses (innovations) Y0 =
{\left[\begin{array}{cc}{y}_{TâKâ1}& {y}_{TâK}\end{array}\right]}^{â²}
to initialize the model. The 1-period-ahead forecast requires both observations, whereas the 2-periods-ahead forecast requires yT – K and the 1-period-ahead forecast V(1). forecast generates all other forecasts by substituting previous forecasts for lagged responses in the model.
[8] Nelson, D. B. “Conditional Heteroskedasticity in Asset Returns: A New Approach.†Econometrica. Vol. 59, 1991, pp. 347–370.
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Parity-check and generator matrices for Hamming code - MATLAB hammgen - MathWorks América Latina
Generate Hamming Code Parity-Check Matrix Using Default Primitive Polynomial
Generate Hamming Code Parity-Check Matrix from Primitive Polynomials
Generate Hamming Code Parity-Check and Generator Matrices
Parity-check and generator matrices for Hamming code
h = hammgen(m)
h = hammgen(m,poly)
[h,g] = hammgen(___)
[h,g,n,k] = hammgen(___)
h = hammgen(m) returns an m-by-n parity-check matrix, h, for a Hamming code of codeword length n = 2m–1. The message length of the Hamming code is n – m. The binary primitive polynomial that the function uses to create the Hamming code is the default primitive polynomial in GF(2^m). For more details of this default polynomial, see the gfprimdf function.
h = hammgen(m,poly) specifies poly, a binary primitive polynomial for GF(2m). The function uses poly to create the Hamming code.
[h,g] = hammgen(___) additionally returns a k-by-n generator matrix, g, that corresponds to the parity-check matrix h. Specify any of the input argument combinations from the previous syntaxes.
[h,g,n,k] = hammgen(___) also returns n, the codeword length and k, the message length, for the Hamming code.
Generate a parity-check matrix, h, for a Hamming code of codeword length 7. The function uses the default primitive polynomial in GF(8) to create the Hamming code.
h = hammgen(3)
Generate the parity-check matrices for the Hamming code of codeword length 15, specifying the primitive polynomials
1+D+{D}^{4}
1+{D}^{3}+{D}^{4}
in GF(16).
h1 = hammgen(4,'1+D+D^4')
h2 = hammgen(4,'1+D^3+D^4')
Remove the embedded 4-by-4 identity matrices that is, the leftmost four columns in each parity-check matrix.
h1 = h1(:,5:end)
Verify that the two resulting matrices differ.
Generate the parity-check matrix, h and the generator matrix, g for the Hamming code of codeword length 7. Also return the codeword length, n, and the message length, k for the Hamming code. The function uses the default primitive polynomial in GF(8) to create the Hamming code.
[h,g,n,k] = hammgen(3)
m — Number of rows in parity-check matrix
integer greater than or equal to two
Number of rows in parity-check matrix, specified as an integer greater than or equal to two. The function uses this value to calculate the codeword length and the message length of the Hamming code.
poly — Binary primitive polynomial in GF(2m)
binary row vector | character vector | string scalar
Binary primitive polynomial in GF(2m), specified as one of these values:
Binary row vector of the polynomial coefficients in order of ascending powers
If poly is specified as a non-primitive polynomial, then the function hammgen displays an error.
h — Parity-check matrix for Hamming code
m-by-n matrix of binary values
Parity-check matrix for Hamming code, returned as an m-by-n matrix of binary values for the Hamming code.
g — Generator matrix for Hamming code
k-by-n matrix of binary values
Generator matrix for Hamming code, returned as a k-by-n matrix of binary values corresponding to the parity-check matrix h.
n — Codeword length of Hamming code
Codeword length of Hamming code, returned as a positive integer. This value is calculated as 2m–1.
k — Message length of Hamming code
Message length of Hamming code, returned as a positive integer. This value is calculated as n–m.
hammgen uses the function gftuple to create the parity-check matrix by converting each element in the Galois field (GF) to its polynomial representation. Unlike gftuple, which performs computations in GF(2m) and processes one m-tuple at a time, the hammgen function generates the entire sequence from 0 to 2m–1. The computation algorithm uses all previously computed values to generate the computation result. If the value of m is less than 25 and the primitive polynomial is the default primitive polynomial for GF(2m), the syntax hammgen(m) might be faster than the syntax hammgen(m,poly).
encode | decode | gen2par | gftuple | gfprimdf
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Transposed 2-D convolution layer - MATLAB - MathWorks India
TransposedConvolution2DLayer
CroppingMode
CroppingSize
Cropping property of TransposedConvolution2DLayer will be removed
This layer is sometimes incorrectly known as a "deconvolution" or "deconv" layer. This layer is the transpose of convolution and does not perform deconvolution.
Create a transposed convolution 2-D layer using transposedConv2dLayer.
Height and width of the filters, specified as a vector of two positive integers [h w], where h is the height and w is the width. FilterSize defines the size of the local regions to which the neurons connect in the input.
If you set FilterSize using an input argument, then you can specify FilterSize as scalar to use the same value for both dimensions.
Example: [5 5] specifies filters of height 5 and width 5.
CroppingMode — Method to determine cropping size
Method to determine cropping size, specified as 'manual' or same.
The software automatically sets the value of CroppingMode based on the 'Cropping' value you specify when creating the layer.
If you set the 'Cropping' option to 'same', then the software automatically sets the CroppingMode property of the layer to 'same' and set the cropping so that the output size equals inputSize .* Stride, where inputSize is the height and width of the layer input.
To specify the cropping size, use the 'Cropping' option of transposedConv2dLayer.
CroppingSize — Output size reduction
Output size reduction, specified as a vector of four nonnegative integers [t b l r], where t, b, l, r are the amounts to crop from the top, bottom, left, and right, respectively.
To specify the cropping size manually, use the 'Cropping' option of transposedConv2dLayer.
Cropping property will be removed in a future release. Use CroppingSize instead. To specify the cropping size manually, use the 'Cropping' option of transposedConv2dLayer.
Output size reduction, specified as a vector of two nonnegative integers [a b], where a corresponds to the cropping from the top and bottom and b corresponds to the cropping from the left and right.
Layer weights for the convolutional layer, specified as a FilterSize(1)-by-FilterSize(2)-by-NumFilters-by-NumChannels array.
Y=CX+B
Y={C}^{\top }X+B
R2019a: Cropping property of TransposedConvolution2DLayer will be removed
Cropping property of TransposedConvolution2DLayer will be removed, use CroppingSize instead. To update your code, replace all instances of the Cropping property with CroppingSize.
averagePooling2dLayer | transposedConv2dLayer | maxPooling2dLayer | convolution2dLayer
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Home : Support : Online Help : Mathematics : Numbers : Type Checking : verify
check for the Boolean of verification results
type(a, verify(bool))
either 'true', 'false', or 'FAIL'
Forcing a result from verify to return a Boolean value is too restrictive, so a more general class of objects may be returned, with the Boolean value of these being checked by the above types.
A result from a call to verify is considered to be true if the result is true. A result from a call to verify is considered to be either false or FAIL if either of those values is returned, or if a list containing either false or FAIL as a first operand is returned, respectively.
The special verification boolean will convert all return values which are lists to return the first operand of the list. This can be used if a boolean value is expected by some procedure.
The special verification truefalse will convert all return values which are lists to return the false. This can be used if a truefalse value is expected by some procedure.
\mathrm{type}\left(\mathrm{true},'\mathrm{verify}'\left('\mathrm{true}'\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(\mathrm{false},'\mathrm{verify}'\left('\mathrm{false}'\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left([\mathrm{false},0.100{10}^{8},'\mathrm{ulps}'],'\mathrm{verify}'\left('\mathrm{false}'\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left(10,10.000001,\mathrm{float}\left(2\right)\right)
[\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{100.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ulps}}]
\mathrm{verify}\left(10,10.000001,\mathrm{boolean}\left(\mathrm{float}\left(2\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left(x,\mathrm{\pi },\mathrm{greater_than}\right)
\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}
\mathrm{verify}\left(x,\mathrm{\pi },\mathrm{truefalse}\left(\mathrm{greater_than}\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{select}\left(\mathrm{verify},[1,2,3,x],\mathrm{\pi },\mathrm{less_than}\right)
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
\mathrm{select}\left(\mathrm{verify},[1,2,3,x],\mathrm{\pi },\mathrm{truefalse}\left(\mathrm{less_than}\right)\right)
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]
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Vector space - Simple English Wikipedia, the free encyclopedia
basic algebraic structure of linear algebra
A vector space is a collection of mathematical objects called vectors, along with some operations you can do on them. Two operations are defined in a vector space: addition of two vectors and multiplication of a vector with a scalar. These operations can change the size of a vector and the direction it points to. The most important thing to understand is that after you do the addition or multiplication, the result is still in the vector space; you have not changed the vector in a way that makes it not a vector anymore. A vector space is often represented using symbols such as
{\displaystyle U}
{\displaystyle V}
{\displaystyle W}
Adding vectors (right side): vector a (blue) is added to vector b (red) by placing them as shown on the right. The sum, a+b, is black. Left side: the sum of two vectors makes the diagonal of a parallelogram.
More formally, a vector space is a special combination of a group and a field. The elements of the group are called vectors, and the elements of the field are called scalars. Vector spaces are important in an area of mathematics called linear algebra, an area which deals with linear functions (functions of straight lines, not curves).
A vector can be represented graphically with an arrow that has a tail and a head. To add two vectors, you place the end of one vector at the head of the other one (see figure). The sum is the vector that goes from the tail of the first vector to the head of the second.
Scalar multiplication means that one vector is made bigger or smaller (it is "scaled"). Scalars are just numbers: if you multiply a vector by 2, you make it twice as long. If you multiply it by 1/2, you make it half as long.
The "vectors" don't have to be vectors in the sense of things that have magnitude and direction. For example, they could be functions, matrices or simply numbers. If they obey the axioms of a vector space (a list of properties a vector space needs to satisfy[2][3]), you can think of them as vectors and the theorems of linear algebra will still apply to them.
There are some combinations of vectors that are special. A minimum set of vectors that—through some combination of addition and multiplication—can reach any point in the vector space is called a basis (of that vector space). It is true that every vector space has a basis. It is also true that all bases of any one vector space have the same number of vectors in them. This is called the dimension theorem. We can then define the dimension of a vector space to be the size of its basis.
↑ "5: Vector Spaces". Mathematics LibreTexts. 2016-02-29. Retrieved 2020-08-23.
↑ Weisstein, Eric W. "Vector Space". mathworld.wolfram.com. Retrieved 2020-08-23.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Vector_space&oldid=7876206"
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There are two universal gates in total: { NAND } and { NOR }.
Since { NAND } is a functional completeness set (as well as { NOR }), you can build any logic gates by using only NAND gate (or NOR gate).
December is a minimalist personal blogging system that natively supports Markdown, LaTeX, and code highlighting.
This project is based on Python(Django), HTML, SQLite and JavaScript(jQuery).
You made it! Welcome to the December blogging system!
This is the first article in the site. You can always delete it and start your writing journey!
Markdown & LaTeX & Code Highlighting
The December system support Markdown & LaTeX & Code Highlighting natively, so you don't need to install extra plugins. This is actually pretty cool.
You can go to the editing page of this article to see its Markdown source code.
Or you can go to this place to learn more about Markdown syntax.
A simple example of LaTeX:
(Of course you are able to use the inline math mode:
e^{ \pm i\theta } = \cos \theta \pm i\sin \theta
And for code highlighting, you can do:
Abstract Divider
Use <!--more--> as abstract divider. When displayed on the home page, the content behind the abstract divider is hidden.
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Long short-term memory - MATLAB lstm - MathWorks Deutschland
Apply LSTM Operation to Sequence Data
Y = lstm(X,H0,C0,weights,recurrentWeights,bias)
[Y,hiddenState,cellState] = lstm(X,H0,C0,weights,recurrentWeights,bias)
[___] = lstm(___,'DataFormat',FMT)
The long short-term memory (LSTM) operation allows a network to learn long-term dependencies between time steps in time series and sequence data.
This function applies the deep learning LSTM operation to dlarray data. If you want to apply an LSTM operation within a layerGraph object or Layer array, use the following layer:
Y = lstm(X,H0,C0,weights,recurrentWeights,bias) applies a long short-term memory (LSTM) calculation to input X using the initial hidden state H0, initial cell state C0, and parameters weights, recurrentWeights, and bias. The input X must be a formatted dlarray. The output Y is a formatted dlarray with the same dimension format as X, except for any 'S' dimensions.
The lstm function updates the cell and hidden states using the hyperbolic tangent function (tanh) as the state activation function. The lstm function uses the sigmoid function given by
\sigma \left(x\right)={\left(1+{e}^{-x}\right)}^{-1}
as the gate activation function.
[Y,hiddenState,cellState] = lstm(X,H0,C0,weights,recurrentWeights,bias) also returns the hidden state and cell state after the LSTM operation.
[___] = lstm(___,'DataFormat',FMT) also specifies the dimension format FMT when X is not a formatted dlarray. The output Y is an unformatted dlarray with the same dimension order as X, except for any 'S' dimensions.
Perform an LSTM operation using three hidden units.
Create the input sequence data as 32 observations with 10 channels and a sequence length of 64
Create the initial hidden and cell states with three hidden units. Use the same initial hidden state and cell state for all observations.
Create the learnable parameters for the LSTM operation.
Perform the LSTM calculation
[dlY,hiddenState,cellState] = lstm(dlX,H0,C0,weights,recurrentWeights,bias);
View the size and dimensions of dlY.
View the size of hiddenState and cellState.
size(cellState)
Check that the output hiddenState is the same as the last time step of output dlY.
if extractdata(dlY(:,:,end)) == hiddenState
disp("The hidden state and the last time step are equal.");
disp("The hidden state and the last time step are not equal.")
The hidden state and the last time step are equal.
You can use the hidden state and cell state to keep track of the state of the LSTM operation and input further sequential data.
Input data, specified as a formatted dlarray, an unformatted dlarray, or a numeric array. When X is not a formatted dlarray, you must specify the dimension label format using 'DataFormat',FMT. If X is a numeric array, at least one of H0, C0, weights, recurrentWeights, or bias must be a dlarray.
The size of the 'C' dimension determines the number of hidden units. The size of the 'C' dimension of H0 must be equal to the size of the 'C' dimensions of C0.
If H0 is a not a formatted dlarray, the size of the first dimension determines the number of hidden units and must be the same size as the first dimension or the 'C' dimension of C0.
C0 — Initial cell state vector
Initial cell state vector, specified as a formatted dlarray, an unformatted dlarray, or a numeric array.
If C0 is a formatted dlarray, it must contain a channel dimension labeled 'C' and optionally a batch dimension labeled 'B' with the same size as the 'B' dimension of X. If C0 does not have a 'B' dimension, the function uses the same cell state vector for each observation in X.
The size of the 'C' dimension determines the number of hidden units. The size of the 'C' dimension of C0 must be equal to the size of the 'C' dimensions of H0.
If C0 is a not a formatted dlarray, the size of the first dimension determines the number of hidden units and must be the same size as the first dimension or the 'C' dimension of H0.
Specify weights as a matrix of size 4*NumHiddenUnits-by-InputSize, where NumHiddenUnits is the size of the 'C' dimension of both C0 and H0, and InputSize is the size of the 'C' dimension of X multiplied by the size of each 'S' dimension of X, where present.
Specify recurrentWeights as a matrix of size 4*NumHiddenUnits-by-NumHiddenUnits, where NumHiddenUnits is the size of the 'C' dimension of both C0 and H0.
Specify bias as a vector of length 4*NumHiddenUnits, where NumHiddenUnits is the size of the 'C' dimension of both C0 and H0.
Y — LSTM output
LSTM output, returned as a dlarray. The output Y has the same underlying data type as the input X.
The size of the 'C' dimension of Y is the same as the number of hidden units, specified by the size of the 'C' dimension of H0 or C0.
cellState — Cell state vector
Cell state vector for each observation, returned as a dlarray or a numeric array. cellState is returned with the same data type as C0.
If the input C0 is a formatted dlarray, the output cellState is returned as a formatted dlarray with the format 'CB'.
functionToLayerGraph does not support the lstm function. If you use functionToLayerGraph with a function that contains the lstm operation, the resulting LayerGraph contains placeholder layers.
The LSTM operation allows a network to learn long-term dependencies between time steps in time series and sequence data. For more information, see the definition of Long Short-Tem Memory Layer on the lstmLayer reference page.
dlarray | fullyconnect | softmax | dlgradient | dlfeval | gru
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4Pi microscope - Wikipedia
(Redirected from 4Pi Microscope)
A 4Pi microscope is a laser scanning fluorescence microscope with an improved axial resolution. With it the typical range of the axial resolution of 500–700 nm can be improved to 100–150 nm, which corresponds to an almost spherical focal spot with 5–7 times less volume than that of standard confocal microscopy.[1]
The improvement in resolution is achieved by using two opposing objective lenses, which both are focused to the same geometrical location. Also the difference in optical path length through each of the two objective lenses is carefully aligned to be minimal. By this method, molecules residing in the common focal area of both objectives can be illuminated coherently from both sides and the reflected or emitted light can also be collected coherently, i.e. coherent superposition of emitted light on the detector is possible. The solid angle
{\displaystyle \Omega }
that is used for illumination and detection is increased and approaches its maximum. In this case the sample is illuminated and detected from all sides simultaneously.
Optical Scheme of 4Pi Microscope
The operation mode of a 4Pi microscope is shown in the figure. The laser light is divided by a beam splitter and directed by mirrors towards the two opposing objective lenses. At the common focal point superposition of both focused light beams occurs. Excited molecules at this position emit fluorescence light, which is collected by both objective lenses, combined by the same beam splitter and deflected by a dichroic mirror onto a detector. There superposition of both emitted light pathways can take place again.
In the ideal case each objective lens can collect light from a solid angle of
{\displaystyle \Omega =2\pi }
. With two objective lenses one can collect from every direction (solid angle
{\displaystyle \Omega =4\pi }
). The name of this type of microscopy is derived from the maximal possible solid angle for excitation and detection. Practically, one can achieve only aperture angles of about 140° for an objective lens, which corresponds to
{\displaystyle \Omega \approx 1.3\pi }
The microscope can be operated in three different ways: In a 4Pi microscope of type A, the coherent superposition of excitation light is used to generate the increased resolution. The emission light is either detected from one side only or in an incoherent superposition from both sides. In a 4Pi microscope of type B, only the emission light is interfering. When operated in the type C mode, both excitation and emission light are allowed to interfere, leading to the highest possible resolution increase (~7-fold along the optical axis as compared to confocal microscopy).
In a real 4Pi microscope light cannot be applied or collected from all directions equally, leading to so-called side lobes in the point spread function. Typically (but not always) two-photon excitation microscopy is used in a 4Pi microscope in combination with an emission pinhole to lower these side lobes to a tolerable level.
In 1971, Christoph Cremer and Thomas Cremer proposed the creation of a perfect hologram, i.e. one that carries the whole field information of the emission of a point source in all directions, a so-called
{\displaystyle 4\pi }
hologram.[2][3] However the publication from 1978 [4] had drawn an improper physical conclusion (i.e. a point-like spot of light) and had completely missed the axial resolution increase as the actual benefit of adding the other side of the solid angle.[5] The first description of a practicable system of 4Pi microscopy, i.e. the setup with two opposing, interfering lenses, was invented by Stefan Hell in 1991.[6] He demonstrated it experimentally in 1994.[7]
In the following years, the number of applications for this microscope has grown. For example, parallel excitation and detection with 64 spots in the sample simultaneously combined with the improved spatial resolution resulted in the successful recording of the dynamics of mitochondria in yeast cells with a 4Pi microscope in 2002.[8] A commercial version was launched by microscope manufacturer Leica Microsystems in 2004[9] and later discontinued.
Up to now, the best quality in a 4Pi microscope was reached in conjunction with super-resolution techniques like the stimulated emission depletion (STED) principle.[10] Using a 4Pi microscope with appropriate excitation and de-excitation beams, it was possible to create a uniformly 50 nm sized spot, which corresponds to a decreased focal volume compared to confocal microscopy by a factor of 150–200 in fixed cells. With the combination of 4Pi microscopy and RESOLFT microscopy with switchable proteins, it is now possible to take images of living cells at low light levels with isotropic resolutions below 40 nm.[11]
Multifocal plane microscopy (MUM)
^ J. Bewersdorf; A. Egner; S.W. Hell (2004). "4Pi-Confocal Microscopy is Coming of Age" (PDF). GIT Imaging & Microscopy (4): 24–25.
^ Cremer C., Cremer T. (1971)
{\displaystyle 4\pi }
Punkthologramme: Physikalische Grundlagen und mögliche Anwendungen. Enclosure to Patent application DE 2116521 „Verfahren zur Darstellung bzw. Modifikation von Objekt-Details, deren Abmessungen außerhalb der sichtbaren Wellenlängen liegen" (Procedure for the imaging and modification of object details with dimensions beyond the visible wavelengths). Filed April 5, 1971; publication date October 12, 1972. Deutsches Patentamt, Berlin. http://depatisnet.dpma.de/DepatisNet/depatisnet?action=pdf&docid=DE000002116521A
^ Considerations on a laser-scanning-microscope with high resolution and depth of field: C. Cremer and T. Cremer in MICROSCOPICA ACTA VOL. 81 NUMBER 1 September, p. 31–44 (1978). Basic design of a confocal laser scanning fluorescence microscope & principle of a confocal laser scanning 4Pi fluorescence microscope, 1978 Archived 2016-03-04 at the Wayback Machine.
^ C. Cremer and T. Cremer (1978): Considerations on a laser-scanning-microscope with high resolution and depth of field Microscopica Acta VOL. 81 NUMBER 1 September, pp. 31—44 (1978)
^ The Nobel Prize in Chemistry 2014 https://www.nobelprize.org/prizes/chemistry/2014/hell/biographical/
^ European patent EP 0491289.
^ S. W. Hell; E. H. K. Stelzer; S. Lindek; C. Cremer (1994). "Confocal microscopy with an increased detection aperture: type-B 4Pi confocal microscopy". Optics Letters. 19 (3): 222–224. Bibcode:1994OptL...19..222H. CiteSeerX 10.1.1.501.598. doi:10.1364/OL.19.000222. PMID 19829598.
^ A. Egner; S. Jakobs; S. W. Hell (2002). "Fast 100-nm resolution three-dimensional microscope reveals structural plasticity of mitochondria in live yeast" (PDF). PNAS. 99 (6): 3370–3375. Bibcode:2002PNAS...99.3370E. doi:10.1073/pnas.052545099. PMC 122530. PMID 11904401.
^ Review article 4Pi microscopy.
^ R. Schmidt; C. A. Wurm; S. Jakobs; J. Engelhardt; A. Egner; S. W. Hell (2008). "Spherical nanosized focal spot unravels the interior of cells". Nature Methods. 5 (6): 539–544. doi:10.1038/nmeth.1214. hdl:11858/00-001M-0000-0012-DBBB-8. PMID 18488034. S2CID 16580036.
^ U. Böhm; S. W. Hell; R. Schmidt (2016). "4Pi-RESOLFT nanoscopy". Nature Communications. 7 (10504): 1–8. Bibcode:2016NatCo...710504B. doi:10.1038/ncomms10504. PMC 4740410. PMID 26833381.
Retrieved from "https://en.wikipedia.org/w/index.php?title=4Pi_microscope&oldid=1071162782"
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Impact factor, an inadequate yardstick | EMS Magazine
Replacement for the impact factor
The role of human assessment
Our aim is twofold. On the one hand we discuss the limitations of the impact factor as a criterion for assessing mathematical journals, and suggest substituting a set of different types of indicators including the SCImago Journal Rank. On the other, we state that scientometrics such as the impact factor cannot be used alone in evaluating researchers’ work: one must have both a package of metrics as an objective measure and peer review by human beings as a subjective judgement.
In the 1960s, the notion of impact factor was introduced to assist libraries in deciding which journals to purchase. Since the late 1990s, it has been employed as a metric for measuring the quality of scholarly journals.
The Web of Science (WOS), a bibliographical database created by Clarivate Analytics, computes the journal impact factor (JIF) to recognize the relative importance of each journal. To be assigned a JIF, a journal first needs to satisfy certain quality criteria in order to be included in the Journal Citation Report (JCR). The JCR is a selective list consisting of more than 11,000 journals. The (2-year-) impact factor of a journal in a specific year measures the average number of citations from that year of the papers published in that journal during the previous 2 years. More precisely, the 2-year-impact factor of a journal in a year
n
is computed by the formula
JIF_{n}=\frac{C_{n}}{P_{n-1}+P_{n-2}},
C_{n}
denotes the number of citations in the year
n
of papers published in the journal in the years
n-1
n-2
P_{m}
stands for the number of papers published in the journal in the year
m
. A citation of a paper given by the author(s) of the paper is called a self-citation.
The SCImago Journal Rank (SJR) of a journal is a 3-year-impact factor reflecting the influence of the journal supported by Scopus. It depends not only on the number of citations of its published papers but also on the prestige of the journals in which the citations appeared; see [4 B. González-Pereira, V. P. Guerrero-Bote, and F. Moya-Anegón, A new approach to the metric of journals’ scientific prestige: The SJR indicator. J. Informetrics4, 379–391 (2010) ]. A drawback, however, has been reported regarding Scopus: namely, the database of Scopus journals with assigned SJR includes about 30,000 journals, which is a very large number of journals of varying quality.
Furthermore, WOS provides the indicator Eigenfactor (EF) that ranks journals in the same manner as that used by Google to rank websites. Based on 5-year citation data, it adjusts for citation differences through various disciplines. Thus the SJR and EF seem to be well-suited for evaluation of the quality of a journal; see [7 H. F. Moed, From Journal Impact Factor to SJR, Eigenfactor, SNIP, CiteScore and Usage Factor. In Applied Evaluative Informetrics. Qualitative and Quantitative Analysis of Scientific and Scholarly Communication. Springer, Cham, 229–244 (2017) ].
Each subject category of JCR journals is divided into four quartiles: Q1, Q2, Q3, and Q4, where Q1 denotes the top 25 percent of all journals in terms of their JIF. There are analogous quartiles for the journals in Scopus according to their SJRs.
Scientometrics indices as found in databases in the year 2020
JIF SJR MCQ EF
Acta Math. 2.458 5.77 3.95 0.007
Iran. J. Fuzzy Syst. 2.276 0.51 0.11
<
J. Funct. Anal. 1.496 2.42 1.61 0.035
J. Funct. Spaces 1.896 0.46 0.43
<
Amer. J. Math. 1.711 3.28 1.67 0.009
Mathematics 1.747 0.3 NA NA
The JIF has received serious criticism for various reasons, such as: lack of statistical significance [9 D. I. Stern, Uncertainty measures for economics journal impact factors. J. Econ. Lit. 51, 173–189 (2013) , 10 J. K. Vanclay, Impact factor: Outdated artefact or stepping-stone to journal certification? Scientometrics92, 211–238 (2012) ], poor representativeness and robustness [5 T. Lando and L. Bertoli-Barsotti, Measuring the citation impact of journals with generalized Lorenz curves. J. Informetrics11, 689–703 (2017) ], insensitivity to field differences [6 H. F. Moed, Measuring contextual citation impact of scientific journals. J. Informetrics4, 265–277 (2010) ], insensitivity to the weight of the citing articles [2 A. Ferrer-Sapena, E. A. Sánchez-Pérez, L. M. González, F. Peset, and R. Aleixandre-Benavent, Mathematical properties of weighted impact factors based on measures of prestige of the citing journals. Scientometrics105, 2089–2108 (2015) ] and manipulability by editorial strategies [8 K. Moustafa, The disaster of the impact factor. Science and Engineering Ethics21, 139–142 (2015) ]. Here is a list of some of the most significant limitations:
it counts citations of articles that are not included in the denominator of the above formula;
its analysis period is 2 years, which is not suitable for evaluation of mathematical research;
it merely counts citations, without considering their quality. Therefore the JIF may force some mathematicians to do research in topics on which a lot of people are working, who can potentially cite their papers. It is easy to find evidence that such topics are mostly outside the mainstream of mathematics;
it includes self-citations;
it is relatively easy to manipulate JIFs and some other scientometrics. There are “mutual citation groups” in which researchers in a certain circle heavily cite each other’s work in order to enhance the JIF of a certain journal and artificially inflate the impact of their own papers.
The SJR aims to fix the above problems by providing a more effective computation formula, including a longer period of 3 years for counting citations, attributing different weight to citations, and limiting self-citations. Some studies show that using the SJR can improve the situation to some extent. It is at any rate a first step towards avoiding some of the limitations of JIF; see [1 M. R. Elkins, C. G. Maher, R. D. Herbert, A. M. Moseley, and C. Sherrington, Correlation between the Journal Impact Factor and three other journal citation indices. Scientometrics85, 81–93 (2010) , 3 E. García-Pachón and R. Arencibia-Jorge, A comparison of the impact factor and the SCImago journal rank index in respiratory system journals. Archivos de Bronconeumología50, 308–309 (2014) ].
To illustrate the drawbacks and inadequacy of JIF in mathematics, let us take a closer look at the JIF numbers. There are mathematical journals in the 2019 list of JCR-Q1 whose impact factors are “unexpectedly large”. For instance, the Iranian Journal of Fuzzy Systems is ranked 15 in the category of Mathematics of the JCR list, while the very prestigious journal Acta Mathematica, launched in 1882, is ranked 13; also, American Journal of Mathematics and Transactions of the American Mathematical Society are ranked 32 and 60, respectively.
However, the SJR for Iranian Journal of Fuzzy Systems is 0.51 but for Acta Mathematica, it is 5.77. Similarly, the Mathematical Citation Quotient (MCQ), a 5-year-impact factor computed by MathSciNet (an online publication of the American Mathematical Society), for Iranian Journal of Fuzzy Systems and Acta Mathematica are 0.11 and 3.95, respectively.
This pattern can be seen in other journals. For example, Journal of Function Spaces is ranked 24, while the leading journal Journal of Functional Analysis is ranked 47! Again both the SJR and the MCQ of Journal of Functional Analysis are much greater than those of Journal of Function Spaces.
There is a similar situation regarding the American Journal of Mathematics, established in 1878, and a recently launched JCR journal named Mathematics.
Some important reasons for such unexpected JIFs are as follows:
a high rate of publication on a topic. For instance, “fixed point theory” is a popular topic that a lot of mathematicians work on;
a considerable number of researchers working on a topic. For example, the number of mathematicians who are working on “fuzzy mathematics” is much greater than those working on “
K
-theory”, and hence the general rate of citations in such topics is high.
the open accessibility of a journal.
Non-ethical ways to increase JIF used by a few journals. While the term “predatory journal” is arguable, the mere appearance of this term shows that the problem does exist.
The backlog between acceptance and publication in some mathematics journals may exceed two years. Journals with such large backlogs, which are usually good journals, may have unexpectedly low JIF. Nowadays, some journals have moved to the continuous article publishing (CAP) model in which every article, after acceptance, is published immediately within the current issue.
We think that Clarivate Analytics should improve its formula for computing JIF. Until then, we suggest that scientific committees should consider a package of indicators such as the JIF, SJR, Citescore, Eigenfactor together.
The scientometric indicators developed for journals, essentially based on citations, should not be applied as a tool to assess the work of individual researchers. In fact, as citation occurs after research, the direction of research should not be affected by any demand for citation. The scientometric data reflect to some extent the quality of a journal, but not so much the actual quality of a single paper, since not all papers in a journal are cited equally.
As we explain in the next section, when a scientific committee uses only scientometric data to evaluate a mathematician’s achievement, without any human assessment, they are using a flawed approach that may result in an unfair judgement.
A large number of universities around the world use scientometrics tools to evaluating the research of academic members, postdoctoral researchers, and Ph.D. candidates for promotion, employment, or funding. It seems that such universities have no other reliable sources, and possibly suffer from lack of any peer-reviewed system in which the content of papers is expected to be evaluated by professional mathematicians. In addition, dealing with scientometric data is much easier than reading papers and assessing their content.
There are mathematicians who believe that scientometric data such as the SJR are reliable instruments for judgments, since they make assessments more objective and free them from the crude or biased judgements of human beings. They argue that quantitative indicators help funding organizations, publishers, and policy-makers to gain strategic intelligence that leads toward fairer outcomes and ensures that their budget is spent in the most effective way.
However, there are others who are against using scientometrics to measure scientific publications, due to the lack of transparency. Scientometrics may cause distortions that have detrimental effects on the development of scientific fields. For example, some supporters of the JIF subscribe to the idea that every paper published in a high-ranked journal must contain excellent mathematics, which is not entirely true in general; one can easily find some counterexamples in the literature. Some mathematicians propose that citations are relevant only when dealing with large numbers. In small numbers, they can be a misuse of statistics. These mathematicians continue to trust in evaluation by human beings, even though it may be subjective in the sense that it is influenced by the human dualities of love and hate, good and bad, as well as true and false. They believe that metrics put the worth and livelihood of our young mathematicians at risk and have undesirable impacts on the scientific life of all mathematicians.
Although citations do not show all the good qualities of a paper, they (in particular, non-self-citations by reputed researchers in prestigious journals) may help experts in evaluating and documentating research work. Papers with no citation over a ‘long period of time’ cannot be regarded as high-level papers. For that matter, not all highly cited papers are necessarily high level papers. However, abuse of scientometric data such as the JIF and games with numbers can happen, and may mislead people instead of being an indicator.
Scientometrics tools can be used, provided that one keeps their disadvantages and distortions in mind, and they are considered together with the judgement of experts based on depth and extent of papers. Such experts could be asked to look at a candidate’s self-selected best papers, research programs, and statements of major achievements. No assessment is complete without a peer review. Furthermore, we need a modification of the policies of universities, funding organizations, and so on to support human assessments.
We hope that the various ideas discussed in this note may help not only mathematicians but the whole of the scientific community to improve their point of view and their assessment guidelines.
Mohammad Sal Moslehian is a Professor of Mathematics at Ferdowsi University of Mashhad and a member of the Academy of Sciences of Iran. He was a member of the Executive Committee of the Iranian Mathematical Society from 2004 to 2012 and a Senior Associate of ICTP in Italy. He is the editor-in-chief of the journals Banach J. Math. Anal., Ann. Funct. Anal., and Adv. Oper. Theory published by Birkhäuser/Springer. moslehian@um.ac.ir
M. R. Elkins, C. G. Maher, R. D. Herbert, A. M. Moseley, and C. Sherrington, Correlation between the Journal Impact Factor and three other journal citation indices. Scientometrics85, 81–93 (2010)
A. Ferrer-Sapena, E. A. Sánchez-Pérez, L. M. González, F. Peset, and R. Aleixandre-Benavent, Mathematical properties of weighted impact factors based on measures of prestige of the citing journals. Scientometrics105, 2089–2108 (2015)
E. García-Pachón and R. Arencibia-Jorge, A comparison of the impact factor and the SCImago journal rank index in respiratory system journals. Archivos de Bronconeumología50, 308–309 (2014)
B. González-Pereira, V. P. Guerrero-Bote, and F. Moya-Anegón, A new approach to the metric of journals’ scientific prestige: The SJR indicator. J. Informetrics4, 379–391 (2010)
T. Lando and L. Bertoli-Barsotti, Measuring the citation impact of journals with generalized Lorenz curves. J. Informetrics11, 689–703 (2017)
H. F. Moed, Measuring contextual citation impact of scientific journals. J. Informetrics4, 265–277 (2010)
H. F. Moed, From Journal Impact Factor to SJR, Eigenfactor, SNIP, CiteScore and Usage Factor. In Applied Evaluative Informetrics. Qualitative and Quantitative Analysis of Scientific and Scholarly Communication. Springer, Cham, 229–244 (2017)
K. Moustafa, The disaster of the impact factor. Science and Engineering Ethics21, 139–142 (2015)
D. I. Stern, Uncertainty measures for economics journal impact factors. J. Econ. Lit. 51, 173–189 (2013)
J. K. Vanclay, Impact factor: Outdated artefact or stepping-stone to journal certification? Scientometrics92, 211–238 (2012)
Mohammad Sal Moslehian, Impact factor, an inadequate yardstick. Eur. Math. Soc. Mag. 120 (2021), pp. 40–42
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Which type of DNA is found in bacteria
Helical DNA
Membrane bound DNA
Straight DNA
Circular free DNA
Bacteria are prokaryotic in nature, in which typical chromosomes are lacking. DNA is circular and naked as it is not surrounded by histones (basic proteins which are responsible for coiled structure of nucleosome).
In split genes the coding sequences are called
Split genes were discovered by Phillip Sharp and Robert Richard. The part of DNA which express itself by making mRNA is called exon. One exon is separated from others by inactive part of DNA called intron. This is called split DNA.
Which one of the following pairs is correctly matched with regard to the codon and the amino acid coded by it
UUA - valine
AUG - cysteine
AAA - lysine
CCC - alanine
DNA polymerase enzyme is required for synthesis of
DNA from RNA
RNA from DNA
DNA from DNA
RNA from RNA
The enzyme needed for replication of DNA i.e. formation of DNA from DNA is DNA polymerase III. DNA replication is initiated by RNA primer which is later removed by DNA polymerase I.
Mutations which alter nucleotide sequence within a gene are :
Gene mutation or point mutation is the result of a change in the nucleotide sequence of the DNA molecule in a particular region of the chromosome. It occurs in the form of duplication, insertion, deletion, inversion or substitution of bases. Frame- shift mutation is the addition or deletion of a base that may change the reading frame of nucleotide sequence.
Base pair substitution is a type of mutation involving replacement or substitution of a single nucleotide base with another in DNA or RNA molecule.
Which one of the following codons codes for the same information as UGC?
UGU codes for the same information as UGC as both code for cystine. UGA and UAG are nonsense codons and UGG codes for tryptophan.
What is true about tRNA?
It binds with an amino acid at it 3' end
It has five double stranded regions
It has a codon at one end which recognizes the anticodon on messenger RNA
It looks like clover leafin the three dimensional structure
tRNA has four recognition sites among these one is the amino acid attachment site. It has the amino acid attachment site with the 3' terminal - CCA sequence.
Lac is obtained from :
Lac is obtained fom Laccifer or Kerria lacca or lac insect that belongs to Phylum- Arthropoda.
Which one of the following correctly represents the manner of replication of DNA?
DNA replication is the process by which DNA makes a copy of itself during cell division. It takes place discontinuously. This process consists of 4 main steps:
Replication Fork formation
Primer binding
Okazaki fragments are short sequences of DNA nucleotides which are synthesized discontinuously. One strand may synthesize a continuous strand. Both new strands are synthesized in 5'
\to
3' direction. Thus one strand is synthesized forwards and the other backwards.
Which one of the following pairs of terms/ names mean one and the same thing?
Gene pool - genome
Codon - gene
Cistron - triplet
DNA fingerprinting - DNA profiling
Gene pool is the total gene present in a population. Genome is the total genetic constitution of an organism. Codon is the basic unit of genetic code, a sequence of three adjacent nucleotide in DNA or mRNA that code for an amino acid. Gene is the basic unit of heredity; a sequence of DNA nucleotide that encodes a protein.
Cistron is a segment of DNA nucleotides that codes for a polypeptide chain. Triplet is a three nucleotides sequence coding for an amino acid.
Therefore, codon
\approx
\approx
DNA fingerprinting is technically called DNA profiling or DNA typing.
https://www.zigya.com/share/MTA=
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For other uses, see plus-minus (disambiguation).
In mathematics, it generally indicates a choice of exactly two possible values, one of which is obtained through addition and the other through subtraction.[1]
In experimental sciences, the sign commonly indicates the confidence interval or uncertainty bounding a range of possible errors in a measurement, often the standard deviation or standard error.[2] The sign may also represent an inclusive range of values that a reading might have.
In medicine, it means "with or without".[3][4]
In engineering, the sign indicates the tolerance, which is the range of values that are considered to be acceptable, safe, or which comply with some standard or with a contract.
In botany, it is used in morphological descriptions to notate "more or less".
In chemistry, the sign is used to indicate a racemic mixture.
In chess, the sign indicates a clear advantage for the white player; the complementary minus-or-plus sign, ∓, indicates the same advantage for the black player.[5]
U+00B1 ± PLUS-MINUS SIGN (±, ±, ±)
U+2213 ∓ MINUS-OR-PLUS SIGN (∓, ∓, ∓)
2.3 In chess
A version of the sign, including also the French word ou ("or"), was used in its mathematical meaning by Albert Girard in 1626, and the sign in its modern form was used as early as 1631, in William Oughtred's Clavis Mathematicae.[6]
In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the plus and minus signs, + or −, allowing the formula to represent two values or two equations.[7]
For example, given the equation x2 = 9, one may give the solution as x = ±3. This indicates that the equation has two solutions, each of which may be obtained by replacing this equation by one of the two equations x = +3 or x = −3. Only one of these two replaced equations is true for any valid solution. A common use of this notation is found in the quadratic formula
{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},}
which describes the two solutions to the quadratic equation ax2 + bx + c = 0.
Similarly, the trigonometric identity
{\displaystyle \sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B)}
can be interpreted as a shorthand for two equations: one with + on both sides of the equation, and one with − on both sides. The two copies of the ± sign in this identity must both be replaced in the same way: it is not valid to replace one of them with + and the other of them with −. In contrast to the quadratic formula example, both of the equations described by this identity are simultaneously valid.
The minus–plus sign, ∓, is generally used in conjunction with the ± sign, in such expressions as x ± y ∓ z, which can be interpreted as meaning x + y − z or x − y + z, but not x + y + z nor x − y − z. The upper − in ∓ is considered to be associated to the + of ± (and similarly for the two lower symbols), even though there is no visual indication of the dependency.
However, the ± sign is generally preferred over the ∓ sign, so if both of them appear in an equation, it is safe to assume that they are linked. On the other hand, if there are two instances of the ± sign in an expression, without a ∓, it is impossible to tell from notation alone whether the intended interpretation is as two or four distinct expressions.
The original expression can be rewritten as x ± (y − z) to avoid confusion, but cases such as the trigonometric identity are most neatly written using the "∓" sign:
{\displaystyle \cos(A\pm B)=\cos(A)\cos(B)\mp \sin(A)\sin(B)}
which represents the two equations:
{\displaystyle {\begin{aligned}\cos(A+B)&=\cos(A)\cos(B)-\sin(A)\sin(B)\\\cos(A-B)&=\cos(A)\cos(B)+\sin(A)\sin(B)\end{aligned}}}
Another example where the minus–plus sign appears is
{\displaystyle x^{3}\pm 1=(x\pm 1)\left(x^{2}\mp x+1\right)}
A third related usage is found in this presentation of the formula for the Taylor series of the sine function:
{\displaystyle \sin \left(x\right)=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \pm {\frac {1}{(2n+1)!}}x^{2n+1}+\cdots ~.}
Here, the plus-or-minus sign indicates that the term may be added or subtracted, in this case depending on whether n is odd or even, the rule can be deduced from the first few terms. A more rigorous presentation of the same formula would multiply each term by a factor of (−1)n, which gives +1 when n is even, and −1 when n is odd. In older texts one occasionally finds (−)n, which means the same.
When the standard presumption that the plus-or-minus signs all take on the same value of +1 or all −1 is not true, then the line of text that immediately follows the equation must contain a brief description of the actual connection, if any, most often of the form “where the ‘±’ signs are independent” or similar. If a brief, simple description is not possible, the equation must be re-written to provide clarity; e.g. by introducing variables such as s1, s2, ... and specifying a value of +1 or −1 separately for each, or some appropriate relation, like
{\displaystyle s_{3}=s_{1}\cdot (s_{2})^{n}\,,}
The use of ± for an approximation is most commonly encountered in presenting the numerical value of a quantity, together with its tolerance or its statistical margin of error.[2] For example, 5.7 ± 0.2 may be anywhere in the range from 5.5 to 5.9 inclusive. In scientific usage, it sometimes refers to a probability of being within the stated interval, usually corresponding to either 1 or 2 standard deviations (a probability of 68.3% or 95.4% in a normal distribution).
Operations involving uncertain values should always try to preserve the uncertainty—in order to avoid propagation of error. If
{\displaystyle ~n=a\pm b\;,}
any operation of the form
{\displaystyle ~m=f(n)~}
must return a value of the form
{\displaystyle ~m=c\pm d~}
, where c is
{\displaystyle \,f(n)\,}
and d is range updated using interval arithmetic.
A percentage may also be used to indicate the error margin. For example, 230 ±10% V refers to a voltage within 10% of either side of 230 V (from 207 V to 253 V inclusive).[citation needed] Separate values for the upper and lower bounds may also be used. For example, to indicate that a value is most likely 5.7, but may be as high as 5.9 or as low as 5.6, one may write 5.7+0.2
In chessEdit
The symbols ± and ∓ are used in chess notation to denote an advantage for white and black, respectively. However, the more common chess notation would be to only use + and –.[5] If several different symbols are used together, then the symbols + and − denote a clearer advantage than ± and ∓. When finer evaluation is desired, three pairs of symbols are used: ⩲ and ⩱ for only a slight advantage; ± and ∓ for a significant advantage; and +– and –+ for a potentially winning advantage, in each case for white or black respectively.[8]
In Unicode: U+00B1 ± PLUS-MINUS SIGN
In ISO 8859-1, -7, -8, -9, -13, -15, and -16, the plus–minus symbol is code 0xB1hex. This location was copied to Unicode.
The symbol also has a HTML entity representations of ±, ±, and ±.
The rarer minus–plus sign is not generally found in legacy encodings, but is available in Unicode as U+2213 ∓ MINUS-OR-PLUS SIGN so can be used in HTML using ∓ or ∓.
In TeX 'plus-or-minus' and 'minus-or-plus' symbols are denoted \pm and \mp, respectively.
Although these characters may also be produced using underlining or overlining + symbol ( + or + ), this is deprecated because the formatting may be stripped at a later date, changing the meaning. It also makes the meaning less accessible to blind users with screen readers.
Windows: Alt+241 or Alt+0177 (numbers typed on the numeric keypad).
Macintosh: ⌥ Option+⇧ Shift+= (equal sign on the non-numeric keypad).
Unix-like systems: Compose,+,- or ⇧ Shift+Ctrl+u B1space (second works on Chromebook)
AutoCAD shortcut string: %%p
Similar charactersEdit
Look up 士, 土, or 干 in Wiktionary, the free dictionary.
The plus–minus sign resembles the Chinese characters 土 (Radical 32) and 士 (Radical 33), whereas the minus–plus sign resembles 干 (Radical 51).
≈ (approximately equal to)
^ Weisstein, Eric W. "Plus or Minus". mathworld.wolfram.com. Retrieved 2020-08-28.
^ a b Brown, George W. (1982). "Standard deviation, standard error: Which 'standard' should we use?". American Journal of Diseases of Children. 136 (10): 937–941. doi:10.1001/archpedi.1982.03970460067015. PMID 7124681.
^ Naess, I. A.; Christiansen, S. C.; Romundstad, P.; Cannegieter, S. C.; Rosendaal, F. R.; Hammerstrøm, J. (2007). "Incidence and mortality of venous thrombosis: a population-based study". Journal of Thrombosis and Haemostasis. 5 (4): 692–699. doi:10.1111/j.1538-7836.2007.02450.x. ISSN 1538-7933. PMID 17367492.
^ Heit, J. A.; Silverstein, M. D.; Mohr, D. N.; Petterson, T. M.; O'Fallon, W. M.; Melton, L. J. (1999-03-08). "Predictors of survival after deep vein thrombosis and pulmonary embolism: a population-based, cohort study". Archives of Internal Medicine. 159 (5): 445–453. doi:10.1001/archinte.159.5.445. ISSN 0003-9926. PMID 10074952.
^ a b Eade, James (2005), Chess For Dummies (2nd ed.), John Wiley & Sons, p. 272, ISBN 9780471774334 .
^ Cajori, Florian (1928), A History of Mathematical Notations, Volume I: Notations in Elementary Mathematics, Open Court, p. 245 .
^ "Definition of PLUS/MINUS SIGN". www.merriam-webster.com. Retrieved 2020-08-28.
^ For details, see Chess annotation symbols#Positions.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Plus–minus_sign&oldid=1083439516"
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Three examples of nonlinear least-squares fitting in Python with SciPy
Least-squares fitting is a well-known statistical technique to estimate parameters in mathematical models. It concerns solving the optimisation problem of finding the minimum of the function
F(\theta) = \sum_{i = 1}^N \rho (f_i(\theta)^2),
\theta=(\theta_1, \ldots, \theta_r)
is a collection of parameters which we want to estimate,
N
is the number of available data points,
\rho
is a loss function to reduce the influence of outliers, and
f_i(\theta)
i
-th component of the vector of residuals.
Given a model function
m(t; \theta)
and some data points
D = \{(t_i, d_i) \mid i = 1,\ldots, N\}
, one normally defines the vector of residuals as the difference between the model prediction and the data, that is:
f_i(\theta) = m(t_i; \theta) - d_i.
If the model is linear, i.e.
