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Coagulation and Fragmentation Models | EMS Press
Historically deterministic equations were used for modelling the dynamics of the density of clusters of different sizes in a medium that evolves by coagulation and/or fragmentation. An alternative stochastic approach uses the distribution of a typical particle in systems undergoing a random evolution. This approach provides tools for deriving limit equations, it easily extends to more general interactions and, finally, it shares the familiar advantages of particle-based methods for computing distributions in high dimensions. The stochastic theory is undergoing intense development, both at a theoretical level, in the construction of models and analysis of their properties, and at a computational level. Major mathematical challenges in the field of coagulation and fragmentation models are related to phase transitions due to fast coagulation (gelation) or fast fragmentation (formation of dust), and to explicit descriptions of the structure of partitions (e.g. allelic partitions induced by random mutations in models for the evolution of populations). A further direction of intensive study is the detailed analysis of particular coagulation and fragmentation processes, where special features (e.g., scaling properties) allow links to be made to Brownian motion and other L\' evy processes, also to some interesting problems in random combinatorics.
Major topics that have been discussed during the workshop include : \begin{itemize} \item[1.] Spatial models of coagulation and fragmentation \\ Models incorporating diffusion; scaling limits; derivation from particle dynamics; equilibrium measures for coagulation-fragmentation processes; formation of structured particles; diffusion limited aggregation \item[2.] Phase transitions in coagulation and fragmentation models\\ Gelation effects; shattering transition (appearance of dust in fragmentation); explosion phenomena; computation of the gelation time; uniqueness issues; convergence of particle systems \item[3.] Aspects of random combinatorics \\ Links to random graphs and trees; genealogy for certain large populations dynamics; mutation and allelic partitions; stochastic coalescents with multiple collisions (
\Lambda
-coalescents) and fragmentations \item[4.] Computational issues \\ Approximation and numerics; analysis of algorithms; Monte Carlo issues around sensitivity in initial conditions and kernel parameters \end{itemize}
Jean Bertoin, James R. Norris, Wolfgang Wagner, Coagulation and Fragmentation Models. Oberwolfach Rep. 4 (2007), no. 4, pp. 2727–2790
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Fish Riddle · xfbs:blog
The internet is full of distractions, and unfortunately, I am not always impervious to all of them. Some of them can lead to interesting results. Today, my distraction came in the shape of a riddle from a riddle from a TED-Ed video, which got me to explore my (rusty, but still somewhat present) math skills.
Now, I was excited to learn about the puzzle to see if I could use programming (I was thinking of a constraint solver or possibly just brute-forcing it) to solve it. But alas, it turns out that it’s just solvable with plain maths 🤷🏽♀️. So let’s dive in and see what we can do here.
You can watch the video to get the story of the puzzle, but it breaks down like this: you have three quadrants, each with a number of fish tanks and sharks in it. You know how many there are in both the first and second quadrants, and you must find out how many there are in the third quadrant.
So let’s first introduce a number of variables to help is keep track of things. We define:
s_i
as the number of creatures in sector
i
h_i
as the number of sharks in sector
i
f_i
as the number of fish in sector
i
t_i
as the number of fish per tank in sector
i
n_i
as the number of fish tanks in sector
i
There are six comstraints given in the puzzle.
There are 50 creatures in total, including sharks and fish.
Each sector has anywhere from one to seven sharks, with no two sectors having the same nmber of sharks.
Each tank has an equal number of fish. Since this is true, we will just use
to refer to any of
t_i
, simce they are all the same.
In total, there are 13 or fewer fish tanks.
Sector Alpha has two sharks and four tanks.
Sector Beta has four sharks and two tanks.
The objective for us is to find both the values of
t
h_3
n_3
. To do this, I started out by using the given constraints to find the number of possible values for each of them.
Applying the given constraint #2, we can limit the search space for
h_3
Similarly, using the given constraint #3, we can limit the search space for
n_3
Finding out the search space for
t
is unfortunately not that simple. First, we need to know how many creatures are currently accounted for.
From constraint 1, we know that there are 50 creatures. Thus, from the amount of creatures we have right now and from that, we can calculate how many are not accounted for yet,
r
(for rest). We also know that the missing creatures must be in our sector (sector Gamma), so we have ourselves a nice simple equation.
Now, given this equation and knowing the search space for both
h_3
n_3
, we can easily restrict the search space for
t
. If we pop in the maximum values for
h_3
n_3
and solve it, we can find the minimum value for
t
, and vice versa for the maximum value.
With this information, we can limit the search space for
t
, because we know that it must be within the bounds of
t_{min}
t_{max}
Now, to actually solve this whole mess, we need to rearrange our equation a little bit.
With this equation, we can see that
44 - h_3
must be divisible by both
(6 + n_3)
t
. So, given that we have a list of candidates for
h_3
, we can simply check their divisors and see if any of them are candidates for
t
h_3
\{x \in \mathbb{N} \vert x \mid (44 - h_3), t_{min} \leq x \leq t_{max}\}
1
43 = (6 + n_3) t
\{\}
3
41 = (6 + n_3) t
\{\}
5
39 = (6 + n_3) t
\{3\}
6
38 = (6 + n_3) t
\{\}
7
37 = (6 + n_3) t
\{\}
Seeing that only
h_3 = 5
produced a valid
t = 3
, these must be our values. Now, all that is left to do is pop them right back into the equation to find
n_3
There we go,
n_3 = 7
. This means that in sector Gamma, there are five sharks and seven fish tanks. Every fish tank contains three fish.
Too bad that this could be solved on paper, I’m hoping that next time I can finally get an excuse to play around with a fancy constraint solver or implement something. But in the meantime, it was fun to do and I hope I didn’t get anything wrong.
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problemJWT_payload HASH(0x56508f21bee0)
The table shows the population of Nepal (in millions) as of June 30 of the given year. Use a linear approximation to estimate the population at midyear in [math]
1984
. Use another linear approximation to predict the population in [math]
2006
. (To approximate a derivative, average the secant slopes "closest" to the point.)
1984
: million
2006
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The Money Flow Index (MFI) is a technical indicator that generates overbought or oversold signals using both prices and volume data.
An MFI reading above 80 is considered overbought and an MFI reading below 20 is considered oversold, although levels of 90 and 10 are also used as thresholds.
A divergence between the indicator and price is noteworthy. For example, if the indicator is rising while the price is falling or flat, the price could start rising.
\begin{aligned} &\text{Money Flow Index}=100-\frac{100}{1+\text{Money Flow Ratio}}\\ &\textbf{where:}\\ &\text{Money Flow Ratio}=\frac{\text{14 Period Positive Money Flow}}{\text{14 Period Negative Money Flow}}\\ &\text{Raw Money Flow}=\text{Typical Price * Volume}\\ &\text{Typical Price}=\frac{\text{High + Low + Close}}{3}\\ \end{aligned}
Money Flow Index=100−1+Money Flow Ratio100where:Money Flow Ratio=14 Period Negative Money Flow14 Period Positive Money FlowRaw Money Flow=Typical Price * VolumeTypical Price=3High + Low + Close
When the price advances from one period to the next Raw Money Flow is positive and it is added to Positive Money Flow. When Raw Money Flow is negative because the price dropped that period, it is added to Negative Money Flow.
Calculate the typical price for each of the last 14 periods.
For each period, mark whether the typical price was higher or lower than the prior period. This will tell you whether raw money flow is positive or negative.
Calculate raw money flow by multiplying the typical price by volume for that period. Use negative or positive numbers depending on whether the period was up or down (see step above).
Calculate the money flow ratio by adding up all the positive money flows over the last 14 periods and dividing it by the negative money flows for the last 14 periods.
Calculate the Money Flow Index (MFI) using the ratio found in step four.
Continue doing the calculations as each new period ends, using only the last 14 periods of data.
One of the primary ways to use the Money Flow Index is when there is a divergence. A divergence is when the oscillator is moving in the opposite direction of price. This is a signal of a potential reversal in the prevailing price trend.
For example, a very high Money Flow Index that begins to fall below a reading of 80 while the underlying security continues to climb is a price reversal signal to the downside. Conversely, a very low MFI reading that climbs above a reading of 20 while the underlying security continues to sell off is a price reversal signal to the upside.
Traders also watch for larger divergences using multiple waves in the price and MFI. For example, a stock peaks at $10, pulls back to $8, and then rallies to $12. The price has made two successive highs, at $10 and $12. If MFI makes a lower higher when the price reaches $12, the indicator is not confirming the new high. This could foreshadow a decline in price.
The overbought and oversold levels are also used to signal possible trading opportunities. Moves below 10 and above 90 are rare. Traders watch for the MFI to move back above 10 to signal a long trade, and to drop below 90 to signal a short trade.
Other moves out of overbought or oversold territory can also be useful. For example, when an asset is in an uptrend, a drop below 20 (or even 30) and then a rally back above it could indicate a pullback is over and the price uptrend is resuming. The same goes for a downtrend. A short-term rally could push the MFI up to 70 or 80, but when it drops back below that could be the time to enter a short trade in preparation for another drop.
The MFI and RSI are very closely related. The main difference is that MFI incorporates volume, while the RSI does not. Proponents of volume analysis believe it is a leading indicator. Therefore, they also believe that MFI will provide signals, and warn of possible reversals, in a more timely fashion than the RSI. One indicator is not better than the other, they are simply incorporating different elements and will, therefore, provide signals at different times.
The MFI is capable of producing false signals. This is when the indicator does something that indicates a good trading opportunity is present, but then the price doesn't move as expected resulting in a losing trade. A divergence may not result in a price reversal, for instance.
The indicator may also fail to warn of something important. For example, while a divergence may result in a price reversing some of the time, divergence won't be present for all price reversals. Because of this, it is recommended that traders use other forms of analysis and risk control and not rely exclusively on one indicator.
Fidelity. "Money Flow Index (MFI)."
How the Money Flow Index and Relative Strength Index Differ
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Independence of ℓ for the supports in the decomposition theorem
15 July 2018 Independence of
\ell
for the supports in the decomposition theorem
Duke Math. J. 167(10): 1803-1823 (15 July 2018). DOI: 10.1215/00127094-2017-0059
In this article, we prove a result on the independence of
\ell
for the supports of irreducible perverse sheaves occurring in the decomposition theorem, as well as for the family of local systems on each support. It generalizes Gabber’s result on the independence of
\ell
of intersection cohomology to the relative case.
Shenghao Sun. "Independence of
\ell
for the supports in the decomposition theorem." Duke Math. J. 167 (10) 1803 - 1823, 15 July 2018. https://doi.org/10.1215/00127094-2017-0059
Received: 30 November 2015; Revised: 3 May 2017; Published: 15 July 2018
Keywords: Decomposition theorem , independence of ℓ , ℓ-adic cohomology , perverse sheaves
Shenghao Sun "Independence of
\ell
for the supports in the decomposition theorem," Duke Mathematical Journal, Duke Math. J. 167(10), 1803-1823, (15 July 2018)
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2-D cross-correlation - MATLAB xcorr2 - MathWorks Korea
Output Matrix Size and Element Computation
Two-Dimensional Cross-Correlation of Arbitrary Complex Matrices
Align Two Images Using Cross-Correlation
Recovery of Template Shift with Cross-Correlation
GPU Acceleration for Cross-Correlation Matrix Computation
c = xcorr2(a,b)
c = xcorr2(a)
c = xcorr2(a,b) returns the cross-correlation of matrices a and b with no scaling. xcorr2 is the two-dimensional version of xcorr.
c = xcorr2(a) is the autocorrelation matrix of input matrix a. This syntax is equivalent to xcorr2(a,a).
Create two matrices, M1 and M2.
M1 = [17 24 1 8 15;
23 5 7 14 16;
4 6 13 20 22;
10 12 19 21 3;
11 18 25 2 9];
M1 is 5-by-5 and M2 is 3-by-3, so their cross-correlation has size (5+3-1)-by-(5+3-1), or 7-by-7. In terms of lags, the resulting matrix is
C=\left(\begin{array}{ccccccc}{c}_{-2,-2}& {c}_{-2,-1}& {c}_{-2,0}& {c}_{-2,1}& {c}_{-2,2}& {c}_{-2,3}& {c}_{-2,4}\\ {c}_{-1,-2}& {c}_{-1,-1}& {c}_{-1,0}& {c}_{-1,1}& {c}_{-1,2}& {c}_{-1,3}& {c}_{-1,4}\\ {c}_{0,-2}& {c}_{0,-1}& {c}_{0,0}& {c}_{0,1}& {c}_{0,2}& {c}_{0,3}& {c}_{0,4}\\ {c}_{1,-2}& {c}_{1,-1}& {c}_{1,0}& {c}_{1,1}& {c}_{1,2}& {c}_{1,3}& {c}_{1,4}\\ {c}_{2,-2}& {c}_{2,-1}& {c}_{2,0}& {c}_{2,1}& {c}_{2,2}& {c}_{2,3}& {c}_{2,4}\\ {c}_{3,-2}& {c}_{3,-1}& {c}_{3,0}& {c}_{3,1}& {c}_{3,2}& {c}_{3,3}& {c}_{3,4}\\ {c}_{4,-2}& {c}_{4,-1}& {c}_{4,0}& {c}_{4,1}& {c}_{4,2}& {c}_{4,3}& {c}_{4,4}\end{array}\right).
As an example, compute the element
{c}_{0,2}
(or C(3,5) in MATLAB®, since M2 is 3-by-3). Line up the two matrices so their (1,1) elements coincide. This placement corresponds to
{c}_{0,0}
{c}_{0,2}
, slide M2 two rows to the right.
Now M2 is on top of the matrix M1(1:3,3:5). Compute the element-by-element products and sum them. The answer should be
1×8+7×3+13×4+8×1+14×5+20×9+15×6+16×7+22×2=585.
[r2,c2] = size(M2);
CC = sum(sum(M1(0+(1:r2),2+(1:c2)).*M2))
Verify the result using xcorr2.
D = xcorr2(M1,M2);
DD = D(0+r2,2+c2)
\mathcal{X}
M×N
\mathcal{H}
P×Q
, their two-dimensional cross-correlation,
\mathcal{C}=\mathcal{X}\star \mathcal{H}
, is a matrix of size
\left(M+P-1\right)×\left(N+Q-1\right)
\mathcal{C}\left(k,l\right)=Tr\phantom{\rule{0.16666666666666666em}{0ex}}\left\{\underset{}{\overset{\sim }{\mathcal{X}}}{\underset{}{\overset{\sim }{\mathcal{H}}}}_{kl}^{†}\right\}\phantom{\rule{2em}{0ex}}\begin{array}{c}1\le k\le M+P-1,\\ 1\le l\le N+Q-1.\end{array}
Tr
is the trace and the dagger denotes Hermitian conjugation. The matrices
\underset{}{\overset{\sim }{\mathcal{X}}}
{\underset{}{\overset{\sim }{\mathcal{H}}}}_{kl}
have size
\left(M+2\left(P-1\right)\right)×\left(N+2\left(Q-1\right)\right)
and nonzero elements given by
\underset{}{\overset{～}{\mathcal{X}}}\left(m,n\right)=\mathcal{X}\left(m-P+1,n-Q+1\right),\phantom{\rule{1em}{0ex}}\begin{array}{c}P\le m\le M+P-1,\\ Q\le n\le N+Q-1\end{array}
{\underset{}{\overset{～}{\mathcal{H}}}}_{kl}\left(p,q\right)=\mathcal{H}\left(p-k+1,q-l+1\right),\phantom{\rule{1em}{0ex}}\begin{array}{c}k\le p\le P+k-1,\\ l\le q\le Q+l-1.\end{array}
Calling xcorr2 is equivalent to this procedure for general complex matrices of arbitrary size.
Create two complex matrices,
\mathcal{X}
7×22
\mathcal{H}
6×17
X = randn([7 22])+1j*randn([7 22]);
H = randn([6 17])+1j*randn([6 17]);
m = 1:M;
[P,Q] = size(H);
p = 1:P;
q = 1:Q;
\underset{}{\overset{\sim }{\mathcal{X}}}
\mathcal{C}
Xt = zeros([M+2*(P-1) N+2*(Q-1)]);
Xt(m+P-1,n+Q-1) = X;
C = zeros([M+P-1 N+Q-1]);
Compute the elements of
\mathcal{C}
by looping over
k
l
. Reset
{\underset{}{\overset{\sim }{\mathcal{H}}}}_{kl}
to zero at each step. Save time and memory by summing element products instead of multiplying and taking the trace.
for k = 1:M+P-1
for l = 1:N+Q-1
Hkl = zeros([M+2*(P-1) N+2*(Q-1)]);
Hkl(p+k-1,q+l-1) = H;
C(k,l) = sum(sum(Xt.*conj(Hkl)));
max(max(abs(C-xcorr2(X,H))))
The answer coincides to machine precision with the output of xcorr2.
Use cross-correlation to find where a section of an image fits in the whole. Cross-correlation enables you to find the regions in which two signals most resemble each other. For two-dimensional signals, like images, use xcorr2.
Load a black-and-white test image into the workspace. Display it with imagesc.
White = max(max(img));
Select a rectangular section of the image. Display the larger image with the section missing.
szx = x:X;
szy = y:Y;
Sect = img(szx,szy);
kimg = img;
kimg(szx,szy) = White;
kumg = White*ones(size(img));
kumg(szx,szy) = Sect;
imagesc(kimg)
imagesc(kumg)
title('Section')
Use xcorr2 to find where the small image fits in the larger image. Subtract the mean value so that there are roughly equal numbers of negative and positive values.
nimg = img-mean(mean(img));
nSec = nimg(szx,szy);
crr = xcorr2(nimg,nSec);
The maximum of the cross-correlation corresponds to the estimated location of the lower-right corner of the section. Use ind2sub to convert the one-dimensional location of the maximum to two-dimensional coordinates.
[ssr,snd] = max(crr(:));
[ij,ji] = ind2sub(size(crr),snd);
plot(crr(:))
plot(snd,ssr,'or')
text(snd*1.05,ssr,'Maximum')
Place the smaller image inside the larger image. Rotate the smaller image to comply with the convention that MATLAB® uses to display images. Draw a rectangle around it.
img(ij:-1:ij-size(Sect,1)+1,ji:-1:ji-size(Sect,2)+1) = rot90(Sect,2);
title('Reconstructed')
plot([y y Y Y y],[x X X x x],'r')
Shift a template by a known amount and recover the shift using cross-correlation.
Create a template in an 11-by-11 matrix. Create a 22-by-22 matrix and shift the original template by 8 along the row dimension and 6 along the column dimension.
template = 0.2*ones(11);
template(6,3:9) = 0.6;
template(3:9,6) = 0.6;
offsetTemplate = 0.2*ones(22);
offsetTemplate((1:size(template,1))+offset(1), ...
(1:size(template,2))+offset(2)) = template;
Plot the original and shifted templates.
imagesc(offsetTemplate)
imagesc(template)
Cross-correlate the two matrices and find the maximum absolute value of the cross-correlation. Use the position of the maximum absolute value to determine the shift in the template. Check the result against the known shift.
cc = xcorr2(offsetTemplate,template);
[max_cc, imax] = max(abs(cc(:)));
[ypeak, xpeak] = ind2sub(size(cc),imax(1));
corr_offset = [(ypeak-size(template,1)) (xpeak-size(template,2))];
isequal(corr_offset,offset)
The shift obtained from the cross-correlation equals the known template shift in the row and column dimensions.
This example requires Parallel Computing Toolbox™ software. Refer to GPU Support by Release (Parallel Computing Toolbox) to see what GPUs are supported.
Put the original and shifted template matrices on your GPU using gpuArray objects.
template = gpuArray(template);
offsetTemplate = gpuArray(offsetTemplate);
Compute the cross-correlation on the GPU.
Return the result to the MATLAB® workspace using gather. Use the maximum absolute value of the cross-correlation to determine the shift, and compare the result with the known shift.
cc = gather(cc);
[max_cc,imax] = max(abs(cc(:)));
[ypeak,xpeak] = ind2sub(size(cc),imax(1));
matrices | gpuArray objects
Input arrays, specified as matrices or gpuArray objects.
See Run MATLAB Functions on a GPU (Parallel Computing Toolbox) and GPU Support by Release (Parallel Computing Toolbox) for details on using xcorr2 with gpuArray (Parallel Computing Toolbox) objects.
Example: sin(2*pi*(0:9)'/10)*sin(2*pi*(0:13)/20) specifies a two-dimensional sinusoidal surface.
Example: gpuArray(sin(2*pi*(0:9)'/10)*sin(2*pi*(0:13)/20)) specifies a two-dimensional sinusoidal surface as a gpuArray object.
c — 2-D cross-correlation or autocorrelation matrix
matrix | gpuArray object
2-D cross-correlation or autocorrelation matrix, returned as a matrix or a gpuArray object.
The 2-D cross-correlation of an M-by-N matrix, X, and a P-by-Q matrix, H, is a matrix, C, of size M+P–1 by N+Q–1. Its elements are given by
C\left(k,l\right)=\sum _{m=0}^{M-1}\sum _{n=0}^{N-1}X\left(m,n\right)\text{\hspace{0.17em}}\overline{H}\left(m-k,n-l\right),\text{ }\text{ }\text{ }\begin{array}{c}-\left(P-1\right)\le k\le M-1,\\ -\left(Q-1\right)\le l\le N-1,\end{array}
where the bar over H denotes complex conjugation.
The output matrix, C(k,l), has negative and positive row and column indices.
A negative row index corresponds to an upward shift of the rows of H.
A negative column index corresponds to a leftward shift of the columns of H.
A positive row index corresponds to a downward shift of the rows of H.
A positive column index corresponds to a rightward shift of the columns of H.
To cast the indices in MATLAB® form, add the size of H: the element C(k,l) corresponds to C(k+P,l+Q) in the workspace.
For example, consider this 2-D cross-correlation:
H = [1 2; 3 4; 5 6]; % H is 3 by 2
C = xcorr2(X,H)
The C(1,1) element in the output corresponds to C(1–3,1–2) = C(–2,–1) in the defining equation, which uses zero-based indexing. To compute the C(1,1) element, shift H two rows up and one column to the left. Accordingly, the only product in the cross-correlation sum is X(1,1)*H(3,2) = 6. Using the defining equation, you obtain
C\left(-2,-1\right)=\sum _{m=0}^{1}\sum _{n=0}^{2}X\left(m,n\right)\text{\hspace{0.17em}}\overline{H}\left(m+2,n+1\right)=X\left(0,0\right)\text{\hspace{0.17em}}\overline{H}\left(2,1\right)=1\text{\hspace{0.17em}}×\text{\hspace{0.17em}}6=6,
with all other terms in the double sum equal to zero.
conv2 | filter2 | xcorr
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General Solutions of Anisotropic Laminated Plates | J. Appl. Mech. | ASME Digital Collection
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the Applied Mechanics Division, June 5, 2002; final revision, Nov. 22, 2002. Associate Editor: J. R. Barber. Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Chair, Department of Mechanics and Environmental Engineering, University of California–Santa Barbara, Santa Barbara, CA 93106-5070, and will be accepted until four months after final publication in the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
Yin, W. (August 25, 2003). "General Solutions of Anisotropic Laminated Plates ." ASME. J. Appl. Mech. July 2003; 70(4): 496–504. https://doi.org/10.1115/1.1576804
Anisotropic laminates with bending-stretching coupling possess eigensolutions that are analytic functions of the complex variables
x+μky,
where the eigenvalues
μk
and the corresponding eigenvectors are determined in the present analysis, along with the higher-order eigenvectors associated with repeated eigenvalues of degenerate laminates. The analysis and the resulting expressions are greatly simplified by using a mixed formulation involving a new set of elasticity matrices A*, B*, and D*. There are 11 distinct types of laminates, each with a different expression of the general solution. For an infinite plate with an elliptical hole subjected to uniform in-plane forces and moments at infinity, closed-form solutions are obtained for all types of anisotropic laminates in terms of the eigenvalues and eigenvectors.
laminates, eigenvalues and eigenfunctions, elasticity, matrix algebra, bending, anisotropic media
Anisotropy, Eigenvalues, Laminates
Lekhnitskii, S. G., 1968, Anisotropic Plates, Gordon and Breach, New York.
Lekhnitskii, S. G., 1963, Theory of Elasticity of an Anisotropic Body, Holden Day, San Francisco.
Muskhelishvili, N. I., 1954, Some Basic Problems of the Mathematical Theory of Elasticity, Nordhoff, Groningen.
Ting, T. C. T., 1966, Anisotropic Elasticity—Theory and Applications, Oxford University Press, New York.
Yin, W.-L., 1997, “A General Analysis Method for Singularities in Composite Structures,” Proc. AIAA/ASME/ASCE/AHS/ASC 38th SDM Conference, Apr. 7–10, Kissimee, FL., AIAA, Washington, DC, pp. 2238–2246.
Yin, W.-L., 2001, “Anisotropic Elasticity and Multi-Material Singularities,” Contemporary Research in Engineering Mechanics, G. A. Kardomateas and V. Birman, eds., ASME, New York, ASME-AMD-249, pp. 117–128.
Banerjee, P. K., and Butterfield, R., 1981, Boundary Element Methods in Engineering Science, McGraw-Hill, London.
A Method for Stress Determination in Plane Anisotropic Bodies
Tsai, S. W., and Hahn, H. T., 1980, Introduction to Composite Materials, Technomic, Lancaster, PA.
Extension of the Stroh Formalism to the Analysis of Bending of Anisotropic Elastic Plates
Structure and Properties of the Solution Space of General Anisotropic Laminates
Hygrothermal Stresses in Unsymmetric Laminates Disturbed by Elliptical Holes
Designing Composites for Graceful Failure
Anisotropic Elasticity Solution of Single Layered Composite Plate Under Self-Equilibrating Cubically Distributed Shear Loading
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Noetherian_ring Knowpia
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence
{\displaystyle I_{1}\subseteq I_{2}\subseteq I_{3}\subseteq \cdots }
of left (or right) ideals has a largest element; that is, there exists an n such that:
{\displaystyle I_{n}=I_{n+1}=\cdots .}
Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian.
Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers), and many general theorems on rings rely heavily on Noetherian property (for example, the Lasker–Noether theorem and the Krull intersection theorem).
Noetherian rings are named after Emmy Noether, but the importance of the concept was recognized earlier by David Hilbert, with the proof of Hilbert's basis theorem (which asserts that polynomial rings are Noetherian) and Hilbert's syzygy theorem.
For noncommutative rings, it is necessary to distinguish between three very similar concepts:
A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals.
A ring is right-Noetherian if it satisfies the ascending chain condition on right ideals.
A ring is Noetherian if it is both left- and right-Noetherian.
For commutative rings, all three concepts coincide, but in general they are different. There are rings that are left-Noetherian and not right-Noetherian, and vice versa.
There are other, equivalent, definitions for a ring R to be left-Noetherian:
Every left ideal I in R is finitely generated, i.e. there exist elements
{\displaystyle a_{1},\ldots ,a_{n}}
in I such that
{\displaystyle I=Ra_{1}+\cdots +Ra_{n}}
Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element.[1]
Similar results hold for right-Noetherian rings.
The following condition is also an equivalent condition for a ring R to be left-Noetherian and it is Hilbert's original formulation:[2]
{\displaystyle f_{1},f_{2},\dots }
of elements in R, there exists an integer
{\displaystyle n}
{\displaystyle f_{i}}
is a finite linear combination
{\textstyle f_{i}=\sum _{j=1}^{n}r_{j}f_{j}}
{\displaystyle r_{j}}
For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.[3] However, it is not enough to ask that all the maximal ideals are finitely generated, as there is a non-Noetherian local ring whose maximal ideal is principal (see a counterexample to Krull’s intersection theorem at Local ring#Commutative case.)
If R is a Noetherian ring, then the polynomial ring
{\displaystyle R[X]}
is Noetherian by the Hilbert's basis theorem. By induction,
{\displaystyle R[X_{1},\ldots ,X_{n}]}
is a Noetherian ring. Also, R[[X]], the power series ring, is a Noetherian ring.
If R is a Noetherian ring and I is a two-sided ideal, then the quotient ring R/I is also Noetherian. Stated differently, the image of any surjective ring homomorphism of a Noetherian ring is Noetherian.
Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. (This follows from the two previous properties.)
A ring R is left-Noetherian if and only if every finitely generated left R-module is a Noetherian module.
If a commutative ring admits a faithful Noetherian module over it, then the ring is a Noetherian ring.[4]
(Eakin–Nagata) If a ring A is a subring of a commutative Noetherian ring B such that B is a finitely generated module over A, then A is a Noetherian ring.[5]
Similarly, if a ring A is a subring of a commutative Noetherian ring B such that B is faithfully flat over A (or more generally exhibits A as a pure subring), then A is a Noetherian ring (see the "faithfully flat" article for the reasoning).
Every localization of a commutative Noetherian ring is Noetherian.
A consequence of the Akizuki–Hopkins–Levitzki theorem is that every left Artinian ring is left Noetherian. Another consequence is that a left Artinian ring is right Noetherian if and only if it is right Artinian. The analogous statements with "right" and "left" interchanged are also true.
A left Noetherian ring is left coherent and a left Noetherian domain is a left Ore domain.
(Bass) A ring is (left/right) Noetherian if and only if every direct sum of injective (left/right) modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct sum of indecomposable injective modules.[6] See also #Implication on injective modules below.
In a commutative Noetherian ring, there are only finitely many minimal prime ideals. Also, the descending chain condition holds on prime ideals.
In a commutative Noetherian domain R, every element can be factorized into irreducible elements (in short, R is a factorization domain). Thus, if, in addition, the factorization is unique up to multiplication of the factors by units, then R is a unique factorization domain.
Any field, including the fields of rational numbers, real numbers, and complex numbers, is Noetherian. (A field only has two ideals — itself and (0).)
Any principal ideal ring, such as the integers, is Noetherian since every ideal is generated by a single element. This includes principal ideal domains and Euclidean domains.
A Dedekind domain (e.g., rings of integers) is a Noetherian domain in which every ideal is generated by at most two elements.
The coordinate ring of an affine variety is a Noetherian ring, as a consequence of the Hilbert basis theorem.
The enveloping algebra U of a finite-dimensional Lie algebra
{\displaystyle {\mathfrak {g}}}
is a both left and right Noetherian ring; this follows from the fact that the associated graded ring of U is a quotient of
{\displaystyle \operatorname {Sym} ({\mathfrak {g}})}
, which is a polynomial ring over a field; thus, Noetherian.[7] For the same reason, the Weyl algebra, and more general rings of differential operators, are Noetherian.[8]
The ring of polynomials in finitely-many variables over the integers or a field is Noetherian.
Rings that are not Noetherian tend to be (in some sense) very large. Here are some examples of non-Noetherian rings:
The ring of polynomials in infinitely-many variables, X1, X2, X3, etc. The sequence of ideals (X1), (X1, X2), (X1, X2, X3), etc. is ascending, and does not terminate.
The ring of all algebraic integers is not Noetherian. For example, it contains the infinite ascending chain of principal ideals: (2), (21/2), (21/4), (21/8), ...
The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let In be the ideal of all continuous functions f such that f(x) = 0 for all x ≥ n. The sequence of ideals I0, I1, I2, etc., is an ascending chain that does not terminate.
The ring of stable homotopy groups of spheres is not Noetherian.[9]
However, a non-Noetherian ring can be a subring of a Noetherian ring. Since any integral domain is a subring of a field, any integral domain that is not Noetherian provides an example. To give a less trivial example,
The ring of rational functions generated by x and y /xn over a field k is a subring of the field k(x,y) in only two variables.
Indeed, there are rings that are right Noetherian, but not left Noetherian, so that one must be careful in measuring the "size" of a ring this way. For example, if L is a subgroup of Q2 isomorphic to Z, let R be the ring of homomorphisms f from Q2 to itself satisfying f(L) ⊂ L. Choosing a basis, we can describe the same ring R as
{\displaystyle R=\left\{\left.{\begin{bmatrix}a&\beta \\0&\gamma \end{bmatrix}}\,\right\vert \,a\in \mathbf {Z} ,\beta \in \mathbf {Q} ,\gamma \in \mathbf {Q} \right\}.}
This ring is right Noetherian, but not left Noetherian; the subset I ⊂ R consisting of elements with a = 0 and γ = 0 is a left ideal that is not finitely generated as a left R-module.
If R is a commutative subring of a left Noetherian ring S, and S is finitely generated as a left R-module, then R is Noetherian.[10] (In the special case when S is commutative, this is known as Eakin's theorem.) However this is not true if R is not commutative: the ring R of the previous paragraph is a subring of the left Noetherian ring S = Hom(Q2, Q2), and S is finitely generated as a left R-module, but R is not left Noetherian.
A unique factorization domain is not necessarily a Noetherian ring. It does satisfy a weaker condition: the ascending chain condition on principal ideals. A ring of polynomials in infinitely-many variables is an example of a non-Noetherian unique factorization domain.
A valuation ring is not Noetherian unless it is a principal ideal domain. It gives an example of a ring that arises naturally in algebraic geometry but is not Noetherian.
Key theoremsEdit
Many important theorems in ring theory (especially the theory of commutative rings) rely on the assumptions that the rings are Noetherian.
Over a commutative Noetherian ring, each ideal has a primary decomposition, meaning that it can be written as an intersection of finitely many primary ideals (whose radicals are all distinct) where an ideal Q is called primary if it is proper and whenever xy ∈ Q, either x ∈ Q or y n ∈ Q for some positive integer n. For example, if an element
{\displaystyle f=p_{1}^{n_{1}}\cdots p_{r}^{n_{r}}}
is a product of powers of distinct prime elements, then
{\displaystyle (f)=(p_{1}^{n_{1}})\cap \cdots \cap (p_{r}^{n_{r}})}
and thus the primary decomposition is a direct generalization of prime factorization of integers and polynomials.[11]
A Noetherian ring is defined in terms of ascending chains of ideals. The Artin–Rees lemma, on the other hand, gives some information about a descending chain of ideals given by powers of ideals
{\displaystyle I\supseteq I^{2}\supseteq I^{3}\supseteq \cdots }
. It is a technical tool that is used to prove other key theorems such as the Krull intersection theorem.
The dimension theory of commutative rings behaves poorly over non-Noetherian rings; the very fundamental theorem, Krull's principal ideal theorem, already relies on the "Noetherian" assumption. Here, in fact, the "Noetherian" assumption is often not enough and (Noetherian) universally catenary rings, those satisfying a certain dimension-theoretic assumption, are often used instead. Noetherian rings appearing in applications are mostly universally catenary.
Non-commutative caseEdit
Goldie's theorem
Implication on injective modulesEdit
Given a ring, there is a close connection between the behaviors of injective modules over the ring and whether the ring is a Noetherian ring or not. Namely, given a ring R, the following are equivalent:
R is a left Noetherian ring.
(Bass) Each direct sum of injective left R-modules is injective.[6]
Each injective left R-module is a direct sum of indecomposable injective modules.[12]
(Faith–Walker) There exists a cardinal number
{\displaystyle {\mathfrak {c}}}
such that each injective left module over R is a direct sum of
{\displaystyle {\mathfrak {c}}}
-generated modules (a module is
{\displaystyle {\mathfrak {c}}}
-generated if it has a generating set of cardinality at most
{\displaystyle {\mathfrak {c}}}
There exists a left R-module H such that every left R-module embeds into a direct sum of copies of H.[14]
The endomorphism ring of an indecomposable injective module is local[15] and thus Azumaya's theorem says that, over a left Noetherian ring, each indecomposable decomposition of an injective module is equivalent to one another (a variant of the Krull–Schmidt theorem).
^ a b Lam (2001), p. 19
^ Eisenbud 1995, Exercise 1.1.
^ Cohen, Irvin S. (1950). "Commutative rings with restricted minimum condition". Duke Mathematical Journal. 17 (1): 27–42. doi:10.1215/S0012-7094-50-01704-2. ISSN 0012-7094.
^ Matsumura, Theorem 3.5. harvnb error: no target: CITEREFMatsumura (help)
^ a b Anderson & Fuller 1992, Proposition 18.13. harvnb error: no target: CITEREFAndersonFuller1992 (help)
^ Bourbaki 1989, Ch III, §2, no. 10, Remarks at the end of the number harvnb error: no target: CITEREFBourbaki1989 (help)
^ Hotta, Takeuchi & Tanisaki (2008, §D.1, Proposition 1.4.6)
^ The ring of stable homotopy groups of spheres is not noetherian
^ Formanek & Jategaonkar 1974, Theorem 3
^ Eisenbud, Proposition 3.11. harvnb error: no target: CITEREFEisenbud (help)
^ Anderson & Fuller 1992, Theorem 25.6. (b) harvnb error: no target: CITEREFAndersonFuller1992 (help)
^ Anderson & Fuller 1992, Theorem 25.8. harvnb error: no target: CITEREFAndersonFuller1992 (help)
^ Anderson & Fuller 1992, Corollary 26.3. harvnb error: no target: CITEREFAndersonFuller1992 (help)
^ Anderson & Fuller 1992, Lemma 25.4. harvnb error: no target: CITEREFAndersonFuller1992 (help)
Atiyah, M. F., MacDonald, I. G. (1969). Introduction to commutative algebra. Addison-Wesley-Longman. ISBN 978-0-201-40751-8
Nicolas Bourbaki, Commutative algebra
Eisenbud, David (1995). Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Vol. 150. Springer-Verlag. doi:10.1007/978-1-4612-5350-1. ISBN 0-387-94268-8.
Formanek, Edward; Jategaonkar, Arun Vinayak (1974). "Subrings of Noetherian rings". Proceedings of the American Mathematical Society. 46 (2): 181–186. doi:10.2307/2039890.
Hotta, Ryoshi; Takeuchi, Kiyoshi; Tanisaki, Toshiyuki (2008), D-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser, doi:10.1007/978-0-8176-4523-6, ISBN 978-0-8176-4363-8, MR 2357361, Zbl 1292.00026
Lam, Tsit Yuen (2001). A first course in noncommutative rings. Graduate Texts in Mathematics. Vol. 131 (2nd ed.). New York: Springer. p. 19. doi:10.1007/978-1-4419-8616-0. ISBN 0387951830. MR 1838439.
Chapter X of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001
"Noetherian ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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§ A natural vector space without an explicit basis
On learning about infinite dimensional vector spaces, one learns that we need to use the axiom of choice to assert that every such vector space has a basis; indeed, it's equivalent to the AoC to assert this. However, I had not known any "natural" examples of such a vector space till I studied the proof of the barvinok algorithm. I produce the example here. Consider a space such as
S \equiv \mathbb R^3
. Now, consider the vector space spanned by the indicator functions of polyhedra in
S
. that's a mouthful, so let's break it down. A polyhedra is defined as a set of points that is defined by linear inequalities:
P \equiv \{ x \in S : a_i \cdot x \leq b_i, i \in [1\dots n] \}
a_i \in S
b \in \mathbb R
. The indicator functions are of the form:
[poly]: S \rightarrow \mathbb R; [poly](x) \equiv \begin{cases} 1 & x \in poly \\ 0 & \text{otherwise} \end{cases}
we can define a vector space of these functions over
\mathbb R
, using the "scaling" action as the action of
\mathbb R
on these functions: The vector space
V
is defined as the span of the indicator functions of all polyhedra. It's clearly a vector space, and a hopefully intuitive one. However, note that the set we generated this from (indicators of polyhedra) don't form a basis since they have many linear dependencies between them. For example, one can write the equation:
*---* *-* *-* *
|###| |#| |#| |
|###| = |#| + |#| - |
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what is the value of (delta) ng for the following reaction :- H2(g) + I2(g) -- 2HI(g) - Chemistry - Thermodynamics - 7033499 | Meritnation.com
∆
ng stands for the change in the number of mole of gaseous species, when a reactant combines to form products.
Mathematically it is given by the formula
∆
ng = (np-nr)g
Where, np stands for the sum of the number of moles present in gaseous species and nr stands for the sum of the number of moles of gaseous species present on reactant side.
For the mentioned reaction, i.e. H2(g) + I2(g) -- 2HI(g)
As all the compounds are present in gaseous form therefore
np = 2 and nr = 1+1
∆
ng = (np-nr)g = 2-2
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Endra Gunawan, Sri Widiyantoro, Shindy Rosalia, Mudrik Rahmawan Daryono, Irwan Meilano, Pepen Supendi, Takeo Ito, Takao Tabei, Fumiaki Kimata, Yusaku Ohta, Nazli Ismail; Coseismic Slip Distribution of the 2 July 2013
Mw
6.1 Aceh, Indonesia, Earthquake and Its Tectonic Implications. Bulletin of the Seismological Society of America 2018;; 108 (4): 1918–1928. doi: https://doi.org/10.1785/0120180035
This study investigates the coseismic slip distribution of the 2 July 2013
Mw
6.1 Aceh earthquake using Global Positioning System (GPS) data, measured geological surface offsets, and an aftershock distribution for a period of four days after the mainshock. We use the aftershock distribution to constrain the fault‐plane strike of a right‐lateral fault identified as the Pantan Terong segment. We estimate the coseismic slip distribution with dip angle information from the Global Centroid Moment Tensor (CMT) (model 1) and U.S. Geological Survey (USGS) (model 2) catalogs. We also estimate the coseismic slip distribution using another two fault models. Model 3 is constructed on a left‐lateral fault, the Celala segment, which is perpendicular to the Aceh segment of the Sumatran fault, and model 4 is constructed using the multiple faults in models 2 and 3. We further estimate the coseismic slip distribution of this earthquake by employing an elastic dislocation model, inverting only the GPS displacements for model 3 and jointly inverting GPS displacements and geological surface offsets for models 1, 2, and 4. Minimum misfit between data and model is obtained with model 3, suggesting that the earthquake slip occurred along a left‐lateral fault. Analysis of stress transfer caused by the 2013 earthquake indicates that the stress level along the Pantan Terong segment is
>0.4 bar
and the southeast part of Aceh segment was brought
∼0.3 bar
closer to failure, suggesting a possible earthquake occurrence in the future. This work demonstrates that the seismicity‐derived fault plane fails to predict the surface displacement, and that the inferred Celala segment produces positive stress on Pantan Terong segment and potentially triggered all the aftershocks.
Mw
Deep Fault Plane Revealed by High‐Precision Locations of Early Aftershocks Following the 12 September 2016
ML
5.8 Gyeongju, Korea, Earthquake
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The Reducibility of a Special Binary Pentanomial
Ryul Kim, Yun Mi Kim, "The Reducibility of a Special Binary Pentanomial", Algebra, vol. 2014, Article ID 482837, 7 pages, 2014. https://doi.org/10.1155/2014/482837
Ryul Kim 1 and Yun Mi Kim 2
1Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea
2Department of Applied Mathematics, Kim Chaek University of Technology, Pyongyang, Democratic People’s Republic of Korea
Academic Editor: Peter Fleischmann
Swan’s theorem determines the parity of the number of irreducible factors of a binary trinomial. In this work, we study the parity of the number of irreducible factors for a special binary pentanomial with even degree , where , and exactly one of , and is odd. This kind of irreducible pentanomials can be used for a fast implementation of trace and square root computations in finite fields of characteristic 2.
Irreducible polynomials of low weight over a finite field are frequently used in many applications such as coding theory and cryptography due to efficient arithmetic implementation in an extension field and, thus, it is important to determine the irreducibility of such polynomials. The weight of a polynomial means the number of its nonzero coefficients.
Characterization of the parity of the number of irreducible factors of a given polynomial is of significance in this context. If a polynomial has an even number of irreducible factors, then it is reducible and, thus, the study on the parity of this number can give a necessary condition for irreducibility. Swan [1] gives the first result determining the parity of the number of irreducible factors of trinomials over . Vishne [2] extends Swan’s theorem to trinomials over an even-dimensional extension of . Many Swan-like results focus on determining the reducibility of higher weight polynomials over ; see for example [3, 4]. Some researchers obtain the results on the reducibility of polynomials over a finite field of odd characteristic. We refer to [5, 6].
On the other hand, Ahmadi and Menezes [7] estimate the number of trace-one elements on the trinomial and pentanomial bases for a fast and low-cost implementation of trace computation. They also present a table of irreducible pentanomials whose corresponding polynomial bases have exactly one trace-one element. Each pentanomial of even degree in this table is of the form , where , and exactly one of , and is odd. In this work, we characterize the parity of the number of irreducible factors of this pentanomial. We describe some preliminary results related to Swan-like results in Section 2 and determine the reducibility of the pentanomial mentioned above in Section 3.
In this section, we recall Swan’s theorem determining the parity of the number of irreducible factors of a polynomial over and some results about the discriminant and the resultant of polynomials.
Let be a field and let , where are the roots of in an extension of . The discriminant of is defined by From the definition, it is clear that has a repeated root if and only if . Since is a symmetric function with respect to the roots of , it is an element of .
The following theorem, due to Swan, relates the parity of the number of irreducible factors of a polynomial over with its discriminant.
Theorem 1 (see [1, 8]). Suppose that the polynomial of degree has no repeated roots and let be a number of irreducible factors of over . Let be any monic lift of to the integers. Then, or , and if and only if .
Let , where are the roots of in an extension of . The resultant of and is defined by It is well known that where denotes the derivative of with respect to . An alternate formula for the discriminant of a monic polynomial is see [9].
Let then, for all , the coefficients of are the elementary symmetric polynomials of . Since each , it follows that for every symmetric polynomial .
The following natation will be used throughout the paper. For all integers and , let We denote simply as and put . Then, the following lemma holds.
Lemma 2 (see [10, 11]). (1) . (2) . (3) .
The following formula, called Newton’s identity, is often used for computation of the discriminant.
Theorem 3 (see [12]). Let and be as above. Then, for every , where .
The reciprocal polynomial of with over a finite field is defined by See Lidl and Niederreiter [12] for more details.
In this section, we characterize the parity of the number of irreducible factors for the pentanomial where is even; and exactly one of , and is odd. For our purpose, we use Swan’s theorem and Newton’s identity. In [10, 11], Newton’s identity has also been used to solve similar problems where it is enough to determine the power sums with indices , but, for (10), one should calculate much more negative indexed power sums. We return this calculation to one of positive indexed power sums by using reciprocals.
It is clear that (10) has no repeated roots because its derivative has a unique root . Let be the monic lift of in (10) to the integers and let denote the roots of in some extension of the rational numbers. The derivative of is Note that . Our work is divided into three cases according to which one of , and is odd.
Case 1 ( is odd). We can write the resultant of and as Since and are even, we have Using Lemma 2 and the fact that the square of every odd integer is congruent to 1 modulo 8, we get Newton’s identity shows that if , then and Therefore, The indices of terms in the above equation have a relation Since , we determine all for by applying Newton’s identity to get Table 1.
The values of for .
Note that does only cover the case of ; that is, . Since , , and are all even, we have Therefore, With reference to Table 1, we can determine all unknown terms in the above equation. We consider two subcases.
Subcase 1 ( is divisible by ). Then, we see easily that hence, the value of modulo depends on a pair . Let .
Theorem 4. Suppose that is odd and is divisible by 4. Then, the pentanomial in (10) has an even number of irreducible factors over if and only if one of the following conditions hold.
Consider(1) :(a);(b) and ;(c) and ;(d) and .(2) :(a) and ;(b) and ;(c) and .
Proof. If , then and, therefore, we have If , then and . So if , that is, , then And if , then (21) holds again. Similarly, if , then and . Thus, if , that is, , then (23) holds and if , then (21) holds. If , then and or can be nonzero only when it is equal to either or . Analyzing the possible cases shows that implies (23) and implies (21). Now, applying Swan’s theorem completes the proof.
Subcase 2 ( is not divisible by ). Then, and we can write where It is clear that if , then and if , then
Now determine and . First, assume that ; that is, . Since and , we have And then because . Next, assume that ; that is, . Clearly, and . Since , if , that is, , then and if , then . Therefore, we get and also , similarly. When and , a similar consideration shows Summarizing the above discussion and applying Swan’s theorem, we have the following theorem.
Theorem 5. Suppose that is odd and . Then, the pentanomial in (10) has an even number of irreducible factors over if and only if one of the following conditions hold.
Consider(1) or :(a);(b) and ;(c) and either or ;(d) and either or ;(2) or :(a) and ;(b) and either or ;(c) and .
Case 2 ( is odd). Similarly, we have From Newton’s identity, we see easily that , , , and are nonzero for and if is even with , then is even. To calculate for negative indices, we observe a monic lift of the reciprocal polynomial of to the integers. Denote the th power sum of the roots of in some extension of the rational numbers by . Then, clearly for every positive integer . We can apply Newton’s identity to to see that is equal to for odd and is even for even . From the above discussion, we have
First consider the case when .
Consider(1):(a) and either or ;(b) and either or ;(c) and either or ;(2):(a), and ;(b), and either or ;(c), and either or ;(d), and either or .
Proof. We determine the unknown terms in (35). Clearly, from . Let again It is easy to see that We also see that is equal to if and equal to , otherwise, since . And, by Newton’s identity, we have With reference to the nonzero coefficients of , we obtain that if and otherwise. Now we can determine modulo . If and , then and thus . Clearly and ; hence . Consideration for the other cases is similar so we describe only the results: if , then if , , then and if , then Next, we compute modulo . If we denote the coefficient of in by , then Since is odd and is even , and is even. It follows, therefore, that . Repeating this process, we get , where . Thus, we obtain Now, Swan’s theorem is used to complete the proof.
The remaining cases when or gives the following theorems, whose proofs follow a similar way and, hence, are omitted.
Theorem 7. Suppose that is odd and . Then, the pentanomial in (10) has an even number of irreducible factors over if and only if one of the following conditions hold.(1), and either or ;(2), and either or .
Consider(1), , and either or ;(2), , and either or ;(3), , and either or ;(4), and either or ;(5), , and either or .
Case 3 ( is odd). Analogously, to Case 2, we can write the resultant of and its derivative as follows: Straightforward calculations show that and and are even. In this case, we have also that is equal to for odd and is even for even . It follows, therefore, that Now, let
We present the following result for the reducibility of in (10) depending on the value of modulo .
Theorem 9. Suppose that is odd. Then, the pentanomial in (10) has an even number of irreducible factors over if and only if one of the following conditions hold.(1) and ;(2), , and either or ;(3), , and either or .
Proof. First, we compute in (45). By Newton’s identity, we get Since , we have and ; hence is equal to if and equal to otherwise. A simple calculation shows that if , then . If , then and, thus, we have Now applying Swan’s theorem completes the proof.
We have determined the parity of the number of irreducible factors of a pentanomial (10) under the condition that and exactly one of , and is odd. Our discussion is based on Swan’s theorem. If is odd, we obtained only a result which depends on modulo instead of exponents of the terms of a given pentanomial. In this case, a complete characterization of the reducibility of the given pentanomial seems to be more difficult.
The authors would like to thank the anonymous referees for their useful comments and suggestions.
R. G. Swan, “Factorization of polynomials over finite fields,” Pacific Journal of Mathematics, vol. 12, pp. 1099–1106, 1962. View at: Publisher Site | Google Scholar | MathSciNet
U. Vishne, “Factorization of trinomials over Galois fields of characteristic 2,” Finite Fields and Their Applications, vol. 3, no. 4, pp. 370–377, 1997. View at: Publisher Site | Google Scholar | MathSciNet
R. Kim, S. Pak, and M. Sin, “Swan-like reducibility for Type I pentanomials over a binary field,” Scientific Studies and Research, vol. 24, no. 2, pp. 249–254, 2014. View at: Google Scholar
Z. Zhao and X. Cao, “A note on the reducibility of binary affine polynomials,” Designs, Codes and Cryptography, vol. 57, no. 1, pp. 83–90, 2010. View at: Publisher Site | Google Scholar | MathSciNet
R. Kim and W. Koepf, “Parity of the number of irreducible factors for composite polynomials,” Finite Fields and Their Applications, vol. 16, no. 3, pp. 137–143, 2010. View at: Publisher Site | Google Scholar | MathSciNet
J. von zur Gathen, “Irreducible trinomials over finite fields,” Mathematics of Computation, vol. 72, no. 244, pp. 1987–2000, 2003. View at: Publisher Site | Google Scholar | MathSciNet
O. Ahmadi and A. Menezes, “On the number of trace-one elements in polynomial bases for
{\mathbb{F}}_{{2}^{n}}
,” Designs, Codes and Cryptography, vol. 37, pp. 493–507, 2005. View at: Google Scholar
O. Ahmadi and G. Vega, “On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields,” Finite Fields and their Applications, vol. 14, no. 1, pp. 124–131, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
B. Hanson, D. Panario, and D. Thomson, “Swan-like results for binomials and trinomials over finite fields of odd characteristic,” Designs, Codes and Cryptography, vol. 61, no. 3, pp. 273–283, 2011. View at: Publisher Site | Google Scholar | MathSciNet
O. Ahmadi and A. Menezes, “Irreducible polynomials of maximum weight,” Utilitas Mathematica, vol. 72, pp. 111–123, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet
W. Koepf and R. Kim, “The parity of the number of irreducible factors for some pentanomials,” Finite Fields and Their Applications, vol. 15, no. 5, pp. 585–603, 2009. View at: Publisher Site | Google Scholar | MathSciNet
Copyright © 2014 Ryul Kim and Yun Mi Kim. 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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Channel beam with elastic properties for deformation - MATLAB - MathWorks 日本
Flexible Channel Beam
Channel beam with elastic properties for deformation
The Flexible Channel Beam block models a slender beam with a C-shaped cross-section, also known as a C-beam. The C-beam consists of two horizontal components, known as flanges, that are connected by one vertical component, which is called a web. The C-beam can have small and linear deformations. These deformations include extension, bending, and torsion. The block calculates the beam cross-sectional properties, such as the axial, flexural, and torsional rigidities, based on the geometry and material properties that you specify.
The geometry of the C-beam is an extrusion of its cross-section. The beam cross-section, defined in the xy-plane, is extruded along the z-axis. To define the cross-section, you can specify its dimensions in the Geometry section of the block dialog box. The figure shows a C-beam and its cross-section. The reference frame of the beam is located at the centroid of the web.
\left[{I}_{x},{I}_{y}\right]=\left[\underset{A}{â«}{\left(yâ{y}_{c}\right)}^{2}dA,\underset{A}{â«}{\left(xâ{x}_{c}\right)}^{2}dA\right]
{I}_{xy}=\underset{A}{â«}\left(xâ{x}_{c}\right)\left(yâ{y}_{c}\right)dA
{I}_{P}={I}_{x}+{I}_{y}
\left[C\right]=\mathrm{α}\left[M\right]+\mathrm{β}\left[K\right]
Flexible Angle Beam | Flexible Cylindrical Beam | Flexible I Beam | Flexible Rectangular Beam | Flexible T Beam | General Flexible Beam | Extruded Solid | Reduced Order Flexible Solid | Rigid Transform
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Full-control H-infinity synthesis - MATLAB hinffc - MathWorks 日本
hinffc
Full-control H-infinity synthesis
[K,CL,gamma] = hinffc(P,nmeas)
[K,CL,gamma] = hinffc(P,nmeas,gamTry)
[K,CL,gamma] = hinffc(P,nmeas,gamRange)
[K,CL,gamma] = hinffc(___,opts)
[K,CL,gamma,info] = hinffc(___)
Full-control synthesis assumes the controller can directly affect both the state vector x and the error signal z. Synthesis with hinffc is the dual of the full-information problem covered by hinffi. For general H∞ synthesis, use hinfsyn.
[K,CL,gamma] = hinffc(P,nmeas) computes the H∞-optimal control law
u=\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right]=Ky
for the plant P. The plant is described by the state-space equations:
\begin{array}{c}dx=Ax+{B}_{1}w+{u}_{1}\\ z={C}_{1}x+{D}_{11}w+{u}_{2}\\ y={C}_{2}x+{D}_{21}w.\end{array}
w represents the disturbance inputs
u1 represents the inputs that affect the state vector
u2 represents the inputs that affect the error
z represents the error outputs to be kept small
y represents the measurement outputs
nmeas is the number of measurements y, which must be the last outputs of P. The gain matrix K minimizes the H∞ norm of the closed-loop transfer function CL from the disturbance signals w to the error signals z.
[K,CL,gamma] = hinffc(P,nmeas,gamTry) calculates a gain matrix for the target performance level gamTry. Specifying gamTry can be useful when the optimal achievable performance is better than you need for your application. In that case, a less-than-optimal solution can have smaller gains and be more numerically well-conditioned. If gamTry is not achievable, hinffc returns [] for K and CL, and Inf for gamma.
[K,CL,gamma] = hinffc(P,nmeas,gamRange) searches the range gamRange for the best achievable performance. Specify the range with a vector of the form [gmin,gmax]. Limiting the search range can speed up computation by reducing the number of iterations performed to test different performance levels.
[K,CL,gamma] = hinffc(___,opts) specifies additional computation options. To create opts, use hinfsynOptions. Specify opts after all other input arguments.
[K,CL,gamma,info] = hinffc(___) returns a structure containing additional information about the H∞ synthesis computation. You can use this argument with any of the previous syntaxes.
Plant, specified as an LTI model such as a state-space (ss) model. If P is a generalized state-space model with uncertain or tunable control design blocks, then hinffc uses the nominal or current value of those elements.
Construct P so that it has the partitioned form
\begin{array}{c}dx=Ax+{B}_{1}w+{u}_{1}\\ z={C}_{1}x+{D}_{11}w+{u}_{2}\\ y={C}_{2}x+{D}_{21}w.\end{array}
Construct P such that the nmeas measurement outputs are the last outputs.
For information about conditions imposed on the plant matrices and how the software addresses them, see hinfsyn.
nmeas — Number of measurements
Number of measurement output signals in the plant, specified as a nonnegative integer. hinffc takes the last nmeas plant outputs as the measurements y. The returned gain matrix K has nmeas inputs.
Target performance level, specified as a positive scalar. hinffc attempts to compute a gain matrix such that the H∞ of the closed-loop system does not exceed gamTry. If this performance level is achievable, then the returned gain matrix has gamma ≤ gamTry. If gamTry is not achievable, hinffc returns an empty matrix.
Performance range for search, specified as a vector of the form [gmin,gmax]. The hinffc command tests only performance levels within that range. It returns a gain matrix with performance:
gamma ≤ gmin, when gmin is achievable.
gamma = Inf when gmax is not achievable. In this case, hinffc returns [] for K and CL.
If you know a range of feasible performance levels, specifying this range can speed up computation by reducing the number of iterations performed by hinffc to test different performance levels.
Additional options for the computation, specified as an options object you create using hinfsynOptions. Available options include displaying algorithm progress at the command line, turning off automatic scaling and regularization, and specifying an optimization method. For more information, see hinfsynOptions.
K — Gain matrix
Gain matrix, returned as a matrix or []. The gain-matrix dimensions are nu-by-nmeas, where nu is the number of states plus the number of error outputs of P (outputs not included in nmeas).
Closed-loop transfer function, returned as a state-space (ss) model object or []. The returned performance level gamma is the H∞ norm of CL.
gamma — Closed-loop performance
Closed-loop performance, returned as a nonnegative scalar value or Inf. This value is the H∞ norm of CL. If you do not provide performance levels to test using gamTry or gamRange, then gamma is the best achievable performance level.
If you provide gamTry or gamRange, then gamma is the actual performance level achieved by the gain matrix computed for the best passing performance level that the function tries. If the specified performance levels are not achievable, then gamma = Inf.
Additional synthesis data, returned as a structure or [] (if the specified performance level is not achievable). info has the following fields.
Performance level used to compute the gain matrix K, returned as a nonnegative scalar. Typically, hinffc tests multiple target performance levels and returns a gain matrix corresponding to the best passing performance level (see the Algorithms section of hinfsyn for details). The value info.gamma is an upper limit on the actual achieved performance returned as the output argument gamma.
Riccati solution Y∞ for the performance level info.gamma, returned as matrix. For more information, see the Algorithms section of hinfsyn.
Regularized plant used for hinffc computation, returned as a state-space (ss) model object. By default, hinffc automatically adds extra disturbances and errors to the plant to ensure that it meets certain conditions (see the Algorithms section of hinfsyn). The field info.Preg contains the resulting plant model.
For information about the algorithms used for H∞ synthesis, see hinfsyn.
hinfsynOptions | hinffi | hinfsyn
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Perform nonlinear least-squares regression using SimBiology models (requires Statistics and Machine Learning Toolbox software) - MATLAB sbionlinfit - MathWorks France
sbionlinfit
pkModelMapObject
pkDataObj
InitEstimates
optionStruct
Perform nonlinear least-squares regression using SimBiology models (requires Statistics and Machine Learning Toolbox software)
sbionlinfit will be removed in a future release. Use sbiofit instead.
results = sbionlinfit(modelObj, pkModelMapObject, pkDataObj, InitEstimates)
results = sbionlinfit(modelObj, pkModelMapObject, pkDataObj, InitEstimates, Name,Value)
results = sbionlinfit(modelObj, pkModelMapObject, pkDataObj, InitEstimates, optionStruct)
[results, SimDataI] = sbionlinfit(...)
results = sbionlinfit(modelObj, pkModelMapObject, pkDataObj, InitEstimates) performs least-squares regression using the SimBiology® model, modelObj, and returns estimated results in the results structure.
results = sbionlinfit(modelObj, pkModelMapObject, pkDataObj, InitEstimates, Name,Value) performs least-squares regression, with additional options specified by one or more Name,Value pair arguments.
Following is an alternative to the previous syntax:
results = sbionlinfit(modelObj, pkModelMapObject, pkDataObj, InitEstimates, optionStruct) specifies optionStruct, a structure containing fields and values used by the options input structure to the nlinfit (Statistics and Machine Learning Toolbox) function.
[results, SimDataI] = sbionlinfit(...) returns simulations of the SimBiology model, modelObj, using the estimated values of the parameters.
SimBiology model object used to fit observed data.
If using a model object containing active doses (that is, containing dose objects created using the adddose method, and specified as active using the Active property of the dose object), be aware that these active doses are ignored by the sbionlinfit function.
PKModelMap object that defines the roles of the model components in the estimation. For details, see PKModelMap object.
If using a PKModelMap object that specifies multiple doses, ensure each element in the Dosed property is unique.
PKData object that defines the data to use in fitting, and the roles of the data columns used for estimation. For details, see PKData object.
For each subset of data belonging to a single group (as defined in the data column specified by the GroupLabel property), the software allows multiple observations made at the same time. If this is true for your data, be aware that:
These data points are not averaged, but fitted individually.
Different numbers of observations at different times cause some time points to be weighted more.
Vector of initial parameter estimates for each parameter estimated in pkModelMapObject.Estimated. The length of InitEstimates must equal at least the length of pkmodelMapObject.Estimated. The elements of InitEstimates are transformed as specified by the ParamTransform name-value pair argument.
Structure containing fields and values used by the options input structure to the nlinfit (Statistics and Machine Learning Toolbox) function. The structure can also use the name-value pairs listed below as fields and values. Defaults for optionStruct are the same as for the options input structure to nlinfit, except for:
DerivStep — Default is the lesser of 1e-4, or the value of the SolverOptions.RelativeTolerance property of the configuration set associated with modelObj, with a minimum of eps^(1/3).
FunValCheck — Default is off.
If you have Parallel Computing Toolbox™, you can enable parallel computing for faster data fitting by setting the name-value pair argument 'UseParallel' to true in the statset options structure as follows:
parpool; % Open a parpool for parallel computing
opt = statset(...,'UseParallel',true); % Enable parallel computing
results = sbionlinfit(...,opt); % Perform data fitting
The Name,Value arguments are the same as the fields and values in the options structure accepted by nlinfit. For a complete list, see the options input argument in the nlinfit (Statistics and Machine Learning Toolbox) reference page in the Statistics and Machine Learning Toolbox™ documentation. The defaults for Name,Value arguments are the same as for the options structure accepted by nlinfit, except for:
Following are additional Name,Value arguments that you can use with sbionlinfit.
Vector of integers specifying a transformation function for each estimated parameter. The transformation function, f, takes estimate as an input and returns beta:
beta = f(estimate)
Each element in the vector must be one of these integers specifying the transformation for the corresponding value of estimate:
0 – beta = estimate
1 – beta = log(estimate) (default)
2 – beta = probit(estimate)
3 – beta = logit(estimate)
Character vector specifying the form of the error term. Default is 'constant'. Each model defines the error using a standard normal (Gaussian) variable e, the function value f, and one or two parameters a and b. Choices are:
'proportional': y = f + b*abs(f)*e
'combined': y = f + (a+b*abs(f))*e
If you specify an error model, the results output argument includes an errorparam property, which has the value:
If you specify an error model, you cannot specify weights.
A matrix of real positive weights, where the number of columns corresponds to the number of responses. That is, the number of columns must equal the number of entries in the DependentVarLabel property of pkDataObj. The number of rows in the matrix must equal the number of rows in the data set.
A function handle that accepts a vector of predicted response values and returns a vector of real positive weights.
If using a function handle, the weights must be a function of the response (dependent variable).
Default is no weights. If you specify weights, you cannot specify an error model.
Logical specifying whether sbionlinfit does fitting for each individual (false) or if it pools all individual data and does one fit (true). If set to true, sbionlinfit uses the same model parameters for each dose level.
1-by-N array of objects, where N is the number of groups in pkDataObj. There is one object per group, and each object contains these properties:
ParameterEstimates — A dataset (Statistics and Machine Learning Toolbox) array containing fitted coefficients and their standard errors.
CovarianceMatrix — Estimated covariance matrix for the fitted coefficients.
beta — Vector of scalars specifying the fitted coefficients in transformed space.
R — Vector of scalars specifying the residual values, where R(i,j) is the residual for the ith time point and the jth response in the group of data. If your model incudes:
A single response, then R is a column vector of residual values associated with time points in the group of data.
Multiple responses, then R is a matrix of residual values associated with time points in the group of data, for each response.
J — Matrix specifying the Jacobian of the model, with respect to an estimated parameter, that is
J\left(i,j,k\right)={\frac{\partial {y}_{k}}{\partial {\beta }_{j}}|}_{{t}_{i}}
If your model incudes:
A single response, then J is a matrix of Jacobian values associated with time points in the group of data.
Multiple responses, then J is a 3-D array of Jacobian values associated with time points in the group of data, for each response.
COVB — Estimated covariance matrix for the transformed coefficients.
mse — Scalar specifying the estimate of the error of the variance term.
errorparam — Estimated parameters of the error model. This property is a scalar if you specify 'constant', 'exponential', or 'proportional' for the error model. This property is a two-element vector if you specify 'combined' for the error model. This property is an empty array if you specify weights using the 'Weights' name-value pair argument.
SimData object containing data from simulating the model using estimated parameter values for individuals. This object includes observed states and logged states.
PKData object | PKModelDesign object | PKModelDesign object | PKModelMap object | Model object | sbionlmefit | nlinfit (Statistics and Machine Learning Toolbox) | sbionlmefitsa
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{x}_{0}
{x}_{0}
{x}_{0}
X\left(t\right)
\mathrm{dX}\left(t\right)=\mathrm{\mu }\left(X\left(t\right),t\right)\mathrm{dt}+\mathrm{\sigma }\left(X\left(t\right),t\right)\mathrm{dW}\left(t\right)
\mathrm{\mu }\left(X\left(t\right),t\right)
\mathrm{\sigma }\left(X\left(t\right),t\right)
W\left(t\right)
{x}_{0}
X
is an
{X}_{1}
{X}_{n}
{\mathrm{\mu }}_{1}
{\mathrm{\mu }}_{n}
{\mathrm{\sigma }}_{1}
{\mathrm{\sigma }}_{n}
be the corresponding drift and diffusion terms. The ItoProcess(X, Sigma) command will create an
Y
{\mathrm{dY}\left(t\right)}_{i}={\mathrm{\mu }}_{i}\left({Y\left(t\right)}_{i},t\right)+{\mathrm{\sigma }}_{i}\left({Y\left(t\right)}_{i},t\right){\mathrm{dW}\left(t\right)}_{i}
W\left(t\right)
is an
\mathrm{with}\left(\mathrm{Finance}\right):
Y≔\mathrm{ItoProcess}\left(1.0,\mathrm{\mu },\mathrm{\sigma },x,t\right)
\textcolor[rgb]{0,0,1}{Y}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X0}}
\mathrm{Drift}\left(Y\left(t\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}
\mathrm{Diffusion}\left(Y\left(t\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}
\mathrm{Drift}\left(\mathrm{exp}\left(Y\left(t\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}\textcolor[rgb]{0,0,1}{+}\frac{{\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{Diffusion}\left(\mathrm{exp}\left(Y\left(t\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}
\mathrm{\mu }≔0.1
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.1}
\mathrm{\sigma }≔0.5
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.5}
\mathrm{PathPlot}\left(\mathrm{exp}\left(Y\left(t\right)\right),t=0..3,\mathrm{timesteps}=100,\mathrm{replications}=10\right)
\mathrm{\mu }≔'\mathrm{\mu }'
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}
\mathrm{\sigma }≔'\mathrm{\sigma }'
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}
\mathrm{X0}≔〈100.0,0.〉
\textcolor[rgb]{0,0,1}{\mathrm{X0}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{100.0}\\ \textcolor[rgb]{0,0,1}{0.}\end{array}]
\mathrm{Μ}≔〈\mathrm{\mu }X[1],\mathrm{\kappa }\left(\mathrm{\theta }-X[2]\right)〉
\textcolor[rgb]{0,0,1}{\mathrm{Μ}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}\right)\end{array}]
\mathrm{\Sigma }≔〈〈\mathrm{sqrt}\left(X[2]\right)X[1]|0.〉,〈0.|\mathrm{\sigma }X[2]〉〉
\textcolor[rgb]{0,0,1}{\mathrm{\Sigma }}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\sqrt{{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
S≔\mathrm{ItoProcess}\left(\mathrm{X0},\mathrm{Μ},\mathrm{\Sigma },X,t\right)
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X2}}
\mathrm{Drift}\left(S\left(t\right)\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\right)\end{array}]
\mathrm{Diffusion}\left(S\left(t\right)\right)
[\begin{array}{cc}\sqrt{{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{\mu }≔0.1
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.1}
\mathrm{\sigma }≔0.5
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.5}
\mathrm{\kappa }≔1.0
\textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{1.0}
\mathrm{\theta }≔0.4
\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.4}
A≔\mathrm{SamplePath}\left(S\left(t\right),t=0..1,\mathrm{timesteps}=100,\mathrm{replications}=10\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}\begin{array}{c}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{100.}& \textcolor[rgb]{0,0,1}{100.670084719786}\\ \textcolor[rgb]{0,0,1}{100.100000000000}& \textcolor[rgb]{0,0,1}{101.034139425728}\\ \textcolor[rgb]{0,0,1}{100.280880089808}& \textcolor[rgb]{0,0,1}{101.924198818577}\\ \textcolor[rgb]{0,0,1}{102.915077811759}& \textcolor[rgb]{0,0,1}{99.6518477031121}\\ \textcolor[rgb]{0,0,1}{103.858818858166}& \textcolor[rgb]{0,0,1}{100.628185358730}\\ \textcolor[rgb]{0,0,1}{104.476699657855}& \textcolor[rgb]{0,0,1}{98.7691518445139}\\ \textcolor[rgb]{0,0,1}{103.737362966326}& \textcolor[rgb]{0,0,1}{95.0859221941374}\\ \textcolor[rgb]{0,0,1}{102.574346549913}& \textcolor[rgb]{0,0,1}{94.1008617878134}\\ \textcolor[rgb]{0,0,1}{101.159282939668}& \textcolor[rgb]{0,0,1}{92.9644135833222}\\ \textcolor[rgb]{0,0,1}{100.709702216007}& \textcolor[rgb]{0,0,1}{93.6061768383076}\end{array}]\\ \hfill \textcolor[rgb]{0,0,1}{\text{slice of 10 × 2 × 101 Array}}\end{array}
\mathrm{PathPlot}\left(A,1,\mathrm{thickness}=3,\mathrm{markers}=\mathrm{false},\mathrm{color}=\mathrm{red}..\mathrm{blue},\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)
\mathrm{PathPlot}\left(A,2,\mathrm{thickness}=3,\mathrm{markers}=\mathrm{false},\mathrm{color}=\mathrm{red}..\mathrm{blue},\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)
\mathrm{ExpectedValue}\left(\mathrm{max}\left(S\left(1\right)[1]-100,0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{21.41114565}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.3390630872}]
X≔\mathrm{GeometricBrownianMotion}\left(100.0,0.05,0.3,t\right)
\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X4}}
Y≔\mathrm{GeometricBrownianMotion}\left(100.0,0.07,0.2,t\right)
\textcolor[rgb]{0,0,1}{Y}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X5}}
\mathrm{\Sigma }≔〈〈1|0.5〉,〈0.5|1〉〉
\textcolor[rgb]{0,0,1}{\mathrm{\Sigma }}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0.5}\\ \textcolor[rgb]{0,0,1}{0.5}& \textcolor[rgb]{0,0,1}{1}\end{array}]
Z≔\mathrm{ItoProcess}\left(〈X,Y〉,\mathrm{\Sigma }\right)
\textcolor[rgb]{0,0,1}{Z}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X6}}
\mathrm{Drift}\left(Z\left(t\right)\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{0.05}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{0.07}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{Diffusion}\left(Z\left(t\right)\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{0.3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0.15}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{0.10}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}& \textcolor[rgb]{0,0,1}{0.2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{ExpectedValue}\left(\mathrm{max}\left(X\left(1\right)-Y\left(1\right),0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{14.32896059}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.2447103632}]
\mathrm{ExpectedValue}\left(\mathrm{max}\left(Z\left(1\right)[1]-Z\left(1\right)[2],0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{8.103315185}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.1520913055}]
|
The workshop \emph{Computational Group Theory} was the fifth of this title held at Oberwolfach. It was attended by
54
participants with broad geographic representation from four continents. Among the participants were~
10
young researchers, visiting Oberwolfach for the first time, some of them still graduate students. We are grateful for the special EU funds making their visits possible. The lecture program was divided into two sections. In the first we had a series of four invited longer lectures, 50 minutes each, given by distinguished, selected participants, surveying recent major developments in their fields. The first of these lectures was held by Eamonn O'Brien (Auckland), who gave a talk on ``The latest developments in the matrix groups computation project''. The second was given by Bill Kantor (Eugene) on ``A presentation of presentations'', introducing the remarkable result that all finite simple groups (except possibly
^2G_2(3^{2k+1})
) can be presented by a bounded number of generators and relations. In the third talk in this series Michael Vaughan-Lee (Oxford) reported on the recently completed classification of all
p
-groups of orders
p^6
p^7
. Finally, Gregor Kemper (M\"unchen) surveyed new developments in Computational Invariant Theory. The other section was made up of
37
short twenty-minute lectures on new work or work in progress. We aimed at giving preference to the younger participants to present their results. This program structure prompted positive feedback by many participants. The short talks reported on various computational aspects---data structures, algorithms, complexity, computer experiments---in a broad range of topics, including matrix groups,
p
-groups, finitely presented groups, permutation groups, representation theory of groups, invariant theory, group cohomology, Lie algebras and combinatorics. The analysis of algorithms requires a thorough theoretical background in the respective fields. The talks also revealed the close interrelationship between the different topics. Work for finite
p
-groups uses Lie rings and algebraic groups, matrix group algorithms rely on methods for finitely presented groups as well as on representation theory, work on permutation groups uses representation theoretic information. The computational solution of open problems such as the construction of Brauer character tables or the cohomology rings of sporadic groups usually requires the application of almost all the known techniques. Methods from other parts of computer algebra, e.g., Gr\"obner bases, come into play in computational cohomology theory as well as in the construction of matrix groups which are factor groups of specific finitely presented groups, e.g., Hurwitz groups. Despite the relatively large number of talks at this workshop, there was plenty of time for discussions. Needless to say that this time was well spent: numerous collaborations were continued and various others were started.
Gerhard Hiß, Derek F. Holt, Michael F. Newman, Computational Group Theory. Oberwolfach Rep. 3 (2006), no. 3, pp. 1795–1878
|
Exponential map - Wikipedia
This article is about the exponential map in differential geometry. For discrete dynamical systems, see Exponential map (discrete dynamical systems).
In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis. Important special cases include:
exponential map (Riemannian geometry) for a manifold with a Riemannian metric,
exponential map (Lie theory) from a Lie algebra to a Lie group,
More generally, in a manifold with an affine connection,
{\displaystyle X\mapsto \gamma _{X}(1)}
{\displaystyle \gamma _{X}}
is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with respect to appropriate affine connections.
Euler's formula forming the unit circle in the complex plane.
This disambiguation page lists articles associated with the title Exponential map.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Exponential_map&oldid=815288096"
|
Home : Support : Online Help : Programming : Maplets : Utilities : Get File
flexible file selection dialog utility
GetFile[refID](opts)
equation(s) of the form option=value; where option is one of approvecaption, approvecheck, directory, filefilter, filename, filterdescription, height, reference, resizable, title, or width; specify options for the Maplet application
The GetFile(opts) calling sequence displays a file dialog that allows the user to choose a file. The value returned is the string corresponding to the chosen file. If a file is not selected, NULL is returned.
This is a front-end to the FileDialog command. As such, leading and trailing spaces in filenames are automatically removed when a file is selected, and quoted filenames are also handled correctly.
Optionally, the GetFile utility validates the existence of a file.
approvecheck = true or false
Indicates whether the existence of the file should be checked before exiting the Maplet application. By default, the value is true. If this is set to true, and the file is not present, an error dialog indicating this is displayed, and the Maplet application does not exit until a file is correctly chosen or the 'Cancel' button is pressed.
directory = string or symbol
Initial directory in which the file dialog opens.
filename = string or symbol
Initial filename when the file dialog opens. By default this is null.
A description of the filefilter. By default, this is all files.
The height of the file dialog in pixels.
Whether the file dialog is resizable. By default this is true.
A reference for the GetFile element.
If the reference is specified by both an index, for example, GetFile[refID], and in the calling sequence, an error results.
The width of the file dialog in pixels.
A file open dialog that checks for the existence of a file before exiting.
\mathrm{with}\left(\mathrm{Maplets}[\mathrm{Utilities}]\right):
\mathrm{GetFile}\left('\mathrm{title}'="Open Maple Worksheet",'\mathrm{directory}'="/user/maple",'\mathrm{filefilter}'="mws",'\mathrm{filterdescription}'="Maple Worksheets"\right)
\textcolor[rgb]{0,0,1}{"/user/maple/example.mws"}
|
Characterization of the Core Properties of a Shock Absorbing Composite | J. Eng. Mater. Technol. | ASME Digital Collection
G. Georgiades,
G. Georgiades
, Manchester M13 9PL, UK
S. O. Oyadiji,
S. O. Oyadiji
e-mail: s.o.oyadiji@manchester.ac.uk
X. Q. Zhu,
Georgiades, G., Oyadiji, S. O., Zhu, X. Q., Wright, J. R., and Turner, J. T. (July 15, 2006). "Characterization of the Core Properties of a Shock Absorbing Composite." ASME. J. Eng. Mater. Technol. October 2007; 129(4): 497–504. https://doi.org/10.1115/1.2772323
This paper is on the characterization of the mechanical properties of Newtonian-type shock absorbing elastomeric composites. This composite material is a blend of elastomeric capsules or beads in a matrix of a Newtonian liquid. The material can be considered as a liquid analogy to elastomeric foams. It exhibits bulk compression characteristics and acts like an elastic liquid during an impact, unlike elastic foams, which exhibit uniaxial compression characteristics. A test cell consisting of an instrumented metal cylinder and a piston was designed. A sample of the material was placed in the instrumented cylinder, which was located at the base of a drop test rig. A drop mass of
17.3kg
was subsequently released from a desired height to impact the piston. From measurements of the acceleration histories of the drop mass and the piston, and from the displacement history of the piston, the force-displacement curves and the associated impact energies absorbed were derived. These are compared to the corresponding characteristics derived from measurements of pressure of the fluid medium inside the cylinder. The results are compared for blends of different bead types, and the different aspects contributing to their performance are discussed. It is shown that the performance curves derived from the accelerometer measurements matched those derived from the pressure measurements. Blends of this composite material of different types of beads showed distinctively different characteristics.
composite materials, compressive strength, elastomers, impact (mechanical), polymer foams, shock absorbers, shock wave effects
Composite materials, Fluids, Shock (Mechanics), Viscosity, Pistons, Cylinders, Stress, Accelerometers, Pressure
The Practicalities of Engineering Cars for Pedestrian Safety
,” 16th International Technical Conference on the Enhanced Safety of Vehicles, Ontario, Paper No. 98-S10-P-16.
Techniques for the Development of Pedestrian Friendly Vehicles
,” Auto Tech 97, NEC, Birmingham, UK.
Device Incorporating Elastic Fluids and Viscous Damping
,” World Intellectual Property Organisation, WO 97/25551.
Improved Impact Absorber With Viscous Damping
,” World Intellectual Property Organisation, PCT/GB98/03594.
Impact Absorbent Building Structures
,” British Patent Office, GB9805887.8.
Elastic Moduli of Hollow Spheres in a Matrix
Preliminary Investigations Into the Mechanical Properties of a Novel Shock Absorbing Elastomeric Composite
Characteristics and Potential Applications of a Novel Shock Absorbing Elastomeric Composite for Enhanced Crashworthiness
Dynamic Mechanical Properties of Polyurethane Elastomers Using a Nonmetalic Hopkinson Bar
Dynamic Compression Characteristics of Flexible Foams—II: Density Variation
Domestic Gas Cylinders Manufactured by Using a Composite Hybrid Steel Glass Reinforced Thermoplastic Matrix Solution
|
The Diabetic Foot Load Monitor: A Portable Device for Real-Time Subject-Specific Measurements of Deep Plantar Tissue Stresses During Gait | J. Med. Devices | ASME Digital Collection
Eran Atlas,
Ziva Yizhar,
e-mail: gefen@eng.tau.ac.ul
Atlas, E., Yizhar, Z., and Gefen, A. (March 10, 2008). "The Diabetic Foot Load Monitor: A Portable Device for Real-Time Subject-Specific Measurements of Deep Plantar Tissue Stresses During Gait." ASME. J. Med. Devices. March 2008; 2(1): 011005. https://doi.org/10.1115/1.2891241
Elevated stresses in deep plantar tissue of diabetic neuropathic patients were associated with an increased risk for foot ulceration, but only interfacial foot pressures are currently measured to evaluate susceptibility to ulcers. The goals of this study were to develop a real-time patient-specific plantar tissue stress monitor based on the Hertz contact theory. The biomechanical model for stress calculations considers the heel and metatarsal head pads, where most ulcers occur. For calculating stress concentrations around the bone-pad interface, plantar tissue is idealized as elastic and incompressible semi-infinite bulk (with properties measured by indentation), which is penetrated by a rigid sphere with the bone’s radius of curvature (from X-ray). Hertz’s theory is used to solve the bone-pad mechanical interactions, after introducing correction coefficients to consider large deformations. Foot-shoe forces are measured to solve and display the principal compressive, tensile, and von Mises plantar tissue stresses in real time. Our system can be miniaturized in a handheld computer, allowing plantar stress monitoring in the patient’s natural environment. Small groups of healthy subjects
(N=6)
and diabetic patients
(N=3)
participated in an evaluation study in which the differences between free walking and treadmill walking were examined. We also compared gait on a flat surface to gait on an ascending/descending slope of
3.5deg
and when ascending/descending stairs. Peak internal compression stress was about threefold greater than the interface pressure at the calcaneus region. Subjects who were inexperience in treadmill walking displayed high gait-cycle variability in the internal stresses as well as poor foot loading. There was no statistical difference between gait on a flat surface and gait when ascending/descending a slope. Internal stresses under the calcaneus during gait on a flat surface, however, were significantly higher than when ascending/descending stairs. We conclude that the present stress monitor is a promising tool for real-time patient-specific evaluation of deep tissue stresses, providing valuable information in the effort to protect diabetic patients from foot ulceration. Clinical studies are now underway to identify which stress parameters can distinguish between diabetic and normal subjects; these parameters may be used for establishing injury threshold criteria.
biomedical equipment, biomedical measurement, bone, diseases, gait analysis, internal stresses, patient monitoring, stress measurement, plantar pressure, heel pad, Hertz contact theory, diabetes, foot ulcers
Diabetes, Stress, Biological tissues, Structural mechanics
Plantar Soft Tissue Loading Under the Medial Metatarsals in the Standing Diabetic Foot
Diffusion of Ulcers in the Diabetic Foot is Promoted by Stiffening of the Plantar Muscular Tissue Under Excessive Bone Compression
Real-Time Subject-Specific Monitoring of Internal Deformations and Stresses in the Soft Tissue of the Foot: A New Approach in Gait Analysis
Biomechanical Analysis of the Three Dimensional Foot Structure During Gait: A Basic Tool for Clinical Applications
Uber die Beruhrunng Fester Elasticher Korper
Analytical Approaches to the Determination of Pressure Distribution Under a Plantar Prominence
Analytical Parametric Analysis of the Contact Problem of Human Buttocks and Negative Poisson’s Ratio Foam Cushions
Zur Theorie der Beruhrung Fester Elastischer Korper
The Effect of Heel-Pad Thickness and Loading Protocol on Measured Heel-Pad Stiffness and Standardized Protocol for Inter-Subject Comparability
Estimating the Effective Young’s Modulus of Soft Tissues From Indentation Test-Nonlinear Finite Element Analysis of the Effects of Friction and Large Deformation
Curumano
Stress Analysis in Three-Dimensional Foot Models of Normal and Diabetic Neuropathy
The Effect of Walking Speed on Peak Plantar Pressure
The Influence of Walking Speed and Footwear on Plantar Pressure in Older Adults
Risk Factors for a Pressure-Related Deep Tissue Injury: A Theoretical Model
|
Mini-Workshop: Arithmetik von Gruppenringen | EMS Press
The mini workshop "Arithmetic of group rings" was attended by 16 participants from Belgium, Brazil, Canada, Germany, Hungary, Israel, Italy, Romania and Spain. The expertise was a good mixture between senior and young researchers. It was a very stimulating experience and thesize of the group allowed excellent discussions amongst all participants. Very fruitful were the problem sessions, resulting in the problems listed at the end of this report.
The main highlights of the conference were:
The complete calculation of the projective Schur subgroup of the Brauer group by Aljadeff and del Rio.
Hertweck's solution of the first Zassenhaus conjecture for finite metacyclic groups.
the description of special subgroups of the unit group of integral group rings, such as the hypercentre and the finite conjugacy centre, and the relation with respect to the normalizer of the trivial units.
discussion of the present state of art via several survey talks presented and the problem sessions
G
determines its integral group ring
\mathbb Z G
and its group
V(\mathbb Z G)
of normalized units. Several talks addressed the interplay of the cohomological properties of these three objects. Further topics included twisted group rings, group rings over local rings, polynomial growth and identities, orders and semigroup rings, Lie structure, representation-theoretic and algorithmic methods.
Eric Jespers, Zbigniew Marciniak, Gabriele Nebe, Wolfgang Kimmerle, Mini-Workshop: Arithmetik von Gruppenringen. Oberwolfach Rep. 4 (2007), no. 4, pp. 3209–3240
|
§ Generating k bitsets of a given length n:
The problem is to generate all bitvectors of length n that have k bits set. For example, generate all bitvectors of length 5 that have 3 bits set. I know that an algorithm exists in Hacker's delight, but I've been too sick to crack open a book, so I decided to discover the algorithm myself. The one I came up with relies on looking at the numbers moving at a certain velocity, and them colliding with each other. For example, let us try to generate all 5C3 combinations of bits. We start wih:
#1 count of position
a b c d e positions
1 1 1 0 0 bitset
< - - - - velocity
Where the < represents that the 1 at position a is moving leftwards. Our arena is circular , so the leftmost 1 can wrap around to the right. This leads to the next state
- - - - <
We continue moving left peacefully.
- - - < -
whoops, we have now collided with a block of 1s. Not to worry, we simply transfer our velocity by way of collision, from the 1 at d to the 1 at b. I denote the transfer as follows:
0 1 1 1 0 original state
- < < < - transfer of velocity
- < - - - final state after transfer of velocity
The 1 at b proceeds along its merry way with the given velocity
< - - - -
Once again, it wraps around, and suffers a collision
- - - - - < (collision, transfer)
- - < < < transfer of velocity
- - < - - final state after transfer of velocity
0 1 0 1 1 #6
- < - - -
< - - - - (collision, transfer velocity)
< - - < <
1 0 1 0 1 #8
- - < - -
- < - - - (colision, transfer velocity
< < - - <
1 1 1 0 0 #11: wrap around to initial state
I don't have a proof of correctness, but I have an intuition that this should generate all states. Does anyone have a proof? EDIT: this algorithm does not work , since it will keep clusters of
k-1
bits next to each other, when a bit hits a cluster of
k - 1
bits. For completeness, I'm going to draft out the usual algorithm in full:
§ Usual Algorithm
Let's consider the same example of 5C3:
1| 0 0 1 1 1 (LSB)
We start with all bits at their lowest position. Now, we try to go to the next smallest number, which still has 3 bits toggled. Clearly, we need the bit at position b to be 1, since that's the next number. Then, we can keep the lower 2 bits d, e set to 1, so that it's still as small a number as possible.
Once again, we now move the digit at d to the digit at c, while keeping the final digit at e to make sure it's still the smallest possible.
Now, we can move the 1 at e to d, since that will lead to the smallest increase:
At this point, we are forced to move to location a, since we have exhausted all smaller locations. so we move the 1 at b to a, and then reset all the other bits to be as close to the LSB as possible:
Continuing this process gives us the rest of the sequence:
8 | 1 1 0 0 1 (note the reset of d!)
|
Introduction to Program Design | Shreyas’ Notes
waterfall method:
design: planning how to build your software to meet functional metrics and software quality metrics
economical: time, space, scalability
future proof: extensibility, reusability, modularity
error-resistant: readability, debuggability, testability
user-friendly: usability, accessibility
bulletproof: reliability, security, robustness
functinoal
Taxonomy of Programming Languages §
abstraction: hiding details to focus on essentials
when is type checking performed?
static: at compile time
dynamic: at run time
are type errors ever permitted? how strict are the rules?
Wrapper Classes §
For each primitive type, Java defines a “wrapper” class that allows primitives to be treated like objects.
Autuboxing and unboxing: automatic conversion from primitive types to the corresponding wrapper classes and vice-versa.
Casting: explicit type conversion.
int b = (float) a;
Foo a = (Foo) o;
Object Design §
principles of OOP:
abstraction via encapsulation
single responsbility and delegation
decoupling and loose coupling
avoid duplication: composition, inheritance
Declaring that one class inherits features and functionalities from another class.
A class may only directly extend one other class.
Subclasses inherit protected, public fields and methods. If the subclass is defined in the same package as the superclass, package-protected fields and methods are also inherited. They inherit these things from all classes they’re descendants of, directly and indirectly.
subclasses can override inherited methods.
subclasses can shadow inherited fields.
Objects §
A class is a template for a custom data type. An object is an instance of a class.
public: accessible outside the class in which they were defined
protected: accessible only in the class and in subclasses
private: accessible only in the class
package-protected: accessible only in the package in which the class is defined
constructor: special method used to create new objects
Java Project Structure §
One file: one class.
One directory: one package.
Each package has its own namespace. Classes in the same namespace can refer to each other without requiring imports.
Parallel package hierarchies for source and tests.
import [package].[class];
import [package].*;
import [package].[class].[staticMethod];
Testing §
white box testing: implementation is known.
coverage. may miss unhandled edge cases. implementation-related edge cases.
black box testing: implmentation is unknown but spec is known.
high-level expectations.
JUnit §
Abstract Classes and Polymorphism §
Abstract Classes §
As opposed to concrete classes.
fields can be declared and defined
methods can be declared and implemented or just declared
constructors can be defined
public abstract Foo { ... }
Abstract methods §
public abstract int exampleMethod();
Abstract methods can’t be private.
All concrete subclasses must implement all abstract methods. In fact, they must implement all abstract methods that were not implemented in a higher level of the hierarchy.
Polymorphism §
polymorphism: the ability for one method to take on multiple forms.
covariance: the ability to use a more specific type than declared
Interfaces §
Encapsulate behaviours.
fields cannot be declared or defined
constructors cannot be defined
All methods must be public. Explicit public and abstract designators are optional.
To implement a method, the default keyword must be used.
default int exampleMethod() {
public class Foo implements Bar {
When a concrete class implements an interface, it must implement all unimplemented methods.
When an abstract class implements an interface, implementing the interface’s methods is optional.
Interfaces vs Abstract classes: Use an abstract class if:
you want to define fields
you want to override methods of Object
public class Foo extends Baz implements Bar {
If a method in a superclass has the same signature as a method in an implemented interface, the superclass hierarchy takes precedence.
Interfaces may implement other interfaces. However, they may not extend classes.
Object class §
A built-in class. Every class that doesn’t explicitly extend another class directly extends Object. Object is at the top of all class hierarchies.
public boolean equals(Object o) §
Returns true iff this and o are equal.
Compares by value if defined appropriately.
The instanceof operator allows checking if an object is an instance of a certain class.
public int hashCode() §
Returns a hash code representing this.
public String toString() §
Containers §
interfaces defined in java.util:
Generics §
Both generic and non-generic types can extend generic types. Generic types can also extend non-generic types.
Generic methods §
public <T> List<T> foo(List<T> bar) {
List<T> baz = new ArrayList<T>();
generic methods: flexibility. good for pure functions.
genertic types: consistency, type safety. better when you want consistent types across methods that share state.