\theta = (m, n)
m(t; m, n) = mt + n
, then the previous approach is reduced to the even more well known linear least-squares fitting problem, in the sense that there exists an explicit formula for finding the optimal value for the parameters
\hat \theta
. In this report we consider more involved problems where the model may be nonlinear and finding the optimal value
\hat \theta
must be done with iterative algorithms. We shall not go into the theoretical details of the algorithms, but rather explore the implementation of the least_squares function available in the scipy.optimize module of the SciPy Python package. In particular, we give examples of how to handle multi-dimensional and multi-variate functions so that they adhere to the least_squares interface.
First example: a scalar function
The first example we will consider is a simple logistic function
y(t) = \frac{K}{1 + e^{-r(t - t_0)}}.
The three parameters in the function are:
K
, the supremum of
y
(think of this as a maximum that is achieved when
t = \infty
r
, the logistic growth rate, or sharpness of the curve, and
t_0
the value at which the midpoint
K/2
is achieved.
Optimising the parameters
\theta = (K, r, t_0)
is straightforward in this case, since least_squares allows us to input the model without any special changes. To do this we first define the model as the Python function
y(theta, t)
and then generate some data in ys by evaluating the model over a sample ts of the independent variable
t
. When generating the test data, we are passing an array of numbers ts directly to the model. We are able to do this because we defined the model y using NumPy’s array object which implements standard element-wise operations between arrays. We add some uniformly distributed noise, make an initial guess for the parameters theta0 and define the vector of residues fun. The vector of residues must be a function of the optimization variables
\theta
so that least_squares can evaluate the fitness of its guesses.
def y(theta, t):
return theta[0] / (1 + np.exp(- theta[1] * (t - theta[2])))
K = 1; r = 10; t0 = 0.5; noise = 0.1
ys = y([K, r, t0], ts) + noise * np.random.rand(ts.shape[0])
def fun(theta):
return y(theta, ts) - ys
theta0 = [1,2,3]
res1 = least_squares(fun, theta0)
We see that the estimated parameters are indeed very close to those of the data.
In some cases, we may want to only optimise some parameters while leaving others fixed. This can be done by removing them from theta and hard-coding them into the model.
Second example: a parametrised circumference
The motivation behind this example is that sometimes we may have data that is described by two dependent quantities which we hypothesize to be a function of a single independent variable. The simplest example I could think of was that of a parametrised circumference:
s(t) = (c_x + r \cos(t), c_y + r \sin(t)).
Here, the parameters are
the centre
C = (c_x, c_y)
r
Like before, we define the model s and generate some random data. Only this time, the model outputs two dependent variables
s(t) = (x(t), y(t))
. As expected, this requires us to define the model differently so that it returns two numbers instead of one as well as adding noise to both outputs. When it comes to defining the vector of residuals, we must take care to match the shape expected by least_squares. Per the documentation, we must provide a vector of
N
elements which least_squares will square before inputing the result into the loss function
\rho
. However, we want to compute the square of the distance between the model prediction and the test data. We would normally do this by calculating
(m_x - d_x)^2 + (m_y - d_y)^2
. But if we did this the result would get squared again by least_squares. The solution is to return a vector of residues of size
2N
where components
1
N
correspond to the differences
m_x - d_x
N + 1
2N
m_y - d_y
. This is easily achieved by taking the difference between the prediction and the data as usual and then flattening the two arrays into a longer one. We are able to do this because least_squares never really sees the raw data and the cost functions presented at the beginning of the report is just there to provided a mathematical background. Also, this approach generalises easily to higher-dimensional model outputs.
def s(theta, t):
x = theta[0] + theta[2] * np.cos(t)
y = theta[1] + theta[2] * np.sin(t)
ts = np.linspace(0, 2 * np.pi)
cx = 1.5; cy = 1.3; r = 0.75; noise = 0.05
ss = s([cx, cy, r], ts)
ss[0] += noise * np.random.rand(ts.shape[0])
return (s(theta, ts) - ss).flatten()
Third example: an elliptic paraboloid
Lastly we focus on an elliptic paraboloid since it can be parametrised as a function of two independent variables with
h(x, y) = a(x - x_0)^2 + b(y - y_0)^2.
and
b
, which determine the eccentricity of the ellipses that the level sets of the paraboloid (
a = b = 1
give circumferences), and
(x_0, y_0)
, which determine the projection of the vertical axis of the paraboloid onto the
xy
In this instance we must also be careful with how we sample the domain of the independent variable. We choose to sample the square
[-1, 1] \times [-1, 1]
20 \times 20
mesh grid, i.e. we evaluate the model at points
(-1 + 0.1k, -1 + 0.1k)
k = 0, \ldots, 20
. This sounds more complicated than it really is, since NumPy provides a method numpy.meshgrid that does this for us. As before, we add noise, taking care to do so for each of the
400 = 20 \times 20
data points we generated. For the model definition, we do not need to do anything special since NumPy also implements binary element-wise operations between the components of a mesh grid.
def h(theta, x, y):
return theta[2] * (x - theta[0])**2 + theta[3] * (y - theta[1])**2
gridx, gridy = np.meshgrid(xs, ys)
x0 = 0.1; y0 = -0.15; a = 1; b = 2; noise = 0.1
hs = h([x0, y0, a, b], gridx, gridy)
hs += noise * np.random.default_rng().random(hs.shape)
return (h(theta, gridx, gridy) - hs).flatten()
theta0 = [0, 0, 1, 2]
Goodness of fit and parameter distribution estimation
Once we have parameter estimates for our model, one question we may ask ourselves is how well does the model fit the data. Since we are doing least-square estimation, one of the easiest errors to calculate is the mean squared error [MSE]. In the context of the previous question, i.e. how well do these parameter estimates fit the data, this error is just the mean of the squares of the residuals
\text{MSE} = \frac{1}{N} \sum_{i = 1}^N f_i(\hat \theta).
Therefore it is a function of both the training data set and the parameters themselves. In the previous three cases the MSE can be calculated easily with Python, by using the result object returned by least_squares:
mse1 = (res1.fun ** 2).mean()
MSE in the logistic equation (ex1): 0.000806
MSE in the parametrised circumference (ex2): 0.000226
MSE in the elliptic paraboloid (ex3): 0.001717
Now suppose we are using the parameter estimates and model for prediction. For instance, think of the first logistic growth model as the cumulative number of cases during an epidemic. Let’s say we are in the middle of the epidemic and only have data available for the first couple of weeks but with to determine when the increase of new cases will start to decline (at
t_0
) or what the total number of cases will be (approximately
K
). We estimate the parameters with the available data and make estimates. Then the epidemic finally stops and we wish to evaluate how well our model did.
In this context the MSE is called the mean square prediction error [MSPE] and is defined as
\text{MSPE} = \frac{1}{Q} \sum_{i = N + 1}^{N + Q} (d_i - m(t_i; \hat \theta))^2,
where we suppose we have made
Q
predictions using the model at the values of the independent variable
t_{N + 1}, \ldots, t_{N + Q}
using the estimated parameters
\hat \theta
# generate data again
K = 1400; r = 0.1; t0 = 50; noise = 50
ys = y([K, r, t0], ts) + noise * (np.random.rand(ts.shape[0]) - 0.5)
# only this time we only use the first 50% of the data
train_limit = 50 # out of 100 datapoints
return y(theta, ts[:train_limit]) - ys[:train_limit]
# run the parameter estimation again
theta0 = [1000, 0.1, 30]
# predict the values for the rest of the epidemic
predict = y(res4.x, ts[train_limit:])
mspe = ((ys[train_limit:] - predict) ** 2).mean()
K = 1266.9713573318118, r = 0.10463262364967481, t_0 = 48.37440994315726
MSPE for a prediction with 50.0% of the data: 9660.411804269846
The previous simulation is extremely sensitive to the amount of training data. In particular, if the training dataset ends much before
t_0
the model can be horribly wrong to the point where the prediction can be that the epidemic will still be in the initial exponential phase by day 100. In any case, this issue is beyond the scope of this report as it has more to do with the high sensitivity to small changes in parameters that characterises exponential models.
However, the idea of measuring the accuracy of a prediction is in the right direction. Another question we might ask ourselves is how can we get error estimates for our predictions before being able to validate them against real data. In other words, can we predict how wrong we will be in addition to predicting how many infected cases there will be? For general models and general techniques of parameter estimation, there are countless answers to this question. In what follows we focus on a very particular approach that, of course, has its flaws.
Error estimates via residual resampling
One common technique for quantifying errors in parameter estimation is the use of confidence intervals. If the underlying distribution of the data is known, it can sometimes be used to derive confidence intervals via explicit formulas. When the underlying distribution is either unknown or too complex to treat analytically, one can try to estimate the distribution of the data itself. One possible way of doing this is to instead estimate the distribution of the residuals and generate new samples from the fitted values.
More formally, one does NLS fitting and retains the fit values
\hat y_i
and the residuals
f_i(\hat \theta)
in addition to the estimated parameters
\hat \theta
. Recall that because of the way we have defined
f
, the relation between these three is
f_i(\hat \theta) = \hat \varepsilon_i = \hat y_i - y_i
. Now we generate new synthetic data,
y_i^* = \hat y_i + \hat \varepsilon_j
\hat \varepsilon_j
is randomly drawn from the collection of residues
\hat \varepsilon_1, \ldots, \hat \varepsilon_N
. We use that synthetic data to refit the model and retain interesting quantities such as the values of the parameter estimates. We repeat this process many times to estimate the distribution of the interesting quantities we picked. We are now in a position to estimate the confidence intervals for the parameters.
# again, we only use the first 50% of the data
# run the parameter estimation
# use the residuals to estimate the error distribution
eps = res5.fun
# residual resampling
M = 500 # number of resamples
theta_est = []
# generate synthetic sample
fit_ys = y(res5.x, ts[:train_limit])
np.random.default_rng().shuffle(eps)
synthetic_ys = fit_ys + eps
# fit the model again
res = least_squares(lambda theta: y(theta, ts[:train_limit]) - synthetic_ys[:train_limit], theta0)
theta_est.append(res.x)
Ks, rs, t0s = np.array(theta_est).transpose()
Here we can see the estimated distributions of the model parameters. The distributions for
t_0
are more or less centred while the distribution for
K
presents a huge variance and is quite skewed. We choose to do the following, fix the parameters
\hat r = \bar r
\hat t_0 = \bar t_0
and plot the predictions for the model with them and the values for the 95% confidence interval for
K
which we estimate from the distribution.
K05, K95 = np.quantile(Ks, [0.05, 0.95])
K = Ks.mean()
r = rs.mean()
t0 = t0s.mean()
y05 = y([K05, r, t0], ts)
print(f'K = {K:.0f} ({K05:.0f}, {K95:.0f}), r = {r:.3f}, t0 = {t0:.0f}')
ax.plot(ts, ys, '.-', label = 'data')
ax.plot(ts, y05, '--', label = '5%')
ax.plot(ts, y([K, r, t0], ts), label = 'fit')
ax.plot(ts, y95, '--', label = '95%')
ax.fill_between(ts, y05, y95, alpha = 0.2, color = 'gray')
ax.set_ylabel('confirmed cases')
ax.set_title('Prediction with confidence intervals')
K = 1258 (1090, 1454), r = 0.105, t0 = 48
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YIQ Knowpia
YIQ is the color space used by the analog NTSC color TV system, employed mainly in North and Central America, and Japan. I stands for in-phase, while Q stands for quadrature, referring to the components used in quadrature amplitude modulation. Other TV systems used different color spaces, such as YUV for PAL or YDbDr for SECAM. Later digital standards use the YCbCr color space. These color spaces are all broadly related, and work based on the principle of adding a color component named chrominance, to a black and white image named luma.
The YIQ color space at Y=0.5. Note that the I and Q chroma coordinates are scaled up to 1.0. See the formulae below in the article to get the right bounds.
An image along with its Y, I, and Q components.
In YIQ the Y component represents the luma information, and is the only component used by black-and-white television receivers. I and Q represent the chrominance information. In YUV, the U and V components can be thought of as X and Y coordinates within the color space. I and Q can be thought of as a second pair of axes on the same graph, rotated 33°; therefore IQ and UV represent different coordinate systems on the same plane.
The YIQ system is intended to take advantage of human color-response characteristics. The eye is more sensitive to changes in the orange-blue (I) range than in the purple-green range (Q)—therefore less bandwidth is required for Q than for I. Broadcast NTSC limits I to 1.3 MHz and Q to 0.4 MHz. I and Q are frequency interleaved into the 4 MHz Y signal, which keeps the bandwidth of the overall signal down to 4.2 MHz. In YUV systems, since U and V both contain information in the orange-blue range, both components must be given the same amount of bandwidth as I to achieve similar color fidelity.
Very few television sets perform true I and Q decoding, due to the high costs of such an implementation. Compared to the cheaper R-Y and B-Y decoding which requires only one filter, I and Q each requires a different filter to satisfy the bandwidth differences between I and Q. These bandwidth differences also require that the 'I' filter include a time delay to match the longer delay of the 'Q' filter. The Rockwell Modular Digital Radio (MDR) was one I and Q decoding set, which in 1997 could operate in frame-at-a-time mode with a PC or in realtime with the Fast IQ Processor (FIQP). Some RCA "Colortrak" home TV receivers made circa 1985 not only used I/Q decoding, but also advertised its benefits along with its comb filtering benefits as full "100 percent processing" to deliver more of the original color picture content. Earlier, more than one brand of color TV (RCA, Arvin) used I/Q decoding in the 1954 or 1955 model year on models utilizing screens about 13 inches (measured diagonally). The original Advent projection television used I/Q decoding. Around 1990, at least one manufacturer (Ikegami) of professional studio picture monitors advertised I/Q decoding.
The YIQ representation is sometimes employed in color image processing transformations. For example, applying a histogram equalization directly to the channels in an RGB image would alter the color balance of the image. Instead, the histogram equalization is applied to the Y channel of the YIQ or YUV representation of the image, which only normalizes the brightness levels of the image.
These formulas allow conversion between YIQ and RGB color spaces, where R, G, and B are gamma-corrected values. Values for the original 1953 NTSC colorimetry and later SMPTE C FCC standard. The following formulas assume:
{\displaystyle R,G,B,Y\in \left[0,1\right],\quad I\in \left[-0.5957,0.5957\right],\quad Q\in \left[-0.5226,0.5226\right]}
The ranges for I and Q [1][2] are a result of the coefficients in the 2nd and 3rd rows of the RGB-to-YIQ equation matrix below, respectively.
NTSC 1953 colorimetryEdit
These formulas approximate the conversion between the original 1953 color NTSC specification and YIQ.[3][4][5]
From RGB to YIQEdit
{\displaystyle {\begin{bmatrix}Y\\I\\Q\end{bmatrix}}\approx {\begin{bmatrix}0.299&0.587&0.114\\0.5959&-0.2746&-0.3213\\0.2115&-0.5227&0.3112\end{bmatrix}}{\begin{bmatrix}R\\G\\B\end{bmatrix}}}
From YIQ to RGBEdit
{\displaystyle {\begin{bmatrix}R\\G\\B\end{bmatrix}}={\begin{bmatrix}1&0.956&0.619\\1&-0.272&-0.647\\1&-1.106&1.703\end{bmatrix}}{\begin{bmatrix}Y\\I\\Q\end{bmatrix}}}
Note that the top row is identical to that of the YUV color space
{\displaystyle {\begin{bmatrix}R\\G\\B\end{bmatrix}}={\begin{bmatrix}1\\1\\1\end{bmatrix}}\implies {\begin{bmatrix}Y\\I\\Q\end{bmatrix}}={\begin{bmatrix}1\\0\\0\end{bmatrix}}}
FCC NTSC Standard (SMPTE C)Edit
In 1987, the Society of Motion Picture and Television Engineers (SMPTE) Committee on Television Technology, Working Group on Studio Monitor Colorimetry, adopted the SMPTE C.[7][8][9] The previous conversion formulas were deprecated, and the NTSC standard contained in the FCC rules for over-the-air analog color TV broadcasting adopted a different matrix:[10][11]
{\displaystyle \left\{{\begin{array}{ccl}E_{Y}^{\prime }&=&0.30E_{R}^{\prime }+0.59E_{G}^{\prime }+0.11E_{B}^{\prime }\\E_{I}^{\prime }&=&-0.27(E_{B}^{\prime }-E_{Y}^{\prime })+0.74(E_{R}^{\prime }-E_{Y}^{\prime })\\E_{Q}^{\prime }&=&0.41(E_{B}^{\prime }-E_{Y}^{\prime })+0.48(E_{R}^{\prime }-E_{Y}^{\prime })\end{array}}\right.}
in matrix notation, that equation system is written as:
{\displaystyle {\begin{bmatrix}E_{Y}^{\prime }\\E_{I}^{\prime }\\E_{Q}^{\prime }\end{bmatrix}}={\begin{bmatrix}0.30&0.59&0.11\\0.599&-0.2773&-0.3217\\0.213&-0.5251&0.3121\end{bmatrix}}{\begin{bmatrix}E_{R}^{\prime }\\E_{G}^{\prime }\\E_{B}^{\prime }\end{bmatrix}}}
{\displaystyle E_{Y}^{\prime }}
is the gamma-corrected voltage of luma.
{\displaystyle E_{R}^{\prime }}
{\displaystyle E_{G}^{\prime }}
{\displaystyle E_{B}^{\prime }}
are the gamma-corrected voltages corresponding to red, green, and blue signals.
{\displaystyle E_{I}^{\prime }}
{\displaystyle E_{Q}^{\prime }}
are the amplitudes of the orthogonal components of the chrominance signal.
To convert from FCC YIQ to RGB:
{\displaystyle E_{R}^{\prime }=E_{Y}^{\prime }+0.9469E_{I}^{\prime }+0.6236E_{Q}^{\prime }}
{\displaystyle E_{G}^{\prime }=E_{Y}^{\prime }-0.2748E_{I}^{\prime }-0.6357E_{Q}^{\prime }}
{\displaystyle E_{B}^{\prime }=E_{Y}^{\prime }-1.1E_{I}^{\prime }+1.7E_{Q}^{\prime }}
From YUV to YIQ and vice versaEdit
{\displaystyle {\begin{bmatrix}Y'\\I\\Q\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&-\sin(33^{\circ })&\cos(33^{\circ })\\0&\cos(33^{\circ })&\sin(33^{\circ })\end{bmatrix}}{\begin{bmatrix}Y'\\U\\V\end{bmatrix}}\approx {\begin{bmatrix}1&0&0\\0&-0.54464&0.83867\\0&0.83867&0.54464\end{bmatrix}}{\begin{bmatrix}Y'\\U\\V\end{bmatrix}}}
Due to Simmetry of the matrix same matrix can be used for YIQ to YUV conversion.[12]
Phase-outEdit
For broadcasting in the United States, it is currently in use only for low-power television stations, as full-power analog transmission was ended by the Federal Communications Commission (FCC) on 12 June 2009. It is still federally mandated for these transmissions as shown in this excerpt of the current FCC rules and regulations part 73 "TV transmission standard":
The equivalent bandwidth assigned prior to modulation to the color difference signals EQ′ and EI′ are as follows:
Q-channel bandwidth: At 400 kHz less than 2 dB down. At 500 kHz less than 6 dB down. At 600 kHz at least 6 dB down.
I-channel bandwidth: At 1.3 MHz less than 2 dB down.
^ "Color Spaces · culori". culorijs.org. Retrieved 27 February 2022.
^ "Built-in Types of Data". introcs.cs.princeton.edu. Retrieved 27 February 2022.
^ "rgb2ntsc: Convert RGB color values to NTSC color space". Image Processing Toolbox Documentation. MathWorks. Retrieved 28 June 2015.
^ "ntsc2rgb: Convert NTSC values to RGB color space". Image Processing Toolbox Documentation. MathWorks. Retrieved 28 June 2015.
^ "ITU-R BT.1700 Characteristics of composite video signals for conventional analogue television systems" (zip/pdf). International Telecommunication Union. 2004-11-30. S170m-2004.pdf: Composite Analog Video Signal NTSC for Studio Applications Page 6. Retrieved 2019-04-16.
^ Society of Motion Picture and Television Engineers (1987–2004): Recommended Practice RP 145-2004. Color Monitor Colorimetry.
^ "TV transmission standards" (PDF). U.S. Government Printing Office. 2013. p. 210. Retrieved 11 July 2014.
^ https://www.itu.int/dms_pubrec/itu-r/rec/bt/R-REC-BT.470-6-199811-S!!PDF-E.pdf#page=9[bare URL PDF]
^ "Chapter 3: Color Spaces" (PDF). Retrieved 2022-03-05.
Buchsbaum, Walter H. Color TV Servicing, third edition. Englewood Cliffs, NJ: Prentice Hall, 1975. ISBN 0-13-152397-X
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Specification - How is RPI ranking its data sets
This document outlines how rug pull index's algorithm is ranking data sets.
§ Data Acquisition
When ranking all data sets sold on Ethereum, the first step is acquiring meta data of all markets on Ocean Protocol and Big Data Protocol. Both projects host a version of the oceanprotocol/market. They take their data from an off-chain meta data cache called Aquarius and a protocol-specific subgraph implementation.
The following sources are used to collect all relevant data to generate a ranking:
From RPI's Ethereum Ergion node, we download the pool's metadata e.g. prices and liquidity.
From the Ocean and BDP subgraph, we get all data token addresses including their names and symbols.
From ethplorer.io and covalent, we download the top token holders for each liquidity pool.
From oceanprotocol/list-purgatory, we download both the list-accounts.json and list-assets.json.
From coingecko, we download the latest price of OCEAN/EUR and BDPToken/EUR.
We call the software that downloads all this meta data the "crawler". In the past, there have been nights where the crawler failed. This was when the main page wasn't showing any reasonable results. When the crawl fails an administrator can manually login to fix the problem by e.g. re-running the crawl.
Once the crawl has finished, we perform a number of calculations that flow into the final ranking.
§ Calculating the Equality of Liquidity Providers in the Pool ("Gini coefficient")
The Ocean Protocol uses Automated Market Makers (or short: "pools") of Balancer's version 1. When staking e.g. QUICRA-0/OCEAN, the balancer pool returns a separate ERC20 token that represents a staker's liquidity in the pool.
As we want to rate a data set's factor of decentralization and hence the potential (or risk) it has of a single user "pulling the rug" - for each data set -, we calculate a Gini coefficent from the pool's population of liquidity providers.
In the following paragraphs, we define the Gini coefficent on an intuitive basis rather than by deriving it from the Lorenz curve [1]. As matrix
M
, we represent the differences of all stake pairs in the pool, with
d_{ij} = | x_i- x_j|
M = \begin{bmatrix} d_{11} & d_{12} & ... & d_{1n} \\ d_{21} & d_{22} & ... & d_{2n} \\ ... & ... & ... & ... \\ d_{n1} & ... & ... & d_{nn} \\ \end{bmatrix}
As a formula, we can represent
M
PD
n
represents the population's size:
PD = \frac{\displaystyle\sum_{i = 1}^n \sum_{j = 1}^n | x_i - x_j |}{n^2}
However, for us to make a statement about the relative inequality of a data set's pool, it requires us to put the absolute difference
PD
in relationship to the a population's worst case inequality. Namely a scenario where one member earns all total income while others earn nothing. We define this case
M_T
T
being the population's total income as:
M_{T} = \begin{bmatrix} 0 & 0 & ... & 0 & T \\ 0 & 0 & ... & 0 & T \\ ... & ... & ... & ...& ... \\ T & T & ... & T & 0\\ \end{bmatrix}
As a formula, we can represent it as
PD_{T}
PD_{T} = \frac{2 (n - 1) T}{n^2}
Where we define the Gini coefficient
G
to be a relation between
PD
PD_{T}
G = \frac{PD}{PD_{T}}
Additionally, since we've discovered an unfair advantage for pools with many small liquidity providers, for data sets with greater than 100 liquidity providers we only consider pool shares of more than 0.1% in the above calculation. For pools with less than 100 LPs, a stake counts towards the Gini coefficient
G
only when its share is more than 1% of the pool.
§ Determining the EURO Value of a Data Set
As outlined in the section "Data Acquisition", a data sets price in OCEAN or BDPToken is downloaded from Acquarius. The latest tickers for OCEAN/EUR and BDPToken/EUR are downloaded from coingecko. The EUR value
v
of an OCEAN data asset which price is
p_{d}
is then calculated by multiplying it by the price of an OCEAN
p_{o}
v = p_{d} \cdot p_{o}
§ Calculating the Daily Ranking of Data Sets
rug pull index fetches and calculates - as previously mentioned - a set of meta data daily. Given an exemplary market of only two data sets
A
B
, over a timespan of two days, a fictional state of rug pull index's data base could look like this:
liquidity (EUR)
A 10 0.5 2021-06-22
B 2 1 2021-06-22
A 5 0.24 2021-06-23
To now calculate a daily ranking, considering the data of both days (June 22 and June 23), rug pull index does the following: In the "liquidity (EUR)" column, the overall highest value ("max value") is selected by searching the data base (it is "10" on June 22). Additionally the smallest Gini coefficient ("min value") is determined ("0.24" on June 23 for). These "min" (Gini coefficient) and "max" (liquidity (EUR)) values are then used as benchmark to evaluate the current day's data sets. Assuming the date to be June 23, 2021 we'd calculate the overall ranking of
A
A = \frac{5}{10} \cdot \frac{0.24}{0.24} \cdot 100 = 50\%
B
B = \frac{7}{10} \cdot \frac{1}{0.24} \cdot 100 = 2,91\%
Hence, the main page's ranking would yield the following table for June 23, 2021:
1 A 50% 5 0.24
2 B 2.91% 7 1
Generally speaking, on a day
d
the rating
r_d
of a data set, can be calculated using the overall minimal Gini coefficient
g_{min}
, the overall maximal liquidity in EUR
l_{max}
, the data set's current liquidity in EUR
l_d
and the data set's current coefficient
g_d
r_d = \frac{l_d}{l_{max}} \cdot \frac{g_d}{g_{min}} \cdot 100
We pledge to update this specification with every update made to rug pull index's algorithm.
2021-09-05: We stopped using Aquarius entirely.
2021-08-09: We stopped retrieving price information from Aquarius and instead now use RPI's Ethereum node.
2021-08-02: We found a more intuitive and, for small populations fitting, derivation of the Gini coefficient from Walter Escudero [1] and replaced it with the original formula from Wikipedia.
2021-06-22: Specification document first published.
§ Feedback and Questions
If you have questions or feedback with regards to this document, please contact us.
ESCUDERO, WALTER, 2018, A NOTE ON A SIMPLE INTERPRETATION OF THE GINI COEFFICIENT. Portal.amelica.org [online]. 2018. [Accessed 2 August 2021]. Available from: http://portal.amelica.org/ameli/jatsRepo/196/196830001/html/index.html
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How hard is it to make a computer find peaks in audio signals?
TL;DR: very hard, it appears.
Feel free to play along with the Jupyter notebook and audio sample used for this experiment:
Jupyter notebook Audio sample (AIFF)
Suppose you have an audio signal and you want to detect peaks in it. Maybe you want to identify the beat in a song or you are trying to parse Morse code. In these situations you have a series of impulses which you can tell are louder than the rest of the audio. Even just by looking at the waveform, it is easy to tell that there is a peak in volume. Is it easy for a computer to do the same?
The pulses we are interested in1 look like the ones above. In this representation we are looking at time on the horizontal axis versus amplitude on the vertical axis. The raw signal is shown in the left graph, but there are so many data points that it almost looks like it has area under it. This is just an artifact produced by the lines that we use to visualize the waveform which join the data points. In the version to the right we have discarded 99% of the samples so that the fact that this is indeed a wave is clearer.
Back to our task of detecting peaks, our first idea is to find the local maxima of the wave function (recall that we are just looking at one peak of many, so we are indeed looking for several maxima). This is relatively easy to do: a sample is a local maximum if and only if it is bigger than the two samples adjacent to itself. Also, before proceeding any further, the function is almost symmetric with respect to the horizontal axis, so we can discard the negative part which will make things easier later 2:
def peaks1(x):
for i, v in enumerate(x[1:-1]):
if x[i] < v and v > x[i + 2]:
# discard negative part
data1 = np.abs(mono)
print(f'{len(peaks1(mono))} out of {nsamples} are peaks')
5921 out of 24863 samples are peaks
This first attempt reveals that around 25% of the samples are considered peaks under this definition of maximum. While this is not very useful in itself, we have uncovered that the signal is indeed very, very wiggly. We can try to smooth it out by computing a moving average and working with that instead. In our context, the audio signal is a sequence of around 25000 values
x_1, x_2, \ldots
A moving average of window size
W \in \mathbb{N}
assigns to each point the average of the last
W
x_m = \frac{1}{W} \sum_{i = m - W}^m x_i \text{ for every } m = 1,\ldots, 25000.
We need to be careful with the first
W
values, since there is not enough history to calculate the average. Similarly, there wont be enough future to calculate the average for the last
W
values. This can be a bit tedious to implement, especially with these last edge cases. We can leverage the fact that convolving the signal with a boxcar function of width
W
1/W
will yield the same results and that NumPy already implements discrete convolutions and properly hadles the aforementioned edge cases.
data2 = np.convolve(data1, np.ones(W), 'valid') / W
In the above plot and the ones that follow, we have also taken the chance and reduced the number of points we are using for the plots, since we have already realised that the data is super wiggly. The original data is still used for the analysis, though.
If we now try to find the peaks we get
That is only a 20% reduction and we have used a window of 1000 samples! Even if we increase the window size (double it, even triple it), we still get thousands of peaks. Increasing the window size is not something desirable, since we can miss peaks that are close together but still “true” peaks that we want to identify. Clearly, convolution is not helpful enough? But… what if we do it once more?
Alright, the peak is slowly shifting to the left, but we don’t care about that so much now, as long as we can find a peak:
31 out of 24863 samples are peaks
Okay, this has improved dramatically, but we still get 31 peaks when we would want to get just one. There must be a better way to do this but for the sake of completeness let’s convolve a third time:
3 out of 24863 samples are peaks
While we are getting somewhere, this is taking way too much effort for our simple test with just one peak. We are still not able to count just one big peak. However, we are throwing away a ton of information: how big are these peaks, how far apart are they… Let’s look at the three biggest peaks reported after each convolution:
1st convolution: [(8046, 5412.995), (8056, 5403.334), (8033, 5394.965)]
2nd convolution: [(7640, 4603.345313), (17143, 245.45059499999996), (18329, 219.05746100000002)]
3rd convolution: [(7174, 4303.5119952939995), (16566, 241.88337264500004), (2165, 189.47858628800003)]
In the above output, each peak is represented by a tuple (time offset, peak height). There is something very interesting going on:
As we expected, there is a shift to the left in the highest reported peak. This is an artifact of the convolution and will not be a problem if we just want to count how many peaks there are.
The first convolution barely yields any results. For one, the first three peaks are equally high and very close together. Of course this tells us that there is a peak around there, but not how it relates to the rest of the signal without looking at the whole of the peak list.
The second convolution clearly calls a dominant peak. The other two who made it to the top three are both far apart and much lower.
Although the last (third) convolution reduces the number of peaks, it probably is not worth it since after we detect a dramatic increase in peak height from the first to the second detected peak after the second convolution we already have a candidate for mount Everest.
One possible reason for the dramatic increase in peak discrimination between the first and the second convolution is that data is much less wiggly (even though it still oscillates quite a lot) after the first convolution. This leads us to the question: do we need such a big window size for the second convolution? The answer appears to be no. Choosing a window size of 50 samples (as opposed to the 1000 we took for the first convolution) yields very similar results but with much less left-shift in the time coordinate where the peak is called:
2nd convolution, smaller window: [(8014, 5386.55954), (10938, 926.6066200000001), (10927, 926.48548)]
This first look at the problem tells us that we cannot use the definition of a maximum common in Maths to find peaks in a waveform right away (precisely because it’s a wave and we are looking for bigger scale features). It also tells us that the problem can be greatly simplified by reducing the wiggly-ness of the data (via convolution or some other method) and that we may not need to go all the way until we find a single peak in a desired region if we introduce some notion of peak prominence. A more detailed characterization of this concept of prominence has the potential to reduce the number and scale of convolutions needed to be able to call peaks. Also, we have encountered an artifact of this method where the time at which a peak is called shifts to the left with the number of convolutions performed. This is due to the asymmetric nature of the peaks we are dealing with, which may also prove to be determinant in our quest for peaks in audio signals.
P.S. Today is Koningsdag here in The Netherlands but sadly we cannot celebrate because of corona. At least Mathematics is always there with you.
This is not just one random pulse. The other day I had an idea which involves counting pops in audio and fitting the time it takes for the things to pop into a normal distribution. I hate eating raw corn. ↩
A more rigorous approach would have calculated the RMS value for tiny slices to reflect sound pressure more accurately. ↩
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powlog - Maple Help
Home : Support : Online Help : Mathematics : Basic Mathematics : Logarithms : powlog
logarithm of an expression
powlog(p)
either formal power series, or polynomial, or function that is acceptable for power series function evalpow
The function powlog(p) returns the formal power series that is the natural logarithm of p.
The power series p must have a nonzero first term (
p\left(0\right)\ne 0
The command with(powseries,powlog) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{powseries}\right):
t≔\mathrm{powpoly}\left(1+x,x\right):
s≔\mathrm{powlog}\left(t\right):
\mathrm{tpsform}\left(s,x,7\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{7}}\right)
u≔\mathrm{powlog}\left(\mathrm{powdiff}\left(\mathrm{powexp}\left(x\right)\right)\right):
\mathrm{tpsform}\left(u,x,6\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\right)
v≔\mathrm{powlog}\left(\mathrm{powadd}\left(\mathrm{powpoly}\left(1+x,x\right),\mathrm{powexp}\left(x\right)\right)\right):
\mathrm{tpsform}\left(v,x,6\right)
\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{32}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{7}}{\textcolor[rgb]{0,0,1}{120}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{O}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\right)
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Solve System of Linear Equations - MATLAB & Simulink - MathWorks Switzerland
\begin{array}{l}{a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\dots +{a}_{1n}{x}_{n}={b}_{1}\\ {a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\dots +{a}_{2n}{x}_{n}={b}_{2}\\ \cdots \\ {a}_{m1}{x}_{1}+{a}_{m2}{x}_{2}+\dots +{a}_{mn}{x}_{n}={b}_{m}\end{array}
A\cdot \stackrel{\to }{x}=\stackrel{\to }{b}
, where A is the coefficient matrix,
A=\left(\begin{array}{ccc}{a}_{11}& \dots & {a}_{1n}\\ ⋮& \ddots & ⋮\\ {a}_{m1}& \cdots & {a}_{mn}\end{array}\right)
\stackrel{\to }{b}
is the vector containing the right sides of equations,
\stackrel{\to }{b}=\left(\begin{array}{c}{b}_{1}\\ ⋮\\ {b}_{m}\end{array}\right)
\begin{array}{l}2x+y+z=2\\ -x+y-z=3\\ x+2y+3z=-10\end{array}
\begin{array}{l}2x+y+z=2\\ -x+y-z=3\\ x+2y+3z=-10\end{array}
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Output Function for Problem-Based Optimization - MATLAB & Simulink - MathWorks Switzerland
Problem-Based Setup
Run Optimization Using Output Function
This example shows how to use an output function to plot and store the history of the iterations for a nonlinear problem. This history includes the evaluated points, the search directions that the solver uses to generate points, and the objective function values at the evaluated points.
For the solver-based approach to this example, see Output Functions for Optimization Toolbox.
Plot functions have the same syntax as output functions, so this example also applies to plot functions.
For both the solver-based approach and the problem-based approach, write the output function according to the solver-based approach. In the solver-based approach, you use a single vector variable, usually denoted x, instead of a collection of optimization variables of various sizes. So to write an output function for the problem-based approach, you must understand the correspondence between your optimization variables and the single solver-based x. To map between optimization variables and x, use varindex. In this example, to avoid confusion with an optimization variable named x, use "in" as the vector variable name.
The problem is to minimize the following function of variables x and y:
f=\mathrm{exp}\left(x\right)\left(4{x}^{2}+2{y}^{2}+4xy+2y+1\right).
In addition, the problem has two nonlinear constraints:
\begin{array}{l}x+y-xy\ge 1.5\\ xy\ge -10.\end{array}
To set up the problem in the problem-based approach, define optimization variables and an optimization problem object.
Define the objective function as an expression in the optimization variables.
f = exp(x)*(4*x^2 + 2*y^2 + 4*x*y + 2*y + 1);
Include the objective function in prob.
To include the nonlinear constraints, create optimization constraint expressions.
cons1 = x + y - x*y >= 1.5;
cons2 = x*y >= -10;
Because this is a nonlinear problem, you must include an initial point structure x0. Use x0.x = –1 and x0.y = 1.
The outfun output function records a history of the points generated by fmincon during its iterations. The output function also plots the points and keeps a separate history of the search directions for the sqp algorithm. The search direction is a vector from the previous point to the next point that fmincon tries. During its final step, the output function saves the history in workspace variables, and saves a history of the objective function values at each iterative step.
For the required syntax of optimization output functions, see Output Function and Plot Function Syntax.
An output function takes a single vector variable as an input. But the current problem has two variables. To find the mapping between the optimization variables and the input variable, use varindex.
idx.x
idx.y
The mapping shows that x is variable 1 and y is variable 2. So, if the input variable is named in, then x = in(1) and y = in(2).
type outfun
function stop = outfun(in,optimValues,state,idx)
persistent history searchdir fhistory
fhistory = [];
% value with history. in must be a row vector.
fhistory = [fhistory; optimValues.fval];
history = [history; in(:)']; % Ensure in is a row vector
optimValues.searchdirection(:)'];
plot(in(idx.x),in(idx.y),'o');
% Add .15 to idx.x to separate label from plotted 'o'
text(in(idx.x)+.15,in(idx.y),...
assignin('base','optimhistory',history);
assignin('base','searchdirhistory',searchdir);
assignin('base','functionhistory',fhistory);
Include the output function in the optimization by setting the OutputFcn option. Also, set the Algorithm option to use the 'sqp' algorithm instead of the default 'interior-point' algorithm. Pass idx to the output function as an extra parameter in the last input. See Passing Extra Parameters.
outputfn = @(in,optimValues,state)outfun(in,optimValues,state,idx);
opts = optimoptions('fmincon','Algorithm','sqp','OutputFcn',outputfn);
Run the optimization, including the output function, by using the 'Options' name-value pair argument.
[sol,fval,eflag,output] = solve(prob,x0,'Options',opts)
Examine the iteration history. Each row of the optimhistory matrix represents one point. The last few points are very close, which explains why the plotted sequence shows overprinted numbers for points 8, 9, and 10.
disp('Locations');disp(optimhistory)
Examine the searchdirhistory and functionhistory arrays.
disp('Search Directions');disp(searchdirhistory)
disp('Function Values');disp(functionhistory)
If your objective function or nonlinear constraint functions are not composed of elementary functions, you must convert the functions to optimization expressions using fcn2optimexpr. See Convert Nonlinear Function to Optimization Expression. For this example, you would enter the following code:
fun = @(x,y)exp(x)*(4*x^2 + 2*y^2 + 4*x*y + 2*y + 1);
f = fcn2optimexpr(fun,x,y);
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Telegram Open Network - Wikipedia
Blockchain network from Telegram
TON, Telegram Open Network
newton-blockchain, Telegram (no longer participation)
For access to wallets uses 24 secret words, on the basis of which calculates 256-bit private key of the wallet
The Open Network (previous Telegram Open Network, both abb. as TON) is a blockchain-based decentralized computer network technology, originally developed by the Telegram team. In May 2020, Telegram withdrew from the project after litigation with the US Securities and Exchange Commission.
1.1 Gram Private Sale
1.3 SEC v. Telegram
1.4 Post Telegram
Since the release of Telegram messenger in 2013, its CEO Pavel Durov emphasized that instant messenger will not include advertising. According to documents related to U.S. Securities and Exchange Commission (SEC) v. Telegram suit (2020), by 2017, the self-funded startup needed money to pay for servers and services. Durov considered venture capital financing but decided against it until the problems are solved.[1] In the mid-December 2017 Bloomberg interview, Durov announced that Telegram would begin monetization in early 2018.[2] TechCrunch had soon confirmed that the company planned launch a blockchain project named “The Open Network” or “Telegram Open Network” (TON) and its native cryptocurrency “Gram”.[3]
In January 2018, a 23-page white paper[4] and a detailed 132-page technical paper[5] shed some light on the project. According to documents, the Durovs planned to attract the existing Telegram user base to TON and promote the mass adoption of cryptocurrencies, turning it into one of the largest blockchains. TON was described as a platform for decentralized apps and services akin to WeChat, Google Play, or App Store,[6][7] or even a decentralized alternative to payment processing services of Visa and MasterCard due to its ability to scale and support millions of transactions per second.[8][9][10] The codebase for TON was created by Nikolai Durov, the developer of Telegram's MTProto protocol, and Pavel became the public figure for the project.[10]
Gram Private Sale[edit]
To fund the development of the messenger and the blockchain project, Telegram attracted investments through a private Gram offering.[11] A two-step legal scheme was employed: Gram purchase agreement was structured as a future contract that allowed investors to receive tokens once TON is launched. As futures were only sold to accredited investors, the offering was exempt from registration as securities under Regulation D of the Securities Act of 1933, on the basis that once TON was operational, the Grams would have utility and would not be considered securities.[12] The contract was drafted by major U.S. law firm Skadden.[13]
The offering ran in two rounds, each taking in $850 million. Russian tech news site calculated the first round as offering 2.25 billion tokens at $0.38 each, with a minimum purchase of $20 million, and the second as offering 640 million tokens at $1.33 each, with a minimum purchase of $1 million.[14][15] As of April 2018, the firm had reported raising $1.7 billion through the ongoing ICO.[16] Company officials have not made any public statements regarding the ICO. As of spring 2018, the only official sources of information were the two Forms D that Telegram filed with the U.S. Securities and Exchange Commission.[17]
The development of TON took place in a completely isolated manner and was opaque for the whole time.[8] The launch of the test network was initially scheduled for Q2 2018 with the main network launch in Q4, but the milestones were postponed several times. The testnet was launched in January 2019 with a half a year deviation from the plan.[18] In May the company released the lite version of the TON blockchain network client.[19] In September the company released the complete source code for TON nodes on GitHub, making it possible to launch a full node and explore the testnet.[20][21] The launch of the TON main network was scheduled for October 31.[22][23]
SEC v. Telegram[edit]
SEC had concerns about Gram private sale and contacted Telegram shortly after, but over the 18 months of communication, the sides didn't come to understanding.[citation needed] On October 11, 2019, a couple of weeks before the planned TON launch, SEC obtained a temporary restrictive order to prevent the distribution of Grams. The regulator contended that initial Gram purchasers would be acting akin to underwriters, and the resale of Grams, once distributed, will be an unregistered distribution of securities.[24]
After a lengthy legal battle between Telegram and the SEC, Judge P. Kevin Castel of the U.S. District Court for the Southern District of New York agreed with the SEC's vision[25] that the sale of Grams, the distribution to initial purchasers, and the highly likely future resale should be viewed as a single "scheme" to distribute Grams to secondary market in an unregistered security offering. The "security" at issue consisted of the whole set of contracts, expectations, and understandings around the sale and the distribution of tokens to interested public, and not the Grams themselves. In this case the initial purchasers acted as underwriters that planned to resell tokens, and not to consume them. The restrictions on Gram distribution remained in force for purchasers based in and outside U.S., as Telegram had no tools to prevent U.S. citizens from purchasing Grams on secondary market.[26][27][24]
Following the court decision, the Telegram Open Network team announced that they would be unable to launch the project by the expected 30 April 2020 deadline. On 12 May 2020, Pavel Durov announced the end of Telegram's active involvement with TON.[28][29] On June 11, Telegram settled with SEC U.S. Securities & Exchange Commission and agreed to return $1.22 billion as "termination amounts" in Gram purchase agreements, and pay an $18.5 million penalty to SEC. The company also agreed to notify SEC of any plans to issue digital assets over the next three years. The judge approved the settlement on June 26. "New and innovative businesses are welcome to participate in our capital markets but they cannot do so in violation of the registration requirements of the federal securities laws," the SEC spokesperson noted. Telegram had subsequently shut down TON test network[30]
In 2020, Telegram repaid TON investors $770 million and converted $443 million into a one-year debt at 10% interest, raising its total liabilities to $625.7 million. On 10 March 2021, the company made the placement of 5-year bonds worth $1 billion to cover the debts it had to return by 30 April 2021.[31][32]
Post Telegram[edit]
By May 2020, when Telegram's commitment to the project was unclear, other projects started to develop the technology.[33]
Launched on May 7, 2020,[34] Free TON is a project to continue development of Telegram Open Network technology using the source code freely available under GNU GPL.[35] By January 2021, the community of the project reached 30,000 people.[36] Free TON's token titled "TON Crystal" or just "TON" is distributed as a reward for contributions to the network.[37] Of five billion tokens issued at the moment of launch, 85% were reserved for users, 5% for validators, and 10% for the developers (including 5% dedicated for TON Labs, the developer of TON OS middleware for TON blockchain, which is the essential part of Free TON).[38][39][34]
On June 29, 2021, the Telegram team said they would consider granting use of the ton.org domain and GitHub repository to the TON Foundation, in response to an open letter. By August 4 the domain was transferred to the TON Foundation.[40][41]
TON was designed to enable processing millions of transactions every second. It also allows building Web3 empowered by decentralized storage, DNS equivalent, instant payments and decentralized services (or DApps).
TON was planned to be a network of blockchains of different types and aims at achieving increased scalability. It was hoped to shard into up to
{\displaystyle 2^{60}}
shard blockchains (shardchains) which themselves would be the shards of
{\displaystyle 2^{32}}
working blockchains (workchains) all validated and committed into one master blockchain (masterchain).
^ "Plaintiff Securities and Exchange Commission's memorandum of law in support of its motion for summary judgement" (PDF). January 13, 2020.
^ Stepan Kravchenko, Nour Al Ali, and Ilya Khrennikov (11 December 2017). "This $5 Billion Encrypted App Isn't for Sale at Any Price". Bloomberg. Retrieved 8 March 2021. {{cite web}}: CS1 maint: multiple names: authors list (link)
^ Butcher, Mike. "Telegram plans multi-billion dollar ICO for chat cryptocurrency". TechCrunch. Retrieved 12 January 2018.
^ "TON.pdf". Google Docs.
^ Mike Butcher, Josh Konstine (January 8, 2018). "Telegram plans multi-billion dollar ICO for chat cryptocurrency". TechCrunch. Retrieved 4 April 2020.
^ Jon Russel (January 15, 2018). "Inside Telegram's ambitious $1.2B ICO to create the next Ethereum". TechCrunch. Retrieved 4 April 2020.
^ a b "Exploring Telegram Open Network". Binance Research. September 27, 2019. Retrieved 4 April 2020.
^ Josh Nadeau (January 15, 2020). "Telegram's Pavel Durov in court over his quest to revolutionize cryptocurrency". Russia Beyond The Headlines. Retrieved 4 April 2020.
^ a b "TON & Gram: how it started and why it ended". PaySpace Magazine. October 9, 2020. Retrieved 5 April 2021.
^ Mix. "Here is the leaked white paper for the massive Telegram ICO". HardFork. Retrieved 6 April 2018.
^ Rachel McIntosh (October 18, 2019). "SEC vs. Telegram: Will Gram Tokens Ever Be Distributed?". Finance Magnates. Retrieved 5 April 2021.
^ Cali Haan (August 20, 2019). "Unauthorized Telegram Token Secondary Market Transferring Risk". Crowdfund Insider. Retrieved 5 April 2021.
^ Korolev, Igor (2 April 2018). "Under what conditions will Telegram return to investors the collected $ 2.55 billion". CNews (in Russian). Retrieved 12 October 2018.
^ Liptak, Andrew (1 April 2018). "Telegram has raised a total of $1.7 billion from its two pre-ICO sales". The Verge. Retrieved 12 October 2018.
^ Shen, Lucinda (31 March 2018). "Even as Bitcoin Languishes, Telegram Raises $1.7 Billion Ahead of Largest ICO Ever". Fortune. Retrieved 31 March 2018.
^ "Telegram Files New Form D with SEC Indicating $1.7 Billion will be Raised | Crowdfund Insider". Crowdfund Insider. 2 April 2018. Retrieved 6 April 2018.
^ "Despite delay, Durov on track to launch his cryptocurrency in March". The Bell. January 23, 2019. Retrieved 17 March 2021.
^ Denis Olshansky (27 May 2019). "Тестовый клиент TON (Telegram Open Network) и новый язык Fift для смарт-контрактов" [TON test client and FIft language for smart contracts] (in Russian). Habr. Retrieved 7 June 2021.
^ "Telegram может начать публичное тестирование своего блокчейна 1 сентября" [Telegram may start testing its blockchain on September 1st] (in Russian). Vedomosti. 28 August 2019. Retrieved 7 June 2021.