Bounded Type Parameters §
upper bound: <T extends Foo> — only subclasses of Foo
lower bound: <T super Bar> — only superclasses of Bar
Wildcards §
(unknown) constraints on the type: homogeneity
Bounded wildcards:
<? extends Foo>
<? super Bar>
Maps §
Map<K, V>, collection of key-value pairs
HashMap (hash table)
TreeMap (red-black tree)
Type Erasure and Invariance §
Type erasure: compiler replaces parameteterized types with raw types and inserts any necessary casts
Because of type erasure, type parameters are not known at runtime.
can’t use type parameter T to construct an object of type T
can’t use type parameter T to construct an array of type T[]
can’t declare static fields using type parameters
can’t use instanceof to check parameterized types
Type Inference §
Type inference: the ability of a compiler to infer the type of a variable when it’s not explicitly defined.
Performed at compile time.
local variable type inference (LVTI)
generic type inference (diamond operator)
code is less self-documenting
can only be used in select cases
easier to make interface changes
Sets §
Set<E>: unordered collection of unique elements of type E
HashSet (hash table)
TreeSet (red-black tree)
boolean contains(Object obj) (by value)
boolean retainAll(Collection<? extends E> c)
boolean removeAll(Collection<? extends E> c)
Errors §
Exceptions §
Throwable hierarchy:
Error: usually originate in the JVM and are not recoverable.
RuntimeException: usually originate in the JVM. Are unchecked. Typically caused by programmer error.
Others: usually originate in the code. Are checked. Typically not caused by programmer error.
catch: try … catch
throw: throws … throw
public double foo(double x) throws Exception {
Declare all checked exceptions that may be thrown.
} catch (FooException err) {
} catch (BarException err) {
In decreasing order of specificity.
I/O §
Data Structures §
Abstract data structures: lists, sets, maps, queues, stacks, priority queues etc
Concrete data structure: array-backed list, linked list, binary heap, treap, hash table etc
Array-backed lists §
append:
O(1)
prepend:
O(n)
insert at arbitrary position:
O(n)
remove by reference:
O(n)
remove by position:
O(n)
O(n)
Linked lists §
append and prepend are
O(1)
. other common operations are
O(n)
Trees §
private Node<T> riit;
contains and remove are
O(n)
Binary Trees §
public class BST<T> {
Trees, except they may have at most two children.
O(n)
Binary Search Trees §
total order:
x.l < x < x.r
total preorder:
x.l < x \leq x.r
insert, contains, and remove are
O(\log n)
in the best case and
O(n)
Binary Heaps §
private ArrayList<T> heap;
shape constraint: complete tree. all layers but the last one must be full.
partial ordering constraint
index math (assuming null at index 0):
children of the node at
i
: at
2i
2i + 1
parent of the node at
i
\frac{i}{2}
remove (priority element elem):
replace roo with the last element (elem) in the heap
while the ordering constraint is not satisfied: swap elem with its parent
insert and remove (priority) are
O(\log n)
. remove (arbitrary) and contains are
O(n)
Treaps §
self-balancing BSTs. combination of BSTs are heaps.
keys are ordered according to the BST property
priorities are ordered according to the heap property
rotations :sparkles:
contains: identical to BSTs
perform a BST insertion
generate a random priority for the new node
perform tree rotations until the heap property is satisfied
traverse a branch to find the node to remove
if it has <2 children, remove
else perform rotations until it has <2 children.
insert, remove and contains are
O(\log n)
Tries §
aka prefix trees.
private String data; // single-character string
private Set<Node> children;
contains (element or prefix): traverse one branch
insert: traverse one branch, adding nodes as necessary, and marking the last one as valid
remove: traverse one branch of the trie. along the way, remove invalid nodes that have no children.
lookup: traverse one branch—from the node to the end of the prefix—and return the tree rooted in that prefix.
O(1)
children per node, these operations are
O(1)
wrt the number of elements in the trie (but
O(m)
m
is the number of characters in the input).
uncompressed tries
Memory and References §
Garbage Collection §
A technique for automatically detecting memory that is no longer in use and automatically freeing it
reference-counting definition of liveness: an object is considered live if there is at least one reference to it; else, it is dead.
trace-based definition of liveness: an object is considered live if it is reachable from the running code; else, it is dead.
JSON §
org.json defines JSONArray and JSONObject. Not generic. Also JSONException.
JSONArray §
JSONArray(Collection<?> copyFrom)
public JSONArray put(Object value)
JSONObject §
JSONObject(Map<?, ?> copyFrom)
JSONObject(Object obj) — uses getter methods defined in obj
public Object get(String key) — throws JSONException if key is not found
public Object opt(String key) — returns null if key is not found
Design Patterns §
|
Universal joint (50348 views - Mechanical Engineering)
A universal joint (universal coupling, U-joint, Cardan joint, Spicer or Hardy Spicer joint, or Hooke's joint) is a joint or coupling connecting rigid rods whose axes are inclined to each other, and is commonly used in shafts that transmit rotary motion. It consists of a pair of hinges located close together, oriented at 90° to each other, connected by a cross shaft. The universal joint is not a constant-velocity joint.
A universal joint (universal coupling, U-joint, Cardan joint, Spicer or Hardy Spicer joint, or Hooke's joint) is a joint or coupling connecting rigid rods whose axes are inclined to each other, and is commonly used in shafts that transmit rotary motion. It consists of a pair of hinges located close together, oriented at 90° to each other, connected by a cross shaft. The universal joint is not a constant-velocity joint.[1]
5 Thompson coupling
The main concept of the universal joint is based on the design of gimbals, which have been in use since antiquity. One anticipation of the universal joint was its use by the ancient Greeks on ballistae.[2] In Europe the universal joint is often called the Cardano joint or Cardan shaft, after the Italian mathematician Gerolamo Cardano; however, in his writings, he mentioned only gimbal mountings, not universal joints.[3]
The mechanism was later described in Technica curiosa sive mirabilia artis (1664) by Gaspar Schott, who mistakenly claimed that it was a constant-velocity joint.[4][5][6] Shortly afterwards, between 1667 and 1675, Robert Hooke analysed the joint and found that its speed of rotation was nonuniform, but that this property could be used to track the motion of the shadow on the face of a sundial.[4] In fact, the component of the equation of time which accounts for the tilt of the equatorial plane relative to the ecliptic is entirely analogous to the mathematical description of the universal joint. The first recorded use of the term universal joint for this device was by Hooke in 1676, in his book Helioscopes.[7][8][9] He published a description in 1678,[10] resulting in the use of the term Hooke's joint in the English-speaking world. In 1683, Hooke proposed a solution to the nonuniform rotary speed of the universal joint: a pair of Hooke's joints 90° out of phase at either end of an intermediate shaft, an arrangement that is now known as a type of constant-velocity joint.[4][11] Christopher Polhem of Sweden later re-invented the universal joint, giving rise to the name Polhemsknut in Swedish.
{\displaystyle \omega _{2}\,}
{\displaystyle \gamma _{1}\,}
{\displaystyle \beta \,}
{\displaystyle \gamma _{2}\,}
, versus input shaft rotation angle,
{\displaystyle \gamma _{1}\,}
, for different bend angles,
{\displaystyle \beta \,}
, of the joint
{\displaystyle \gamma _{1}}
{\displaystyle \gamma _{2}}
{\displaystyle \beta }
{\displaystyle {\hat {\mathbf {x} }}}
{\displaystyle {\hat {\mathbf {y} }}}
{\displaystyle {\hat {\mathbf {x} }}_{1}}
{\displaystyle {\hat {\mathbf {x} }}_{2}}
{\displaystyle {\hat {\mathbf {x} }}_{1}}
{\displaystyle \gamma _{1}}
{\displaystyle {\hat {\mathbf {x} }}_{2}}
{\displaystyle \gamma _{2}}
{\displaystyle {\hat {\mathbf {x} }}_{1}}
{\displaystyle \gamma _{1}}
{\displaystyle {\hat {\mathbf {x} }}_{1}=[\cos \gamma _{1}\,,\,\sin \gamma _{1}\,,\,0]}
{\displaystyle {\hat {\mathbf {x} }}_{2}}
{\displaystyle {\hat {x}}=[1,0,0]}
{\displaystyle [\pi \!/2\,,\,\beta \,,\,0}
{\displaystyle {\hat {\mathbf {x} }}_{2}=[-\cos \beta \sin \gamma _{2}\,,\,\cos \gamma _{2}\,,\,\sin \beta \sin \gamma _{2}]}
{\displaystyle {\hat {\mathbf {x} }}_{1}}
{\displaystyle {\hat {\mathbf {x} }}_{2}}
{\displaystyle {\hat {\mathbf {x} }}_{1}\cdot {\hat {\mathbf {x} }}_{2}=0}
{\displaystyle \tan \gamma _{1}=\cos \beta \tan \gamma _{2}\,}
{\displaystyle \gamma _{2}}
{\displaystyle \gamma _{2}=\tan ^{-1}[\tan \gamma _{1}/\cos \beta ]\,}
{\displaystyle \gamma _{2}}
{\displaystyle \gamma _{2}}
be continuous over the angles of interest. For example, the following explicit solution using the atan2(y,x) function will be valid for
{\displaystyle -\pi <\gamma _{1}<\pi }
{\displaystyle \gamma _{2}=\mathrm {atan2} (\sin \gamma _{1},\cos \beta \,\cos \gamma _{1})}
{\displaystyle \gamma _{1}}
{\displaystyle \gamma _{2}}
{\displaystyle \omega _{1}=d\gamma _{1}/dt}
{\displaystyle \omega _{2}=d\gamma _{2}/dt}
{\displaystyle \omega _{2}={\frac {\omega _{1}\cos \beta }{1-\sin ^{2}\beta \cos ^{2}\gamma _{1}}}}
{\displaystyle a_{1}}
{\displaystyle a_{2}}
{\displaystyle a_{2}={\frac {a_{1}\cos \beta }{1-\sin ^{2}\beta \,\cos ^{2}\gamma _{1}}}-{\frac {\omega _{1}^{2}\cos \beta \sin ^{2}\beta \sin 2\gamma _{1}}{(1-\sin ^{2}\beta \cos ^{2}\gamma _{1})^{2}}}}
{\displaystyle \gamma _{1}\,}
{\displaystyle \gamma _{2}\,}
{\displaystyle \gamma _{3}\,}
{\displaystyle \gamma _{4}\,}
{\displaystyle \beta \,}
{\displaystyle \tan \gamma _{2}=\cos \beta \,\tan \gamma _{1}\qquad \tan \gamma _{4}=\cos \beta \,\tan \gamma _{3}}
{\displaystyle \gamma _{3}=\gamma _{2}+\pi /2}
{\displaystyle \tan(\gamma +\pi /2)=1/\tan \gamma }
{\displaystyle \tan \gamma _{4}=\cos \beta /\tan \gamma _{2}=1/\tan \gamma _{1}=\tan(\gamma _{1}+\pi /2)\,}
A double Cardan joint consists of two universal joints mounted back to back with a center yoke; the center yoke replaces the intermediate shaft. Provided that the angle between the input shaft and center yoke is equal to the angle between the center yoke and the output shaft, the second Cardan joint will cancel the velocity errors introduced by the first Cardan joint and the aligned double Cardan joint will act as a CV joint.
Bolt (fastener)Rag jointMechanical engineeringCrimp (joining)Slider crank chain inversionFour-bar linkageDrive shaft
This article uses material from the Wikipedia article "Universal joint", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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Invariance of $o$-minimal cohomology with definably compact supports
o
Mário J. Edmundo1, 2; Luca Prelli2
1 Universidade Aberta, Rua Braamcamp 90, 1250-052 Lisboa, Portugal, and
2 CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
In this paper we find general criteria for invariance and finiteness results for
o
-minimal cohomology in an arbitrary
o
-minimal structure. We apply our criteria and obtain new invariance and finiteness results for
o
-minimal cohomology in
o
-minimal expansions of ordered groups and for the
o
-minimal cohomology of definably compact definable groups in arbitrary
o
-minimal structures.
Classification: 03C64, 55N30
o
-minimal structures,
o
-minimal cohomology.
Mário J. Edmundo 1, 2; Luca Prelli 2
author = {M\'ario J. Edmundo and Luca Prelli},
title = {Invariance of $o$-minimal cohomology with definably compact supports},
TI - Invariance of $o$-minimal cohomology with definably compact supports
%T Invariance of $o$-minimal cohomology with definably compact supports
Mário J. Edmundo; Luca Prelli. Invariance of $o$-minimal cohomology with definably compact supports. Confluentes Mathematici, Volume 7 (2015) no. 1, pp. 35-53. doi : 10.5802/cml.17. https://cml.centre-mersenne.org/articles/10.5802/cml.17/
[1] A. Berarducci and A. Fornasiero O-minimal cohomology: finiteness and invariance results J. Math. Logic 9 (2) (2009) 167–182.
[2] A. Berarducci and M. Otero O-minimal fundamental group, homology and manifolds J. London Math. Soc. 65 (2) (2002) 257–270.
[3] J. Bochnak, M. Coste and M-F. Roy Real algebraic geometry Springer-Verlag 1998.
[4] G. Bredon Sheaf theory Second Edition Springer-Verlag 1997.
[5] M. Coste An introduction to
-minimal geometry Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa (2000).
[6] M. Carral and M. Coste Normal spectral spaces and their dimensions J. Pure and Appl. Algebra 30 (3) (1983) 227–235.
[7] M. Coste and M.-F. Roy La topologie du spectre réel in Ordered fields and real algebraic geometry, Contemporary Mathematics 8 (1982) 27–59.
[8] H. Delfs Homology of locally semialgebraic spaces LNM 1484 Springer-Verlag 1991.
[9] L. van den Dries Tame topology and
o
-minimal structures Cambridge University Press 1998.
[10] M. Edmundo, G. Jones and N. Peatfield Sheaf cohomology in
o
-minimal structures J. Math. Logic 6 (2) (2006) 163–179.
[11] M. Edmundo, M. Mamino and L. Prelli On definably proper maps Fund. Math. (to appear).
[12] M. Edmundo and M. Otero Definably compact abelian groups J. Math. Logic 4 (2) (2004) 163–180.
[13] M. Edmundo and L. Prelli Poincaré - Verdier duality in
o
-minimal structures Ann. Inst. Fourier Grenoble 60 (4) (2010) 1259–1288.
[14] M. Edmundo and L. Prelli Sheaves on T-topologies J. Math. Soc. Japan (to appear).
[15] M. Edmundo and G. Terzo A note on generic subsets of definable groups Fund. Math. 215 (1) (2011) 53–65.
[16] M. Edmundo and A. Woerheide Comparision theorems for
o
-minimal singular (co)homology Trans. Amer. Math. Soc. 360 (9) (2008) 4889–4912.
[17] P. Eleftheriou A semi-linear group which is not affine Ann. Pure Appl. Logic 156 (2008) 287 – 289.
[18] P. Eleftheriou, Y. Peterzil and J. Ramakrishnan Interpretable groups are definable J. Math. Log. 14 1450002 (2014) [47 pages].
[19] A. Fornasiero O-minimal spectrum Unpublished, 33pp, 2006.
[20] R. Godement Théorie des faisceaux Hermann 1958.
[21] B. Iversen Cohomology of sheaves Springer Verlag 1986.
[22] M. Kashiwara and P. Schapira Sheaves on manifolds Springer Verlag 1990.
[23] M. Otero A survey on groups definable in
o
-minimal structures in Model Theory with Applications to Algebra and Analysis, vol. 2, Editors: Z. Chatzidakis, D. Macpherson, A. Pillay and A. Wilkie, LMS LNS 350 Cambridge Univ. Press (2008) 177–206
[24] Y. Peterzil and C. Steinhorn Definable compactness and definable subgroups of
o
-minimal groups J. London Math. Soc. 59 (2) (1999) 769–786.
[25] A. Pillay On groups and fields definable in
o
-minimal structures J. Pure Appl. Algebra 53 (1988) 239 – 255.
[26] A. Pillay Sheaves of continuous definable functions J. Symb. Logic 53 (4) (1988) 1165–1169.
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§ A walkway of lanterns (TODO)
§ Semidirect products
(\alpha \equiv \{ a, b, \dots\}, +, 0)
(\omega \equiv \{ X, Y, \dots\}, \times, 1)
\cdot ~: ~\omega \rightarrow Automorphisms(\alpha)
rotations:
\mathbb Z 5
\mathbb Z 2
D_5 = \mathbb Z5 \rtimes \mathbb Z2
\begin{aligned} \begin{bmatrix} 1 & 0 \\ a & X \end{bmatrix} \begin{bmatrix} 1 & 0 \\ b & Y \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ a + X \cdot b & XY \end{bmatrix} \end{aligned}
(Y \mapsto b) \xrightarrow{act} (X \mapsto a)
XY \mapsto a + X \cdot b
§ A walkway of lanterns
\mathbb Z
as a long walkway. you start at 0. You are but a poor lamp lighter.
Where are the lamps? At each
i \in \mathbb Z
, you have a lamp that is either on, or off. So you have
\mathbb Z2
L \equiv \mathbb Z \rightarrow \mathbb Z2
is our space of lanterns. You can act on this space by either moving using
\mathbb Z
, or toggling a lamp using
\mathbb Z2
\mathbb Z2^{\mathbb Z} \rtimes \mathbb Z
g = (lights:\langle-1, 0, 1\rangle, loc:10)
move_3: (lights: \langle \rangle, loc: 3)
move_3 \cdot g = (lights:\langle-1, 0, 1\rangle, loc:13)
togglex = (lights:\langle 0, 2 \rangle, loc: 0)
togglex \cdot g = (lights: \langle -1, 0, 1, 13, 15 \rangle, loc:13)
toggley = (lights: \langle -13, -12 \rangle, loc:0)
toggley\cdot g= (lights:\langle -1 \rangle, loc:13)
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§ An invitation to homology and cohomology, Part 2 --- Cohomology
Once again, we have our humble triangle with vertices
V = \{r, g, b\}
E = \{o, m, c \}
, faces
F = \{ f \}
with boundary maps
\partial_{EV}
\partial_{FE}
\partial_{FE}(f)= o + m + c
\partial_{EV}(o) = r - g
\partial_{EV}(m) = b - r
\partial_{EV}(c)= g - b
h_v: V \rightarrow \mathbb R
on the vertices as:
h_v(r) = 3
h_v(g) = 4
h_v(b) = 10
We now learn how to extend this function to the higher dimensional objects, the edges and the faces of the triangle. To extend this function to the edges, we define a new function:
h_e: E \rightarrow R
h_e(e) \equiv \sum_i \alpha_i h_v(v_i)
\partial_{EV} e = \sum_i \alpha_i v_i
Expanded out on the example, we evaluate
h_v
h_e(o) \equiv d h_v(o) = h_v(r) - h_v(g) = 3 - 4 = -1
h_e(m) \equiv d h_v(m) = h_v(b) - h_v(r) = 10 - 3 = +7
h_e(c) \equiv d h_v(c) = h_v(g) - h_v(b) = 4 - 10 = -6
More conceptually, we have created an operator called
(the coboundary operator ) which takes functions defined on vertices to functions defined on edges. This uses the boundary map on the edges to "lift" a function on the vertices to a function on the edges. It does so by assigning the "potential difference" of the vertices to the edges.
d: (V \rightarrow \mathbb R) \rightarrow (E \rightarrow \mathbb R)
d(h_v) \equiv h_e
h_e(e) \equiv \sum_i \alpha_i f(v_i)
\partial_{EV} e = \sum_i \alpha_i v_i
We can repeat the construction we performed above, to construct another operator
d : (E \rightarrow \mathbb R) \rightarrow (F \rightarrow \mathbb R)
, defined in exactly the same way as we did before. For example, we can evaluate:
h_f \equiv d(h_e)
h_f(f) \equiv d h_e(f) = h_e(o) + h_e(m) + h_e(c) = -1 + 7 -6 = 0
What we have is a chain:
h_v \xrightarrow{d} h_e \xrightarrow{d} h_f
Where we notice that
d^2 = d \circ d = 0
, since the function
h_f
that we have gotten evaluates to zero on the face
f
. We can prove this will happen in general , for any choice of
h_v
. (it's a good exercise in definition chasing). Introducing some terminology, A differential form
f
is said to be a closed differential form iff
df = 0
. In our case,
h_e
is closed , since
d h_e = h_f = 0
h_v
is not closed , since
d h_v = h_e \neq 0
. The intuition for why this is called "closed" is that its coboundary vanishes.
§ Exploring the structure of functions defined on the edges
Here, we try to understand what functions defined on the edges can look like, and their relationship with the
d
operator. We discover that there are some functions
g_e: E \rightarrow \mathbb R
which can be realised as the differential of another function
g_v: V \rightarrow \mathbb R
. The differential forms such as
g_e
which can be generated a
g_v
d
operator are called as exact differential forms . That is,
g_e = d g_v
exactly , such that there is no "remainder term" on applying the
d
operator. We take an example of a differential form that is not exact , which has been defined on the edges of the triangle above. Let's call it
h_e
. It is defined on the edges as:
h_e(c) = 3
h_e(m) = 2
h_e(o) = 1
We can calcuate
h_f = d h_e
the same way we had before:
h_f(f) \equiv d h_e(f) = h_e(o) + h_e(m) + h_e(c) = 3 + 1 + 2 = 6
d h_e \neq 0
, this form is not exact. Let's also try to generate
h_e
from a potential. We arbitrarily fix the potential of
b
0
. That is, we fix
h_v(b) = 0
, and we then try to see what values we are forced to values of
h_v
across the rest of the triangle.
h_v b = 0
h_e(c) = h_v(g) - h_v(b)
h_v(g) = h_v(b) + h_e(c) = 0 + 3 = 3
h_e(o) = h_v(r) - h_v(g)
h_v(r) = h_v(g) + h_e(o) = 3 + 1 = 4
h_e(m) = h_v(b) - h_v(r)
2 = 0 - 4
. This is a contradiction!
Ideally, we need
h_v(b) = 6
for the values to work out.
Hence, there can exist no such
h_v
h_e \equiv d h_v
. The interesting thing is, when we started out by assigning
h_v(b) = 0
, we could make local choices of potentials that seemed like they would fit together, but they failed to fit globally throughout the triangle. This failure of locally consistent choices to be globally consistent is the essence of cohomology.
§ Cohomology of half-filled butterfly
Here, we have vertices
V \equiv \\{ r, g, b, b, p \\}
E \equiv \\{rb, gr, bg, m, o, c \\}
and faces
F \equiv \\{ f \\}
. Here, we see a differential form
h_e
that is defined on the edges, and also obeys the equation
dh_e = 0
(Hence is closed). However, it does not have an associated potential energy to derive it from. That is, there cannot exist a certain
h_v
d h_v = h_e
. So, while every exact form is closed, not every closed form is exact. Hence, this
g
that we have found is a non-trivial element of
Kernel(d_{FE}) / Image(d_{EV})
dh_e = 0
h_e \in Kernel(d_{FE})
, while there does not exist a
h_v
d h_v = h_e
, hence it is not quotiented by the image of
d_{EV}
. So the failure of the space to be fully filled in (ie, the space has a hole), is measured by the existence of a function
h_e
that is closed but not exact! This reveals a deep connection between homology and cohomology, which is made explicit by the Universal Coefficient Theorem
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Head Kinematics in Mini-Sled Tests of Foam Padding: Relevance of Linear Responses From Free Motion Headform (FMH) Testing to Head Angular Responses | J. Biomech Eng. | ASME Digital Collection
J. Ivarsson,
UVa Center for Applied Biomechanics, 1011 Linden Avenue, Charlottesville, VA 22902
D. C. Viano,
Crash Safety Division, Department of Machine and Vehicle Systems, Chalmers University of Technology, SE-412 96 Go¨teborg, Sweden
General Motors Research and Development Center, Warren, MI 48090-9055
Saab Automobile AB, SE-461 80 Trollha¨ttan, Sweden
P. Lo¨vsund,
P. Lo¨vsund
Y. Parnaik
Bioengineering Center, Wayne State University, Detroit, MI 48202
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division February 12, 2002; revision received April 1, 2003. Associate Editor: C. L. Vaughan.
Ivarsson, J., Viano, D. C., Lo¨vsund, P., and Parnaik, Y. (August 1, 2003). "Head Kinematics in Mini-Sled Tests of Foam Padding: Relevance of Linear Responses From Free Motion Headform (FMH) Testing to Head Angular Responses ." ASME. J Biomech Eng. August 2003; 125(4): 523–532. https://doi.org/10.1115/1.1590360
The revised Federal Motor Vehicle Safety Standard (FMVSS) No. 201 specifies that the safety performance of vehicle upper interiors is determined from the resultant linear acceleration response of a free motion headform (FMH) impacting the interior at 6.7 m/s. This study addresses whether linear output data from the FMH test can be used to select an upper interior padding that decreases the likelihood of rotationally induced brain injuries. Using an experimental setup consisting of a Hybrid III head-neck structure mounted on a mini-sled platform, sagittal plane linear and angular head accelerations were measured in frontal head impacts into foam samples of various stiffness and density with a constant thickness (51 mm) at low (∼5.0 m/s), intermediate (∼7.0 m/s), and high (∼9.6 m/s) impact speeds. Provided that the foam samples did not bottom out, recorded peak values of angular acceleration and change in angular velocity increased approximately linearly with increasing peak resultant linear acceleration and value of the Head Injury Criterion
HIC36.
The results indicate that the padding that produces the lowest possible peak angular acceleration and peak change in angular velocity without causing high peak forces is the one that produces the lowest possible
HIC36
without bottoming out in the FMH test.
angular velocity, impact (mechanical), biomechanics, kinematics, safety, automobiles, foams, brain, acceleration, health hazards
Accelerometers, Brain, Cushioning materials, Foams (Chemistry), Kinematics, Testing, Safety, Wounds, Vehicles
Sounik, D. F., Gansen, P., Clemons, J. L., and Liddle, J. W., 1997, “Head-Impact Testing of Polyurethane Energy-Absorbing (EA) Foams,” SAE International Congress and Exposition, SAE Technical Paper No. 970160.
Myers, B. S., and Nightingale, R. W., 1997, “The Dynamics of Head and Neck Impact and its Role in Injury Prevention and the Complex Clinical Presentation of Cervical Spine Injury,” in Proceedings of the 1997 International IRCOBI Conference on the Biomechanics of Impact, pp. 15–33.
Nightingale, R. W., McElhaney, J. H., Camacho, D. L., Kleinberger, M., Winkelstein, B. A., and Myers, B. S., 1997, “The Dynamic Responses of the Cervical Spine: Buckling, End Conditions, and Tolerance in Compressive Impacts,” in Proceedings of the 41st Stapp Car Crash Conference, SAE Technical Paper No. 973344, pp. 451–471.
Mechanical Behavior of Foamed Materials Under Dynamic Compression
J. Cell Plast.
Monk, M. W., and Sullivan, L. K., 1986, “Energy Absorption Material Selection Methodology for Head/A-pillar,” in Proceedings of the 30th Stapp Car Crash Conference, SAE Technical Paper No. 861887, pp. 185–198.
Rossio, R. C., Vecchio, M., and Abramczyk, J., 1993, “Polyurethane Energy Absorbing Foams for Automotive Applications,” SAE International Congress and Exposition, SAE Technical Paper No. 930433.
Sounik, D. F., McCullough, D. W., Clemons, J. L., and Liddle, J. L., 1994, “Dynamic Impact Testing of Polyurethane Energy Absorbing (EA) Foams,” SAE International Congress and Exposition, SAE Technical Paper No. 940879.
Chou, C. C., Zhao, Y., Lim, G. G., Patel, R. N., Shahab, S. A., and Patel, P. J., 1995, “Comparative Analysis of Different Energy Absorbing Materials for Interior Head Impact,” SAE International Congress and Exposition, SAE Technical Paper No. 950332.
Faruque, O., Liu, N., and Chou, C. C., 1997, “Strain Rate Dependent Foam-Constitutive Modeling and Applications,” SAE International Congress and Exposition, SAE Technical Paper No. 971076.
Ullrich, J., Emanuel, D., Fong, W., Nusholtz, G., Chaudhry, M., and Williams, S., 1997, “Comparison of Energy Management Materials for Head Impact Protection,” SAE International Congress and Exposition, SAE Technical Paper No. 970159.
Yu, L. C., Kowalski, E. L., and Elchison, B. K., 1997, “Material Comparison using Free Motion Headform (FMH) Impact and Alternative Test Method,” SAE International Congress and Exposition, SAE Technical Paper No. 970165.
Hodgson, V. R., and Thomas, L. M., 1979, “Acceleration Induced Shear Strains in a Monkey Brain Hemisection,” in Proceedings of the 23rd Stapp Car Crash Conference, SAE Technical Paper No. 791023, pp. 589–611.
Lo¨wenhielm
Mathematical Simulation of Gliding Contusions
Viano, D. C., Melvin, J. W., McCleary, J. D., Madeira, R. G., Shee, T. R., and Horsch, J. D., 1986, “Measurement of Head Dynamics and Facial Contact Forces in the Hybrid III dummy,” in Proceedings of the 30th Stapp Car Crash Conference, SAE Technical Paper No. 861891, pp. 269–289.
Computing Body Segment Trajectories in the Hybrid III Dummy using Linear Accelerometer Data
Amori, R. T., Armitage, R. R., Chou, C. C., Lim, G. G., Patel, R. N., and Shahab, S. A., 1995, “Influence of System Variables on Interior Head Impact Testing,” SAE International Congress and Exposition, SAE Technical Paper No. 950882.
Mathematical Modeling of Seat and Occupant Interaction in Rear Impact
Structural Foams – Materials with Millistructure
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Square degree - Wikipedia
Measure of solid angle
1 deg2 in ... ... is equal to ...
3.04617×10−4 sr
A square degree (deg2) is a non-SI unit measure of solid angle. Other denotations include sq. deg. and (°)2. Just as degrees are used to measure parts of a circle, square degrees are used to measure parts of a sphere. Analogous to one degree being equal to π/180 radians, a square degree is equal to (π/180)2 steradians (sr), or about 1/3283 sr or about 3.046×10−4 sr.
The whole sphere has a solid angle of 4πsr which is approximately 41253 deg2:
{\displaystyle 4\pi \left({\frac {180}{\pi }}\right)^{2}\,{\deg }^{2}={\frac {360^{2}}{\pi }}\,\,{\deg }^{2}={\frac {129\,600}{\pi }}\,\,{\deg }^{2}\approx 41\,252.96\,\,{\deg }^{2}}
The full moon covers only about 0.2 deg2 of the sky when viewed from the surface of the Earth. The Moon is only a half degree across (i.e. a circular diameter of roughly 0.5°), so the moon's disk covers a circular area of: π(0.5°/2)2, or 0.2 square degrees. The moon varies from 0.188 to 0.244 deg2 depending on its distance to the Earth.
Viewed from Earth, the Sun is roughly half a degree across (the same as the full moon) and covers only 0.2 deg2 as well.
It would take 210100 times the full moon (or the Sun) to cover the entire celestial sphere.
Conversely, an average full moon (or the Sun) covers a 2 / 210100 fraction, or less than 1/1000 of a percent (0.00000952381) of the celestial hemisphere, or above-the-horizon sky.
Assuming the Earth to be a sphere with a surface area of 510 million km2, the area of Northern Ireland (14130 km2) and Connecticut (14357 km2) represent a solid angle of 1.14 deg2 and 1.16 deg2, respectively.
The largest constellation, Hydra, covers a solid angle of 1303 deg2, whereas the smallest, Crux, covers only 68 deg2.[1]
Spat (unit)
^ "RASC Calgary Centre - The Constellations". calgary.rasc.ca. Retrieved 2022-02-16.
"Square Degrees - the Area of something on the sky". The RASC Calgary Centre. 2018-11-05. Retrieved 2022-01-21.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Square_degree&oldid=1080933463"
Units of solid angle
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Parseval's theorem - Wikipedia
Theorem in mathematics
In mathematics, Parseval's theorem[1] usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh.[2]
Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem.[3]
1 Statement of Parseval's theorem
2 Notation used in engineering
Statement of Parseval's theorem[edit]
{\displaystyle A(x)}
{\displaystyle B(x)}
are two complex-valued functions on
{\displaystyle \mathbb {R} }
{\displaystyle 2\pi }
that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series
{\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{inx}}
{\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{inx}}
{\displaystyle \sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }A(x){\overline {B(x)}}\,\mathrm {d} x,}
{\displaystyle i}s the imaginary unit and horizontal bars indicate complex conjugation. Substituting
{\displaystyle A(x)}
{\displaystyle {\overline {B(x)}}}
{\displaystyle {\begin{aligned}\sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(\sum _{n=-\infty }^{\infty }a_{n}e^{inx}\right)\left(\sum _{n=-\infty }^{\infty }{\overline {b_{n}}}e^{-inx}\right)\,\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}e^{i1x}+a_{2}e^{i2x}+\cdots \right)\left({\overline {b_{1}}}e^{-i1x}+{\overline {b_{2}}}e^{-i2x}+\cdots \right)\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}e^{i1x}{\overline {b_{1}}}e^{-i1x}+a_{1}e^{i1x}{\overline {b_{2}}}e^{-i2x}+a_{2}e^{i2x}{\overline {b_{1}}}e^{-i1x}+a_{2}e^{i2x}{\overline {b_{2}}}e^{-i2x}+\cdots \right)\mathrm {d} x\\[6pt]&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(a_{1}{\overline {b_{1}}}+a_{1}{\overline {b_{2}}}e^{-ix}+a_{2}{\overline {b_{1}}}e^{ix}+a_{2}{\overline {b_{2}}}+\cdots \right)\mathrm {d} x\end{aligned}}}
As is the case with the middle terms in this example, many terms will integrate to
{\displaystyle 0}
over a full period of length
{\displaystyle 2\pi }
(see harmonics):
{\displaystyle {\begin{aligned}\sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}&={\frac {1}{2\pi }}\left[a_{1}{\overline {b_{1}}}x+ia_{1}{\overline {b_{2}}}e^{-ix}-ia_{2}{\overline {b_{1}}}e^{ix}+a_{2}{\overline {b_{2}}}x+\cdots \right]_{-\pi }^{+\pi }\\[6pt]&={\frac {1}{2\pi }}\left(2\pi a_{1}{\overline {b_{1}}}+0+0+2\pi a_{2}{\overline {b_{2}}}+\cdots \right)\\[6pt]&=a_{1}{\overline {b_{1}}}+a_{2}{\overline {b_{2}}}+\cdots \\[6pt]\end{aligned}}}
More generally, given an abelian locally compact group G with Pontryagin dual G^, Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces L2(G) and L2(G^) (with integration being against the appropriately scaled Haar measures on the two groups.) When G is the unit circle T, G^ is the integers and this is the case discussed above. When G is the real line
{\displaystyle \mathbb {R} }
, G^ is also
{\displaystyle \mathbb {R} }
and the unitary transform is the Fourier transform on the real line. When G is the cyclic group Zn, again it is self-dual and the Pontryagin–Fourier transform is what is called discrete Fourier transform in applied contexts.
Parseval's theorem can also be expressed as follows: Suppose
{\displaystyle f(x)}
is a square-integrable function over
{\displaystyle [-\pi ,\pi ]}
{\displaystyle f(x)}
{\displaystyle f^{2}(x)}
are integrable on that interval), with the Fourier series
{\displaystyle f(x)\simeq {\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }(a_{n}\cos(nx)+b_{n}\sin(nx)).}
Then[4][5][6]
{\displaystyle {\frac {1}{\pi }}\int _{-\pi }^{\pi }f^{2}(x)\,\mathrm {d} x={\frac {a_{0}^{2}}{2}}+\sum _{n=1}^{\infty }\left(a_{n}^{2}+b_{n}^{2}\right).}
Notation used in engineering[edit]
In electrical engineering, Parseval's theorem is often written as:
{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,\mathrm {d} t={\frac {1}{2\pi }}\int _{-\infty }^{\infty }|X(\omega )|^{2}\,\mathrm {d} \omega =\int _{-\infty }^{\infty }|X(2\pi f)|^{2}\,\mathrm {d} f}
{\displaystyle X(\omega )={\mathcal {F}}_{\omega }\{x(t)\}}
represents the continuous Fourier transform (in normalized, unitary form) of
{\displaystyle x(t)}
{\displaystyle \omega =2\pi f}
is frequency in radians per second.
The interpretation of this form of the theorem is that the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency.
For discrete time signals, the theorem becomes:
{\displaystyle \sum _{n=-\infty }^{\infty }|x[n]|^{2}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }|X_{2\pi }({\phi })|^{2}\mathrm {d} \phi }
{\displaystyle X_{2\pi }}
is the discrete-time Fourier transform (DTFT) of
{\displaystyle x}
{\displaystyle \phi }
represents the angular frequency (in radians per sample) of
{\displaystyle x}
Alternatively, for the discrete Fourier transform (DFT), the relation becomes:
{\displaystyle \sum _{n=0}^{N-1}|x[n]|^{2}={\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}}
{\displaystyle X[k]}
{\displaystyle x[n]}
, both of length
{\displaystyle N}
We show the DFT case below. For the other cases, the proof is similar. By using the definition of inverse DFT of
{\displaystyle X[k]}
, we can derive
{\displaystyle {\frac {1}{N}}\sum _{k=0}^{N-1}|X[k]|^{2}={\frac {1}{N}}\sum _{k=0}^{N-1}X[k]\cdot X^{*}[k]={\frac {1}{N}}\sum _{k=0}^{N-1}\left[\sum _{n=0}^{N-1}x[n]\,\exp \left(-j{\frac {2\pi }{N}}k\,n\right)\right]\,X^{*}[k]={\frac {1}{N}}\sum _{n=0}^{N-1}x[n]\left[\sum _{k=0}^{N-1}X^{*}[k]\,\exp \left(-j{\frac {2\pi }{N}}k\,n\right)\right]={\frac {1}{N}}\sum _{n=0}^{N-1}x[n](N\cdot x^{*}[n])=\sum _{n=0}^{N-1}|x[n]|^{2},}
{\displaystyle *}
represents complex conjugate.
Parseval's theorem is closely related to other mathematical results involving unitary transformations:
^ Parseval des Chênes, Marc-Antoine Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaire du second ordre, à coefficients constants" presented before the Académie des Sciences (Paris) on 5 April 1799. This article was published in Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savants, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savants étrangers.), vol. 1, pages 638–648 (1806).
^ Rayleigh, J.W.S. (1889) "On the character of the complete radiation at a given temperature," Philosophical Magazine, vol. 27, pages 460–469. Available on-line here.
^ Plancherel, Michel (1910) "Contribution à l'etude de la representation d'une fonction arbitraire par les integrales définies," Rendiconti del Circolo Matematico di Palermo, vol. 30, pages 298–335.
^ Arthur E. Danese (1965). Advanced Calculus. Vol. 1. Boston, MA: Allyn and Bacon, Inc. p. 439.
^ Wilfred Kaplan (1991). Advanced Calculus (4th ed.). Reading, MA: Addison Wesley. p. 519. ISBN 0-201-57888-3.
^ Georgi P. Tolstov (1962). Fourier Series. Translated by Silverman, Richard. Englewood Cliffs, NJ: Prentice-Hall, Inc. p. 119.
Parseval, MacTutor History of Mathematics archive.
George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001).
Hubert Kennedy, Eight Mathematical Biographies (Peremptory Publications: San Francisco, 2002).
Alan V. Oppenheim and Ronald W. Schafer, Discrete-Time Signal Processing 2nd Edition (Prentice Hall: Upper Saddle River, NJ, 1999) p 60.
William McC. Siebert, Circuits, Signals, and Systems (MIT Press: Cambridge, MA, 1986), pp. 410–411.
David W. Kammler, A First Course in Fourier Analysis (Prentice–Hall, Inc., Upper Saddle River, NJ, 2000) p. 74.
Parseval's Theorem on Mathworld
Retrieved from "https://en.wikipedia.org/w/index.php?title=Parseval%27s_theorem&oldid=1074630422"
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J | Special Issue : Sustainable and Resource – Efficient Homes and Communities
Sustainable and Resource – Efficient Homes and Communities
Submit to J Review for J Edit a Special Issue
Special Issue "Sustainable and Resource – Efficient Homes and Communities"
A special issue of J (ISSN 2571-8800). This special issue belongs to the section "Environmental Sciences".
Over the past half century, homes and communities in many nations have been built with disregard to nature while exhausting natural resources during construction and after occupancy. A much talked-about term that casts a framework for new design thinking is sustainability. The fundamental thrust is thought processes and practices about the future consequences of present development actions.
This Special Issue looks for papers that address urban planning and design of communities and homes with smaller environmental footprints and efficient resource consumption. In particular, the guest editor is looking for papers on urban planning of sustainable communities with attention to higher density and walkability. The issue will also welcome papers on technology-oriented subjects such as active and passive heating and cooling systems, healthy indoor environments, energy-efficient windows, net-zero buildings, sustainable building materials selection, water recycling and efficiency, waste management and disposal, xeriscaping, edible landscapes, and green roofs.
Papers selected for this Special Issue will be subject to a peer review procedure with the aim of rapid and wide dissemination of research results, developments, and applications.
active and passive heating and cooling systems
sustainable building materials selection
edible landscapes and green roofs
J 2021, 4(4), 645-663; https://doi.org/10.3390/j4040047 - 25 Oct 2021
The urban climatology consists not only of the urban canopy temperature but also of wind regime and boundary layer evolution among other secondary variables. The energetic input and response of urbanized areas is rather different to rural or forest areas. In this paper, [...] Read more.
h/d
h/d
h/d\approx 4
h/d>4
; the temperature at the pedestrian level follows similar behavior; the urban boundary layer grows slowly, which in combination with low wind, can worsen pollution dispersion. Full article
J 2021, 4(2), 116-130; https://doi.org/10.3390/j4020010 - 07 May 2021
Current social and environmental challenges have led to the rethinking of residential designs. Global warming, food insecurity, and, as a result, costly fresh produce are some of the causes of the reconsideration. Moreover, with obligatory isolation following the global COVID-19 pandemic, some are [...] Read more.
Current social and environmental challenges have led to the rethinking of residential designs. Global warming, food insecurity, and, as a result, costly fresh produce are some of the causes of the reconsideration. Moreover, with obligatory isolation following the global COVID-19 pandemic, some are realizing the importance of nature and air quality in homes. This paper explores the potential integration of indoor living walls (ILWs) in Canadian homes for agricultural and air purification purposes. By reviewing a number of case studies, this paper investigates how the development of such walls can alter the traditional food production chain, while reducing environmental threats. The findings show that current indoor living wall practices can be transformed into a useful source of fresh food, and, to some degree, alter traditional food supply. They can also help in creating inexpensive methods of air purification. Full article
Khushal Matai
J 2020, 3(3), 343-357; https://doi.org/10.3390/j3030027 - 18 Sep 2020
The solar photovoltaic (SPV) market is growing at a rapid pace with ambitious targets being set worldwide. India is not far behind with an overall solar target of 100 gigawatts (GW) to be achieved by 2022, out of which 40 gigawatts is to [...] Read more.
The solar photovoltaic (SPV) market is growing at a rapid pace with ambitious targets being set worldwide. India is not far behind with an overall solar target of 100 gigawatts (GW) to be achieved by 2022, out of which 40 gigawatts is to be achieved by solar rooftop. Additionally, the depleting non-renewable energy sources and the extensive pollution being done by the aforementioned sources are fueling the renewable energy drive. The threat of climate change, which is fast becoming a reality with effects seen globally, is another contributing factor. The effect of SPV installations on the temperature profiles of their surroundings and the urban thermal environment (UTE) is being studied at a global level, which has arrived at contradictory results, positive as well as negative. However, no such study has been done in the Indian context, which is crucial considering the country’s targets for rooftop installation specifically. The thermal environment of the vicinity is affected by the installations, as seen in the various global studies; the question is how this heat–energy balance is occurring in the Indian context. This review paper looks critically at studies focusing on the relation between SPV installation and the urban heat island (UHI) effect. It is a compilation and analysis of 22 different studies done so far at the global level to gain a thorough understanding of the diverse results. In conclusion, this review highlights the absence of any comprehensive study on the interaction of SPV installations with the built environment at a micro-level and establishes the need for region-based complete studies on the thermal behavior of SPV technology. Full article
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Ally Salim Jr
Probabilities for Physicians Part I: Randomness
At Elsa Health we work with many healthcare providers with very different levels of exposure to probabilities - from the community healthcare providers with absolutely no experience to health researchers with a good familiarity of confidence intervals and p-values - more on these later! From all these encounters we have noticed a common trend, there is lack of an intuitive understanding of probabilities and the randomness of the world. This is particularly true when the probabilities seem to be counter intuitive (see the Monty Hall Problem and examples here and here).
We are starting a simple, casual, and intuitive series of short posts on probabilities for healthcare providers who wish to get a better understanding or dust off their probability know-how! The posts will include content that is needed to develop your own health algorithm using the Elsa Open Health Algorithm Platform. We will try as much as possible to keep the content approachable and free of the dense mathematics that might turn many away from applying the beautiful concepts of probabilities.
Many of us have some basic understanding of probabilities as we go through life and put numbers on how likely certain things are to happen. We all see weather forecasts, election polls, sports odds, and even in casual conversations where statements like "It is likely that ..." or "I'm pretty sure that..." or even the one friend who is always certain and says: "I am 1,000% sure that...". All these are different ways we express probabilities and certainties/uncertainties in our daily lives.
It is worth noting that anyone who is one thousand percent sure of anything either has insider information or is trying to sell you something. Either way, something is fishy!
With that said, we should move forward with the definition: "A probability is how likely something is to happen", and that probabilities range from 0 (impossible event) to 1 (guaranteed event).
Often we deal with probabilities in the form of percentages between 0% and 100%. However, we will stick to the more appropriate scale of 0 - 1 for the rest of this series.
Probabilities and Randomness
The world around us is full of random events that happen from the quantum level to the migration patterns of wildebeests in the Serengeti. If you pick up the phone and call anyone in your contacts there is a probability value for whether or not they will pick up. Furthermore, that probability might depend on who you call, what time you call, the status of your relationship, or even whether or not they owe you money. We can think of these as factors that affect how "likely" the person is to pick up when you call.
In a more clinical scenario, imagine a world where half (1/2 or 0.5 or 50%) of the population have a certain disease. Let's call this disease Probabitis. We are also going to assume people either have the disease or not. In this strange world, the doctor can just randomly diagnose a patient as either having the disease or not, even without talking to or even seeing the patient and the doctor would be right around 50% of the time in the long run!
There are countless more examples of randomness at work, and even more examples of how humans have internalized randomness into simple mental models that frequently come in handy. Mathematically, probabilities are the tools we use to handle and deal with randomness in the world. Probabilities allow us to describe a world that is vague constantly changing.
But What About Statistics?
There is a blurry but significant difference between statistics and probability. Probability deals with predicting the likelihood of future events, while statistics involves the analysis of the frequency of past events[reference].
Statistics can tell us what percent of the population with headaches had relief after taking aspirin while Probabilities can tell us how likely it is that you will have relief from your headache after taking aspirin.
Random and Not Random Things
We can either describe things as being random or not random. When things are random they are called Stochastic and when they are not they are called Deterministic. We will dive deeper into these in a later post, it is enough to just know the terms for now.
Additionally, we are going to use the term "variable" to mean a "thing" from now on. So don't let that throw you off.
An example of a random variable is a persons age. If you walk into a bar and ask everyone how old they are, you are likely to receive a wide range of answers. Those answers are random. However, we can use probabilities to describe the randomness in the answers, for instance, we know that:
No one is below 0 years of age
It is pretty safe to assume there are no neonates or children (unless it's a child friendly bar??here)
It's also unlikely that there will be many elderly people (again, depends on context)
No one is going to be 200 years old
Using all this information, we can make pretty good guesses for the ages we will hear in the responses. This is our internal probability intuition at work, and, when we add mathematical rigor to this intuition we end up with more reliable and scalable "guesses".
The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which ofttimes they are unable to account."
Interactive Example: The Gardener & the Statistician
Let's imagine there are two people, a gardener and a statistician. The gardener grows two kinds of flowers, purple and red flowers. The statistician works for a fancy research group and is trying to study different gardens to quantify the ratio of purple to red flowers.
For this we scenario we will introduce our first type of random variable: The Bernoulli. We will cover this in more detail in the next post, but for now we describe this distribution as that of a coin toss. When both outcomes are equal, i.e: a fair coin, the outcomes are equally likely to be heads or tails. However, If the coin is heavier on once side, then the outcomes will favor one outcome over the other. Here the probability of a success (heads) is called "p", and it ranges from 0 to 1, where 0.5 is a fair coin.
Back to our example and simulation. Let's describe everything we need to know/assume.
Let's say our farmer has 80 purple flower seeds
She also has 20 red flower seeds
Sometimes, out of random bad luck, some seeds do not grow into flowers
Given the assumption 1 and 2, we expect our statistician to observer 80 purple flowers and 20 red flowers. When we include assumption 3, the scenario is more realistic and now the statistician might observe different results that are random around a certain number (80 purple vs 20 red). This type of randomness that is present in the world can be described by probability and random variables. In this case we represent the probability of a given flower being purple as:
\begin{align} Bernoulli(p = 0.8) \end{align}
We can simulate what the Statistician is likely to observer with different p values below:
Bernoullis' Garden
Probability of purple is: Bernoulli ( )
Statistics Summary: There are 0 Purple flowers and 100 Red flowers. Therefore, 0% of the flowers are Purple.
Try re-running the experiment without changing the probability of purple.
Since probabilities describe random events and the likelihood of those events happening, even with a fixed probability, it is possible to observe different results. You can test this for yourself by setting the value of p to a specific value, and rerun the simulation to see what the statistician would observe in an alternate universe. You will notice that the summary statistics vary slightly every time you rerun the simuation. That is randomness, and probabilities let us tame this feature of the universe.
Randomness in Medicine
In clinical and medical research we often encounter randomness in the results of our experiments; this is clear from the small differences we observe in literature findings. For example, one study can find that 80% of patients with malaria have a headache and another can find that only 66% have a headache. Both results are correct from a statistical perspective, and the more studies we do, the more observations we make, the more likely we are to get the true probability value.
Check out the open health platform we are building!
Open the Health Algorithm Platform
More on the Bernoulli distribution & random variables
The Beta distribution & random variables
The Normal distribution & random variables
To learn more about our work, or if you are interested in working together, please reach out to us through our website, or follow us on social media!
To contact us directly visit our site: elsa.health/contact
Collaborative Healthcare with Technology
Smarter Medical Records - Hypertension
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Diophantische Approximationen | EMS Press
Moscow Lomonosov State University, Russian Federation
Peter Schlickewei
The workshop Diophantische Approximationen (Diophantine approximations), organised by Yuri V. Nesterenko (Moscow) and Hans-Peter Schlickewei (Marburg) was held April 15th - April 21st, 2007. This meeting was well attended with over 40 participants with broad geographic representation. This workshop was a nice blend of researchers with various backgrounds. All the participants were inspired by the fact that the conference immediately followed the 300 anniversary of Euler birth (15.04.1707). Loosely speaking Diophantine approximation is a branch of Number Theory that can be described as a study of the solvability of inequalities in integers, though this main theme of the subject is often unbelievably generalized. As an example, one can be interested in properties of rational points of algebraic varieties defined over an algebraic number field. The conference was concerned with a variety of problems of this kind. Below we briefly recall some of the results presented at this conference, thus outlining some modern lines of investigation in Diophantine approximation. More details can be found in the corresponding abstracts. The classical Subspace Theorem claims that all integer solutions
{\bf x}\in \mathbb{Z}^n
of a special system of linear inequalities with algebraic coefficients belong to a finite number of linear subspaces of
\mathbb{R}^n
. This theorem proved by W.Schmidt in 70-th of 20-th century is a far reaching generalization of the famous theorem of Roth about approximation of algebraic numbers by rationals. Subsequently Schmidt gave an estimate for the number of such subspaces. This result was improved and extended by H.P.~Schlickewei and J.H.~Evertse. Another approach to the proof of Schmidt's theorem was proposed by G.~Faltings and G.~W\"ustholz. In the joint talk of J.H.~Evertse and R.~Ferretti the upper bound for the number of the subspaces in question was significantly improved by combining ideas of Schmidt, Faltings and W\"ustholz. Results of this kind have many applications. For example Y.~Bugeaud in his talk announced joint with J.H.~Evertse theorem that for any real algebraic number
\xi
b>1
the number of distinct blocks of
n
letters occurring in the
b
-ary expansion of
\xi
asymptotically exceed
n(\log n)^\eta
\eta<1/14
. Another example is connected to the classical theorem of Siegel about integer points on curves of genus
g\geq 1
. In the survey talk of Yu.~Bilu another proof of this theorem based on quantitative version of Subspace Theorem was presented. This proof belongs to P.~Corvaja and U.~Zannier (2002) who applied their arguments to integral points on surfaces. Corresponding results were presented in the talk of Bilu the same as more precise statement of A.~Levin and P.~Autissier. Talks of P.~Habegger, A.~Galateau were devoted to the problem of lower bounds of heights on subvarieties of group varieties that is analogous to the classical Lehmer problem. Earlier works in this direction belong to E.~Bombieri, D.~Masser, U.~Zannier, F.~Amoroso, S.~David and P.~Philippon. P.~Mihailescu discussed in his talk so called Fermat-Catalan equation. In particular he gave some sufficient conditions on prime numbers
p, q
providing existence only trivial rational solutions for the equation
x^p+y^q=1
. The methods used by Mihailescu have a cyclotomic nature and they combine class field conditions with some new approximation techniques. The well-known Khintchine Transference Principle relates the measure of simultaneous rational approximation of the real numbers
\theta_1,\ldots ,\theta_n
with the measure of linear independence over
\bf Q
1, \theta_1,\ldots,\theta_n
. M.~Laurent introduced in his talk exponents which measure the sharping of the approximation to the point
\Theta=(\theta_1,\ldots ,\theta_n)
by rational linear varieties of dimension
d
d, 0\leq d <n,
and proved some inequalities connecting these exponents. The Khintchine's inequality follows as a special case. Another kind of transference ideas were used in the joint talk of V.~Beresnevich and S.~Vilani to state some metric Diophantine approximation results. The transference lemma in functional domain directed to applications in multiplicity estimates and algebraic independence theory was reported by P.~Philippon. The determination of the arithmetic nature of values of the Riemann zeta function
\zeta(s)
at odd values
s\in\bf Z
s>3
, is one of the most challenging problems in number theory. After Apery's celebrated proof of the irrationality of
\zeta(3)
, it took over twenty years until T.~Rivoal proved that there are infinitely many numbers among
\zeta(3), \zeta(5), \zeta(7),\ldots
that are linearly independent over
\bf Q
and W.~Zudilin stated that at least one of
\zeta(5), \zeta(7), \zeta(9), \zeta(11)
is irrational number. The difficulties are connected to constructions of good rational approximations to the corresponding values of zeta function. All known constructions have a hypergeometric nature. In a joint talk C.~Krattenthaler and T.~Rivoal gave a survey of recent constructions and explained the proof of so called Denominator conjecture that is based on some identities between a very-well-poised hypergeometric series and a multiple sum due to G.~Andrews. Some constructions of approximations to zeta-values with multiple real integrals were discussed in the talk of C.~Viola. T.~Rivoal presented a new proof of the irrationality of
\zeta(3)
that uses the expansion of the Hurwitz zeta function in interpolation series of rational functions. Such an interpolation process was first studied by Rene Lagrange in 1935. The arithmetic properties of values of the Tschakaloff function
T_q(z)
have been investigated in many works. One of the open problems is to prove linear independence of values of
T_q(z)
for rational
z
with different values of the parameter
q
. In the joint talk of W.~Zudilin and K.~V\"a\"an\"anen some results of this kind were presented. In 2005 C.~Fuchs and A.~Dujella gave a negative answer on the question of Euler about existence of four positive integers with the property that the product of any two of them plus sum of multipliers is a perfect square. In his lecture C.~Fuchs discussed analogous question for any four integer numbers. The new result of A.~Dujella, A.~Filipin and C.~Fuchs is the finiteness of the number of quadruples satisfying this condition. Moreover an effective bound for the size of the integers was proved. A.~Dujella in his talk discussed another analogous problem: to find a set of positive distinct integers
S
such that for any pair
x, y\in S
xy+1
is a square. The set
S=\{1, 3, 8, 120\}
was found by Fermat. It is proved that
\# S\leq 5
and there exists not more than finitely many sets with 5 elements. But no example has ever been found. These problems are connected to lower bounds for linear forms in logarithms of algebraic numbers. Diophantine equations of another type were discussed in the talk of M.~Bennett. B.~Adamczewski surveyed some results connected to some problem of Mahler and Mend\'es France involving tools from automata theory, combinatorics on words and Diophantine approximation. Another excellent survey of results and open questions connected to Hilbert's tenth problem about universal algorithm for solution of Diophantine equations was given by Yu.~Matiyasevich.
Yuri V. Nesterenko, Peter Schlickewei, Diophantische Approximationen. Oberwolfach Rep. 4 (2007), no. 2, pp. 1115–1190
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Affine Algebraic Geometry | EMS Press
K. Peter Russell
Affine geometry deals with algebro-geometric questions of affine varieties. In the last decades this area has developed into a systematic discipline with a sizeable international group of researchers, and with methods coming from commutative and non-commutative algebra, algebraic, complex analytic and differential geometry, singularity theory and topology. The meeting was attended by 48 participants, among them the most important senior researchers in this field and many promising young mathematicians. Especially helpful were the programs for young researchers: the NSF Oberwolfach program, the EU grant and JAMS grant. They allowed to increase the number of young participants considerably. The conference took place in a very lively atmosphere, made possible by the excellent facilities of the institute. There were 24 talks with a considerable number of lectures given by young researchers at the beginning of their careers. Moreover there were 4 invited lectures that gave an overview over some of the most vivid subfields of Affine Geometry. The program left plenty of time for cooperation and discussion among the participants. We highlight the areas in which new results were presented by the lecturers: \begin{itemize} \item Jacobian problem, especially its connections with the Dixmier conjecture (Belov-Kanel and Kontsevich) and possible algebraic approaches and reductions. \item Log algebraic varieties; in particular log algebraic surfaces. \item Automorphisms of
\mathbb A^n
, in particular tame and wild automorphisms of
\mathbb A^3
, Hilbert's 14
th
problem and locally nilpotent derivations. \item Automorphism groups of affine and non-affine varieties, especially in dimension 2 and 3. \item Cancellation problem and embedding problem. \end{itemize} In the first 4 areas there were overview talks given by D.\ Wright, M.\ Miyanishi, D.\ Daigle and Sh.\ Kaliman. Finally, in a problem session there were presented a number of open questions and problems, which are listed in at the end of this report.
Hubert Flenner, K. Peter Russell, Mikhail Zaidenberg, Affine Algebraic Geometry. Oberwolfach Rep. 4 (2007), no. 1, pp. 5–82
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Design of Flexure Pivot Tilting Pads Gas Bearings for High-speed Oil-Free Microturbomachinery | J. Tribol. | ASME Digital Collection
Sim, K., and Kim, D. (July 27, 2006). "Design of Flexure Pivot Tilting Pads Gas Bearings for High-speed Oil-Free Microturbomachinery." ASME. J. Tribol. January 2007; 129(1): 112–119. https://doi.org/10.1115/1.2372763
This paper introduces flexure pivot tilting pad gas bearings with pad radial compliance for high-speed oil-free microturbomachinery. The pad radial compliance was for accommodation of rotor centrifugal growth at high speeds. Analytical equation for the rotor centrifugal growth based on plane stress model agreed very well with finite element method results. Parametric studies on pivot offset, preload, and tilting stiffness were performed using nonlinear orbit simulations and coast-down simulations. Higher preload and pivot offset increased both critical speeds of the rotor-bearing system and onset speeds of instability due to the increased wedge effect. Pad radial stiffness and nominal bearing clearance were very important design parameters for high-speed applications due to the physically existing rotor centrifugal growth. From the series of parametric studies, the maximum achievable rotor speed was limited by the minimum clearance at the pad pivot calculated from the rotor growth and radial deflection of pads due to hydrodynamic pressure. Pad radial stiffness also affects the rotor instability significantly. Small radial stiffness could accommodate rotor growth more effectively but deteriorated rotor instability. From parametric studies on a bearing with
28.5mm
in diameter and
33.2mm
in length, optimum pad radial stiffness and bearing clearance are
1-2×107N∕m
35μm
, respectively, and the maximum achievable speed appears
180krpm
. The final design with suggested optimum design variables could be also stable under relatively large destabilizing forces.
design engineering, machine bearings, turbomachinery, micromechanical devices, finite element analysis, rotors, mechanical stability, tilting pad gas bearings, microturbomachinery, stability, shaft growth
Bearings, Clearances (Engineering), Design, Gas bearings, Rotors, Stiffness, Simulation, Bending (Stress), Shorelines
Coulomb-Friction Damping Effects in Elastically Supported Gas Foil Bearings
Advancements in the Performance of Aerodynamic Foil Journal Bearings—High-Speed and Load Capability
Mil. Eng.
, New York, Chaps. 1.5 and 3.6.
Proceedings of ASME Turbo Expo 2004, Power for Land, Sea, and Air
, Vienna, Austria, ASME Paper No. GT-53621.
On line source: http://www.kmcbearings.com/products/journal-bearings/fp-gas-brg.htmlhttp://www.kmcbearings.com/products/journal-bearings/fp-gas-brg.html.
Stability Analysis of Gas-Lubricated, Self Acting, Plain, Cylindrical, Journal Bearings of Finite Length Using Galerkin’s Method
A Study on the Characteristics of Externally Pressurized Gas Bearings
Finite Element Analysis of Gas Bearings for Oil-Free Turbomachinery
, 2004, Design and Fabrication of Sub-Millimeter Scale Gas Bearings with Tungsten-Containing Diamond Like Carbon Coatings, PhD. thesis, University of Texas at Austin, Austin, TX.
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Sociable number - Wikipedia
In mathematics, sociable numbers are numbers whose aliquot sums form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918.[1] In a sociable sequence, each number is the sum of the proper divisors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point.
If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3, and searches for them have been made up to
{\displaystyle 5\times 10^{7}}
as of 1970.[2]
It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1), or, equivalently, whether there exist numbers whose aliquot sequence never terminates, and hence grows without bound.
2 List of known sociable numbers
3 Searching for sociable numbers
4 Conjecture of the sum of sociable number cycles
As an example, the number 1,264,460 is a sociable number whose cyclic aliquot sequence has a period of 4:
The sum of the proper divisors of
{\displaystyle 1264460}
{\displaystyle =2^{2}\cdot 5\cdot 17\cdot 3719}
1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860,
{\displaystyle 1547860}
{\displaystyle =2^{2}\cdot 5\cdot 193\cdot 401}
1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636,
{\displaystyle 1727636}
{\displaystyle =2^{2}\cdot 521\cdot 829}
1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184, and
{\displaystyle 1305184}
{\displaystyle =2^{5}\cdot 40787}
List of known sociable numbersEdit
The following categorizes all known sociable numbers as of July 2018[update] by the length of the corresponding aliquot sequence:
in sequence[3]
(Perfect number)
(Amicable number)
1225736919[4] 220
6 5 21,548,919,483
It is conjectured that if n is congruent to 3 modulo 4 then there are no such sequence with length n.
The 5-cycle sequence is: 12496, 14288, 15472, 14536, 14264
The only known 28-cycle is: 14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716. (sequence A072890 in the OEIS).
These two sequences provide the only sociable numbers below 1 million (other than the perfect and amicable numbers).
Searching for sociable numbersEdit
{\displaystyle G_{n,s}}
{\displaystyle n}
{\displaystyle s(k)}
{\displaystyle k}
.[5]Cycles in
{\displaystyle G_{n,s}}
{\displaystyle [1,n]}
Conjecture of the sum of sociable number cyclesEdit
It is conjectured that as the number of sociable number cycles with length greater than 2 approaches infinity, the percentage of the sums of the sociable number cycles divisible by 10 approaches 100%. (sequence A292217 in the OEIS).
^ P. Poulet, #4865, L'Intermédiaire des Mathématiciens 25 (1918), pp. 100–101. (The full text can be found at ProofWiki: Catalan-Dickson Conjecture.)
^ Bratley, Paul; Lunnon, Fred; McKay, John (1970). "Amicable numbers and their distribution". Mathematics of Computation. 24 (110): 431–432. doi:10.1090/S0025-5718-1970-0271005-8. ISSN 0025-5718.
^ https://oeis.org/A003416 cross referenced with https://oeis.org/A052470
^ Sergei Chernykh Amicable pairs list
H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423–429
A list of known sociable numbers
Extensive tables of perfect, amicable and sociable numbers
Weisstein, Eric W. "Sociable numbers". MathWorld.
A003416 (smallest sociable number from each cycle) and A122726 (all sociable numbers) in OEIS
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sociable_number&oldid=1088258095"
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Mass - Simple English Wikipedia, the free encyclopedia
For other uses, see Mass (disambiguation).
The mass of an object is a measure of the amount of matter in a body.[1] A mountain has typically more mass than a rock, for instance. Mass should not be confused with the related but quite different concept of weight.
We can measure the mass of an object if a force acts on the object. If the mass is greater, the object will have less acceleration (change in its motion). This measure of mass is called inertial mass because it measures inertia.[2]
A large mass like the Earth will attract a small mass like a human being with enough force to keep the human being from floating away. "Mass attraction" is another word for gravity, a force that exists between all matter. When we measure the force of gravity from an object, we can find its gravitational mass. Tests of inertial and gravitational mass show that they are the same or almost the same.[2]
2 Conservation of mass and relativity
Units of mass[change | change source]
The unit of mass in the International System of Units is the kilogram, which is represented by the symbol 'kg'. Fractions and multiples of this basic unit include the gram (one thousandth of a kg, symbol 'g') and the tonne (one thousand kg), amongst many others.
In some fields or applications, it is convenient to use different units to simplify the discussions or writings. For instance,
Atomic physicists deal with the tiny masses of individual atoms and measure them in atomic mass units.
Jewelers normally work with small jewels and precious stones where masses are traditionally measured in carats, which correspond to 200 mg or 0.2 g.
The masses of stars are very large and are sometimes expressed in units of solar masses.
Traditional units are still in encountered in some countries: imperial units such as the ounce or the pound were in widespread use within the British Empire. Some of them are still popular in the United States, which also uses units like the short ton (2,000 pounds, 907 kg) and the long ton (2,240 pounds), not to be confused with the metric tonne (1,000 kg).
Conservation of mass and relativity[change | change source]
Main article: Conservation of mass
Mass is an intrinsic property of the object: it does not depend on its volume, or position in space, for instance. For a long time (at least since the works of Antoine Lavoisier in the second half of the eighteen century), it has been known that the sum of the masses of objects that interact or of the chemicals that react remain conserved throughout these processes. This remains an excellent approximation for everyday life and even most laboratory work.
However, Einstein has shown through his special theory of relativity that the mass m of an object moving at speed v with respect to an observer must be higher than the mass of the same object observed at rest m0 with respect to the observer. The applicable formula is
{\displaystyle m={\frac {m_{0}}{\sqrt {1-(v^{2}/c^{2})}}}}
where c stands for the speed of light. This change in mass is only important when the speed of the object with respect to the observer becomes a large fraction of c.
↑ Tsokos, K. A. (2005). Physics for the IB Diploma. Cambridge, United Kingdom: Cambridge University Press. p. 63. ISBN 9780521604055.
↑ 2.0 2.1 Knight, Randall Dewey (2003). Physics for scientists and engineers with modern physics : a strategic approach. San Francisco: Pearson/Addison-Wesley. p. 349. ISBN 0-321-24329-3. OCLC 54427199.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Mass&oldid=8109973"
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Mini-Workshop: Topology of closed one-forms and Cohomology Jumping Loci | EMS Press
This Mini-Workshop was organized by M. Farber (Durham), A. Suciu (Boston) and S. Yuzvinsky (Eugene). It brought together researchers working on two distinct, yet related topics: \begin{itemize} \item The {\em topology of closed one-forms} is a field of research initiated in 1981 by S.~P.~Novikov. In this version of Morse theory, one studies closed
1
-forms and their zeroes instead of smooth functions and their critical points. \item The {\em cohomology jumping loci} are the support varieties for cohomology with coefficients in rank one local systems, and the related resonance varieties. In recent years, these varieties have emerged as a central object of study in the theory of hyperplane arrangements and related spaces. \end{itemize} Even though these two fields share some common roots, so far they have developed in parallel, with not much overlap or interaction. Nevertheless, it is becoming increasingly apparent that there are deep connections between the two theories, with potentially fruitful applications going both ways: \begin{itemize} \item An example is provided by the Lusternik-Schnirelmann category, and the related notions of category weight and topological complexity of robot motion planning. Such notions are amenable to being studied via closed
1
-forms, and have applications to dynamical systems and motion planning in robotics. A good understanding of the cohomology ring and resonance varieties yields useful bounds. \item The Bieri-Neumann-Strebel invariants, which generalize the Thurston\linebreak norm from
3
-dimensional topology, are directly related to Novikov - Sikorav homology, Alexander invariants, and the resonance varieties. \item Undergirding some of this theory is a spectral sequence, introduced by Farber and Novikov in the mid 1980s. Recently, this machinery has been extended in a way that connects it to the cohomology jumping loci. \end{itemize} Given the multifaceted nature of these topics, the meeting brought together people with a variety of backgrounds, including topology, algebra, discrete geometry, geometric analysis, and singularity theory. Several participants were recent Ph.D.'s, most of them on their first visit to Oberwolfach. In all, there were 16 people attending the workshop (including the organizers), coming from the United States, Great Britain, France, Romania, Canada, and Germany. The Mini-Workshop provided a lively forum for discussing a host of questions related to the themes listed above. The day-by-day schedule was kept flexible, and was agreed upon on short notice, making it possible to shape the program on-site, and in response to the interests expressed by the participants. The borderline between problem sessions and formal lectures were often blurred. Spending a concentrated and highly intense week in a relatively small group allowed for in-depth and continuing conversations, in particular with new acquaintances. These opportunities were enhanced by the diversity of backgrounds of the participants. A basic objective of the Mini-Workshop was to bring together some of the people most actively working in two related fields, and to seek common ground for further advances and collaborations. In the ideally suited research atmosphere at Oberwolfach, participants had the opportunity to explain their respective approaches, and the variety of techniques they use. The lively atmosphere and the free-flow of ideas led to a deeper understanding of the subject, to progress in solving several open problems, and to fruitful insights on how to attack new problems.
Alexander I. Suciu, Michael Farber, Sergey Yuzvinsky, Mini-Workshop: Topology of closed one-forms and Cohomology Jumping Loci. Oberwolfach Rep. 4 (2007), no. 3, pp. 2321–2360
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Error Dialog - Maple Help
Home : Support : Online Help : Programming : Maplets : Utilities : Error Dialog
flexible error message display utility
ErrorDialog(opts)
ErrorDialog[refID](opts)
equation(s) of the form option=value; where option is one of caption, errormessage, reference, title, textboxmin, background, charperheight, maxheight, minheight, or width; specify options for the Maplet application
The ErrorDialog(opts) calling sequence displays a message dialog that is formatted for displaying large error messages. There is no return value, simply a button to dismiss the error dialog.
The error dialog is displayed in one of two forms. For small error messages a Maplet MessageDialog is used to display the error. For larger messages a read-only TextBox is used to display the error. The error dialog automatically chooses which form to use based on the length of the error message, but the criteria and formatting are adjustable through options.
The opts argument can contain one or more of the following equations that set Maplet application options. These are not in alphabetical order, as the latter options are specific to the TextBox form of the dialog.
The text that appears preceding the error message. Intended to describe when/how the error occurred. This is optional.
errormessage = string
The formatted error message to be displayed. This field is required, but the errormessage = part can be omitted leaving only the string argument. Note: StringTools:-FormatMessage must be used to convert a message with parameters into a text representation.
A reference for the ErrorDialog element.
If the reference is specified by both an index, for example, ErrorDialog[refID], and in the calling sequence, an error results.
The title that appears in the error dialog title bar. The default title is Error.
textboxmin = nonnegint
The minimum number of characters in the error message that will cause the error dialog to switch from a message dialog style to displaying the error in a separate text box. The default is 60.
The remaining options are only concerned with the text box error dialog style.
The color of the highlights of the text box in which the error is displayed. This can be a recognized color name, an RGB color structure, or a string of the form "#RRGGBB" where each pair is a two-digit hexadecimal number.
charperheight = posint
The number of characters that corresponds to a line in the text error message. Ignoring the minheight and maxheight settings below, this is used to compute the number of lines high the text box will be, based on the length of the error message. The default is 60, meaning an error with less than 60 characters represents a single line, less than 120 represents 2, and so on.
maxheight = posint
The maximum number of vertical text lines in the text box. The default is 5.
minheight = posint
The minimum number of vertical text lines in the text box. The default is 3.
The width of the text box (in characters) in which the error is displayed. The default is 60.
An error dialog with the default settings displays as a message dialog when the error is less than 60 characters long.
\mathrm{with}\left(\mathrm{Maplets}[\mathrm{Utilities}]\right):
\mathrm{ErrorDialog}\left('\mathrm{title}'="Error Example",'\mathrm{caption}'="Sample error:",'\mathrm{errormessage}'="Short test error message"\right)
Another example with a larger error message uses the text box form.
\mathrm{ErrorDialog}\left('\mathrm{title}'="Error Example",'\mathrm{caption}'="Sample error:",'\mathrm{errormessage}'="Longer test error message to force the text box form for the Error Dialog. Just a few more characters..."\right)
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Constrained Minimization Using Pattern Search, Problem-Based - MATLAB & Simulink - MathWorks Nordic
If your objective or nonlinear constraint functions are not Supported Operations for Optimization Variables and Expressions, use fcn2optimexpr to convert them to a form suitable for the problem-based approach. For example, suppose that instead of the constraint
xy\ge 10
you have the constraint
{I}_{1}\left(x\right)+{I}_{1}\left(y\right)\ge 10
{I}_{1}\left(x\right)
is the modified Bessel function besseli(1,x). (The Bessel functions are not supported functions.) Create this constraint using fcn2optimexpr as follows. First create an optimization expression for
{I}_{1}\left(x\right)+{I}_{1}\left(y\right)
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State coordinate transformation for state-space model - MATLAB ss2ss - MathWorks United Kingdom
Continuous-time or discrete-time numeric LTI models, such as ss or dss models.
Generalized or uncertain LTI models, such as genss or uss (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)
For such models, the state transformation is applied only to the state vectors of the numeric portion of the model. For more information about decomposition of these models, see getLFTModel and Internal Structure of Generalized Models.
Identified state-space idss (System Identification Toolbox) models. (Using identified models requires System Identification Toolbox™ software.)
\overline{x}=Tx
\begin{array}{l}\stackrel{˙}{x}=Ax+Bu\\ y=Cx+Du\end{array}
\begin{array}{l}\stackrel{˙}{\overline{x}}=TA{T}^{-1}\overline{x}+TBu\\ y=C{T}^{-1}\overline{x}+Du\end{array}
\begin{array}{c}E\stackrel{˙}{x}=Ax+Bu\\ y=Cx+Du\end{array}
\begin{array}{c}E{T}^{-1}\stackrel{˙}{\overline{x}}=A{T}^{-1}\overline{x}+Bu\\ y=C{T}^{-1}\overline{x}+Du\end{array}
\begin{array}{c}\frac{dx}{dt}=Ax+Bu+Ke\\ y=Cx+Du+e\end{array}
\begin{array}{l}\stackrel{˙}{\overline{x}}=TA{T}^{-}{}^{1}\overline{x}+TBu+TKe\\ y=C{T}^{-}{}^{1}\overline{x}+Du+e\end{array}
\begin{array}{c}E\stackrel{˙}{x}=Ax+Bu\\ y=Cx+Du,\end{array}
\begin{array}{c}E{T}^{-1}\stackrel{˙}{\overline{x}}=A{T}^{-1}\overline{x}+Bu\\ y=C{T}^{-1}\overline{x}+Du.\end{array}
\begin{array}{c}TE{T}^{-1}\stackrel{˙}{\overline{x}}=TA{T}^{-1}\overline{x}+TBu\\ y=C{T}^{-1}\overline{x}+Du\end{array}
balreal | canon | balance
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Positive integers divisible by the product of their nonzero digits | EMS Press
{\cal N}_0(x)
denote the set of positive integers
n\le x
which are divisible by the product of their nonzero digits. In this note, we show that if
x
x^{.495}<\#{\cal N}_0(x)<x^{.654}
Florian Luca, Jean-Marie De Koninck, Positive integers divisible by the product of their nonzero digits. Port. Math. 64 (2007), no. 1, pp. 75–85
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problemJWT_payload HASH(0x565089e510c0)
The pressure, measured in Pascals (abbreviated Pa) on a certain object B is proportional to the temperature measured in degrees Kelvin (degK), and inversely proportional to the volume of B in cubic meters: [math] where [math]
c
is a physical constant that depends on B and converts the units to Pascals. In this problem it is [math] Suppose the temperature and the volume both depend on time, hence so the does the pressure: [math]
P(t) = \frac{T(t)}{V(t)} c
. At time [math]
t=10
seconds the temperature of B is [math]
34 \; \text{degK}
and is increasing at a rate of [math]
2 \; \frac{\text{degK}}{\text{s}}
, and the volume is [math]
7\; \text{m}^3
and is increasing at the rate [math]
6\; \frac{\text{m}^3}{\text{s}}
Using the quotient rule, the pressure on the object is changing at the rate
Your answer should include units.
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x^3 - 9x^2 + 19x + 5
by the factor that you chose in the preceding problem. If it is a factor, use it and the resulting factor to find all the zeros of the polynomial. If it is not a factor, reconsider your answer to the preceding problem and try a different factor.
2 by 3 rectangle, labeled as follows: left edge, x, minus 5, interior top, left, x, cubed.
Labels added: Top edge, left, x, squared, interior bottom, left, negative 5, x, squared, interior top, middle, negative 4, x squared.
Labels added: Top edge, middle, negative 4, x, interior bottom, middle, 20, x, interior top, right, negative, x.
x^3 − 9x^2 + 19x + 5 = (x - 5)(x^2 - 4x - 1)
x − 5 = 0
x^2 - 4x − 1 = 0
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problemJWT_payload HASH(0x56508af88418)
f(x) = 4+3x-x^{3}
. Find (a) the intervals on which [math]
f
is increasing, (b) the intervals on which [math]
f
is decreasing, (c) the open intervals on which [math]
f
is concave up, (d) the open intervals on which [math]
f
is concave down, and (e) the [math]
x
-coordinates of all inflection points.
f
is increasing on the interval(s)
f
is decreasing on the interval(s)
f
is concave up on the open interval(s)
f
is concave down on the open interval(s)
(e) the [math]
x
coordinate(s) of the points of inflection are
Notes: In the first four boxes, your answer should either be a single interval, such as [0,1), a comma separated list of intervals, such as (-inf, 2), (3,4], or the word "none".
In the last box, your answer should be a comma separated list of [math]
x
values or the word "none".
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5x^2 - 7x - 6 = 0
as you answer the questions in parts (a) through (d) below.
5x^2 - 7x - 6
(5x + 3)(x - 2)
Explain the relationship between the factors of the polynomial expression and the solutions to the equation.
The solutions are the values that make the factors equal to
0
How are the solutions to the equation related to the lead coefficient and constant term in the original polynomial?
3
2
6
5
is a factor of the lead coefficient.
Watch the video below if you need help remembering how to factor quadratics. If you only need a little help, jump to 2:28 in the video. Click the following link for a full version: Factoring Quadratics Video
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This problem is a checkpoint for solving and graphing inequalities. It will be referred to as Checkpoint 8A.
Graph the inequality in part (a) and the system of inequalities in part (b).
| x + 1 |≥ 3
y ≤ -2x + 3
y ≥ x
x ≥ -1
Answers and extra practice for the Checkpoint problems are located in the back of your printed textbook or in the Reference Tab of your eBook. If you have an eBook for CCA2, login and then click the following link: Checkpoint 8A: Solving and Graphing Inequalities
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The number of clusters in Hierarchical Clustering | R-bloggers
The number of clusters in Hierarchical Clustering
Posted on January 22, 2014 by chenangen in R bloggers | 0 Comments
[This article was first published on Chen-ang Statistics » R, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Cluster analysis is widely applied in data analysis. Obviously hierarchical clustering is the simple and important method to do clustering. In brief, hierarchical clustering methods use the elements of a proximity matrix to generate a tree diagram or dendogram. From the tree diagram, we can draw our own conclusions about the results of clustering. However, when the cluster analysis solution is given, the question is how to determine the number of clusters k. For some value of k, we want to determine whether the clusters are sufficiently separated so as to illustrate minimal overlap. There is no doubt that we can choose an appropriate threshold value or use scatter diagram to determine that. Furthermore, statistic value is also very useful to determine the value of k. There are some valuable test statistics or pseudo test statistics as follow. In addition, I also provide a corresponding R function to implement.
R_k^2
R_k^2
for k clusters is defined as
R_k^2=\frac{B_k}{T}=1-\frac{P_k}{T}
T, P_k means total sum of squares, within cluster sum of squares, respectively. For n clusters, obviously each
P_k=0
R^2=1
. As the number of clusters decreases from n to 1 they should become more widely separated. A large decrease in
R_k^2
would represent a distinct join. Actually, we also can use semipartial R^2 statistic to reach our goal.
2 semipartial
R_k^2
The semipartial
R_k^2
SR_k^2=\frac{B_{KL}^2}{T}=R_{k+1}^2-R_k^2
B_{KL}^2
W_M-(W_K+W_L)
W_t
means the sum of squares in cluster
G_t
F_k
F_k
statistic for k clusters is defined as
F_k=\frac{(T-P_k)/(k-1)}{(P_k)/(n-k)}=1-\frac{B_k(n-k)}{P_k(k-1)}
If pseudo
F_k
decreases with k and reaches a maximum value, the value of k at the maximum or immediately prior to the point may be a candidate for the value of k.
t^2
t^2
t^2=\frac{B_{KL}^2}{(W_L+W_K)/(n_K+n_L-2)}
for joining cluster
G_L
G_K
n_L
n_K
As a matter of fact, SAS enables us to get the value of these statistics easily through the PROC CLUSTER and PROC TREE. However, it is not convenient to calculate them in R. Last semester, as a teaching assistant of the course of multivariate statistical analysis, the professor gave these assignments(writing R funtions to calculate one of these test statistics) to the students. In order to correct their codes, I also write a R function which can calculate all of these test statistics at the same time. The output of this function is similar with the SAS output. If your want to view the source code, please click this link.
Besides writing function, a package called NbClust offers a simper and better way to determine the number of clusters. It provides 30 popular indices and also proposes to user a recommended number of clusters. More details could be found from the reference manual of this package.
library(NbClust);
data(USArrests);
NbClust(USArrests,diss="NULL",distance="euclidean",
min.nc=2,max.nc=8,method="ward",index="pseudot2",
alphaBeale=0.1);
Please note that the output is a little different from the SAS output.
Timm, Neil H. Applied multivariate analysis. Springer, 2002.
To leave a comment for the author, please follow the link and comment on their blog: Chen-ang Statistics » R.
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§ Complex orthogonality in terms of projective geometry
If we think of complex vectors
p = [p_1, p_2]
q = [q_1, q_2]
as belonging to projective space : that is,
p \simeq p_1/p_2
q \simeq q_1 / q_2
, we can interpret orthogonality as:
\begin{aligned} p . q = 0 \\ p_1 \overline q_1 + p_2 \overline q_2 = 0 \\ p_1 / p_2 = - \overline{q_2} / \overline{q_1} \\ p = -1/\overline{q} = -q/|q| \\ \end{aligned}
If we imagine these as points on the Riemann sphere, TODO
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Shanti has enough money to buy 90 pumps worth Rs 500 each How many pumps can he buy, if - Maths - Comparing Quantities - 7056609 | Meritnation.com
Shanti has enough money to buy 90 pumps worth Rs 500 each. How many pumps can he buy, if he gets a discount of Rs 50 on each pump.
Cost of each pump = Rs.500
So cost of 90 pump =
90×500
= Rs.45000
So total money that Shanti had = Rs.45000
Discount on each pump = Rs.50
So cost of each pump after discount = Rs.500 - Rs.50 = Rs.450
So, number of pumps in Rs.45000 =
\frac{45000}{450}
Therefore Shanti can buy 10 pumps.
Devansh Jain answered this
Total money that Shanti has = 500*90 = Rs 45000
Cost of each pump after discount = 500 - 50 = Rs 450
Number of pumps that can be bought after discount = 45000/450 = 100 pumps
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The city of Waynesboro is trying to decide whether to initiate a composting project where each residence would be provided with a dumpster for garden and yard waste. The city manager needs some measure of assurance that the citizens will participate before launching the project, so he chooses a random sample of
25
homes and provides them with the new dumpster for yard and garden waste. After one week the contents of each dumpster is weighed (in pounds) before processing. The sorted data is shown below:
0
0
0
0
1.7
2.6
2.9
4.2
4.4
5.1
5.6
6.4
8.0
8.9
9.7
10.1
11.2
13.5
15.1
16.3
17.7
21.4
22.0
22.2
36.5
245.5
Create a combination boxplot and histogram. Use an interval of
0
42
pounds on the x-axis and a bin width of
6
The distribution has a right skew and an outlier at
36.5
pounds so the center is best described by the median of
8.0
pounds and the spread by the IQR of
12.95
Describe the center, shape, spread and outliers.
What is a better measure of center for this distribution the mean or median and why?
The median is better in this case because it is not affected by skewing and outliers.
What is a better measure of spread the standard deviation or IQR and why?
The IQR is better in this case because it is less affected by skewing and outliers than the standard deviation.
The city can sell the compost, and engineers estimate the program will be profitable if each home averages at least
9
pounds of material. The city manager sees the mean is nearly
10
pounds and is ready to order dumpsters for every residence. What advice would you give him?
Removing the outlier from the data drops the mean to
8.7
pounds which is below the profitable minimum. Suggest running the test a few more weeks. Perhaps as people get used to the composting program, they will participate more.
Click on the link at right for the full eTool version: CCA2 8-126 HW eTool
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维基教科书:格式手册 - 维基教科书，自由的教学读本
维基教科书:格式手册
The Wikibooks Manual of Style serves to illustrate good structural and stylistic practices to help editors produce higher quality wikibooks.
1 Wikibook titles
2.9 Print versions
Wikibook titles[编辑]
See also Wikibooks:Naming policy for information on how to name books and their chapters.
Wikibooks should be titled based on their aspect. This is a combination of the target audience, scope, and depth of the book. Several books on one topic but with differing aspects can exist (see Wikibooks:Forking policy for details). For example, a book on woodwork aimed at the general public may be entitled simply "Woodworking", and a mathematics text for commerce students may be called "Mathematics for Commerce" or "Commercial Mathematics" instead of simply "Mathematics".
Some people prefer to use title casing like books often do, while other people prefer to use sentence casing like Wikipedia does. Title casing can reduce the potential for conflict with categories. For this reason title casing is often preferred more by Wikibookians. Whichever scheme you use please be consistent and follow the existing style for books you are editing.
Structure[编辑]
See also book design for some helpful tips and ideas.
Main page[编辑]
The main page is generally the first page a new reader sees. It should give a quick overview of the scope, intended audience and layout of the book. Splash pages should be avoided. Often the layout is simply the full table of contents. Collections, printable versions and PDF files should be easily accessible from this page.
Links to the book from the card catalog office and external websites, should point to a book's main page. The subject index on the card catalog office requires the {{subject}} template to be placed on the main page. The book's main page or category should also be placed in any other categories that it belongs to. If you still require help with categorizing a book, please request assistance in the reading room.
Interlingual links should be placed on the book's main page. Books across language projects may be dissimilar even when about the same subject. Be wary about placing interlingual links on any other pages.
Table of contents[编辑]
In general, the table of contents should reside on the main page, giving readers an overview of the available material. In cases where this is impractical, a special page can be created for it. Common names for such pages are Contents, Table of contents or similar.
An introduction page is the first page where learning begins. A book's introduction page commonly delves into purpose and goals of the book; what the book aims to teach, who the audience is, what the book's scope is, what topics the book covers, history of the subject, reasons to learn the subject, any conventions used in the book, or any other information that might make sense in an introductory chapter. Common names for such pages are Introduction or About. The later is more commonly used when information about the book is split from the introduction of the subject matter.
Navigation[编辑]
Navigation aids are links included on pages to make navigating between pages easier. Navigation aids can help make reading books easier for readers, but can also make maintaining and contributing to books harder. Most web browsers can back track through pages already visited, and the wiki software also adds links for navigating back to the table of contents if pages use the slash convention in their name and a book's main page is the table of contents as suggested. Using a template on pages can help reduce some of the maintenance issues, since only one page must be edited if things change. There is no standard for navigation aids and are optional due to their potential drawbacks.
Bibliography[编辑]
A bibliography is useful for collecting resources that were cited in the book, for linking to other useful resources on the subject that go beyond the scope of the book, and for including works that can aid in verifying the factual accuracy of what is written in the book. When used, such pages are commonly named Further Reading, References, or similar.
Glossary[编辑]
A glossary lists terms in a particular domain of knowledge with the definitions for those terms. A glossary is completely optional and is most useful in books aimed at people new to a field of study. Glossary should always be used for such a page.
Appendices[编辑]
An appendix includes important information that does not quiet fit into the flow of a book. For example a list of math formulas might be used as an appendix in a math book. Appendices are optional and books may have more than one appendix. Examples of common ways to name appendices are Appendix/Keywords and Appendix:Forumlas.
Cover pages[编辑]
Cover pages are useful for print versions. These should be separated from the main page (remember: Wikibooks is not paper) but can be used to make print versions.
Print versions[编辑]
See Wikibooks:Print versions for more on print versions.
Control Systems great introduction describing (and linking to some of) the prerequisites for this book.
Spanish uses a splash page, but the table of contents is well annotated and accessible.
Haskell has a very nice layout sectioned off for audiences of different levels.
Style[编辑]
Where appropriate the first use a word or phrase akin to the title of the page should be marked in bold and when a new term is introduced, it is italicised.
Headers[编辑]
Headers should be used to divide page sections and provide content layout. Primary information should be marked with a "==", and information secondary to that should be marked with a "===" header, and so on, for example:
=== Cats ===
There is no need to provide systematised navigation between headers as the Linking section describes; only provide navigation between pages. A list of sections is automatically provided on the page when more then 2 headers are used.
Linking[编辑]
See also: Help:Links
Books with a deep sub-page hierarchy should include links on module pages to help navigate within its branch of the hierarchy. Templates can help maintain a navigation aid by including it on the necessary pages.
Footnotes and references[编辑]
See also Help:References.
Wikibooks has a couple of really simple implementations for footnotes and references. One uses {{note}} to place notes at the bottom of the page and {{ref}} to reference these at appropriate places in the text. The other places the references between <ref> and </ref> tags rigth in the text, and then uses the {{reflist}} to automatically generate a list of these at the bottom of the page.
Mathematics[编辑]
Format using HTML or in-line Mediawiki markup when using variables or other simple mathematical notation within a sentence. Use <math></math> tags for more complicated forms.
Italicise variables: a+b, etc. Do not italicise Greek letter variables, function names or their brackets.
To introduce mathematics notation in a display format, move to the start of the new line, add a single colon, and then the notation within the <math></math> tags. If punctuation follows the notation, do so inside the tags.
: <math>\int_0^\infty {1 \over {x^2 + 1}}\,dx = \frac{\pi}{2},</math>
{\displaystyle \int _{0}^{\infty }{1 \over {x^{2}+1}}\,dx={\frac {\pi }{2}},}
is correctly used for "display" guidelines.
If a notation does not render as a PNG, you may force it to do so by adding a "\," at the end of the formula.
教科书一般可以分为“主页面”、“目录”和“内容”三部分。
“目录”和“内容”必须是“主页面”的子页面；例如创建一本“示例教科书”，其目录的标题应为“示例教科书/目录”。
“主页”和“目录”可以合并。
主页用于简要介绍该教科书和其他的一些基本信息。
主页必须包含的内容有——该教科书的分类、指向目录或内容的超链接等；若教科书参考了某些资料，也应在主页中说明。
主页中可以包含的内容有——教科书的封面、此教科书的描述等。
目录用于陈列教科书内容的链接。
目录内容的第一级目录应该用: 、 、 、 、 来表示该部分内容的完成情况。
建议将教科书内容作为目录页的子页面，以方便读者返回目录。
教科书中的内容应保持格式统一。
格式的内容可以由教科书的创建者建立；若已有多人参与了该教科书的编写时，统一格式的建立应在该教科书主页面的讨论页征得共识之后。
对现有统一格式的大幅更改亦需在主页面的讨论页征得共识。
若存在统一格式，应在主页面说明或加入指向说明统一格式的页面的超链接。
若教科书的结构设计确实美观合理的，不必拘泥于以上指引。
取自“https://zh.wikibooks.org/w/index.php?title=Wikibooks:格式手册&oldid=85509”
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Paul_Sabatier_(chemist) Knowpia
Prof Paul Sabatier FRS(For)[2] HFRSE (French: [sabatje]; 5 November 1854 – 14 August 1941) was a French chemist, born in Carcassonne. In 1912, Sabatier was awarded the Nobel Prize in Chemistry along with Victor Grignard. Sabatier was honoured for his work improving the hydrogenation of organic species in the presence of metals.