^ Vladislav Voytenko (7 September 2019). "Исходный код TON Blockchain теперь доступен на GitHub" [TON blockchain source code now available on GitHub] (in Russian). Kod. Retrieved 7 June 2021.
^ Sean Hollister (27 August 2019). "Telegram will launch its Gram cryptocurrency by October 31 or bust". The Verge. Retrieved 21 March 2021.
^ Mikhail Tetkin (26 June 2020). "Самолет в никуда. Весь путь развития Telegram Open Network" [A plane to nowhere. The history of Telegram Open Network] (in Russian). RBC. Retrieved 7 June 2021.
^ a b Robert A. Schwinger (May 26, 2020). "Blockchain law. A "Telegram" to SAFTs: "Beware!"" (PDF). New York Law Journal. Retrieved 15 March 2021.
^ "SEC v. Telegram: Key Takeaways and Implications". www.cooley.com. Retrieved 13 May 2020.
^ "Is There Life for SAFTs After the Telegram Case?". JD Supra. July 21, 2020. Retrieved 10 March 2021.
^ David I. Miller, Charlie Berk (2020-04-01). "SEC v. Telegram: A Groundbreaking Decision in Cryptocurrency Enforcement?". Greenberg Traurig. Retrieved 10 March 2021.
^ "What Was TON And Why It Is Over". Pavel Durov. 12 May 2020. Retrieved 12 May 2020.
^ Durov, Pavel (2020-05-12). "What Was TON And Why It Is Over". Telegraph – Pavel Durov. Retrieved 2020-05-22.
^ "Telegram to pay SEC fine of $18.5 million and return $1.2 billion to investors as it dissolves TON". Techcrunch. 26 June 2020. Retrieved 28 June 2020.
^ Павел Казарновский, Владислав Скобелев, Анна Балашова (15 March 2021). "Telegram разместил бонды на $1 млрд. На каких условиях инвесторы рискнули вложиться во внебиржевые бумаги" [What were the terms of Telegram's $1 billion bonds placement] (in Russian). RBC. Retrieved 15 March 2021. {{cite news}}: CS1 maint: multiple names: authors list (link)
^ Schechner, Sam (2021-03-15). "Telegram App Is Booming but Needs Advertisers—and $700 Million Soon". Wall Street Journal. ISSN 0099-9660. Retrieved 2021-03-16.
^ Ivan Sychev (14 January 2021). "Free TON вырос до $200 млн и планирует дойти до $1 млрд" [Free TON capitalisation reached $200 million and aims for $1 billion] (in Russian). VC.ru. Retrieved 8 June 2021.
^ a b "Недооцененный блокчейн: как разработчики платформы TON запустили ее под другим названием и привлекли $6 млн" [Underrated blockchain: how the developers of TON launched it under the other name and raised $6 million] (in Russian). Forbes. 20 May 2021. Retrieved 30 May 2021.
^ "ton/ton-blockchain". GitHub. Retrieved 8 June 2021.
^ Vladislav Skobelev, Anastasia Skrinnikova (7 May 2020). "Блокчейн-платформу TON запустили без Павла Дурова и Telegram. Но без аудитории мессенджера проект рискует остаться нишевым" [TON blockchain was launched without Pavel Durov and Telegram. But without Telegram users it risks to remain a niche product] (in Russian). RBC. Retrieved 7 June 2021.
^ Elyas Kasmi (8 May 2020). "Основателя Telegram не взяли в его собственный блокчейн-проект" [Telegram founder wasn't accepted into his own blockchain project] (in Russian). CNews. Retrieved 7 June 2021.
^ Mikhail Kabanov (22 May 2021). "TON: каким стал один из самых громких блокчейн-проектов без Дурова" [TON: What one of the most famous blockchain projects became without Pavel Durov] (in Russian). RBC. Retrieved 30 May 2021.
^ "Telegram response to an open letter". 2021-06-29.
^ "TON Legacy: How Telegram's Crypto Coin Lives On". Nasdaq. 2021-08-17.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Telegram_Open_Network&oldid=1079110040"
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An iron bar (L1 = 0 1 m, A1= 0 02 m2 , K1 = 80 W m 1 K - Physics - Mechanical Properties Of Solids - 16911833 | Meritnation.com
An iron bar (L1 = 0.1 m, A1= 0.02 m2
, K1 = 80 W m?1 K?1) and a brass bar (L2 = 0.1 m, A2 = 0.02
, K2 = 120 W m?1K?1) are soldered end to end. The free ends of the iron bar and brass bar are
maintained at 400K and 300 K respectively. What is the the temperature of the junction of the two
Let emperature of the junction be {T}_{0}. As both the bars are connected in series so heat current through them is equal .\phantom{\rule{0ex}{0ex}}Heat current through iron rod =\frac{{K}_{iron}{A}_{iron}\left({T}_{1}-T\right)}{{L}_{1}} \phantom{\rule{0ex}{0ex}}=\frac{{K}_{iron}{A}_{iron}\left(400-T\right)}{0.1}=\frac{80×0.02\left(400-T\right)}{0.1}\phantom{\rule{0ex}{0ex}}Heat current through brass rod =\frac{{K}_{brass}{A}_{brass}\left(T-{T}_{2}\right)}{{L}_{2}}=\phantom{\rule{0ex}{0ex}}\frac{{K}_{brass}{A}_{brass}\left(T-300\right)}{0.1}=\frac{120×0.02×\left(T-300\right)}{0.1}\phantom{\rule{0ex}{0ex}}So\phantom{\rule{0ex}{0ex}}\frac{80×0.02\left(400-T\right)}{0.1} =\frac{120×0.02×\left(T-300\right)}{0.1}\phantom{\rule{0ex}{0ex}}80\left(400-T\right) =120\left(T-300\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{\left(400-T\right)}{\left(T-300\right)}=\frac{120}{80}=1.5\phantom{\rule{0ex}{0ex}}400- T=1.5 T-450\phantom{\rule{0ex}{0ex}}950= 2.5 T\phantom{\rule{0ex}{0ex}}T =380 K\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
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Functional analysis - Wikiversity
This is translation Work in Progress based on the german Functional Analysis course as pilot of Wiki2Reveal translation of a lecture into another language. At the same time the Wikibook about Functional Analysis provides already contain from the mathematical domain. The challenge is to combine the concept of template based generation of content in Wikibooks and Wiki2Reveal-slide in a join concept of workflow, so that a mathematical theory (Lemma of Zorn) can be (non compulsory) maintained at a single place in a MediaWiki and an update or correction e.g. in a mathematical expression is visible in the WikiBook, in Wikiversity and in the dynamic generation of slides. Wiki2Reveal and the corresponding Wikiversity learning resource are already in sync and if the underlying Wikiversity article was updated or small typo was corrected then the next start of the Wiki2Reveal slide the show the correction. Furthermore mathematical expressions can be language agnostic, which means that the mathematical expression might be used in multiple languages without any alteration, i.e. the following formula looks the same in the english, german, italian and french language.
{\displaystyle \sum _{k=1}^{n}={\frac {n\cdot (n+1)}{2}}}
The reference might require to use an ID in the math-tags for the mathematical expression and even a reference to a permanent link might be available. It would be kind to keep the section "Wiki2Reveal - Versions" for provided alternative previews of the mathematical content and the regular chapter structure as usable for a real application in lecture setting simular to Wiki2Reveal-pilot in the german Wikiversity course for Functional Analysis.
1 Chapter 1 - Introduction - Basics
2 Wiki2Reveal - Versions
3 Use of materials for lectures
4 Origin of materials
6 Wiki Books
Chapter 1 - Introduction - Basics[edit | edit source]
Wiki2Reveal - Versions[edit | edit source]
Use of materials for lectures[edit | edit source]
The lecture is provided in a PanDoc slide format (PanDocElectron-SLIDE) in Wikiversity, which can be transferred into annotatable slides using the Wiki2Reveal tool or using PanDocElectron to load the Wikiversity source available online and convert it into presentation slides that can be used offline. You can also use Wiki2Reveal to create a RevealJS or DZSlides presentation directly from Wikiversity articles.
Origin of materials[edit | edit source]
In the spirit of OER (Open Educational Resources), the lecture content should be made freely available. Initially, the slides created from the customizable wiki content were made available in a GitHib repository to facilitate download and use. However, maintaining and updating the content in a repository is very costly. Therefore, Wiki2Reveal was developed for the lecture slides, which allows to generate lecture slides directly from the wiki content and to annotate this online as well. The articles are usually annotated with as little text as possible, so that the generation of the slides also does not exceed the space on a slide. All slide pages in Wikiversity therefore have a note PanDocElectron-SLIDE at the bottom of the pages and are assigned to the Wiki2Reveal category. When editing these pages, please be careful not to overfill the slides. More detailed text about the slides is usually created in separate articles. If the explanation pages explicitly refer to a slide, the explanation page gets a marker PanDocElectron-TEXT and SLIDE or TEXT version refer to each other reciprocally.
Course:Topological Criteria of Regularity - german source for translation
Course:Spatial modeling - deterministic contact models
Definition: metric
Quiz for lecture content
Video conferencing/Oral exam
Tailored WikiBook
Wiki Books[edit | edit source]
Retrieved from "https://en.wikiversity.org/w/index.php?title=Functional_analysis&oldid=2368132"
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54E17 Nearness spaces
p
M
\sigma
54E55 Bitopologies
A few topological problems
Gary Gruenhage, Paul Szeptycki (2011)
We introduce a two player topological game and study the relationship of the existence of winning strategies to base properties and covering properties of the underlying space. The existence of a winning strategy for one of the players is conjectured to be equivalent to the space have countable network weight. In addition, connections to the class of D-spaces and the class of hereditarily Lindelöf spaces are shown.
A lattice of conditions on topological spaces II
P. Moody, G. Reed, A. Roscoe, P. Collins (1991)
A note about
{M}_{1}
-spaces and stratifiable spaces
Jurij H. Bregman (1983)
A note on k-c-semistratifiable spaces and strong
\beta
Li-Xia Wang, Liang-Xue Peng (2011)
Recall that a space
X
is a c-semistratifiable (CSS) space, if the compact sets of
X
{G}_{\delta }
-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a
{T}_{2}
X
is a k-c-semistratifiable space if and only if
X
g
function which satisfies the following conditions: (1) For each
x\in X
\left\{x\right\}=\bigcap \left\{g\left(x,n\right):n\in ℕ\right\}
g\left(x,n+1\right)\subseteq g\left(x,n\right)
n\in ℕ
. (2) If a...
A note on monotone countable paracompactness
Ge Ying, Chris Good (2001)
We show that a space is MCP (monotone countable paracompact) if and only if it has property
\left(*\right)
, introduced by Teng, Xia and Lin. The relationship between MCP and stratifiability is highlighted by a similar characterization of stratifiability. Using this result, we prove that MCP is preserved by both countably biquotient closed and peripherally countably compact closed mappings, from which it follows that both strongly Fréchet spaces and q-space closed images of MCP spaces are MCP. Some results on...
A remark on the weak topology of the Hilbert space
Małgorzata Wójcicka (1987)
Another generalization of the Dugundji extension theorem
Carlos R. Borges (1988)
\delta
-stratifiable spaces
Kedian Li (2009)
Characterizations of σ-spaces
R. Heath, Richard Hodel (1973)
Concerning certain minimal cover refineable spaces
U. Christian (1972)
Convexité topologique et prolongement des fonctions continues
R. Cauty (1973)
Correction to the paper: Pairwise monotonically normal spaces
Josefa Marin, Salvador Romaguera (1992)
Correction to the paper: Sets invariant under projections onto one dimensional subspaces
Simon Fitzpatrick, Bruce Calvert (1992)
Decreasing (G) spaces
Ian Stares (1998)
We consider the class of decreasing (G) spaces introduced by Collins and Roscoe and address the question as to whether it coincides with the class of decreasing (A) spaces. We provide a partial solution to this problem (the answer is yes for homogeneous spaces). We also express decreasing (G) as a monotone normality type condition and explore the preservation of decreasing (G) type properties under closed maps. The corresponding results for decreasing (A) spaces are unknown.
Carlos Borges (1978)
Olena Karlova, Volodymyr Maslyuchenko, Volodymyr Mykhaylyuk (2012)
We investigate the Baire classification of mappings f: X × Y → Z, where X belongs to a wide class of spaces which includes all metrizable spaces, Y is a topological space, Z is an equiconnected space, which are continuous in the first variable. We show that for a dense set in X these mappings are functions of a Baire class α in the second variable.
Hyperspaces of CW-complexes
Bao-Lin Guo, Katsuro Sakai (1993)
It is shown that the hyperspace of a connected CW-complex is an absolute retract for stratifiable spaces, where the hyperspace is the space of non-empty compact (connected) sets with the Vietoris topology.
Les stratifiés topologiques et les polyèdres
Laurent Siebenmann (1973)
On conjecture que certains espaces localement étoilés admettent toujours une jolie stratification naturelle, et deviennent ainsi ce qu’on appelle des
CS
ensembles. On cite quelques propriétés agréables des
CS
ensembles, et quelques exemples exotiques qui distinguent les
CS
ensembles, les espaces triangulables, et les espaces localement triangulables.
Metrizability, generalized metric spaces and
g
Jun-iti Nagata (1988)
|
No-cloning_theorem Knowpia
In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park,[1] in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by Wootters and Zurek[2] as well as Dieks[3] the same year). The aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem (as generally understood) concerns only pure states whereas the generalized statement regarding mixed states is known as the no-broadcast theorem.
The no-cloning theorem has a time-reversed dual, the no-deleting theorem. Together, these underpin the interpretation of quantum mechanics in terms of category theory, and, in particular, as a dagger compact category.[4][5] This formulation, known as categorical quantum mechanics, allows, in turn, a connection to be made from quantum mechanics to linear logic as the logic of quantum information theory (in the same sense that intuitionistic logic arises from Cartesian closed categories).
According to Asher Peres[6] and David Kaiser,[7] the publication of the 1982 proof of the no-cloning theorem by Wootters and Zurek[2] and by Dieks[3] was prompted by a proposal of Nick Herbert[8] for a superluminal communication device using quantum entanglement, and Giancarlo Ghirardi[9] had proven the theorem 18 months prior to the published proof by Wootters and Zurek in his referee report to said proposal (as evidenced by a letter from the editor[9]). However, Ortigoso[10] pointed out in 2018 that a complete proof along with an interpretation in terms of the lack of simple nondisturbing measurements in quantum mechanics was already delivered by Park in 1970.[1]
Theorem and proofEdit
Suppose we have two quantum systems A and B with a common Hilbert space
{\displaystyle H=H_{A}=H_{B}}
. Suppose we want to have a procedure to copy the state
{\displaystyle |\phi \rangle _{A}}
of quantum system A, over the state
{\displaystyle |e\rangle _{B}}
of quantum system B, for any original state
{\displaystyle |\phi \rangle _{A}}
(see bra–ket notation). That is, beginning with the state
{\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}}
, we want to end up with the state
{\displaystyle |\phi \rangle _{A}\otimes |\phi \rangle _{B}}
. To make a "copy" of the state A, we combine it with system B in some unknown initial, or blank, state
{\displaystyle |e\rangle _{B}}
{\displaystyle |\phi \rangle _{A}}
, of which we have no prior knowledge.
The state of the initial composite system is then described by the following tensor product:
{\displaystyle |\phi \rangle _{A}\otimes |e\rangle _{B}.}
(in the following we will omit the
{\displaystyle \otimes }
symbol and keep it implicit).
There are only two permissible quantum operations with which we may manipulate the composite system:
We can perform an observation, which irreversibly collapses the system into some eigenstate of an observable, corrupting the information contained in the qubit(s). This is obviously not what we want.
Alternatively, we could control the Hamiltonian of the combined system, and thus the time-evolution operator U(t), e.g. for a time-independent Hamiltonian,
{\displaystyle U(t)=e^{-iHt/\hbar }}
. Evolving up to some fixed time
{\displaystyle t_{0}}
yields a unitary operator U on
{\displaystyle H\otimes H}
, the Hilbert space of the combined system. However, no such unitary operator U can clone all states.
The no-cloning theorem answers the following question in the negative: Is it possible to construct a unitary operator U, acting on
{\displaystyle H_{A}\otimes H_{B}=H\otimes H}
, under which the state the system B is in always evolves into the state the system A is in, regardless of the state system A is in?
Theorem: There is no unitary operator U on
{\displaystyle H\otimes H}
such that for all normalised states
{\displaystyle |\phi \rangle _{A}}
{\displaystyle |e\rangle _{B}}
{\displaystyle H}
{\displaystyle U(|\phi \rangle _{A}|e\rangle _{B})=e^{i\alpha (\phi ,e)}|\phi \rangle _{A}|\phi \rangle _{B}}
{\displaystyle \alpha }
{\displaystyle \phi }
{\displaystyle e}
The extra phase factor expresses the fact that a quantum-mechanical state defines a normalised vector in Hilbert space only up to a phase factor i.e. as an element of projectivised Hilbert space.
To prove the theorem, we select an arbitrary pair of states
{\displaystyle |\phi \rangle _{A}}
{\displaystyle |\psi \rangle _{A}}
{\displaystyle H}
. Because U is supposed to be unitary, we would have
{\displaystyle \langle \phi |\psi \rangle \langle e|e\rangle \equiv \langle \phi |_{A}\langle e|_{B}|\psi \rangle _{A}|e\rangle _{B}=\langle \phi |_{A}\langle e|_{B}U^{\dagger }U|\psi \rangle _{A}|e\rangle _{B}=e^{-i(\alpha (\phi ,e)-\alpha (\psi ,e))}\langle \phi |_{A}\langle \phi |_{B}|\psi \rangle _{A}|\psi \rangle _{B}\equiv e^{-i(\alpha (\phi ,e)-\alpha (\psi ,e))}\langle \phi |\psi \rangle ^{2}.}
Since the quantum state
{\displaystyle |e\rangle }
is assumed to be normalized, we thus get
{\displaystyle |\langle \phi |\psi \rangle |^{2}=|\langle \phi |\psi \rangle |.}
{\displaystyle |\langle \phi |\psi \rangle |=1}
{\displaystyle |\langle \phi |\psi \rangle |=0}
. Hence by the Cauchy–Schwarz inequality either
{\displaystyle \phi =e^{i\beta }\psi }
{\displaystyle \phi }
{\displaystyle \psi }
. However, this cannot be the case for two arbitrary states. Therefore, a single universal U cannot clone a general quantum state. This proves the no-cloning theorem.
Take a qubit for example. It can be represented by two complex numbers, called probability amplitudes (normalised to 1), that is three real numbers (two polar angles and one radius). Copying three numbers on a classical computer using any copy and paste operation is trivial (up to a finite precision) but the problem manifests if the qubit is unitarily transformed (e.g. by the Hadamard quantum gate) to be polarised (which unitary transformation is a surjective isometry). In such a case the qubit can be represented by just two real numbers (one polar angle and one radius equal to 1), while the value of the third can be arbitrary in such a representation. Yet a realisation of a qubit (polarisation-encoded photon, for example) is capable of storing the whole qubit information support within its "structure". Thus no single universal unitary evolution U can clone an arbitrary quantum state according to the no-cloning theorem. It would have to depend on the transformed qubit (initial) state and thus would not have been universal.
In the statement of the theorem, two assumptions were made: the state to be copied is a pure state and the proposed copier acts via unitary time evolution. These assumptions cause no loss of generality. If the state to be copied is a mixed state, it can be purified.[clarification needed] Alternately, a different proof can be given that works directly with mixed states; in this case, the theorem is often known as the no-broadcast theorem.[11][12] Similarly, an arbitrary quantum operation can be implemented via introducing an ancilla and performing a suitable unitary evolution.[clarification needed] Thus the no-cloning theorem holds in full generality.
The no-cloning theorem prevents the use of certain classical error correction techniques on quantum states. For example, backup copies of a state in the middle of a quantum computation cannot be created and used for correcting subsequent errors. Error correction is vital for practical quantum computing, and for some time it was unclear whether or not it was possible. In 1995, Shor and Steane showed that it is by independently devising the first quantum error correcting codes, which circumvent the no-cloning theorem.
Similarly, cloning would violate the no-teleportation theorem, which says that it is impossible to convert a quantum state into a sequence of classical bits (even an infinite sequence of bits), copy those bits to some new location, and recreate a copy of the original quantum state in the new location. This should not be confused with entanglement-assisted teleportation, which does allow a quantum state to be destroyed in one location, and an exact copy to be recreated in another location.
The no-cloning theorem is implied by the no-communication theorem, which states that quantum entanglement cannot be used to transmit classical information (whether superluminally, or slower). That is, cloning, together with entanglement, would allow such communication to occur. To see this, consider the EPR thought experiment, and suppose quantum states could be cloned. Assume parts of a maximally entangled Bell state are distributed to Alice and Bob. Alice could send bits to Bob in the following way: If Alice wishes to transmit a "0", she measures the spin of her electron in the z direction, collapsing Bob's state to either
{\displaystyle |z+\rangle _{B}}
{\displaystyle |z-\rangle _{B}}
. To transmit "1", Alice does nothing to her qubit. Bob creates many copies of his electron's state, and measures the spin of each copy in the z direction. Bob will know that Alice has transmitted a "0" if all his measurements will produce the same result; otherwise, his measurements will have outcomes
{\displaystyle |z+\rangle _{B}}
{\displaystyle |z-\rangle _{B}}
with equal probability. This would allow Alice and Bob to communicate classical bits between each other (possibly across space-like separations, violating causality).
Quantum states cannot be discriminated perfectly.[13]
The no cloning theorem prevents an interpretation of the holographic principle for black holes as meaning that there are two copies of information, one lying at the event horizon and the other in the black hole interior. This leads to more radical interpretations, such as black hole complementarity.
The no-cloning theorem applies to all dagger compact categories: there is no universal cloning morphism for any non-trivial category of this kind.[14] Although the theorem is inherent in the definition of this category, it is not trivial to see that this is so; the insight is important, as this category includes things that are not finite-dimensional Hilbert spaces, including the category of sets and relations and the category of cobordisms.
Imperfect cloningEdit
Even though it is impossible to make perfect copies of an unknown quantum state, it is possible to produce imperfect copies. This can be done by coupling a larger auxiliary system to the system that is to be cloned, and applying a unitary transformation to the combined system. If the unitary transformation is chosen correctly, several components of the combined system will evolve into approximate copies of the original system. In 1996, V. Buzek and M. Hillery showed that a universal cloning machine can make a clone of an unknown state with the surprisingly high fidelity of 5/6.[15]
Imperfect quantum cloning can be used as an eavesdropping attack on quantum cryptography protocols, among other uses in quantum information science.
^ a b Park, James (1970). "The concept of transition in quantum mechanics". Foundations of Physics. 1 (1): 23–33. Bibcode:1970FoPh....1...23P. CiteSeerX 10.1.1.623.5267. doi:10.1007/BF00708652. S2CID 55890485.
^ a b Wootters, William; Zurek, Wojciech (1982). "A Single Quantum Cannot be Cloned". Nature. 299 (5886): 802–803. Bibcode:1982Natur.299..802W. doi:10.1038/299802a0. S2CID 4339227.
^ a b Dieks, Dennis (1982). "Communication by EPR devices". Physics Letters A. 92 (6): 271–272. Bibcode:1982PhLA...92..271D. CiteSeerX 10.1.1.654.7183. doi:10.1016/0375-9601(82)90084-6. hdl:1874/16932.
^ Baez, John; Stay, Mike (2010). "Physics, Topology, Logic and Computation: A Rosetta Stone" (PDF). New Structures for Physics. Berlin: Springer. pp. 95–172. ISBN 978-3-642-12821-9.
^ Coecke, Bob (2009). "Quantum Picturalism". Contemporary Physics. 51: 59–83. arXiv:0908.1787. doi:10.1080/00107510903257624. S2CID 752173.
^ Peres, Asher (2003). "How the No-Cloning Theorem Got its Name". Fortschritte der Physik. 51 (45): 458–461. arXiv:quant-ph/0205076. Bibcode:2003ForPh..51..458P. doi:10.1002/prop.200310062. S2CID 16588882.
^ Kaiser, David (2011). How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival. W. W. Norton. ISBN 978-0-393-07636-3.
^ Herbert, Nick (1982). "FLASH—A superluminal communicator based upon a new kind of quantum measurement". Foundations of Physics. 12 (12): 1171–1179. Bibcode:1982FoPh...12.1171H. doi:10.1007/BF00729622. S2CID 123118337.
^ a b Ghirardi, GianCarlo (2013), "Entanglement, Nonlocality, Superluminal Signaling and Cloning", in Bracken, Paul (ed.), Advances in Quantum Mechanics, IntechOpen (published April 3, 2013), arXiv:1305.2305, doi:10.5772/56429, ISBN 978-953-51-1089-7, S2CID 118778014
^ Ortigoso, Juan (2018). "Twelve years before the quantum no-cloning theorem". American Journal of Physics. 86 (3): 201–205. arXiv:1707.06910. Bibcode:2018AmJPh..86..201O. doi:10.1119/1.5021356. S2CID 119192142.
^ Barnum, Howard; Caves, Carlton M.; Fuchs, Christopher A.; Jozsa, Richard; Schumacher, Benjamin (1996-04-08). "Noncommuting Mixed States Cannot Be Broadcast". Physical Review Letters. 76 (15): 2818–2821. arXiv:quant-ph/9511010. Bibcode:1996PhRvL..76.2818B. doi:10.1103/PhysRevLett.76.2818. PMID 10060796. S2CID 11724387.
^ Kalev, Amir; Hen, Itay (2008-05-29). "No-Broadcasting Theorem and Its Classical Counterpart". Physical Review Letters. 100 (21): 210502. arXiv:0704.1754. Bibcode:2008PhRvL.100u0502K. doi:10.1103/PhysRevLett.100.210502. PMID 18518590. S2CID 40349990.
^ Bae, Joonwoo; Kwek, Leong-Chuan (2015-02-27). "Quantum state discrimination and its applications". Journal of Physics A: Mathematical and Theoretical. 48 (8): 083001. arXiv:1707.02571. Bibcode:2015JPhA...48h3001B. doi:10.1088/1751-8113/48/8/083001. ISSN 1751-8113. S2CID 119199057.
^ S. Abramsky, "No-Cloning in categorical quantum mechanics", (2008) Semantic Techniques for Quantum Computation, I. Mackie and S. Gay (eds), Cambridge University Press. arXiv:0910.2401
^ Bužek, V.; Hillery, M. (1996). "Quantum Copying: Beyond the No-Cloning Theorem". Phys. Rev. A. 54 (3): 1844–1852. arXiv:quant-ph/9607018. Bibcode:1996PhRvA..54.1844B. doi:10.1103/PhysRevA.54.1844. PMID 9913670. S2CID 1446565.
V. Buzek and M. Hillery, Quantum cloning, Physics World 14 (11) (2001), pp. 25–29.
|
{\displaystyle H_{n}(a,b)}
{\displaystyle a[n]b}
{\displaystyle a[n]b}
{\displaystyle \uparrow }
{\displaystyle [n]}
{\displaystyle n-2}
{\displaystyle \uparrow }
{\displaystyle 2\uparrow 4=H_{3}(2,4)=2[3]4=2\times (2\times (2\times 2))=2^{4}=16}
{\displaystyle \uparrow \uparrow }
{\displaystyle 2\uparrow \uparrow 4=H_{4}(2,4)=2[4]4=2\uparrow (2\uparrow (2\uparrow 2))=2^{2^{2^{2}}}=2^{16}=65,536}
{\displaystyle \uparrow \uparrow \uparrow }
{\displaystyle {\begin{aligned}2\uparrow \uparrow \uparrow 4=H_{5}(2,4)=2[5]4&=2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow \uparrow 2))\\&=2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow 2))\\&=2\uparrow \uparrow (2\uparrow \uparrow 4)\\&=\underbrace {2\uparrow (2\uparrow (2\uparrow \dots ))} \;=\;\underbrace {\;2^{2^{\cdots ^{2}}}} \\&\;\;\;\;\;2\uparrow \uparrow 4{\mbox{ copies of }}2\;\;\;\;\;{\mbox{65,536 2s}}\\\end{aligned}}}
{\displaystyle a\geq 0,n\geq 1,b\geq 0}
{\displaystyle a\uparrow ^{n}b=H_{n+2}(a,b)=a[n+2]b}
{\displaystyle \uparrow ^{n}}
{\displaystyle 2\uparrow \uparrow \uparrow \uparrow 3=2\uparrow ^{4}3.}
{\displaystyle {\begin{matrix}H_{1}(a,b)=a[1]b=a+b=&a+\underbrace {1+1+\dots +1} \\&b{\mbox{ copies of }}1\end{matrix}}}
{\displaystyle {\begin{matrix}H_{2}(a,b)=a[2]b=a\times b=&\underbrace {a+a+\dots +a} \\&b{\mbox{ copies of }}a\end{matrix}}}
{\displaystyle {\begin{matrix}4\times 3&=&\underbrace {4+4+4} &=&12\\&&3{\mbox{ copies of }}4\end{matrix}}}
{\displaystyle b}
{\displaystyle {\begin{matrix}a\uparrow b=H_{3}(a,b)=a[3]b=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix}}}
{\displaystyle {\begin{matrix}4\uparrow 3=4^{3}=&\underbrace {4\times 4\times 4} &=&64\\&3{\mbox{ copies of }}4\end{matrix}}}
{\displaystyle {\begin{matrix}a\uparrow \uparrow b=H_{4}(a,b)=a[4]b=&\underbrace {a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}} &=&\underbrace {a\uparrow (a\uparrow (\dots \uparrow a))} \\&b{\mbox{ copies of }}a&&b{\mbox{ copies of }}a\end{matrix}}}
{\displaystyle {\begin{matrix}4\uparrow \uparrow 3=&\underbrace {4^{4^{4}}} &=&\underbrace {4\uparrow (4\uparrow 4)} &=&4^{256}&\approx &1.34078079\times 10^{154}&\\&3{\mbox{ copies of }}4&&3{\mbox{ copies of }}4\end{matrix}}}
{\displaystyle 3\uparrow \uparrow 2=3^{3}=27}
{\displaystyle 3\uparrow \uparrow 3=3^{3^{3}}=3^{27}=7,625,597,484,987}
{\displaystyle 3\uparrow \uparrow 4=3^{3^{3^{3}}}=3^{3^{27}}=3^{7625597484987}\approx 1.2580143\times 10^{3638334640024}}
{\displaystyle 3\uparrow \uparrow 5=3^{3^{3^{3^{3}}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}}\approx 3^{1.2580143\times 10^{3638334640024}}}
{\displaystyle {\begin{matrix}a\uparrow \uparrow \uparrow b=H_{5}(a,b)=a[5]b=&\underbrace {a_{}\uparrow \uparrow (a\uparrow \uparrow (\dots \uparrow \uparrow a))} \\&b{\mbox{ copies of }}a\end{matrix}}}
{\displaystyle {\begin{matrix}a\uparrow \uparrow \uparrow \uparrow b=H_{6}(a,b)=a[6]b=&\underbrace {a_{}\uparrow \uparrow \uparrow (a\uparrow \uparrow \uparrow (\dots \uparrow \uparrow \uparrow a))} \\&b{\mbox{ copies of }}a\end{matrix}}}
and so on. The general rule is that a{\displaystyle n}
{\displaystyle n-1}
{\displaystyle {\begin{matrix}a\ \underbrace {\uparrow _{}\uparrow \!\!\dots \!\!\uparrow } _{n}\ b=\underbrace {a\ \underbrace {\uparrow \!\!\dots \!\!\uparrow } _{n-1}\ (a\ \underbrace {\uparrow _{}\!\!\dots \!\!\uparrow } _{n-1}\ (\dots \ \underbrace {\uparrow _{}\!\!\dots \!\!\uparrow } _{n-1}\ a))} _{b{\text{ copies of }}a}\end{matrix}}}
{\displaystyle 3\uparrow \uparrow \uparrow 2=3\uparrow \uparrow 3=3^{3^{3}}=3^{27}=7,625,597,484,987}
{\displaystyle {\begin{matrix}3\uparrow \uparrow \uparrow 3=3\uparrow \uparrow (3\uparrow \uparrow 3)=3\uparrow \uparrow (3\uparrow 3\uparrow 3)=&\underbrace {3_{}\uparrow 3\uparrow \dots \uparrow 3} \\&3\uparrow 3\uparrow 3{\mbox{ copies of }}3\end{matrix}}{\begin{matrix}=&\underbrace {3_{}\uparrow 3\uparrow \dots \uparrow 3} \\&{\mbox{7,625,597,484,987 copies of 3}}\end{matrix}}{\begin{matrix}=&\underbrace {3^{3^{3^{3^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{3}}}}}}}}} \\&{\mbox{7,625,597,484,987 copies of 3}}\end{matrix}}}
{\displaystyle a^{b}}
{\displaystyle b}
{\displaystyle a}
{\displaystyle a\uparrow b}
{\displaystyle a^{b}}
{\displaystyle a\uparrow b}
{\displaystyle a\uparrow ^{n}b}
{\displaystyle a\uparrow ^{4}b=a\uparrow \uparrow \uparrow \uparrow b}
{\displaystyle a\uparrow \uparrow b}
{\displaystyle a\uparrow \uparrow 4=a\uparrow (a\uparrow (a\uparrow a))=a^{a^{a^{a}}}}
{\displaystyle a\uparrow \uparrow b=\underbrace {a^{a^{.^{.^{.{a}}}}}} _{b}}
{\displaystyle a\uparrow \uparrow \uparrow b}
{\displaystyle a\uparrow \uparrow \uparrow 4=a\uparrow \uparrow (a\uparrow \uparrow (a\uparrow \uparrow a))=\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{a}}}}
{\displaystyle a\uparrow \uparrow \uparrow b=\left.\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}b}
{\displaystyle a\uparrow \uparrow \uparrow \uparrow b}
{\displaystyle a\uparrow \uparrow \uparrow \uparrow 4=a\uparrow \uparrow \uparrow (a\uparrow \uparrow \uparrow (a\uparrow \uparrow \uparrow a))=\left.\left.\left.\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}a}
{\displaystyle a\uparrow \uparrow \uparrow \uparrow b=\underbrace {\left.\left.\left.\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\cdots \right\}a} _{b}}
{\displaystyle a\uparrow ^{n}b}
{\displaystyle ^{b}a}
{\displaystyle a\uparrow \uparrow b={}^{b}a}
{\displaystyle a\uparrow \uparrow \uparrow b=\underbrace {^{^{^{^{^{a}.}.}.}a}a} _{b}}
{\displaystyle a\uparrow \uparrow \uparrow \uparrow b=\left.\underbrace {^{^{^{^{^{a}.}.}.}a}a} _{\underbrace {^{^{^{^{^{a}.}.}.}a}a} _{\underbrace {\vdots } _{a}}}\right\}b}
{\displaystyle 4\uparrow ^{4}4}
{\displaystyle \underbrace {^{^{^{^{^{4}.}.}.}4}4} _{\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{4}}}=\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{^{^{^{4}4}4}4}}}
{\displaystyle \uparrow ^{n}}
{\displaystyle {\begin{matrix}a\uparrow ^{n}b&=&a[n+2]b&=&a\to b\to n\\{\mbox{(Knuth)}}&&{\mbox{(hyperoperation)}}&&{\mbox{(Conway)}}\end{matrix}}}
{\displaystyle 6\uparrow \uparrow 4}
{\displaystyle \underbrace {6^{6^{.^{.^{.^{6}}}}}} _{4}}
{\displaystyle 6\uparrow \uparrow 4}
{\displaystyle 6^{6^{6^{6}}}}
{\displaystyle 6^{6^{46,656}}}
{\displaystyle \underbrace {6^{6^{.^{.^{.^{6}}}}}} _{4}}
{\displaystyle 10\uparrow (3\times 10\uparrow (3\times 10\uparrow 15)+3)}
{\displaystyle \underbrace {100000...000} _{\underbrace {300000...003} _{\underbrace {300000...003} _{15}}}}
{\displaystyle 10^{3\times 10^{3\times 10^{15}}+3}}
{\displaystyle f(x)}
{\displaystyle f_{0}(x)=x+1}
{\displaystyle f_{3}(x)}
{\displaystyle f_{4}(x)}
{\displaystyle f_{\omega }(x)}
{\displaystyle f_{\omega +1}(x)}
{\displaystyle f_{\omega ^{2}}(x)}
{\displaystyle a\uparrow ^{n}b={\begin{cases}a^{b},&{\text{if }}n=1;\\1,&{\text{if }}n>1{\text{ and }}b=0;\\a\uparrow ^{n-1}(a\uparrow ^{n}(b-1)),&{\text{otherwise }}\end{cases}}}
{\displaystyle a,b,n}
{\displaystyle a\geq 0,n\geq 1,b\geq 0}
{\displaystyle (a\uparrow ^{1}b=a\uparrow b=a^{b})}
{\displaystyle (a\uparrow ^{2}b=a\uparrow \uparrow b)}
{\displaystyle (a\uparrow ^{0}b=a\times b)}
{\displaystyle a\uparrow ^{n}b={\begin{cases}a\times b,&{\text{if }}n=0;\\1,&{\text{if }}n>0{\text{ and }}b=0;\\a\uparrow ^{n-1}(a\uparrow ^{n}(b-1)),&{\text{otherwise }}\end{cases}}}
{\displaystyle a,b,n}
{\displaystyle a\geq 0,n\geq 0,b\geq 0}
{\displaystyle \uparrow ^{0}}
{\displaystyle H_{n}(a,b)=a[n]b=a\uparrow ^{n-2}b{\text{ for }}n\geq 0.}
{\displaystyle a\uparrow b\uparrow c}
{\displaystyle a\uparrow (b\uparrow c)}
{\displaystyle (a\uparrow b)\uparrow c}
{\displaystyle 0\uparrow ^{n}b=H_{n+2}(0,b)=0[n+2]b}
{\displaystyle 2\uparrow ^{n}b}
{\displaystyle 2^{b}}
{\displaystyle 2\uparrow ^{n}b}
{\displaystyle H_{n+2}(2,b)}
{\displaystyle 2[n+2]b}
{\displaystyle 2^{b}}
{\displaystyle 2^{65\,536}\approx 2.0\times 10^{19\,728}}
{\displaystyle 2^{2^{65\,536}}\approx 10^{6.0\times 10^{19\,727}}}
{\displaystyle 2\uparrow \uparrow b}
{\displaystyle {\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65536{\mbox{ copies of }}2\end{matrix}}}
{\displaystyle {\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65536{\mbox{ copies of }}2\end{matrix}}}
{\displaystyle {\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65536{\mbox{ copies of }}2\end{matrix}}}
{\displaystyle 2\uparrow \uparrow \uparrow b}
{\displaystyle {\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\65536{\mbox{ copies of }}2\end{matrix}}}
{\displaystyle 2\uparrow \uparrow \uparrow \uparrow b}
The table is the same as that of the Ackermann function, except for a shift i{\displaystyle n}
{\displaystyle b}
{\displaystyle 3^{b}}
{\displaystyle 3\uparrow ^{n}b}
{\displaystyle H_{n+2}(3,b)}
{\displaystyle 3[n+2]b}
{\displaystyle 3^{b}}
{\displaystyle 3^{7{,}625{,}597{,}484{,}987}}
{\displaystyle 3^{3^{7{,}625{,}597{,}484{,}987}}}
{\displaystyle 3\uparrow \uparrow b}
{\displaystyle {\begin{matrix}\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\7{,}625{,}597{,}484{,}987{\mbox{ copies of }}3\end{matrix}}}
{\displaystyle 3\uparrow \uparrow \uparrow b}
{\displaystyle {\begin{matrix}\underbrace {3_{}^{3^{{}^{.\,^{.\,^{.\,^{3}}}}}}} \\7{,}625{,}597{,}484{,}987{\mbox{ copies of }}3\end{matrix}}}
{\displaystyle 3\uparrow \uparrow \uparrow \uparrow b}
{\displaystyle 4^{b}}
{\displaystyle 4\uparrow ^{n}b}
{\displaystyle H_{n+2}(4,b)}
{\displaystyle 4[n+2]b}
{\displaystyle 4^{b}}
{\displaystyle 1.3407807930\times 10^{154}}
{\displaystyle 4^{1.3407807930\times 10^{154}}}
{\displaystyle 4^{4^{1.3407807930\times 10^{154}}}}
{\displaystyle 4\uparrow \uparrow b}
{\displaystyle 4^{1.3407807930\times 10^{154}}}
{\displaystyle {\begin{matrix}\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\4^{1.3407807930\times 10^{154}}{\mbox{ copies of }}4\end{matrix}}}
{\displaystyle 4\uparrow \uparrow \uparrow b}
{\displaystyle {\begin{matrix}\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\\underbrace {4_{}^{4^{{}^{.\,^{.\,^{.\,^{4}}}}}}} \\4^{1.3407807930\times 10^{154}}{\mbox{ copies of }}4\end{matrix}}}
{\displaystyle 4\uparrow \uparrow \uparrow \uparrow b}
{\displaystyle 10^{b}}
{\displaystyle 10\uparrow ^{n}b}
{\displaystyle H_{n+2}(10,b)}
{\displaystyle 10[n+2]b}
{\displaystyle 10^{b}}
{\displaystyle 10^{10,000,000,000}}
{\displaystyle 10^{10^{10,000,000,000}}}
{\displaystyle 10^{10^{10^{10,000,000,000}}}}
{\displaystyle 10\uparrow \uparrow b}
{\displaystyle {\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}}
{\displaystyle {\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}}
{\displaystyle {\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}}
{\displaystyle 10\uparrow \uparrow \uparrow b}
{\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\10{\mbox{ copies of }}10\end{matrix}}}
{\displaystyle {\begin{matrix}\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\10{\mbox{ copies of }}10\end{matrix}}}
{\displaystyle 10\uparrow \uparrow \uparrow \uparrow b}
{\displaystyle 10\uparrow ^{n}b}
{\displaystyle \uparrow ^{0}}
|
Correction to “L∞-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds,” Duke Math. J. 77 (1995), 799–817
1 February 2016 Correction to “
{L}^{\infty }
-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds,” Duke Math. J. 77 (1995), 799–817
Shin-Ya Koyama "Correction to “
{L}^{\infty }
-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds,” Duke Math. J. 77 (1995), 799–817," Duke Mathematical Journal, Duke Math. J. 165(2), 413-415, (1 February 2016)
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Cryosurgery of Normal and Tumor Tissue in the Dorsal Skin Flap Chamber: Part II—Injury Response | J. Biomech Eng. | ASME Digital Collection
Nathan E. Hoffmann,
Nathan E. Hoffmann
Department of Biomedical Engineering, University of Minnesota, Minneapolis, MN 55455
Departments of Biomedical Engineering, Mechanical Engineering, and Urologic Surgery, University of Minnesota, Minneapolis, MN 55455
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division July 14, 1999; revised manuscript received February 27, 2001. Associate Editor: J. J. McGrath.
Hoffmann, N. E., and Bischof, J. C. (February 27, 2001). "Cryosurgery of Normal and Tumor Tissue in the Dorsal Skin Flap Chamber: Part II—Injury Response ." ASME. J Biomech Eng. August 2001; 123(4): 310–316. https://doi.org/10.1115/1.1385839
It has been hypothesized that vascular injury may be an important mechanism of cryosurgical destruction in addition to direct cellular destruction. In this study, we report correlation of tissue and vascular injury after cryosurgery to the temperature history during cryosurgery in an in vivo microvascular preparation. The dorsal skin flap chamber, implanted in the Copenhagen rat, was chosen as the cryosurgical model. Cryosurgery was performed in the chamber on either normal skin or tumor tissue propagated from an AT-1 Dunning rat prostate tumor, as described in a companion paper (Hoffmann and Bischof, 2001). The vasculature was then viewed at 3 and 7 days after cryoinjury under brightfield and FITC-labeled dextran contrast enhancement to assess the vascular injury. The results showed that there was complete destruction of the vasculature in the center of the lesion and a gradual return to normal patency moving radially outward. Histologic examination showed a band of inflammation near the edge of a large necrotic region at both 3 and 7 days after cryosurgery. The area of vascular injury observed with FITC-labeled dextran quantitatively corresponded to the area of necrosis observed in histologic section, and the size of the lesion for tumor and normal tissue was similar at 3 days post cryosurgery. At 7 days after cryosurgery, the lesion was smaller for both tissues, with the normal tissue lesion being much smaller than the tumor tissue lesion. A comparison of experimental injury data to the thermal model validated in a companion paper (Hoffmann and Bischof, 2001) suggested that the minimum temperature required for causing necrosis was
−15.6±4.3°C
in tumor tissue and
−19.0±4.4°C
in normal tissue. The other thermal parameters manifested at the edge of the lesion included a cooling rate of ∼28°C/min, 0 hold time, and a ∼9°C/min thawing rate. The conditions at the edge of the lesion are much less severe than the thermal conditions required for direct cellular destruction of AT-1 cells and tissues in vitro. These results are consistent with the hypothesis that vascular-mediated injury is responsible for the majority of injury at the edge of the frozen region in microvascular perfused tissue.
blood vessels, skin, freezing, biothermics, surgery, physiological models, Cryoinjury, Cell Injury, Cryosurgery, Vascular Injury, Intravital Microscopy, Skin Flap Chamber, Microvasculature
Biological tissues, Skin, Tumors, Wounds, Freezing, Temperature
, this issue, pp.
Cryosurgery of Dunning AT-1 Rat Prostate Tumor: Thermal, Biophysical and Viability Response at the Cellular and Tissue Level
An Assessment of Tumor Cell Viability After in Vitro Freezing
The Mechanism of the Protective Action of Glycerol Against Haemolysis by Freezing and Thawing
Biochemical Alterations and Tissue Viability in AT-1 Tumor Tissue After in Vitro Cryo-Ablation
A Parametric Study of Freezing Injury in AT-1 Rat Prostate Tumor Cells
Effect of Thermal Variables on Frozen Human Primary Prostatic Adenocarcinoma Cells
The Observation of Freeze–Thaw Cycles Upon Cancer Cell Suspensions
Vascular Reactions of the Skin to Injury. III: Some Effects of Freezing, of Cooling, and of Warming
Kreyberg
Eine Methode zum Experimentelen Nachweis Von Stase Mittels Spezieller Praeparate
The Immediate Vascular Changes in True Frostbite
Effect of Freezing and Thawing on the Microcirculation and Capillary Endothelium of the Hamster Cheek Pouch
Studies on Frost-Bite With Special Reference to Treatment and the Effect on Minute Blood Vessels
Necrosis of Whole Mouse Skin in Situ and Survival of Transplanted Epithelium After Freezing to −78°C and −196°C
A Transparent Access Chamber for the Rat Dorsal Skin Fold
Establishment and Characterization of Seven Dunning Rat Prostatic Cancer Cell Lines and Their Use in Developing Methods for Predicting Metastatic Abilities of Prostatic Cancers
Fluorescence Digital Microscopy of Interstitial Macromolecular Diffusion in Burn Injury
Microvascular Permeability of Albumin, Vascular Surface Area, and Vascular Volume Measured in Human Adenocarcinoma LS174T Using Dorsal Chamber in SCID Mice
Laprell-Moschner
Effects of Prolonged Cold Injury on the Subcutaneous Microcirculation of the Hamster. I. Technique, Morphology and Tissue Oxygenation
Angiogenesis, Microvascular Architecture, Microhemodynamics, and Interstitial Fluid Pressure During Early Growth of Human Adenocarcinoma LS174T in SCID Mice
Bevington, P. R., and Robinson, D. K., 1992, “Error Analysis,” in: Data Reduction and Error Analysis for the Physical Sciences, 2nd ed., McGraw-Hill, New York, pp. 38–48.
Moore, D. S., and McCabe, G. P., 1993, “Inference for Distributions,” in: Introduction to the Practice of Statistics, W. H. Freeman, New York, pp. 498–574.
Freezing Bone Without Excision: An Experimental Study of Bone-Cell Destruction and Manner of Regrowth in Dogs
Cryosurgery of Breast Cancer
Le Pivert, P., 1980, “Basic Considerations of the Cryolesion,” in: Handbook of Cryosurgery, R. Ablin, ed., Marcel Dekker, New York. pp. 15–68.
A Light and Electron Microscopic Analysis of Vascular Injury
Burn-Induced Alterations in Vasoactive Function of the Peripheral Cutaneous Microcirculation
Identification of Vascular System in Experimental Carcinoma for Cryosurgery—Histochemical Observations of Lectin UEA-1 and Alkaline Phosphatase Activity in Vascular Endothelium
Morphologic Characterization of Acute Injury to Vascular Endothelium of Skin After Frostbite
Analysis of Microvascular Changes in Frostbite Injury
Evidence for an Early Free Radical-Mediated Reperfusion Injury in Frostbite
Free Radic Biol. Med.
The Effect of Superoxide Dismutase on the Skin Microcirculation After Ischemia and Reperfusion
Prog. Appl. Microcirculation
Vascular Reactions After Experimental Cold Injury
Cotran, R. S., Kumar, V., and Collins, T., 1999, “Cellular Injury and Cell Death,” in: Pathologic Basis of Disease, W. B. Saunders, Philadelphia, pp. 1–31.