Marcellin Berthelot[1]
Sabatier studied at the École Normale Supérieure, starting in 1874. Three years later, he graduated at the top of his class.[3] In 1880, he was awarded a Doctor of Science degree from the College de France.[3]
In 1883 Sabatier succeeded Édouard Filhol at the Faculty of Science, and began a long collaboration with Jean-Baptiste Senderens, so close that it was impossible to distinguish the work of either man. They jointly published 34 notes in the Accounts of the Academy of Science, 11 memoirs in the Bulletin of the French Chemical Society and 2 joint memoirs to the Annals of Chemistry and Physics.[4] The methanation reactions of COx were first discovered by Sabatier and Senderens in 1902.[5] Sabatier and Senderen shared the Academy of Science's Jecker Prize in 1905 for their discovery of the Sabatier–Senderens Process.[4]
After 1905–06 Senderens and Sabatier published few joint works, perhaps due to the classic problem of recognition of the merit of contributions to joint work.[4] Sabatier taught science classes most of his life before he became Dean of the Faculty of Science at the University of Toulouse in 1905.
The reduction of carbon dioxide using hydrogen at high temperature and pressure is another use of nickel catalyst to produce methane. This is called the Sabatier reaction and is used in the International Space Station to produce the necessary water without relying on stock from the earth.[6]
{\displaystyle {\begin{matrix}{}\\{\ce {{CO2}+4H2->[400\ ^{\circ }{\ce {C}}][{\ce {pressure}}]{CH4}+2H2O}}\\{}\end{matrix}}}
Sabatier's office desk and collection of chemicals at the University of Toulouse
Sabatier was married with four daughters, one of whom wed the Italian chemist Emilio Pomilio.[3]
The Paul Sabatier University in Toulouse is named in honour of Paul Sabatier, as is one of Carcassonne's high schools. Paul Sabatier was a co-founder of the Annales de la Faculté des Sciences de Toulouse, together with the mathematician Thomas Joannes Stieltjes.
^ Fechete, Ioana (2016). "Paul Sabatier – The father of the chemical theory of catalysis". Comptes Rendus Chimie. Elsevier BV. 19 (11–12): 1374–1381. doi:10.1016/j.crci.2016.08.006. ISSN 1631-0748.
^ Rideal, E. K. (1942). "Paul Sabatier. 1859-1941". Obituary Notices of Fellows of the Royal Society. 4 (11): 63–66. doi:10.1098/rsbm.1942.0006. S2CID 137424552.
^ a b c d "Paul Sabatier - Biography". The Nobel Foundation. Retrieved 2013-12-07.
^ a b c Alcouffe 2006, p. 10.
^ Rönsch et al. 2016.
^ Administrator, NASA Content (2015-08-17). "The Sabatier System: Producing Water on the Space Station". NASA. Retrieved 2018-09-06.
Alcouffe, Alain (December 2006), La loi de 1905 et l'université de Toulouse ou la La laïcité au bon sens du terme (in French), Iesr – Toulouse, retrieved 2017-07-26
Rönsch, Stefan; Schneider, Jens; Matthischke, Steffi; Schlüter, Michael; Götz, Manuel; Lefebvre, Jonathan; Prabhakaran, Praseeth; Bajohr, Siegfried (2016-02-15), "Review on methanation – From fundamentals to current projects", Fuel, 166: 276–296, doi:10.1016/j.fuel.2015.10.111
"Paul Sabatier (to 150th anniversary of his birthday)". Russian Journal of Applied Chemistry. 77 (11): 1909–1912. 2004. doi:10.1007/s11167-005-0190-6. S2CID 195233988.
Rideal, E. K. (1951). "Presidential address. Concepts in catalysis. The contributions of Paul Sabatier and of Max Bodenstein". Journal of the Chemical Society (Resumed): 1640–1647. doi:10.1039/JR9510001640.
Taylor, H. (1944). "Paul Sabatier 1854–1941". Journal of the American Chemical Society. 66 (10): 1615–1617. doi:10.1021/ja01238a600.
Paul Sabatier on Nobelprize.org
including the Nobel Lecture, December 11, 1912 The Method of Direct Hydrogenation by Catalysis
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Electricity and Magnetism | Shreyas’ Notes
Electricity and magnetism are related to the existence and motion of electric charges.
e = 1.6 \times {10}^{-19} C
k = 9 \times {10}^9 N/C^2
\vec F_{1 \text{ on } 2} = \frac{k q_1 q_2}{r^2} \hat{r}
\vec F_{1 \text{ on } 2} = -\vec F_{2 \text{ on } 1}
\vec F_P = \sum_i \vec F_{i \text{ on }P} = \int_{q_0}^{q_f} \frac{k q_P}{r^2} \cdot dq \hat{r}
Law of conservation of charge: the amount of charge in a region cannot change except by charges flowing across the boundary of the region.
A flow of charges is called current.
Both insulators and conductors can be charged. When charges are placed on conductors, they repel each other and spread out to minimize electrostatic potential energy.
The electric field of a point charge
q
\vec E = \frac{kq}{r^2} \hat{r}
\hat{r}
points from the point charge to the point of interest, and is placed at the point of interest. Electric fields exist regardless of the presence of a second charge.
F_{1 \text{ on } 2} = q_2 \vec E_{1 \text{ at } q_2}
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Quantization - Neural Network Distiller
Compression Scheduling
Preparing a Model for Quantization
Compressing Models
Quantization Algorithms
Range-Based Linear Quantization
Pruning Filters and Channels
Pruning a Language Model
Quantizing a Language Model
Quantizing GNMT
For any of the methods below that require quantization-aware training, please see here for details on how to invoke it using Distiller's scheduling mechanism.
Let's break down the terminology we use here:
Linear: Means a float value is quantized by multiplying with a numeric constant (the scale factor).
Range-Based: Means that in order to calculate the scale factor, we look at the actual range of the tensor's values. In the most naive implementation, we use the actual min/max values of the tensor. Alternatively, we use some derivation based on the tensor's range / distribution to come up with a narrower min/max range, in order to remove possible outliers. This is in contrast to the other methods described here, which we could call clipping-based, as they impose an explicit clipping function on the tensors (using either a hard-coded value or a learned value).
In this method we can use two modes - asymmetric and symmetric.
In asymmetric mode, we map the min/max in the float range to the min/max of the integer range. This is done by using a zero-point (also called quantization bias, or offset) in addition to the scale factor.
Let us denote the original floating-point tensor by
x_f
, the quantized tensor by
x_q
, the scale factor by
q_x
, the zero-point by
zp_x
and the number of bits used for quantization by
n
. Then, we get:
In practice, we actually use
zp_x = round(min_{x_f}q_x)
. This means that zero is exactly representable by an integer in the quantized range. This is important, for example, for layers that have zero-padding. By rounding the zero-point, we effectively "nudge" the min/max values in the float range a little bit, in order to gain this exact quantization of zero.
Note that in the derivation above we use unsigned integer to represent the quantized range. That is,
x_q \in [0, 2^n-1]
. One could use signed integer if necessary (perhaps due to HW considerations). This can be achieved by subtracting
2^{n-1}
Let's see how a convolution or fully-connected (FC) layer is quantized in asymmetric mode: (we denote input, output, weights and bias with
x, y, w
b
respectively)
We can see that the bias has to be re-scaled to match the scale of the summation.
In a proper integer-only HW pipeline, we would like our main accumulation term to simply be
\sum{x_q w_q}
. In order to achieve this, one needs to further develop the expression we derived above. For further details please refer to the gemmlowp documentation
In symmetric mode, instead of mapping the exact min/max of the float range to the quantized range, we choose the maximum absolute value between min/max. In addition, we don't use a zero-point. So, the floating-point range we're effectively quantizing is symmetric with respect to zero, and so is the quantized range.
There's a nuance in the symmetric case with regards to the quantized range. Assuming
N_{bins}=2^n-1
, we can use either a "full" or "restricted" quantized range:
Quantized Range
\left[-\frac{N_{bins}}{2}, \frac{N_{bins}}{2} - 1\right]
\left[-\left(\frac{N_{bins}}{2} - 1\right), \frac{N_{bins}}{2} - 1\right]
8-bit example
[-128, 127]
(As shown in image above)
[-127,127]
q_x = \frac{(2^n-1)/2}{\max(abs(x_f))}
q_x = \frac{2^{n-1}-1}{\max(abs(x_f))}
The restricted range is less accurate on-paper, and is usually used when specific HW considerations require it. Implementations of quantization "in the wild" that use a full range include PyTorch's native quantization (from v1.3 onwards) and ONNX. Implementations that use a restricted range include TensorFlow, NVIDIA TensorRT and Intel DNNL (aka MKL-DNN). Distiller can emulate both modes.
Using the same notations as above, we get (regardless of full/restricted range):
Again, let's see how a convolution or fully-connected (FC) layer is quantized, this time in symmetric mode:
Comparing the Two Modes
The main trade-off between these two modes is simplicity vs. utilization of the quantized range.
When using asymmetric quantization, the quantized range is fully utilized. That is because we exactly map the min/max values from the float range to the min/max of the quantized range. Using symmetric mode, if the float range is biased towards one side, could result in a quantized range where significant dynamic range is dedicated to values that we'll never see. The most extreme example of this is after ReLU, where the entire tensor is positive. Quantizing it in symmetric mode means we're effectively losing 1 bit.
On the other hand, if we look at the derviations for convolution / FC layers above, we can see that the actual implementation of symmetric mode is much simpler. In asymmetric mode, the zero-points require additional logic in HW. The cost of this extra logic in terms of latency and/or power and/or area will of course depend on the exact implementation.
Scale factor scope: For weight tensors, Distiller supports per-channel quantization (per output channel).
Removing outliers (post-training only): As discussed here, in some cases the float range of activations contains outliers. Spending dynamic range on these outliers hurts our ability to represent the values we actually care about accurately.
Currently, Distiller supports clipping of activations during post-training quantization using the following methods:
Averaging: Global min/max values are replaced with an average of the min/max values of each sample in the batch.
Mean +/- N*Std: Take N standard deviations for the tensor's mean, and in any case don't exceed the tensor's actual min/max. N is user configurable.
ACIQ - Analytical calculation of clipping values assuming either a Gaussian or Laplace distribution. As proposed in Post training 4-bit quantization of convolutional networks for rapid-deployment.
Scale factor approximation (post-training only): This can be enabled optionally, to simulate an execution pipeline with no floating-point operations. Instead of multiplying with a floating-point scale factor, we multiply with an integer and then do a bit-wise shift:
Q \approx {A}/{2^n}
Q
denotes the FP32 scale factor,
A
denotes the integer multiplier and
n
denotes the number of bits by which we shift after multiplication. The number of bits assigned to
A
is usually a parameter of the HW, and in Distiller it is configured by the user. Let us denote that with
m
Q
m
A
n
Implementation in Distiller
For post-training quantization, this method is implemented by wrapping existing modules with quantization and de-quantization operations. The wrapper implementations are in range_linear.py.
The following operations have dedicated implementations which consider quantization:
torch.nn.Conv2d/Conv3d
distiller.modules.Concat
distiller.modules.EltwiseAdd
distiller.modules.EltwiseMult
distiller.modules.Matmul
distiller.modules.BatchMatmul
Any existing module will likely need to be modified to use the distiller.modules.* modules. See here for details on how to prepare a model for quantization.
To automatically transform an existing model to a quantized model using this method, use the PostTrainLinearQuantizer class. For details on ways to invoke the quantizer see here.
When using PostTrainLinearQuantizer, by default, any operation not in the list above is "fake"-quantized, meaning it is executed in FP32 and its output is quantized. Quantization for specific layers (or groups of layers) can be disabled using Distiller's override mechanism (see example here).
For weights and bias the scale factor and zero-point are determined once at quantization setup ("offline" / "static"). For activations, both "static" and "dynamic" quantization is supported. Static quantization of activations requires that statistics be collected beforehand. See details on how to do that here.
The calculated quantization parameters are stored as buffers within the module, so they are automatically serialized when the model checkpoint is saved.
To apply range-based linear quantization in training, use the QuantAwareTrainRangeLinearQuantizer class. As it is now, it will apply weights quantization to convolution, FC and embedding modules. For activations quantization, it will insert instances FakeLinearQuantization module after ReLUs. This module follows the methodology described in Benoit et al., 2018 and uses exponential moving averages to track activation ranges.
Note that the current implementation of QuantAwareTrainRangeLinearQuantizer supports training with single GPU only.
Similarly to post-training, the calculated quantization parameters (scale factors, zero-points, tracked activation ranges) are stored as buffers within their respective modules, so they're saved when a checkpoint is created.
Note that converting from a quantization-aware training model to a post-training quantization model is not yet supported. Such a conversion will use the activation ranges tracked during training, so additional offline or online calculation of quantization parameters will not be required.
(As proposed in DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients)
In this method, we first define the quantization function
quantize_k
, which takes a real value
a_f \in [0, 1]
and outputs a discrete-valued
a_q \in \left\{ \frac{0}{2^k-1}, \frac{1}{2^k-1}, ... , \frac{2^k-1}{2^k-1} \right\}
k
is the number of bits used for quantization.
Activations are clipped to the
[0, 1]
range and then quantized as follows:
For weights, we define the following function
f
, which takes an unbounded real valued input and outputs a real value in
[0, 1]
Now we can use
quantize_k
to get quantized weight values, as follows:
This method requires training the model with quantization-aware training, as discussed here. Use the DorefaQuantizer class to transform an existing model to a model suitable for training with quantization using DoReFa.
Gradients quantization as proposed in the paper is not supported yet.
The paper defines special handling for binary weights which isn't supported in Distiller yet.
(As proposed in PACT: Parameterized Clipping Activation for Quantized Neural Networks)
This method is similar to DoReFa, but the upper clipping values,
\alpha
, of the activation functions are learned parameters instead of hard coded to 1. Note that per the paper's recommendation,
\alpha
is shared per layer.
This method requires training the model with quantization-aware training, as discussed here. Use the PACTQuantizer class to transform an existing model to a model suitable for training with quantization using PACT.
(As proposed in WRPN: Wide Reduced-Precision Networks)
In this method, activations are clipped to
[0, 1]
and quantized as follows (
k
is the number of bits used for quantization):
Weights are clipped to
[-1, 1]
and quantized as follows:
k-1
bits are used to quantize weights, leaving one bit for sign.
This method requires training the model with quantization-aware training, as discussed here. Use the WRPNQuantizer class to transform an existing model to a model suitable for training with quantization using WRPN.
The paper proposed widening of layers as a means to reduce accuracy loss. This isn't implemented as part of WRPNQuantizer at the moment. To experiment with this, modify your model implementation to have wider layers.
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A mixer has a marked price of Rs 2575 There are two options for the customer: a - Maths - Comparing Quantities - 12997469 | Meritnation.com
A mixer has a marked price of Rs. 2575. There are two options for the customer:
a. SP is Rs. 2575 along with an attractive gift worth Rs. 300.
b. Discount of 5% but no free gifts.
Given: MP=Rs 2575\phantom{\rule{0ex}{0ex}}\left(a\right) SP=2575\phantom{\rule{0ex}{0ex}}Gifts worth =Rs 300\phantom{\rule{0ex}{0ex}}So , Final price=2575-300\phantom{\rule{0ex}{0ex}} =2275 Rs\phantom{\rule{0ex}{0ex}}\left(b\right) 5% discount on Rs 2575\phantom{\rule{0ex}{0ex}}So , \phantom{\rule{0ex}{0ex}}Final price=2575-\left(2575×5%\right)\phantom{\rule{0ex}{0ex}} =2575-128.75\phantom{\rule{0ex}{0ex}} =2446.25 Rs\phantom{\rule{0ex}{0ex}}\mathbit{T}\mathbit{h}\mathbit{e}\mathbit{r}\mathbit{e}\mathbit{f}\mathbit{o}\mathbit{r}\mathbit{e}\mathbf{ }\mathbit{o}\mathbit{f}\mathbit{f}\mathbit{e}\mathbit{r}\mathbf{ }\left(\mathbf{a}\right)\mathbf{ }\mathbit{i}\mathbit{s}\mathbf{ }\mathbit{b}\mathbit{e}\mathbit{t}\mathbit{t}\mathbit{e}\mathbit{r}\mathbf{ }\mathbit{t}\mathbit{h}\mathbit{e}\mathbit{n}\mathbf{ }\mathbit{o}\mathbit{f}\mathbit{f}\mathbit{e}\mathbit{r}\mathbf{ }\left(\mathbf{b}\right)\phantom{\rule{0ex}{0ex}}
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Shoshichi_Kobayashi Knowpia
Shoshichi Kobayashi (小林 昭七, Kobayashi Shōshichi, born on January 4, 1932, in Kōfu, Japan, died on 29 August 2012)[1] was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi.[2] His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie algebras.
Shōshichi Kobayashi in Berkeley
Geometry prize (1987)
Kobayashi graduated from the University of Tokyo in 1953. In 1956, he earned a Ph.D. from the University of Washington under Carl B. Allendoerfer. His dissertation was Theory of Connections.[3] He then spent two years at the Institute for Advanced Study and two years at MIT. He joined the faculty of the University of California, Berkeley in 1962 as an assistant professor, was awarded tenure the following year, and was promoted to full professor in 1966.
Kobayashi served as chairman of the Berkeley Mathematics Department for a three-year term from 1978 to 1981 and for the 1992 Fall semester. He chose early retirement under the VERIP plan in 1994.
The two-volume book Foundations of Differential Geometry, which he coauthored with Katsumi Nomizu, has been known for its wide influence. In 1970 he was an invited speaker for the section on geometry and topology at the International Congress of Mathematicians in Nice.
Technical contributionsEdit
Kobayashi's earliest work dealt with the geometry of connections on principal bundles. Many of these results, along with others, were later absorbed into Foundations of Differential Geometry.
As a consequence of the Gauss–Codazzi equations and the commutation formulas for covariant derivatives, James Simons discovered a formula for the Laplacian of the second fundamental form of a submanifold of a Riemannian manifold.[4] As a consequence, one can find a formula for the Laplacian of the norm-squared of the second fundamental form. This "Simons formula" simplifies significantly when the mean curvature of the submanifold is zero and when the Riemannian manifold has constant curvature. In this setting, Shiing-Shen Chern, Manfredo do Carmo, and Kobayashi studied the algebraic structure of the zeroth-order terms, showing that they are nonnegative provided that the norm of the second fundamental form is sufficiently small.
As a consequence, the case in which the norm of the second fundamental form is constantly equal to the threshold value can be completely analyzed, the key being that all of the matrix inequalities used in controlling the zeroth-order terms become equalities. As such, in this setting, the second fundamental form is uniquely determined. As submanifolds of space forms are locally characterized by their first and second fundamental forms, this results in a complete characterization of minimal submanifolds of the round sphere whose second fundamental form is constant and equal to the threshold value. Chern, do Carmo, and Kobayashi's result was later improved by An-Min Li and Jimin Li, making use of the same methods.[5]
On a Kähler manifold, it is natural to consider the restriction of the sectional curvature to the two-dimensional planes which are holomorphic, i.e. which are invariant under the almost-complex structure. This is called the holomorphic sectional curvature. Samuel Goldberg and Kobayashi introduced an extension of this quantity, called the holomorphic bisectional curvature; its input is a pair of holomorphic two-dimensional planes. Goldberg and Kobayashi established the differential-geometric foundations of this object, carrying out many analogies with the sectional curvature. In particular they established, by the Bochner technique, that the second Betti number of a connected closed manifold must equal one if there is a Kähler metric whose holomorphic bisectional curvature is positive. Later, Kobayashi and Takushiro Ochiai proved some rigidity theorems for Kähler manifolds. In particular, if M is a closed Kähler manifold and there exists α in H1, 1(M, ℤ) such that
{\displaystyle c_{1}(M)\geq (n+1)\alpha ,}
then M must be biholomorphic to complex projective space. This, in combination with the Goldberg–Kobayashi result, forms the final part of Yum-Tong Siu and Shing-Tung Yau's proof of the Frankel conjecture.[6] Kobayashi and Ochiai also characterized the situation of c1(M) = nα as M being biholomorphic to a quadratic hypersurface of complex projective space.
Kobayashi is also notable for having proved that a hermitian–Einstein metric on a holomorphic vector bundle over a compact Kähler manifold has deep algebro-geometric implications, as it implies semistability and decomposability as a direct sum of stable bundles.[7] This establishes one direction of the Kobayashi–Hitchin correspondence. Karen Uhlenbeck and Yau proved the converse result, following well-known partial results by Simon Donaldson.
In the 1960s, Kobayashi introduced what is now known as the Kobayashi metric. This associates a pseudo-metric to any complex manifold, in a holomorphically invariant way.[8] This sets up the important notion of Kobayashi hyperbolicity, which is defined by the condition that the Kobayashi metric is a genuine metric (and not only a pseudo-metric). With these notions, Kobayashi was able to establish a higher-dimensional version of the Alhfors–Schwarz lemma from complex analysis.
Kobayashi, Shoshichi (1959). "Geometry of bounded domains". Transactions of the American Mathematical Society. 92 (2): 267–290. doi:10.1090/S0002-9947-1959-0112162-5. MR 0112162. Zbl 0136.07102.
Goldberg, Samuel I.; Kobayashi, Shoshichi (1967). "Holomorphic bisectional curvature". Journal of Differential Geometry. 1 (3–4): 225–233. doi:10.4310/jdg/1214428090. MR 0227901. Zbl 0169.53202.
Chern, S. S.; do Carmo, M.; Kobayashi, S. (1970). "Functional Analysis and Related Fields". In Browder, Felix E. (ed.). Functional Analysis and Related Fields. Conference in honor of Professor Marshall Stone, held at the University of Chicago (May 1968). New York: Springer. pp. 59–75. doi:10.1007/978-3-642-48272-4_2. ISBN 978-3-642-48274-8. MR 0273546. Zbl 0216.44001.
Kobayashi, Shoshichi; Ochiai, Takushiro (1973). "Characterizations of complex projective spaces and hyperquadrics". Journal of Mathematics of Kyoto University. 13 (1): 31–47. doi:10.1215/kjm/1250523432. MR 0316745. Zbl 0261.32013.
Kobayashi, Shoshichi (1976). "Intrinsic distances, measures and geometric function theory". Bulletin of the American Mathematical Society. 82 (3): 357–416. doi:10.1090/S0002-9904-1976-14018-9. MR 0414940. Zbl 0346.32031.
Kobayashi, Shoshichi; Nomizu, Katsumi (1963). Foundations of differential geometry. Vol I. Interscience Tracts in Pure and Applied Mathematics. Vol. 15. Reprinted in 1996. New York–London: John Wiley & Sons, Inc. ISBN 0-471-15733-3. MR 0152974. Zbl 0119.37502. [9]
Kobayashi, Shoshichi; Nomizu, Katsumi (1969). Foundations of differential geometry. Vol II. Interscience Tracts in Pure and Applied Mathematics. Vol. 15. Reprinted in 1996. New York–London: John Wiley & Sons, Inc. ISBN 0-471-15732-5. MR 0238225. Zbl 0175.48504.
Kobayashi, Shoshichi (1972). Transformation groups in differential geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 70. New York–Heidelberg: Springer-Verlag. ISBN 0-387-05848-6. MR 0355886. Zbl 0246.53031.
Kobayashi, Shoshichi (1984). An introduction to the theory of connections. Seminar on Mathematical Sciences. Vol. 8. Notes by Kotaro Yamada. Yokohama: Keio University, Department of Mathematics. MR 0856760. Zbl 0547.53018.
Kobayashi, Shoshichi (1987). Differential geometry of complex vector bundles. Publications of the Mathematical Society of Japan. Vol. 15. Reprinted in 2014. Princeton, NJ: Princeton University Press. ISBN 0-691-08467-X. MR 0909698. Zbl 0708.53002. [10]
Kobayashi, Shoshichi (1998). Hyperbolic complex spaces. Grundlehren der mathematischen Wissenschaften. Vol. 318. Berlin: Springer-Verlag. ISBN 3-540-63534-3. MR 1635983. Zbl 0917.32019.
Kobayashi, Shoshichi (2005). Hyperbolic manifolds and holomorphic mappings. An introduction (Second edition of 1970 original ed.). Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd. ISBN 981-256-496-9. MR 2194466. Zbl 1084.32018. [11]
Kobayashi, Shoshichi (2019). Differential geometry of curves and surfaces. Springer Undergraduate Mathematics Series. Translated by Nagmo, Eriko Shinozaki; Tanaka, Makiki Sumi (Revised edition of 1977 original ed.). Singapore: Springer. doi:10.1007/978-981-15-1739-6. ISBN 978-981-15-1738-9. S2CID 210246198. Zbl 1437.53001.
Kobayashi was also the author of several textbooks which (as of 2022) have only been published in Japanese.[12]
^ ＵＣバークリー校名誉教授・小林昭七さん死去 (in Japanese). Asahi Shimbun. 2012-09-06. Retrieved 2012-09-16.
^ Jensen, Gary R (2014). "Remembering Shoshichi Kobayashi". Notices of the American Mathematical Society. 61 (11): 1322–1332. doi:10.1090/noti1184.
^ S. Kobayashi (1957). "Theory of Connections". Annali di Matematica Pura ed Applicata. 43: 119–194. doi:10.1007/bf02411907. S2CID 120972987.
^ James Simons. Minimal varieties in Riemannian manifolds. Ann. of Math. (2) 88 (1968), 62–105.
^ Li An-Min and Li Jimin. An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. Math. (Basel) 58 (1992), no. 6, 582–594.
^ Yum Tong Siu and Shing Tung Yau. Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59 (1980), no. 2, 189–204.
^ Kobayashi 1987, Theorem 5.8.3.
^ Kobayashi 2005.
^ Hermann, Robert (1964). "Review: Foundations of differential geometry by Shoshichi Kobayashi and Katsumi Nomizu". Bulletin of the American Mathematical Society. 70 (2): 232–235. doi:10.1090/S0002-9904-1964-11094-6. MR 1566282.
^ Okonek, Christian (1988). "Review: Differential geometry of complex vector bundles, by S. Kobayashi". Bulletin of the American Mathematical Society. 19 (2): 528–530. doi:10.1090/s0273-0979-1988-15731-x.
^ Griffiths, P. (1972). "Review: Hyperbolic manifolds and holomorphic mappings, by S. Kobayashi". Bulletin of the American Mathematical Society. 78 (4): 487–490. doi:10.1090/s0002-9904-1972-12966-5.
^ Books authored by Shoshichi Kobayashi
特集◎小林昭七 [Feature: Kobayashi Shoshichi]. 数学セミナー (in Japanese). February 2013. Archived from the original on 2013-01-26.
Shoshichi Kobayashi – In Memoriam
Publications of Shoshichi Kobayashi
Shoshichi Kobayashi Department of Mathematics UC Berkeley
Shoshichi Kobayashi at the Mathematics Genealogy Project
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Earlier in the year I switching from being a long time Comcast account holder to CenturyLink. They brought fiber to the home (FTTH) and offered up near-symmetrical gig internet for just
65 month including taxes and fees! Compared to the
100+ I paid Comcast for 50 Mbps it was a no brainer. The experience has been good so far but I have been held back by their included 3000A modem. There was nothing majorly wrong with it it just didn't have the best range, nor enough ethernet ports for me plus I already had a Nighthawk X8 R8500 sitting unused so I decided to figure how to remove the 3000A. It turned out to be a pain in the ass so I thought I woud share it with you so save you some pain.
Before you read further make sure you are on a CenturyLink fiber plan and your router can set the VLAN. If your router can't do this then don't even bother as this guide won't be for you.
With that out of the way. You may want to take a screenshot or download this page because you will be without internet for a few minutes.
Call CenturyLink and get your PPP username and password. You may think its the password on your paperwork you got when you got your internet installed. This is not the case! You need to call to get it. While you are on the line make sure you confirm your VLAN. It likely will be 201, but yours may be different.
Step 2 (this is where you lose internet)
Unplug your CenturyLink provided router and add your new one using the same cable/ports as the old one. Turn off the ONT (where the fiber gets converted to ethernet) wait a minute or so and turn on the ONT and the new router.
Setup the router as you would so you can access it. I will use Netgear here as an example
Navigate to http://192.168.1.1/start.htm and enter the default user/pass: admin/password. Change it if you would like
Navigate to Advanced > Setup > Internet Setup and make sure the following information is present:
ISP: PPPoE Login: The username you got from CL Password: The password you got from CL Connection Mode: Dial on Demand Idle Timeout: 5
Navigate to Advanced > Advanced Setup > VLAN/IPTV Setup and duplicate the setup in my screenshot below. If you have Prism TV this will be different.
Enable VLAN/IPTV Setup Check VLAN tag group VLAN ID: Whatever CL gave you, likely 201 Priority: 0 Wireless and Wired: All
← The Easiest Way to Backup your Raspberry Pi SD card on OS X
2 ISPs 1 router - Ensuring your home is always connected. →
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Scaling and wavelet filter - MATLAB qmf - MathWorks Deutschland
Create Quadrature Mirror Filter
Controlling Phase of a Quadrature Mirror Filter
Scaling and wavelet filter
Y = qmf(X)
Y = qmf(X,P)
Y = qmf(X) changes the signs of the even-indexed elements of the reversed vector filter coefficients X.
Y = qmf(X,P) changes the signs of the even-indexed elements of the reversed vector filter coefficients X if P is 0. If P is 1, the signs of the odd-indexed elements are reversed. Changing P changes the phase of the Fourier transform of the resulting wavelet filter by π radians.
This example shows how to create a quadrature mirror filter associated with the db10 wavelet.
Obtain the scaling filter associated with the db10 wavelet.
sF = dbwavf("db10");
dbwavf normalizes the filter coefficients so that the norm is equal to
1/\sqrt{2}
. Normalize the coefficients so that the filter has norm equal to 1.
G = sqrt(2)*sF;
Obtain the wavelet filter coefficients by using qmf. Plot the filters.
H = qmf(G);
stem(G)
title("Scaling (Lowpass) Filter G")
title("Wavelet (Highpass) Filter H")
Save the current extension mode. Set the extension mode to Periodization. Generate a random signal of length 64. Perform a single-level wavelet decomposition of the signal using G and H. For purposes of reproducibility, set the random seed to the default value.
origmode = dwtmode("status","nodisplay");
dwtmode("per","nodisplay")
sig = randn(1,n);
[a,d] = dwt(sig,G,H);
The lengths of the approximation and detail coefficients are both 32. Confirm that the filters preserve energy.
[sum(sig.^2) sum(a.^2)+sum(d.^2)]
Compute the frequency responses of G and H. Zeropad the filters when taking the Fourier transform.
F = 0:1/n:1-1/n;
Plot the magnitude of each frequency response.
plot(F(1:n/2+1),abs(Gdft(1:n/2+1)),"r")
plot(F(1:n/2+1),abs(Hdft(1:n/2+1)),"b")
xlabel("Normalized Frequency")
legend("Lowpass Filter","Highpass Filter","Location","east")
Confirm the sum of the squared magnitudes of the frequency responses of G and H at each frequency is equal to 2.
sumMagnitudes = abs(Gdft).^2+abs(Hdft).^2;
[min(sumMagnitudes) max(sumMagnitudes)]
Confirm that the filters are orthonormal.
df = [G;H];
id = df*df'
id = 2×2
dwtmode(origmode,"nodisplay")
This example shows the effect of setting the phase parameter of the qmf function.
Obtain the decomposition lowpass filter associated with a Daubechies wavelet.
lowfilt = wfilters("db4");
Use the qmf function to obtain the decomposition lowpass filter for a wavelet. Then, compare the signs of the values when the qmf phase parameter is set to 0 or 1. The reversed signs indicates a phase shift of
\pi
radians, which is the same as multiplying the DFT by
{e}^{i\pi }
p0 = qmf(lowfilt,0)
0.2304 -0.7148 0.6309 0.0280 -0.1870 -0.0308 0.0329 0.0106
-0.2304 0.7148 -0.6309 -0.0280 0.1870 0.0308 -0.0329 -0.0106
Compute the magnitudes and display the difference between them. Unlike the phase, the magnitude is not affected by the sign reversals.
abs(p0)-abs(p1)
X — Filter coefficients
Filter coefficients, specified as a vector.
P — Phase parameter
Phase parameter, specified as follows.
0 — Change signs of even-indexed elements of the reversed vector X
1 — Change signs of odd-indexed elements of the reversed vector X
Let x be a finite energy signal. Two filters F0 and F1 are quadrature mirror filters (QMF) if, for any x,
{‖{y}_{0}‖}^{2}+{‖{y}_{1}‖}^{2}={‖x‖}^{2}
where y0 is a decimated version of the signal x filtered with F0, so y0 is defined by x0 = F0(x) and y0(n) = x0(2n). Similarly, y1 is defined by x1 = F1(x) and y1(n) = x1(2n). This property ensures a perfect reconstruction of the associated two-channel filter banks scheme (see [1] p. 103).
For example, if F0 is a Daubechies scaling filter with norm equal to 1 and F1 = qmf(F0), then the transfer functions F0(z) and F1(z) of the filters F0 and F1 satisfy the condition:
|{F}_{0}\left(z\right){|}^{2}+|{F}_{1}\left(z\right){|}^{2}=2.
wfilters | dwtfilterbank
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Carlos is always playing games with his graphing calculator, but now his calculator has contracted a virus. The
\boxed{\text{TRACE}}
\boxed{\text{ZOOM}}
\boxed{\text{WINDOW}}
functions on his calculator are not working. He needs to solve
x^3 + 5x^2 - 16x - 14 = 0
, so he graphs
y = x^3 + 5x^2 - 16x - 14
and sees the graph at right in the standard window.
From the graph, what appears to be an integer solution to the equation?
x = -7
Check your answers from part (a) in the equation.
Substitute your answer from part (a) into the original equation.
x=-7
is a solution to the question, what is the factor associated with this solution?
(x + 7)
Use polynomial division to find the other factor.
x^³ + 5x^² − 16x − 14
x + 7
using a generic rectangle. 2 by 3 rectangle, labeled as follows: left edge, x, & 7, top edge left, x squared, interior top left, x cubed.
Labels added to the rectangle, interior top middle, negative 2, x squared, interior bottom left, 7, x squared.
Labels added to rectangle, top edge, middle, negative 2, x, interior, bottom middle, negative 14, x, interior top left, negative 2, x.
Labels added to rectangle, top edge, right, negative 2, interior bottom right, negative 14.
x^2 - 2x - 2
Use your new factor to complete this equation:
x^3 + 5x^2 - 16x - 14 =
x+7
)(other factor)
=0
The “other factor” leads to two other solutions to the equation. Find these two new solutions and give all three solutions to the original equation.
Use the Quadratic Formula with the answer found in part (d) that you used for part (e).
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The ratio of Neha's age to that of Jane is 4:7 and the ratio of Jane's age to that of - Maths - Comparing Quantities - 6877702 | Meritnation.com
The ratio of Neha's age to that of Jane is 4:7 and the ratio of Jane's age to that of Smita is 4:3. If Smita is 21 years old, how old is Neha?
Ratio of Neha's age to that of Jane is 4:7. This means;
\frac{\mathrm{Neha}\text{'}\mathrm{s}\quad \mathrm{age}}{\mathrm{Jane}\text{'}\mathrm{s}\quad \mathrm{age}}=\frac{4}{7}
So, Neha's age =
\frac{4}{7}×
Jane's age.........(i)
Now, Ratio of Jane's age to that of Smita is 4:3. This means;
\frac{\mathrm{Jane}\text{'}\mathrm{s}\quad \mathrm{age}}{\mathrm{Smita}\text{'}\mathrm{s}\quad \mathrm{age}}=\frac{4}{3}
So, Jane's age =
\frac{4}{3}×
Smita's age =
\frac{4}{3}\times 21\quad =\quad 28
Putting Jane's age in (i) we get;
Neha's age =
\frac{4}{7}\times 28\quad =\quad 16
Therefore Neha is 16 years old.
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ISO metric screw thread (44909 views - Mechanical Engineering)
The ISO metric screw threads are the world-wide most commonly used type of general-purpose screw thread. They were one of the first international standards agreed when the International Organization for Standardization was set up in 1947. The "M" designation for metric screws indicates the nominal outer diameter of the screw, in millimeters (e.g., an M6 screw has a nominal outer diameter of 6 millimeters).
3D CAD Models - ISO metric screw thread
The ISO metric screw threads are the world-wide most commonly used type of general-purpose screw thread.[1] They were one of the first international standards agreed when the International Organization for Standardization was set up in 1947.[citation needed]
The "M" designation for metric screws indicates the nominal outer diameter of the screw, in millimeters (e.g., an M6 screw has a nominal outer diameter of 6 millimeters).
The design principles of ISO general-purpose metric screw threads ("M" series threads) are defined in international standard ISO 68-1.[2] Each thread is characterized by its major diameter, D (Dmaj in the diagram), and its pitch, P. ISO metric threads consist of a symmetric V-shaped thread. In the plane of the thread axis, the flanks of the V have an angle of 60° to each other. The thread depth is 0.614 × pitch. The outermost 1⁄8 and the innermost 1⁄4 of the height H of the V-shape are cut off from the profile.
{\displaystyle \theta }
{\displaystyle H={\frac {1}{2\tan \theta }}\cdot P={\frac {\sqrt {3}}{2}}\cdot P\approx 0.866\cdot P}
{\displaystyle P=2\tan \theta \cdot H={\frac {2}{\sqrt {3}}}\cdot H\approx 1.155\cdot H}
{\displaystyle {\begin{aligned}D_{\text{min}}&=D_{\text{maj}}-2\cdot {\frac {5}{8}}\cdot H=D_{\text{maj}}-{\frac {5{\sqrt {3}}}{8}}\cdot P\approx D_{\text{maj}}-1.082532\cdot P\\D_{\text{p}}&=D_{\text{maj}}-2\cdot {\frac {3}{8}}\cdot H=D_{\text{maj}}-{\frac {3{\sqrt {3}}}{8}}\cdot P\approx D_{\text{maj}}-0.649519\cdot P\end{aligned}}}
A metric ISO screw thread is designated by the letter M followed by the value of the nominal diameter D (Dmaj in the diagram above) and the pitch P, both expressed in millimetres and separated by the multiplication sign, × (e.g., M8×1.25). If the pitch is the normally used "coarse" pitch listed in ISO 261 or ISO 262, it can be omitted (e.g., M8). Tolerance classes defined in ISO 965-1 can be appended to these designations, if required (e.g., M500– 6g in external threads). If, for instance, only M20 is given then it is coarse pitch thread.
External threads are designated by lowercase letter, g or h. Internal threads are designated by upper case letters, G or H.
ISO 261 specifies a detailed list of preferred combinations of outer diameter D and pitch P for ISO metric screw threads.[4]
ISO 262 specifies a shorter list of thread dimensions – a subset of ISO 261.[5]
1 0.25 0.2 16 2 1.5
1.2 0.25 0.2 18 2.5 2 or 1.5
1.4 0.3 0.2 20 2.5 2 or 1.5
1.8 0.35 0.2 24 3 2
2 0.4 0.25 27 3 2
2.5 0.45 0.35 30 3.5 2
3 0.5 0.35 33 3.5 2
3.5 0.6 0.35 36 4 3
6 1 0.75 45 4.5 3
7 1 0.75 48 5 3
8 1.25 1 or 0.75 52 5 4
10 1.5 1.25 or 1 56 5.5 4
12 1.75 1.5 or 1.25 60 5.5 4
In addition to coarse and fine threads, there is another division of extra fine, or “superfine” threads, with a very fine pitch thread. Superfine pitch metric threads are occasionally used in automotive components, such as suspension struts, and are commonly used in the aviation manufacturing industry. This is because extra fine threads are more resistant to coming loose from vibrations.[6]
Below are some common wrench sizes for metric screw threads. Hex head widths (width across flats, wrench size) are for DIN 934 hex nuts and hex head bolts. Other (usually smaller) sizes may occur for reasons of weight and cost reduction.
• Flat head counter-
sunk cap screw
M1.6 3.2 3.2 1.5 - 0.7
M2 4 4 1.5 1.25 0.9
M2.5 5 5 2 1.5 1.3
M3.5 6 6 - - -
M4 7 7 3 2.5 2
M7 11 11 - - -
M39 60 60 - - -
Screw threadThreaded insertCaptive fastener기계공학List of screw drives육각형Threaded pipe
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Convex and Algebraic Geometry | EMS Press
Institut Mathématique de Jussieu-Paris Rive Gauche, France
The workshop \emph{Convex and Algebraic Geometry} was organized by Klaus Altmann (Berlin), Victor Batyrev (T\"ubingen), and Bernard Teissier (Paris). Both title subjects meet primarily in the theory of toric varieties. These constitute the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Thus, toric geometry and its several generalizations provide a kind of section from polyhedral into algebraic geometry. While this reflects only a thin slice of algebraic geometry, it is general enough to display many important phenomena, techniques, and methods. It serves as a wonderful testing ground for general theories such as the celebrated mirror symmetry in its different flavours. In particular, much of the popularity of toric geometry originates in mathematical physics.
The meeting was attended by almost 50 participants from many European countries, Canada, the USA, and Japan. The program consisted of talks by 23 speakers, among them many young researchers. Most subjects fit more or less into the following main areas:
\topic{Derived categories, quivers, and (homological) mirror symmetry} {Bondal, Craw, Horja, Maclagan, Perling, Siebert, Ueda}
One of the major discussions during the meeting concerned the existence of strongly exceptional sequences on toric varieties which consist of line bundles. A full exceptional sequence provides a kind of ``basis'' for the derived category. While Hille and Perling presented an example that does not carry such a sequence of full length, Bondal suggested a method to link this question to sheaves on the dual real torus that are constructible with respect to a certain stratification.
In general, one expects to gain exceptional sequences from the universal bundles on moduli spaces. Using this method, Craw constructs those sequences on smooth toric Fano threefolds. In this context, Maclagan and Ueda consider the case of three-dimensional abelian quotient singularities. Ueda investigates the Fukaya category of the corresponding potential on the dual torus explicitly.
Using mirror symmetry, Horja establishes a connection between the orbifold
K
-theory of toric Deligne-Mumford stacks and solutions to GKZ-hypergeometric
D
\topic{Degenerations and deformations} {Brown, Hausen, Siebert, S\"uss, Vollmert}
Gross and Siebert have developed a program to understand mirror symmetry as the duality of certain degeneration data. The special fibers split into toric components, and the degeneration is encoded in a topological manifold
B
with an affine and a polytopal structure. Duality is now inherited from discrete geometry, and the topology of
B
reflects the topology of the general fiber. In particular, if
B
\Q
-homology)
\PP^n_\C
, then this construction might lead to (compact) Hyperk\"ahler varieties.
Considering, in a special case, a certain contraction of the total space of these families leads to a description of torus actions on algebraic varieties via divisors on their Chow quotients. These divisors carry polytopes or even polyhedral complexes as their coefficients, compare the talks of Hausen, S\"uss, and Vollmert. In a similar setting, but with an explicit manipulation of Pfaffians, Brown and Reid construct smoothings of certain non-isolated singularities giving rise to four-dimensional flips.
\topic{Tropical geometry and Welschinger invariants} {Itenberg, Shustin, Siebert}
The most rigorous degeneration of a variety is the tropical one. Here, everything takes place over the so-called tropical semiring, and one ends up with piecewise linear spaces. In fact, Siebert's degeneration data mentioned above correspond to these objects.
Itenberg and Shustin use this approach to calculate the Welschinger invariants, which are a kind of real version of Gromov-Witten invariants. Along the lines of the method of Gathmann and Markwig, there is a recursive formula for theses invariants. In the case of del Pezzo surfaces, it turns out that both invariants are (log-)asymptotically equivalent.
\topic{Commutative algebra, GKZ-systems, and polytopes} {Bruns, Haase, Hering, Hor\-ja, Miller, Pasquier, Stienstra}
A generalization of toric varieties in a different direction from the torus actions mentioned above is given by the notion of spherical varieties. Pasquier considers horospherical Fano varieties and comes up with an adapted notion of (generalized, coloured) reflexive polytopes. Bruns, Haase, and Hering deal with ordinary polytopes and their relations to syzygies of toric varieties.
For an integral matrix
A
one obtains a semigroup algebra
\C[\N A]
(leading to the usual affine toric variety) and a GKZ-hypergeometric system of differential equations. The latter depends on a parameter
\beta
, and Miller has reported on a result that relates the set of
\beta
where the rank of the system jumps to the set of those multidegrees where the semigroup algebra
\C[\N A]
carries local cohomology. In particular, the Cohen-Macaulay property is equivalent to the constant rank condition, answering an old question of Sturmfels.
One of the nighttime discussions gave rise to the suggestion to not include normality in the definition of a toric variety, thus overcoming the cumbersome term of a ``not necessarily normal toric variety''.
The workshop was closed on Friday night by an informal piano recital by Benjamin Nill and Milena Hering featuring Strawinsky, Liszt, and Chopin.
Victor V. Batyrev, Bernard Teissier, Klaus Altmann, Convex and Algebraic Geometry. Oberwolfach Rep. 3 (2006), no. 1, pp. 253–316
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Mr Kapur withdrew rs 25000 from an ATM If he received 150 notes in denominations of rs 500 and rs - Maths - Linear Equations in One Variable - 6863680 | Meritnation.com
Mr.Kapur withdrew rs.25000 from an ATM.If he received 150 notes in denominations of rs.500 and rs.100,find the no. of notes of each denomination?
although i know how to do it, but it is a 2 variable method...can u suggest a 1 variable method?
ans-25,125
Suppose the number of Rs.500 notes be x.
Total number of notes = 150
Then, number of Rs.100 notes =
\left(150-x\right)
Amount of 150 notes =
500x+100\left(150-x\right)
Total amount withdrew from an ATM = Rs.25000
500x+100\left(150-x\right)=25000\phantom{\rule{0ex}{0ex}}⇒500x+15000-100x=25000\phantom{\rule{0ex}{0ex}}⇒400x=25000-15000\phantom{\rule{0ex}{0ex}}⇒x=\frac{10000}{400}=25
Therefore number of Rs.500 notes = 25 and number of Rs.100 notes =
150-25=125
Pradyumna answered this
let the number 500 rs notes be x
and number of 100rs notes be (150-x)
25000 = x(500) + (150 - x)(100)
25000 = 500x + 15000 - 100x
25000 = 500x - 100x +15000
25000 -15000 = 400x
number of 500 rs notes = x=25 notes
number of 100rs notes = 150 - x=150-25 = 125 notes
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§ Diameter of a tree
§ Key property of the diameter
p
be a path of maximum diameter, which starts at
p
q
. Consider a tree where the diameter is shown in golden:
We claim that a node at distance
d
from the left can have a subtree of height at most
d
Suppose this were not the case. Then, we can build a longer diameter (in pink) that is longer than the "supposed diameter" (in gold):
§ Algorithm to find the diameter:
First perform DFS to find a vertex "on the edge", say
v
. Then perform DFS again starting from this vertex
v
. The farthest vertex from
v
w
gives us the diameter (the distance from
v
w
§ Proof by intuition/picture:
first imagine the tree lying flat on the table.
Hold the tree up at node
c
. It's going to fall by gravity and arrange as shown below. This is the same as performing a DFS.
Pick one of the lowest nodes (we pick
g
). Now hold the entire tree from this lowest node, and once again allow gravity to act.
This will give us new lowest nodes such as
b
. This node
b
is going to be diameter, "because" it's the distance from a lowest node to another lowest node.
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Envelope (mathematics) - Wikipedia
For the envelope of an oscillating signal, see Envelope (waves). For the abstract concept, see Envelope (category theory).
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.
To have an envelope, it is necessary that the individual members of the family of curves are differentiable curves as the concept of tangency does not apply otherwise, and there has to be a smooth transition proceeding through the members. But these conditions are not sufficient – a given family may fail to have an envelope. A simple example of this is given by a family of concentric circles of expanding radius.
1 Envelope of a family of curves
3 Envelope of a family of surfaces
5.4 Huygens's principle
Envelope of a family of curvesEdit
The envelope of the family Ct is then defined as the set
{\displaystyle {\mathcal {D}}}
of points (x,y) for which, simultaneously,
{\displaystyle F(t,x,y)=0~~{\mathsf {and}}~~{\partial F \over \partial t}(t,x,y)=0}
for some value of t, where
{\displaystyle \partial F/\partial t}
is the partial derivative of F with respect to t.[1]
If t and u, t≠u are two values of the parameter then the intersection of the curves Ct and Cu is given by
{\displaystyle F(t,x,y)=F(u,x,y)=0\,}
{\displaystyle F(t,x,y)=0~~{\mathsf {and}}~~{\frac {F(u,x,y)-F(t,x,y)}{u-t}}=0.}
Letting u → t gives the definition above.
An important special case is when F(t, x, y) is a polynomial in t. This includes, by clearing denominators, the case where F(t, x, y) is a rational function in t. In this case, the definition amounts to t being a double root of F(t, x, y), so the equation of the envelope can be found by setting the discriminant of F to 0 (because the definition demands F=0 at some t and first derivative =0 i.e. its value 0 and it is min/max at that t).
For example, let Ct be the line whose x and y intercepts are t and 11−t, this is shown in the animation above. The equation of Ct is
{\displaystyle {\frac {x}{t}}+{\frac {y}{11-t}}=1}
{\displaystyle x(11-t)+yt-t(11-t)=t^{2}+(-x+y-11)t+11x=0.\,}
{\displaystyle (-x+y-11)^{2}-44x=(x-y)^{2}-22(x+y)+121=0.\,}
Often when F is not a rational function of the parameter it may be reduced to this case by an appropriate substitution. For example, if the family is given by Cθ with an equation of the form u(x, y)cos θ+v(x, y)sin θ=w(x, y), then putting t=eiθ, cos θ=(t+1/t)/2, sin θ=(t-1/t)/2i changes the equation of the curve to
{\displaystyle u{1 \over 2}(t+{1 \over t})+v{1 \over 2i}(t-{1 \over t})=w}
{\displaystyle (u-iv)t^{2}-2wt+(u+iv)=0.\,}
{\displaystyle (u-iv)(u+iv)-w^{2}=0\,}
{\displaystyle u^{2}+v^{2}=w^{2}.\,}
{\displaystyle E_{1}\subseteq {\mathcal {D}}}
{\displaystyle E_{2}\subseteq {\mathcal {D}}}
{\displaystyle E_{3}\subseteq {\mathcal {D}}}
{\displaystyle {\mathcal {D}}}
is the set of points defined at the beginning of this subsection's parent section.
These definitions E1, E2, and E3 of the envelope may be different sets. Consider for instance the curve y = x3 parametrised by γ : R → R2 where γ(t) = (t,t3). The one-parameter family of curves will be given by the tangent lines to γ.
First we calculate the discriminant
{\displaystyle {\mathcal {D}}}
. The generating function is
{\displaystyle F(t,(x,y))=3t^{2}x-y-2t^{3}.}
Calculating the partial derivative Ft = 6t(x – t). It follows that either x = t or t = 0. First assume that x = t and t ≠ 0. Substituting into F:
{\displaystyle F(t,(t,y))=t^{3}-y\,}
and so, assuming that t ≠ 0, it follows that F = Ft = 0 if and only if (x,y) = (t,t3). Next, assuming that t = 0 and substituting into F gives F(0,(x,y)) = −y. So, assuming t = 0, it follows that F = Ft = 0 if and only if y = 0. Thus the discriminant is the original curve and its tangent line at γ(0):
{\displaystyle {\mathcal {D}}=\{(x,y)\in \mathbb {R} ^{2}:y=x^{3}\}\cup \{(x,y)\in \mathbb {R} ^{2}:y=0\}\ .}
{\displaystyle L:=F(t,(x,y))-F(t+\varepsilon ,(x,y))=2\varepsilon ^{3}+6\varepsilon t^{2}+6\varepsilon ^{2}t-(3\varepsilon ^{2}+6\varepsilon t)x=0.}
{\displaystyle \lim _{\varepsilon \to 0}{\frac {1}{\varepsilon }}L=6t(t-x)\ .}
{\displaystyle \lim _{\varepsilon \to 0}{\frac {1}{\varepsilon ^{2}}}L=3x\ .}
{\displaystyle E_{1}=\{(x,y)\in \mathbb {R} ^{2}:y=x^{3}\}\ .}
{\displaystyle E_{2}=\{(x,y)\in \mathbb {R} ^{2}:y=x^{3}\}\ .}
{\displaystyle F(t,(x_{0},y_{0}))=3t^{2}x_{0}-y_{0}-2t^{3}=0\ .}
{\displaystyle E_{3}=\varnothing .}
This plot gives the envelope of the family of lines connecting points (t,0), (0,k - t), in which k takes the value 1.
{\displaystyle F(x,y,t)=-{\frac {kx}{t}}-t+x+k-y=0\,}
{\displaystyle {\frac {\partial F(x,y,t)}{\partial t}}={\frac {kx}{t^{2}}}-1=0\,}
These two equations jointly define the equation of the envelope. From (2) we have:
{\displaystyle t={\sqrt {kx}}\,}
Substituting this value of t into (1) and simplifying gives an equation for the envelope:
{\displaystyle y=({\sqrt {x}}-{\sqrt {k}})^{2}\,}
Or, rearranging into a more elegant form that shows the symmetry between x and y:
{\displaystyle {\sqrt {x}}+{\sqrt {y}}={\sqrt {k}}}
We can take a rotation of the axes where the b axis is the line y=x oriented northeast and the a axis is the line y=-x oriented southeast. These new axes are related to the original x-y axes by x=(b+a)/√2 and y=(b-a)/√2 . We obtain, after substitution into (4) and expansion and simplification,
{\displaystyle b={\frac {1}{k{\sqrt {2}}}}a^{2}+{\frac {k}{2{\sqrt {2}}}}}
which is apparently the equation for a parabola with axis along a=0, or y=x.
{\displaystyle F(t,{\mathbf {x} })=({\mathbf {x} }-\gamma (t))\cdot {\mathbf {T} }(t)\ .}
{\displaystyle L_{t_{0}}:=\{{\mathbf {x} }\in \mathbb {R} ^{2}:F(t_{0},{\mathbf {x} })=0\}}
{\displaystyle {\frac {\partial F}{\partial t}}(t,{\mathbf {x} })=\kappa (t)({\mathbf {x} }-\gamma (t))\cdot {\mathbf {N} }(t)-1\ ,}
{\displaystyle {\frac {\partial F}{\partial t}}=\lambda \kappa (t)-1\ .}
{\displaystyle {\mathcal {D}}=\gamma (t)+{\frac {1}{\kappa (t)}}{\mathbf {N} }(t)\ .}
The following example shows that in some cases the envelope of a family of curves may be seen as the topologic boundary of a union of sets, whose boundaries are the curves of the envelope. For
{\displaystyle s>0}
{\displaystyle t>0}
consider the (open) right triangle in a Cartesian plane with vertices
{\displaystyle (0,0)}
{\displaystyle (s,0)}
{\displaystyle (0,t)}
{\displaystyle T_{s,t}:=\left\{(x,y)\in \mathbb {R} _{+}^{2}:\ {\frac {x}{s}}+{\frac {y}{t}}<1\right\}.}
Fix an exponent
{\displaystyle \alpha >0}
, and consider the union of all the triangles
{\displaystyle T_{s,t}}
subjected to the constraint
{\displaystyle \textstyle s^{\alpha }+t^{\alpha }=1}
, that is the open set
{\displaystyle \Delta _{\alpha }:=\bigcup _{s^{\alpha }+t^{\alpha }=1}T_{s,t}.}
To write a Cartesian representation for
{\displaystyle \textstyle \Delta _{\alpha }}
, start with any
{\displaystyle \textstyle s>0}
{\displaystyle \textstyle t>0}
{\displaystyle \textstyle s^{\alpha }+t^{\alpha }=1}
{\displaystyle \textstyle (x,y)\in \mathbb {R} _{+}^{2}}
. The Hölder inequality in
{\displaystyle \textstyle \mathbb {R} ^{2}}
with respect to the conjugated exponents
{\displaystyle p:=1+{\frac {1}{\alpha }}}
{\displaystyle \textstyle q:={1+\alpha }}
{\displaystyle x^{\frac {\alpha }{\alpha +1}}+y^{\frac {\alpha }{\alpha +1}}\leq \left({\frac {x}{s}}+{\frac {y}{t}}\right)^{\frac {\alpha }{\alpha +1}}{\Big (}s^{\alpha }+t^{\alpha }{\Big )}^{\frac {1}{\alpha +1}}=\left({\frac {x}{s}}+{\frac {y}{t}}\right)^{\frac {\alpha }{\alpha +1}}}
{\displaystyle \textstyle s:\,t=x^{\frac {1}{1+\alpha }}:\,y^{\frac {1}{1+\alpha }}}
. In terms of a union of sets the latter inequality reads: the point
{\displaystyle (x,y)\in \mathbb {R} _{+}^{2}}
{\displaystyle \textstyle \Delta _{\alpha }}
, that is, it belongs to some
{\displaystyle \textstyle T_{s,t}}
{\displaystyle \textstyle s^{\alpha }+t^{\alpha }=1}
, if and only if it satisfies
{\displaystyle x^{\frac {\alpha }{\alpha +1}}+y^{\frac {\alpha }{\alpha +1}}<1.}
Moreover, the boundary in
{\displaystyle \mathbb {R} _{+}^{2}}
{\displaystyle \textstyle \Delta _{\alpha }}
is the envelope of the corresponding family of line segments
{\displaystyle \left\{(x,y)\in \mathbb {R} _{+}^{2}:\ {\frac {x}{s}}+{\frac {y}{t}}=1\right\}\ ,\qquad s^{\alpha }+t^{\alpha }=1}
{\displaystyle x^{\frac {\alpha }{\alpha +1}}+y^{\frac {\alpha }{\alpha +1}}=1.}
Notice that, in particular, the value
{\displaystyle \alpha =1}
gives the arc of parabola of the Example 2, and the value
{\displaystyle \alpha =2}
(meaning that all hypotenuses are unit length segments) gives the astroid.
{\displaystyle {\frac {d^{2}y}{dt^{2}}}=-g,\;{\frac {d^{2}x}{dt^{2}}}=0,}
{\displaystyle {\frac {dx}{dt}}{\bigg |}_{t=0}=v\cos \theta ,\;{\frac {dy}{dt}}{\bigg |}_{t=0}=v\sin \theta ,\;x{\bigg |}_{t=0}=y{\bigg |}_{t=0}=0.}
{\displaystyle F(x,y,\theta )=x\tan \theta -{\frac {gx^{2}}{2v^{2}\cos ^{2}\theta }}-y=0.}
{\displaystyle {\frac {\partial F}{\partial \theta }}={\frac {x}{\cos ^{2}\theta }}-{\frac {gx^{2}\tan \theta }{v^{2}\cos ^{2}\theta }}=0.}
{\displaystyle y={\frac {v^{2}}{2g}}-{\frac {g}{2v^{2}}}x^{2}.}
Envelope of a family of surfacesEdit
{\displaystyle F(x,y,z,a)=0}
depending on a real parameter a.[2] For example, the tangent planes to a surface along a curve in the surface form such a family.
{\displaystyle F(x,y,z,a)=0,\,\,{F(x,y,z,a^{\prime })-F(x,y,z,a) \over a^{\prime }-a}=0.}
{\displaystyle F(x,y,z,a)=0,\,\,{\partial F \over \partial a}(x,y,z,a)=0.}
Ordinary differential equationsEdit
{\displaystyle t^{2}-2tx+y(x)=0.\ }
{\displaystyle t=\left({\frac {dy}{dx}}\right)/2}
{\displaystyle \left({\frac {dy}{dx}}\right)^{2}\!\!-4x{\frac {dy}{dx}}+4y=0.}
{\displaystyle D_{a}u(x;a)=0\,}
{\displaystyle v(x)=u(x;\varphi (x)),\quad x\in \Omega ,}
CausticsEdit
{\displaystyle L[\gamma ]=\int _{a}^{b}|\gamma '(t)|\,dt}
Huygens's principleEdit
^ Eisenhart, Luther P. (2008), A Treatise on the Differential Geometry of Curves and Surfaces, Schwarz Press, ISBN 1-4437-3160-9
^ Forsyth, Andrew Russell (1959), Theory of differential equations, Six volumes bound as three, New York: Dover Publications, MR 0123757 , §§100-106.
^ Evans, Lawrence C. (1998), Partial differential equations, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0772-9 .
^ John, Fritz (1991), Partial differential equations (4th ed.), Springer, ISBN 978-0-387-90609-6 .
^ Born, Max (October 1999), Principle of Optics, Cambridge University Press, ISBN 978-0-521-64222-4 , Appendix I: The calculus of variations.
^ Arnold, V. I. (1997), Mathematical Methods of Classical Mechanics, 2nd ed., Berlin, New York: Springer-Verlag, ISBN 978-0-387-96890-2 , §46.
"Envelope of a family of plane curves" at MathCurve.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Envelope_(mathematics)&oldid=1033094941"
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Dynamics of Cocycles and One-Dimensional Spectral Theory | EMS Press
The mini-workshop \emph{Dynamics of Cocycles and One-Di\-mensional Spectral Theory}, organised by David Damanik (Pasadena), Russell Johnson (Firenze) and Daniel Lenz (Chemnitz), was held November 13--19, 2005.
There have been a number of recent breakthroughs in the spectral theory of one-dimensional Schr\"odinger operators with quasi-periodic potentials that were accomplished using sophisticated dynamical systems methods; especially by establishing reducibility properties of certain quasi-periodic
\mathrm{SL}(2,\R)
-valued cocycles.
The most popular example of a one-dimensional Schr\"odinger operators with quasi-periodic potential is given by the almost Mathieu operator,
[H u]_n = u_{n+1} + u_{n-1} + 2\lambda \cos (2\pi (n\alpha + \theta)) u_n,
\lambda \not= 0
\alpha
is irrational. Using the connection with dynamics, it was recently shown for all parameter values that the spectrum of
H
is a Cantor set of Lebesgue measure
|4 - 4 |\lambda||
It was the objective of the mini-workshop to bring together experts from both spectral theory and dynamical systems to learn from each other and to further explore potential applications of dynamical systems methods in the context of quasi-periodic Schr\"odinger operators. Special attention was paid to having many young participants. This was made easy by the fact that currently there are many excellent graduate students and postdocs working on problems located at the interface between the two areas. Consequently, about two thirds of the participants belonged to the age group 35~years or younger.
The talks presented by the participants reflected the current developments in this area. Among other things, there is now an improved understanding of analytic potentials with non-perturbatively small coupling, there are extensions of some results known for analytic potentials to certain classes of non-analytic potentials, while phenomena different from those in the analytic case may occur for potentials of low regularity, and there is improved understanding of the case of Liouville frequencies.
Among the highlights of the discussions outside the talks one could mention the ``joining'' of independent and closely related work of Bjerkl\"ov and J\"ager and the solution by Avila and Damanik of one of the few questions about the almost Mathieu operator that were still left open after the recent advances triggering the mini-workshop.
David Damanik, Russell Johnson, H. Daniel Lenz, Dynamics of Cocycles and One-Dimensional Spectral Theory. Oberwolfach Rep. 2 (2005), no. 4, pp. 2933–2978
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Deflection of Piezoelectric Actuator - MATLAB & Simulink - MathWorks India
Finite Element and Analytical Solutions
This example shows how to solve a coupled elasticity-electrostatics problem.
Piezoelectric materials deform under an applied voltage. Conversely, deforming a piezoelectric material produces a voltage. Therefore, analysis of a piezoelectric part requires the solution of a set of coupled partial differential equations with deflections and electrical potential as dependent variables.
In this example, the model is a two-layer cantilever beam, with both layers made of the same polyvinylidene fluoride (PVDF) material. The polarization direction points down (negative y-direction) in the top layer and points up in the bottom layer. The typical length to thickness ratio is 100. When you apply a voltage between the lower and upper surfaces of the beam, the beam deflects in the y-direction because one layer shortens and the other layer lengthens.
The equilibrium equations describe the elastic behavior of the solid:
-\nabla \cdot \sigma =f
\sigma
is the stress tensor, and
f
is the body force vector. Gauss's Law describes the electrostatic behavior of the solid:
\nabla \cdot D=\rho
D
is the electric displacement, and
\rho
is the distributed free charge. Combine these two PDE systems into this single system:
-\nabla \cdot \left\{\begin{array}{c}\sigma \\ D\end{array}\right\}=\left\{\begin{array}{c}f\\ -\rho \end{array}\right\}
For a 2-D analysis,
\sigma
has the components
{\sigma }_{11},{\sigma }_{22},
{\sigma }_{12}={\sigma }_{21}
D
{D}_{1}
{D}_{2}
The constitutive equations for the material define the stress tensor and electric displacement vector in terms of the strain tensor and electric field. For a 2-D analysis of an orthotropic piezoelectric material under plane stress conditions, you can write these equations as
\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{12}\\ {D}_{1}\\ {D}_{2}\end{array}\right\}=\left[\begin{array}{ccccc}{C}_{11}& {C}_{12}& & {e}_{11}& {e}_{31}\\ {C}_{12}& {C}_{22}& & {e}_{13}& {e}_{33}\\ & & {G}_{12}& {e}_{14}& {e}_{34}\\ {e}_{11}& {e}_{13}& {e}_{14}& -{\mathcal{E}}_{1}& \\ {e}_{31}& {e}_{33}& {e}_{34}& & -{\mathcal{E}}_{2}\end{array}\right]\left\{\begin{array}{c}{ϵ}_{11}\\ {ϵ}_{22}\\ {\gamma }_{12}\\ -{E}_{1}\\ -{E}_{2}\end{array}\right\}
{C}_{ij}
are the elastic coefficients,
{\mathcal{E}}_{i}
are the electrical permittivities, and
{e}_{ij}
are the piezoelectric stress coefficients. The piezoelectric stress coefficients in these equations conform to conventional notation in piezoelectric materials where the z-direction (the third direction) aligns with the "poled" direction of the material. For the 2-D analysis, align the "poled" direction with the y-axis. Write the strain vector in terms of the x-displacement
u
and y-displacement
v
\left\{\begin{array}{c}{ϵ}_{11}\\ {ϵ}_{22}\\ {\gamma }_{12}\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial u}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\end{array}\right\}
Write the electric field in terms of the electrical potential
\varphi
\left\{\begin{array}{c}{E}_{1}\\ {E}_{2}\end{array}\right\}=-\left\{\begin{array}{c}\frac{\partial \varphi }{\partial x}\\ \frac{\partial \varphi }{\partial y}\end{array}\right\}
You can substitute the strain-displacement equations and electric field equations into the constitutive equations and get a system of equations for the stresses and electrical displacements in terms of displacement and electrical potential derivatives. Substituting the resulting equations into the PDE system equations yields a system of equations that involve the divergence of the displacement and electrical potential derivatives. As the next step, arrange these equations to match the form required by the toolbox.
Partial Differential Equation Toolbox™ requires a system of elliptic equations to be expressed in a vector form:
-\nabla \cdot \left(c\otimes \nabla u\right)+au=f
or in a tensor form:
-\frac{\partial }{\partial {x}_{k}}\left({c}_{ijkl}\frac{\partial {u}_{j}}{\partial {x}_{l}}\right)+{a}_{ij}{u}_{j}={f}_{i}
where repeated indices imply summation. For the 2-D piezoelectric system in this example, the system vector
u
u=\left\{\begin{array}{c}u\\ v\\ \varphi \end{array}\right\}
This is an
N=3
system. The gradient of
u
\nabla u=\left\{\begin{array}{c}\frac{\partial u}{\partial x}\\ \phantom{\rule{1em}{0ex}}\frac{\partial u}{\partial y}\phantom{\rule{1em}{0ex}}\\ \frac{\partial v}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial \varphi }{\partial x}\\ \frac{\partial \varphi }{\partial y}\end{array}\right\}
For details on specifying the coefficients in the format required by the toolbox, see:
The c coefficient in this example is a tensor. You can represent it as a 3-by-3 matrix of 2-by-2 blocks:
\left[\begin{array}{cccccc}c\left(1\right)& c\left(2\right)& c\left(4\right)& c\left(6\right)& c\left(11\right)& c\left(13\right)\\ \cdot & c\left(3\right)& c\left(5\right)& c\left(7\right)& c\left(12\right)& c\left(14\right)\\ \cdot & \cdot & c\left(8\right)& c\left(9\right)& c\left(15\right)& c\left(17\right)\\ \cdot & \cdot & \cdot & c\left(10\right)& c\left(16\right)& c\left(18\right)\\ \cdot & \cdot & \cdot & \cdot & c\left(19\right)& c\left(20\right)\\ \cdot & \cdot & \cdot & \cdot & \cdot & c\left(21\right)\end{array}\right]
To map terms of constitutive equations to the form required by the toolbox, write the c tensor and the solution gradient in this form:
\left[\begin{array}{cccccc}{c}_{1111}& {c}_{1112}& {c}_{1211}& {c}_{1212}& {c}_{1311}& {c}_{1312}\\ \cdot & {c}_{1122}& {c}_{1221}& {c}_{1222}& {c}_{1321}& {c}_{1322}\\ \cdot & \cdot & {c}_{2211}& {c}_{2212}& {c}_{2311}& {c}_{2312}\\ \cdot & \cdot & \cdot & {c}_{2222}& {c}_{2321}& {c}_{2322}\\ \cdot & \cdot & \cdot & \cdot & {c}_{3311}& {c}_{3312}\\ \cdot & \cdot & \cdot & \cdot & \cdot & {c}_{3322}\end{array}\right]\left\{\begin{array}{c}\frac{\partial u}{\partial x}\\ \phantom{\rule{1em}{0ex}}\frac{\partial u}{\partial y}\phantom{\rule{1em}{0ex}}\\ \frac{\partial v}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial \varphi }{\partial x}\\ \frac{\partial \varphi }{\partial y}\end{array}\right\}
From this equation, you can map the traditional constitutive coefficients to the form required for the c matrix. The minus sign in the equations for the electric field is incorporated into the c matrix to match the toolbox's convention.
\left[\begin{array}{cccccc}{C}_{11}& \cdot & \cdot & {C}_{12}& {e}_{11}& {e}_{31}\\ \cdot & {G}_{12}& {G}_{12}& \cdot & {e}_{14}& {e}_{34}\\ \cdot & \cdot & {G}_{12}& \cdot & {e}_{14}& {e}_{34}\\ \cdot & \cdot & \cdot & {C}_{22}& {e}_{13}& {e}_{33}\\ \cdot & \cdot & \cdot & \cdot & -{\mathcal{E}}_{1}& \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & -{\mathcal{E}}_{2}\end{array}\right]\left\{\begin{array}{c}\frac{\partial u}{\partial x}\\ \phantom{\rule{1em}{0ex}}\frac{\partial u}{\partial y}\phantom{\rule{1em}{0ex}}\\ \frac{\partial v}{\partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial \varphi }{\partial x}\\ \frac{\partial \varphi }{\partial y}\end{array}\right\}
Create a PDE model. The equations of linear elasticity have three components, so the model must have three equations.
L = 100e-3; % Beam length in meters
H = 1e-3; % Overall height of the beam
H2 = H/2; % Height of each layer in meters
topLayer = [3 4 0 L L 0 0 0 H2 H2];
bottomLayer = [3 4 0 L L 0 -H2 -H2 0 0];
gdm = [topLayer;bottomLayer]';
g = decsg(gdm,'R1+R2',['R1';'R2']');
Plot the geometry with the face and edge labels.
pdegplot(model,'EdgeLabels','on', ...
axis([-.1*L,1.1*L,-4*H2,4*H2])
Specify the material properties of the beam layers. The material in both layers is polyvinylidene fluoride (PVDF), a thermoplastic polymer with piezoelectric behavior.
E = 2.0e9; % Elastic modulus, N/m^2
NU = 0.29; % Poisson's ratio
G = 0.775e9; % Shear modulus, N/m^2
d31 = 2.2e-11; % Piezoelectric strain coefficients, C/N
d33 = -3.0e-11;
Specify relative electrical permittivity of the material at constant stress.
relPermittivity = 12;
Specify the electrical permittivity of vacuum.
permittivityFreeSpace = 8.854187817620e-12; % F/m
C11 = E/(1 - NU^2);
C12 = NU*C11;
c2d = [C11 C12 0; C12 C11 0; 0 0 G];
pzeD = [0 d31; 0 d33; 0 0];
Specify the piezoelectric stress coefficients.
pzeE = c2d*pzeD;
D_const_stress = [relPermittivity 0;
0 relPermittivity]*permittivityFreeSpace;
Convert the dielectric matrix from constant stress to constant strain.
D_const_strain = D_const_stress - pzeD'*pzeE;
You can view the 21 coefficients as a 3-by-3 matrix of 2-by-2 blocks. The cij matrices are the 2-by-2 blocks in the upper triangle of this matrix.
c11 = [c2d(1,1) c2d(1,3) c2d(3,3)];
c12 = [c2d(1,3) c2d(1,2); c2d(3,3) c2d(2,3)];
c13 = [pzeE(1,1) pzeE(1,2); pzeE(3,1) pzeE(3,2)];
c33 = [D_const_strain(1,1)
D_const_strain(2,1)
D_const_strain(2,2)];
ctop = [c11(:); c12(:); c22(:); -c13(:); -c23(:); -c33(:)];
cbot = [c11(:); c12(:); c22(:); c13(:); c23(:); -c33(:)];
f = [0 0 0]';
specifyCoefficients(model,'m',0,'d',0,'c',ctop,'a',0,'f',f,'Face',2);
specifyCoefficients(model,'m',0,'d',0,'c',cbot,'a',0,'f',f,'Face',1);
Set the voltage (solution component 3) on the top of the beam (edge 1) to 100 volts.
voltTop = applyBoundaryCondition(model,'mixed', ...
'Edge',1,...
'u',100,...
Specify that the bottom of the beam (edge 2) is grounded by setting the voltage to 0.
voltBot = applyBoundaryCondition(model,'mixed', ...
'u',0,...
Specify that the left side (edges 6 and 7) is clamped by setting the x- and y-displacements (solution components 1 and 2) to 0.
clampLeft = applyBoundaryCondition(model,'mixed', ...
'Edge',6:7,...
'u',[0 0],...
'EquationIndex',1:2);
The stress and charge on the right side of the beam are zero. Accordingly, use the default boundary condition for edges 3 and 4.
msh = generateMesh(model,'Hmax',5e-4);
ZGradients: [0x3 double]
Access the solution at the nodal locations. The first column contains the x-deflection. The second column contains the y-deflection. The third column contains the electrical potential.
rs = result.NodalSolution;
Find the minimum y-deflection.
feTipDeflection = min(rs(:,2));
fprintf('Finite element tip deflection is: %12.4e\n',feTipDeflection);
Finite element tip deflection is: -3.2900e-05
Compare this result with the known analytical solution.
tipDeflection = -3*d31*100*L^2/(8*H2^2);
fprintf('Analytical tip deflection is: %12.4e\n',tipDeflection);
Analytical tip deflection is: -3.3000e-05
Plot the deflection components and the electrical potential.
varsToPlot = char('X-Deflection, meters', ...
'Y-Deflection, meters', ...
'Electrical Potential, Volts');
for i = 1:size(varsToPlot,1)
pdeplot(model,'XYData',rs(:,i),'Contour','on')
title(varsToPlot(i,:))
% scale the axes to make it easier to view the contours
axis([0, L, -4*H2, 4*H2])
Hwang, Woo-Seok, and Hyun Chul Park. "Finite Element Modeling of Piezoelectric Sensors and Actuators." AIAA Journal 31, no.5 (May 1993): 930-937.
Pieford, V. "Finite Element Modeling of Piezoelectric Active Structures." PhD diss., Universite Libre De Bruxelles, 2001.
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Tolerance interval - Wikipedia
Not to be confused with Engineering tolerance.
A tolerance interval is a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls. "More specifically, a 100×p%/100×(1−α) tolerance interval provides limits within which at least a certain proportion (p) of the population falls with a given level of confidence (1−α)."[1] "A (p, 1−α) tolerance interval (TI) based on a sample is constructed so that it would include at least a proportion p of the sampled population with confidence 1−α; such a TI is usually referred to as p-content − (1−α) coverage TI."[2] "A (p, 1−α) upper tolerance limit (TL) is simply a 1−α upper confidence limit for the 100 p percentile of the population."[2]
A tolerance interval can be seen as a statistical version of a probability interval. "In the parameters-known case, a 95% tolerance interval and a 95% prediction interval are the same."[3] If we knew a population's exact parameters, we would be able to compute a range within which a certain proportion of the population falls. For example, if we know a population is normally distributed with mean
{\displaystyle \mu }
{\displaystyle \sigma }
, then the interval
{\displaystyle \mu \pm 1.96\sigma }
includes 95% of the population (1.96 is the z-score for 95% coverage of a normally distributed population).
However, if we have only a sample from the population, we know only the sample mean
{\displaystyle {\hat {\mu }}}
and sample standard deviation
{\displaystyle {\hat {\sigma }}}
, which are only estimates of the true parameters. In that case,
{\displaystyle {\hat {\mu }}\pm 1.96{\hat {\sigma }}}
will not necessarily include 95% of the population, due to variance in these estimates. A tolerance interval bounds this variance by introducing a confidence level
{\displaystyle \gamma }
, which is the confidence with which this interval actually includes the specified proportion of the population. For a normally distributed population, a z-score can be transformed into a "k factor" or tolerance factor[4] for a given
{\displaystyle \gamma }
via lookup tables or several approximation formulas.[5] "As the degrees of freedom approach infinity, the prediction and tolerance intervals become equal."[6]
2 Relation to other intervals
This section needs expansion with: mathematical equations. You can help by adding to it. (talk) (July 2014)
Normal case[edit]
Relation to other intervals[edit]
The tolerance interval is less widely known than the confidence interval and prediction interval, a situation some educators have lamented, as it can lead to misuse of the other intervals where a tolerance interval is more appropriate.[7][8]
The tolerance interval differs from a confidence interval in that the confidence interval bounds a single-valued population parameter (the mean or the variance, for example) with some confidence, while the tolerance interval bounds the range of data values that includes a specific proportion of the population. Whereas a confidence interval's size is entirely due to sampling error, and will approach a zero-width interval at the true population parameter as sample size increases, a tolerance interval's size is due partly to sampling error and partly to actual variance in the population, and will approach the population's probability interval as sample size increases.[7][8]
The tolerance interval is related to a prediction interval in that both put bounds on variation in future samples. However, the prediction interval only bounds a single future sample, whereas a tolerance interval bounds the entire population (equivalently, an arbitrary sequence of future samples). In other words, a prediction interval covers a specified proportion of a population on average, whereas a tolerance interval covers it with a certain confidence level, making the tolerance interval more appropriate if a single interval is intended to bound multiple future samples.[8][9]
[7] gives the following example:
So consider once again a proverbial EPA mileage test scenario, in which several nominally identical autos of a particular model are tested to produce mileage figures
{\displaystyle y_{1},y_{2},...,y_{n}}
. If such data are processed to produce a 95% confidence interval for the mean mileage of the model, it is, for example, possible to use it to project the mean or total gasoline consumption for the manufactured fleet of such autos over their first 5,000 miles of use. Such an interval, would however, not be of much help to a person renting one of these cars and wondering whether the (full) 10-gallon tank of gas will suffice to carry him the 350 miles to his destination. For that job, a prediction interval would be much more useful. (Consider the differing implications of being "95% sure" that
{\displaystyle \mu \geq 35}
as opposed to being "95% sure" that
{\displaystyle y_{n+1}\geq 35}
.) But neither a confidence interval for
{\displaystyle \mu }
nor a prediction interval for a single additional mileage is exactly what is needed by a design engineer charged with determining how large a gas tank the model really needs to guarantee that 99% of the autos produced will have a 400-mile cruising range. What the engineer really needs is a tolerance interval for a fraction
{\displaystyle p=.99}
of mileages of such autos.
Another example is given by:[9]
The air lead levels were collected from
{\displaystyle n=15}
different areas within the facility. It was noted that the log-transformed lead levels fitted a normal distribution well (that is, the data are from a lognormal distribution. Let
{\displaystyle \mu }
{\displaystyle \sigma ^{2}}
, respectively, denote the population mean and variance for the log-transformed data. If
{\displaystyle X}
denotes the corresponding random variable, we thus have
{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}
{\displaystyle \exp(\mu )}
is the median air lead level. A confidence interval for
{\displaystyle \mu }
can be constructed the usual way, based on the t-distribution; this in turn will provide a confidence interval for the median air lead level. If
{\displaystyle {\bar {X}}}
{\displaystyle S}
denote the sample mean and standard deviation of the log-transformed data for a sample of size n, a 95% confidence interval for
{\displaystyle \mu }
{\displaystyle {\bar {X}}\pm t_{n-1,0.975}S/{\sqrt {n}}}
{\displaystyle t_{m,1-\alpha }}
{\displaystyle 1-\alpha }
quantile of a t-distribution with
{\displaystyle m}
degrees of freedom. It may also be of interest to derive a 95% upper confidence bound for the median air lead level. Such a bound for
{\displaystyle \mu }
{\displaystyle {\bar {X}}+t_{n-1,0.95}S/{\sqrt {n}}}
. Consequently, a 95% upper confidence bound for the median air lead is given by
{\displaystyle \exp {\left({\bar {X}}+t_{n-1,0.95}S/{\sqrt {n}}\right)}}
. Now suppose we want to predict the air lead level at a particular area within the laboratory. A 95% upper prediction limit for the log-transformed lead level is given by
{\displaystyle {\bar {X}}+t_{n-1,0.95}S{\sqrt {\left(1+1/n\right)}}}
. A two-sided prediction interval can be similarly computed. The meaning and interpretation of these intervals are well known. For example, if the confidence interval
{\displaystyle {\bar {X}}\pm t_{n-1,0.975}S/{\sqrt {n}}}
is computed repeatedly from independent samples, 95% of the intervals so computed will include the true value of
{\displaystyle \mu }
, in the long run. In other words, the interval is meant to provide information concerning the parameter
{\displaystyle \mu }
only. A prediction interval has a similar interpretation, and is meant to provide information concerning a single lead level only. Now suppose we want to use the sample to conclude whether or not at least 95% of the population lead levels are below a threshold. The confidence interval and prediction interval cannot answer this question, since the confidence interval is only for the median lead level, and the prediction interval is only for a single lead level. What is required is a tolerance interval; more specifically, an upper tolerance limit. The upper tolerance limit is to be computed subject to the condition that at least 95% of the population lead levels is below the limit, with a certain confidence level, say 99%.
One-sided normal tolerance intervals have an exact solution in terms of the sample mean and sample variance based on the noncentral t-distribution.[10] Two-sided normal tolerance intervals can be obtained based on the noncentral chi-squared distribution.[10]
^ D. S. Young (2010), Book Reviews: "Statistical Tolerance Regions: Theory, Applications, and Computation", TECHNOMETRICS, FEBRUARY 2010, VOL. 52, NO. 1, pp.143-144.
^ a b Krishnamoorthy, K. and Lian, Xiaodong(2011) 'Closed-form approximate tolerance intervals for some general linear models and comparison studies', Journal of Statistical Computation and Simulation, First published on: 13 June 2011 doi:10.1080/00949655.2010.545061
^ Thomas P. Ryan (22 June 2007). Modern Engineering Statistics. John Wiley & Sons. pp. 222–. ISBN 978-0-470-12843-5. Retrieved 22 February 2013.
^ "Statistical interpretation of data — Part 6: Determination of statistical tolerance intervals". ISO 16269-6. 2014. p. 2.
^ "Tolerance intervals for a normal distribution". Engineering Statistics Handbook. NIST/Sematech. 2010. Retrieved 2011-08-26.
^ De Gryze, S.; Langhans, I.; Vandebroek, M. (2007). "Using the correct intervals for prediction: A tutorial on tolerance intervals for ordinary least-squares regression". Chemometrics and Intelligent Laboratory Systems. 87 (2): 147. doi:10.1016/j.chemolab.2007.03.002.
^ a b c Stephen B. Vardeman (1992). "What about the Other Intervals?". The American Statistician. 46 (3): 193–197. doi:10.2307/2685212. JSTOR 2685212.
^ a b c Mark J. Nelson (2011-08-14). "You might want a tolerance interval". Retrieved 2011-08-26.
^ a b K. Krishnamoorthy (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons. pp. 1–6. ISBN 978-0-470-38026-0.
^ a b Derek S. Young (August 2010). "tolerance: An R Package for Estimating Tolerance Intervals". Journal of Statistical Software. 36 (5): 1–39. ISSN 1548-7660. Retrieved 19 February 2013. , p.23
Hahn, Gerald J.; Meeker, William Q.; Escobar, Luis A. (2017). Statistical Intervals: A Guide for Practitioners and Researchers (2nd ed.). John Wiley & Sons, Incorporated. ISBN 978-0-471-68717-7.
K. Krishnamoorthy (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons. ISBN 978-0-470-38026-0. ; Chap. 1, "Preliminaries", is available at http://media.wiley.com/product_data/excerpt/68/04703802/0470380268.pdf
Derek S. Young (August 2010). "tolerance: An R Package for Estimating Tolerance Intervals". Journal of Statistical Software. 36 (5): 1–39. ISSN 1548-7660. Retrieved 19 February 2013.
ISO 16269-6, Statistical interpretation of data, Part 6: Determination of statistical tolerance intervals, Technical Committee ISO/TC 69, Applications of statistical methods. Available at http://standardsproposals.bsigroup.com/home/getpdf/458
Retrieved from "https://en.wikipedia.org/w/index.php?title=Tolerance_interval&oldid=1082335473"
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GlobalSolve (Matrix Form) - Maple Help
Home : Support : Online Help : Mathematics : Optimization : GlobalOptimization Package : GlobalSolve (Matrix Form)
GlobalOptimization[GlobalSolve](Matrix Form)
find a global solution to a nonlinear program in Matrix form
GlobalSolve(n, p, nc, nlc, bd, opts)
GlobalSolve(n, p, bd, opts)
positive integer; number of variables
non-negative integer or list of 2 non-negative integers; number of nonlinear constraints
procedure; nonlinear constraints
list; bounds
(optional) equation(s) of the form option = value where option is one of evaluationlimit, feasibilitytolerance, initialpoint, maximize, merittarget, method, noimprovementlimit, objectivetarget, optimalitytolerance, penaltymultiplier or timelimit; specify options for the GlobalSolve command
The GlobalSolve command attempts to compute a global solution to a nonlinear program (NLP) over a bounded region. See the following Notes section for a detailed explanation of the solution obtained. An NLP has the following form:
f\left(x\right)
v\left(x\right)\le 0
w\left(x\right)=0
\mathrm{bl}\le x\le \mathrm{bu}
x
f\left(x\right)
x
v\left(x\right)
w\left(x\right)
x
\mathrm{bl}
\mathrm{bu}
are vectors. The relations involving vectors are element-wise.
The global solver optimizes a merit function, incorporating a penalty term for the constraints, and provides both branch-and-bound and adaptive stochastic search methods. The global search phase is followed by a local search phase to refine the solution. No derivatives are required.
This help page describes how to specify the problem in Matrix form. For details about the exact format of the objective function and the constraints, see the GlobalOptimization/MatrixForm help page. The algebraic and operator forms for specifying an NLP are described in the GlobalOptimization[GlobalSolve] help page. The Matrix form is more complex, but leads to more efficient computation.
The solver is designed to search the specified region for a global solution to a non-convex optimization problem. If the optimization problem is convex (for example, a linear program) or a local solution is acceptable, it is recommended that you use the commands for local optimization in the Optimization package. The Optimization package commands, which are more efficient, can compute global solutions to convex problems.
Consider the first calling sequence. The first parameter n is the number of problem variables. The second parameter p is a procedure that takes one input Vector parameter of size n, representing x, and returns the value of f(x).
The third parameter nc is a list of two non-negative integers representing the number of nonlinear inequality constraints and the number of nonlinear equality constraints. If there are no inequality constraints, nc can be a single integer value. The fourth parameter nlc is a procedure,
\mathrm{proc}\left(x,y\right)\mathrm{...}\mathrm{end proc}
, that computes the values of the nonlinear constraints. The current point is passed as the Vector x, and the values of v(x) followed by the values of w(x) are returned using the Vector parameter y.
The fifth parameter bd is a list
[\mathrm{bl},\mathrm{bu}]
of lower and upper bounds. In general, bl and bu must be n-dimensional Vectors. The GlobalOptimization/MatrixForm help page describes an alternate way of specifying the Vectors.
If there are no nonlinear constraints, the second calling sequence, in which parameters nc and nlc are omitted, should be used.
Maple returns the solution as a list containing the final minimum (or maximum) value and a Vector representing a point (the extremum).
The opts argument can contain one or more of the following options. These options are described in more detail in the GlobalOptimization/Options help page.
Some options apply to only the local search phase that follows the global search phase. The descriptions for those options specifically mention local search. Otherwise, the option applies to the global search.
evaluationlimit = posint -- Set the maximum number of merit function evaluations performed during the global search. The global search phase terminates if this limit is reached.
feasibilitytolerance = positive and numeric -- Set the maximum absolute allowable constraint violation in the local search phase.
initialpoint = set(equation), list(equation), or list(numeric) -- Use the provided initial point, which is an n-dimensional Vector.
maximize = truefalse -- Maximize the objective function when m is 'true' and minimize when m is 'false'. The option 'maximize' is equivalent to 'maximize'='true'. The default is 'maximize'='false'.
merittarget = numeric -- Set an acceptable target value for the merit function. If the merit function achieves this value, the global search phase terminates.
method = branchandbound, singlestart, multistart, or reducedgradient -- Set the global search algorithm: branch-and-bound (method=branchandbound), adaptive random search with a single starting point (method = singlestart), or adaptive random search with multiple starting points (method = multistart). Specifying method = reducedgradient omits the global search phase, performing the local search only. The default is method = multistart.
noimprovementlimit = posint -- Set the maximum number of merit function evaluations performed with no improvement in the merit function. The global search phase terminates if this limit is reached.
objectivetarget = numeric -- Set an acceptable target value for the objective function in the local search phase. If the objective function achieves this value, then the local search phase terminates.
optimalitytolerance = positive and numeric -- Set the tolerance for the Kuhn-Tucker optimality conditions in the local search phase.
penaltymultiplier = positive and numeric -- Set the constraint penalty multiplier. The constraint violations in the merit function are weighted by this value, which must evaluate to a positive numeric value.
timelimit = posint -- Set the maximum computation time, in seconds, for the global solver.
For more information on the methods used by the global solver, with suggestions for achieving best performance, see the GlobalOptimization/Computation help page.
The global solver searches for the optimal solution until one of the termination criteria is met. Then either the best available solution is returned or an error is issued stating that a solution could not be obtained. The termination criteria can be set using the options. Otherwise, default values for these options are applied. In particular, the evaluationlimit option must be set to a sufficiently high value for difficult optimization problems or unexpected answers may be produced.
In the case that an error is issued because the time limit has been exceeded, the last solution computed may be retrieved using the GetLastSolution command.
It is highly recommended that you set infolevel[GlobalOptimization] to
1
or higher to display messages describing the progress of the solver. These messages can give indications that the solution might not be optimal and that option values should be adjusted.
The computation is performed in floating-point. Therefore, all data provided must have type realcons and all returned solutions are floating-point, even if the problem is specified with exact values. The solver uses externally called code that works with hardware floats, but it is possible to evaluate the objective function and the constraints in Maple with higher precision. See the GlobalOptimization/Computation help page for details.
\mathrm{with}\left(\mathrm{GlobalOptimization}\right):
Find the global solution to an unconstrained nonlinear minimization problem in Matrix form.
p := proc(V) V[1]^2-V[1]+1 end proc:
\mathrm{bl}≔\mathrm{Vector}\left([0.],\mathrm{datatype}=\mathrm{float}\right):
\mathrm{bu}≔\mathrm{Vector}\left([1.],\mathrm{datatype}=\mathrm{float}\right):
\mathrm{bd}≔[\mathrm{bl},\mathrm{bu}]:
\mathrm{GlobalSolve}\left(1,p,\mathrm{bd}\right)
[\textcolor[rgb]{0,0,1}{0.750000000000015099}\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{0.500000122749046}\end{array}]]
xy-4x-5y
3{x}^{2}+2y=6
4{y}^{2}+5x\le 20
Express the objective function as a procedure.
p:= proc(V) -4*V[1]+V[1]*V[2]-5*V[2] end proc:
Express the constraints in a single procedure with the parameters V and W.
nlc := proc(V, W)
W[1] := 5*V[1]+4*V[2]^2-20;
W[2] := 3*V[1]^2+2*V[2]-6
Express the bounds in Matrix form.
\mathrm{bl}≔\mathrm{Vector}\left([0.,0.],\mathrm{datatype}=\mathrm{float}\right):
\mathrm{bu}≔\mathrm{Vector}\left([5.,5.],\mathrm{datatype}=\mathrm{float}\right):
\mathrm{bd}≔[\mathrm{bl},\mathrm{bu}]:
Find the global minimum with GlobalSolve.
\mathrm{GlobalSolve}\left(2,p,[1,1],\mathrm{nlc},\mathrm{bd}\right)
[\textcolor[rgb]{0,0,1}{-8.44181784096971910}\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{1.15685868788946}\\ \textcolor[rgb]{0,0,1}{0.992517000973123}\end{array}]]
GlobalOptimization/Computation
GlobalOptimization[GetLastSolution]
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Arrr - Uncyclopedia, the content-free encyclopedia
Avast, cur!