Cardiovascular Effects of a Ketamine-Medetomidine Combination That Produces Deep Sedation in Yucatan Mini Swine
Lab Anim. Sci.
The Cardiorespiratory Effects of Diazepam–Ketamine and Xylazine–Ketamine Anesthetic Combinations in Sheep
Evaluation of Anesthetic Effects on Parameters for the in Situ Rat Brain Perfusion Technique
Effect of Injectable or Inhalational Anesthetics and of Neuroleptic, Neuroleptanalgesic, and Sedative Agents on Tumor Blood Flow
Investigating the Mechanism and Effect of Cryoimmunology in the Copenhagen Rat
Monitoring Renal Cryosurgery: Predictors of Tissue Necrosis in Swine
Cryonecrosis of Normal and Tumor-Bearing Rat Liver Potentiated by Inflow Occlusion
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The value of spearman's rank correlation coefficent no of observations was found to be 2/3 The sum of the - Economics - Correlation - 13313489 | Meritnation.com
{r}_{k }= 1 - \frac{6\underset{}{\sum {D}^{2}}}{{N}^{3 }- N}
rk = Coefficient of rank correlation
D = Rank differences
rk = 2/3
ΣD2 = 55
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Ask Answer - Articles, My Mother at Sixty-six, Theme 1 From the Beginning of Time, Lost Spring, Continuity and Differentiability, History, Theme 6 The Three Orders, Theme 8 Confrontation of Cultures, Major Themes, Government Budget and the Economy - Popular Questions for School Students
Ayantika Dey
Eaplain the statement" I saw my mother......her face ashen like that of a corpse"
Kritanjali Kalita
"Western writers saw the infancy of society in india".Justify the words with regard to origin of sociology in India.
Hema Munjal
How does the author focus on the 'perpetual state of poverty' of the children not wearing not wearing footwear?
Taru Kishore
f(x)= tan5x sin2x , when x is not equal to zero
k , x=0
continuous at x=0?
What are the modes of disposal of the dead prevalent at present? To what extent do these represent social differences?
Do you think the child labour law is enforced? If the child labour law is enforced, approximately how many ragpickers and how many bangle makers would be freed? Envisage the life Saheb and Mukesh would enjoy if they were free.
Abhimanyu Rajbhor
What were the factors affecting social and economic relations ?
10 - marks question
How do the anthropologists feel that the information about living hunter gatherer can be used to under stand past society ? Discuss
What reason does Saheb give for not going to school?
discuss the difference between the arawaks and the Spanish. which is these differences would you consider most significant and why
Chapter name : THETHIRD LEVEL (VISTAS) CHAPTER 1
Distinguish between propensity to consume and propensity to save with the help ofnumerical example
Read the source and answer the question that is given below the source
Prabhavati Gupta and the village of Danguna
This is what Prabhavati Gupta states in her inscription:
Prabhavati Gupta... commands the gramakutumbinas (householders/peasants living in the village).
Brahmanas and others living in the village of Danguna...
"Be it known to you that on the twelfth (lunar day) of the bright (fortnight) of Karttika, we have, in order to increase out religious merit donated this village with the pouring out of water, to the Acharya (teacher) Chanalasvamin... You should obey all (his) commands....
We confer on (him) the following exemptions typical of an agarhara... (this village is) not to be entered by soldiers and policemen; (it is) exempt from (the obligation to provide) grass, (animal) hides as seats, and charcoal (to touring royal officers); exempt form (the royal prerogative of) purchasing fermenting liquors and digging (salt); exempt from (the right to) mines and khadira trees; exempt form (the obligation to supply) flowers and milk; (it id donated) together with (the right to) hidden treasures and deposits (and) together with major and minor taxes...."
This charter has been written in the thirteenth (regnal) year. (It has been) engraved by Chakradasa.
⇒
What were the things produced in the village?
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Support vector machine template - MATLAB templateSVM - MathWorks í•œêµ
\mathrm{α}
\mathrm{α}
Suppose that the alpha coefficient for observation j is αj and the box constraint of observation j is Cj, j = 1,...,n, where n is the training sample size.
true At each iteration, if αj is near 0 or near Cj, then MATLAB® sets αj to 0 or to Cj, respectively.
MATLAB stores the final values of α in the Alpha property of the trained SVM model object.
G\left({x}_{j},{x}_{k}\right)=\mathrm{exp}\left(â{â{x}_{j}â{x}_{k}â}^{2}\right)
G\left({x}_{j},{x}_{k}\right)={x}_{j}â²{x}_{k}
G\left({x}_{j},{x}_{k}\right)={\left(1+{x}_{j}â²{x}_{k}\right)}^{q}
SaveSupportVectors — Store support vectors, their labels, and the estimated α coefficients
Store support vectors, their labels, and the estimated α coefficients as properties of the resulting model, specified as the comma-separated pair consisting of 'SaveSupportVectors' and true or false.
If SaveSupportVectors is true, the resulting model stores the support vectors in the SupportVectors property, their labels in the SupportVectorLabels property, and the estimated α coefficients in the Alpha property of the compact, SVM learners.
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(Redirected from Maya (software))
www.autodesk.com/products/maya/overview
Autodesk Maya, commonly shortened to just Maya (/ˈmaɪə/ MY-ə[3][4]), is a 3D computer graphics application that runs on Windows, macOS and Linux, originally developed by Alias and currently owned and developed by Autodesk. It is used to create assets for interactive 3D applications (including video games), animated films, TV series, and visual effects.
3 Industry usage
Maya was originally an animation product based on code from The Advanced Visualizer by Wavefront Technologies, Thomson Digital Image (TDI) Explore, PowerAnimator by Alias, and Alias Sketch!. The IRIX-based projects were combined and animation features were added; the project codename was Maya.[5] Walt Disney Feature Animation collaborated closely with Maya's development during its production of Dinosaur.[6] Disney requested that the user interface of the application be customizable so that a personalized workflow could be created. This was a particular influence in the open architecture of Maya, and partly responsible for it becoming popular in the animation industry.
After Silicon Graphics Inc. acquired both Alias and Wavefront Technologies, Inc., Wavefront's technology (then under development) was merged into Maya. SGI's acquisition was a response to Microsoft Corporation acquiring Softimage 3D. The new wholly owned subsidiary was named "Alias
{\displaystyle |}
Wavefront".[7]
In the early days of development Maya started with Tcl as the scripting language, in order to leverage its similarity to a Unix shell language, but after the merger with Wavefront it was replaced with Maya Embedded Language (MEL). Sophia, the scripting language in Wavefront's Dynamation, was chosen as the basis of MEL.[8]
Maya 1.0 was released in February 1998. Following a series of acquisitions, Maya was bought by Autodesk in 2005.[9][10] Under the name of the new parent company, Maya was renamed Autodesk Maya. However, the name "Maya" continues to be the dominant name used for the product.
Maya is an application used to generate 3D assets for use in film, television, games, and commercials. The software was initially released for the IRIX operating system. However, this support was discontinued in August 2006 after the release of version 6.5. Maya was available in both "Complete" and "Unlimited" editions until August 2008, when it was turned into a single suite.[11]
Users define a virtual workspace (scene) to implement and edit media of a particular project. Scenes can be saved in a variety of formats, the default being .mb (Maya D). Maya exposes a node graph architecture. Scene elements are node-based, each node having its own attributes and customization. As a result, the visual representation of a scene is based entirely on a network of interconnecting nodes, depending on each other's information. For the convenience of viewing these networks, there is a dependency and a directed acyclic graph.
Nowadays, the 3D models can be imported to game engines such as Unreal Engine and Unity.
Industry usage[edit]
The widespread use of Maya in the film industry is usually associated with its development on the film Dinosaur, released by Disney and The Secret Lab on May 19, 2000.[12] In 2003, when the company received an Academy Award for Technical Achievement, it was noted to be used in films such as The Lord of the Rings: The Two Towers, Spider-Man (2002), Ice Age, and Star Wars: Episode II – Attack of the Clones.[13] By 2015, VentureBeat Magazine stated that all ten films in consideration for the Best Visual Effects Academy Award had used Autodesk Maya and that it had been "used on every winning film since 1997."[14]
On March 1, 2003, Alias was given an Academy Award for Technical Achievement by the Academy of Motion Picture Arts and Sciences for scientific and technical achievement for their development of Maya software.[13]
In 2005, while working for Alias|Wavefront, Jos Stam shared an Academy Award for Technical Achievement with Edwin Catmull and Tony DeRose for their invention and application of subdivision surfaces.[15]
On February 8, 2008, Duncan Brinsmead, Jos Stam, Julia Pakalns and Martin Werner received an Academy Award for Technical Achievement for the design and implementation of the Maya Fluid Effects system.[16][17]
List of Maya plugins
^ "C++ Applications". stroustrup.com. Retrieved December 16, 2016.
^ Baas, Matthias (May 8, 2006). "Python/Maya: Introductory tutorial". cgkit.sourceforge.net. Archived from the original on November 15, 2010. Retrieved December 10, 2010.
^ "Maya 2017 Overview". YouTube. Autodesk. Archived from the original on 2021-11-17. Retrieved May 18, 2018.
^ "Maya LT 2018 – Overview". YouTube. Autodesk. Archived from the original on 2021-11-17. Retrieved May 18, 2018.
^ "History". Maya books. Archived from the original on November 25, 2010. Retrieved December 11, 2010.
^ Muwanguzi, Michael J (July 1, 2010). "Maya 2011". Microfilmmaker Magazine. Archived from the original (Software Review) on July 20, 2011. Retrieved December 11, 2010.
^ Weisbard, Sam (December 13, 2002). "Wavefront Discontinued Products and Brands". Alias. Design engine. Archived from the original on August 22, 2009. Retrieved December 10, 2010.
^ Sharpe, Jason; Lumsden, Charles J; Woolridge, Nicholas (2008), In silico: 3D animation and simulation of cell biology with Maya and MEL, Morgan Kaufmann Martin, p. 263, ISBN 978-0-12-373655-0
^ Autodesk (October 4, 2005). "Autodesk Signs Definitive Agreement to Acquire Alias". Archived from the original on January 10, 2016. Retrieved October 23, 2015.
^ "Autodesk Maya Features – Compare". Autodesk. Archived from the original on 2010-10-06. Retrieved 2010-10-02.
^ Warren, Scott (16 June 2017). Learning Games: The Science and Art of Development. Springer. p. 77.
^ a b Sellers, Dennis (14 January 2003). "Maya gets Oscar for Technical Achievement". Macworld. Retrieved 8 January 2019.
^ Terdiman, Daniel (15 January 2015). "And the Oscar for Best Visual Effects Goes to… Autodesk's Maya". media. VentureBeat.
^ "PIXAR Awards". Archived from the original on September 27, 2011. Retrieved November 15, 2011.
^ "Scientific & Technical Awards Winners". January 6, 2003. Archived from the original on February 16, 2009. Retrieved December 10, 2010.
^ "Technical Achievement Award". January 6, 2003. Retrieved December 10, 2010.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Autodesk_Maya&oldid=1088337905"
3D graphics software that uses Qt
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Logarithm - Simple English Wikipedia, the free encyclopedia
Logarithms or logs are a part of mathematics. They are related to exponential functions. A logarithm tells what exponent (or power) is needed to make a certain number, so logarithms are the inverse (opposite) of exponentiation. Historically, they were useful in multiplying or dividing large numbers.
An opened nautilus shell. Its chambers make a logarithmic spiral
An example of a logarithm is
{\displaystyle \log _{2}(8)=3\ }
. In this logarithm, the base is 2, the argument is 8 and the answer is 3. In this case, the exponentiation function would be:
{\displaystyle 2^{3}=2\times 2\times 2=8\,}
The most common types of logarithms are common logarithms, where the base is 10, binary logarithms, where the base is 2, and natural logarithms, where the base is e ≈ 2.71828.[1][2]
2 Relationship with exponential functions
3 Difference to roots
5 Common logarithms
6 Natural logarithms
7 Common bases for logarithms
8 Properties of logarithms
8.1 Properties from the definition of a logarithm
8.2 Operations within logarithm arguments
8.3 Logarithm tables, slide rules, and historical applications
Logarithms were first used in India in the 2nd century BC. The first to use logarithms in modern times was the German mathematician Michael Stifel (around 1487–1567). In 1544, he wrote down the following equations:
{\displaystyle q^{m}q^{n}=q^{m+n}}
{\displaystyle {\tfrac {q^{m}}{q^{n}}}=q^{m-n}}
. This is the basis for understanding logarithms. For Stifel,
{\displaystyle m}
{\displaystyle n}
had to be whole numbers. John Napier (1550–1617) did not want this restriction, and wanted a range for the exponents.
John Napier worked on logarithms
According to Napier, logarithms express ratios:
{\displaystyle a}
has the same ratio to
{\displaystyle b}
{\displaystyle c}
{\displaystyle d}
if the difference of their logarithms matches. Mathematically:
{\displaystyle \log(a)-\log(b)=\log(c)-\log(d)}
. At first, base e was used (even though the number had not been named yet). Henry Briggs proposed to use 10 as a base for logarithms, such logarithms are very useful in astronomy.[2]
Relationship with exponential functionsEdit
A logarithm tells what exponent (or power) is needed to make a certain number, so logarithms are the inverse (opposite) of exponentiation.
Just as an exponential function has three parts, a logarithm has three parts as well: a base, an argument and an answer (also called power).
The following is an example of an exponential function:
{\displaystyle 2^{3}=8\ }
In this function, the base is 2, the argument is 3 and the answer is 8.
This exponential equation has an inverse, its logarithmic equation:
{\displaystyle \log _{2}(8)=3\ }
In this equation, the base is 2, the argument is 8 and the answer is 3.
Difference to rootsEdit
Addition has one inverse operation: the subtraction. Also, multiplication has one inverse operation: the division. However, exponentiation actually has two inverse operations: root and logarithm. The reason why this is the case has to do with the fact that the exponentiation is not commutative.
If x+2=3, then one can use subtraction to find out that x=3−2. This is the same if 2+x=3: one also gets x=3−2. This is because x+2 is the same as 2+x.
If x · 2=3, then one can use division to find out that x=
{\textstyle {\frac {3}{2}}}
. This is the same if 2 · x=3: one also gets x=
{\textstyle {\frac {3}{2}}}
. This is because x · 2 is the same as 2 · x.
If x²=3, then one can use the (square) root to find out that x =
{\textstyle {\sqrt {3}}}
. However, if 2x=3, then one cannot use the root to find out x. Rather, one has to use the (binary) logarithm to find out that x=log2(3).
This is because 2x usually is not the same as x2 (for example, 25=32 but 5²=25).
Logarithms can make multiplication and division of large numbers easier, because adding logarithms is the same as multiplying, and subtracting logarithms is the same as dividing.
Before calculators became popular and common, people used logarithm tables in books to multiply and divide.[2] The same information in a logarithm table was available on a slide rule, a tool with logarithms written on it.
Aside from computations, logarithm also has many other real-life applications:
Logarithmic spirals are common in nature. Examples include the shell of a nautilus, or the arrangement of seeds on a sunflower.
In chemistry, the negative of the base-10 logarithm of the activity of hydronium ions (H3O+, the form H+ takes in water) is the measure known as pH. The activity of hydronium ions in neutral water is 10−7 mol/L at 25 °C, hence a pH of 7. (This is a result of the equilibrium constant, the product of the concentration of hydronium ions and hydroxyl ions, in water solutions being 10−14 M2.)
The Richter scale measures earthquake intensity on a base-10 logarithmic scale.[1]
Musical intervals are measured logarithmically as semitones. The interval between two notes in semitones is the base-21/12 logarithm of the frequency ratio (or equivalently, 12 times the base-2 logarithm). Fractional semitones are used for non-equal temperaments. Especially to measure deviations from the equal tempered scale, intervals are also expressed in cents (hundredths of an equally-tempered semitone). The interval between two notes in cents is the base-21/1200 logarithm of the frequency ratio (or 1200 times the base-2 logarithm). In MIDI, notes are numbered on the semitone scale (logarithmic absolute nominal pitch with middle C at 60). For microtuning to other tuning systems, a logarithmic scale is defined filling in the ranges between the semitones of the equal tempered scale in a compatible way. This scale corresponds to the note numbers for whole semitones. (see microtuning in MIDI Archived 2008-02-12 at the Wayback Machine).
Common logarithmsEdit
Logarithms to base 10 are called common logarithms. They are usually written without the base. For example:
{\displaystyle \log(100)=2\ }
{\displaystyle 10^{2}=100\ }
Natural logarithmsEdit
Logarithms to base e are called natural logarithms. The number e is nearly 2.71828, and is also called the Eulerian constant after the mathematician Leonhard Euler.
The natural logarithms can take the symbols
{\displaystyle \log _{e}(x)\,}
{\displaystyle \ln(x)\,}
.[3] Some authors prefer the use of natural logarithms as
{\displaystyle \log(x)}
,[4] but usually mention this on preface pages.
Common bases for logarithmsEdit
{\displaystyle \operatorname {ld} }
Very common in computer science (binary)
{\displaystyle \ln }
{\displaystyle \log }
The base of this is the Eulerian constant e. This is the most common logarithm used in pure mathematics.
{\displaystyle \log _{10}}
{\displaystyle \log }
(sometimes also written as
{\displaystyle \lg }
) Used in some sciences like chemistry and biology.
any number, n
{\displaystyle \log _{n}}
This is the general way to write logarithms
Properties of logarithmsEdit
Logarithms have many properties. For example:
Properties from the definition of a logarithmEdit
This property is straight from the definition of a logarithm:
{\displaystyle \log _{n}(n^{a})=a}
{\displaystyle \log _{2}(2^{3})=3}
{\displaystyle \log _{2}{\bigg (}{\frac {1}{2}}{\bigg )}=-1}
{\displaystyle {\frac {1}{2}}=2^{-1}}
The logarithm to base b of a number a, is the same as the logarithm of a divided by the logarithm of b. That is,
{\displaystyle \log _{b}(a)={\frac {\log(a)}{\log(b)}}}
For example, let a be 6 and b be 2. With calculators we can show that this is true (or at least very close):
{\displaystyle \log _{2}(6)={\frac {\log(6)}{\log(2)}}}
{\displaystyle \log _{2}(6)\approx 2.584962}
{\displaystyle 2.584962\approx {\frac {0.778151}{0.301029}}\approx 2.584970}
The results above had a small error, but this was due to the rounding of numbers.
Since it is hard to picture the natural logarithm, we find that, in terms of a base-ten logarithm:
{\displaystyle \ln(x)={\frac {\log(x)}{\log(e)}}\approx {\frac {\log(x)}{0.434294}}}
, where 0.434294 is an approximation for the logarithm of e.
Operations within logarithm argumentsEdit
Logarithms which multiply inside their argument can be changed as follows:
{\displaystyle \log(ab)=\log(a)+\log(b)}
{\displaystyle \log(1000)=\log(10\cdot 10\cdot 10)=\log(10)+\log(10)+\log(10)=1+1+1=3}
Similarly, logarithm which divides inside the argument can be turned into a difference of logarithm (because it is the inverse operation of multiplication):
{\displaystyle \log {\bigg (}{\frac {a}{b}}{\bigg )}=\log(a)-\log(b)}
Logarithm tables, slide rules, and historical applicationsEdit
Before electronic computers, logarithms were used every day by scientists. Logarithms helped scientists and engineers in many fields such as astronomy.
Before computers, the table of logarithms was an important tool.[5] In 1617, Henry Briggs printed the first logarithm table. This was soon after Napier's basic invention. Later, people made tables with better scope and precision. These tables listed the values of logb(x) and bx for any number x in a certain range, at a certain precision, for a certain base b (usually b = 10). For example, Briggs' first table contained the common logarithms of all integers in the range 1–1000, with a precision of 8 digits.
Since the function f(x) = bx is the inverse function of logb(x), it has been called the antilogarithm.[6] People used these tables to multiply and divide numbers. For example, a user looked up the logarithm in the table for each of two positive numbers. Adding the numbers from the table would give the logarithm of the product. The antilogarithm feature of the table would then find the product based on its logarithm.
For manual calculations that need precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier ways.
Many logarithm tables give logarithms by separately providing the characteristic and mantissa of x, that is to say, the integer part and the fractional part of log10(x).[7] The characteristic of 10 · x is one plus the characteristic of x, and their significands are the same. This extends the scope of logarithm tables: given a table listing log10(x) for all integers x ranging from 1 to 1000, the logarithm of 3542 is approximated by
{\displaystyle \log _{10}(3542)=\log _{10}(10\cdot 354.2)=1+\log _{10}(354.2)\approx 1+\log _{10}(354).\,}
Another critical application was the slide rule, a pair of logarithmically divided scales used for calculation, as illustrated here:
Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to x is proportional to the logarithm of x.
Numbers are marked on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms. For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. Many engineers and scientists used slide rules until the 1970s. Scientists can work faster using a slide rule than using a logarithm table.[8]
Wikimedia Commons has media related to Logarithm.
↑ 1.0 1.1 "The Ultimate Guide to Logarithm — Theory & Applications". Math Vault. 2016-05-08. Retrieved 2020-08-29.
↑ 2.0 2.1 2.2 "logarithm | Rules, Examples, & Formulas". Encyclopedia Britannica. Retrieved 2020-08-29.
↑ Weisstein, Eric W. "Natural Logarithm". mathworld.wolfram.com. Retrieved 2020-08-29.
↑ Abramowitz, Milton; Stegun, Irene A., eds. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (10th ed.), New York: Dover Publications, ISBN 978-0-486-61272-0 , section 4.7., p. 89
↑ Maor, Eli (2009), E: The Story of a Number, Princeton University Press, ISBN 978-0-691-14134-3, sections 1, 13
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Logarithm&oldid=7315594"
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Group 2 Compounds: Meaning, Uses, Examples & Properties
Group 2 metals, also known as alkaline earth metals, have the ability to form a wide range of compounds with a wide range of uses. From chemical analysis to medical diagnosis, these compounds have several uses in daily life. But how do these compounds form, and why do their chemical properties make them suitable for such uses?
Firstly, we will look at what group 2 elements are, and summarise their properties.
After that, we will look at some key examples of group 2 compounds to get an idea of what they are.
Then, we shall explore the reactions of group 2 compounds with water and acid.
We will then look into the solubility of group 2 compounds.
Finally, we will outline some of the uses of group 2 compounds.
Group 2 elements also known as the 'alkaline earth metals' are a part of the s-block on the periodic table, having two electrons in the outermost shell. When the metal atoms react to form ions, they lose the two outer electrons and form 2+ ions.
Periodic table highlighting the group 2 metals. Sahraan Khowaja, StudySmarter Originals.
· Group 2 elements have two electrons in their outer shell and these electrons are found in the outer s-orbital.
· When these group 2 elements react, they lose their outer electrons and gain a charge of 2+.
· The melting points of group 2 elements generally decrease as you go down the group.
· First ionisation energy decreases as you go down group 2.
· Electronegativity also decreases as you go down group 2.
· The reactivity of group 2 elements increases as you go down the group.
Check out the 'group 2' explanation for more information.
What are group 2 compounds?
Elements of group 2 can form compounds that form when elements are combined together. Let's recall some common group 2 compounds.
Calcium hydroxide, Ca(OH)2, is one well-known example. Calcium hydroxide is a colourless crystal produced by mixing water and quicklime.
Magnesium carbonate, MgCO3, is another example. This is a white solid and a lot of its forms exist as minerals.
Strontium oxide, SrO, is formed by the reaction between strontium and oxygen. This compound is useful in ceramic applications.
The metals magnesium to barium react with water.
Beryllium does not react with water.
The rest of the group 2 metals are able to react with cold water and form metal hydroxides and hydrogen.
the metal dissolving and it dissolves faster going down the group
the solution heating up
with calcium, the solution becomes more saturated, and a white precipitate forms.
The oxides of group 2 metals are basic oxides apart from beryllium oxide. They react with water to form metal hydroxides and the solutions are strongly alkaline due to the hydroxide ions (OH-).
CaO(s) + H20(l) —› Ca(OH)2(aq)
The hydroxides of group 2 elements magnesium to barium are soluble in water and form alkaline solutions. The solutions are strongly alkaline due to the presence of the hydroxide ions, OH-. The products of this reaction always has the same formula of M(OH)2, where M is a Group 2 metal.
The carbonates of the group 2 metals are insoluble in water except for BeCO3.
The oxides of group 2 metals react with acids to form salts.
When group 2 hydroxides react with dilute acids then the product is a colourless solution of metal salts.
The group 2 carbonates react with dilute acid to produce a salt, water and carbon dioxide. For example,
{\mathrm{MgCO}}_{3}+2\mathrm{HCl}\to {\mathrm{MgCl}}_{2}+{\mathrm{H}}_{2}\mathrm{O}+{\mathrm{CO}}_{2}
The group 2 carbonates decompose when heated to form the oxide and carbon dioxide.
CaCO3(s) —› CaO(s) + CO2(g)
Solubility of Group 2 Compounds
The group 2 hydroxides become more soluble as you go down the group. Due to this, the pH of the solution increases as the concentration of the OH- ions increases.
An equation that represents the hydroxide dissolving in water is:
X(OH)2(aq) —› X2+(aq) + 2OH-(aq)
In this instance, the X represents a group 2 element.
Solubility Of Sulphates
As you go down group 2, the solubility of the sulfates decreases. Magnesium is the most soluble in the group and barium is the least which makes it insoluble.
Uses Of Group 2 Compounds
Let’s talk about some uses of common group 2 compounds.
Magnesium oxide is a white solid and can be used as a heat-resistant ceramic to line furnaces due to its very high melting temperature. Magnesium oxide turns to magnesium hydroxide in water.
Since magnesium hydroxide is insoluble in water, it is the active ingredient in milk of magnesia, used as an antacid and laxative.
Calcium hydroxide forms an alkaline solution as it is slightly soluble in water and the alkaline solution is normally called limewater.
Barium hydroxide is sometimes used as an alkali in chemical analysis.
Barium sulfate (BaSO4) is insoluble and is used in medicine. It absorbs X-rays strongly and is used to diagnose disorders of the intestines and stomach. Due to its insolubility, it is not absorbed into the bloodstream from the gut.
Calcium hydroxide, Ca(OH)2, is used in agriculture in order to increase the pH of soil.
Calcium oxide, CaO, can be used to remove SO2 from flue gases. SO2 is formed when fossil fuels are burnt to produce electricity.
Acidified barium chloride, BaCl2, is used to test for sulfate ions. This is done by adding dilute hydrochloric acid to the solution followed by barium chloride solution. If a white precipitate is formed, then this shows that the sulfate ions are present in the compound.
Group 2 Compounds - Key takeaways
When the group 2 metal atoms react to form ions, they lose the two outer electrons and form 2+ ions.
Elements of group 2 can form compounds that form when elements are combined together.
Beryllium does not react with water, but the rest of the group 2 metals are able to react with cold water and form metal hydroxides and hydrogen gas.
The group 2 sulfates become less soluble as you go down the group.
Barium sulfate (BaSO4) is insoluble and therefore has uses in the medical sector.
When the group 2 metals react to form ions, they lose the two outer electrons and form 2+ ions. Elements of group 2 can form compounds when elements are combined together.
What are some uses of Group 2 compounds?
Barium sulfate (BaSO4), which is insoluble, is used in medicine. It absorbs X-rays strongly and is used to diagnose disorders of the intestines and stomach. Due to its insolubility, it is not absorbed into the bloodstream from the gut.
Magnesium hydroxide is the active ingredient in milk of magnesia, which is used as an antacid and laxative.
What is the solubility of Group 2 compounds?
- The group 2 hydroxides become more soluble as you go down the group.
- The group 2 sulfates become less soluble as you go down the group.
What are some common properties of group 2 compounds?
- The oxides of group 2 metals are basic oxides apart from beryllium oxide.
- The oxides of group 2 metals react with water to form metal hydroxides which are strongly alkaline solutions.
- The carbonates of the group 2 metals are insoluble in water except for BeCO3.
-The oxides of group 2 metals react with acids to form salts.
What are some common reactions of group 2 compounds?
- The oxides of group 2 metals, apart from beryllium oxide, react with water to form metal hydroxides.
- The oxides of group 2 metals react with acids to form salts.
- Group 2 hydroxides react with dilute acids. The product is a colourless solution of metal salts.
- The group 2 carbonates decompose when heated to form the oxide and carbon dioxide.
Final Group 2 Compounds Quiz
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Quantile predictions for out-of-bag observations from bag of regression trees - MATLAB - MathWorks Deutschland
oobQuantilePredict
Predict Out-of-Bag Medians Using Quantile Regression
Estimate Out-of-Bag Prediction Intervals Using Percentiles
Estimate Out-of-Bag Conditional Cumulative Distribution Using Quantile Regression
Quantile predictions for out-of-bag observations from bag of regression trees
YFit = oobQuantilePredict(Mdl)
YFit = oobQuantilePredict(Mdl,Name,Value)
[YFit,YW] = oobQuantilePredict(___)
YFit = oobQuantilePredict(Mdl) returns a vector of medians of the predicted responses at all out-of-bag observations in Mdl.X, the predictor data, and using Mdl, which is a bag of regression trees. Mdl must be a TreeBagger model object and Mdl.OOBIndices must be nonempty.
YFit = oobQuantilePredict(Mdl,Name,Value) uses additional options specified by one or more Name,Value pair arguments. For example, specify quantile probabilities or trees to include for quantile estimation.
[YFit,YW] = oobQuantilePredict(___) also returns a sparse matrix of response weights using any of the previous syntaxes.
Quantile probability, specified as the comma-separated pair consisting of 'Quantile' and a numeric vector containing values in the interval [0,1]. For each observation (row) in Mdl.X, oobQuantilePredict estimates corresponding quantiles for all probabilities in Quantile.
For 'all', oobQuantilePredict uses the indices 1:Mdl.NumTrees.
Estimated quantiles for out-of-bag observations, returned as an n-by-numel(tau) numeric matrix. n is the number of observations in the training data (numel(Mdl.Y)) and tau is the value of the Quantile name-value pair argument. That is, YFit(j,k) is the estimated 100*tau(k) percentile of the response distribution given X(j,:) and using Mdl.
Response weights, returned as an n-by-n sparse matrix. n is the number of responses in the training data (numel(Mdl.Y)). YW(:,j) specifies the response weights for the observation in Mdl.X(j,:).
oobQuantilePredict predicts quantiles using linear interpolation of the empirical cumulative distribution function (cdf). For a particular observation, you can use its response weights to estimate quantiles using alternative methods, such as approximating the cdf using kernel smoothing.
Load the carsmall data set. Consider a model that predicts the fuel economy (in MPG) of a car given its engine displacement.
Train an ensemble of bagged regression trees using the entire data set. Specify 100 weak learners and save out-of-bag indices.
Mdl = TreeBagger(100,Displacement,MPG,'Method','regression',...
Perform quantile regression to predict the out-of-bag median fuel economy for all training observations.
oobMedianMPG = oobQuantilePredict(Mdl);
oobMedianMPG is an n-by-1 numeric vector of medians corresponding to the conditional distribution of the response given the sorted observations in Mdl.X. n is the number of observations, size(Mdl.X,1).
Sort the observations in ascending order. Plot the observations and the estimated medians on the same figure. Compare the out-of-bag median and mean responses.
[sX,idx] = sort(Mdl.X);
oobMeanMPG = oobPredict(Mdl);
plot(sX,oobMedianMPG(idx));
plot(sX,oobMeanMPG(idx),'r--');
legend('Data','Out-of-bag median','Out-of-bag mean');
Load the carsmall data set. Consider a model that predicts the fuel economy of a car (in MPG) given its engine displacement.
Perform quantile regression to predict the out-of-bag 2.5% and 97.5% percentiles.
oobQuantPredInts = oobQuantilePredict(Mdl,'Quantile',[0.025,0.975]);
oobQuantPredInts is an n-by-2 numeric matrix of prediction intervals corresponding to the out-of-bag observations in Mdl.X. n is number of observations, size(Mdl.X,1). The first column contains the 2.5% percentiles and the second column contains the 97.5% percentiles.
Plot the observations and the estimated medians on the same figure. Compare the percentile prediction intervals and the 95% prediction intervals, assuming the conditional distribution of MPG is Gaussian.
[oobMeanMPG,oobSTEMeanMPG] = oobPredict(Mdl);
STDNPredInts = oobMeanMPG + [-1 1]*norminv(0.975).*oobSTEMeanMPG;
h2 = plot(sX,oobQuantPredInts(idx,:),'b');
h3 = plot(sX,STDNPredInts(idx,:),'r--');
Estimate the out-of-bag response weights.
[~,YW] = oobQuantilePredict(Mdl);
YW is an n-by-n sparse matrix containing the response weights. n is the number of training observations, numel(Y). The response weights for the observation in Mdl.X(j,:) are in YW(:,j). Response weights are independent of any specified quantile probabilities.
Estimate the out-of-bag, conditional cumulative distribution function (ccdf) of the responses by:
ccdf(:,j) is the empirical out-of-bag ccdf of the response, given observation j.
Choose a random sample of four training observations. Plot the training sample and identify the chosen observations.
[randX,idx] = datasample(Mdl.X,4);
plot(randX,Mdl.Y(idx),'*','MarkerSize',10);
text(randX-10,Mdl.Y(idx)+1.5,{'obs. 1' 'obs. 2' 'obs. 3' 'obs. 4'});
Plot the out-of-bag ccdf for the four chosen responses in the same figure.
plot(sortY,ccdf(:,idx));
legend('ccdf given obs. 1','ccdf given obs. 2',...
'ccdf given obs. 3','ccdf given obs. 4',...
title('Out-of-Bag Conditional Cumulative Distribution Functions')
L={Q}_{1}-1.5*IQR
U={Q}_{3}+1.5*IQR,
{w}_{tj}\left(x\right)=\frac{I\left\{{X}_{j}\in {S}_{t}\left(x\right)\right\}}{\sum _{k=1}^{{n}_{\text{train}}}I\left\{{X}_{k}\in {S}_{t}\left(x\right)\right\}},
{w}_{j}^{\ast }\left(x\right)=\frac{1}{T}\sum _{t=1}^{T}{w}_{tj}\left(x\right).
oobQuantilePredict estimates out-of-bag quantiles by applying quantilePredict to all observations in the training data (Mdl.X). For each observation, the method uses only the trees for which the observation is out-of-bag.
For observations that are in-bag for all trees in the ensemble, oobQuantilePredict assigns the sample quantile of the response data. In other words, oobQuantilePredict does not use quantile regression for out-of-bag observations. Instead, it assigns quantile(Mdl.Y,tau), where tau is the value of the Quantile name-value pair argument.
[2] Breiman, L. “Random Forests.” Machine Learning. Vol. 45, 2001, pp. 5–32.
predict | quantilePredict | oobQuantileError | TreeBagger
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Decile Definition
A decile is a quantitative method of splitting up a set of ranked data into 10 equally large subsections. This type of data ranking is performed as part of many academic and statistical studies in the finance and economics fields. The data may be ranked from largest to smallest values, or vice versa.
A decile, which has 10 categorical buckets may be contrasted with percentiles that have 100, quartiles that have four, or quintiles that have five.
A decile is a quantitative method of splitting up a set of ranked data into 10 equally large subsections.
A decile rank arranges the data in order from lowest to highest and is done on a scale of one to 10 where each successive number corresponds to an increase of 10 percentage points.
This type of data ranking is performed as part of many academic and statistical studies in the finance and economics fields.
Understanding a Decile
In descriptive statistics, a decile is used to categorize large data sets from the highest to lowest values, or vice versa. Like the quartile and the percentile, a decile is a form of a quantile that divides a set of observations into samples that are easier to analyze and measure.
While quartiles are three data points that divide an observation into four equal groups or quarters, a decile consists of nine data points that divide a data set into 10 equal parts. When an analyst or statistician ranks data and then splits them into deciles, they do so in an attempt to discover the largest and smallest values by a given metric.
For example, by splitting the entire S&P 500 Index into deciles (50 firms in each decile) using the P/E multiple, the analyst will discover the companies with the highest and lowest P/E valuations in the index.
A decile is usually used to assign decile ranks to a data set. A decile rank arranges the data in order from lowest to highest and is done on a scale of one to 10 where each successive number corresponds to an increase of 10 percentage points. In other words, there are nine decile points. The 1st decile, or D1, is the point that has 10% of the observations below it, D2 has 20% of the observations below it, D3 has 30% of the observations falling below it, and so on.
How to Calculate a Decile
There is no one way of calculating a decile; however, it is important that you are consistent with whatever formula you decide to use to calculate a decile. One simple calculation of a decile is:
\begin{aligned} &\text{D1} = \text{Value of } \left [ \frac{ n + 1 }{ 10 } \right ] \text{th Data} \\ \end{aligned}
D1=Value of [10n+1]th Data
\begin{aligned} &\text{D2} = \text{Value of } \left [ \frac{ 2 \times ( n + 1 ) }{ 10 } \right ] \text{th Data} \\ \end{aligned}
D2=Value of [102×(n+1)]th Data
\begin{aligned} &\text{D3} = \text{Value of } \left [ \frac{ 3 \times ( n + 1 ) }{ 10 } \right ] \text{th Data} \\ \end{aligned}
\begin{aligned} &\text{D9} = \text{Value of } \left [ \frac{ 9 \times ( n + 1 ) }{ 10 } \right ] \text{th Data} \\ \end{aligned}
From this formula, it is given that the 5th decile is the median since 5 (n+1) / 10 is the data point that represents the halfway point of the distribution.
Deciles in Finance and Economics
Deciles are used in the investment field to assess the performance of a portfolio or a group of mutual funds. The decile rank acts as a comparative number that measures the performance of an asset against similar assets.
For example, say an analyst is evaluating the performance of a set of mutual funds over time, a mutual fund that is ranked five on a decile scale of one to 10 means it’s in the top 50%. By splitting the mutual funds into deciles, the analyst can review the best and worst-performing mutual funds for a given time period, arranged from the smallest to highest average return on investment.
The government also uses deciles to determine the level of income inequality in the country, that is, how income is distributed. For example, if the top 20 wage earners of a country of 50,000 citizens fall in the 10th decile and earn more than 50% of the total income in the country, one can conclude that there is a very high degree of income inequality in that country. In this case, the government can introduce measures to decrease the wage gap, such as increasing the income tax of the rich and creating estate taxes to limit how much wealth can be passed on to beneficiaries as an inheritance.
Example of a Decile
The table below shows the ungrouped scores (out of 100) for 30 exam takers:
Using the information presented in the table, the 1st decile can be calculated as:
= Value of [(30 + 1) / 10]th data
= Value of 3.1st data, which is 0.1 of the way between scores 55 and 57
= 55 + 2 (0.1) = 55.2 = D1
D1 means that 10% of the data set falls below 55.2.
Let’s calculate the 3rd decile:
D3 = Value of 3 (30 + 1) / 10
D3 = Value of 9.3rd position, which is 0.3 between the scores of 65 and 66
Thus, D3 = 65 + 1 (0.3) = 65.3
30% of the 30 scores in the observation fall below 65.3.
What would we get if we were to calculate the 5th decile?
D5 = Value of 15.5th position, halfway between scores 76 and 78
50% of the scores fall below 77.
Also, notice how the 5th decile is also the median of the observation. Looking at the data set in the table, the median, which is the middle data point of any given set of numbers, can be calculated as (76 + 78) / 2 = 77 = median = D5. At this point, half of the scores lie above and below the distribution.
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Home : Support : Online Help : Education : Student Packages : Statistics : Median
Median(A, numeric_options, output_option)
Median(M, numeric_options, output_option)
Median(X, numeric_options, output_option)
The Median function computes the median of the specified random variable or data sample.
The first parameter can be a Vector, a list, a Matrix, a random variable, or an algebraic expression involving random variables (see Student[Statistics][RandomVariable]).
In the second calling sequence, if X is a discrete random variable, then the median is defined as the first point
such that the CDF at t is greater than or equal to
\frac{1}{2}
If the median is determined by only one data point in the data set, then the median equals that data point.
If the median is determined by two data points in the data set, and at least one of them has floating point values or the option numeric is included , then the computation is done in floating point. Otherwise the computation is exact.
By default, the median is computed according to the rules mentioned above. To always compute the median numerically, specify the numeric or numeric = true option.
\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):
Compute the median of the Normal distribution with parameters
p
q
\mathrm{Median}\left(\mathrm{NormalRandomVariable}\left(p,q\right)\right)
\textcolor[rgb]{0,0,1}{p}
\mathrm{Median}\left(\mathrm{NormalRandomVariable}\left(3,5\right)\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{Median}\left(\mathrm{NormalRandomVariable}\left(\mathrm{\pi },5\right),\mathrm{numeric}\right)
\textcolor[rgb]{0,0,1}{3.141592654}
\mathrm{Median}\left(\mathrm{NormalRandomVariable}\left(\mathrm{\pi },5\right),\mathrm{numeric},\mathrm{output}=\mathrm{plot}\right)
Compute the median of the given data. Since
\mathrm{\pi }
4
are the numbers that are eventually used to compute the median, and they are both exact values (not using floating point values), the result is an exact expression.
\mathrm{Median}\left(\mathrm{Vector}[\mathrm{row}]\left([\mathrm{\pi },1,4,6.01]\right)\right)
\frac{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}
\frac{26}{3}
22.0
are the numbers that are eventually used to compute the median, and
22.0
is a floating point value, the result will be a floating point value.
\mathrm{Median}\left([34.2,22.0,2,\frac{26}{3}]\right)
\textcolor[rgb]{0,0,1}{15.33333333}
If the output=both option is included, then both the value of the median and its plot will be returned.
\mathrm{median},\mathrm{graph}≔\mathrm{Median}\left([34.2,22.0,2,\frac{26}{3}],\mathrm{output}=\mathrm{both}\right)
\textcolor[rgb]{0,0,1}{\mathrm{median}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{graph}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{15.33333333}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PLOT}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{...}}\right)
\mathrm{median}
\textcolor[rgb]{0,0,1}{15.33333333}
\mathrm{graph}
M≔\mathrm{Matrix}\left([[3,5.0,\mathrm{ln}\left(300\right),3],[2,2\mathrm{\pi },9,3],[1,\mathrm{sqrt}\left(26\right),4,3.0],[5,7,7.0,3]]\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{5.0}& \textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{300}\right)& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\pi }}& \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{1}& \sqrt{\textcolor[rgb]{0,0,1}{26}}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{3.0}\\ \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{7}& \textcolor[rgb]{0,0,1}{7.0}& \textcolor[rgb]{0,0,1}{3}\end{array}]
Compute the median of each of the columns according to the computation rules.
\mathrm{Median}\left(M\right)
[\begin{array}{cccc}\frac{\textcolor[rgb]{0,0,1}{5}}{\textcolor[rgb]{0,0,1}{2}}& \frac{\sqrt{\textcolor[rgb]{0,0,1}{26}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\pi }}& \textcolor[rgb]{0,0,1}{6.351891238}& \textcolor[rgb]{0,0,1}{3.000000000}\end{array}]
The Student[Statistics][Median] command was introduced in Maple 18.
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what is gravity please explain - Physics - Nuclei - 16910733 | Meritnation.com
what is gravity? please explain
Gravity is the force by which two bodies attract themselves by the virtue of their mass. This force is directly proportional to the product of the mass of the two bodies and is inversely proportional to the square of the distance between them.
F=G\frac{{m}_{1}{m}_{2}}{{R}^{2}}\phantom{\rule{0ex}{0ex}}where,\phantom{\rule{0ex}{0ex}}G = Gravitational cons\mathrm{tan}t=6.67 × {10}^{-11} {m}^{3} k{g}^{-1} {s}^{-2}\phantom{\rule{0ex}{0ex}}{m}_{1} and {m}_{2} are the masses of the two bodies\phantom{\rule{0ex}{0ex}}R is the dis\mathrm{tan}ce between them.
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What Is the International Fisher Effect?
The International Fisher Effect (IFE) is an economic theory stating that the expected disparity between the exchange rate of two currencies is approximately equal to the difference between their countries' nominal interest rates.
The International Fisher Effect (IFE) states that differences in nominal interest rates between countries can be used to predict changes in exchange rates.
According to the IFE, countries with higher nominal interest rates experience higher rates of inflation, which will result in currency depreciation against other currencies.
In practice, evidence for the IFE is mixed and in recent years direct estimation of currency exchange movements from expected inflation is more common.
Understanding the International Fisher Effect (IFE)
The IFE is based on the analysis of interest rates associated with present and future risk-free investments, such as Treasuries, and is used to help predict currency movements. This is in contrast to other methods that solely use inflation rates in the prediction of exchange rate shifts, instead functioning as a combined view relating inflation and interest rates to a currency's appreciation or depreciation.
The theory stems from the concept that real interest rates are independent of other monetary variables, such as changes in a nation's monetary policy, and provide a better indication of the health of a particular currency within a global market. The IFE provides for the assumption that countries with lower interest rates will likely also experience lower levels of inflation, which can result in increases in the real value of the associated currency when compared to other nations. By contrast, nations with higher interest rates will experience depreciation in the value of their currency.
This theory was named after U.S. economist Irving Fisher.
Calculating the International Fisher Effect
IFE is calculated as:
\begin{aligned}&E=\frac{i_1-i_2}{1+i_2}\ \approx\ i_1-i_2\\&\textbf{where:}\\&E=\text{the percent change in the exchange rate}\\&i_1=\text{country A's interest rate}\\&i_2=\text{country B's interest rate}\end{aligned}
E=1+i2i1−i2 ≈ i1−i2where:E=the percent change in the exchange ratei1=country A’s interest rate
For example, if country A's interest rate is 10% and country B's interest rate is 5%, country B's currency should appreciate roughly 5% compared to country A's currency. The rationale for the IFE is that a country with a higher interest rate will also tend to have a higher inflation rate. This increased amount of inflation should cause the currency in the country with a higher interest rate to depreciate against a country with lower interest rates.
The Fisher Effect and the International Fisher Effect
The Fisher Effect and the IFE are related models but are not interchangeable. The Fisher Effect claims that the combination of the anticipated rate of inflation and the real rate of return are represented in the nominal interest rates. The IFE expands on the Fisher Effect, suggesting that because nominal interest rates reflect anticipated inflation rates and currency exchange rate changes are driven by inflation rates, then currency changes are proportionate to the difference between the two nations' nominal interest rates.
Application of the International Fisher Effect
Empirical research testing the IFE has shown mixed results, and it is likely that other factors also influence movements in currency exchange rates. Historically, in times when interest rates were adjusted by more significant magnitudes, the IFE held more validity. However, in recent years inflation expectations and nominal interest rates around the world are generally low, and the size of interest rate changes is correspondingly relatively small. Direct indications of inflation rates, such as consumer price indexes (CPI), are more often used to estimate expected changes in currency exchange rates.
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Ljung-Box Q-Test - MATLAB & Simulink - MathWorks 日本
{H}_{0}:{\mathrm{Ï}}_{1}={\mathrm{Ï}}_{2}=â¦={\mathrm{Ï}}_{m}=0.
mâ\mathrm{ln}\left(N\right)
Q\left(m\right)=N\left(N+2\right){â}_{h=1}^{m}\frac{{\stackrel{^}{\mathrm{Ï}}}_{h}^{2}}{Nâh}.
This is a modification of the Box-Pierce Portmanteau “Q†statistic [3]. Under the null hypothesis, Q(m) follows a
{\mathrm{Ï}}_{m}^{2}
{\mathrm{Ï}}^{2}
You can also test for conditional heteroscedasticity by conducting a Ljung-Box Q-test on a squared residual series. An alternative test for conditional heteroscedasticity is Engle’s ARCH test (archtest).
[1] Ljung, G. and G. E. P. Box. “On a Measure of Lack of Fit in Time Series Models.†Biometrika. Vol. 66, 1978, pp. 67–72.
[3] Box, G. E. P. and D. Pierce. “Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models.†Journal of the American Statistical Association. Vol. 65, 1970, pp. 1509–1526.
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RESOLFT - Wikipedia
RESOLFT, an acronym for REversible Saturable OpticaL Fluorescence Transitions, denotes a group of optical fluorescence microscopy techniques with very high resolution. Using standard far field visible light optics a resolution far below the diffraction limit down to molecular scales can be obtained.
With conventional microscopy techniques, it is not possible to distinguish features that are located at distances less than about half the wavelength used (i.e. about 200 nm for visible light). This diffraction limit is based on the wave nature of light. In conventional microscopes the limit is determined by the used wavelength and the numerical aperture of the optical system. The RESOLFT concept surmounts this limit by temporarily switching the molecules to a state in which they cannot send a (fluorescence-) signal upon illumination. This concept is different from for example electron microscopy where instead the used wavelength is much smaller.
2 Resolution below the diffraction-limit
3.1 STED Microscopy
3.2 GSD microscopy
3.3 SPEM and SSIM
3.4 RESOLFT with switchable proteins
3.5 RESOLFT with switchable organic dyes
Switching fluorescence on and off in specific areas results in enabling fluorescence in areas smaller than the diffraction limit. Rasterisation of the whole sample results in a pixel image with extremely high resolution.