“Arrr! I nay support the expression of Arr within society! Be ye a member! BE YE! ”
~ Oscar Wilde on Arrr'ing
Arrr, the new age response to many the question. Such as "Do I look fat in these pants?", or "What's the first letter of the word wrong?" "Give me your lunch money!" and in Family Feud "The incorrect response if your girlfriend asks 'are you ever going to propose to me?'". Usually connected with the old and new pirates Arr is a dying expression that some people have argued, especially those from The Psycho Friends Network who believe in the right of piracy and "Arr"acy, should be brought back into today's language.
1 Psycho Friends Network and Piracy
2 Every Day Usage of Arrr
3 Experiments of Arrr
4 Mathematics of Arr-ness
Psycho Friends Network and Piracy[edit]
Long ago when Oscar Wilde wasn't heading The Psycho Friends Network the world was free to to express themselves in all sorts of four letter descriptions of curled language. There are those who claim weee and yaaay in cries of joy, and other who are often of the brrr and eep type. Unknowingly we have gone throughout history expression our emotions through 'arr'ness, but once the laws of The Psycho Friends Network came into play the accessibility of simply spilling ones guts had become illegal, unless you became a member of the Network. This was excruciatingly difficult, as their hazing process consisted of several chickens, a cow prod and a kilo of sugar. The term Arr was then not to be used for the first six months in a brainwashing process called "cowardice". Until the Governator took over the new presidential position many people had been hunted and cow prodded in protest for having used 'arrr' and not being a member.
Arnold Schwarzenwhatever created new laws in the Network and it came to pass that anyone who muttered, or yelled, the words "arrr" automatically became a member of The Psycho Friends Network.There was still the hazing process involved, though it was quick and frightening, usually leaving life scars in ones soul until death, and possibly even after.
Every Day Usage of Arrr[edit]
Arr can be used every day in quite normal, natural circumstances. Suggested more for those who easily and effortlessly release their anger even the more timid of people can mutter an 'arrr' in times of confusion. Do not, at any stage, confuse this with muttering 'ooooh', for the consequences of that have come to be catastrophic.
In 1927 when the world was new and round Oscar Wilde was reported to have come into a fight with one of his drinking fiends in a bar. He was challenged over the bar, and asked if defeat and submission was inevitable for himself to which he did, indeed, reply "Arrr". His fiend mistook this to be hearing the softer expression of ooooh and three days later Oscar Wilde committed himself to writing slash flash fiction and claiming that hetrosexuality was a "simple misconception and should be eaten at all family restaraunts, along with a nice side of salad. Preferably ceasar. Arrr." His own process of self hazing back into society consisted of a cow prod, and several buckets of cheese.
In this instance Oscar Wilde had been turned from Hetro to Homo in a matter of days simply because of a miscommunication error. In turn he had produced mass amounts of gay erotica literature (which has never been recovered) and so the expression of "Arrr" has always been conveyed as a lightly used term. And though experiments were conducted in the mid 1990s more proof is needed as to the extent of how bad the consequences can get.
Although more complex versions of the word have found their place in the modern and pre-post-maybe-but-you-say-it-before-I-do-just-in-case-It-isn’t-cool-because-you’re-not-trying-to-score-a-date-with-the-captain-of-the-team Lexicon, such as: Yarrrn (as in “a ball of”), Yarrrrrrd (as in “the longest”) and Naarrrr (as in Johnny Depp’s response to any question involving his eyeliner the film “The curse of the black pearl”) most still find that nothing compares (or fits so neatly with “…Me mateys”) to the original. In it’s simplest form, the term ‘Arrr’ has the distinct privilege of being the only word now spoken by such intellectual greats as Physicist Stephen Hawking, and Legendary biscuit Pirate -Keanu Reeves as well as being one of the top five guttural noises overheard coming out of the men’s room at the annual Academy pre-awards chilli cook-off, though the latter is believed to have little to do with piracy.
Arrr currently (and eternally) resides in Hope Springs Vir. With his wife and 18 children we can’t be arsed to name.
Experiments of Arrr[edit]
A total of three experiments were conducted both inside and outside The Psycho Friends Network.
The first took place on a pirate ship in the new islands of the world, where three men on a cruise boat took the opportunity (and the money) to show the world what the meaning of 'Arrr' is to them. Their expressiveness through the week (totally five days, as their weekends dissolve into one day of binge drinking and whoring on the corners of cities) was remarkable. Each person they plundered and threatened were all reported to have gone into deep shock and given over all their belongings. Whether this was due to their extreme talent at "Arrr"ing or their machetes it is not yet known.
The second experiment was conducted in a country bumpkin kindergarten where the children had all been brought up on pigs and dirt, and could not state "Arrr" without it sounding like "Ya'll". The only thing this proved was that piracy was not for kids, and neither was English.
Finally, in 2035, about the time Emperor Schwarzenegger's encased head finally achieved sentiency and turned his robot armies against Manhattan (later destroyed by Admiral Will Smith). Sput -the Dog first launched into space by the russians, finally re-entered the atmosphere. Upon landing the locals (having never seen such an animal, attempted to communicate with the wayward canine through a series of clicks and beeps native to their home planet (plukesi IV) -to which the now long-since deceased animal failed to reply. Amazed at such wantonly callous disregard for common courtesy, the Plukesi attempted to rename their widely used word for "Etiquette" with the only decipherable Plukesi words embossed on the doomed russian ship: "Made" -Plukesian for "A", "In" -plukesian for "r" and "Canada" -Plukesian for a long or "double R" sound.
The trend did not catch on. we say arr to be cool. cauzee pirates "arr" cool! :D
Mathematics of Arr-ness[edit]
{\displaystyle {\frac {Arrr^{2}\}}{hazing^{cowprod}cheese}}+\left[{\frac {\gamma }{cowprod}}+{\frac {\delta }{chicken}}+{\frac {\epsilon }{1kgsugar}}\right]{\frac {dw}{dz}}-{\frac {\alpha \beta z-ThePsychoFriendsNetwork}{\left(OscarWilde\right)\left(z-d\right)}}w}
{\displaystyle =TheCoreof\int \left(scarring^{8}+purgatory\right)'Manhattanx=\int CaptainAroset^{4}cheesex\left({\frac {{\mathfrak {sacrifice\bigodot }}^{7}}{{\frac {x}{666}}Arr^{r}+C}}\right).}
Look up Angry pirate in Undictionary, the twisted dictionary
Retrieved from "https://uncyclopedia.com/w/index.php?title=Arrr&oldid=6132640"
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Coincidence algorithm - MATLAB coincidence - MathWorks 日本
Coincidence Algorithm
Staggered PRF Radar with Maximum Range
x = coincidence(res,div,maxval)
x = coincidence(res,div,maxval,tol)
x = coincidence(res,div,maxval) returns the scalar x that is less than or equal to maxval and is congruent to each remainder in res for the corresponding divisor in div. x satisfies
x = coincidence(res,div,maxval,tol) also specifies the tolerance. In practice, there may be no value that satisfies all constraints in res and div exactly. In that case, coincidence identifies a set of candidates that approximately satisfy the constraints and are within an interval of width 2 × tol centered at the candidates' median. The function then returns the median as x.
Find a number smaller than 1000 that has a remainder of 2 when divided by 9, a remainder of 3 when divided by 10.4, and a remainder of 6.3 when divided by 11.
There is no number that satisfies the constraints exactly, so specify a tolerance of 1. coincidence identifies a set of numbers that approximately satisfy the constraints and are within
2Ãtol=2
from their median. The function then outputs the median.
x = coincidence([2 3 6.3],[9 10.4 11],1000,tol)
Increase the tolerance to 2.
Specify a tolerance of 3.3. Any tolerance larger than this value results in the same answer.
In a staggered pulse repetition frequency (PRF) radar system, the first PRF corresponds to 70 range bins and the second PRF corresponds to 85 range bins. The target is detected at bin 47 for the first PRF and bin 12 for the second PRF. Assuming each range bin is 50 meters, compute the target range from these two measurements. Assume the farthest target can be 50 km away.
idx = coincidence([47 12],[70 85],50e3/50);
row vector of nonnegative numbers
Remainder array, specified as a row vector of nonnegative numbers. res must have the same number of elements as div.
maxval — Upper bound
Upper bound, specified as a positive scalar.
Tolerance, specified as a nonnegative scalar.
x — Congruent value
Congruent value, returned as a scalar.
crt | iscoprime
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Lychrel number - Wikipedia
Do any base-10 Lychrel numbers exist?
A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the 196-algorithm, after the most famous number associated with the process. In base ten, no Lychrel numbers have been yet proved to exist, but many, including 196, are suspected on heuristic[1] and statistical grounds. The name "Lychrel" was coined by Wade Van Landingham as a rough anagram of "Cheryl", his girlfriend's first name.[2]
1 Reverse-and-add process
1.1 Formal definition of the process
2 Proof not found
3 Threads, seed and kin numbers
4 196 palindrome quest
Reverse-and-add process[edit]
The reverse-and-add process produces the sum of a number and the number formed by reversing the order of its digits. For example, 56 + 65 = 121. As another example, 125 + 521 = 646.
Some numbers become palindromes quickly after repeated reversal and addition, and are therefore not Lychrel numbers. All one-digit and two-digit numbers eventually become palindromes after repeated reversal and addition.
About 80% of all numbers under 10,000 resolve into a palindrome in four or fewer steps; about 90% of those resolve in seven steps or fewer. Here are a few examples of non-Lychrel numbers:
56 becomes palindromic after one iteration: 56+65 = 121.
57 becomes palindromic after two iterations: 57+75 = 132, 132+231 = 363.
59 becomes a palindrome after three iterations: 59+95 = 154, 154+451 = 605, 605+506 = 1111
89 takes an unusually large 24 iterations (the most of any number under 10,000 that is known to resolve into a palindrome) to reach the palindrome 8813200023188.
10,911 reaches the palindrome 4668731596684224866951378664 (28 digits) after 55 steps.
1,186,060,307,891,929,990 takes 261 iterations to reach the 119-digit palindrome 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544, which was a former world record for the Most Delayed Palindromic Number. It was solved by Jason Doucette's algorithm and program (using Benjamin Despres' reversal-addition code) on November 30, 2005.
On January 23, 2017 a Russian schoolboy, Andrey S. Shchebetov, announced on his web site that he had found a sequence of the first 126 numbers (125 of them never reported before) that take exactly 261 steps to reach a 119-digit palindrome. This sequence was published in OEIS as A281506. This sequence started with 1,186,060,307,891,929,990 - by then the only publicly known number found by Jason Doucette back in 2005. On May 12, 2017 this sequence was extended to 108864 terms in total and included the first 108864 delayed palindromes with 261-step delay. The extended sequence ended with 1,999,291,987,030,606,810 - its largest and its final term.
On 26 April 2019, Rob van Nobelen computed a new World Record for the Most Delayed Palindromic Number: 12,000,700,000,025,339,936,491 takes 288 iterations to reach a 142 digit palindrome.
On 5 January 2021, Anton Stefanov computed two new Most Delayed Palindromic Numbers: 13968441660506503386020 and 13568441660506503386420 takes 289 iterations to reach the same 142 digit palindrome as the Rob van Nobelen number.
On December 14, 2021, Dmitry Maslov computed a new World Record for the Most Delayed Palindromic Number: 1,000,206,827,388,999,999,095,750 takes 293 iterations to reach 132 digit palindrome.
The OEIS sequence A326414 contains 19353600 terms with 288-step delay known at present.
Any number from A281506 could be used as a primary base to construct higher order 261-step palindromes. For example, based on 1,999,291,987,030,606,810 the following number 199929198703060681000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001999291987030606810 also becomes a 238-digit palindrome 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544 after 261 steps.
The smallest number that is not known to form a palindrome is 196. It is the smallest Lychrel number candidate.
The number resulting from the reversal of the digits of a Lychrel number not ending in zero is also a Lychrel number.
Formal definition of the process[edit]
{\displaystyle n}
be a natural number. We define the Lychrel function for a number base
{\displaystyle b>1}
{\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} }
{\displaystyle F_{b}(n)=n+\sum _{i=0}^{k-1}d_{i}b^{k-i-1}}
{\displaystyle k=\lfloor \log _{b}{n}\rfloor +1}
is the number of digits in the number in base
{\displaystyle b}
{\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}}
is the value of each digit of the number. A number is a Lychrel number if there does not exist a natural number
{\displaystyle i}
{\displaystyle F_{b}^{i+1}(n)=2F_{b}^{i}(n)}
{\displaystyle F^{i}}
{\displaystyle i}
{\displaystyle F}
Proof not found[edit]
In other bases (these bases are power of 2, like binary and hexadecimal), certain numbers can be proven to never form a palindrome after repeated reversal and addition,[3] but no such proof has been found for 196 and other base 10 numbers.
It is conjectured that 196 and other numbers that have not yet yielded a palindrome are Lychrel numbers, but no number in base ten has yet been proven to be Lychrel. Numbers which have not been demonstrated to be non-Lychrel are informally called "candidate Lychrel" numbers. The first few candidate Lychrel numbers (sequence A023108 in the OEIS) are:
The numbers in bold are suspected Lychrel seed numbers (see below). Computer programs by Jason Doucette, Ian Peters and Benjamin Despres have found other Lychrel candidates. Indeed, Benjamin Despres' program has identified all suspected Lychrel seed numbers of less than 17 digits.[4] Wade Van Landingham's site lists the total number of found suspected Lychrel seed numbers for each digit length.[5]
The brute-force method originally deployed by John Walker has been refined to take advantage of iteration behaviours. For example, Vaughn Suite devised a program that only saves the first and last few digits of each iteration, enabling testing of the digit patterns in millions of iterations to be performed without having to save each entire iteration to a file.[6] However, so far no algorithm has been developed to circumvent the reversal and addition iterative process.
Threads, seed and kin numbers[edit]
The term thread, coined by Jason Doucette, refers to the sequence of numbers that may or may not lead to a palindrome through the reverse and add process. Any given seed and its associated kin numbers will converge on the same thread. The thread does not include the original seed or kin number, but only the numbers that are common to both, after they converge.
Seed numbers are a subset of Lychrel numbers, that is the smallest number of each non palindrome producing thread. A seed number may be a palindrome itself. The first three examples are shown in bold in the list above.
Kin numbers are a subset of Lychrel numbers, that include all numbers of a thread, except the seed, or any number that will converge on a given thread after a single iteration. This term was introduced by Koji Yamashita in 1997.
196 palindrome quest[edit]
Because 196 (base-10) is the lowest candidate Lychrel number, it has received the most attention.
In the 1980s, the 196 palindrome problem attracted the attention of microcomputer hobbyists, with search programs by Jim Butterfield and others appearing in several mass-market computing magazines.[7][8][9] In 1985 a program by James Killman ran unsuccessfully for over 28 days, cycling through 12,954 passes and reaching a 5366-digit number.[9]
John Walker began his 196 Palindrome Quest on 12 August 1987 on a Sun 3/260 workstation. He wrote a C program to perform the reversal and addition iterations and to check for a palindrome after each step. The program ran in the background with a low priority and produced a checkpoint to a file every two hours and when the system was shut down, recording the number reached so far and the number of iterations. It restarted itself automatically from the last checkpoint after every shutdown. It ran for almost three years, then terminated (as instructed) on 24 May 1990 with the message:
196 had grown to a number of one million digits after 2,415,836 iterations without reaching a palindrome. Walker published his findings on the Internet along with the last checkpoint, inviting others to resume the quest using the number reached so far.
In 1995, Tim Irvin and Larry Simkins used a multiprocessor computer and reached the two million digit mark in only three months without finding a palindrome. Jason Doucette then followed suit and reached 12.5 million digits in May 2000. Wade VanLandingham used Jason Doucette's program to reach 13 million digits, a record published in Yes Mag: Canada's Science Magazine for Kids. Since June 2000, Wade VanLandingham has been carrying the flag using programs written by various enthusiasts. By 1 May 2006, VanLandingham had reached the 300 million digit mark (at a rate of one million digits every 5 to 7 days). Using distributed processing,[10] in 2011 Romain Dolbeau completed a billion iterations to produce a number with 413,930,770 digits, and in February 2015 his calculations reached a number with billion digits.[11] A palindrome has yet to be found.
Other potential Lychrel numbers which have also been subjected to the same brute force method of repeated reversal addition include 879, 1997 and 7059: they have been taken to several million iterations with no palindrome being found.[12]
In base 2, 10110 (22 in decimal) has been proven to be a Lychrel number, since after 4 steps it reaches 10110100, after 8 steps it reaches 1011101000, after 12 steps it reaches 101111010000, and in general after 4n steps it reaches a number consisting of 10, followed by n+1 ones, followed by 01, followed by n+1 zeros. This number obviously cannot be a palindrome, and none of the other numbers in the sequence are palindromes.
Lychrel numbers have been proven to exist in the following bases: 11, 17, 20, 26 and all powers of 2.[13][14][15]
No base contains any Lychrel numbers smaller than the base. In fact, in any given base b, no single-digit number takes more than two iterations to form a palindrome. For b > 4, if k < b/2, then k becomes palindromic after one iteration: k + k = 2k, which is single-digit in base b (and thus a palindrome). If k > b/2, k becomes palindromic after two iterations.
The smallest number in each base which could possibly be a Lychrel number are (sequence A060382 in the OEIS):
Smallest possible Lychrel number in base b
written in base b (base 10)
11 83A (1011)
13 12CA (2701)
14 1BB (361)
15 1EC (447)
16 19D (413)
17 B6G (3297)
18 1AF (519)
19 HI (341)
20 IJ (379)
21 1CI (711)
22 KL (461)
23 LM (505)
24 MN (551)
25 1FM (1022)
26 OP (649)
27 PQ (701)
28 QR (755)
29 RS (811)
31 TU (929)
32 UV (991)
33 VW (1055)
34 1IV (1799)
35 1JW (1922)
36 YZ (1259)
Lychrel numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
^ O'Bryant, Kevin (26 December 2012). "Reply to Status of the 196 conjecture?". MathOverflow.
^ "FAQ". Archived from the original on 2006-12-01.
^ Brown, Kevin. "Digit Reversal Sums Leading to Palindromes". MathPages.
^ VanLandingham, Wade. "Lychrel Records". p196.org. Archived from the original on 2016-04-28. Retrieved 2011-08-29.
^ VanLandingham, Wade. "Identified Seeds". p196.org. Archived from the original on 2016-04-28. Retrieved 2011-08-29.
^ "On Non-Brute Force Methods". Archived from the original on 2006-10-15.
^ "Bits and Pieces". The Transactor. Transactor Publishing. 4 (6): 16–23. 1984. Retrieved 26 December 2014.
^ Rupert, Dale (October 1984). "Commodares: Programming Challenges". Ahoy!. Ion International (10): 23, 97–98.
^ a b Rupert, Dale (June 1985). "Commodares: Programming Challenges". Ahoy!. Ion International (18): 81–84, 114.
^ Swierczewski, Lukasz; Dolbeau, Romain (June 23, 2014). The p196_mpi Implementation of the Reverse-And-Add Algorithm for the Palindrome Quest. International Supercomputing Conference. Leipzig, Germany.
^ Dolbeau, Romain. "The p196_mpi page". www.dolbeau.name.
^ "Lychrel Records". Archived from the original on December 5, 2003. Retrieved September 2, 2016.
^ See comment section in OEIS: A060382
^ "Digit Reversal Sums Leading to Palindromes".
^ "Letter from David Seal". Archived from the original on 2013-05-30. Retrieved 2017-03-08.
OEIS sequence A023108 (Positive integers which apparently never result in a palindrome under ...)
John Walker – Three years of computing
Tim Irvin – About two months of computing
Jason Doucette – World records – 196 Palindrome Quest, Most Delayed Palindromic Number
196 and Other Lychrel Numbers by Wade VanLandingham
Weisstein, Eric W. "196-Algorithm". MathWorld.
MathPages – Digit Reversal Sums Leading to Palindromes
All known delayed palindromic numbers – Dmitry Maslov, MDPN project
Testing a delayed palindrome – Dmitry Maslov, MDPN project
Retrieved from "https://en.wikipedia.org/w/index.php?title=Lychrel_number&oldid=1084890889"
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Model Theory and Groups | EMS Press
Frank Olaf Wagner
The workshop \emph{Model Theory and Groups}, organised by Andreas Baudisch (Berlin), David Marker (Chicago), Katrin Tent (Bielefeld) and Frank Wagner (Lyon) was held January 14th--20th, 2007. This meeting focused on interactions between classical model theoretic investigations of groups and their applications to geometric group theory and vice versa. It was well attended with 55 scientists, both model theorists as well as geometric group theorists, including
11
women and a relatively large number of young researchers and students. Needless to say that participants came from a broad geographical background. For many years groups have played a central role in model theory, both in applied model theory where one is focused on understanding algebraic structures and, more surprisingly, in pure model theory where one is studying structures from an abstract viewpoint. At first, only the most basic tools from the general theory were needed in applications, but, over the last ten years, some of the most sophisticated ideas from pure model theory have played an important role in applications, most notably Hrushovski's proof of the Mordell-Lang Conjecture for function fields. The investigation of variations of Mordell-Lang like theorems in different situations played an important role in a number of talks. Geometric group theory and model theory have started interacting in the context of free groups and surface groups as well as in the study of the asymptotic behaviour of geometric properties on groups. This was a second main topic of the conference which particularly profited from the fact that researchers from different areas attended the meeting and presented their results. At the core of model theoretic investigations of groups were the reports on groups of finite Morley rank around the Cherlin-Zilber Conjecture which states that every simple group of finite Morley rank is an algebraic group over an algebraically closed field. While originally attempts at proving this conjecture have followed the lines for the classification of algebraic groups, more recent advances have been made by adapting and generalising ideas from the classification of finite simple groups, in particular the study of the 2-Sylow subgroup, which has allowed a distinction into three cases: even characteristic, odd characteristic (including
0
) and degenerate (no involutions). The even case is solved, and important progress has been made in the other cases. The recent construction of so-called bad fields, i.e.\ fields of finite Morley rank with a distinguished multiplicative subgroup also added new impetus to the search for new proofs not involving assumptions on the non-existence of such fields. The organisers asked Dugald Macpherson and Charles Steinhorn before the conference to give a three-lecture tutorial on asymptotic classes and measurable structures. This is a new development in model theory generalising results on finite and pseudofinite fields. In addition, 27 participants were invited to report on their research (18 long and 9 short talks). Altogether it was a very successful workshop which inspired a number of new cooperations and further projects. The reader may find here extended abstracts of all talks (in the order in which the talks were given).
Andreas Baudisch, David Marker, Katrin Tent, Frank Olaf Wagner, Model Theory and Groups. Oberwolfach Rep. 4 (2007), no. 1, pp. 83–138
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The Deepwater Horizon oil spill in the Gulf of Mexico on April 20, 2010 was "an environmental disaster of unprecedented proportions" and was a "devastating blow to the resource-dependent economy of the region" [1]. According to the Washington Post, it is estimated that [math]
9857
cubic meters of oil per day spilled into the Gulf of Mexico on August 2, 2010 [2].
Assume that an average of [math]
9857 \ \mathrm{m^3/day}
of oil spilled into the Gulf of Mexico every day and that it formed a hemispherical dome of radius [math]
r
on the ocean floor.
Find a formula for the volume of oil spilled, in cubic meters, [math]
t
days after the start of the oil spill. Include units in your answer.
V(t) =
help (units)
Find the derivative of your volume formula with respect to time. Include units in your answer.
\displaystyle \frac{dV}{dt} =
When the volume of the oil spill is [math]
50,000 \ \mathrm{m^3}
, what is the radius of the hemisphere of oil? Include units in your answer. Give your answer accurate to 3 decimal places.
r =
Find the rate of change of the radius with respect to time when the volume of the oil spill is [math]
50,000 \ \mathrm{m^3}
. Include units in your answer.
\displaystyle \frac{dr}{dt} =
Since only the oil that is in contact with seawater can mix with seawater, it is important to know how much of the surface area of the oil spill is in contact with seawater. Find the rate of change of the hemisphere's surface area with respect to time when the volume of the oil spill is [math]
50,000 \ \mathrm{m^3}
. Include units in your answer. You should assume that only the top of the hemispherical dome of oil comes in contact with seawater (not the flat bottom, which is incontact with the ocean floor).
\displaystyle \frac{dA}{dt} =
\displaystyle A'(t) = \frac{dA}{dt}
a constant function? If it is not, is it an increasing or decreasing function of time? In other words, as [math]
t
increases, does [math]
\displaystyle \frac{dA}{dt}
stay the same, increase, or decrease? Explain and justify your answer using complete sentences with correct grammar, spelling, and punctuation. Please type your answer in the box below.
[1] Deepwater Horizon Oil Spill, Phase I Early Restoration Plan and Environmental Assessment, Prepared by the Deepwater Horizon Natural Resource Trustees from: State of Alabama, State of Florida, State of Louisiana, State of Mississippi, State of Texas, Department of the Interior, National Oceanic and Atmospheric Administration. link Retrieved Sept. 15, 2012.
[2] Joel Achenbach and David Fahrenthold (2010-08-02). "Oil well spilled out 4.9 million barrels, new numbers reveal". Washington Post. Retrieved 2010-05-25.
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Whitehead's_theory_of_gravitation Knowpia
In theoretical physics, Whitehead's theory of gravitation was introduced by the mathematician and philosopher Alfred North Whitehead in 1922.[1] While never broadly accepted, at one time it was a scientifically plausible alternative to general relativity. However, after further experimental and theoretical consideration, the theory is now generally regarded as obsolete.
Principal featuresEdit
Whitehead developed his theory of gravitation by considering how the world line of a particle is affected by those of nearby particles. He arrived at an expression for what he called the "potential impetus" of one particle due to another, which modified Newton's law of universal gravitation by including a time delay for the propagation of gravitational influences. Whitehead's formula for the potential impetus involves the Minkowski metric, which is used to determine which events are causally related and to calculate how gravitational influences are delayed by distance. The potential impetus calculated by means of the Minkowski metric is then used to compute a physical spacetime metric
{\displaystyle g_{\mu \nu }}
, and the motion of a test particle is given by a geodesic with respect to the metric
{\displaystyle g_{\mu \nu }}
.[2][3] Unlike the Einstein field equations, Whitehead's theory is linear, in that the superposition of two solutions is again a solution. This implies that Einstein's and Whitehead's theories will generally make different predictions when more than two massive bodies are involved.[4]
Following the notation of Chiang and Hamity[5] , introduce a Minkowski spacetime with metric tensor
{\displaystyle \eta _{ab}=\mathrm {diag} (1,-1,-1,-1)}
, where the indices
{\displaystyle a,b}
run from 0 through 3, and let the masses of a set of gravitating particles be
{\displaystyle m_{a}}
The Minkowski arc length of particle
{\displaystyle A}
{\displaystyle \tau _{A}}
. Consider an event
{\displaystyle p}
{\displaystyle \chi ^{a}}
. A retarded event
{\displaystyle p_{A}}
{\displaystyle \chi _{A}^{a}}
on the world-line of particle
{\displaystyle A}
is defined by the relations
{\displaystyle (y_{A}^{a}=\chi ^{a}-\chi _{A}^{a},y_{A}^{a}y_{Aa}=0,y_{A}^{0}>0)}
. The unit tangent vector at
{\displaystyle p_{A}}
{\displaystyle \lambda _{A}^{a}=(dx_{A}^{a}/d\tau _{A})p_{A}}
. We also need the invariants
{\displaystyle w_{A}=y_{A}^{a}\lambda _{Aa}}
. Then, a gravitational tensor potential is defined by
{\displaystyle g_{ab}=\eta _{ab}-h_{ab},}
{\displaystyle h_{ab}=2\sum _{A}{\frac {m_{A}}{w_{A}^{3}}}y_{Aa}y_{Ab}.}
It is the metric
{\displaystyle g}
that appears in the geodesic equation.
Whitehead's theory is equivalent with the Schwarzschild metric[4] and makes the same predictions as general relativity regarding the four classical solar system tests (gravitational red shift, light bending, perihelion shift, Shapiro time delay), and was regarded as a viable competitor of general relativity for several decades. In 1971, Will argued that Whitehead's theory predicts a periodic variation in local gravitational acceleration 200 times longer than the bound established by experiment.[6][7] Misner, Thorne and Wheeler's textbook Gravitation states that Will demonstrated "Whitehead's theory predicts a time-dependence for the ebb and flow of ocean tides that is completely contradicted by everyday experience".[8]: 1067
Fowler argued that different tidal predictions can be obtained by a more realistic model of the galaxy.[9][2] Reinhardt and Rosenblum claimed that the disproof of Whitehead's theory by tidal effects was "unsubstantiated".[10] Chiang and Hamity argued that Reinhardt and Rosenblum's approach "does not provide a unique space-time geometry for a general gravitation system", and they confirmed Will's calculations by a different method.[5] In 1989, a modification of Whitehead's theory was proposed that eliminated the unobserved sidereal tide effects. However, the modified theory did not allow the existence of black holes.[11]
Subrahmanyan Chandrasekhar wrote, "Whitehead's philosophical acumen has not served him well in his criticisms of Einstein."[12]
Philosophical disputesEdit
Clifford M. Will argued that Whitehead's theory features a prior geometry.[13] Under Will's presentation (which was inspired by John Lighton Synge's interpretation of the theory[14][15]), Whitehead's theory has the curious feature that electromagnetic waves propagate along null geodesics of the physical spacetime (as defined by the metric determined from geometrical measurements and timing experiments), while gravitational waves propagate along null geodesics of a flat background represented by the metric tensor of Minkowski spacetime. The gravitational potential can be expressed entirely in terms of waves retarded along the background metric, like the Liénard–Wiechert potential in electromagnetic theory.
A cosmological constant can be introduced by changing the background metric to a de Sitter or anti-de Sitter metric. This was first suggested by G. Temple in 1923.[16] Temple's suggestions on how to do this were criticized by C. B. Rayner in 1955.[17][18]
Will's work was disputed by Dean R. Fowler, who argued that Will's presentation of Whitehead's theory contradicts Whitehead's philosophy of nature. For Whitehead, the geometric structure of nature grows out of the relations among what he termed "actual occasions". Fowler claimed that a philosophically consistent interpretation of Whitehead's theory makes it an alternate, mathematically equivalent, presentation of general relativity.[9] In turn, Jonathan Bain argued that Fowler's criticism of Will was in error.[2]
^ Whitehead, A. N. (2011-06-16) [1922]. The Principle of Relativity: With Applications to Physical Science. Cambridge University Press. ISBN 978-1-107-60052-2.
^ a b c Bain, Jonathan (1998). "Whitehead's Theory of Gravity". Stud. Hist. Phil. Mod. Phys. 29 (4): 547–574. Bibcode:1998SHPMP..29..547B. doi:10.1016/s1355-2198(98)00022-7.
^ Synge, J. L. (1952-03-06). "Orbits and rays in the gravitational field of a finite sphere according to the theory of A. N. Whitehead". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 211 (1106): 303–319. Bibcode:1952RSPSA.211..303S. doi:10.1098/rspa.1952.0044. ISSN 0080-4630. S2CID 121363087.
^ a b Eddington, Arthur S. (1924). "A comparison of Whitehead's and Einstein's formulas". Nature. 113 (2832): 192. Bibcode:1924Natur.113..192E. doi:10.1038/113192a0. S2CID 36114166.
^ a b Chiang, C. C.; Hamity, V. H. (August 1975). "On the local newtonian gravitational constant in Whitehead's theory". Lettere al Nuovo Cimento. Series 2. 13 (12): 471–475. doi:10.1007/BF02745961. ISSN 1827-613X. S2CID 121832243.
^ Will, Clifford M. (1971). "Relativistic Gravity in the Solar System. II. Anisotropy in the Newtonian Gravitational Constant". The Astrophysical Journal. IOP Publishing. 169: 141. Bibcode:1971ApJ...169..141W. doi:10.1086/151125. ISSN 0004-637X.
^ Gibbons, Gary; Will, Clifford M. (2008). "On the multiple deaths of Whitehead's theory of gravity". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 39 (1): 41–61. arXiv:gr-qc/0611006. Bibcode:2008SHPMP..39...41G. doi:10.1016/j.shpsb.2007.04.004. ISSN 1355-2198. S2CID 17017857.
^ Misner, Charles W.; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 978-0-7167-0344-0.
^ a b Fowler, Dean (Winter 1974). "Disconfirmation of Whitehead's Relativity Theory -- A Critical Reply". Process Studies. 4 (4): 288–290. doi:10.5840/process19744432. Archived from the original on 2013-01-08.
^ Reinhardt, M.; Rosenblum, A. (1974). "Whitehead contra Einstein". Physics Letters A. Elsevier BV. 48 (2): 115–116. Bibcode:1974PhLA...48..115R. doi:10.1016/0375-9601(74)90425-3. ISSN 0375-9601.
^ Hyman, Andrew (1989). "A New Interpretation of Whitehead's Theory" (PDF). Il Nuovo Cimento. 387 (4): 387–398. Bibcode:1989NCimB.104..387H. doi:10.1007/bf02725671. S2CID 122670014. Archived from the original (PDF) on 2012-02-04.
^ Chandrasekhar, S. (March 1979). "Einstein and general relativity: Historical perspectives". American Journal of Physics. 47 (3): 212–217. doi:10.1119/1.11666. ISSN 0002-9505.
^ Will, Clifford (1972). "Einstein on the Firing Line". Physics Today. 25 (10): 23–29. Bibcode:1972PhT....25j..23W. doi:10.1063/1.3071044.
^ Synge, John (1951). Relativity Theory of A. N. Whitehead. Baltimore: University of Maryland.
^ Tanaka, Yutaka (1987). "Einstein and Whitehead-The Comparison between Einstein's and Whitehead's Theories of Relativity". Historia Scientiarum. 32.
^ Temple, G. (1924). "Central Orbit in Relativistic Dynamics Treated by the Hamilton-Jacobi Method". Philosophical Magazine. 6. 48 (284): 277–292. doi:10.1080/14786442408634491.
^ Rayner, C. (1954). "The Application of the Whitehead Theory of Relativity to Non-static Spherically Symmetrical Systems". Proceedings of the Royal Society of London. 222 (1151): 509–526. Bibcode:1954RSPSA.222..509R. doi:10.1098/rspa.1954.0092. S2CID 123355240.
^ Rayner, C. (1955). "The Effects of Rotation in the Central Body on its Planetary Orbits after the Whitehead Theory of Gravitation". Proceedings of the Royal Society of London. 232 (1188): 135–148. Bibcode:1955RSPSA.232..135R. doi:10.1098/rspa.1955.0206. S2CID 122796647.
Will, Clifford M. (1993). Was Einstein Right?: Putting General Relativity to the Test (2nd ed.). Basic Books. ISBN 978-0-465-09086-0.
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The 12cc Penn State Pulsatile Pediatric Ventricular Assist Device: Fluid Dynamics Associated With Valve Selection | J. Biomech Eng. | ASME Digital Collection
12cc
Penn State Pulsatile Pediatric Ventricular Assist Device: Fluid Dynamics Associated With Valve Selection
Benjamin T. Cooper,
Benjamin T. Cooper
, 205 Hallowell Building, University Park, PA 16802
Tobias C. Long,
e-mail: kbm10@psu.edu
Cooper, B. T., Roszelle, B. N., Long, T. C., Deutsch, S., and Manning, K. B. (June 24, 2008). "The
12cc
Penn State Pulsatile Pediatric Ventricular Assist Device: Fluid Dynamics Associated With Valve Selection." ASME. J Biomech Eng. August 2008; 130(4): 041019. https://doi.org/10.1115/1.2939342
The mortality rate for infants awaiting a heart transplant is 40% because of the extremely limited number of donor organs. Ventricular assist devices (VADs), a common bridge-to-transplant solution in adults, are becoming a viable option for pediatric patients. A major obstacle faced by VAD designers is thromboembolism. Previous studies have shown that the interrelated flow characteristics necessary for the prevention of thrombosis in a pulsatile VAD are a strong inlet jet, a late diastolic recirculating flow, and a wall shear rate greater than
500s−1
. Particle image velocimetry was used to compare the flow fields in the chamber of the
12cc
Penn State pediatric pulsatile VAD using two mechanical heart valves: Björk–Shiley monostrut (BSM) tilting disk valves and CarboMedics (CM) bileaflet valves. In conjunction with the flow evaluation, wall shear data were calculated and analyzed to help quantify wall washing. The major orifice inlet jet of the device containing BSM valves was more intense, which led to better recirculation and wall washing than the three jets produced by the CM valves. Regurgitation through the CM valve served as a significant hindrance to the development of the rotational flow.
biomedical equipment, cardiology, diseases, haemodynamics, jets, paediatrics, prosthetics, rotational flow, shear flow, pediatric ventricular assist device, thrombosis, fluid dynamics, mechanical heart valves, pulsatile, wall shear, particle image velocimetry
Flow (Dynamics), Pediatrics, Rotational flow, Shear (Mechanics), Valves, Ventricular assist devices, Shear rate, Jets
,” Dallas, TX.
Pierce-Donarchy Pediatric VAD: Progress in Development
Health Resources and Services Administration, Healthcare Systems Bureau, Division of Transplantation, 2006, “
Annual Report of the U.S. Organ Procurement and Transplantation Network and the Scientific Registry of Transplant Recipients: Transplant Data 1996–2005
Sakopoulus
Scand. J. Thorac. Cardiovasc. Surg.
Options for Mechanical Support in Pediatric Patients
Gruenweld
Determinants of Successes in Pediatric Cardiac Patients Undergoing Extracorporeal Membrane Oxygenation
Hemolytic Characteristics of Oxygenators During Clinical Extracorporeal Membrane Oxygenation
Comparison of Novacor and HeartMate Vented Electric Left Ventricular Assist Devices in a Single Institution
Clinical Outcomes in Patients With Chronic Congestive Heart Failure Who Undergo Left Ventricular Assist Device Implantation
Mechanical Circulatory Support: Reality and Dreams Experience of a Single Center
Total Artificial Hearts: Bridge to Transplantation
Health Technol. Assess
Randomized Evaluation of Mechanical Assistance for the Treatment of Congestive Heart Failure (REMATCH) Study Group. Long-Term Use of a Left Ventricular Assist Device for End-Stage Heart Failure
Rosenbergk
Fontainek
Gainesk
Piercek
The Pennsylvania State University Paracorporeal Ventricular Assist Device Assist Pump: Optimal Methods of Use
Testing of a 50cc Stroke Volume Completely Implantable Artificial Heart: Expanding Chronic Mechanical Circulatory Support to Women, Adolescents, and Small Stature Men
Mechanical Circulatory Support in Pediatric Patients
Current Clinical Status of Pulsatile Pediatric Support
Avarhami
The Hemodynamics of the Berlin Pulsatile VAD and the Role of Its MHV Configuration
Nagdyman
Heart Transplantation in Children After Mechanical Circulatory Support With Pulsatile Pneumatic Assist Device
Visualization and Analysis of Mural Thrombogenesis on Collage, Polyurethane and Nylon
Laser Doppler Velocimetry and Flow Visualization Studies in the Regurgitant Leakage Flow Region of Three Mechanical Mitral Valves
Numerical Model of Flow in a Sac-Type Ventricular Assist Device
The National Heart, Lung, and Blood Institute Pediatric Circulatory Support Program
The Fluid Mechanics of the Penn State 12cc Pulsatile Pediatric Ventricular Assist Device
Super-Resolution, PIV Processing by Recursive Local-Correlation
Off Design Considerations of the 50cc Penn State Ventricular Assist Device
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§ Coarse structures
A coarse structure on the set
X
is a collection of relations on
X
E \subseteq 2^{X \times X}
(called as controlled sets / entourages ) such that:
(\delta \equiv \{ (x, x) : x \in X \}) \in E
Closed under subsets:
\forall e \in E, f \subset e \implies f \in E
Closed under transpose: if
e \in E
(e^T \equiv \{ (y, x) : (x, y) \in e \}) \in E
Closed under finite unions.
Closed under composition:
\forall e, f \in E, e \circ f \in E
\circ
is composition of relations.
The sets that are controlled are "small" sets. The bounded coarse structure on a metric space
(X, d)
is the set of all relations such that there exists a uniform bound such that all related elements are within that bounded distance.
(e \subset X \times X) \in E \iff \exists \delta \in \mathbb R, \forall (x, y) \in E, d(x, y) < \delta
We can check that the functions:
f: \mathbb Z \rightarrow \mathbb R, f(x) \equiv x
g: \mathbb R \rightarrow \mathbb Z, g(x) \equiv \lfloor x \rfloor
are coarse inverses to each other. I am interested in this because if topology is related to semidecidability, then coarse structures (which are their dual) are related to..?
What is a.. coarse structure by AMS
|
§ Symplectic version of classical mechanics
§ Basics, symplectic mechanics as inverting
\omega
I've never seen this kind of "inverting
\omega
" perspective written down anywhere. Most of them start by using the inteior product
i_X \omega
without ever showing where the thing came from. This is my personal interpretation of how the symplectic version of classical mecanics comes to be. If we have a non-degenerate, closed two-form
\omega: T_pM \times T_pM \rightarrow \mathbb R
. Now, given a hamiltonian
H: M \rightarrow \mathbb R
, we can construct a vector field
X_H: M \rightarrow TM
under the definition:
\begin{aligned} &\text{partially apply $\omega$ to see $\omega$ as a mapping from $T_p$ to $T_p^*M$} \\ &\omega2: T_p M \rightarrow T_p*M \equiv \lambda (v: T_p M). \lambda (w: T_p M) . \omega(v, w) \\ &\omega2^{-1}: T_p^*M \rightarrow T_p M; dH: M \rightarrow T_p* M \\ &X_H \equiv \lambda (p: M) \rightarrow \omega2^{-1} (dH(p)) \\ &(p: M) \xrightarrow{dH} dH(p) : T_p*M \xrightarrow{\omega2^{-1}} \omega2^{-1}(dH(p)): T_pM \\ &X_H = \omega2^{-1} \circ dH \end{aligned}
This way, given a hamiltonian
H: M \rightarrow \mathbb R
, we can construct an associated vector field
X_H
, in a pretty natural way. We can also go the other way. Given the
X
, we can build the
dH
under the equivalence:
\begin{aligned} &\omega2^{-1} \circ dH = X_H\\ &dH = \omega2(X_H) \\ &\int dH = \int \omega2(X_H) \\ &H = \int \omega2(X_H) \end{aligned}
This needs some demands, like the one-form
dH
being integrable. But this works, and gives us a bijection between
X_H
H
as we wanted. We can also analyze the definition we got from the previous manipulation:
\begin{aligned} &\omega2(X_H) = dH \\ &\lambda (w: T_p M) \omega(X_H, w) = dH \\ &\omega(X_H, \cdot) = dH \\ \end{aligned}
We can take this as a relationship between
X_H
dH
. Exploiting this, we can notice that
dH(X_H) = 0
. That is, moving along
X_H
does not modify
dH
\begin{aligned} &\omega2(X_H) = dH \\ &\lambda (w: T_p M) \omega(X_H, w) = dH \\ &dH(X_H) = \omega(X_H, X_H) = 0 ~ \text{$\omega$ is anti-symmetric} \end{aligned}
§ Preservation of
\omega
X_H^*(\omega) = \omega
. That is, pushing forward
\omega
along the vector field
X_H
\omega
. TODO.
§ Moment Map
Now that we have a method of going from a vector field
X_H
to a Hamiltonian
H
, we can go crazier with this. We can generate vector fields using Lie group actions on the manifold, and then look for hamiltonians corresponding to this lie group. This lets us perform "inverse Noether", where for a given choice of symmetry, we can find the Hamiltonian that possesses this symmetry. We can create a map from the Lie algebra
\mathfrak{g} \in \mathfrak{G}
to a vector field
X_{\mathfrak g}
, performing:
\begin{aligned} &t : \mathbb R \mapsto e^{t\mathfrak g} : G \\ &t : \mathbb R \mapsto \phi(e^{t\mathfrak g}) : M \\ &X_{\mathfrak g} \equiv \frac{d}{dt}(\phi(e^{t\mathfrak g}))|_{t = 0}: TM \end{aligned}
We can then attempt to recover a hamiltonian
H_{\mathfrak g}
X_{\mathfrak g}
. If we get a hamiltonian from this process, then it will have the right symmetries.
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Compute positive-sequence active and reactive powers - Simulink - MathWorks United Kingdom
Power (PLL-Driven, Positive-Sequence)
The Power (PLL-Driven, Positive-Sequence) block computes the positive-sequence active power P (in watts) and reactive power Q (in vars) of a periodic set of three-phase voltages and currents. To perform this computation, the block first computes the positive sequence of the input voltages and currents with a running window over one cycle of the fundamental frequency given by input 1. The reference frame required for the computation is given by the input 2. The first two inputs are normally connected to the outputs of a PLL block.
These formulas are then evaluated:
\begin{array}{c}P=3×\frac{|{V}_{1}|}{\sqrt{2}}×\frac{|{I}_{1}|}{\sqrt{2}}×\mathrm{cos}\left(\phi \right)\\ Q=3×\frac{|{V}_{1}|}{\sqrt{2}}×\frac{|{I}_{1}|}{\sqrt{2}}×\mathrm{sin}\left(\phi \right)\\ \phi =\angle {V}_{1}-\angle {I}_{1}\end{array}
Specify the minimum frequency value to determine the buffer size of the Variable Time Delay block used by the block to compute the phasors. Default is 45.
Specify the initial positive-sequence magnitude and phase (relative to the PLL phase), in degrees, of the voltage signals. Default is [1, 0].
Specify the initial positive-sequence magnitude and phase (relative to the PLL phase), in degrees, of the current signals. Default is [1, 0].
The vectorized signal of the three [a b c ] voltage sinusoidal signals. Typical input signals are voltages measured by the Three-Phase V-I Measurement block.
The vectorized signal of the three [a b c ] current sinusoidal signals. Typical input signals are currents measured by the Three-Phase V-I Measurement block.
Positive-sequence active power (watts)
Positive-sequence reactive power (vars)
The power_PowerPLLDrivenPositiveSequence model shows how the block evaluates the positive-sequence active and reactive powers of a voltage source connected to a three-phase load. It shows that the block outputs accurate values for P and Q even if the fundamental frequency of the voltage supply (containing harmonics) varies during the simulation.
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Online derivative calculator with steps and function graphing
\dfrac{d}{dx}
`sin(sqrt(e^x + a)/2)`
Differentiate w.r.t to
Our calculator will calculate the derivative of any function using the common rules of differentiation (product rule, quotient rule, chain rule, etc.). It can show the steps and interactive graphing for both input and result function. It can handle polynomial, rational, irrational, exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic and inverse hyperbolic functions. Also, it will evaluate the derivative at the given point, if needed. You can pick a random function using the 'Random function' button. You can also choose the order of the derivative by entering a value from min 1 to max 5 in 'Times?'.