RESOLFT microscopy is an optical microscopy with very high resolution that can image details in samples that cannot be imaged with conventional or confocal microscopy. Within RESOLFT the principles of STED microscopy[1][2] and GSD microscopy are generalized. Structures that are normally too close to each other to be distinguished are read out sequentially.
Within this framework all methods can be explained that operate on molecules that have at least two distinguishable states, where reversible switching between the two states is possible, and where at least one such transition can be optically induced.
In most cases fluorescent markers are used, where one state (A) which is bright, that is, generates a fluorescence signal, and the other state (B) is dark, and gives no signal. One transition between them can be induced by light (e.g. A→B, bright to dark).
The sample is illuminated inhomogeneously with the illumination intensity at one position being very small (zero under ideal conditions). Only at this place are the molecules never in the dark state B (if A is the pre-existing state) and remain fully in the bright state A. The area where molecules are mostly in the bright state can be made very small (smaller than the conventional diffraction limit) by increasing the transition light intensity (see below). Any signal detected is thus known to come only from molecules in the small area around the illumination intensity minimum. A high resolution image can be constructed by scanning the sample, i.e., shifting the illumination profile across the surface.[3]
The transition back from B to A can be either spontaneous or driven by light of another wavelength. The molecules have to be switchable several times in order to be present in state A or B at different times during scanning the sample. The method also works if the bright and the dark state are reversed, one then obtains a negative image.
Resolution below the diffraction-limit[edit]
RESOLFT principle: The sample is inhomogeneously illuminated (red line). The intensity is reduced in an area comparable to the classical resolution limit, but is (ideally) zero at only one position. The threshold intensity (blue line) required to switch the molecules to the dark state is only a fraction of the maximum intensity applied, so only molecules very close to the position of minimum intensity can be in the bright state. The green area in the right frame illustrates the size of the area where molecules are able to generate a signal.
In RESOLFT the area where molecules reside in state A (bright state) can be made arbitrarily small despite the diffraction-limit.
One has to illuminate the sample inhomogeneously so that an isolated zero intensity point is created. This can be achieved e.g. by interference.
At low intensities (lower than the blue line in the image) most marker molecules are in the bright state, if the intensity is above, most markers are in the dark state.
Upon weak illumination we see that the area where molecules remain in state A is still quite large because the illumination is so low that most molecules reside in state A. The shape of the illumination profile does not need to be altered. Increasing the illumination brightness already results in a smaller area where the intensity is below the amount for efficient switching to the dark state. Consequently, also the area where molecules can reside in state A is diminished. The (fluorescence) signal during a following readout originates from a very small spot and one can obtain very sharp images.
In the RESOLFT concept, the resolution can be approximated by
{\displaystyle \Delta d\simeq {\frac {\lambda }{\pi n{\sqrt {\frac {I}{Isat}}}}}}
, whereby
{\displaystyle I_{sat}}
is the characteristic intensity required for saturating the transition (half of the molecules remain in state A and half in state B), and
{\displaystyle I}
denotes the intensity applied. If the minima are produced by focusing optics with a numerical aperture
{\displaystyle {\mbox{NA}}=n\sin \alpha }
, the minimal distance at which two identical objects can be discerned is
{\displaystyle \Delta d={\frac {\lambda }{2n\cdot \sin \alpha \cdot {\sqrt {1+{\frac {I}{Isat}}}}}}}
which can be regarded as an extension of Abbe’s equation. The diffraction-unlimited nature of the RESOLFT family of concepts is reflected by the fact that the minimal resolvable distance
{\displaystyle \Delta d}
can be continuously decreased by increasing
{\displaystyle \varsigma ={\frac {I}{Isat}}}
. Hence the quest for nanoscale resolution comes down to maximizing this quantity. This is possible by increasing
{\displaystyle I}
or by lowering
{\displaystyle I_{sat}}
Different processes are used when switching the molecular states. However, all have in common that at least two distinguishable states are used. Typically the fluorescence property used marks the distinction of the states, however this is not essential, as absorption or scattering properties could also be exploited.[4]
STED Microscopy[edit]
(Main article STED microscopy)
Within the STED microscopy (STimulated Emission Depletion microscopy)[1][2] a fluorescent dye molecule is driven between its electronic ground state and its excited state while sending out fluorescence photons. This is the standard operation mode in fluorescence microscopy and depicts state A. In state B the dye is permanently kept in its electronic ground state through stimulated emission. If the dye can fluoresce in state A and not in state B, the RESOLFT concept applies.
GSD microscopy[edit]
(Main article GSD microscopy)
GSD microscopy (Ground State Depletion microscopy) also uses fluorescent markers. In state A, the molecule can freely be driven between the ground and the first excited state and fluorescence can be sent out. In the dark state B the ground state of the molecule is depopulated, a transition to a long lived excited state takes place from which fluorescence is not emitted. As long as the molecule is in the dark state, it's not available for cycling between ground and excited state, fluorescence is hence turned off.
SPEM and SSIM[edit]
SPEM (Saturated Pattern Excitation Microscopy)[5] and SSIM (Saturated Structured Illumination Microscopy)[6] are exploiting the RESOLFT concept using saturated excitation to produce "negative" images, i.e. fluorescence occurs from everywhere except at a very small region around the geometrical focus of the microscope. Also non point-like patterns are used for illumination. Mathematical image reconstruction is necessary to obtain positive images again.
RESOLFT with switchable proteins[edit]
Some fluorescent proteins can be switched on and off by light of an appropriate wavelength. They can be used in a RESOLFT-type microscope.[7] During illumination with light, these proteins change their conformation. In the process they gain or lose their ability to emit fluorescence. The fluorescing state corresponds to state A, the non-fluorescing to state B and the RESOLFT concept applies again. The reversible transition (e.g. from B back to A) takes place either spontaneously or again driven by light. Inducing conformational changes in proteins can be achieved already at much lower switching light intensities as compared to stimulated emission or ground state depletion (some W/cm²). In combination with 4Pi microscopy images with isotropic resolution below 40 nm have been taken of living cells at low light levels.[8]
RESOLFT with switchable organic dyes[edit]
Just as with proteins, also some organic dyes can change their structure upon illumination.[9][10] The ability to fluoresce of such organic dyes can be turned on and off through visible light. Again the applied light intensities can be quite low (some 100 W/cm²).
^ a b Stefan W. Hell & Jan Wichmann (1994). "Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy". Optics Letters. 19 (11): 780–2. Bibcode:1994OptL...19..780H. doi:10.1364/OL.19.000780. PMID 19844443.
^ a b Thomas A. Klar; Stefan W. Hell (1999). "Subdiffraction resolution in far-field fluorescence microscopy". Optics Letters. 24 (14): 954–956. Bibcode:1999OptL...24..954K. doi:10.1364/OL.24.000954. PMID 18073907.
^ See also Confocal laser scanning microscopy.
^ Stefan W. Hell (2004). "Strategy for far-field optical imaging and writing without diffraction limit". Physics Letters A. 326 (1–2): 140–145. Bibcode:2004PhLA..326..140H. doi:10.1016/j.physleta.2004.03.082.
^ Rainer Heintzmann; Thomas M. Jovin; Christoph Cremer (2002). "Saturated patterned excitation microscopy a concept for optical resolution improvement". Journal of the Optical Society of America A. 19 (8): 1599–2109. Bibcode:2002JOSAA..19.1599H. doi:10.1364/JOSAA.19.001599. hdl:11858/00-001M-0000-0029-2C24-9. PMID 12152701.
^ Mats G. L. Gustafsson (2005). "Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution". Proceedings of the National Academy of Sciences of the United States of America. 102 (37): 13081–6. Bibcode:2005PNAS..10213081G. doi:10.1073/pnas.0406877102. PMC 1201569. PMID 16141335.
^ Michael Hofmann; Christian Eggeling; Stefan Jakobs; Stefan W. Hell (2005). "Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins". Proceedings of the National Academy of Sciences of the United States of America. 102 (49): 17565–9. Bibcode:2005PNAS..10217565H. doi:10.1073/pnas.0506010102. PMC 1308899. PMID 16314572.
^ Ulrike Böhm; Stefan W. Hell; Roman Schmidt (2016). "4Pi-RESOLFT nanoscopy". Nature Communications. 7 (10504): 1–8. Bibcode:2016NatCo...710504B. doi:10.1038/ncomms10504. PMC 4740410. PMID 26833381.
^ Mariano Bossi; Jonas Fölling; Marcus Dyba; Volker Westphal; Stefan W. Hell (2006). "Breaking the diffraction resolution barrier in far-field microscopy by molecular optical bistability". New Journal of Physics. 8 (11): 275. Bibcode:2006NJPh....8..275B. doi:10.1088/1367-2630/8/11/275.
^ Jiwoong Kwon; Jihee Hwang; Jaewan Park; Gi Rim Han; Kyu Young Han; Seong Keun Kim (2015). "RESOLFT nanoscopy with photoswitchable organic fluorophores". Scientific Reports. 5: 17804. Bibcode:2015NatSR...517804K. doi:10.1038/srep17804. PMC 4671063. PMID 26639557.
Retrieved from "https://en.wikipedia.org/w/index.php?title=RESOLFT&oldid=1061989037"
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TrivialGroup - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : TrivialGroup
TrivialGroup()
TrivialGroup( form = F )
The trivial group is the unique (up to isomorphism) group of order
1
The TrivialGroup() command returns a trivial group.
The form = F option controls the type of group returned. By default, a permutation group (with empty generating set) is returned; this is the same as using the form = "permgroup" option. If the form = "fpgroup" option is passed, then a trivial finitely presented group is returned, with empty sets of generators and relators. Finally, you can explicitly request a symbolic trivial group by using the form = "symbolic" option.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
\mathrm{TrivialGroup}\left(\right)
〈〉
\mathrm{GroupOrder}\left(\mathrm{TrivialGroup}\left(\right)\right)
\textcolor[rgb]{0,0,1}{1}
T≔\mathrm{TrivialGroup}\left('\mathrm{form}'="fpgroup"\right)
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{≔}⟨\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{∣}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{}⟩
\mathrm{Generators}\left(T\right)
[]
\mathrm{Relators}\left(T\right)
\textcolor[rgb]{0,0,1}{\varnothing }
The GroupTheory[TrivialGroup] command was introduced in Maple 17.
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Find Global or Multiple Local Minima - MATLAB & Simulink - MathWorks í•œêµ
Single Global Minimum Via GlobalSearch
Multiple Local Minima Via MultiStart
This example illustrates how GlobalSearch finds a global minimum efficiently, and how MultiStart finds many more local minima.
The objective function for this example has many local minima and a unique global minimum. In polar coordinates, the function is
f\left(r,t\right)=g\left(r\right)h\left(t\right)
\begin{array}{l}g\left(r\right)=\left(\mathrm{sin}\left(r\right)-\frac{\mathrm{sin}\left(2r\right)}{2}+\frac{\mathrm{sin}\left(3r\right)}{3}-\frac{\mathrm{sin}\left(4r\right)}{4}+4\right)\frac{{r}^{2}}{r+1}\\ h\left(t\right)=2+\mathrm{cos}\left(t\right)+\frac{\mathrm{cos}\left(2t-\frac{1}{2}\right)}{2}.\end{array}
Plot the functions
g
h
, and create a surface plot of the function
f
fplot(@(r)(sin(r) - sin(2*r)/2 + sin(3*r)/3 - sin(4*r)/4 + 4) .* r.^2./(r+1), [0 20])
title(''); ylabel('g'); xlabel('r');
fplot(@(t)2 + cos(t) + cos(2*t-1/2)/2, [0 2*pi])
title(''); ylabel('h'); xlabel('t');
fsurf(@(x,y) sawtoothxy(x,y), [-20 20])
% sawtoothxy is defined in the first step below
xlabel('x'); ylabel('y'); title('sawtoothxy(x,y)');
The global minimum is at
r = 0
, with objective function 0. The function
g\left(r\right)
grows approximately linearly in
r
, with a repeating sawtooth shape. The function
h\left(t\right)
has two local minima, one of which is global.
The sawtoothxy.m file converts from Cartesian to polar coordinates, then computes the value in polar coordinates.
type sawtoothxy
To search for the global minimum using GlobalSearch, first create a problem structure. Use the 'sqp' algorithm for fmincon,
optimoptions(@fmincon,'Algorithm','sqp','Display','off'));
The start point is [100,-50] instead of [0,0] so GlobalSearch does not start at the global solution.
Validate the problem structure by running fmincon.
Create the GlobalSearch object, and set iterative display.
300 1564 1.547e-15 5.858e+04 1.074 Stage 2 Search
400 1664 1.547e-15 1.84e+05 4.16 Stage 2 Search
500 1764 1.547e-15 2.683e+04 11.84 Stage 2 Search
800 2064 1.547e-15 6.249e+04 163.8 Stage 2 Search
950 2356 1.547e-15 477 589.7 387 2 Stage 2 Local
952 2420 1.547e-15 368.4 477 250.7 2 Stage 2 Local
1000 2468 1.547e-15 4.031e+04 530.9 Stage 2 Search
The solver finds three local minima, including the global minimum near [0,0].
To search for multiple minima using MultiStart, first create a problem structure. Because the problem is unconstrained, use the fminunc solver. Set options not to show any display at the command line.
optimoptions(@fminunc,'Display','off'));
Validate the problem structure by running it.
Create a default MultiStart object.
Run the solver for 50 iterations, recording the local minima.
[x,fval,eflag,output,manymins] = run(ms,problem,50)
message: 'MultiStart completed some of the runs from the start points....'
manymins=1×10 object
1x10 GlobalOptimSolution array with properties:
The solver does not find the global minimum near [0,0]. The solver finds 10 distinct local minima.
Plot the function values at the local minima:
histogram([manymins.Fval],10)
Plot the function values at the three best points:
bestf = [manymins.Fval];
histogram(bestf(1:3),10)
MultiStart starts fminunc from start points with components uniformly distributed between –1000 and 1000. fminunc often gets stuck in one of the many local minima. fminunc exceeds its iteration limit or function evaluation limit 40 times.
GlobalSearch | MultiStart | createOptimProblem
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Transition Year in the Irish school system: Challenges and opportunities for teaching and learning mathematics | EMS Magazine
2 The Transition Year curriculum
3 Scope of TY mathematics
4 Divisibility of integers – an interesting topic for TY
This article gives an overview of the Irish secondary school Transition Year programme and in particular how the teaching and learning of mathematics is situated within it.
Second-level education in Ireland caters for students between 12 and 18 years of age. It consists of two cycles, the Junior Cycle and Senior Cycle. The 3-year Junior Cycle prepares students for state examinations in the core subjects of Mathematics, Irish and English, as well as in students’ individually chosen subjects. Following the Junior Cycle, students have the option of enrolling on a 1-year Transition Year (TY) programme or progressing directly to the Senior Cycle. The 2-year programme of the Senior Cycle culminates in the terminal state examination, the Leaving Certificate, a qualification which heavily influences the destination of students for further education and training.
In this article, we give an overview of the TY programme and consider the potential for introducing some interesting mathematics within it. TY is an optional extra year of study that is taken up by the majority of students before embarking on two years of study for the Leaving Certificate1According to [2 A. Clerkin, Personal development in secondary education: the Irish Transition Year. Education Policy Analysis Archives 20.38 (2012) ], about 55 % of students took TY in 2010/11. Although there is consensus that there is an upward trend in participation rates, no data is currently available to confirm this.. The official mission of TY is [6 National Council for Curriculum and Assessment, Transition Year programmes – guidelines for schools, ncca.ie/en/resources/ty_transition_year_school_guidelines/ (2021) , p. 3]:
It is government policy that each school design its own TY programme to meet the needs of its students. However, the Department of Education (the government ministry responsible for education) has produced guidelines as to what such a programme should entail. Four areas of learning are suggested [8 Professional Development Service for Teachers, TY Curriculum, pdst.ie/TY/curriculum (2021) ]:
Core: This includes Mathematics along with English, Irish, Information Technology and Physical Education
Subject sampling: These may be taster modules in subjects to help students decide on subject choices for Leaving Certificate. Students may also be exposed to subjects that they have not studied before e.g. Environmental Studies, Japanese, Arabic, Drama, Photography, Global Development
TY-specific: Study and activities which allow students to pursue their own interests and to contribute to their local communities e.g. social entrepreneurship, charitable activities
Calendar: Work experience, foreign exchange visits, sporting activities, visiting speakers, drama/musical production
Over the years the programme has been very well received amongst students, parents, guardians and teachers alike and is considered unique in that there is no similar programme offered in education systems of other countries [2 A. Clerkin, Personal development in secondary education: the Irish Transition Year. Education Policy Analysis Archives 20.38 (2012) ]. In its 2020 report on education in Ireland, the OECD commended the programme as it “allows students to sample different subjects and undertake work experience and other projects, and helps guide them in choosing their upper secondary education subjects and future career path” [7 Organisation for Economic Co-operation and Development, Education in Ireland: an OECD assessment of the senior cycle review, oecd.org/ireland/education-in-ireland-636bc6c1-en.htm (2020) ].
For students, TY may be viewed as an academically low-risk interlude between state examinations because there is no state-level assessment of student performance. Many schools do issue individual assessment certificates that rate student attainment across the programme but these marks generally do not count towards future progression.
There is no prescribed syllabus for mathematics in TY and so, theoretically at least, teachers have great flexibility regarding the mathematics they present to their students. Government does not give specific recommendations as to what should be taught but a 2011 circular called for schools “to provide innovative learning opportunities and increased mathematics teaching hours to the extent feasible” in Transition Year [3 Department of Education, Circular 0058/0011, education.ie/ (2011) ]. Up to 2012, there had been very little information available on the content of TY mathematics programmes being delivered in schools, how they were delivered and the time spent on delivery. Then, as part of PISA 2012, a national survey of TY mathematics teachers was conducted.
3.1 PISA 2012 National Survey of TY Mathematics Teachers
Mathematics teachers were invited to complete questionnaires which focussed on concerns about the quality of TY mathematics offerings. In total, 1321 questionnaires were completed, an 80 % response rate, and the findings were reported in [5 G. Moran et al., Mathematics in Transition Year: insights of teachers from PISA 2012, erc.ie/documents/p12ty_maths_report.pdf (2013) ]. In the survey, teachers were asked to rate their level of agreement or disagreement with statements about the purpose of TY mathematics. It was found that only 31 % of teachers strongly agreed that one purpose of TY mathematics is to increase students’ confidence in their problem-solving ability, and only 34 % strongly agreed that one purpose was to encourage greater interest in mathematics. Almost 40 % disagreed or strongly disagreed that one purpose of TY mathematics is to familiarise students with the history of mathematics. These results were certainly out of kilter with government guidelines, as set out in [4 Department of Education, Transition Year programmes – guidelines for schools, education.ie/en/Schools-Colleges/Information/Curriculum-and-Syllabus/Transition-Year-/ty_transition_year_school_guidelines.pdf (1993) ], which state that these particular aspects of mathematics should be emphasised during TY. More importantly perhaps, they indicate a strong lack of agreement amongst teachers as to the purpose of teaching mathematics during TY.
3.2 Teaching resources and challenges
The national survey found that there was a great deal of variation in terms of resources that TY maths teachers used. Some teachers used textbooks from the Junior Cycle to consolidate previously covered content while others mined the Internet for engaging and challenging material. There was also evidence of teaching the Leaving Certificate syllabus, something the government guidance expressly discourages.
In the period from 2012 to 2018, we have had the opportunity of meeting with many TY teachers during outreach activities in secondary schools across Ireland. Teachers regularly commented on their struggle to engage students with mathematics, especially during TY. They reported that one view commonly held by students was that maths would not be relevant to their lives once their secondary schooling came to an end. Teachers agreed on the need for a book or module which would focus on the importance of the subject in the everyday world, introduce genuinely interesting ideas and also include some history of mathematics. The challenge of stoking the interest of teenagers who were sceptical about the global importance of maths (as well as a concern for students with genuine mathematical curiosity), encouraged us to write a book of projects to engage the interest of TY students2MIghTY Maths was published in October 2021 by Curriculum Development Unit, Mary Immaculate College, Limerick. Each topic can be investigated either in teacher-led lessons or in a guided project by the individual student or groups of students. The book is available for purchase at www.curriculumdevelopmentunit.com.. Exercises would need to be recreational as well as challenging in order to enthuse the target audience. In the next section, we present one of the topics presented in the book.
At the teacher training college where we work, it sometimes happens that a student teacher will express their delight to learn for the first time, during an elementary number theory course, that divisibility of a number by 3 can be determined by checking if the sum of its digits is divisible by 3. They may wonder why they had not learned this rule earlier in their careers or why their school knowledge was restricted to rules for 2, 5 and 10. In any case, the topic of divisibility appeared to be a good one for TY students and student teachers alike.
Like the rule for 3, there exists a similar rule for the divisibility of a number by 9 and it forms the basis of a wonderful trick called Mindreader, which appeared in [1 C. Budd and C. Sangwin, Mathematics galore! Oxford University Press (2001) ] and is used in a project on number tricks in our book.
The “mindreader” asks a volunteer to
pick any five-digit number where not all digits are the same and keep this number secret;
jumble up the digits to get a new number;
subtract the smaller of these numbers from the larger;
hide one of the digits,
H
, in the answer, making sure
H\neq 0
call out the other digits in the answer.
The “mindreader” pronounces the hidden digit to be
H
In order to know what the hidden digit
H
is, we need to understand some basic ideas.
Definition 1. The digital root of a number is the sum of all its digits.
Example 1. The digital root of
3046178
29
3+0+4+6+1+7+8=29
There follow two mathematical facts.
Lemma 1 (Divisibility rule for
9
). Take any number
x
d
be the digital root of
x
x
9\Leftrightarrow d
9
Lemma 2. For any number
x
, with digital root
d
x-d
9
31572869
41
31572869-41=31572828
has digital root
36=4\times 9
Armed with Lemmata 1 and 2, we can now look at understanding the details of how the trick Mindreader works. We follow the given steps for the volunteer:
Pick any five-digit number where not all digits are the same. Let
x
have digits
ABCDE
\begin{split}x&=10000A+1000B+100C+10D+E\\ &=(A+B+C+D+E)+9999A+999B+99C+9D\\ &=d+9k,\quad\textrm{where}\ d\ \textrm{is the digital root of}\ x\\[-3.0pt] &\hphantom{{}=d+9k,{}}\quad\textrm{and}\ k=1111A+111B+11C+D.\end{split}
Indeed, Lemma 2 tells us that any number equals its digital root plus some multiple of
9
Jumble up the digits to get a new number. Call the new number
y
y
CEDAB
, for example, so
\begin{split}y&=10000C+1000E+100D+10A+B\\ &=(C+E+D+A+B)+9999C+999E+99D+9A\\ &=(A+B+C+D+E)+9\ell\\ &=d+9\ell,\quad\textrm{where}\ d\ \textrm{is the digital root of}\ y\ (\textrm{and}\ x)\\[-3.0pt] &\hphantom{{}=d+9\ell,{}}\quad\textrm{and}\ \ell=1111C+111E+11D+A.\end{split}
Subtract the smaller of these numbers from the larger. Let us suppose that
x>y
x-y=(d+9k)-(d+9\ell)=9(k-\ell).
N.B. The number
x-y
9
. Therefore, by Lemma 1, its digital root is a multiple of
9
Hide one of the digits in your answer,
H
, making sure the hidden digit is not a zero.
Call out the remaining digits of the number
x-y
At this point, the “mindreader” has all but one of the digits that make up the digital root. All they need to do is add them and see what digit is required to sum to the next greatest multiple of
9
. For example, if they are told the digits are
0
1
3
8
. These sum to
12
. The next highest multiple of
9
18
so the hidden digit
H
18-12=6
. Or, if they are told the digits are
7
5
6
9
27
, already a multiple of
9
. Since it was specified that zero should not be hidden, they conclude that the hidden digit
H
9
Example 3. Here is a worked example. The volunteer
picks the number
x=47523
jumbles the digits of
x
to get the number
y=37254
x-y=47523-37254=10269
hides the digit
2
calls out the digits
1,0,6
9
Now the “mindreader” knows that
x-y
, being a multiple of 9, has digital root
d
a multiple of 9. They know that
d-H=1+0+6+9=16\Rightarrow H=d-16
. Since the next multiple of 9 greater than 16 is 18, they conclude that
H=18-16=2
. The “mindreader” pronounces that the hidden digit is
2
Remark 1. The Mindreader trick shown here uses 5-digit numbers. Is there anything special about
5
or would the trick work for any number of digits?
Finally, we give Table 4 as a sample detailed lesson plan for teaching the content of Mindreader.
The Transition Year programme in Irish secondary schools is a highly regarded programme. Maintaining the position of mathematics, a core subject in the programme, requires a great effort on the part of schools because there is no curriculum and guidelines on teaching content are rather vague. Individual teachers and schools go to great lengths to provide useful and interesting material, yet gaps clearly remain. We give an example of divisibility of integers as a topic that may engage TY students’ interest and we encourage the use of stimulating and recreational mathematical exercises and projects across the programme.
Ronan Flatley is Lecturer in Mathematics at Mary Immaculate College, Limerick, Ireland. He currently serves on the Committee of the Irish Mathematical Society. He is the author of MIghTY Maths, a book of maths projects for Transition Year students. ronan.flatley@mic.ul.ie
According to [2 A. Clerkin, Personal development in secondary education: the Irish Transition Year. Education Policy Analysis Archives 20.38 (2012) ], about 55 % of students took TY in 2010/11. Although there is consensus that there is an upward trend in participation rates, no data is currently available to confirm this.
MIghTY Maths was published in October 2021 by Curriculum Development Unit, Mary Immaculate College, Limerick. Each topic can be investigated either in teacher-led lessons or in a guided project by the individual student or groups of students. The book is available for purchase at www.curriculumdevelopmentunit.com.
C. Budd and C. Sangwin, Mathematics galore! Oxford University Press (2001)
A. Clerkin, Personal development in secondary education: the Irish Transition Year. Education Policy Analysis Archives 20.38 (2012)
Department of Education, Circular 0058/0011, education.ie/ (2011)
Department of Education, Transition Year programmes – guidelines for schools, education.ie/en/Schools-Colleges/Information/Curriculum-and-Syllabus/Transition-Year-/ty_transition_year_school_guidelines.pdf (1993)
G. Moran et al., Mathematics in Transition Year: insights of teachers from PISA 2012, erc.ie/documents/p12ty_maths_report.pdf (2013)
National Council for Curriculum and Assessment, Transition Year programmes – guidelines for schools, ncca.ie/en/resources/ty_transition_year_school_guidelines/ (2021)
Organisation for Economic Co-operation and Development, Education in Ireland: an OECD assessment of the senior cycle review, oecd.org/ireland/education-in-ireland-636bc6c1-en.htm (2020)
Professional Development Service for Teachers, TY Curriculum, pdst.ie/TY/curriculum (2021)
Ronan Flatley, Transition Year in the Irish school system: Challenges and opportunities for teaching and learning mathematics. Eur. Math. Soc. Mag. 123 (2022), pp. 41–44
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Genesis Settings
With Altair, beacon chain genesis is long behind us. Nevertheless, the ability to spin-up testnets is useful in all sorts of scenarios, so the Altair spec retains genesis functionality, now called initialisation.
The following parameters refer to the actual mainnet beacon chain genesis and I'll explain them in that context. When starting up new testnets, these will of course be changed. For example, see the configuration file for the Prater testnet.
GENESIS_DELAY uint64(604800) (7 days)
MIN_GENESIS_ACTIVE_VALIDATOR_COUNT is the minimum number of full validator stakes that must have been deposited before the beacon chain can start producing blocks. The number is chosen to ensure a degree of security. It allows for four 128 member committees per slot, rather than the 64 committees per slot eventually desired to support fully operational data shards. Fewer validators means higher rewards per validator, so it is designed to attract early participants to get things bootstrapped.
In the actual event of beacon chain genesis, there were 21,063 participating validators, comfortably exceeding the minimum necessary count.
Having a MIN_GENESIS_TIME allows us to start the chain with fewer validators than was previously thought necessary. The previous plan was to start the chain as soon as there were MIN_GENESIS_ACTIVE_VALIDATOR_COUNT validators staked. But there were concerns that with a lowish initial validator count, a single entity could form the majority of them and then act to prevent other validators from entering (a "gatekeeper attack"). A minimum genesis time allows time for all those who wish to make deposits to do so before they could be excluded by a gatekeeper attack.
The beacon chain actually started at 12:00:23 UTC on the 1st of December 2020. The extra 23 seconds comes from the timestamp of the first Eth1 block to meet the genesis criteria, block 11320899. I like to think of this as a little remnant of proof of work forever embedded in the beacon chain's history.
Unlike Ethereum 1.0, the beacon chain gives in-protocol versions to its forks. See the Version custom type for more explanation.
GENESIS_FORK_VERSION is the fork version the beacon chain starts with at its "genesis" event: the point at which the chain first starts producing blocks. In Altair, this value is used only when computing the cryptographic domain for deposit messages, which are valid across all forks.
ALTAIR_FORK_VERSION is defined elsewhere.
Seven days' notice was regarded as sufficient to allow client dev teams time to make a release once the genesis parameters were known, and for node operators to upgrade to that release. And, of course, to organise some parties. It was increased from 2 days over time due to lessons learned on some of the pre-genesis testnets.
ETH1_FOLLOW_DISTANCE uint64(2**11) (= 2,048) Eth1 blocks ~8 hours
This was originally six seconds, but is now twelve, and has been other values in between.
Network delays are the main limiting factor in shortening the slot length. Three communication activities need to be accomplished within a slot, and it is supposed that four seconds is enough for the vast majority of nodes to have participated in each:
Blocks are proposed at the start of a slot and should have propagated to most of the network within the first four seconds.
At four seconds into a slot, committee members create and broadcast attestations, including attesting to this slot's block. During the next four seconds, these attestations are collected by aggregators in each committee.
At eight seconds into the slot, the aggregators broadcast their aggregate attestations which then have four seconds to reach the validator who is proposing the next block.
This slot length has to account for shard blocks as well in later phases. There was some discussion around having the beacon chain and shards on differing cadences, but the latest sharding design tightly couples the beacon chain with the shards. Shard blocks under this design will be much larger, which led to the extension of the slot to 12 seconds.
There is a general intention to shorten the slot time in future, perhaps to [8 seconds](https://github.com/ethereum/consensus-specs/issues/1890#issue-638024803, if it proves possible to do this in practice. Or perhaps to lengthen it to 16 seconds.
The assumed block interval on the Eth1 chain, used in conjunction with ETH1_FOLLOW_DISTANCE when considering blocks on the Eth1 chain, either at genesis, or when voting on the deposit contract state.
A validator can stop participating once it has made it through the exit queue. However, its funds remain locked for the duration of MIN_VALIDATOR_WITHDRAWABILITY_DELAY. Initially, this is to allow some time for any slashable behaviour to be detected and reported so that the validator can still be penalised (in which case the validator's withdrawable time is pushed EPOCHS_PER_SLASHINGS_VECTOR into the future). When data shards are introduced this delay will also allow for shard rewards to be credited and for proof of custody challenges to be mounted.
Note that, for the time being, there is no mechanism to withdraw a validator's balance in any case. Nonetheless, being in a "withdrawable" state means that a validator has now fully exited from the protocol.
This really anticipates the implementation of data shards. The idea is that it's bad for the stability of longer-lived committees if validators can appear and disappear very rapidly. Therefore, a validator cannot initiate a voluntary exit until SHARD_COMMITTEE_PERIOD epochs after it is activated. Note that it could still be ejected by slashing before this time.
This is used to calculate the minimum depth of block on the Ethereum 1 chain that can be considered by the Eth2 chain: it applies to the Genesis process and the processing of deposits by validators. The Eth1 chain depth is estimated by multiplying this value by the target average Eth1 block time, SECONDS_PER_ETH1_BLOCK.
Validator Cycle
If a validator's effective balance falls to 16 Ether or below then it is exited from the system. This is most likely to happen as a result of the "inactivity leak", which gradually reduces the balances of inactive validators in order to maintain the liveness of the beacon chain.
In concrete terms, this means that up to four validators can enter or exit the active validator set each epoch (900 per day) until we have 327,680 active validators, at which point the limit rises to five.
The rate at which validators can exit is strongly related to the concept of weak subjectivity, and the weak subjectivity period.
CHURN_LIMIT_QUOTIENT
INACTIVITY_SCORE_BIAS uint64(2**2) (= 4) score points per inactive epoch
INACTIVITY_SCORE_RECOVERY_RATE uint64(2**4) (= 16) score points per leak-free epoch
INACTIVITY_SCORE_BIAS
If the beacon chain hasn't finalised an epoch for longer than MIN_EPOCHS_TO_INACTIVITY_PENALTY epochs, then it enters "leak" mode. In this mode, any validator that does not vote (or votes for an incorrect target) is penalised an amount each epoch of (effective_balance * inactivity_score) // (INACTIVITY_SCORE_BIAS * INACTIVITY_PENALTY_QUOTIENT_ALTAIR). See INACTIVITY_PENALTY_QUOTIENT_ALTAIR for discussion of the inactivity leak itself.
The per-validator inactivity-score is new in Altair. During Phase 0, inactivity penalties were an increasing global amount applied to all validators that did not participate in an epoch, regardless of their individual track records of participation. So a validator that was able to participate for a significant fraction of the time nevertheless could be quite severely penalised due to the growth of the per-epoch inactivity penalty. Vitalik gives a simplified example: "if fully [off]line validators get leaked and lose 40% of their balance, someone who has been trying hard to stay online and succeeds at 90% of their duties would still lose 4% of their balance. Arguably this is unfair."
In addition, if many validators are able to participate intermittently, it indicates that whatever event has befallen the chain is potentially recoverable (unlike a permanent network partition, or a super-majority network fork, for example). The inactivity leak is intended to bring finality to irrecoverable situations, so prolonging the time to finality if it's not irrecoverable is likely a good thing.
With Altair, each validator has an individual inactivity score in the beacon state which is updated by process_inactivity_updates() as follows.
Every epoch, irrespective of the inactivity leak,
decrease the score by one when the validator makes a correct timely target vote, and
increase the score by INACTIVITY_SCORE_BIAS otherwise.
When not in an inactivity leak
decrease every validator's score by INACTIVITY_SCORE_RECOVERY_RATE.
There is a floor of zero on the score. So, outside a leak, validators' scores will rapidly return to zero and stay there, since INACTIVITY_SCORE_RECOVERY_RATE is greater than INACTIVITY_SCORE_BIAS.
When in a leak, if
p
is the participation rate between
0
1
\lambda
is INACTIVITY_SCORE_BIAS, then the expected score after
N
epochs is
\max (0, N((1-p)\lambda - p))
\lambda = 4
\max (0, N(4 - 5p))
. So a validator that is participating 80% of the time or more can maintain a score that is bounded near zero. With less than 80% average participation, its score will increase unboundedly.
INACTIVITY_SCORE_RECOVERY_RATE
When not in an inactivity leak, validators' inactivity scores are reduced by INACTIVITY_SCORE_RECOVERY_RATE + 1 per epoch when they make a timely target vote, and by INACTIVITY_SCORE_RECOVERY_RATE - INACTIVITY_SCORE_BIAS when they don't. So, even for non-performing validators, scores decrease three times faster than they increase.
The new scoring system means that some validators will continue to be penalised due to the leak, even after finalisation starts again. This is intentional. When the leak causes the beacon chain to finalise, at that point we have just 2/3 of the stake online. If we immediately stop the leak (as we used to), then the amount of stake online would remain close to 2/3 and the chain would be vulnerable to flipping in and out of finality as small numbers of validators come and go. We saw this behaviour on some of the testnets prior to launch. Continuing the leak after finalisation serves to increase the balances of participating validators to greater than 2/3, providing a margin that should help to prevent such behaviour.
See the section on the Inactivity Leak for some more analysis of the inactivity score and some graphs of its effect.
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Combinatorial Games: Level 3 Challenges Practice Problems Online | Brilliant
Combinatorial Games: Level 2 Challenges
An intelligent trader travels from 1 place to another carrying 3 sacks having 30 coconuts in each of them. No sack can hold more than 30 coconuts and he can move the coconuts between sacks. On the way, he passes through 30 checkpoints and on each checkpoint he has to give 1 coconut from each sack he is carrying.
What is the maximum number of coconuts he can have after passing through every checkpoint?
by sandeep devarapalli
Alice and Bob are playing Tic-Tac-Toe on the above triangular board. They play as in in normal Tic-Tac-Toe: taking turns, each player claims one square. The first to claim three squares in a row (vertically, horizontally, or diagonally) wins.
Alice plays first. Alice and Bob both want to win and play optimally. Which of the following is true?
Alice's first move is in square A and Alice wins. Neither Alice nor Bob get three squares in a row. Alice's first move is in square B and Alice wins. Bob wins. Alice's first move is in square C and Alice wins.
Written on a blackboard is the polynomial
{ x }^{ 2 }+x+2014
Calvin and Hobbes take turns alternatively (starting with Calvin) in the following game. During his turn, Calvin should either increase or decrease the coefficient of x by 1. And during his turn, Hobbes should either increase or decrease the constant coefficient by 1. If Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots, then Who Has A Winning Strategy?
The Game Is Endless Hobbes Can't Say Calvin
Alice and Carla are playing a game often learned in elementary school known as Twenty One. The rules for the game are as follows:
Each player takes turns saying between
1
3
consecutive numbers, with the first player starting with the number
1
. For example, Player
1
could say the numbers
1
2
, then Player
2
can say "
3
4
5
", then Player
1
6
" and so on.
The goal of the game is to get the other person to say "
21
", meaning that you have to be the one to say "
20
Carla begins to get bored with the game, so she decides to make it a little bit more challenging. The name of the new game is Two Hundred Thirty Nine, where the goal is now to get the other person to say "
239
". Carla decides that she'll go first and that Alice will go second. Also, each player is now able to say up to "
6
" numbers per turn. Is there a way to tell which player is going to win before the game even starts?
Assume that each player plays "perfectly", meaning that if there was an optimal way of playing, both players would be playing the best that the game allows them to play.
Alice Either Player Carla
X
O
X
a
b
c
d
e
a
b
b
e
by Jeremy Bansil
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Solve Differential Equations Using Laplace Transform - MATLAB & Simulink - MathWorks Deutschland
Definition: Laplace Transform
Workflow: Solve RLC Circuit Using Laplace Transform
Solve differential equations by using Laplace transforms in Symbolic Math Toolbox™ with this workflow. For simple examples on the Laplace transform, see laplace and ilaplace.
f\left(t\right)
\mathit{F}\left(\mathit{s}\right)={\int }_{0}^{\infty }\mathit{f}\left(\mathit{t}\right){\mathit{e}}^{-\mathit{ts}}\mathit{dt}.
Declare Equations
You can use the Laplace transform to solve differential equations with initial conditions. For example, you can solve resistance-inductor-capacitor (RLC) circuits, such as this circuit.
Resistances in ohm:
{R}_{1},{R}_{2},{R}_{3}
Currents in ampere:
{I}_{1},{I}_{2},{I}_{3}
Inductance in henry:
L
Capacitance in farad:
C
AC voltage source in volts:
E\left(t\right)
Capacitor charge in coulomb:
Q\left(t\right)
Apply Kirchhoff's voltage and current laws to get the following equations.
\begin{array}{l}{\mathit{I}}_{1}={\mathit{I}}_{2}+{\mathit{I}}_{3}\\ \mathit{L}\frac{{\mathit{dI}}_{1}}{\mathit{dt}}+{\mathit{I}}_{1}{\mathit{R}}_{1}+{\mathit{I}}_{2}{\mathit{R}}_{2}=0\\ \mathit{E}\left(\mathit{t}\right)+{\mathit{I}}_{2}{\mathit{R}}_{2}-\frac{\mathit{Q}}{\mathit{C}}-{\mathit{I}}_{3}{\mathit{R}}_{3}=0\end{array}
Substitute the relation
{I}_{3}=dQ/dt
(which is the rate of the capacitor being charged) to the above equations to get the differential equations for the RLC circuit.
\begin{array}{l}\frac{{\mathit{dI}}_{1}}{\mathit{dt}}-\frac{{\mathit{R}}_{2}}{\mathit{L}}\frac{\mathit{dQ}}{\mathit{dt}}=-\frac{{\mathit{R}}_{1}+{\mathit{R}}_{2}}{\mathit{L}}{\mathit{I}}_{1}\\ \frac{\mathit{dQ}}{\mathit{dt}}=\frac{1}{{\mathit{R}}_{2}+{\mathit{R}}_{3}}\left(\mathit{E}\left(\mathit{t}\right)-\frac{\mathit{Q}}{\mathit{C}}\right)+\frac{{\mathit{R}}_{2}}{{\mathit{R}}_{2}+{\mathit{R}}_{3}}{\mathit{I}}_{1}\end{array}
Declare the variables. Because the physical quantities have positive values, set the corresponding assumptions on the variables. Let
E\left(t\right)
be an alternating voltage of 1 V.
syms L C I1(t) Q(t) s
R = sym('R%d',[1 3]);
assume([t L C R] > 0)
E(t) = 1*sin(t); % AC voltage = 1 V
Declare the differential equations.
dI1 = diff(I1,t);
dQ = diff(Q,t);
eqn1 = dI1 - (R(2)/L)*dQ == -(R(1)+R(2))/L*I1
eqn1(t) =
\frac{\partial }{\partial t}\mathrm{ }{I}_{1}\left(t\right)-\frac{{R}_{2} \frac{\partial }{\partial t}\mathrm{ }Q\left(t\right)}{L}=-\frac{{I}_{1}\left(t\right) \left({R}_{1}+{R}_{2}\right)}{L}
eqn2 = dQ == (1/(R(2)+R(3))*(E-Q/C)) + R(2)/(R(2)+R(3))*I1
\frac{\partial }{\partial t}\mathrm{ }Q\left(t\right)=\frac{\mathrm{sin}\left(t\right)-\frac{Q\left(t\right)}{C}}{{R}_{2}+{R}_{3}}+\frac{{R}_{2} {I}_{1}\left(t\right)}{{R}_{2}+{R}_{3}}
Compute the Laplace transform of eqn1 and eqn2.
eqn1LT = laplace(eqn1,t,s)
eqn1LT =
s \mathrm{laplace}\left({I}_{1}\left(t\right),t,s\right)-{I}_{1}\left(0\right)+\frac{{R}_{2} \left(Q\left(0\right)-s \mathrm{laplace}\left(Q\left(t\right),t,s\right)\right)}{L}=-\frac{\left({R}_{1}+{R}_{2}\right) \mathrm{laplace}\left({I}_{1}\left(t\right),t,s\right)}{L}
s \mathrm{laplace}\left(Q\left(t\right),t,s\right)-Q\left(0\right)=\frac{{R}_{2} \mathrm{laplace}\left({I}_{1}\left(t\right),t,s\right)}{{R}_{2}+{R}_{3}}+\frac{\frac{C}{{s}^{2}+1}-\mathrm{laplace}\left(Q\left(t\right),t,s\right)}{C \left({R}_{2}+{R}_{3}\right)}
The function solve solves only for symbolic variables. Therefore, to use solve, first substitute laplace(I1(t),t,s) and laplace(Q(t),t,s) with the variables I1_LT and Q_LT.
syms I1_LT Q_LT
eqn1LT = subs(eqn1LT,[laplace(I1,t,s) laplace(Q,t,s)],[I1_LT Q_LT])
{I}_{1,\mathrm{LT}} s-{I}_{1}\left(0\right)+\frac{{R}_{2} \left(Q\left(0\right)-{Q}_{\mathrm{LT}} s\right)}{L}=-\frac{{I}_{1,\mathrm{LT}} \left({R}_{1}+{R}_{2}\right)}{L}
{Q}_{\mathrm{LT}} s-Q\left(0\right)=\frac{{I}_{1,\mathrm{LT}} {R}_{2}}{{R}_{2}+{R}_{3}}-\frac{{Q}_{\mathrm{LT}}-\frac{C}{{s}^{2}+1}}{C \left({R}_{2}+{R}_{3}\right)}
Solve the equations for I1_LT and Q_LT.
eqns = [eqn1LT eqn2LT];
vars = [I1_LT Q_LT];
[I1_LT, Q_LT] = solve(eqns,vars)
I1_LT =
\frac{L {I}_{1}\left(0\right)-{R}_{2} Q\left(0\right)+C {R}_{2} s+L {s}^{2} {I}_{1}\left(0\right)-{R}_{2} {s}^{2} Q\left(0\right)+C L {R}_{2} {s}^{3} {I}_{1}\left(0\right)+C L {R}_{3} {s}^{3} {I}_{1}\left(0\right)+C L {R}_{2} s {I}_{1}\left(0\right)+C L {R}_{3} s {I}_{1}\left(0\right)}{\left({s}^{2}+1\right) \left({R}_{1}+{R}_{2}+L s+C L {R}_{2} {s}^{2}+C L {R}_{3} {s}^{2}+C {R}_{1} {R}_{2} s+C {R}_{1} {R}_{3} s+C {R}_{2} {R}_{3} s\right)}
Q_LT =
\frac{C \left({R}_{1}+{R}_{2}+L s+L {R}_{2} {I}_{1}\left(0\right)+{R}_{1} {R}_{2} Q\left(0\right)+{R}_{1} {R}_{3} Q\left(0\right)+{R}_{2} {R}_{3} Q\left(0\right)+L {R}_{2} {s}^{2} {I}_{1}\left(0\right)+L {R}_{2} {s}^{3} Q\left(0\right)+L {R}_{3} {s}^{3} Q\left(0\right)+{R}_{1} {R}_{2} {s}^{2} Q\left(0\right)+{R}_{1} {R}_{3} {s}^{2} Q\left(0\right)+{R}_{2} {R}_{3} {s}^{2} Q\left(0\right)+L {R}_{2} s Q\left(0\right)+L {R}_{3} s Q\left(0\right)\right)}{\left({s}^{2}+1\right) \left({R}_{1}+{R}_{2}+L s+C L {R}_{2} {s}^{2}+C L {R}_{3} {s}^{2}+C {R}_{1} {R}_{2} s+C {R}_{1} {R}_{3} s+C {R}_{2} {R}_{3} s\right)}
{I}_{1}
Q
by computing the inverse Laplace transform of I1_LT and Q_LT. Simplify the result. Suppress the output because it is long.
I1sol = ilaplace(I1_LT,s,t);
Qsol = ilaplace(Q_LT,s,t);
I1sol = simplify(I1sol);
Qsol = simplify(Qsol);
Before plotting the result, substitute symbolic variables by the numeric values of the circuit elements. Let
{R}_{1}=4\phantom{\rule{0.2777777777777778em}{0ex}}\Omega
{R}_{2}=2\phantom{\rule{0.2777777777777778em}{0ex}}\Omega
{R}_{3}=3\phantom{\rule{0.2777777777777778em}{0ex}}\Omega
C=1/4\phantom{\rule{0.2777777777777778em}{0ex}}F
L=1.6\phantom{\rule{0.2777777777777778em}{0ex}}H
. Assume that the initial current is
{I}_{1}\left(0\right)=2\phantom{\rule{0.2777777777777778em}{0ex}}A
and the initial charge is
Q\left(0\right)=2\phantom{\rule{0.2777777777777778em}{0ex}}C
vars = [R L C I1(0) Q(0)];
values = [4 2 3 1.6 1/4 2 2];
I1sol = subs(I1sol,vars,values)
I1sol =
\frac{200 \mathrm{cos}\left(t\right)}{8161}+\frac{405 \mathrm{sin}\left(t\right)}{8161}+\frac{16122 {\mathrm{e}}^{-\frac{81 t}{40}} \left(\mathrm{cosh}\left(\frac{\sqrt{1761} t}{40}\right)-\frac{742529 \sqrt{1761} \mathrm{sinh}\left(\frac{\sqrt{1761} t}{40}\right)}{14195421}\right)}{8161}
Qsol = subs(Qsol,vars,values)
Qsol =
\frac{924 \mathrm{sin}\left(t\right)}{8161}-\frac{1055 \mathrm{cos}\left(t\right)}{8161}+\frac{17377 {\mathrm{e}}^{-\frac{81 t}{40}} \left(\mathrm{cosh}\left(\frac{\sqrt{1761} t}{40}\right)+\frac{1109425 \sqrt{1761} \mathrm{sinh}\left(\frac{\sqrt{1761} t}{40}\right)}{30600897}\right)}{8161}
Plot the current I1sol and charge Qsol. Show both the transient and steady state behavior by using two different time intervals:
0\le \mathit{t}\le 15
2\le \mathit{t}\le 25
fplot(I1sol,[0 15])
ylabel('I1(t)')
fplot(Qsol,[0 15])
title('Charge')
ylabel('Q(t)')
text(3,-0.1,'Transient')
text(15,-0.07,'Steady State')
text(3,0.35,'Transient')
text(15,0.22,'Steady State')
Initially, the current and charge decrease exponentially. However, over the long term, they are oscillatory. These behaviors are called "transient" and "steady state", respectively. With the symbolic result, you can analyze the result's properties, which is not possible with numeric results.
Visually inspect I1sol and Qsol. They are a sum of terms. Find the terms by using children. Then, find the contributions of the terms by plotting them over [0 15]. The plots show the transient and steady state terms.
I1terms = children(I1sol);
I1terms = [I1terms{:}];
Qterms = children(Qsol);
Qterms = [Qterms{:}];
fplot(I1terms,[0 15])
title('Current terms')
fplot(Qterms,[0 15])
title('Charge terms')
The plots show that I1sol has a transient and steady state term, while Qsol has a transient and two steady state terms. From visual inspection, notice I1sol and Qsol have a term containing the exp function. Assume that this term causes the transient exponential decay. Separate the transient and steady state terms in I1sol and Qsol by checking terms for exp using has.
I1transient = I1terms(has(I1terms,'exp'))
I1transient =
\frac{16122 {\mathrm{e}}^{-\frac{81 t}{40}} \left(\mathrm{cosh}\left(\frac{\sqrt{1761} t}{40}\right)-\frac{742529 \sqrt{1761} \mathrm{sinh}\left(\frac{\sqrt{1761} t}{40}\right)}{14195421}\right)}{8161}
I1steadystate = I1terms(~has(I1terms,'exp'))
I1steadystate =
\left(\begin{array}{cc}\frac{200 \mathrm{cos}\left(t\right)}{8161}& \frac{405 \mathrm{sin}\left(t\right)}{8161}\end{array}\right)
Similarly, separate Qsol into transient and steady state terms. This result demonstrates how symbolic calculations help you analyze your problem.
Qtransient = Qterms(has(Qterms,'exp'))
Qtransient =
\frac{17377 {\mathrm{e}}^{-\frac{81 t}{40}} \left(\mathrm{cosh}\left(\frac{\sqrt{1761} t}{40}\right)+\frac{1109425 \sqrt{1761} \mathrm{sinh}\left(\frac{\sqrt{1761} t}{40}\right)}{30600897}\right)}{8161}
Qsteadystate = Qterms(~has(Qterms,'exp'))
Qsteadystate =
\left(\begin{array}{cc}-\frac{1055 \mathrm{cos}\left(t\right)}{8161}& \frac{924 \mathrm{sin}\left(t\right)}{8161}\end{array}\right)
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An annuity table is a tool for determining the present value of an annuity or other structured series of payments. Such a tool, used by accountants, actuaries, and other insurance personnel, takes into account how much money has been placed into an annuity and how long it has been there to determine how much money would be due to an annuity buyer or annuitant.
Figuring the present value of any future amount of an annuity may also be performed using a financial calculator or software built for such a purpose.
An annuity table is a tool used to determine the present value of an annuity.
An annuity table calculates the present value of an annuity using a formula that applies a discount rate to future payments.
An annuity table uses the discount rate and number of period for payment to give you an appropriate factor.
Using an annuity table, you will multiply the dollar amount of your recurring payment by the given factor.
How an Annuity Table Works
An annuity table provides a factor, based on time, and a discount rate (interest rate) by which an annuity payment can be multiplied to determine its present value. For example, an annuity table could be used to calculate the present value of an annuity that paid $10,000 a year for 15 years if the interest rate is expected to be 3%.
According to the concept of the time value of money, receiving a lump sum payment in the present is worth more than receiving the same sum in the future. As such, having $10,000 today is better than being given $1,000 per year for the next 10 years because the sum could be invested and earn interest over that decade. At the end of the 10-year period, the $10,000 lump sum would be worth more than the sum of the annual payments, even if invested at the same interest rate.
Annuity Table Uses
A lottery winner could use an annuity table to determine whether it makes more financial sense to take his lottery winnings as a lump-sum payment today or as a series of payments over many years. Lottery winnings are a rare form of an annuity. More commonly, annuities are a type of investment used to provide individuals with a steady income in retirement.
Annuity Table and the Present Value of an Annuity
Present Value of an Annuity Formulas
The formula for the present value of an ordinary annuity, as opposed to an annuity due, is as follows:
\begin{aligned}&\text{P} =\text{PMT}\times\frac{ 1 - (1 + r) ^ -n}{r}\\&\textbf{where:}\\&\text{P} = \text{Present value of an annuity stream}\\&\text{PMT} =\text{Dollar amount of each annuity payment}\\&r = \text{Interest rate (also known as the discount rate)}\\&n = \text{Number of periods in which payments will be made}\end{aligned}
P=PMT×r1−(1+r)−nwhere:P=Present value of an annuity streamPMT=Dollar amount of each annuity paymentr=Interest rate (also known as the discount rate)
Assume an individual has an opportunity to receive an annuity that pays $50,000 per year for the next 25 years, with a discount rate of 6%, or a lump sum payment of $650,000. He needs to determine the more rational option. Using the above formula, the present value of this annuity is:
\begin{aligned}&\text{PVA} = \$50,000 \times \frac{1 - (1 + 0.06) ^ -25}{0.06} = \$639,168\\&\textbf{where:}\\&\text{PVA}=\text{Present value of annuity}\end{aligned}
PVA=$50,000×0.061−(1+0.06)−25=$639,168where:
Given this information, the annuity is worth $10,832 less on a time-adjusted basis, and the individual should choose the lump sum payment over the annuity.
Note, this formula is for an ordinary annuity where payments are made at the end of the period in question. In the above example, each $50,000 payment would occur at the end of the year, each year, for 25 years. With an annuity due, the payments are made at the beginning of the period in question. To find the value of an annuity due, simply multiply the above formula by a factor of (1 + r):
\begin{aligned}&\text{P} = \text{PMT} \times\left(\frac{1 - (1 + r) ^ -n}{r}\right) \times (1 + r)\end{aligned}
P=PMT×(r1−(1+r)−n)×(1+r)
If the above example of an annuity due, its value would be:
\begin{aligned}&\text{P}= \$50,000\\&\quad \times\left( \frac{1 - (1 + 0.06) ^ -25}{0.06}\right)\times (1 + 0.06) = \$677,518\end{aligned}
P=$50,000
In this case, the individual should choose the annuity due, because it is worth $27,518 more than the lump sum payment.
Rather than working through the formulas above, you could alternatively use an annuity table. An annuity table simplifies the math by automatically giving you a factor for the second half of the formula above. For example, the present value of an ordinary annuity table would give you one number (referred to as a factor) that is pre-calculated for the (1 - (1 + r) ^ - n) / r) portion of the formula.
The factor is determined by the the interest rate (r in the formula) and the number of periods in which payments will be made (n in the formula). In an annuity table, the number of periods is commonly depicted down the left column. The interest rate is commonly depicted across the top row. Simply select the correct interest rate and number of periods to find your factor in the intersecting cell. That factor is then multiplied by the dollar amount of the annuity payment to arrive at the present value of the ordinary annuity.
Below is an example of a present value of an ordinary annuity table:
n 1% 2% 3% 4% 5% 6%
If we take the example above with a 6% interest rate and a 25 year period, you will find the factor = 12.7834. If you multiply this 12.7834 factor from the annuity table by the $50,000 payment amount, you will get $639,168. Notice, this is the same as the result of the formula above.
There is a separate table for the present value of an annuity due, and it will give you the correct factor based on the second formula.
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#Dec 22, 2021 🐛 Jupyter Output Captions
Captions on Jupyter Outputs now render correctly in exported PDFs or LaTeX files, fixing this issue.
#Dec 22, 2021 - Chrome Extension Update (v1.5.0)
We have released a update with a umber of fixes and improvements to align with the Curvenote webapp
@mentions - mention people in comments and they will be notified in Curvenote and by email. Mentions in existing comments now also show up properly.
After commenting, all new comments should immediately show up, previously where was an intermittent bug where comments would not show until a refresh.
Math now correctly renders in comments
y=f(x)
Previously, when scrolling through cells of a Jupyter notebook using arrow keys, it was possible for the Output section of the panel to open and steal focus. This has been fixed.
The title link in the panel now always points to the latest version of your notebook, while a new version link is pinned to the specific version you are working with
#Dec 22, 2021 - Improved section numbering
We have improved the section numbering to work across different blocks. Previously the a section with a number 1.2, for example, would not show up it if was in the next block. All fixed up now!
#Dec 21, 2021 - Mention collaborators in comments with notifications
You can now mention collaborators in comments and they will receive a notification. This notification comes instantly as an “in-app” notification in Curvenote, and if you don’t see this within a few minutes, a notification will also be sent by email. Clicking on the link in a notification or a comment will scroll you to the right place in a document and highlight the comment.
#🐛 Bug Fixes & Quality of Life
Improvements to the comment placement when content loads as well as when Jupyter outputs are rendered (changing the height) ↕️
Clicking on a notification no longer require a reload of the page. ⚡
Header sizes have been visually improved to make them stand out a bit more. 🔍
#Dec 14, 2021 - Figures from Jupyter Outputs
Jupyter Outputs blocks can now be added to an article as a fully fledged Figure with the same caption, numbering and referencing controls available on normal images added to content blocks in an article.
Once a Jupyter output is added to an article and selected, a menu will appear and you can start typing a caption immediately. The caption stays attached to the output as long as it remains in the article, and when the output itself can is updated from Jupyter, the caption text, settings and references will be preserved both in the Curvenote app and when exporting to LaTeX.
#Dec 1, 2021 - Local Backup 🤖
We’ve added a local backup feature that will keep a copy of the draft of each block in the browser's local storage until a user saves that draft as a version. This backup is cleared once a successful save is confirmed by the server responding with an exact copy of the content saved.
Users can see local backups and use them to download or restore content as shown below. In normal operation, this will be empty except for a backup of the current draft.
Figure 4:Local backups are in the tray on the right hand side.
#Nov 30, 2021 - Rich text comments
You can now add rich-text comment to any block. 🎉 This includes being able to add references, citations, and links as well as standard formatting like code, bold, italics, etc. You can even have inline math and equations!
#Quality of Life improvements 🚀
There were a number of small improvements around syncing comments in the editor, around speed and performance. ⚡
There was a bug that resolved comments wouldn’t always resolve on other clients, that has been fixed! 🐛
#Nov 25, 2021 - Synchronization Warnings 🐛
We have added improved warnings to inform users when the Curvenote client fails to sync to the server. The syncing issue is something that we are actively working on (tracking this here) and this notification is a step to ensure that users know when something has gone wrong, whilst we make further fixes.
📃Improved server-side logging around the synchronization api
#Nov 24, 2021 - Codemark & Comment Speedups
The comment selection/placement algorithm has been sped up as we get ready for some other improvements around commenting workflows.
We have released a new version of codemark that works with collaborative input! Before your cursor would jump inside of the code mark.
There was a bug around tables where they didn’t resize/work properly. 🐛
#Nov 19, 2021 - Quality of life 🐛
The export form checkboxes are now initialized to their defaults and work as expected
Form options remembered properly (#8)
arXiv templates have the correct character count advisory note
Removed color coding for the summary count of optional options and tagged blocks
#Nov 17, 2021 - Project & Team Add Notifications
Existing users will now get an in-app notification when they are added as a Collaborator on a project or a member on a Team. Click on the notification to visit the project or team.
Improvements to the emails sent when someone is added/invites to a project or team, which now also include links directly to the project or team they were added to.
When a user is added to a Team, they will only receive one notification email with a link to the team instead of an email for each team project.
In app notifications for exports have been re-enabled.
The Abstract tag no longer shows up in the Article header, if it is added as a tag on the article block.
#Nov 16, 2021 - Project & Team Invite Improvements
We have added the ability to both search for an existing user and invite a new user via an email when you are adding them to a project or a team. The user suggestions will be a helpful feature as we start to bring this into our editor in the coming months!
You can also specify an email directly, or copy and paste from Gmail or Outlook. Invited users who do not have an account will show up as pending in your access list, these will get resolved when the user signs up!
#Nov 11, 2021 - Export 🚚
Improvements and fixes to the AGU2019 and Plain Latex export templates.
MPDI Energies renamed to MDPI and additional options added. All 350+ journals can now be access via the options available during export.
#Nov 8, 2021 - Quality of Life 🐛
Affiliation now has a max length of 160 characters (up from 50) and that is shown in the user interface, rather than error-ing without a very helpful message.
We have improved the messaging around saving a draft, previously other collaborators saw this as the draft being “discarded” 👻, that is now changed to showing that it is saved.
#Nov 2, 2021 - Inline Code Editing
We have improved the handling of inline code to show you where you input cursor is and if the next character that you type will be “marked” as inline code. You can now also create inline code marks in two additional ways to typing forward `code as previously, (1) going back in the editor and filling in the mark code`, and (2) selecting text and hitting the `. The changes also improve when you select on the editor, if there is an option, the cursor defaults to being outside of the code mark!
Figure 10:Improvements to allow arrowing into/out of inline code.
Let us know what you think of the new experience! 🚀
#Oct 27, 2021 Figure and Table Captions
We have added the ability to put figure, table and code captions on elements, and directly edit them in the editor. Previously you could add captions to figures, but they were only editable in the settings menu and were did not let you collaboratively edit.
Figure 11:A recording of adding a caption to a figure!
This works also works for tables and code blocks, for example:
Table 1:This is a simple table of equations!
a^2 + b^2 = c^2
Pythagoras, 530 BC
\log(xy) = \log(x) + \log(y)
John Napier, 1610
\frac{df}{dt} = \lim_{h \to 0}\frac{f(t+h) - f(t)}{h}
F=G\frac{m_1m_2}{r^2}
i^2 = -1
console.log(\`${p.x}, ${p.y}\`);
You can reference these Figure 11, Table 1, Program %s, using the [[ command.
#🚀 Improvements
You can now reference tables! For example in, Table 1! Use the [[table: command to bring up a list of all tables in the document.
You can now center or align tables!
The inline-actions for editor now no longer are shown when you open a modal (e.g. image settings). Previously this would leave a hanging image actions, for example, on top of the modal which was difficult to close.
#Oct 22, 2021
#⚡ Quality of life improvements!
Better fractions ½ when you type fractions in Curvenote, they are turned into the typographic equivalent that are supported — ½ ⅓ ⅔ ¼ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅐ ⅛ ⅜ ⅝ ⅞ ⅑ ⅒! These were a little too ambitious when you started a 321/2 would turn into 32½, this no longer happens! 🙂
The copyright and registered trademark symbols © & ® are also inserted when you type, however, in a list when you talk about (a) something; (b) something else; and finally © … that would also turn into a copyright symbol. It doesn’t anymore!
We have also improved the handling of wrapping math in $ which already didn’t happen if you were talking about
1.00 and
2.00, but if you inserted in a bracket ($1.00) then it would match. That is no longer the case. 🎉
Strikethrough is less sensitive! especially when you are writing ~5 and ~7, before that would strike it out before.
We have removed the accelerator for dynamic text, which was previously , this confused with writing html in many cases.
Keyboard shortcuts for block copy paste and selecting between blocks now work again, was not working since yesterday’s word-deployment. ⏭️
#📘 Export to MS Word
We can now export Curvenote articles to MS Word (docx) format.
Our aim when exporting is to bring over your all of content to docx with all the styling and features (links, citations, images, captions) you have in the Curvenote article. This feature is still in beta as we’ve not quite added all of those yet! Give it a try, and we’d love to have your feedback.
#⚡ Quality of life improvements
We have fixed heading numbering issues. Previously subheadings would not inherit their parent numbering so section 1.1.0 would show up as 0.1.0
#🖼️ Image Upload
#🧙 Chrome Extension Fixes 🐛 v0.3.2
Updated the Curvenote logos in the deployed version!
#🪐 Chrome Extension Fixes 🐛 v0.3.1
We’ve fixed a number of issues with the Chrome extension
Fixed - In Jupyterlab 3.1+, the Panel and toolbar controls for the extension were not initializing or only appearing intermittently.
Busy spinners we’re no longer visible on individual cells when they were saving, these have been reintroduced.
Updated some internal calls that were resulting in console error messages
#Oct 12, 2021 - Footnotes(When you want to say something, but also kinda don’t?)
We have added the ability to add footnotes to your documents!
Using the /footnote command, or insert from the top menu
These will also show up in your
\LaTeX
and PDF exports as well!
Figure 14:You can now add footnotes in the editor directly!
Bug fixed 🐛 with citation view in Firefox 🦊. There was a problem with the date-parsing of citations, leading to showing ???? instead of the actual year 📅.
Occasionally some references would not load for projects that are partially public (advanced publishing mode), this should now be fixed.
#📗Updates to
\LaTeX
We’ve made updates to three templates (default, Arxiv[nips] and Arxiv[two-column]) to add various options including - corresponding author, backlink, show date, affiliation, location, keywords.
#Oct 7, 2021
#📋 New Export Template Options
Our export process for
\LaTeX
& PDFs has been extended to include screens for Options and Tagged Content. This allows you export your paper or report to a template while taking advantage of all their features and meeting their minimum requirements.
There are two additional screens, the first is a form that will list all of the configurable options for the template you chose.
The second is a listing of Tagged Content accepted by the selected template. Tagged Content is our way of marking blocks from in your article as the “special content” that the template expects and treats differently.
For example, a block tagged as abstract will be separated from the main content and used in the appropriate abstract section in the Journal template.
Each template has different required and optional blocks of content, so this is a uniform way to include those in your paper and then let the chosen template typeset accordingly.
Remember that all of our templates are open source here — we’re actively porting more and are more than happy to accept and help with contributions — if there is a template that you’d like to see, open an issue or PR.
#⌨️ New code editor and highlight experience
You can now have a syntax-highlighted experience when you are creating code snippets in Curvenote. To create a code block:
Type ``` to create a syntax highlighted editor
Or use the command menu and type /code
To exit the code environment, either hit Esc or use Cmd/Ctrl-Enter depending on your platform. When you are in the code editor, it will allow you to choose the language. By default we have chosen Python.
For example, in Python 🐍
from pooch import retrieve
\# Download the file and save it locally.
fname = retrieve(
url="https://some-data-server.org/a-data-file.nc",
known\_hash="md5:70e2afd3fd7e336ae478b1e740a5f08e",
Or in Typescript 🚀
#✨ Updates
The thesis template has been updated with content, and the paper template has been improved to include the latest tables.
The issues with backticks having zero-width has been fixed.
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Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees
October 2021 Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees
Florian Henning, Christof Külske
Florian Henning,1 Christof Külske1
1Faculty of Mathematics, Ruhr University Bochum
We study gradient models for spins taking values in the integers (or an integer lattice), which interact via a general potential depending only on the differences of the spin values at neighboring sites, located on a regular tree with
\mathit{d}+1
neighbors. We first provide general conditions in terms of the relevant p-norms of the associated transfer operator Q which ensure the existence of a countable family of proper Gibbs measures, describing localization at different heights. Next we prove existence of delocalized gradient Gibbs measures, under natural conditions on Q. We show that the two conditions can be fulfilled at the same time, which then implies coexistence of both types of measures for large classes of models including the SOS-model, and heavy-tailed models arising for instance for potentials of logarithmic growth.
Florian Henning is partially supported by the Research Training Group 2131 High-dimensional phenomena in probability-Fluctuations and discontinuity of German Research Council (DFG).
The authors thank an anonymous referee for clarifying remarks and suggestions.
Florian Henning. Christof Külske. "Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees." Ann. Appl. Probab. 31 (5) 2284 - 2310, October 2021. https://doi.org/10.1214/20-AAP1647
Received: 1 February 2020; Revised: 1 August 2020; Published: October 2021
Keywords: boundary law , delocalization , Gibbs measures , gradient Gibbs measures , heavy tails , Localization , regular tree
Florian Henning, Christof Külske "Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees," The Annals of Applied Probability, Ann. Appl. Probab. 31(5), 2284-2310, (October 2021)
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Medical Entrance Exam Question and Answers | Laws of Motion - Zigya
A body of mass 0.1 kg attains a velocity of 10 ms' in 0.1 sec. The force acting on the body is
\mathrm{F}=\mathrm{ma}=\mathrm{m}\times \frac{∆\mathrm{V}}{∆\mathrm{t}}\phantom{\rule{0ex}{0ex}}=0.1\times \frac{10}{0.1}=10\mathrm{N}
A body is moving in a circular path with acceleration 'a'. If its velocity gets doubled, find the ratio of acceleration after and before the change
\frac{1}{4}:1
In a circular motion
\mathrm{\alpha }=\frac{\quad {\mathrm{\nu }}^{2}}{\mathrm{r}}
\frac{{\mathrm{\alpha }}_{2}}{{\mathrm{\alpha }}_{1}}=\left(\frac{{\mathrm{\nu }}_{2}}{{\mathrm{\nu }}_{1}}\right)={\left(\frac{2{\mathrm{\nu }}_{1}}{{\mathrm{\nu }}_{1}}\right)}^{2}=4\phantom{\rule{0ex}{0ex}}\Rightarrow {\mathrm{\alpha }}_{2}:{\mathrm{\alpha }}_{1}\quad =4:1
The angle through which a cyclist bends when he covers a circular path of 34.3 m circumference in
\sqrt{22}
sec is (g=9.8 m/s)
.Given:-Speed of particle V=
\frac{\mathrm{S}}{\mathrm{t}}
\frac{34.3}{\sqrt{22}}
radius r=
\frac{\mathrm{S}}{2\mathrm{\pi }}
\frac{34.3}{2\mathrm{\pi }}
\mathrm{\theta }
\frac{{\mathrm{V}}^{2}}{\mathrm{rg}}=\frac{{\displaystyle \frac{(34.3{)}^{2}}{({\sqrt{22)}}^{2}}}}{{\displaystyle \frac{34.3}{2\mathrm{\pi }}\times 9.8}}
\frac{34.3\times 2\times {\displaystyle \frac{22}{7}}}{22\times 9.8}
\frac{68.6}{68.6}=1
\mathrm{\theta }
= 1 .....(tan450=1)
Assertion: It is difficult to move a cycle along the road with its brakes on.
Reason: Sliding friction is greater than rolling friction.
If both the assertion and reason are true and reason is a correct explanation of the assertion.
If both assertion and reason are true but assertion is not a correct explanation of the assertion.
When brakes are on, there is no rolling of the wheels and the wheels slide. The sliding friction is greater than the rolling friction. Thus it is difficult to move a cycle along the road with its breaks on.
Rolling friction is the resistance to motion experienced by a body when it rolls upon another. It is much less than sliding friction for same pair of bodies. When one body rolls upon another, there is theoretically no sliding or slip between them. And if both are perfectly rigid, there is no surface of contact.
Above figure of rolling friction.
A particle is revolving in a circle of radius R. If the force acting on it is inversely proportional to R, then the time period is proportional to
Given that the force is inversly proportional to R.
F ∝
\frac{1}{\mathrm{R}}
⇒ F =
\frac{\mathrm{k}}{\mathrm{R}}
......(i)
According to centripetal force
F = mRω2
⇒ mRω2 =
\frac{\mathrm{k}}{\mathrm{R}}
⇒ ω2 =
\frac{\mathrm{k}}{\mathrm{m}\quad {\mathrm{R}}^{2}}
∴ ω2 ∝
\frac{1}{{\mathrm{R}}^{2}}
⇒ ω ∝
\frac{1}{\mathrm{R}}
\frac{2\mathrm{\pi }}{\mathrm{T}}\quad \propto \quad \frac{1}{\mathrm{R}}
where ω =
\frac{2\mathrm{\pi }}{\mathrm{T}}
where ω is the angular frequency or angular speed
⇒ T ∝ R
Three different objects m1 , m2 and m3 to fall from rest and from the same point O along three different frictionless paths. The speeds of the three objects, on reaching the ground, will be in the ratio of
m1 : m2 : m3
m1 : 2 m2 : 3m3
\frac{1}{{\mathrm{m}}_{1}}\quad :\quad \frac{1}{{\mathrm{m}}_{2}}\quad :\quad \frac{1}{{\mathrm{m}}_{3}}
As the speed of an object, falling freely under gravity, depends only upon its height from which it is allowed to fall and not upon its mass. Since the paths are frictionless and all objects are falling through the same vertical height, therefore their speeds on reaching the ground must be same or ratio of their speeds = 1 : 1 : 1.
Assertion: In an elastic collision of two billiard balls, the total kinetic energy is conserved during the short time of oscillation of the balls (i.e. when they are in contact).
Reason: Energy spent against friction does not follow the law of conservation of energy.
A billiard ball is a small, hard ball used in cue sports such as carom billiards, pool and snooker.
The billiard balls in an elastic collision are in a deformed state. And their total energy is in the form of P.E. and K.E. Thus K.E. is less than the total energy. The energy spent against friction is dissipated in the form of heat which is not available for doing work.
Assertion: Work done in uniform circular motion is zero.
Reason: Force is always directed along displacement.
Uniform motion is the object which is moving with a constant angular velocity. So the angular acceleration of the object is zero.
Displacement use only the force componeent along the object's displacement. The force component perpendicular to the displacement direction does zero work.
In uniform circular motion, the force is always directed perpendicular to the displacement.
If mass of an atom M moving with speed v, what will be its speed after the emission of an
\mathrm{\alpha }
-particle is zero?
\frac{\mathrm{Mv}}{\mathrm{M}\quad +\quad 2}
\frac{\mathrm{Mv}}{\mathrm{M}\quad -\quad 4}
\frac{\mathrm{M}\quad \mathrm{v}}{\mathrm{M}\quad +\quad 4}
\frac{\mathrm{M}\quad -\quad 4}{\mathrm{Mv}}
\frac{\mathrm{Mv}}{\mathrm{M}\quad -\quad 4}
Applying the principle of momentum conservation,
Mv = ( M
-
4 ) v'
[ since the speed of the
\mathrm{\alpha }
-particle is zero ]
⇒ v' =
\frac{\mathrm{M}\quad \mathrm{v}}{\mathrm{M}\quad -\quad 4}
A car of mass 1000 kg moves on a circular track of radius 20 m. If the coefficient of friction of 0.64. The maximum velocity with which the car can be moved is
\frac{0.64\times 20}{1000\times 100}\mathrm{m}/\mathrm{s}
\frac{1000}{64\times 20}\mathrm{m}/\mathrm{s}
{\mathrm{V}}_{\mathrm{max}}=\sqrt{\mathrm{\mu rg}}=\sqrt{0.64\times 20\times 9.8}\phantom{\rule{0ex}{0ex}}\quad \mathrm{where},\mathrm{\mu }-\mathrm{coefficient}\quad \mathrm{of}\quad \mathrm{friction}\phantom{\rule{0ex}{0ex}}\quad {\mathrm{V}}_{\mathrm{max}}=11.2\quad \mathrm{m}/\mathrm{s}
|
{\displaystyle {\hat {H}}|\psi _{n}(t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi _{n}(t)\rangle }
{\displaystyle {\frac {1}{{c}^{2}}}{\frac {{\partial }^{2}{\phi }_{n}}{{\partial t}^{2}}}-{{\nabla }^{2}{\phi }_{n}}+{\left({\frac {mc}{\hbar }}\right)}^{2}{\phi }_{n}=0}
{\displaystyle \Xi _{b}^{-}}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Physics&oldid=1084598275"
|
random polynomial over a finite field
random monic prime polynomial over a finite field
Randpoly(n, x) mod p
Randpoly(n, x, alpha) mod p
Randprime(n, x) mod p
Randprime(n, x, alpha) mod p
Randpoly(n, x) mod p returns a polynomial of degree n in the variable x whose coefficients are selected at random from the integers mod p.
Randprime(n, x) mod p returns a random monic irreducible polynomial of degree
0<n
in the variable x over the integers mod p where p must be a prime integer.
\mathrm{GF}\left({p}^{k}\right)
. The field extension alpha is specified by a RootOf a monic univariate polynomial of degree k which must be irreducible.
Thus Randprime(n, x, alpha) mod p creates a random monic irreducible polynomial of degree
0<n
in the variable x over
\mathrm{GF}\left({p}^{k}\right)
\mathrm{Randpoly}\left(4,x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}
\mathrm{Randprime}\left(4,x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({y}^{2}+y+1\right)\right):
f≔\mathrm{Randpoly}\left(2,x,\mathrm{\alpha }\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}
\mathrm{Factor}\left(f\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2
{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}
g≔\mathrm{Randprime}\left(2,x,\mathrm{\alpha }\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{Irreduc}\left(g\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Book Review: “Number Theory” by Andrej Dujella | EMS Magazine
Book Review: “Number Theory” by Andrej Dujella
As a student of Number Theory, I really appreciated the famous book of G. H. Hardy and E. M. Wright, while some of my friends frequently mentioned the book of Z. I. Borevich and I. R. Shafarevich. Of course, even these two books did not cover the whole (huge) field of Number Theory, and several other excellent books could be cited as well. More recently, many books have been devoted to (parts of) this vast field whose characteristic is to be both primary and not primary (pun intended). The meaning of the expression “Number Theory” itself has changed over time – partly because the domain has exploded – to the point that some contemporary authors now refer to “Modern Number Theory” …
A very recent book, entitled Number Theory and based on teaching materials, has been written by A. Dujella. Devoted to several subfields of this domain, this book is both extremely nice to read and to work from. It starts from primary results given in the first three chapters, ranging from the Peano axioms to the principle of induction, from the Fibonacci numbers to Euclid’s algorithm, from prime numbers to congruences, and so on. Chapter 3 ends with primitive roots, decimal representation of rationals, and pseudoprimes. Chapters 4 and 5 then deal with quadratic residues (including the computation of square roots modulo a prime number) and quadratic forms (including the representation of integers as sums of
2
4
3
squares). Chapters 6 and 7 are devoted to arithmetic functions (in particular, multiplicative functions, asymptotic behaviour of the summatory function of classical arithmetic functions, and the Dirichlet product), and to the distribution of primes (elementary estimates for the number of primes less than a given number, the Riemann function, Dirichlet characters, and a proof that an infinite number of primes are congruent to
\ell
k
\gcd(\ell,k)=1
). Chapter 8 deals with first results on Diophantine approximation, from continued fractions to Newton approximations and the LLL algorithm, while Chapter 9 studies applications of Diophantine approximation to cryptography (RSA, attacks on RSA, etc.). Actually two more chapters are devoted to Diophantine approximation, Chapter 10 (linear Diophantine approximation, Pythagorean triangles, Pellian equations, the Local-global principle, …) and Chapter 14 (Thue equations, the method of Tzanakis, linear forms in logarithms, Baker–Davenport reduction, …). Chapters 11, 12, and 13 deal with polynomials, algebraic numbers, and approximation of algebraic numbers. The book ends with Chapters 15 and 16 which cover elliptic curves and Diophantine problems.
This quick and largely incomplete description clearly shows that this book addresses many jewels of number theory. This is done in a particularly appealing way, mostly elementary when possible, with many well-chosen examples and attractive exercises. I arbitrarily choose two delightful examples, the kind of “elementary” statements that a beginner could attack, but whose proofs require some ingenuity, namely the unexpected statements 4.6 and 4.7:
p>5
be a prime number. Prove that there are two consecutive positive integers that are both quadratic residues and two consecutive positive integers that are both quadratic nonresidues modulo
p
n
be an integer of the form
16\,k+12
\{b_{1},b_{2},b_{3},b_{4}\}
be a set of integers such that
b_{i}\cdot b_{j}+n
is a perfect square for all
i,j
i\neq j
. Prove that all numbers
b_{i}
are even.
The book also comprises short historical indications and 426 references. It really made me think of my first reading of Hardy and Wright, and I almost felt regret that I cannot start studying Number Theory again from scratch, but using this book! I highly recommend it not only to neophytes, but also to more “established” scientists who would like to start learning Number Theory, or to refresh and increase their knowledge of the field in an entertaining and subtle way.
Textbook of the University of Zagreb, Školska knjiga, Zagreb, 2021, 621 pages, translated by Petra Švob, ISBN 978-953-0-30897-8
Jean-Paul Allouche, Book Review: “Number Theory” by Andrej Dujella. Eur. Math. Soc. Mag. 121 (2021), pp. 54–55
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Initialize new track in tracker - MATLAB initializeTrack - MathWorks Nordic
Initialize Track in Radar Tracker
Initialize new track in tracker
trackID = initializeTrack(tracker,track)
trackID = initializeTrack(tracker,track,filter)
trackID = initializeTrack(tracker,track) initializes a new track in the tracker. The tracker must be updated at least once before initializing a track. If the track is initialized successfully, the tracker assigns the output trackID to the track, sets the UpdateTime of the track equal to the last step time in the tracker, and synchronizes the data in the input track to the initialized track.
A warning is issued if the tracker already maintains the maximum number of tracks specified by itsMaxNumTracks property. In this case, the trackID is returned as 0, which indicates a failure to initialize the track.
trackID = initializeTrack(tracker,track,filter) initializes a new track in the tracker, using a specified tracking filter, filter.
Create a radar tracker and update the tracker with detections at
\mathit{t}=0
\mathit{t}=1
radarTracker with properties:
Filter object, specified as a trackingKF, trackingEKF, or trackingUKF object.
radarTracker | deleteTrack
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Linearize Nonlinear Models - MATLAB & Simulink - MathWorks í•œêµ
Applications of Linearization
Linearization in Simulink Control Design
Model Requirements for Exact Linearization
Operating Point Impact on Linearization
Linearization is a linear approximation of a nonlinear system that is valid in a small region around an operating point.
For example, suppose that the nonlinear function is
y={x}^{2}
. Linearizing this nonlinear function about the operating point x = 1, y = 1 results in a linear function
y=2xâ1
Near the operating point,
y=2xâ1
y={x}^{2}
. Away from the operating point, the approximation is poor.
The next figure shows a possible region of good approximation for the linearization of
y={x}^{2}
. The actual region of validity depends on the nonlinear model.
Extending the concept of linearization to dynamic systems, you can write continuous-time nonlinear differential equations in this form:
\begin{array}{l}\stackrel{Ë}{x}\left(t\right)=f\left(x\left(t\right),u\left(t\right),t\right)\\ y\left(t\right)=g\left(x\left(t\right),u\left(t\right),t\right).\end{array}
In these equations, x(t) represents the system states, u(t) represents the inputs to the system, and y(t) represents the outputs of the system.
To represent the linearized model, define new variables centered about the operating point:
\begin{array}{l}\mathrm{δ}x\left(t\right)=x\left(t\right)â{x}_{0}\\ \mathrm{δ}u\left(t\right)=u\left(t\right)â{u}_{0}\\ \mathrm{δ}y\left(t\right)=y\left(t\right)â{y}_{0}\end{array}
The linearized model in terms of δx, δu, and δy is valid when the values of these variables are small:
\begin{array}{l}\mathrm{δ}\stackrel{Ë}{x}\left(t\right)=A\mathrm{δ}x\left(t\right)+B\mathrm{δ}u\left(t\right)\\ \mathrm{δ}y\left(t\right)=C\mathrm{δ}x\left(t\right)+D\mathrm{δ}u\left(t\right)\end{array}
Linearization is useful in model analysis and control design applications.
Exact linearization of the specified nonlinear Simulink® model produces linear state-space, transfer-function, or zero-pole-gain equations that you can use to:
Plot the Bode response of the Simulink model.
Evaluate loop stability margins by computing open-loop response.
Analyze and compare plant response near different operating points.
Design linear controller
Classical control system analysis and design methodologies require linear, time-invariant models. Simulink Control Design™ automatically linearizes the plant when you tune your compensator. See Choose a Control Design Approach.
Analyze closed-loop stability.
Measure the size of resonances in frequency response by computing closed-loop linear model for control system.
Generate controllers with reduced sensitivity to parameter variations and modeling errors.
You can use Simulink Control Design software to linearize continuous-time, discrete-time, or multirate Simulink models. The resulting linear time-invariant model is in state-space form.
By default, Simulink Control Design linearizes models using a block-by-block approach. This block-by-block approach individually linearizes each block in your Simulink model and combines the results to produce the linearization of the specified system.
You can also linearize your system using full-model numerical perturbation, where the software computes the linearization of the full model by perturbing the values of the root-level inputs and states. For each input and state, the software perturbs the model by a small amount and computes a linear model based on the model response to these perturbations. You can perturb the model using either forward differences or central differences.
The block-by-block linearization approach has several advantages to full-model numerical perturbation:
Exact linearization supports most Simulink blocks.
However, Simulink blocks with strong discontinuities or event-based dynamics linearize (correctly) to zero or large (infinite) gain. Models that include event-based or discontinuous behavior require special handling by Simulink Control Design software. Such event-based or discontinuous behavior can come from blocks such as:
Blocks from Discontinuities library
Pulse width modulation (PWM) signals
For most applications, the states in your Simulink model should be at steady state. Otherwise, your linear model is only valid over a small time interval.
Choosing the right operating point for linearization is critical for obtaining an accurate linear model. The linear model is an approximation of the nonlinear model that is valid only near the operating point at which you linearize the model.
Although you specify which Simulink blocks to linearize, all blocks in the model affect the operating point.
A nonlinear model can have two very different linear approximations when you linearize about different operating points.
The linearization result for this model is shown next, with the initial condition for the integration x0 = 0.
This table summarizes the different linearization results for two different operating points.
Linearization Result
Initial Condition = 5, State x1 = 5 30/s
Initial Condition = 0, State x1 = 0 0
You can linearize your Simulink model at three different types of operating points:
Trimmed operating point — Linearize at Trimmed Operating Point
Simulation snapshot — Linearize at Simulation Snapshot
Triggered simulation event — Linearize at Triggered Simulation Events
Trimming and Linearization Part 1: What is Linearization?
Trimming and Linearization Part 2: The Practical Side of Linearization
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(13E)-(15S)-15-hydroxy-9-oxoprosta-10,13-dienoate Wikipedia
(13E)-(15S)-15-hydroxy-9-oxoprosta-10,13-dienoate
In enzymology, a prostaglandin-A1 Delta-isomerase (EC 5.3.3.9) is an enzyme that catalyzes the chemical reaction
{\displaystyle \rightleftharpoons }
Hence, this enzyme has one substrate, (13E)-(15S)-15-hydroxy-9-oxoprosta-10,13-dienoate (Prostaglandin A1 or PGA1), and one product, (13E)-(15S)-15-hydroxy-9-oxoprosta-11,13-dienoate (Prostaglandin C1).
This enzyme belongs to the family of isomerases, specifically those intramolecular oxidoreductases transposing C=C bonds. The systematic name of this enzyme class is (13E)-(15S)-15-hydroxy-9-oxoprosta-10,13-dienoate Delta10-Delta11-isomerase. This enzyme is also called prostaglandin A isomerase.
Polet H, Levine L (1975). "Metabolism of prostaglandins E, A, and C in serum". J. Biol. Chem. 250 (2): 351–7. PMID 234423.
|
Solution of Perfect Path Patrol (ICPC NAQ 2020) - Tofu's Blog
This solution is using:
The "community" is actually a tree structure, so we can DFS it.
For the root, since we still didn't assign any patroller yet, so for every child edge, we need to assign some patrollers to satisfy their
p
s (number of patrollers).
Consider there are only two children. The one with
p = 2
a
, and the one with
p = 3
b
. The best way to assign patrollers is to assign
2
people to patrol the path between
and
b
, and one person to patrol the path between the root and
b
Then when we DFS
b
, we know there are
3
people whose patrolling task include the street (edge) between
b
and its parent node (which is root). We do not care where those people start from. Whether they depart from the root or
a
, we both can extend their "routes" to
b
's children.
Thus, the costs of satisfying every node's children are independent. Firstly we assign those people who come from its parent node to cover its children. And then, if needed, we allocate more people to meet the needs of all children. The root is just a little bit different: no one comes from its parent since it doesn't have a parent.
Well, this is the core part of this problem.
To simplify this part, suppose our current node is
a
k
people come from its parent node. We put the
p
s of all
a
's children into a sequence. Now you have 3 options:
Choose one element in the sequence and subtract it by
1
. This action is free, but you can only do it
k
times. (This action means extending one person's route who comes from the parent node).
Choose two elements in the sequence and subtract them by
1
. This action cost
1
. (This action means choose two children and assign a new patroller for the path between them).
1
1
. (This action means choose one child and assign a new patroller for the path between it and the current node
a
Our goal is to use these actions to make the whole sequence to
0
, and minimalize the cost.
Since the first action is free, so we want to do that as much as possible.
And because the second action can subtract two elements together, so compare with action three, we want to do action two more.
Now, consider the root, which does not have opportunity to use action one. If we want to do this process by hand, we can use this method:
select the biggest and the second biggest element in the sequence, subtract them by
1
, and then put them back to the sequence.
repeat, unless there's only one non-zero element.
At the end, if the sequence is still not
0
, we have to subtract that non-zero element one by one.
This is a nice method to find the minimal cost, but it is super slow: it is
O(P \log N)
P
is up to
10^9
Okay, now this is the most interesting part:
The biggest element of the sequence is bigger than the sum of remaining elements.
The biggest element of the sequence is less than or equal to the sum of remaining elements.
Suppose the biggest element is
x_{max}
, and the sum of remaining elements is
s_{remain}
. Also, suppose the sum of all elements is
s = x_{max} + s_{remain}
For case one, the solution is quite simple: subtract the biggest element every time together with another element. After all other elements are zero, the biggest element can only be subtracted one by one. In this case, the cost is just
x_{max}
For case two, I want to firstly write down the conclusion: The cost is
\frac s 2
s
\lceil \frac s 2 \rceil
s
Now, I will prove this by induction:
s
is odd, then we can substract the biggest element once to make
s
even. This will not increase the cost, because substracting two elements will not change the parity of
s
So, we asume
s
n
is the length of the sequence.
n = 2
n = 2
and "the biggest element of the sequence is less than or equal to the sum of remaining elements", we know that those two elements are equal. Thus, the cost is
\frac s 2
Induction steps:
Suppose for
n = k - 1
, the conclusion holds.
WTP: The conclusion holds for
n = k
x_{max} \leq s_{remain}
, everytime we substract
x_{max}
, we can subtract a smaller element together, until
x_{max}
is substracted to
0
x_{max}
is the biggest one, so it is definitely bigger or equal to the biggest element in the remaining elements.
So, we can ensure that after
x_{max}
0
k - 1
remaining elements, the biggest one is not bigger than the sum of other elements (since we have the ability to subtract that one to zero too).
Hence, according to our hypothesis, the conclusion holds for
n = k
This is how to assign people for the root. For other arbitrary nodes, we need to do "extending" as well.
To make sure our cost is minimalized, we want the biggest element of the sequence after applying "extending" is as small as possible. So, we can just use a balanced BST to maintain the sequence, and we pick up the biggest element, try to subtract it to make it equal to the second biggest element. If there are multiple biggest element, then just try to make the assignment evenly distributed. Repeat this process until you don't have opportunity to repeat more or all elements are zero.
Time complexity of this algorithm is
O(n \log n)
This is my code for this problem:
#define NS (500005)
template<typename _Tp> inline void IN(_Tp &dig)
char c; bool flag = 0; dig = 0;
while (c = getchar(), !isdigit(c)) if (c == '-') flag = 1;
if (flag) dig = -dig;
vector<PII> graph[NS];
LL dfs(int a, int f, int r)
static map<int, int, greater<int> > t;
t.clear(), t[0] = 0;
for (int i = 0, p; i < graph[a].size(); i += 1)
if ((p = graph[a][i].second, graph[a][i].first) != f)
if (!t[p]) t[p] = 1;
else t[p] += 1;
int v = (*t.begin()).first, k = (*t.begin()).second;
int v0 = (*t.begin()).first;
if ((LL)(v - v0) * k <= r)
t[v0] += k;
r -= (LL)(v - v0) * k;
int v1 = v - r / k, v2 = v1 - 1;
int k1 = k - r % k, k2 = k - k1;
if (!t[v1]) t[v1] = k1;
else t[v1] += k1;
if (v2 > 0 && k2 > 0)
map<int, int, greater<int> >::iterator i = t.begin();
int mx = i->first;
t[mx] -= 1;
LL rsum = (LL)i->first * i->second;
while ((++i) != t.end()) rsum += (LL)i->first * i->second;
if (mx <= rsum) res = (rsum + mx) / 2 + ((rsum + mx) & 1);
else res = mx;
for (int i = 0, u, p; i < graph[a].size(); i += 1)
if ((p = graph[a][i].second, u = graph[a][i].first) != f)
res += dfs(u, a, p);
IN(u), IN(v), IN(p);
graph[u].push_back(PII(v, p)), graph[v].push_back(PII(u, p));
printf("%lld\n", dfs(0, -1, 0));
|
Grzegorczyk_hierarchy Knowpia
The Grzegorczyk hierarchy (/ɡrɛˈɡɔːrtʃək/, Polish pronunciation: [ɡʐɛˈɡɔrt͡ʂɨk]), named after the Polish logician Andrzej Grzegorczyk, is a hierarchy of functions used in computability theory (Wagner and Wechsung 1986:43). Every function in the Grzegorczyk hierarchy is a primitive recursive function, and every primitive recursive function appears in the hierarchy at some level. The hierarchy deals with the rate at which the values of the functions grow; intuitively, functions in lower levels of the hierarchy grow slower than functions in the higher levels.
First we introduce an infinite set of functions, denoted Ei for some natural number i. We define
{\displaystyle E_{0}(x,y)=x+y}
{\displaystyle E_{1}(x)=x^{2}+2}
. I.e., E0 is the addition function, and E1 is a unary function which squares its argument and adds two. Then, for each n greater than 2, we define
{\displaystyle E_{n}(x)=E_{n-1}^{x}(2)}
, i.e. the x-th iterate of
{\displaystyle E_{n-1}}
evaluated at 2.
From these functions we define the Grzegorczyk hierarchy.
{\displaystyle {\mathcal {E}}^{n}}
, the n-th set in the hierarchy, contains the following functions:
Ek for k < n
the zero function (Z(x) = 0);
the successor function (S(x) = x + 1);
the projection functions (
{\displaystyle p_{i}^{m}(t_{1},t_{2},\dots ,t_{m})=t_{i}}
the (generalized) compositions of functions in the set (if h, g1, g2, ... and gm are in
{\displaystyle {\mathcal {E}}^{n}}
{\displaystyle f({\bar {u}})=h(g_{1}({\bar {u}}),g_{2}({\bar {u}}),\dots ,g_{m}({\bar {u}}))}
is as well)[note 1]; and
the results of limited (primitive) recursion applied to functions in the set, (if g, h and j are in
{\displaystyle {\mathcal {E}}^{n}}
{\displaystyle f(t,{\bar {u}})\leq j(t,{\bar {u}})}
{\displaystyle {\bar {u}}}
{\displaystyle f(0,{\bar {u}})=g({\bar {u}})}
{\displaystyle f(t+1,{\bar {u}})=h(t,{\bar {u}},f(t,{\bar {u}}))}
, then f is in
{\displaystyle {\mathcal {E}}^{n}}
as well)[note 1]
{\displaystyle {\mathcal {E}}^{n}}
is the closure of set
{\displaystyle B_{n}=\{Z,S,(p_{i}^{m})_{i\leq m},E_{k}:k<n\}}
with respect to function composition and limited recursion (as defined above).
These sets clearly form the hierarchy
{\displaystyle {\mathcal {E}}^{0}\subseteq {\mathcal {E}}^{1}\subseteq {\mathcal {E}}^{2}\subseteq \cdots }
because they are closures over the
{\displaystyle B_{n}}
{\displaystyle B_{0}\subseteq B_{1}\subseteq B_{2}\subseteq \cdots }
They are strict subsets (Rose 1984; Gakwaya 1997). In other words
{\displaystyle {\mathcal {E}}^{0}\subsetneq {\mathcal {E}}^{1}\subsetneq {\mathcal {E}}^{2}\subsetneq \cdots }
because the hyper operation
{\displaystyle H_{n}}
{\displaystyle {\mathcal {E}}^{n}}
{\displaystyle {\mathcal {E}}^{n-1}}
{\displaystyle {\mathcal {E}}^{0}}
includes functions such as x+1, x+2, ...
{\displaystyle {\mathcal {E}}^{1}}
provides all addition functions, such as x+y, 4x, ...
{\displaystyle {\mathcal {E}}^{2}}
provides all multiplication functions, such as xy, x4
{\displaystyle {\mathcal {E}}^{3}}
provides all exponentiation functions, such as xy, 222x, and is exactly the elementary recursive functions.
{\displaystyle {\mathcal {E}}^{4}}
provides all tetration functions, and so on.
Notably, both the function
{\displaystyle U}
and the characteristic function of the predicate
{\displaystyle T}
from the Kleene normal form theorem are definable in a way such that they lie at level
{\displaystyle {\mathcal {E}}^{0}}
of the Grzegorczyk hierarchy. This implies in particular that every recursively enumerable set is enumerable by some
{\displaystyle {\mathcal {E}}^{0}}
Relation to primitive recursive functionsEdit
{\displaystyle {\mathcal {E}}^{n}}
is the same as that of the primitive recursive functions, PR, except that recursion is limited (
{\displaystyle f(t,{\bar {u}})\leq j(t,{\bar {u}})}
for some j in
{\displaystyle {\mathcal {E}}^{n}}
) and the functions
{\displaystyle (E_{k})_{k<n}}
are explicitly included in
{\displaystyle {\mathcal {E}}^{n}}
. Thus the Grzegorczyk hierarchy can be seen as a way to limit the power of primitive recursion to different levels.
It is clear from this fact that all functions in any level of the Grzegorczyk hierarchy are primitive recursive functions (i.e.
{\displaystyle {\mathcal {E}}^{n}\subseteq {\mathsf {PR}}}
) and thus:
{\displaystyle \bigcup _{n}{{\mathcal {E}}^{n}}\subseteq {\mathsf {PR}}}
It can also be shown that all primitive recursive functions are in some level of the hierarchy (Rose 1984; Gakwaya 1997), thus
{\displaystyle \bigcup _{n}{{\mathcal {E}}^{n}}={\mathsf {PR}}}
and the sets
{\displaystyle {\mathcal {E}}^{0},{\mathcal {E}}^{1}-{\mathcal {E}}^{0},{\mathcal {E}}^{2}-{\mathcal {E}}^{1},\dots ,{\mathcal {E}}^{n}-{\mathcal {E}}^{n-1},\dots }
partition the set of primitive recursive functions, PR.
The Grzegorczyk hierarchy can be extended to transfinite ordinals. Such extensions define a fast-growing hierarchy. To do this, the generating functions
{\displaystyle E_{\alpha }}
must be recursively defined for limit ordinals (note they have already been recursively defined for successor ordinals by the relation
{\displaystyle E_{\alpha +1}(n)=E_{\alpha }^{n}(2)}
). If there is a standard way of defining a fundamental sequence
{\displaystyle \lambda _{m}}
, whose limit ordinal is
{\displaystyle \lambda }
, then the generating functions can be defined
{\displaystyle E_{\lambda }(n)=E_{\lambda _{n}}(n)}
. However, this definition depends upon a standard way of defining the fundamental sequence. Rose (1984) suggests a standard way for all ordinals α < ε0.
The original extension was due to Martin Löb and Stan S. Wainer (1970) and is sometimes called the Löb–Wainer hierarchy.
{\displaystyle {\bar {u}}}
represents a tuple of inputs to f. The notation
{\displaystyle f({\bar {u}})}
means that f takes some arbitrary number of arguments and if
{\displaystyle {\bar {u}}=(x,y,z)}
{\displaystyle f({\bar {u}})=f(x,y,z)}
. In the notation
{\displaystyle f(t,{\bar {u}})}
, the first argument, t, is specified explicitly and the rest as the arbitrary tuple
{\displaystyle {\bar {u}}}
{\displaystyle {\bar {u}}=(x,y,z)}
{\displaystyle f(t,{\bar {u}})=f(t,x,y,z)}
. This notation allows composition and limited recursion to be defined for functions, f, of any number of arguments.
Cichon, E. A.; Wainer, S. S. (1983), "The slow-growing and the Grzegorczyk hierarchies", Journal of Symbolic Logic, 48 (2): 399–408, doi:10.2307/2273557, ISSN 0022-4812, MR 0704094
Gakwaya, J.–S. (1997), A survey on the Grzegorczyk Hierarchy and its Extension through the BSS Model of Computability
Grzegorczyk, A (1953). "Some classes of recursive functions" (PDF). Rozprawy matematyczne. 4: 1–45.
Löb, M.H.; Wainer, S.S. (1970). "Hierarchies of Number Theoretic Functions I, II: A Correction". Archiv für mathematische Logik und Grundlagenforschung. 14: 198–199. doi:10.1007/bf01991855.
Rose, H.E., Subrecursion: Functions and hierarchies, Oxford University Press, 1984. ISBN 0-19-853189-3
Wagner, K. and Wechsung, G. (1986), Computational Complexity, Mathematics and its Applications v. 21. ISBN 978-90-277-2146-4
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Seasonal adjustment of debt securities issues
Seasonal adjustment estimates are available for euro area securities issues by sector and maturity (short-term and long-term). Data refer to end-of-month outstanding amounts, net issues, indexes of notional stocks and growth rates.
Three month annualised growth rates
Six month annualised growth rates
Background, approach and formulas
Seasonal adjustment is the process of estimating and removing seasonal effects from a time series. Seasonally adjusted data therefore facilitate analysis of short-term dynamics and the identification of changes in trends. The general principles followed by the ECB in the seasonal adjustment of time series are laid down in the ECB publication Seasonal adjustment of monetary aggregates and HICP for the euro area.
The approach used to seasonally adjust the securities issues statistics utilises a multiplicative decomposition using the Census X-12-ARIMA method, Version 0.2.10. Outliers are taken into consideration in order to minimise distortions to the estimated seasonal components. (Technical notes on the seasonal adjustment of securities issues series)
To ensure the additivity of the seasonally adjusted components to the seasonally adjusted aggregates, the seasonally adjusted series for the securities issues total are derived indirectly from the breakdowns by sector and maturity. A direct adjustment of the total is regularly carried out for monitoring purposes. The difference between direct and indirect estimates is generally negligible.
For seasonal adjustments during the year, forecast seasonal factors are used and seasonal factors are reestimated once a year. A review of the seasonal factors in use based on concurrent adjustments which might lead to a revision of seasonal factors takes place once a quarter. In addition, seasonal factors can be reviewed in the case of large data revisions.
Growth rate formulas
The three-month annualised growth rate is calculated with the formula below:
{R}_{3}^{t}=\left[{\left(\frac{{X}^{t}}{{X}^{t-3}}\right)}^{4}-1\right]\ast 100
where R3t represents the 3-month annualised growth rate for the reference month t and Xt is the securities issues index in month t.
The six-month annualised growth rate is calculated with the formula below:
{R}_{6}^{t}=\left[{\left(\frac{{X}^{t}}{{X}^{t-6}}\right)}^{2}-1\right]\ast 100
Monthly net issues of debt securities by euro area residents, broken down by sector, seasonally adjusted and non-adjusted
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On the M-Projective Curvature Tensor of -Contact Metric Manifolds
R. N. Singh, Shravan K. Pandey, "On the M-Projective Curvature Tensor of -Contact Metric Manifolds", International Scholarly Research Notices, vol. 2013, Article ID 932564, 6 pages, 2013. https://doi.org/10.1155/2013/932564
R. N. Singh 1 and Shravan K. Pandey 1
1Department of Mathematical Sciences, APS University, Rewa, Madhya Pradesh 486003, India
Academic Editor: J. Keesling
The object of the present paper is to study some curvature conditions on -contact metric manifolds.
The notion of the odd dimensional manifolds with contact and almost contact structures was initiated by Boothby and Wong in 1958 rather from topological point of view. Sasaki and Hatakeyama reinvestigated them using tensor calculus in 1961. Tanno [1] classified the connected almost contact metric manifolds whose automorphism groups possess the maximum dimension. For such a manifold, the sectional curvature of plain sections containing is a constant, say . He showed that they can be divided into three classes: (i) homogeneous normal contact Riemannian manifolds with , (ii) global Riemannian products of line or a circle with a Kähler manifold of constant holomorphic sectional curvature if , and (iii) a warped product space if . It is known that the manifolds of class (i) are characterized by admitting a Sasakian structure. Kenmotsu [2] characterized the differential geometric properties of the manifolds of class (iii); so the structure obtained is now known as Kenmotsu structure. In general, these structures are not Sasakian [2].
On the other hand in Pokhariyal and Mishra [3] defined a tensor field on a Riemannian manifold as where and . Such a tensor field is known as m-projective curvature tensor. Later, Ojha [4] defined and studied the properties of m-projective curvature tensor in Sasakian and Khler manifolds. He also showed that it bridges the gap between the conformal curvature tensor, conharmonic curvature tensor, and concircular curvature tensor on one side and H-projective curvature tensor on the other. Recently m-projective curvature tensor has been studied by Chaubey and Ojha [5], Singh et al. [6], Singh [7], and many others. Motivated by the above studies, in the present paper, we study flatness and symmetry property of -contact metric manifolds regarding m-projective curvature tensor. The present paper is organized as follows.
In this paper, we study the m-projective curvature tensor of -contact metric manifolds. In Section 2, some preliminary results are recalled. In Section 3, we study m-projectively semisymmetric -contact metric manifolds. Section 4 deals with m-projectively flat -contact metric manifolds. -m-projectively flat -contact metric manifolds are studied in Section 5 and obtained necessary and sufficient condition for an -contact metric manifold to be -m-projectively flat. In Section 6, m-projectively recurrent -contact metric manifolds are studied. Section 7 is devoted to the study of -contact metric manifolds satisfying . The last section deals with an -contact metric manifolds satisfying .
2. Contact Metric Manifolds
An odd dimensional differentiable manifold is said to admit an almost contact structure if there exist a tensor field of type (1, 1), a vector field , and a 1-form satisfying An almost contact metric structure is said to be normal if the induced almost complex structure on the product manifold defined by is integrable, where is tangent to , is coordinate of , and is smooth function on . Let be a compatible Riemannian metric with almost contact structure , that is, Then, becomes an almost contact metric manifold equipped with an almost contact metric structure . From (2) and (7), it can be easily seen that for all vector fields and . An almost contact metric structure becomes a contact metric structure if for all vector fields and . The 1-form is then a contact form, and is its characteristic vector field. We define a (1, 1)-tensor field by , where denotes the Lie-differentiation. Then, is symmetric and satisfies . We have and . Also, holds in a contact metric manifold.
A normal contact metric manifold is a Sasakian manifold. An almost contact metric manifold is Sasakian if and only if for all vector fields and , where is the Levi-Civita connection of the Riemannian metric . A contact metric manifold for which is a killing vector is said to be a K-contact manifold. A Sasakian manifold is K-contact, but the converse needs not be true. However, a 3-dimensional K-contact manifold is Sasakian [8]. It is well known that the tangent sphere bundle of a flat Riemannian manifold admits a contact metric structure satisfying [9]. On the other hand, on a Sasakian manifold the following holds: As a generalization of both and the Sasakian case; Blair et al. [11] considered the -nullity condition on a contact metric manifold and gave several reasons for studying it.
The -nullity distribution ([10, 11]) of contact metric manifold is defined by for all , where . A contact metric manifold with is called a -manifold. In particular, on a -manifold, we have On a -manifold, . If , the structure is Sasakian ( and is indeterminate), and if , the -nullity condition determines the curvature of completely [11]. In fact, for a -manifold, the conditions of being a Sasakian manifold, a K-contact manifold, and are all equivalent.
In a -manifold, the following relations hold ([11, 12]): where is the Ricci tensor of type (0, 2), is the Ricci operator, that is, , and is the scalar curvature of the manifold. From (11), it follows that Also in a -manifold, holds.
The -nullity distribution of a Riemannian manifold [13] is defined by for all and being a constant. If the characteristic vector field , then we call a contact metric manifold an -contact metric manifold [14]. If , then -contact metric manifold is Sasakian, and if , then -contact metric manifold is locally isometric to the product for and flat for . If , the scalar curvature is . If , then a -contact metric manifold reduces to a -contact metric manifolds. In [9], -contact metric manifold were studied in some detail.
In -contact metric manifolds the following relations hold ([15, 16]):
For a -dimensional almost contact metric manifold, m-projective curvature tensor is given by [3] for arbitrary vector fields , , and , where is the Ricci tensor of type and is the Ricci operator, that is, .
The m-projective curvature tensor for an -contact metric manifold is given by
3. M-Projectively Semisymmetric -Contact Metric Manifolds
Definition 1. A -dimensional -contact metric manifold is said to be -projectively semisymmetric [17] if it satisfies , where is the Riemannian curvature tensor and is the m-projective curvature tensor of the manifold.
Theorem 2. An m-projectively semisymmetric -contact metric manifold is an Einstein manifold.
Proof. Suppose that an -contact metric manifold is -projectively semisymmetric. Then, we have The above equation can be written as follows: In view of (22), the above equation reduces to Now, taking the inner product of the above equation with and using (3) and (9), we get which on using (30), (32), (34), and (35), gives Putting in the above equation and taking summation over , , we get which shows that is an Einstein manifold. This completes the proof.
4. M-Projectively Flat -Contact Metric Manifolds
Theorem 3. An m-projectively flat -contact metric manifold is an Einstein manifold.
Proof. Let . Then, from (30), we have Let be an orthonormal basis of the tangent space at any point. Putting in the above equation and summing over , , we get which shows that is an Einstein manifold. This completes the proof.
5. -M-Projectively Flat -Contact Metric Manifolds
Definition 4. A -dimensional -contact metric manifold is said to be -m-projectively flat [18] if for all .
Theorem 5. A -dimensional -contact metric manifold is -m-projectively flat if and only if it is an -Einstein manifold.
Proof. Let . Then, in view if (30), we have By virtue of (9), (23), and (28), the above equation reduces to which by putting , gives Now, taking the inner product of above equation with , we get which shows that -contact metric manifold is an -Einstein manifold. Conversely, suppose that (47) is satisfied. Then, by virtue of (46) and (31), we have . This completes the proof.
6. M-Projectively Recurrent -Contact Metric Manifolds
Definition 6. A nonflat Riemannian manifold is said to be m-projectively recurrent if its m-projective curvature tensor satisfies the condition where is nonzero 1-form.
Theorem 7. If an -contact metric manifold is m-projectively recurrent, then it is an -Einstein manifold.
Proof. We define a function on , where the metric is extended to the inner product between the tensor fields. Then, we have This can be written as From the above equation, we have Since the left-hand side of the above equation is identically zero and on , then that is, 1-form is closed.
Now from we have In view of (52) and (54), we have Thus by virtue of Theorem 3, the above equation shows that is an -Einstein manifold. This completes the proof.
7. -Contact Metric Manifolds Satisfying
Theorem 8. If on an -contact metric manifold , then .
Proof. Let . In this case, we can write In view of (34), the above equation reduces to Now, putting in above equation and using (3), (9), and (23), we get This completes the proof.
Theorem 9. On an -contact metric manifold, if , then .
Proof. Suppose that , then it can be written as which on using (33), takes the form Taking the inner product of above equation with , we get Now using (22), (28), and (29) in the above equation, we get Putting in the above equation and summing over , , we get This completes the proof.
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Copyright © 2013 R. N. Singh and Shravan K. Pandey. 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.
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Partial order reduction - Wikipedia
In computer science, partial order reduction is a technique for reducing the size of the state-space to be searched by a model checking or Automated planning and scheduling algorithm. It exploits the commutativity of concurrently executed transitions, which result in the same state when executed in different orders.
In explicit state space exploration, partial order reduction usually refers to the specific technique of expanding a representative subset of all enabled transitions. This technique has also been described as model checking with representatives.[1] There are various versions of the method, the so-called stubborn set method,[2] ample set method,[1] and persistent set method.[3]
1 Ample sets
2 Stubborn sets
Ample sets[edit]
Ample sets are an example of model checking with representatives. Their formulation relies on a separate notion of dependency. Two transitions are considered independent only if whenever they are mutually enabled, they cannot disable another and the execution of both results in a unique state regardless of the order in which they are executed. Transitions that are not independent, are dependent. In practice dependency is approximated using static analysis.
Ample sets for different purposes can be defined by giving conditions as to when a set of transitions is "ample" in a given state.
{\displaystyle {ample(s)=\varnothing }\iff {enabled(s)=\varnothing }}
C1 If a transition
{\displaystyle \alpha }
depends on some transition relation in
{\displaystyle ample(s)}
, this transition cannot be invoked until some transition in the ample set is executed.
Conditions C0 and C1 are sufficient for preserving all the deadlocks in the state space. Further restrictions are needed in order to preserve more nuanced properties. For instance, in order to preserve properties of linear temporal logic, the following two conditions are needed:
C2 If
{\displaystyle enabled(s)\neq ample(s)}
, each transition in the ample set is invisible
C3 A cycle is not allowed if it contains a state in which some transition
{\displaystyle \alpha }
is enabled, but is never included in ample(s) for any states s on the cycle.
These conditions are sufficient for an ample set, but not necessary conditions.[4]
Stubborn sets[edit]
Stubborn sets make no use of an explicit independence relation. Instead they are defined solely through commutativity over sequences of actions. A set
{\displaystyle T(s)}
is (weakly) stubborn at s, if the following hold.
{\displaystyle \forall a\in T(s)\forall b_{1},...,b_{n}\notin T(s)}
, if execution of the sequence
{\displaystyle b_{1},...,b_{n},a}
is possible and leads to the state
{\displaystyle s'}
, then execution of the sequence
{\displaystyle a,b_{1},...,b_{n}}
is possible and will lead to state
{\displaystyle s'}
D1 Either
{\displaystyle s}
is a deadlock, or
{\displaystyle \exists a\in T(s)}
{\displaystyle \forall b_{1},...,b_{n}\notin T(s)}
, the execution of
{\displaystyle b_{1},...,b_{n},a}
These conditions are sufficient for preserving all deadlocks, just like C0 and C1 are in the ample set method. They are, however, somewhat weaker, and as such may lead to smaller sets. The conditions C2 and C3 can also be further weakened from what they are in the ample set method, but the stubborn set method is compatible with C2 and C3.
There are also other notations for partial order reduction. One of the commonly used is the persistent set/sleep set algorithm. Detailed information can be found in Patrice Godefroid's thesis.[3]
In symbolic model checking, partial order reduction can be achieved by adding more constraints (guard strengthening). Further applications of partial order reduction involve automated planning.
^ a b (Peled 1993)
^ (Valmari 1990)
^ a b (Godefroid 1994)
^ (Clarke, Grumberg & Peled 1999)
Clarke, Edmund M.; Grumberg, Orna; Peled, Doron A. (1999). Model Checking. MIT Press.
Flanagan, Cormac; Godefroid, Patrice (2005). "Dynamic partial-order reduction for model checking software". Proceedings of POPL ’05, 32nd ACM Symp. on Principles of Programming Languages. pp. 110–121.
Godefroid, Patrice (1994). Partial-Order Methods for the Verification of Concurrent Systems -- An Approach to the State-Explosion Problem (PostScript) (PhD). University of Liege, Computer Science Department.
Holzmann, Gerard J (1993). The Spin Model Checker: Primer and Reference Manual. Addison-Wesley. ISBN 978-0-321-22862-8.
Peled, Doron A. (1993). "All from One, One for All: Model Checking Using Representatives". Proceedings of CAV'93, LNCS 697, Springer 1993. pp. 409–423. doi:10.1007/3-540-56922-7_34.
Valmari, Antti (1990). "Stubborn sets for reduced state space generation". Advances in Petri Nets 1990, LNCS 483, Springer 1991. pp. 491–515. doi:10.1007/3-540-53863-1_36.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Partial_order_reduction&oldid=1070270250"
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Normalize across each channel for each observation independently - MATLAB instancenorm - MathWorks Deutschland
Apply Instance Normalization
Normalize across each channel for each observation independently
Y = instancenorm(X,offset,scaleFactor)
Y = instancenorm(X,offset,scaleFactor,'DataFormat',FMT)
Y = instancenorm(___Name,Value)
The instance normalization operation normalizes the input data across each channel for each observation independently. To improve the convergence of training the convolutional neural network and reduce the sensitivity to network hyperparameters, use instance normalization between convolution and nonlinear operations such as relu.
The instancenorm function applies the layer normalization operation to dlarray data. Using dlarray objects makes working with high dimensional data easier by allowing you to label the dimensions. For example, you can label which dimensions correspond to spatial, time, channel, and batch dimensions using the "S", "T", "C", and "B" labels, respectively. For unspecified and other dimensions, use the "U" label. For dlarray object functions that operate over particular dimensions, you can specify the dimension labels by formatting the dlarray object directly, or by using the DataFormat option.
To apply instance normalization within a layerGraph object or Layer array, use instanceNormalizationLayer.
Y = instancenorm(X,offset,scaleFactor) applies the instance normalization operation to the input data X and transforms using the specified offset and scale factor.
The function normalizes over grouped subsets of the 'S' (spatial), 'T' (time), and 'U' (unspecified) dimensions of X for each observation in the 'C' (channel) and 'B' (batch) dimensions, independently.
Y = instancenorm(X,offset,scaleFactor,'DataFormat',FMT) applies the instance normalization operation to the unformatted dlarray object X with format specified by FMT using any of the previous syntaxes. The output Y is an unformatted dlarray object with dimensions in the same order as X. For example, 'DataFormat','SSCB' specifies data for 2-D image input with format 'SSCB' (spatial, spatial, channel, batch).
Y = instancenorm(___Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in previous syntaxes. For example, 'Epsilon',3e-5 sets the variance offset to 3e-5.
Create randomized input data with two spatial, one channel, and one observation dimension.
X = randn(width,height,channels,numObservations);
offset = dlarray(zeros(channels,1));
scaleFactor = dlarray(ones(channels,1));
Calculate the instance normalization.
dlZ = instancenorm(dlX,offset,scaleFactor);
View the size and format of the normalized data.
dims(dlZ)
\stackrel{^}{{x}_{i}}=\frac{{x}_{i}-{\mu }_{I}}{\sqrt{{\sigma }_{I}^{2}+ϵ}},
{y}_{i}=\gamma {\stackrel{^}{x}}_{i}+\beta ,
relu | fullyconnect | dlconv | dlarray | dlgradient | dlfeval | batchnorm | layernorm | groupnorm
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The current ratio is a popular metric used across the industry to assess a company's short-term liquidity with respect to its available assets and pending liabilities. In other words, it reflects a company's ability to generate enough cash to pay off all its debts once they become due. It's used globally as a way to measure the overall financial health of a company.
While the range of acceptable current ratios varies depending on the specific industry type, a ratio between 1.5 and 3 is generally considered healthy. A ratio value lower than 1 may indicate liquidity problems for the company, though the company may still not face an extreme crisis if it's able to secure other forms of financing. A ratio over 3 may indicate that the company is not using its current assets efficiently or is not managing its working capital properly.
The current ratio is calculated using two standard figures that a company reports in it's quarterly and annual financial results which are available on a company's balance sheet: current assets and current liabilities. The formula to calculate the current ratio is as follows:
\begin{aligned} \text{Current Ratio} = \frac{\text{Current Assets} }{\text{Current Liabilities} } \\ \end{aligned}
Current Ratio=Current LiabilitiesCurrent Assets
Current assets can be found on a company's balance sheet and represent the value of all assets it can reasonably expect to convert into cash within one year. The following are examples of current assets:
For instance, a look at the annual balance sheet of leading American retail giant Walmart Inc. (WMT) for the fiscal year ending January 2018 reveals that the company had $6.76 billion worth of cash and short-term investments, $5.61 billion in total accounts receivable, $43.78 billion in inventory, and $3.51 billion in other current assets. Walmart's current assets for the period are the sum of these items on the balance sheet: $59.66 billion.
The current assets figure is different from a similar figure called total assets, which also includes net property, equipment, long-term Investments, long-term notes receivable, intangible assets, and other tangible assets.
Current liabilities are a company's debts or obligations that are due within one year, appearing on the company's balance sheet. The following are examples of current liabilities:
Accrued liabilities like dividend, income tax, and payroll
For instance, for the fiscal year ending January 2018, Walmart had short-term debt of $5.26 billion, accounts payable worth $46.09 billion, other current liabilities worth $22.12 billion, and income tax payable worth $645 million. That brings Walmart's total current liabilities to $78.53 billion for the period.
Real-Life Examples of the Current Ratio
Based on the above-mentioned figures for Walmart, the current ratio for the retail giant is calculated as $59.66 / $78.52 = 0.76.
Similarly, technology leader Microsoft Corp. (MSFT) reported total current assets of $169.66 billion and total current liabilities of $58.49 billion for the fiscal year ending June 2018. Its current ratio comes to 2.90 ($169.66 / $58.49).
Investors and analysts would consider Microsoft's current ratio of 2.90 to be financially superior and healthy compared to Walmart's 0.76 because it indicates that the technology giant is better placed to pay off its obligations.
However, one must note that both companies belong to different industrial sectors and have different operating models, business processes, and cash flows that impact the current ratio calculations. Like with other financial ratios, the current ratio should be used to compare companies to their industry peers that have similar business models. Comparing the current ratios of companies across different industries may not lead to productive insights.
The current ratio is one of several measures that indicate the financial health of a company, but it's not the single and conclusive one. One must use it along with other liquidity ratios, as no single figure can provide a comprehensive view of a company.
Walmart. "2018 Annual Report," Page 63. Accessed March 4, 2020.
Walmart. "Annual Report 2019," Pages 46, 57. Accessed March 4, 2020.
Microsoft. "Annual Report 2018." Accessed March 4, 2020.
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Cohomological goodness and the profinite completion of Bianchi groups
15 July 2008 Cohomological goodness and the profinite completion of Bianchi groups
F. Grunewald, A. Jaikin-Zapirain, P. A. Zalesskii
F. Grunewald,1 A. Jaikin-Zapirain,2 P. A. Zalesskii3
1Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf
2Departamento de Matemáticas, Facultad de Ciencias Módulo C-XV, Universidad Autónoma de Madrid
3Departamento de Matemática, Universidade de Brasília
Duke Math. J. 144(1): 53-72 (15 July 2008). DOI: 10.1215/00127094-2008-031
The concept of cohomological goodness was introduced by J.-P. Serre in his book on Galois cohomology [31]. This property relates the cohomology groups of a group to those of its profinite completion. We develop properties of goodness and establish goodness for certain important groups. We prove, for example, that the Bianchi groups (i.e., the groups
\mathrm{PSL}\left(2,O\right)
O
is the ring of integers in an imaginary quadratic number field) are good. As an application of our improved understanding of goodness, we are able to show that certain natural central extensions of Fuchsian groups are residually finite. This result contrasts with examples of P. Deligne [5], who shows that the analogous central extensions of
\mathrm{Sp}\left(4,\mathbb{Z}\right)
do not have this property
F. Grunewald. A. Jaikin-Zapirain. P. A. Zalesskii. "Cohomological goodness and the profinite completion of Bianchi groups." Duke Math. J. 144 (1) 53 - 72, 15 July 2008. https://doi.org/10.1215/00127094-2008-031
Secondary: 14G32 , 19B37 , 57N10
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Estimation Input Signals - MATLAB & Simulink - MathWorks í•œêµ
Resp=\frac{FFT\left({y}_{est}\left(t\right)\right)\text{â}}{FFT\left({u}_{est}\left(t\right)\right)}.
Superposition signals are available only for online estimation with the Frequency Response Estimator block. For frequency response estimation at a vector of frequencies ω = [ω1, … , ωN] at amplitudes A = [A1, … , AN], the superposition signal is given by:
\mathrm{Î}u=\underset{i}{â}{A}_{i}\mathrm{sin}\left({\mathrm{Ï}}_{i}t\right).
The block supplies the perturbation Δu for the duration of the experiment (while the start/stop signal is positive). The block determines how long to wait for system transients to die away and how many cycles to use for estimation as shown in the following illustration.
Texp is the experiment duration that you specify with your configuration of the start/stop signal (See the start/stop port description on the block reference page for more information). For the estimation computation, the block uses only the data collected in a window of NlongestP. Here, P is the period of the slowest frequency in the frequency vector ω, and Nlongest is the value of the Number of periods of the lowest frequency used for estimation block parameter. Any cycles before this window are discarded. Thus, the settling time Tsettle = Texp – NlongestP. If you know that your system settles quickly, you can shorten Texp without changing Nlongest to effectively shorten Tsettle. If your system is noisy, you can increase Nlongest to get more averaging in the data-collection window. Either way, always choose Texp long enough for sufficient settling and sufficient data-collection. The recommended Texp = 2NlongestP.
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How to Prepare a Statement of Cash Flows: 13 Steps (with Pictures)
1 Calculating Beginning Cash and Cash Equivalents
2 Calculating Cash Generated from Operations
3 Calculating Cash Flows from Investing and Financing Activities
4 Calculating Ending Cash and Cash Equivalents
A statement of cash flows is one of the four major financial statements prepared by corporations at the end of each accounting period (the others being a balance sheet, income statement, and statement of retained earnings). The goal of the cash flow statement is to provide an accurate picture of the cash inflows, outflows, and net changes of cash during the accounting period. The statement is prepared by calculating net changes to cash from operating, investing, and financing activities. The total increase or decrease in cash for the current year is added to the ending cash from the prior year to calculate the ending cash and cash equivalents for the current year. Keep in mind that the ending cash amount on the statement of cash flows should be equal to the ending cash amount on the balance sheet. If the amounts are not equal, then there has been an error.
Calculating Beginning Cash and Cash Equivalents Download Article
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Determine the ending cash balance from the prior year. If the company prepared a statement of cash flows for the prior year, you can find this information there. If not, you will have to find information from the prior year's ending balance sheet and calculate the ending cash balance. Include cash and cash equivalents that can be converted into cash within one year. Cash equivalents include money market funds, certificates of deposit and savings accounts.[1] X Research source
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Add up the value of all of the cash and cash equivalents. On the balance sheet, find the value of the cash and cash equivalents. Suppose, for example, at the end of the prior year, the company had $800,000 in cash. In addition, it had money market funds worth $2,500,000 and CDs worth $1,500,000. Finally, there were savings accounts worth $1,200,000.
Add all of these amounts together to determine the ending cash balance for the prior year.
$800,000 (cash) + $2,500,000 (money market funds) + $1,500,000 (CDs) + $1,200,000 (savings) = $6,000,000 (prior year ending balance).
Establish the beginning cash balance for the current year. The ending balance from the prior year becomes the beginning balance for the current year. Using the above example, the ending balance from the prior year was $6,000,000. Use this as the beginning balance for the current year.
The beginning balance of cash and cash equivalents for the current year is $6,000,000.
Calculating Cash Generated from Operations Download Article
Start with net income. Net income is total revenues less operating expenses, depreciation, amortization and taxes. It is the company’s profit for the year. It includes all of the money that is left over after expenses have been paid. It is found on the company’s income statement.[2] X Research source
The company in the above example reported a net income of $8,000,000.
Adjust for depreciation and amortization. Depreciation and amortization are non-cash expenses that record the decrease in value of assets over time. They are calculated based on the original value of the asset and its useful life. But, since these expenses do not require an expenditure or receipt of cash, the amounts must be added back to the cash balances.[3] X Research source
The company in the above example reported $4,000,000 in depreciation and amortization expenses. As a consequence, $4,000,000 would be added back to the cash balance.
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Make adjustments for accounts payable and accounts receivable. Accounts payable is money the company owes to pay its creditors. Accounts receivable is money owed to the company for goods and services. For the income statement, accruals for accounts payable and accounts receivable are entered for the time period in which they occurred, whether or not cash has actually been paid or received. However, these accruals are non-cash transactions, so they must be adjusted for the statement of cash flows.[4] X Research source
Make sure to check the balance sheet for accrued liability accounts, such as Accrued Taxes or Accrued Payroll. These are expenses that will occur in the future, but that are not cash expenses right now. However, you will still need to adjust for these on the statement of cash flows. On the other hand, if you have any Prepaid Assets on the balance sheet, then these are expenses that have already been paid but that have not been incurred. You do not need to adjust these.
The Accounts receivable balance at the end of the prior year is the beginning balance of the current year. For example, imagine that the beginning balance was $6 million. At the end of the period, the accounts receivable balance is $8 million, an increase of $2 million during the year. Accounts receivable is income that has been earned, but not transferred into cash.
Accordingly, an increase in AR during the period means that the company has used cash during the year to finance its sales and requires deducting the increase from the cash balance. A decrease in AR means that customers have paid down the amounts previously owed and requires adding the decrease back in the cash balance.
For the company in the above example, net change to accounts receivable was $2,000,000. The money is still owed by customers, but it has not been paid. So, this must be subtracted.
Net change to accounts payable was $1,000,000. This is money the company owes but has not yet paid. So this must be added.
Calculate net cash generated from operations. Start with net income. Add back in depreciation and amortization expense. Reverse accruals for accounts payable and accounts receivable.
$8 million (net income) + $4 million (depreciation & Amortization expense) - $2 million (increase in Accounts Receivable) + $1 million (increase in Accounts Payable) = $11 million (net cash generated from operations).
The net cash flow provided by operating activities is $11,000,000.
Calculating Cash Flows from Investing and Financing Activities Download Article
Review investments in capital. Capital investments are all of the funds the company used to buy equipment that can produce goods or services.[5] X Research source When a company purchases equipment, it exchanges one asset (cash) for another asset (capital equipment). As a consequence, the purchase of the equipment is a use of cash. Similarly, if a company sold capital equipment, it would also be an exchange of one asset for another (receiving cash or an account receivable for the equipment). If a company purchases capital equipment with cash during the time period for which it is preparing the statement of cash flows, this outflow of cash must be included.[6] X Research source
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Determine the impact of financing activities. Financing activities include issuing and redemption of long and short term debt, issuing and retirement of stock and payment of stock dividends. These activities can have positive and negative effects on cash flow. Issuing debt and stock increase the company’s cash. Redeeming debt and paying stock dividends decreases cash.
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Make adjustments for investments and financing. Deduct cash paid for purchasing capital equipment. Subtract cash paid to redeem debt or pay dividends. Add in cash raised by issuing stock or new debt. Imagine that the example company performed the following transactions:
They purchased new computer equipment and assembly line machinery for a total of $4,000,000. This must be subtracted.
They increased short term debt by $500,000 and issued $250,000 in stock. These must be added.
Finally, they redeemed $3,000,000 in long-term debt and paid $2,000,000 in dividends. These must be subtracted.
-$4 million (equipment purchases for cash) + $0.5 million (sale of debt for cash) + $0.25 million (sale of stock for cash) - $3 million (redemption of long-term debt) - $2 million (payment of dividends) = $8.25 million (reduction in cash during period due to investment and financing activities).
The net adjustment to cash for investing and financing activities is -$8,250,000.
Calculating Ending Cash and Cash Equivalents Download Article
Determine the net increase or decrease to cash. This means figuring out if there was a net increase or decrease to cash for the current year. Start with total cash flows from operating activities. Add adjustments to cash flows for investment and financing activities. The end result is the total net increase or decrease to cash for the year.
In the above example, net cash flow from operating activities was $11,000,000.
The net change to cash from investing and financing activities was -$8,250,000.
The net increase or decrease to cash is
{\displaystyle \$11,000,000-\$8,250,000=\$2,750,000}
Calculate ending cash and cash equivalents. Start with the ending cash balance from the prior year. Add the net increase or decrease to cash from the current year. The end result is the total ending cash and cash equivalents for this year.
For the company in the above example, the ending cash balance from the prior year was $6,000,000.
The net increase or decrease to cash for the current year was $2,750,000.
The ending cash and cash equivalents for the current year is
{\displaystyle \$6,000,000+\$2,750,000=\$8,750,000}
Use the cash flow statement to evaluate the company’s financial health. The cash flow statement removes accounting methods such as accruals, depreciation and amortization. Therefore, it provides a more accurate statement of how cash is flowing in and out of the company. This allows investors to get a clear picture of the company’s earning power and operating success.[7] X Research source
A net increase in cash usually means that the company is running its operations efficiently and responsibly managing its investing and financing activities.
A net decrease in cash might indicate problems with the company’s operating, investing or financing activities. It would signal that the company needs to decrease expenses somewhere in order to improve its financial health.
Keep in mind that cash flow analysis is only a small part of analyzing a company's financial health. A net decrease in cash might also be concurrent with a major investment in the company's future growth. Similarly, a net increase in cash might reflect that management is getting lazy about reinvesting in the company.
How do I adjust for WHT and VAT in cash flow statements?
WHT is withholding tax, so if this is a liability on the balance sheet, it would be employees' withholding taxes that have not yet been paid to local/state/federal agencies. The taxes are taken out of the employees gross pay, which was a cash expense, so nothing needs to be added back. If it is the employer's portion of payroll tax, it would be a payable account and this would be added back because it was recorded as an expense but not yet paid. VAT is value added tax and this would be treated like any other expense if it was accrued or booked as a payable.
What can you do when ending cash is less than opening cash in the cash flow statement?
If ending cash is less than opening cash, it means that the cash balance decreased through the year, which is a possibility. What matters is if the ending cash balance on the cash flow statement ties to the actual ending cash balance. This is how you will know if an error has been made.
↑ http://www.money-zine.com/investing/investing/building-a-cash-flow-statement/
↑ http://www.money-zine.com/definitions/investing-dictionary/capital/
↑ http://www.money-zine.com/investing/investing/understanding-cash-flow/
To prepare a statement of cash flows, find out how much money the company had last year by checking the prior year’s ending balance sheet. Then, add the company’s net income, which is its revenue minus its expenses, taxes, and the depreciation of its assets. Make sure you include the amount the company owes other, and what others owe the company. Record those debts now, even though they haven’t been paid yet. Add those figures up to get your ending cash. Keep reading for tips from our Accounting reviewer on including investing expenses and income.
Italiano:Predisporre un Prospetto dei Flussi di Cassa
Español:preparar un estado de flujos de efectivo
Bahasa Indonesia:Menyiapkan Laporan Arus Kas
Français:créer un tableau de financement
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Oscillations-Quiz-1
Home » Unlabelled » Oscillations-Quiz-1
Oscillations-Quiz-1 Nabajyoti Ghosh 4.9 of 5
OSCILLATIONS QUIZ - 1 ...
OSCILLATIONS QUIZ - 1
Q1. A spring balance has a scale that can read from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this balance when displaced and released oscillates harmonically with a time period of 0.6 s. The mass of the body is (take
d=\mathrm{10\quad }\mathrm{m}/{\mathrm{s^2}}^{}
Q2. Two particles P and Q describe SHM of same amplitude a and frequency v along the same straight line. The maximum distance between the two particles is . The initial phase difference between them is
\textcolor[rgb]{0,0,0}{\pi }\textcolor[rgb]{0,0,0}{/}\textcolor[rgb]{0,0,0}{2}
\textcolor[rgb]{0,0,0}{\pi }\textcolor[rgb]{0,0,0}{/}\textcolor[rgb]{0,0,0}{6}
\textcolor[rgb]{0,0,0}{\pi }\textcolor[rgb]{0,0,0}{/}\textcolor[rgb]{0,0,0}{3}
Q3. A particle of mass m moving along the x -axis has a potential energy U(x) = a + bx2 where a and b are positive constants. It will execute simple harmonic motion with a frequency determined by the value of
b and a alone
b and m alone
b, a and m alone
Q4. The time taken by a particle executing simple harmonic motion to pass from point A to B where its velocities are same, is 2 s. After another 2 s, it returns to (b). The time period of oscillation is
Q5. A block is resting on a piston which executes simple harmonic motion in vertical plain with a period of 2.0 s in vertical plane at an amplitude just sufficient for the block to separate from the piston. The maximum velocity of the piston is
\frac{\textcolor[rgb]{0,0,0}{5/}}{\textcolor[rgb]{0,0,0}{\pi }}\textcolor[rgb]{0,0,0}{\mathrm{m}}\textcolor[rgb]{0,0,0}{/}\textcolor[rgb]{0,0,0}{\mathrm{s}}
\frac{\textcolor[rgb]{0,0,0}{10/}}{\textcolor[rgb]{0,0,0}{\pi }}\textcolor[rgb]{0,0,0}{\mathrm{m}}\textcolor[rgb]{0,0,0}{/}\textcolor[rgb]{0,0,0}{\mathrm{s}}
\frac{\textcolor[rgb]{0,0,0}{\mathrm{\pi /}}}{\textcolor[rgb]{0,0,0}{2}}\textcolor[rgb]{0,0,0}{\mathrm{m}}\textcolor[rgb]{0,0,0}{/}\textcolor[rgb]{0,0,0}{\mathrm{s}}
\frac{\textcolor[rgb]{0,0,0}{20/}}{\textcolor[rgb]{0,0,0}{\pi }}\textcolor[rgb]{0,0,0}{\mathrm{m}}\textcolor[rgb]{0,0,0}{/}\textcolor[rgb]{0,0,0}{\mathrm{s}}
Q6. A point mass is subjected to two simultaneous sinusoidal displacements in x - direction, and Adding a third sinusoidal displacement brings the mass to a complete rest. The values of B and ϕ
Q7. A particle executing SHM has velocities u and v and accelerations a and b in two of its positions. Find the distance between these two positions
Q8. Two masses m1 and m2 are suspended together by a massless spring of constant k. When the masses are in equilibrium, m1 is removed without disturbing the system; the amplitude of vibration is:
Q9. A metal rod of length L and mass m is pivoted at one end. A thin disk of mass M and radius R( < L) is attached at its centre to the free end of the rod. Consider two ways the disc is attached case A- the disc is not free to rotate about its centre and case B– the disc is free to rotate about its centre. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is/are true?
Restoring torque in case A = Restoring torque in case B
Restoring torque in case A < Restoring torque in case B
Angular frequency for case A < Angular frequency for case B
Angular frequency for case A > Angular frequency for case B
Q10. A particle, free to move along the x-axis, has potential energy given by U(x) = K[ 1 - exp(-x2 ) for
-\infty <x<+\propto \quad
where k is a positive constant of appropriate dimensions. Then
For small displacement from x=0, the motion is simple harmonic
If its total mechanical energy is k/2 , it has its minimum kinetic energy at the origin
At points away from the origin, the particle is in unstable equilibrium
BEST NEET COACHING CENTER | BEST IIT JEE COACHING INSTITUTE | BEST NEET, IIT JEE COACHING INSTITUTE: Oscillations-Quiz-1
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https://www.cleariitmedical.com/2021/07/oscillations-quiz-1.html
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Validators that do not fulfil their assigned duties are penalised by losing small amounts of stake.
Receiving a penalty is not the same as being slashed!
Break-even uptime for a validator is around 43%.
Incentivisation of validators on the beacon chain is a combination of carrot and stick. Validators are rewarded for contributing to the chain's security, and penalised for failing to contribute. As we shall see, penalties are quite mild. Nonetheless they provide good motivation for stakers to ensure that their validator deployments are running well.
It's common to hear of the penalties for being offline being referred to as "getting slashed". This is incorrect. Being slashed is a severe punishment for very specific misbehaviours, and results in the validator being ejected from the protocol in addition to some or all of its stake being removed.
Penalties are subtracted from validators' balances on the beacon chain and effectively burned, so they reduce the net issuance of the beacon chain.
Attestation penalties
Attestations are penalised for being missing, late, or incorrect. We'll lump these together as "missed" for conciseness.
Attesters are penalised for missed CASPER FFG votes, that is, missed source or target votes. But there is no penalty for a missed head vote. If a source vote is incorrect, then the target vote is missed; if the source or target vote is incorrect then the head vote is missed.
Let's update our rewards matrix to give the full picture of penalties and rewards for attestations. Recall that this shows the weights; we need to multiply by
\frac{nb}{W_{\Sigma}}
to get the actual reward.
<= 5 slots
<= 32 slots
> 32 Slots (missing)
Wrong source
-W_s-W_t
-W_s-W_t
-W_s-W_t
-W_s-W_t
Correct source only
W_s-W_t
W_s-W_t
-W_s-W_t
-W_s-W_t
Correct source and target only
W_s+W_t
W_s+W_t
-W_s+W_t
-W_s-W_t
Correct source, target and head
W_s+W_t+W_h
W_s+W_t
-W_s+W_t
-W_s-W_t
For more intuition, we can put in the numbers,
W_s = 14
W_t = 26
W_h = 14
, and normalise with
W_{\Sigma} = 64
-0.625
-0.625
-0.625
-0.625
-0.188
-0.188
-0.625
-0.625
+0.625
+0.625
+0.188
-0.625
+0.844
+0.625
+0.188
-0.625
Break-even uptime
Stakers sometimes worry that downtime will be very expensive. To examine this, we can estimate the break-even uptime. We'll ignore sync committee participation since that is so rare, so only attestations are relevant for the calculation.
We'll assume that, when online, the validator's performance is perfect, and that the rest of the validators are performing well (both of which are pretty good approximations to the beacon chain's actual performance over its first year).
p
is the proportion of time the validator is online, then its net income is,
0.844p - 0.625(1-p) = 1.469p - 0.625
. This is positive for
p > 42.5\%
. So, if your validator is online more than 42.5% of the time, you will be earning a positive return.
A useful rule of thumb is that it takes about a day of uptime to recover from a day of downtime.
Sync committee penalties
The small group of validators currently on sync committee duty receive a reward in each slot that they sign off on the correct head block (correct from the proposer's point of view).
Validators that don't participate (sign the wrong head block or don't show up at all) receive a penalty exactly equal to the reward they would have earned for being correct. And the block proposer receives nothing for the missing contribution.
Historical note: Since sync committee participation is rare for any given validator, and since rewards are significant, there were concerns with earlier designs that the resulting variance in rewards for validators would be quite unfair. Small stakers might prefer to join staking pools rather than solo stake in order to smooth out the variance, similarly to how proof of work mining pools have sprung up.
One suggested approach to reducing the variance was not to reward sync committee participation at all, but rather to raise overall reward levels for everyone and to penalise the sync committee validators if they did not participate. Ultimately the approach adopted was to reduce the length of sync committees (meaning lower rewards, but more often), reduce the proportion of total reward for participation, and introduce a penalty for non-participation – kind of half-way to the other proposal.
The main reasons1 for not adopting the former proposal, although it is elegant, seem to be around the psychology of being explicitly penalised but never explicitly rewarded. The penalty for not participating in a sync committee would be substantially bigger than the attestation reward over an epoch. In addition, participation is not entirely in the validator's own hands: it depends on the next block proposer being on the right fork. There were also concerns about changing the clean relationship between proposer rewards and the value of the duties they include in blocks.
Remarks on penalties
There are no explicit penalties related to block proposers.
In particular, there is no explicit penalty for failing to include deposits from the Eth1 chain, nor any direct incentive for including them. However, if a block proposer does not include deposits that the rest of the network knows about, then its block is invalid. This provides a powerful incentive to include outstanding deposits.
Also note that penalties are not scaled with participation as rewards are.
The detailed penalty calculations are defined in the spec in these functions:
Penalties for missed attestations are calculated in get_flag_index_deltas() as part of epoch processing.
Penalties for missed sync committee participation are calculated in process_sync_aggregate() as part of block processing.
The quite interesting discussion remains on the Ethereum R&D Discord.↩
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Issuance is the amount of new Ether created by the protocol in order to incentivise its participants.
An ideally running beacon chain issues a set amount of Ether per epoch, which is a multiple of the base reward per increment.
Total issuance is proportional to the square root of the number of validators. This is not a completely arbitrary choice.
There are three views we can take of the rewards given to validators to incentivise their correct participation in the protocol.
First, there is "issuance", which is the overall amount of new Ether generated by the protocol to pay rewards. Second there is the expected reward a validator might earn over the long run. And, third, there is the actual reward that any particular validator earns.
In this section we will look at issuance, and in the next we'll look at rewards. There is a strong relationship between these, though, so the separation is not totally clean.
First we must define the fundamental unit of reward, which is the "base reward per increment".
The base reward per increment
All rewards are calculated in terms of a "base reward per increment". This is in turn calculated as
Gwei(EFFECTIVE_BALANCE_INCREMENT * BASE_REWARD_FACTOR // integer_squareroot(get_total_active_balance(state)))
We will call the base reward per increment
b
for brevity. An increment is one unit of effective balance, which is 1 ETH (EFFECTIVE_BALANCE_INCREMENT), so active validators have up to 32 increments.
The BASE_REWARD_FACTOR is the big knob that we could turn if we wished to change the issuance rate of Ether on the beacon chain. So far it's always been set at 64 which results in the issuance graph we see below. This seems to be working very well and there are no plans to change it.
Issuance is the amount of new Ether created by the protocol in order to incentivise its participants. The net issuance, after accounting for penalties, burned transaction fees and so forth is sometimes referred to as inflation, or supply growth.
The Eth1 chain issues new Ether in the form of block and uncle rewards. Since the London upgrade this issuance has been offset in part, or even at times exceeded by the burning of transaction base fees due to EIP-1559.
During the current Altair period, issuance on the beacon chain is additional to that on the Eth1 chain, but is much smaller (around 10% as much). After The Merge, there will no longer be any block or uncle rewards on the Eth1 chain. But the base fee burn will remain, and it is very likely that the net issuance will become negative – more Ether will be destroyed than created1 – at least in the short to medium term. In the longer term, Anders Elowsson argues that an equilibrium will be reached between Ether issuance from proof of stake and Ether destruction due to EIP-1559, leading to a steady overall supply of Ether.
In the following we will be assuming that the beacon chain is running optimally, that is, with all validators performing their duties perfectly. In reality this is impossible to achieve on a permissionless, globally distributed, peer-to-peer network, although the beacon chain has been performing within a few percent of optimally for most of its history. In any case, actual validator rewards and net issuance will certainly be a little or a lot lower, depending on participation rates in the network.
Under the ideal conditions we are assuming, the beacon chain is designed to issue a total of exactly
Tb
Gwei in rewards per epoch. Here,
T
is the total number of increments held by active validators, or in other words the total of all their effective balances in Ether. This is the maximum issuance – the maximum amount of new Ether – that the beacon chain can generate. If all
N
validators have the maximum 32 ETH effective balance, then this works out to be
32Nb
Gwei per epoch in total.
365.25 \times 225 = 82181.25
epochs per year, and BASE_REWARD_FACTOR
= 64
\begin{aligned} \text{Max issuance per year} &= 82181.25 \times \frac{32 \times 64 \times N}{\sqrt{32 \times 10^9 \times N}} \text{ETH} \\ &= 940.87 \sqrt{N} \\ \end{aligned}
With 300,000 validators this equates to 515,333 ETH per year, plus change. For comparison, the Eth1 block and uncle rewards currently amount to almost five million ETH per year.
We can graph the maximum issuance as a function of the number of validators. It's just a scaled square root curve.
Maximum annual protocol issuance on the beacon chain as a function of the number of active validators.
The goal is to distribute these rewards evenly among validators (continuing to assume that things are running optimally), so that, on a long term average, each validator
i
n_ib
Gwei per epoch, where
n_i
is the number of increments it possesses, equivalently its effective balance in Ether. In these terms
T = \sum^{N-1}_{i=0}{n_i}
Thus, a well-performing validator with a 32 ETH effective balance can expect to earn a long-term average of
32b
Gwei per epoch. Of course,
b
changes over time as the total active balance changes, but absent a mass slashing event that change will be slow.
Similarly to the issuance calculation, we can calculate the expected annual percentage reward for a validator due to participating in the beacon chain protocol:
\begin{aligned} \text{APR} &= 100 \times 82181.25 \times \frac{64}{\sqrt{32 \times 10^9 \times N}} \% \\ &= \frac{2940.21}{\sqrt{N}} \% \\ \end{aligned}
For example, with 300,000 validators participating, this amounts to an expected return of 5.37% on a validator's effective balance.
Graphing this give us an inverse square root curve.
The expected annual percentage rewards for stakers as a function of the number of active validators.
Inverse square root scaling
The choice to scale the per-validator expected reward with
\frac{1}{\sqrt{N}}
is not obvious, and we can imagine different scenarios.
If we model the per-validator reward as
r \propto N^{-p}
, then some options are as follows.
p = 0
: each validator earns a constant return regardless of the total number of validators. Issuance is proportional to
N
p = \frac{1}{2}
: issuance scales like
\sqrt{N}
, the formula we are using.
p = 1
: each validator's expected reward is inversely proportional to the total number of validators. Issuance is independent of the total number of validators.
Adopting a concave function is attractive as it allows an equilibrium number of validators to be discovered without constantly fiddling with parameters. Ideally, if more validators join, we want the per-validator reward to decrease to disincentivise further joiners; if validators drop out we want the per-validator reward to increase to encourage new joiners. Eventually, an equilibrium number of validators will be found that balances the staking reward against the perceived risk and opportunity cost of staking. Assuming that the protocol is not overly sensitive to the total number of validators, this seems to be a nice feature to have.
That would rule out the first,
p=0
, option. The risk with
p = 0
is that, if the reward rate is set lower than the perceived risk, then all rational validators will exit. If we set it too high, then we end up paying for more security than we need (too many over-incentivised validators). Frequent manual tuning via hard-forks could be required to adjust the rate.
The arguments for selecting
p = \frac{1}{2}
p = 1
are quite subtle and relate to discouragement attacks. With
p \ne 0
, a set of validators may act against other validators by censoring them, or performing other types of denial of service, in order to persuade them to exit the system, thus increasing the rewards for themselves. Subject to various assumptions and models, we find that we require
p \le \frac{1}{2}
for certain kinds of attack to be profitable. Essentially, we don't want to increase rewards too much for validators that succeed in making other validators exit the beacon chain.
Note that after the Merge, validators' income will include a significant component from transaction tips and MEV, which will have the effect of pushing
p
1
, and much of this reasoning will become moot. Discouragement attacks in that regime are an unsolved problem.
For more background to the
\frac{1}{\sqrt{N}}
reward curve, see
Casper: The Fixed Income Approach,
Vitalik's Serenity Design Rationale, and
the Discouragement Attacks paper.
You can see Ethereum's current issuance and play with various scenarios at ultrasound.money.↩
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University of Florida/Eml4500/f08.FEABBQ/HW1/Notes - Wikiversity
University of Florida/Eml4500/f08.FEABBQ/HW1/Notes
< University of Florida | Eml4500 | f08.FEABBQ/HW1(Redirected from Eml4500.f08.FEABBQ/HW1/Notes)
Lecture Notes - Weeks 1,2[edit | edit source]
1 Lecture Notes - Weeks 1,2
3.1 Labeling Convention
4 Force-Displacement Relation
5 Recipe: Steps to Solving Simple Truss System Problems
The Following is a summary of the topics and information discussed during the first two weeks of EML4500 - Finite Element Analysis (FEA) at the University of Florida in the fall of 2008. A summary of all administrative information can be found here.
Wikiversity is just one of many projects formed under the parent organization of Wikimedia. Since Wikiversity is the platform in which the Professor, Dr. Loc Vu Quoc, plans on using for group collaboration and homework submission, most of the first week was dedicated to familiarizing the students with its structure.
Free Body Diagrams[edit | edit source]
Although a review from statics, the free body diagram (FBD) is an essential tool in FEA. In addition, because the systems for which FEA is used can be complex, it is necessary to form a rigorous convention for the labeling of all FBDs. Figures 1-3 below are the examples used in class to illustrate this convention.
Labeling Convention[edit | edit source]
In breaking a structure apart, it is important to differentiate between global FBDs (Fig. 2) and local FBDs (Fig. 3). For this purpose a circle is used when describing global nodes, while a square is used for local nodes. In addition, it is necessary to distinguish each piece, or element. A triangle is used for this purpose.
It is also important to assign a convention for the naming of forces and displacements. In global FBDs, as shown in Fig. 2, each force is titled R where the subscript denotes which node the force is being applied to, and the superscript shows the direction of the force. For local FBDs, as shown in Fig. 3, f denotes a force while d indicates a displacement. Here the subscript defines the particular local force or displacement, while the superscript indicates the element it corresponds to.
Force-Displacement Relation[edit | edit source]
The relationship between forces and displacements can be modeled by a combination of springs. The most simple example is a 1-D spring with one fixed end. In this case, the relationship between the force applied to the free end and the displacement of that end of the spring is given by:
{\displaystyle f=k\cdot d}
To illustrate how matrices can be used for systems with more degrees of freedom (dof), a simple example is a 1-D spring with both ends being free. Here the force-displacement relation is given by:
{\displaystyle \left[{\begin{array}{c}f_{1}\\f_{2}\end{array}}\right]=\left[{\begin{array}{cc}k&-k\\-k&k\end{array}}\right]\left[{\begin{array}{c}d_{1}\\d_{2}\end{array}}\right]}
Recipe: Steps to Solving Simple Truss System Problems[edit | edit source]
The first step of analyzing a truss system is to look at the entire system as a whole, also known as the global picture. The preliminary procedures in this analysis are first creating the global coordinate system, numbering the nodes in the system, as well as noting the global forces and displacement degrees of freedom (dofs) in the system. As a convention, both the global forces and displacement dofs should be named starting with the first global node in the directions of the global coordinate system (x and y directions) and continued in this fashion for each successive node. The aforementioned FBDs are good tools for the visualization of systems and the forces and dofs acting upon them. For both the global dofs and the global forces, each contain some unknown and some known quantities. The known values are usually given in the problem as contraints or parameters, such as a fixed node (zero displacement) or an applied force. The unknown values are the values that will be solved for using finite element method (FEM).
After the system is correctly identified, the next step is to analyze each element in the system separately. A FBD should be created for each element of the system, identifying the dofs and forces acting upon each particular element. The nodes, dofs, and forces should be identified by the same convention mentioned in step 1, however either global or local element coordinates may be used.
Once all the dofs and forces are identified for each element in the system, the next step is to create the element stiffness and force matrices. These matrices are assembled into the global force-displacement (FD) relation:
{\displaystyle {\underline {K}}\cdot {\underline {d}}={\underline {F}}}
where K is a nxn stiffness matrix, d is a nx1 displacement dof matrix, F is a nx1 force matrix, and n is the number of both known and unknown dofs.
Next, the known dofs are eliminated to reduce the global force displacement relations. The K,d, and F matrices above - that were originally nxn, nx1, and nx1 matrices, respectively - become mxm, mx1, and mx1, respectively, where m is the number of unknown only dofs and is less than n. Assuming the new stiffness, K, matrix is an invertible matrix, the above equation can be rearranged to solve for the displacement matrix.
Compute the element forces from the now known dofs and from this also calculate the element stress.
Use these to compute the remaining unknown (reaction) forces.
Retrieved from "https://en.wikiversity.org/w/index.php?title=University_of_Florida/Eml4500/f08.FEABBQ/HW1/Notes&oldid=2224773"
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Rank features for unsupervised learning using Laplacian scores - MATLAB fsulaplacian - MathWorks América Latina
fsulaplacian
Rank Features Using Specified Similarity Matrix
Rank features for unsupervised learning using Laplacian scores
idx = fsulaplacian(X)
idx = fsulaplacian(X,Name,Value)
[idx,scores] = fsulaplacian(___)
idx = fsulaplacian(X) ranks features (variables) in X using the Laplacian scores. The function returns idx, which contains the indices of features ordered by feature importance. You can use idx to select important features for unsupervised learning.
idx = fsulaplacian(X,Name,Value) specifies additional options using one or more name-value pair arguments. For example, you can specify 'NumNeighbors',10 to create a similarity graph using 10 nearest neighbors.
[idx,scores] = fsulaplacian(___) also returns the feature scores scores, using any of the input argument combinations in the previous syntaxes. A large score value indicates that the corresponding feature is important.
Rank the features based on importance.
[idx,scores] = fsulaplacian(X);
Create a bar plot of the feature importance scores.
xlabel('Feature rank')
ylabel('Feature importance score')
Select the top five most important features. Find the columns of these features in X.
The 15th column of X is the most important feature.
Compute a similarity matrix from Fisher's iris data set and rank the features using the similarity matrix.
Find the distance between each pair of observations in meas by using the pdist and squareform functions with the default Euclidean distance metric.
D = pdist(meas);
Z = squareform(D);
Construct the similarity matrix and confirm that it is symmetric.
S = exp(-Z.^2);
issymmetric(S)
Rank the features.
idx = fsulaplacian(meas,'Similarity',S)
Ranking using the similarity matrix S is the same as ranking by specifying 'NumNeighbors' as size(meas,1).
idx2 = fsulaplacian(meas,'NumNeighbors',size(meas,1))
idx2 = 1×4
Input data, specified as an n-by-p numeric matrix. The rows of X correspond to observations (or points), and the columns correspond to features.
The software treats NaNs in X as missing data and ignores any row of X containing at least one NaN.
Example: 'NumNeighbors',10,'KernelScale','auto' specifies the number of nearest neighbors as 10 and the kernel scale factor as 'auto'.
Similarity — Similarity matrix
[] (empty matrix) (default) | symmetric matrix
Similarity matrix, specified as the comma-separated pair consisting of 'Similarity' and an n-by-n symmetric matrix, where n is the number of observations. The similarity matrix (or adjacency matrix) represents the input data by modeling local neighborhood relationships among the data points. The values in a similarity matrix represent the edges (or connections) between nodes (data points) that are connected in a similarity graph. For more information, see Similarity Matrix.
If you specify the 'Similarity' value, then you cannot specify any other name-value pair argument. If you do not specify the 'Similarity' value, then the software computes a similarity matrix using the options specified by the other name-value pair arguments.
Standardized Euclidean distance. Each coordinate difference between observations is scaled by dividing by the corresponding element of the standard deviation computed from X. Use the Scale name-value pair argument to specify a different scaling factor.
Mahalanobis distance using the sample covariance of X, C = cov(X,'omitrows'). Use the Cov name-value pair argument to specify a different covariance matrix.
Minkowski distance. The default exponent is 2. Use the P name-value pair argument to specify a different exponent, where P is a positive scalar value.
One minus the cosine of the included angle between observations (treated as vectors)
One minus the sample correlation between observations (treated as sequences of values)
Example: 'Distance','minkowski','P',3 specifies to use the Minkowski distance metric with an exponent of 3.
Covariance matrix for the Mahalanobis distance metric, specified as the comma-separated pair consisting of 'Cov' and a positive definite matrix.
std(X,'omitnan') (default) | numeric vector of nonnegative values
Scaling factors for the standardized Euclidean distance metric, specified as the comma-separated pair consisting of 'Scale' and a numeric vector of nonnegative values.
Scale has length p (the number of columns in X), because each dimension (column) of X has a corresponding value in Scale. For each dimension of X, fsulaplacian uses the corresponding value in Scale to standardize the difference between observations.
log(size(X,1)) (default) | positive integer
Number of nearest neighbors used to construct the similarity graph, specified as the comma-separated pair consisting of 'NumNeighbors' and a positive integer.
Example: 'NumNeighbors',10
KernelScale — Scale factor
Scale factor for the kernel, specified as the comma-separated pair consisting of 'KernelScale' and 'auto' or a positive scalar. The software uses the scale factor to transform distances to similarity measures. For more information, see Similarity Graph.
The 'auto' option is supported only for the 'euclidean' and 'seuclidean' distance metrics.
If you specify 'auto', then the software selects an appropriate scale factor using a heuristic procedure. This heuristic procedure uses subsampling, so estimates can vary from one call to another. To reproduce results, set a random number seed using rng before calling fsulaplacian.
idx — Indices of features ordered by feature importance
Indices of the features in X ordered by feature importance, returned as a numeric vector. For example, if idx(3) is 5, then the third most important feature is the fifth column in X.
scores — Feature scores
Feature scores, returned as a numeric vector. A large score value in scores indicates that the corresponding feature is important. The values in scores have the same order as the features in X.
A similarity graph models the local neighborhood relationships between data points in X as an undirected graph. The nodes in the graph represent data points, and the edges, which are directionless, represent the connections between the data points.
If the pairwise distance Disti,j between any two nodes i and j is positive (or larger than a certain threshold), then the similarity graph connects the two nodes using an edge [2]. The edge between the two nodes is weighted by the pairwise similarity Si,j, where
{S}_{i,j}=\mathrm{exp}\left(-{\left(\frac{Dis{t}_{i,j}}{\sigma }\right)}^{2}\right)
fsulaplacian constructs a similarity graph using the nearest neighbor method. The function connects points in X that are nearest neighbors. Use 'NumNeighbors' to specify the number of nearest neighbors.
S={\left({S}_{i,j}\right)}_{i,j=1,\text{\hspace{0.17em}}\dots ,\text{\hspace{0.17em}}n}
A degree matrix Dg is an n-by-n diagonal matrix obtained by summing the rows of the similarity matrix S. That is, the ith diagonal element of Dg is
{D}_{g}\left(i,i\right)=\sum _{j=1}^{n}{S}_{i,j}.
A Laplacian matrix, which is one way of representing a similarity graph, is defined as the difference between the degree matrix Dg and the similarity matrix S.
L={D}_{g}-S.
The fsulaplacian function ranks features using Laplacian scores[1] obtained from a nearest neighbor similarity graph.
fsulaplacian computes the values in scores as follows:
For each data point in X, define a local neighborhood using the nearest neighbor method, and find pairwise distances
Dis{t}_{i,j}
for all points i and j in the neighborhood.
Convert the distances to the similarity matrix S using the kernel transformation
{S}_{i,j}=\mathrm{exp}\left(-{\left(\frac{Dis{t}_{i,j}}{\sigma }\right)}^{2}\right)
, where σ is the scale factor for the kernel as specified by the 'KernelScale' name-value pair argument.
Center each feature by removing its mean.
{\stackrel{˜}{x}}_{r}={x}_{r}-\frac{{x}_{r}^{T}{D}_{g}1}{{1}^{T}{D}_{g}1}1,
where xr is the rth feature, Dg is the degree matrix, and
{1}^{T}={\left[1,\cdots ,1\right]}^{T}
Compute the score sr for each feature.
{s}_{r}=\frac{{\stackrel{˜}{x}}_{r}^{T}S{\stackrel{˜}{x}}_{r}}{{\stackrel{˜}{x}}_{r}^{T}{D}_{g}{\stackrel{˜}{x}}_{r}}.
Note that [1] defines the Laplacian score as
{L}_{r}=\frac{{\stackrel{˜}{x}}_{r}^{T}L{\stackrel{˜}{x}}_{r}}{{\stackrel{˜}{x}}_{r}^{T}{D}_{g}{\stackrel{˜}{x}}_{r}}=1-\frac{{\stackrel{˜}{x}}_{r}^{T}S{\stackrel{˜}{x}}_{r}}{{\stackrel{˜}{x}}_{r}^{T}{D}_{g}{\stackrel{˜}{x}}_{r}},
where L is the Laplacian matrix, defined as the difference between Dg and S. The fsulaplacian function uses only the second term of this equation for the score value of scores so that a large score value indicates an important feature.
Selecting features using the Laplacian score is consistent with minimizing the value
\frac{{\sum }_{i,j}{\left({x}_{ir}-{x}_{jr}\right)}^{2}{S}_{i,j}}{Var\left({x}_{r}\right)},
where xir represents the ith observation of the rth feature. Minimizing this value implies that the algorithm prefers features with large variance. Also, the algorithm assumes that two data points of an important feature are close if and only if the similarity graph has an edge between the two data points.
[1] He, X., D. Cai, and P. Niyogi. "Laplacian Score for Feature Selection." NIPS Proceedings. 2005.
[2] Von Luxburg, U. “A Tutorial on Spectral Clustering.” Statistics and Computing Journal. Vol.17, Number 4, 2007, pp. 395–416.
fscmrmr | relieff | sequentialfs
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Gram-Schmidt Calculator - Maple Help
Home : Support : Online Help : Math Apps : Algebra and Geometry : Matrices : Gram-Schmidt Calculator
Inner product spaces are one of the most important concepts in linear algebra.
What is an Inner Product?
An inner product is an operation defined in a vector space that takes two vectors as parameters and produces a scalar (usually a real or a complex number) as a result. In general, inner products are denoted as C ·, · D. Specifically, the inner product of the elements a and b of the vector space V is written as: Ca, bD. For an operation to be an inner product in a vector space V, it must satisfy the following axioms.
For all x, y, z ∈ V and α a scalar of the field where the vector space is defined:
Bilinearity in the first argument: Cα⋅x, yD = α⋅Cx, yD and Cx + z, yD = Cx, yD + Cz, yD.
Conjugate Symmetry: Cx, yD =
\stackrel{‾}{〈\mathrm{y}, \mathrm{x}〉}
\stackrel{‾}{〈\mathrm{y}, \mathrm{x}〉}
denotes the complex conjugate of Cx, yD.
Positive Definiteness: For all non-zero x ∈ V, Cx, xD > 0.
A complex inner product is an inner product where the resulting scalar is a complex number.
A real inner product is an inner product where the resulting scalar is a real number.
What is Gram-Schmidt Orthogonalization?
A set of non-zero vectors from a vector space is said to be orthogonal if the inner product between any two vectors in the set is equal to 0. A set of vectors is said to be orthonormal if the set is orthogonal and if for any vector v in the set we have: Cv,vD = 1.
The Gram-Schmidt theorem states that given any set of linearly independent vectors from a vector space, it is always possible to generate an orthogonal set with the same number of vectors as the original set. The way to generate this set is by constructing it from the original set of vectors by using Gram-Schmidt's orthogonalization process:
If the projection of v onto u is given by
\mathrm{proj__u}\left(v\right)=\frac{〈v,u〉}{〈u,u〉}u
, then form a sequence of vectors as follows:
\mathrm{u__1}=\mathrm{v__1}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\mathrm{u__2}=\mathrm{v__2}-\mathrm{proj__u__1}\left(\mathrm{v__2}\right)
\mathrm{u__3}=\mathrm{v__3}-\mathrm{proj__u__1}\left(\mathrm{v__3}\right)-\mathrm{proj__u__2}\left(\mathrm{v__3}\right)
⋮
\mathrm{u__1},\mathrm{u__2},...\mathrm{u__k}
will be the required set of orthogonal vectors.
This calculator applies the Gram-Schmidt orthogonalization process to the columns of a matrix or to a set of vectors. By default, it performs the exact computation (as opposed to decimal approximations), and performs orthonormalization.
Select the Orthogonalization option if you want to orthogonalize your input instead of orthonormalizing it.
Select Complex Inner Product if your entries include complex numbers instead of real numbers.
If the input matrix or vectors contains floating point numbers, or if the Floating-Point Calculations option is selected, the Gram-Schmidt process will be carried out using floating point arithmetic, which necessarily introduces round-off error. As a result, linear dependence of the vectors (or less than full rank of the matrix) is not likely to be detected. In the floating-point domain, the singular value decomposition is a much superior method for obtaining an orthogonal basis for the span of a set of vectors.
Problem Entry
To enter your matrix or vectors, you can use palettes or keyboard shortcuts.
To use palettes, right-click in the entry box and select the Matrix button: (in a web browser), or open the Matrix palette (Maple and the Maple Player). Use the palette to enter your matrix (the process will be applied to the columns) or column vectors. When entering vectors, separate your vectors by commas:
\left[\begin{array}{c}1 \\ 2\end{array}\right]
\left[\begin{array}{c}4\\ -1\end{array}\right]
To enter vectors using the keyboard, use angle brackets to define your vector, and separate vectors by commas.
For example, enter <1,2>, <4,4> or <1,2,3>, <4,-1,2>, <11, 3/2, 0>
To enter a matrix using the keyboard, enter each column as a vector, separate columns by vertical bars, and encase the whole thing in angle brackets.
For example, to enter
\left[\begin{array}{cc}1& 3\\ -5& -1\end{array}\right]
, use: < <1, -5> | <3, -1> >.
Enter vectors for Gram-Schmidt:
Calculation and output form:
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Axiom_of_adjunction Knowpia
In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x ∪ {y} given by "adjoining" the set y to the set x.
{\displaystyle \forall x\,\forall y\,\exists w\,\forall z\,[z\in w\leftrightarrow (z\in x\lor z=y)].}
Bernays (1937, page 68, axiom II (2)) introduced the axiom of adjunction as one of the axioms for a system of set theory that he introduced in about 1929. It is a weak axiom, used in some weak systems of set theory such as general set theory or finitary set theory. The adjunction operation is also used as one of the operations of primitive recursive set functions.
Tarski and Smielew showed that Robinson arithmetic can be interpreted in a weak set theory whose axioms are extensionality, the existence of the empty set, and the axiom of adjunction (Tarski 1953, p.34).
In fact, empty set and adjunction alone (without extensionality) suffice to interpret Robinson arithmetic.[1]
^ Mancini, Antonella; Montagna, Franco (Spring 1994). "A minimal predicative set theory". Notre Dame Journal of Formal Logic. 35 (2): 186–203. doi:10.1305/ndjfl/1094061860. Retrieved 23 November 2021.
Bernays, Paul (1937), "A System of Axiomatic Set Theory--Part I", The Journal of Symbolic Logic, Association for Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862
Kirby, Laurence (2009), "Finitary Set Theory", Notre Dame J. Formal Logic, 50 (3): 227–244, doi:10.1215/00294527-2009-009, MR 2572972
Tarski, Alfred (1953), Undecidable theories, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland Publishing Company, MR 0058532
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A Simplified Method to Account for Plastic Rate Sensitivity With Large Deformations | J. Appl. Mech. | ASME Digital Collection
A Simplified Method to Account for Plastic Rate Sensitivity With Large Deformations
Structural Mechanics Program, Office of Naval Research, Code 474, Arlington, Va. 22217
Westinghouse Hanford Co., P.O. Box 1970, Richland, Wash. 99352
Perrone, N., and Bhadra, P. (December 1, 1979). "A Simplified Method to Account for Plastic Rate Sensitivity With Large Deformations." ASME. J. Appl. Mech. December 1979; 46(4): 811–816. https://doi.org/10.1115/1.3424659
A string supported impulsively loaded mass is used to study large deformation rate sensitivity effects where membrane action is dominant. It is found that an overall correction factor can be devised using physical properties associated with the average strain rate. Maximum strain rate occurs with a velocity field corresponding to the deformation state wherein half the initial kinetic energy has been dissipated. (If V0 is initial velocity, V0/
2
is associated with maximum strain rate.) Exact and approximate solutions for a broad range of parameters serve to illustrate and verify the procedure. A discussion is presented to show how the same methodology could also be applied via a modal approach to an arbitrary three-dimensional structure undergoing large deformations, if the primary mechanism for energy absorption is from membrane action.
Deformation, Membranes, Absorption, Kinetic energy, String
Nonlinear Vibrations of Beams, Strings, Plates, and Membranes Without Initial Tension
Chaotic Vibrations on a Coupled Vibrating System of a Post-Buckled Cantilevered Beam and an Axial Vibrating Body Connected With a Stretched String
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Pixelated Butterfly: Simple and Efficient Sparse Training for Neural Network Models · Hazy Research
Beidi Chen, Tri Dao and Chris Ré.
Our paper Pixelated Butterfly: Simple and Efficient Sparse Training for Neural Network Models is available on arXiv, and our code is available on GitHub.
Why Sparsity?
Recent results suggest that overparameterized neural networks generalize well (Belkin et al. 2019). We've witnessed the rise and success of large models (e.g., AlphaFold, GPT-3, DALL-E, DLRM), but they are expensive to train and becoming economically, technically, and environmentally unsustainable (Thompson et al. 2020). For example, a single training run of GPT-3 takes a month on 1,024 A100 GPUs and costs $12M. An ideal model should use less computation and memory while retaining the generalization benefits of large models.
The simplest and most popular direction is to sparsify these models -- sparsify matrix multiplications, the fundamental building blocks underlying neural networks. Sparsity is not new! It has a long history in machine learning (Lecun et al. 1990) and has driven fundamental progress in other fields such as statistics (Tibshirani et al. 1996), neuroscience (Foldiak et al. 2003), and signal processing (Candes et al. 2005). What is new is that in the modern overparameterized regime, sparsity is no longer used to regularize models -- sparse models should behave as close as possible to a dense model, only smaller and faster.
Sparse training is an active research area, but why has sparsity not been adopted widely? Below we summarize a few challenges that motivate our work:
Choice of Sparse Parameterization: Many existing methods, e.g., pruning (Lee et al. 2018, Evci et al. 2020), lottery tickets (Frankle et al. 2018), hashing (Chen et al. 2019, Kitaev et al. 2020) maintain dynamic sparsity masks. However, the overhead of evolving the sparsity mask often slows down (instead of speeds up!) training.
Hardware Suitability: Most existing methods adopt unstructured sparsity, which may be efficient in theory, but not on hardware such as GPUs (highly optimized for dense computation). An unstructured sparse model with 1% nonzero weights can be as slow as a dense model (Hooker et al. 2020).
Memory access for a hardware with block size 16: one (red) location access means full 16
\times
(blue) access.
Layer Agnostic Sparsity: Most existing work targets a single type of operation such as attention (Child et al. 2019, Zaheer et al. 2020), whereas neural networks often compose different modules (attention, MLP). In many applications the MLP layers are the main training bottleneck (Wu et al. 2020).
Therefore, we are looking for simple static sparsity patterns that are hardware-suitable and widely applicable to most NN layers.
Our Approach: Pixelated Butterfly
Intuition: In our early exploration, we observe that one sparsity pattern: butterfly + low-rank, consistently outperforms the others. This "magic" sparsity pattern closely connects to two lines of work in matrix structures (see Figure 1):
Sparse + low-rank matrices (Candes et al. 2011, Udell et al. 2019, Chen et al. 2021): can capture global and local information,
Butterfly matrices (Parker et al. 1995, Dao et al. 2019): their products can tightly represent any sparse matrices with near-optimal space and time complexity. With butterfly, we can avoid the combinatorial problem of searching over all the possible sparsity pattern!
Butterfly + low-rank is static and applicable to most NN layers.
Figure 1: Observation - Butterfly + Low-rank is a simple an effective fixed sparsity pattern.
However, butterfly matrices are inefficient on modern hardware because (1) their sparsity patterns are not block-aligned, and (2) they are products of many factors -- hard to parallelize.
We address the issues with two simple changes (see Figure 2).
Block butterfly matrices operate at the block level, yielding a block-aligned sparsity pattern.
Flat butterfly matrices are first-order approximation of butterfly with residual connection, that turns the original product of factors into a sum.
Figure 2: Visualization of Flat, Block, and Flat Block butterfly.
One last problem is that Flat butterfly matrices are necessarily high-rank and cannot represent low-rank matrices. The good news is we have a low-rank term that can increase expressiveness of Flat butterfly!
Putting everything together: Our proposal, Pixelated Butterfly, combines Flat Block butterfly and low-rank matrices to yield a simple and efficient sparse training method (see Figure 3). It applies to most major network layers that rely on matrix multiplication.
Figure 3: Pixelfly targets GEMM-based networks, which it views as a series of matmul. For each matmul from Schema, it (1) allocates compute budget based on dimension and layer type, (2) decides a mapping to our proposed flat block butterfly sparsity patterns, (3) outputs a hardware-aware sparse mask.
In short: up to 2.5
\times
faster training MLP-Mixer, ViT, and GPT-2 medium from scratch with no drop in accuracy.
Details: Pixelfly can improve training speed of different model architectures while retaining model quality on a wide range of domains and tasks (both upstream and downstream).
Image classification: We train both MLP-Mixer and ViT models from scratch up to 2.3
\times
faster on wall-clock time with no drop in accuracy compared to the dense model and up to 4
\times
compared to RigL and BigBird sparse baselines.
(top-1 acc. ↑)
Mixer-B/16 75.6 - 59.9M 12.6G
Pixelfly-Mixer-B/16 76.3 2.3
\times
17.4M 4.3G
ViT-B/16 78.5 - 86.6M 17.6G
Pixelfly-ViT-B/16 78.6 2.0
\times
Language modeling & text classification: We speed up GPT-2 medium dense model training by 2.5
\times
, achieving a perplexity of 22.5 on wikitext-103 and 5.2
\times
speed-up on Long Range Arena (LRA) benchmark.
(ppl ↓)
GPT-2-medium 20.9 - 345M 168G
Pixelfly-GPT-2-medium 21.0 2.5
\times
203M 27G
Downstream tasks: We train Pixelfly-GPT2-small on a larger scale dataset, OpenWebText, and evaluate the downstream quality on zero-shot generation and classification tasks (EleutherAI Repo, Zhao et al. 2021), achieving comparable and even better performance to the dense model.
(avg acc ↑)
(acc ↑)
GPT-2-small 31.8 38.3 30.5 117M
Pixelfly-GPT-2-small 32.1 38.3 32.5 82M
GPT-2-medium 33.2 31.87 35.4 345M
Pixelfly-GPT-2-medium 33.4 30.5 38.9 203M
Sparsity: The Way Forward
Our method is a first step towards the goal of making sparse models train faster than dense models and make them more accessible to the general machine learning community. We are excited about several future directions.
Pixelfly 2.0: Pixelfly is a simple first order approximation of the rich class of butterfly matrices, and there could be better approximations or even an exact hardware-efficient re-parameterization.
Going beyond dense models: Many algorithms for exploiting model sparsity have long been used in scientific computing, e.g., locality sensitive hashing, butterfly factorization. Sparse models might unlock tremendous performance improvements in applications such as PDE solving and MRI reconstruction.
Algorithm-Hardware Co-design: Inspired by the remarkable success of model pruning for inference, it is possible that dynamic block sparse mask could be made efficient yet still accurate on the next generation of ML accelerators.
Data Sparsity: In the near future (or now), there will be a swing from model-centric to data-centric AI, where the focus is to study how data can shape and systematically change what a model learns (see this blog). It is possible that a subset of training data is necessary for a model to maintain accuracy or quality, which can speed up training from a different angle!
Made by Hazy Research. Learn more about the lab ↗
|
Give initial conditions or initial solution - MATLAB setInitialConditions - MathWorks í•œêµ
Set initial conditions for the L-shaped membrane geometry to be
{x}^{2}+{y}^{2}
, except in the lower left square where it is
{x}^{2}-{y}^{4}
Set the initial conditions to
{x}^{2}+{y}^{2}
Set the initial conditions on region 2 to
{x}^{2}-{y}^{4}
. This setting overrides the first setting because you apply it after the first setting.
ut0=\left[\begin{array}{c}4+\frac{x}{{x}^{2}+{y}^{2}+{z}^{2}}\\ 5-\mathrm{tanh}\left(z\right)\\ 10\frac{y}{{x}^{2}+{y}^{2}+{z}^{2}}\end{array}\right]
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Time-Frequency Gallery - MATLAB & Simulink - MathWorks América Latina
Audio signal processing: Fundamental frequency estimation, cross synthesis, spectral envelope extraction, time-scale modification, time-stretching, and pitch shifting. (See Phase Vocoder with Different Synthesis and Analysis Windows for more details.)
stft computes the short-time Fourier transform. To invert the short-time Fourier transform, use the istft function.
dlstft computes the deep learning short-time Fourier transform. You must have Deep Learning Toolbox™ installed.
pspectrum or spectrogram computes the spectrogram.
xspectrogram computes the cross-spectrogram of two signals.
You can also use the spectrogram view in Signal Analyzer to view the spectrogram of a signal.
Use the persistence spectrum option in pspectrum or Signal Analyzer to identify signals hidden in other signals.
\beta =30
Deep learning: The CWT can be used to create time-frequency representations that can be used to train a convolutional neural network. Classify Time Series Using Wavelet Analysis and Deep Learning (Wavelet Toolbox) shows how to classify ECG signals using scalograms and transfer learning.
cwt (Wavelet Toolbox) computes the continuous wavelet transform and displays the scalogram. Alternatively, create a CWT filter bank using cwtfilterbank (Wavelet Toolbox) and apply the wt (Wavelet Toolbox) function. Use this method to run in parallel applications or when computing the transform for several functions in a loop.
icwt (Wavelet Toolbox) inverts the continuous wavelet transform.
Signal Analyzer has a scalogram view to visualize the CWT of a time series.
For more details on otoacoustic emissions, see "Determining Exact Frequency Through the Analytic CWT" in CWT-Based Time-Frequency Analysis (Wavelet Toolbox).
Deep learning: Synchrosqueezed transforms can be used to extract time-frequency features and fed into a network that classifies time-series data. Waveform Segmentation Using Deep Learning shows how fsst outputs can be fed into an LSTM network that classifies ECG signals.
Use the 'reassigned' option in spectrogram, set the 'Reassigned' argument to true in pspectrum, or check the Reassign box in the spectrogram view of Signal Analyzer to compute reassigned spectrograms.
fsst computes the Fourier synchrosqueezed transform. Use the ifsst function to invert the Fourier synchrosqueezed transform. (See Fourier Synchrosqueezed Transform of Speech Signal for reconstruction of speech signals using ifsst.)
wsst (Wavelet Toolbox) computes the wavelet synchrosqueezed transform. Use the iwsst (Wavelet Toolbox) function to invert the wavelet synchrosqueezed transform. (See Inverse Synchrosqueezed Transform of Chirp (Wavelet Toolbox) for reconstruction of a quadratic chirp using iwsst (Wavelet Toolbox).)
\beta =20
cqt (Wavelet Toolbox) computes the constant-Q Gabor transform.
icqt (Wavelet Toolbox) inverts the constant-Q Gabor transform.
ewt (Wavelet Toolbox) computes the empirical wavelet transform.
modwt (Wavelet Toolbox) computes the maximal overlap discrete wavelet transform. To obtain the MRA analysis, use modwtmra (Wavelet Toolbox).
tqwt (Wavelet Toolbox) computes the tunable Q-factor wavelet transform. To obtain the MRA analysis, use tqwtmra (Wavelet Toolbox).
Load the vibration signal from a defective bearing generated in the Compute Hilbert Spectrum of Vibration Signal example. The signal is sampled at a rate 10 kHz.
Signal Analyzer | Signal Multiresolution Analyzer (Wavelet Toolbox)
cqt (Wavelet Toolbox) | cwt (Wavelet Toolbox) | cwtfilterbank (Wavelet Toolbox) | dlstft | emd | ewt (Wavelet Toolbox) | fsst | hht | icqt (Wavelet Toolbox) | icwt (Wavelet Toolbox) | ifsst | istft | iwsst (Wavelet Toolbox) | kurtogram | modwt (Wavelet Toolbox) | modwtmra (Wavelet Toolbox) | pkurtosis | pspectrum | spectrogram | stft | tqwt (Wavelet Toolbox) | tqwtmra (Wavelet Toolbox) | vmd | wsst (Wavelet Toolbox) | wt (Wavelet Toolbox) | xspectrogram | wvd | xwvd
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Interview Query | Nightly Job - Python Problem
Every night between 7 pm and midnight, two computing jobs from two different sources are randomly started with each one lasting an hour.
Unfortunately, when the jobs simultaneously run, they cause a failure in some of the company’s other nightly jobs, resulting in downtime for the company that costs
\$1000
The CEO, who has enough time today to hear one word, needs a single number representing the annual (365 days) cost of this problem.
Note: Write a function to simulate this problem and output an estimated cost.
Bonus - How would you solve this using probability?
simulate_overlap(n) -> 0.4
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Electrochemical Series: Explanation & Table | StudySmarter
One thing that makes the periodic table so special is that each of the elements is different in one way or another. Although there are trends in some of their properties, they all differ ever so slightly. A consequence of this is that they all have different reactivities. We can see this in terms of how easily they give up their electrons - or in other words, how easily they are oxidised. The electrochemical series is a handy form of sorting species in this way, and is the basis behind redox reactions, fuel cells, and batteries.
This article is about the electrochemical series in physical chemistry.
We'll learn what the electrochemical series is, which will involve an introduction to half-cells, electrochemical cells, and standard electrode potential.
You'll then see the electrochemical series in the form of a table.
Lastly, we'll explore the applications of the electrochemical series.
Electrochemical series explanation
At the start, we put across the idea that some elements are better at giving up their electrons than others. This is a fundamental principle of chemistry, and is the basis behind all redox reactions. To explore this in more detail, we need to look at the electrochemical series. But first, you need a bit of background knowledge. For that, we'll start by learning about half-cells, electrochemical cells, and standard electrode potential.
Imagine putting a rod of zinc in a solution of zinc ions. You eventually form an equilibrium, in which some of the zinc atoms give up their electrons to form zinc ions. Here's the equation. You'll notice that it is conventional to write the ions and electrons on the left-hand side and the metal atom on the right:
{\mathrm{Zn}}^{2+}\left(\mathrm{aq}\right)+2{\mathrm{e}}^{-}⇌\mathrm{Zn}\left(\mathrm{s}\right)
The zinc ions move into solution whilst the released electrons gather on the rod, giving the rod a negative charge. This creates a potential difference between the zinc rod and the zinc ion solution, known as an electrode potential. The exact position of this equilibrium and the exact potential difference of the system depends on the reactivity of zinc and how easily it gives up its electrons.
The more negative a potential difference, the further to the left the equilibrium lies, and the more easily a metal gives up its electrons. For example, a metal that creates a potential difference of -1.2 V gives up its electrons more easily than a metal that creates a difference of -0.3 V.
Now, consider what would happen if you put a rod of copper in a solution of copper ions. Some of the copper atoms react, giving up their electrons to form copper ions. Once again, you'll eventually form an equilibrium. But copper is less reactive than zinc and is not so good at giving up its electrons - we can say that copper is a worse reducing agent. It means that the potential difference of this copper system is less negative than the potential difference of the zinc system. We call the system created by putting a metal in a solution of its own ions a half -cell.
Reducing agents are themselves oxidised. This means that a better reducing agent is oxidised more easily.
Finally, take a moment to think about what would happen if you joined the zinc half-cell and the copper half-cell together with a wire and a salt bridge. Zinc is a better reducing agent than copper - it is more reactive and gives up its electrons more easily. This means that it has a more negative potential difference than copper does. It creates an overall potential difference between the two cells, which we can also call an electrode potential, showing the difference in how readily the two metals give up their electrons. The potential difference is measured by a voltmeter connected to the system.
The combination of two half-cells is known as an electrochemical cell. It is based on nothing more than simple redox reactions. Because zinc has a more negative potential difference than copper, there is a greater build-up of electrons on the zinc rod. If we allow these electrons to flow, then they will travel through the wire from zinc, the better reducing agent, to copper, the worse reducing agent. Meanwhile, positive ions from the solution will flow across the salt bridge in the same direction to balance the charge. Zinc atoms turn into zinc ions, losing electrons; copper ions turn into copper atoms, gaining electrons. Here are the two equations:
\mathrm{At}\mathrm{the}\mathrm{zinc}\mathrm{rod}:\mathrm{Zn}\left(\mathrm{s}\right)\to {\mathrm{Zn}}^{2+}\left(\mathrm{aq}\right)+2{\mathrm{e}}^{-}\mathrm{At}\mathrm{the}\mathrm{copper}\mathrm{rod}:{\mathrm{Cu}}^{2+}\left(\mathrm{aq}\right)+ 2{\mathrm{e}}^{-}\to \mathrm{Cu}\left(\mathrm{s}\right)
In general, electrons always travel from the better reducing agent (the more reactive metal, which gives up its electrons more easily) to the worse reducing agent (the less reactive metal, which is worse at giving up its electrons).
We know that if you put a metal in solution, it forms an equilibrium of metal atoms and metal ions. This creates a potential difference, the value of which depends on the position of the equilibrium. We can't directly measure the potential difference generated by a single half-cell. However, we can measure the potential difference generated when you connect two half-cells together in an electrochemical cell. If we record the potential differences generated when you connect a whole range of different half-cells to one particular reference half-cell, we can create a table comparing these values, and thus rank the metals from the most reactive (the best reducing agent) to the least reactive (the worst reducing agent).
In fact, scientists have done this. The reference half-cell used is the hydrogen electrode. We call the potential difference between a half-cell and the reference hydrogen electrode under standard conditions the cell's standard electrode potential. It is a measurement of the element's reducing ability.
Standard electrode potential, E°, is the potential difference generated when a half-cell is connected to a hydrogen half-cell under standard conditions. It is also known as electromotive force or standard reduction potential.
Standard conditions are 298 K, 1.00 mol dm-3 and 100 kPa. You can look at both the hydrogen electrode and the importance of standard conditions in Electrode Potential.
Representing standard electrode potentials
We represent standard electrode potentials (E°) using half equations involving the element and its ions. Note the following:
Although we've emphasised how standard electrode potentials tell you how easily an element gives up its electrons, they are written as reduction potentials - in other words, how easily an ion gains electrons. This means that we write the ions and electrons on the left-hand side and the combined element on the right-hand side.
Hydrogen always has a value of zero, as it is the standard with which we measure all other electrode potentials.
A more negative electrode potential means the element is more likely to give up its electrons. It is a better reducing agent and is more easily oxidised.
A more positive electrode potential means the element is less likely to give up its electrons. It is a worse reducing agent and is less easily oxidised.
Here is the standard electrode potential for zinc, Zn:
{\mathrm{Zn}}^{2+}\left(\mathrm{aq}\right)+2{\mathrm{e}}^{-}\to \mathrm{Zn}\left(\mathrm{s}\right)\mathrm{E}°=-0.76\mathrm{V}
The standard electrode potential value is negative. This means that zinc is a better reducing agent than hydrogen and is more easily oxidised.
Amassing the standard electrode potentials of different elements creates the electrochemical series.
The electrochemical series is a list of elements ordered by their standard electrode potentials. It tells us how easily each element is oxidised compared to a reference half-cell, the hydrogen electrode.
The electrochemical series is the basis behind all kinds of modern fuel cells and batteries. But before we look at these applications, let's take a look at the electrochemical series itself in the form of a table.
Electrochemical series table
The moment you've been waiting for - here's the electrochemical series. We've shown it as a table of different half-cells and their standard electrode potentials.
Note that the electrochemical series can either run from positive to negative, or negative to positive. Here, we've shown it from negative to positive, with the most-easily oxidised element (lithium, Li) at the top. This means that lithium is the strongest reducing agent.
Electrochemical series applications
We've learned what the electrochemical series is. Now let's consider some of its applications.
Combining two half-cells with different electrode potentials creates an electrochemical cell. We can use this to generate electricity. Because of this, the electrochemical series is the basis of fuel cells and batteries.
We can also use electrode potentials to calculate an electrochemical cell's overall cell potential.
In addition, the electrochemical series allows us to predict the direction of a redox reaction and predict whether disproportionation reactions can occur. In general, electrons travel from a species with a more negative standard electrode potential to a species with a more positive standard electrode potential.
We can also use it to identify strong and weak oxidising and reducing agents. In general, species with a negative electrode potential are good reducing agents and tend to lose electrons themselves.
You'll look at electrochemical cells in Electrochemical Cells, where you'll also be able to calculate the cell's cell potential and work out the direction of redox reactions. In Electrode Potential, you'll explore what electrode potentials can tell us about a species' oxidising or reducing ability. There, you'll also look at the hydrogen electrode and the importance of standard conditions in calculating electrode potentials. Last of all, Applications of Electrochemical Cells will explore how the electrochemical series is used in fuel cells and batteries.
Electrochemical Series - Key takeaways
A negative electrode potential means that an element is more easily oxidised than hydrogen, whilst a positive electrode potential means that an element is less easily oxidised than hydrogen.
Standard electrode potentials are measured under standard conditions of 298 K, 1.00 mol dm-3, and 100 kPa.
The electrochemical series can be used to create electrochemical cells, predict the direction of redox reactions, and identify strong and weak oxidising and reducing agents.
What is the electrochemical series and what is its significance?
The electrochemical series is a list of elements ordered by their standard electrode potentials. It gives us important information about which substances are good oxidising agents and which ones are good reducing agents, and also helps us predict the direction of redox reactions.
The electrochemical series is a list of elements that are ordered by their standard electrode potentials.
What is the order of the electrochemical series?
In an electrochemical series, the species are arranged in order of their standard electrode potentials. The electrochemical series can either run from positive to negative, or negative to positive.
What are some applications of the electrochemical series?
The electrochemical series can be used to identify good oxidising and reducing agents, calculate the cell potential of electrochemical cells, and predict the direction of redox reactions.
What is an example of the electrochemical series?
Consider zinc and copper. Zinc is better at giving up its electrons than copper and so has a more negative electrode potential: its standard electrode potential is -0.76 V, compared to copper's +0.34 V. In an electrochemical series running from negative to positive, zinc is therefore found higher up in the series than copper.
Final Electrochemical Series Quiz
In the electrochemical series, elements are ordered according to their ____.
What do we call the system created by putting a metal rod in a solution of its own ions?
A half-cell
A combination of two half-cells.
What are the components of an electrochemical cell?
Half cells connected by a wire
Standard electrode potential is the potential difference generated when a half-cell is connected to a hydrogen half-cell under standard conditions.
What is the symbol for standard electrode potential?
What is standard electrode potential measured in?
Species with a more negative electrode potential are better ____.
True or false? A metal forming a solution of ions with a charge of 2+ always has a more negative electrode potential than a metal forming a solution of ions with a charge of 1+.
Give three applications of the electrochemical series.
Predicting the direction of redox reactions.
Identifying strong and weak oxidising and reducing agents.
Creating electrochemical cells.
Which half-cell has a value of 0 in the electrochemical series?
of the users don't pass the Electrochemical Series quiz! Will you pass the quiz?
Buffer Solutions Learn
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Wideband LOS propagation channel - MATLAB - MathWorks Switzerland
{L}_{fsp}=\frac{{\left(4\pi R\right)}^{2}}{{\lambda }^{2}},
\gamma ={\gamma }_{o}\left(f\right)+{\gamma }_{w}\left(f\right)=0.1820f{N}^{″}\left(f\right).
{N}^{″}\left(f\right)=\sum _{i}{S}_{i}{F}_{i}+{{N}^{″}}_{D}^{}\left(f\right)
{S}_{i}={a}_{1}×{10}^{-7}{\left(\frac{300}{T}\right)}^{3}\mathrm{exp}\left[{a}_{2}\left(1-\left(\frac{300}{T}\right)\right]P.
{S}_{i}={b}_{1}×{10}^{-1}{\left(\frac{300}{T}\right)}^{3.5}\mathrm{exp}\left[{b}_{2}\left(1-\left(\frac{300}{T}\right)\right]W.
W=\frac{\rho T}{216.7}.
{\gamma }_{c}={K}_{l}\left(f\right)M,
{\gamma }_{R}=k{R}^{\alpha },
r=\frac{1}{0.477{d}^{0.633}{R}_{0.01}^{0.073\alpha }{f}^{0.123}-10.579\left(1-\mathrm{exp}\left(-0.024d\right)\right)}
{f}_{m}=\left\{\begin{array}{c}{f}_{c}-\frac{{f}_{s}}{2}+\left(m-1\right)\Delta f\text{, }{N}_{B}\text{ even}\\ {f}_{c}-\frac{\left({N}_{B}-1\right){f}_{s}}{2{N}_{B}}+\left(m-1\right)\Delta f\text{, }{N}_{B}\text{ odd}\end{array},\text{ }m=1,\dots ,{N}_{B}
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