A derivative of a function refers to the expression that, generally, represents the slope of a function at any point within its domain.
If a function is defined as
y=f(x)+c
, then the derivative of the function is denoted by
\frac{df}{dx}
f^{\prime}(x)
y^{\prime}
For instance; if a straight line
y=f(x)+c
passes through points (1,0) and (2,4), then its derivative which is the slope of the line is
y^{\prime}=f^{\prime}(x)=\frac{d f}{d x}=\frac{\Delta y}{\Delta x}=\frac{4-0}{2-1}=3
When we find the derivative of a function, we say, we are differentiating the function. If a function allows, we can repeatedly find the derivative of the derivatives. The first derivative is usually termed as slope or gradient or simply derivative. However, if we are interested in other derivatives other than the first one, it is always specified as second, third, or fourth derivatives, and so on.
The second derivative implies differentiating the function twice or simply, the derivative of the first derivative. We denote it as
y^{\prime \prime}=f^{\prime \prime}(x)=\frac{d f^{2}}{d x^{2}}=\frac{d}{d x}\left(\frac{d f}{d x}\right)
The third derivative is the derivative of the second derivative. It is also similar to differentiating the function thrice. We have
y^{\prime \prime \prime}=f^{\prime \prime \prime}(x)=\frac{d f^{3}}{d x^{3}}=\frac{d}{d x}\left(\frac{d^{2} f}{d x^{2}}\right)
Significance of derivative
The first derivative represents the rate of change of one quantity with respect to another. The first derivative is used in finance to quantify the rate of investment, the rate of growth of an investment, the rate of inflation, the rate of growth on production among others. It is also used in mechanics to quantify the speed, velocity, and rate of change of momentum among others. It is also used to measure the rate of change of energy. In the chemical field, it is used to measure the rate of change of chemical concentrations, the rate of reaction among
In mechanics: Velocity
Velocity or speed is the rate of change of displacement of distance respectively. It is used to measure the rate at which a body moves.
A train covered a distance of 186 miles in two hours. Find the average speed of the train.
\frac{\text { change in distance }}{\text { Change in time }}=\frac{186}{2}=93 \text { miles } / \text { hour }
In financial matters: Rate of inflation:
The rate of inflation is a measure of how the price increases or decreases on the market over a given period. It is expressed as a percentage.
The price of one gallon of gas was $3.034 a year ago. If the current price is 3.190, find the inflation rate during that year.
\frac{\mid \text { Current price }-\text { Old price } \mid}{\text { Old price }} \times 100 \%=\frac{|3.034-3.190|}{3.034}=\frac{0.156}{3.034} \times 100 \%=5.14 \%
In financial matters: Rate of income growth:
The rate of income growth in a country can be measured in terms of the derivative, or gradient of the average income of the person in that country. This can be measured using per-capita income.
The per-capita income of the US was in 1990 and in 2000. Determine the rate of income growth over the 10 years.
\begin{array}{c} \frac{\text { |Current per capita income }-\text { Old per capita income } \mid}{\text { Old per capita income }} \times 100 \%=\frac{|36,000-23,640|}{36,000}=\frac{12,360}{36,000} \times 100 \%=34.33 \% \end{array}
The derivatives are categorized into two main types, ordinary derivative or partial derivative. However, we use the term derivative to simply imply the first ordinary derivative of a function. When we have an expression that is a function of one variable only, then the derivative is an ordinary derivative. For instance, we have y = f(x) or h = g(t), then the first derivatives will simply be total and will be given as
y^{\prime}=\frac{d f}{d x} \quad \text { or } h^{\prime}=\frac{d g}{d t}
f(x) = 3x^2 + 2
with respect to x is
f^{\prime}(x)=2 \times 3\left(x^{2-1}\right)=6 x
If the function has more than one variable, then we can find the derivative with respect to one variable as we make another or others constant. This type of derivative is said to be partial. For instance, when the function is y = f(t,s) where t and s are other variables, then
i. The partial derivative of y with respect to t is
y_{t}=\frac{\partial f}{\partial t}
ii. The partial derivative of y with respect to s is
y_{s}=\frac{\partial f}{\partial s}
f(x) = 3x^2 + 2y
with respect to x and y is
f_{x}=\frac{\partial f}{\partial x}=6x+0=6x
f_{y}=\frac{\partial f}{\partial y}=0+2=2
If the function has three variables, such as f = f(r, θ, z), then we have
i. The partial derivative of y with respect to r is
f_{r}=\frac{\partial f}{\partial r}
ii. The partial derivative of y with respect to θ is
f_{θ}=\frac{\partial f}{\partial θ}
iii. The partial derivative of y with respect to z is
f_{z}=\frac{\partial f}{\partial z}
f(x, y, z) = 3x^2z^2 + 2yx
with respect to x, y and z is
f_{x}=\frac{\partial f}{\partial x}=6x(z^2) + 1(2y) = 6xz^2+2y
f_{y}=\frac{\partial f}{\partial y}=0 + 1(2x) = 2x
f_{z}=\frac{\partial f}{\partial z}=2z(3x^2) + 0 = 6zx^2
Basic rules of derivative
In this section, we consider the derivative rules for ordinary derivatives which also apply for partial derivatives.
a. Derivatives of power functions and polynomials
c. Quotient rule
d. Power rule
e. Chain rule
f. Derivative of trigonometric functions
g. Derivative of exponential functions
h. Derivative of logarithmic functions
Derivatives of power functions and polynomials.
Power functions are generally written as
f(x)=kx^n
where k and n are constants while x is a variable. The derivative of the function is
f^{\prime}(x)=\frac{d f}{d x}=k n x^{n-1}
f(x)=8x^3
24x^2
The polynomial is composed of terms that are power functions, therefore, its derivative is determined by differentiating term by term. If the term is a constant, for instance, h(x) = k where is a constant, then h'(x) = 0 because
h(x)=kx^0
h^{\prime}(x)=0*kx^{0-1}=0
f(x) = 5x^4 - 6x^3 + 4x - 10
f^{\prime}(x) = 5*4x^{4-1} - 6*3x^{3-1} + 4*x^{1-1} - 0 = 20x^3 - 18x^2+4
If a function f(x) is a product of two functions, g(x) and h(x) that are differentiable, then the derivative of f(x) = g(x) * h(x) is
f^{\prime}(x)=\frac{d f}{d x}=g^{\prime}(x) * h(x) + g(x) * h^{\prime}(x)
f(x)=\frac{d f}{d x}=h(x) \frac{d g(x)}{d x}+g(x) \frac{d h(x)}{d x}
In simple terms, if
f^{\prime} = g'h + gh'
If a function f(x) is a quotient of two functions, g(x) and h(x) that are differentiable, then the derivative of is
f(x) = \frac{g(x)}{h(x)}
f^{\prime}(x)=\frac{d f}{d x}=\frac{g^{\prime}(x) \cdot h(x)-g(x) \cdot h^{\prime}(x)}{h^{2}(x)}
f=\frac{g}{h} \text { then } f^{\prime}=\frac{g^{\prime} h-g h^{\prime}}{h^{2}}
f(x)=\frac{d f}{d x}=h(x) \frac{d g(x)}{d x}+g(x) \frac{d h(x)}{d x}
If a function f(x) is expressed as
f(x) = [g(x)]^n
, then its derivative is
\frac{d f}{d x}=n[g(x)]^{n-1} \cdot \frac{d g}{d x}=[g(x)]^{n-1} g^{\prime}(x)
f(x) = (5x^3 + 2)^4
\begin{aligned} f^{\prime}(x) &=4\left(5 x^{3}+2\right)^{4-1} \frac{d}{d x}\left(5 x^{3}+2\right) \\ &=4\left(5 x^{3}+2\right)^{3} \cdot 15 x^{2} \\ &=60 x^{2}\left(5 x^{3}+2\right)^{3} \end{aligned}
The chain rule is a method for differentiation that works for composite functions. It can also be used to differentiate a power function because such function can be expressed as a composite function. If f(x) = g(h(x)) then f=g(h) and h=h(x) hence
\frac{d f}{d x}=g^{\prime}(h(x)) \cdot h^{\prime}(x)=\frac{d f}{d h} \cdot \frac{d h}{d x}
f(x) = (5x^3 + 2)^4
h(x) = 5x^3+2
f(h)=h^4
\frac{df}{dh}=4h^3
\frac{dh}{dx}=15x^2
\frac{d f}{d x}=\frac{d f}{d h} \cdot \frac{d h}{d x}=4 h^{3} \cdot 15 x^{2}=60 x^{2}\left(5 x^{3}+2\right)
We provide a table showing the derivatives of trigonometric functions
f(x)=sin(x)
f(x)=cos(x)
f(x)=cos(x)
f(x)=-sin(x)
f(x)=tan(x)
f(x)=sec^2(x)
f(x)=sec(x)
f(x)=sec(x)tan(x)
f(x)=csc(x)
f(x)=-csc(x)cot(x)
f(x)=cot(x)
f(x)=-csc^2(x)
sin(2x^2+ \pi)
We use the chain rule: Let
h(x) = 2x^2 + \pi
f(h) = sin h
. We have h'(x) = 4x and f'(h) = cos h
\frac{d f}{d x}=\frac{d f}{d h} \cdot \frac{d h}{d x}=\cos h \cdot 4 x=4 x \cos h=4 x \cos 2 x^{2}+\pi
f(x)=\sqrt{\sec (4 x)}
Let $$h(x)=\sec (4 x)$$ and $$f(h)=\sqrt{h}=h^{\frac{1}{2}}$$. We have by chain rule $$h^{\prime}(x)=4 \sec (4 x) \tan (4 x)$$ and $$ f^{\prime}(h)=\frac{1}{2} h^{\frac{1}{2}-1}=\frac{1}{2} h^{-\frac{1}{2}}=\frac{1}{2 \sqrt{h}} $$ Thus, $$ \frac{d f}{d x}=\frac{d f}{d h} \cdot \frac{d h}{d x}=\frac{1}{2 \sqrt{h}} \cdot 4 \sec (4 x) \tan (4 x)=\frac{4 \sec (4 x) \tan (4 x)}{2 \sqrt{\sec (4 x)}}=2 \sqrt{\sec (4 x)} \tan (4 x) $$
f(x) = a^{g(x)}
f'(x) = a^{g(x)}*g(x)ln a
If a = e = 2.71..., then a = ln e = 1, hence we have derivative of
f(x) = e^{g(x)}
f'(x) = e^{g(x)} * g'(x)
f(x)=e^{x^{2} \tan x}.
g(x)=x^2tan^2(x)
g^{\prime}(x)
, we use the product rule. $$ g^{\prime}(x)=2 x(\tan x)+\left(x^{2}\right) \sec ^{2} x=2 x \tan x+x^{2} \sec ^{2} x $$ Therefore, $$ f^{\prime}(x)=e^{g(x)} \cdot g^{\prime}(x)=\left(2 x \tan x+x^{2} \sec ^{2} x\right) e^{x^{2} \tan x} $$
f(x)=2^{\frac{x}{x^{2}+1}}
Let $$g(x)=\frac{x}{x^{2}+1}$$, using the quotient rule, $$ g^{\prime}(x)=\frac{1\left(x^{2}+1\right)-2 x(x)}{\left(x^{2}+1\right)^{2}}=\frac{x^{2}+1-2 x^{2}}{\left(x^{2}+1\right)^{2}}=\frac{1-x^{2}}{\left(x^{2}+1\right)^{2}} $$ Therefore, the derivative is $$ f^{\prime}(x)=a^{g(x)} \cdot g^{\prime}(x) \ln a=2^{\frac{x}{x^{2}+1}} \cdot \frac{1-x^{2}}{\left(x^{2}+1\right)^{2}} \cdot \ln 2=\frac{\left(1-x^{2}\right) 2^{\frac{x}{x^{2}+1}} \ln 2}{\left(x^{2}+1\right)^{2}} $$
f(x)=log_a(g(x))
f^{\prime}(x)=\frac{g^{\prime}(x)}{g(x) \ln a}.
If a=e=2.71..., then ln a=ln e=1, hence we have the derivative of
f(x)=log_e(g(x)) = ln g(x)
f^{\prime}(x)=\frac{g^{\prime}(x)}{g(x)}
f(x)=\ln \left(x^{2}+3 x\right)
g(x)=x^{2}+3 x
f(g)=\ln g
g^{\prime}(x)=2 x+3
f^{\prime}(g)=\frac{1}{g}
f^{\prime}(x)=\frac{g^{\prime}(x)}{g(x)}=\frac{1}{g} \cdot(2 x+3)=\frac{2 x+3}{x^{2}+3 x}
f(x)=\log_{3}(\cos (2 x))
Let $$g(x)=\cos (2 x)$$, then $$f(g)=\log _{3} g$$. By the chain rule, we have, $$g^{\prime}(x)=-2 \sin (2 x)$$ and $$f^{\prime}(g)=\frac{1}{g \ln 3}$$ Therefore, $$ f^{\prime}(x)=\frac{g^{\prime}(x)}{g(x) \ln a}=\frac{1}{g \ln 3} \cdot(2 \cos (2 x))=\frac{2 \sin (2 x)}{\cos (2 x) \ln 3}=\frac{2 \tan (2 x)}{\ln 3} $$
Constant Multiple Rule
\frac{d}{dx} (kf) = kf\prime
Sum (or Difference) Rule
\frac{d}{dx}(f+g)=f\prime + g\prime \enspace or \enspace \frac{d}{dx}(f-g)=f\prime - g\prime
\frac{d}{dx}(x^n) = nx^{n-1}
\frac{d}{dx} (fg)=f\prime g + fg\prime
\frac{d}{dx}(\frac{f}{g})= \frac{f\prime g - fg\prime}{g^2}
\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}
\frac{d}{dx}(e^x) = e^x
\frac{d}{dx}(\text{ln}x) = \frac{1}{x}
\frac{d}{dx}(sin x) = cos x
\frac{d}{dx}(cos x) = -sin x
© 2021 onlinederivativecalculator.com
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Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : OneForm : AsOperator
OneForm Object as Operator
omega( X)
a OneForm object
A OneForm object omega can act as an operator on a VectorField X, by contraction.
\mathrm{\omega }
is a 1-form. and if
X=\underset{i=1}{\overset{n}{∑}}{\mathrm{\xi }}_{i}
is a vector field (both on a space with coordinates
\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)
), then their contraction is
\mathrm{\omega }\left(X\right)=\underset{i=1}{\overset{n}{∑}}{\mathrm{\theta }}_{i}{\mathrm{\xi }}_{i}
Because it can act as an operator, a OneForm object is of type appliable. See Overview of OneForm Overloaded Builtins for more detail.
When a OneForm is acting as an operator, it will distribute itself over indexable types such as Vectors, Matrices, lists, and tables.
This method is associated with the OneForm object. For more detail, see Overview of the OneForm object.
\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):
R[x]≔\mathrm{VectorField}\left(y\mathrm{D}[z]-z\mathrm{D}[y],\mathrm{space}=[x,y,z]\right)
{\textcolor[rgb]{0,0,1}{R}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\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}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\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}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}
R[y]≔\mathrm{VectorField}\left(-x\mathrm{D}[z]+z\mathrm{D}[x],\mathrm{space}=[x,y,z]\right)
{\textcolor[rgb]{0,0,1}{R}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{z}\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}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\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}{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}
R[z]≔\mathrm{VectorField}\left(x\mathrm{D}[y]-y\mathrm{D}[x],\mathrm{space}=[x,y,z]\right)
{\textcolor[rgb]{0,0,1}{R}}_{\textcolor[rgb]{0,0,1}{z}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\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}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\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}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}
\mathrm{\omega }≔\mathrm{OneForm}\left(xd[x]+yd[y]+zd[z],\mathrm{space}=[x,y,z]\right)
\textcolor[rgb]{0,0,1}{\mathrm{\omega }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dx}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dy}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{dz}}
\mathrm{\omega }\left(R[x]\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{\omega }\left([R[x],R[y],R[z]]\right)
[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]
\mathrm{\omega }\left(R\right)
\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}]\right)
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problemJWT_payload HASH(0x56508d1e6e20)
Use the Squeeze Theorem to evaluate the following limit. If the answer is positive infinite, type "I"; if negative infinite, type "N"; and if it does not exist, type "D".
[By using a graphing calculator, one can see that [math]
f(x)=\displaystyle\frac{\sin{x}}{x}
crosses its horizontal asymptote infinitely many times.]
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Computation | Free Full-Text | TS Fuzzy Robust Sampled-Data Control for Nonlinear Systems with Bounded Disturbances
Principal Components Analysis of EEG Signals for Epileptic Patient Identification
DC Drive Adaptive Speed Controller Based on Hyperstability Theory
Nonlinear Dynamics and Performance Analysis of a Buck Converter with Hysteresis Control
Poongodi, T.
Mishra, P. Prakash
Lim, C. Peng
Saravanakumar, T.
Boonsatit, N.
Hammachukiattikul, P.
Thangavel Poongodi
Thangavel Saravanakumar
Nattakan Boonsatit
Department of Mathematics, National Institute of Technology, Nagaland 797103, India
Institute for Intelligent Systems Research and Innovation, Deakin University, Waurn Ponds, VIC 3216, Australia
Department of Mathematics, Anna University Regional Campus, Coimbatore 641046, India
Department of Mathematics, Faculty of Science and Technology, Rajamangala University of Technology Suvarnabhumi, Nonthaburi 11000, Thailand
Department of Mathematics, Faculty of Science, Phuket Rajabhat University (PKRU), 6 Thepkasattree Road, Raddasa, Phuket 83000, Thailand
Department of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand
We investigate robust fault-tolerant control pertaining to Takagi–Sugeno (TS) fuzzy nonlinear systems with bounded disturbances, actuator failures, and time delays. A new fault model based on a sampled-data scheme that is able to satisfy certain criteria in relation to actuator fault matrix is introduced. Specifically, we formulate a reliable controller with state feedback, such that the resulting closed-loop-fuzzy system is robust, asymptotically stable, and able to satisfy a prescribed
{H}_{\infty }
performance constraint. Linear matrix inequality (LMI) together with a proper construction of the Lyapunov–Krasovskii functional is leveraged to derive delay-dependent sufficient conditions with respect to the existence of robust
{H}_{\infty }
controller. It is straightforward to obtain the solution by using the MATLAB LMI toolbox. We demonstrate the effectiveness of the control law and less conservativeness of the results through two numerical simulations. View Full-Text
Keywords: Takagi–Sugeno (TS) fuzzy models; H∞ control; fault-tolerant control; bounded disturbances Takagi–Sugeno (TS) fuzzy models; H∞ control; fault-tolerant control; bounded disturbances
Poongodi, T.; Mishra, P.P.; Lim, C.P.; Saravanakumar, T.; Boonsatit, N.; Hammachukiattikul, P.; Rajchakit, G. TS Fuzzy Robust Sampled-Data Control for Nonlinear Systems with Bounded Disturbances. Computation 2021, 9, 132. https://doi.org/10.3390/computation9120132
Poongodi T, Mishra PP, Lim CP, Saravanakumar T, Boonsatit N, Hammachukiattikul P, Rajchakit G. TS Fuzzy Robust Sampled-Data Control for Nonlinear Systems with Bounded Disturbances. Computation. 2021; 9(12):132. https://doi.org/10.3390/computation9120132
Poongodi, Thangavel, Prem P. Mishra, Chee P. Lim, Thangavel Saravanakumar, Nattakan Boonsatit, Porpattama Hammachukiattikul, and Grienggrai Rajchakit. 2021. "TS Fuzzy Robust Sampled-Data Control for Nonlinear Systems with Bounded Disturbances" Computation 9, no. 12: 132. https://doi.org/10.3390/computation9120132
Chee Peng Lim received a Ph.D. degree in intelligent systems from the University of Sheffield, Sheffield, U.K., in 1997. He is currently Professor of Computational Intelligence with the Institute for Intelligent Systems Research and Innovation, Deakin University, Australia. His research interests include computational intelligence-based systems for data analytics, condition monitoring, optimization, and decision support. He has authored/coauthored over 450 technical papers in journals, conference proceedings, and books.
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FMP100 Trip/Odometer settings - Wiki Knowledge Base | Teltonika GPS
Main Page > Easy Trackers > FMP100 > FMP100 Manual > FMP100 Configuration > FMP100 Trip/Odometer settings
{\displaystyle ECOscore={\cfrac {10}{\cfrac {Egen}{d{\cfrac {Eallowed}{100}}}}}}
{\displaystyle ECOscore={\cfrac {10}{\cfrac {10}{100{\cfrac {10}{100}}}}}=10,00}
{\displaystyle ECOscore={\cfrac {10}{\cfrac {15}{100{\cfrac {10}{100}}}}}=6,66}
Remember iButton functionality. If Remember iButton and Trip parameters are enabled, ignition is on and iButton is attached then FMP100 remembers iButton ID. iButton ID is saved and sent to the server with every record. If new iButton is attached during the Trip, FMP100 remembers new iButton ID. FMP100 forgets iButton ID after ignition is off and ignition timeout is reached.
Private/Business Mode feature will allow for FMP100 users to mask and secure privacy while using business vehicle for personal uses during not working hours.
In order to manually select between Private or Business mode for FMP100 device, 4 different sources can be selected, which can be combined together:
FMP100 Button - by configured Keyboard part, the physical FMP100 button can be used to choose trip mode.
In order to set automatic mode changes between Private or Business modes, the whole week can be manually preconfigured. By default device is using the GMT+0 time zone. In order to match time zone where FMP100 device is located, the time zone can be modified.
In order to automatically match changed times without changing configuration of FMP100 device, the Daylight saving can be Enabled and configure the Start on and End on points by configuring the Week, Day, Month and Time.
If Daylight saving is not needed or FMP100 device is used in countries where time doesn't change, the option can be Disabled.
Retrieved from "https://wiki.teltonika-gps.com/wikibase/index.php?title=FMP100_Trip/Odometer_settings&oldid=61790"
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The workshop ``Calculus of Variations'' took place from July 9 to 15, 2006, and was attended by almost fifty participants, mostly from European and North American universities and research institutes. There were 24 lectures on recent research topics, plus a review lecture on the Lieb-Thirring inequalities by Michael Loss (Georgia Tech, Atlanta). As the workshop had no specific focus, talks covered a wide range of topics, with the aim of featuring different research trends, bringing new problems to the fore, and stimulating interaction between mathematicians from different backgrounds. \smallskip Five lectures were focused on problems related to Continuum Mechanics and Materials Science. Gero Friesecke (Munich and Warwick) presented some results on a simplified model for molecules, where the aim is to give a rigorous explanation of the screening effect (i.e., the fact that the interaction of electrically balanced molecules due to electrostatic forces is short ranged); this problem is still open and presumably quite challenging in case of `realistic' models. L\'aszl\'o Sz\'ekelyhidy (ETH Z\"urich) presented new result about the structure of quasiconvex hulls for sets of
2\times2
matrices, perhaps the most interesting development on this topic in recent years. Sergio Conti (Duisburg) considered the asymptotic behaviour of an energy functional that appears in the modeling of different physical problems, such as blistering in elastic films, magnetic thin films, etc.; the main result presented in his lecture adds one more piece to the work of many authors towards the proof of a conjecture by Aviles and Giga on the variational limit of such functional. The lecture by Felix Otto (Bonn) was focused on the rigorous analysis of pattern formation in micromagnetics: this type of pattern formation is particularly interesting because of the complexity of the observed behaviours -- not yet fully explained in rigorous terms -- and of the relative simplicity of the underlying continuum model. Related to this topic was also the lecture of Hans Kn\"upfer (Bonn). \smallskip Four lectures dealt with regularity problems of different sorts. G.~Rosario Mingione (Parma) reviewed some recent developments on the regularity of solutions of nonlinear parabolic systems. Michael Struwe (ETH Z\"urich) presented a new approach to regularity for harmonic maps valued in a hypersurfaces, yielding new results when the domain dimension is larger than
2
. The regularity of harmonic maps valued in Riemannian manifolds was also considered by Ernst Kuwert (Freiburg); these results stemmed from other results on the conformal structure of surfaces with suitable bounds on the Willmore energy. Mariel Saez (MPI for Gravitationl Physics, Potsdam) presented a Lipschitz regularity result for the pseudo-infinity Laplacian. \smallskip A certain number of lectures were related to shape optimization and optimal transport problems. Almut Burchard (Toronto) presented some partial results about the shape of closed curves in the three-dimensional space that minimize the first eigenvalue of the associated one-dimensional Schr\"odinger operator; it is conjectured that these curves are circles (among other things, the conjecture is related to the optimal constant in a particular Lieb-Thirring inequality). Jochen Denzler (Knoxville) and Giuseppe Buttazzo (Pisa) considered other optimization problems related to the first eigenvalue of (variants of) the Laplace operator on a given domain. Alexander Plakhov (Aveiro) studied bodies of minimal resistance moving through a rarefied particle gas. Francesco Maggi (Duisburg) and Aldo Pratelli (Pavia) presented some recent quantitative versions with optimal exponents of the classical isoperimetric inequality in the
n
-dimensional Euclidean space. Qinglan Xia (UC Davis) proposed a model for the shape formation in tree leaves which postulates a step-by-step optimized growth for the associated transport system (the venation of the leaf), where ``optimized'' refers to a given transport cost. Numerical simulations based on this simple model show that varying the two built-in parameters generates a wide variety of leaf shapes. Vladimir Oliker (Emory University, Atlanta) described a variational approach to the Aleksandrov problem about the existence of closed convex hypersurfaces with prescribed integral Gauss curvature. A similar approach is also used to design reflecting surfaces with prescribed irradiance properties; the functional underlying this variational principle is related to Monge-Kantorovich optimal transport theory. \smallskip Yann Brenier (Nice) considered the problem of foliating the three-dimensional Euclidean space and the four-dimensional Minkowski space by extremal surfaces (which in Minkowski space can be interpreted as classical relativistic strings). One way of obtaining such foliations is finding minimizers or critical points of suitable energy functionals, subject to certain nonlinear constraints; due to these constraints, standard methods do not apply in this case, and the existence of such minimizers is open. Pierre Cardaliaguet (Brest) studied a non-local geometric evolution problem for sets in the
n
-dimensional Euclidean space, which can be formally viewed as the gradient flow of a linear combination of volume and capacity. Since this flow preserves inclusion, it allows for a notion of weak solutions in the sense of viscosity; it is shown that such solutions agree with the limits of the the minimizing movements obtained by time discretization. Diogo Gomes (Instituto Superior Tecnico, Lisbon) reviewed some recent results on the viscosity solution of Hamilton-Jacobi equations and the relations with the associated Hamiltonian dynamics, and Aubrey-Mather theory. Olvier Druet (ENS Lyon) presented new results on the bubbling phenomenon for the solutions (and also the Palais-Smale sequences) of sequences of variational elliptic equations in dimension two with critical nonlinearities. Robert Jerrard (Toronto) described a version of the
\Gamma
-convergence method designed for saddle points instead of minima, and used this abstract tool to obtain non-trivial solutions to the Ginzburg-Landau system in dimension three. Reiner Sch\"atzle (T\"ubingen) gave a proof of (a modified version of) a conjecture by De Giorgi on the approximation of the Willmore functional for hypersurfaces in dimension three; the conjecture is still open in higher dimensions. Keith Ball (University College London) presented the proof of a long-standing conjecture (due to Lieb) on the entropy gap between the normalized sum of
N
independent copies of a given random variable
X
and its limit as
N\to\infty
, i.e., the Gaussian distribution. A key role in the proof is played by a new variational characterization of Fisher information. The lecture by Gerhard Huisken (MPI for Gravitationl Physics, Potsdam) was focused on the problem of defining mass in general relativity; in particular, he presented a new definition based on the isoperimetric inequality (more precisely, on the asymptotic behaviour of the isoperimetric profile), and some results on the properties of this mass. One of the advantages of this definition, compared to others based on the notion of curvature, is the relatively simple calculus that is required for handling it. Furthermore, it can be adapted so as to obtain a notion of localized mass.
Tristan Rivière, Giovanni Alberti, Snigdhayan Mahanta, Calculus of Variations. Oberwolfach Rep. 3 (2006), no. 3, pp. 1879–1940
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Trisectrix_of_Maclaurin Knowpia
In geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a sectrix of Maclaurin. The curve is named after Colin Maclaurin who investigated the curve in 1742.
Maclaurin's Trisectrix as the locus of the intersection of two rotating lines
Let two lines rotate about the points
{\displaystyle P=(0,0)}
{\displaystyle P_{1}=(a,0)}
so that when the line rotating about
{\displaystyle P}
has angle
{\displaystyle \theta }
with the x axis, the rotating about
{\displaystyle P_{1}}
{\displaystyle 3\theta }
{\displaystyle Q}
be the point of intersection, then the angle formed by the lines at
{\displaystyle Q}
{\displaystyle 2\theta }
. By the law of sines,
{\displaystyle {r \over \sin 3\theta }={a \over \sin 2\theta }\!}
so the equation in polar coordinates is (up to translation and rotation)
{\displaystyle r=a{\frac {\sin 3\theta }{\sin 2\theta }}={a \over 2}{\frac {4\cos ^{2}\theta -1}{\cos \theta }}={a \over 2}(4\cos \theta -\sec \theta )\!}
The curve is therefore a member of the Conchoid of de Sluze family.
In Cartesian coordinates the equation of this curve is
{\displaystyle 2x(x^{2}+y^{2})=a(3x^{2}-y^{2})\!}
If the origin is moved to (a, 0) then a derivation similar to that given above shows that the equation of the curve in polar coordinates becomes
{\displaystyle r=2a\cos {\theta \over 3}\!}
making it an example of a limacon with a loop
The trisection propertyEdit
The Trisectrix of Maclaurin showing the angle trisection property
Given an angle
{\displaystyle \phi }
, draw a ray from
{\displaystyle (a,0)}
whose angle with the
{\displaystyle x}
{\displaystyle \phi }
. Draw a ray from the origin to the point where the first ray intersects the curve. Then, by the construction of the curve, the angle between the second ray and the
{\displaystyle x}
{\displaystyle \phi /3}
Notable points and featuresEdit
The curve has an x-intercept at
{\displaystyle 3a \over 2}
and a double point at the origin. The vertical line
{\displaystyle x={-{a \over 2}}}
is an asymptote. The curve intersects the line x = a, or the point corresponding to the trisection of a right angle, at
{\displaystyle (a,{\pm {1 \over {\sqrt {3}}}a})}
. As a nodal cubic, it is of genus zero.
Relationship to other curvesEdit
The trisectrix of Maclaurin can be defined from conic sections in three ways. Specifically:
It is the inverse with respect to the unit circle of the hyperbola
{\displaystyle 2x=a(3x^{2}-y^{2})}
It is cissoid of the circle
{\displaystyle (x+a)^{2}+y^{2}=a^{2}}
{\displaystyle x={a \over 2}}
It is the pedal with respect to the origin of the parabola
{\displaystyle y^{2}=2a(x-{\tfrac {3}{2}}a)}
The inverse with respect to the point
{\displaystyle (a,0)}
is the Limaçon trisectrix.
The trisectrix of Maclaurin is related to the Folium of Descartes by affine transformation.
J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 36, 95, 104–106. ISBN 0-486-60288-5.
Weisstein, Eric W. "Maclaurin Trisectrix". MathWorld.
"Trisectrix of Maclaurin" at MacTutor's Famous Curves Index
Maclaurin Trisectrix at mathcurve.com
"Trisectrix of Maclaurin" at Visual Dictionary Of Special Plane Curves
Wikimedia Commons has media related to Maclaurin's Trisectrix.
Loy, Jim "Trisection of an Angle", Part VI
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Quadratic Chabauty and rational points, I: p-adic heights
15 August 2018 Quadratic Chabauty and rational points, I:
p
-adic heights
Jennifer S. Balakrishnan, Netan Dogra
We give the first explicit examples beyond the Chabauty–Coleman method where Kim’s nonabelian Chabauty program determines the set of rational points of a curve defined over
\mathbb{Q}
or a quadratic number field. We accomplish this by studying the role of
p
-adic heights in explicit non-Abelian Chabauty.
Jennifer S. Balakrishnan. Netan Dogra. "Quadratic Chabauty and rational points, I:
p
-adic heights." Duke Math. J. 167 (11) 1981 - 2038, 15 August 2018. https://doi.org/10.1215/00127094-2018-0013
Received: 3 May 2016; Revised: 9 March 2018; Published: 15 August 2018
Keywords: non-Abelian Chabauty , p-adic heights , rational points on higher genus curves
Jennifer S. Balakrishnan, Netan Dogra "Quadratic Chabauty and rational points, I:
p
-adic heights," Duke Mathematical Journal, Duke Math. J. 167(11), 1981-2038, (15 August 2018)
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Mini-Workshop: Attraction to Solitary Waves and Related Aspects of Physics | EMS Press
Alexander I. Komech
The workshop \emph{Attraction to Solitary Waves and Related Aspects of Physics}, organised by Vladimir Buslaev (St. Petersburg University), Andrew Comech (Texas A\&M), Alexander Komech (Universit\"at Wien), and Boris Vainberg (UNC -- Charlotte) was held February 10--16, 2008. This meeting was attended with 15 participants with broad geographic representation from Europe and America. This workshop was a blend of researchers with backgrounds in Partial Differential Equations, Harmonic Analysis, and Quantum Field Theory.
The aim of the miniworkshop has been the discussion of current state of the long-time asymptotics for nonlinear Hamiltonian partial differential equations and relation to mathematical problems of Quantum Physics.
The central themes were the orbital and asymptotic stability of solitary waves, quantum scattering, renormalization, and global attraction to solitary waves.
\subsection*{Bohr's transitions as global attraction to solitary waves}
According to Bohr's postulates \cite{bohr1913}, an unperturbed electron runs forever along certain \emph{stationary orbit}, which we denote
\vert E\rangle
and call \emph{quantum stationary state}. Once in such a state, the electron has a fixed value of energy
E
, not losing the energy via emitting radiation. The electron can jump from one quantum stationary state to another, \begin{gather}\label{transitions} \vert E\sb{-}\rangle \longmapsto \vert E\sb{+}\rangle, \end{gather} emitting or absorbing a quantum of light with the energy equal to the difference of the energies
E\sb{+}
E\sb{-}
. Bohr's second postulate states that the electrons can jump from one quantum stationary state (Bohr's \emph{stationary orbit}) to another.
Bohr's \emph{stationary orbits} were interpreted by Schr\"odinger as \emph{quasistationary solitary wave solutions} of the form \begin{gather}\label{sw00} \psi(x,t)=\phi(x)e^{-i\omega t}, \qquad {with} \quad \omega\in\R, \quad \lim\sb{\abs{x}\to\infty}\phi(x)=0. \end{gather} We will call such solutions \emph{solitary waves}. Other appropriate names are \emph{nonlinear eigenfunctions} and \emph{quantum stationary states} (the solution \eqref{sw00} is not exactly stationary, but certain observable quantities, such as the charge and current densities, are time-independent indeed). As a consequence, the electron in such a state does not emit the energy and ``circles'' forever around the nucleus in an atom.
Bohr's \emph{quantum jumps} can be interpreted dynamically as long-time asymptotics \begin{gather}\label{ga} \Psi(t)\longrightarrow\vert E\sb\pm\rangle, \qquad t\to\pm\infty, \end{gather} for any trajectory
\Psi(t)
of the corresponding dynamical system, where the limiting states
\vert E\sb\pm\rangle
generally depend on the trajectory. Then the quantum stationary states should be viewed as the points of the \emph{global attractor}
\mathscr{A}
The attraction \eqref{ga} takes the form of the long-time asymptotics \begin{gather}\label{asymptotics} \psi(x,t) \sim \phi\sb{\omega\sb\pm}(x)e\sp{-i\omega\sb{\pm}t}, \qquad t\to\pm\infty, \end{gather} that hold for each finite energy solution.
Now let us describe the existing results on solitary waves in the context of dispersive Hamiltonian systems.
\subsection*{Nonlinear wave equations. Well-posedness in the energy space}
The nonlinear wave equations take their origin in Quantum Field Theory from the articles by Schiff \cite{PhysRev.84.1,PhysRev.84.10}, who considered the nonlinear Klein--Gordon equation in his research on the classical nonlinear meson theory of nuclear forces. The mathematical analysis of this equation is started by J\"orgens \cite{MR0130462} and Segal \cite{MR0153967,MR0152908}, who studied its global well-posedness in the energy space. Since then, this equation (alongside with the nonlinear Schr\"odinger equation) has been the main playground for developing tools to handle more general nonlinear Hamiltonian systems.
\subsection*{Local attraction to zero}
The asymptotics of type \eqref{asymptotics} were discovered first with
\psi\sb\pm=0
in the scattering theory. Segal \cite{MR0217453} and then Morawetz and Strauss \cite{MR0233062,MR0303097} studied the (nonlinear) scattering for solutions of nonlinear Klein--Gordon equation in
\R^3
. We may interpret these results as \emph{local} (referring to small initial data) attraction to zero: \begin{gather}\label{attraction-0} \psi(x,t)\sim\psi\sb\pm=0,\qquad t\to\pm\infty. \end{gather} The asymptotics \eqref{attraction-0} hold on an arbitrary compact set and represent the well-known local energy decay. These results were further extended in \cite{MR535231,MR654553,MR824083,MR1120284}.
\subsection*{Solitary waves}
Apparently, there could be no \emph{global} attraction to zero (\emph{global} referring to arbitrary initial data) if there are solitary wave solutions of the form
\phi\sb\omega(x)e\sp{-i\omega t}
. The existence of solitary wave solutions \begin{gather*} \psi\sb\omega(x,t)=\phi\sb\omega(x)e\sp{-i\omega t}, \qquad \omega\in\R, \quad\phi\sb\omega\in H\sp 1(\R^n), \end{gather*} with
H\sp{1}(\R^n)
being the Sobolev space, to the nonlinear Klein--Gordon equation (and nonlinear Schr\"odinger equation) in
\R^n
, in a rather generic situation, was established in \cite{MR0454365} (a more general result was obtained in \cite{MR695535,MR695536}). Typically, such solutions exist for
\omega
from an interval or a collection of intervals of the real line. We denote the set of all solitary waves by
{\mathcal{S}\sb{0}}
While all localized stationary solutions to the nonlinear wave equations in spatial dimensions
n\ge 3
turn out to be unstable (the result known as ``Derrick's theorem'' \cite{MR30:4510}), \emph{quasistationary} solitary waves can be orbitally stable. Stability of solitary waves takes its origin from \cite{VaKo} and has been extensively studied by Strauss and his school in \cite{MR901236,MR723756,MR792821,MR804458}.
\subsection*{Local attraction to solitary waves}
The asymptotic stability of solitary waves (convergence to a solitary wave for the initial data sufficiently close to it) has been studied by Soffer and Weinstein \cite{MR1071238,MR1170476} in the context of nonlinear
\mathbf{U}(1)
-invariant Schr\"odinger equation with a potential. This theory has been further developed by Buslaev and Perelman \cite{MR1199635e,MR1334139} and then by others in \cite{MR1488355,MR1681113,MR1893394,MR1835384,MR2027616,MR1972870} and other papers. Up to date, there are many open problems. While generically we expect that any orbitaly stable solitary wave is also asymptotically stable, the proof of this statement is only available for just a few cases and under very strong assumptions.
The existing results on orbital and asymptotic stability suggest that the set of orbitally stable solitary waves typically forms a \emph{local attractor}, that is to say, attracts any finite energy solutions that were initially close to it. Moreover, a natural hypothesis is that the set of all solitary waves forms a \emph{global attractor} of all finite energy solutions.
\subsection*{Global attraction to solitary waves}
The \emph{global attraction} of type \eqref{asymptotics} with
\psi\sb\pm\ne 0
\omega\sb{\pm}=0
was established in certain models in \cite{MR1203302e,MR1359949,MR1412428,MR1434147,MR1726676,MR1748357} for a number of nonlinear wave problems. There the attractor is the set of all \emph{static} stationary states. Let us mention that this set could be infinite and contain continuous components.
In \cite{MR2032730} and \cite{ubk-arma}, the attraction to the set of solitary waves (see Fig.~\ref{fig-attractor}) is proved for the Klein--Gordon field coupled to a nonlinear oscillator. In \cite{ukk-mpi}, this result has been generalized for the Klein--Gordon field coupled to several oscillators. The paper \cite{ukr-hp} gives the extension to the higher-dimensional setting for a model with the nonlinear self-interaction of the mean field type.
\begin{figure}[htbp] \begin{latexonly} \input fig-attractor.tex \end{latexonly} \caption{For
t\to\pm\infty
, a finite energy solution
\Psi(t)
approaches (in local energy seminorms) the global attractor
\mathscr{A}
which coincides with the set of all solitary waves
\mathcal{S}\sb 0
.} \label{fig-attractor} \end{figure}
Let us mention one more recent advance, \cite{MR2304091}, in the field of nontrivial (nonzero) global attractors for Hamiltonian PDEs. In that paper, the global attraction for the nonlinear Schr\"odinger equation in dimensions
n\ge 5
was considered. The dispersive (outgoing) wave was explicitly specified using the rapid decay of local energy in higher dimensions. The global attractor was proved to be compact, but it was not identified with the set of solitary waves \cite[Remark 1.18]{MR2304091}.
\subsection*{Relation to Quantum Physics}
The Quantum Mechanics is formulated in terms of partial differential equations: coupled nonlinear Maxwell-Schrodinger, Maxwell-Dirac, Maxwell-Yang-Mills equations, etc. The Quantum Field Theory is formulated in terms of the correponding second quantized equations. The main goal of our workshop was to achieve the critical concentration of experts in Quantum Theory on one side and in PDEs on another side, to have a thorough discussion of recent advances in both areas and an exchange which could stimulate further progress.
Vladimir Buslaev, Andrew Comech, Alexander I. Komech, Boris Vainberg, Mini-Workshop: Attraction to Solitary Waves and Related Aspects of Physics. Oberwolfach Rep. 5 (2008), no. 1, pp. 367–400
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4-aminobutyrate transaminase - Wikipedia
(Redirected from 4-Aminobutyrate aminotransferase)
4-Aminobutyrate transaminase homodimer, Pig
In enzymology, 4-aminobutyrate transaminase (EC 2.6.1.19), also called GABA transaminase or 4-aminobutyrate aminotransferase, or GABA-T, is an enzyme that catalyzes the chemical reaction:
4-aminobutanoate + 2-oxoglutarate
{\displaystyle \rightleftharpoons }
succinate semialdehyde + L-glutamate
Thus, the two substrates of this enzyme are 4-aminobutanoate (GABA) and 2-oxoglutarate. The two products are succinate semialdehyde and L-glutamate.
This enzyme belongs to the family of transferases, specifically the transaminases, which transfer nitrogenous groups. The systematic name of this enzyme class is 4-aminobutanoate:2-oxoglutarate aminotransferase. This enzyme participates in 5 metabolic pathways: alanine and aspartate metabolism, glutamate metabolism, beta-alanine metabolism, propanoate metabolism, and butanoate metabolism. It employs one cofactor, pyridoxal phosphate.
This enzyme is found in prokaryotes, plants, fungi, and animals (including humans).[1] Pigs have often been used when studying how this protein may work in humans.[2]
1 Enzyme Commission number
2 Reaction pathway
3 Cellular and metabolic role
Enzyme Commission number[edit]
GABA-T is Enzyme Commission number 2.6.1.19. This means that it is in the transferase class of enzymes, the nitrogenous transferase sub-class and the transaminase sub-subclass.[3] As a nitrogenous transferase, its role is to transfer nitrogenous groups from one molecule to another. As a transaminase, GABA-T's role is to move functional groups from an amino acid and a α-keto acid, and vice versa. In the case of GABA-T, it takes a nitrogen group from GABA and uses it to create L-glutamate.
Reaction pathway[edit]
In animals, fungi, and bacteria, GABA-T helps facilitate a reaction that moves an amine group from GABA to 2-oxoglutarate, and a ketone group from 2-oxoglutarate to GABA.[4][5][6] This produces succinate semialdehyde and L-glutamate.[4] In plants, pyruvate and glyoxylate can be used in the place of 2-oxoglutarate.[7] catalyzed by the enzyme 4-aminobutyrate—pyruvate transaminase:
(1) 4-aminobutanoate (GABA) + pyruvate ⇌ succinate semialdehyde + L-alanine
(2) 4-aminobutanoate (GABA) + glyoxylate ⇌ succinate semialdehyde + glycine
Cellular and metabolic role[edit]
The primary role of GABA-T is to break down GABA as part of the GABA-Shunt.[2] In the next step of the shunt, the semialdehyde produced by GABA-T will be oxidized to succinic acid by succinate-semialdehyde dehydrogenase, resulting in succinate. This succinate will then enter mitochondrion and become part of the citric acid cycle.[8] The critic acid cycle can then produce 2-oxoglutarate, which can be used to make glutamate, which can in turn be made into GABA, continuing the cycle.[8]
GABA is a very important neurotransmitter in animal brains, and a low concentration of GABA in mammalian brains has been linked to several neurological disorders, including Alzheimer's disease and Parkinson's disease.[9][10] Because GABA-T degrades GABA, the inhibition of this enzyme has been the target of many medical studies.[9] The goal of these studies is to find a way to inhibit GABA-T activity, which would reduce the rate that GABA and 2-oxoglutarate are converted to semialdehyde and L-glutamate, thus raising GABA concentration in the brain. There is also a genetic disorder in humans which can lead to a deficiency in GABA-T. This can lead to developmental impairment or mortality in extreme cases.[11]
In plants, GABA can be produced as a stress response.[5] Plants also use GABA to for internal signaling and for interactions with other organisms near the plant.[5] In all of these intra-plant pathways, GABA-T will take on the role of degrading GABA. It has also been demonstrated that the succinate produced in the GABA shunt makes up a significant proportion of the succinate needed by the mitochondrion.[12]
In fungi, the breakdown of GABA in the GABA shunt is key in ensuring maintaining a high level of activity in the critic acid cycle.[13] There is also experimental evidence that the breakdown of GABA by GABA-T plays a role in managing oxidative stress in fungi.[13]
There have been several structures solved for this class of enzymes, given PDB accession codes, and published in peer reviewed journals. At least 4 such structures have been solved using pig enzymes: 1OHV, 1OHW, 1OHY, 1SF2, and at least 4 such structures have been solved in Escherichia coli: 1SFF, 1SZK, 1SZS, 1SZU. There are actually some differences between the enzyme structure for these organisms. E. coli enzymes of GABA-T lack an iron-sulfur cluster that is found in the pig model.[14]
Amino acid residues found in the active site of 4-aminobutyrate transaminase include Lys-329, which are found on each of the two subunits of the enzyme.[15] This site will also bind with a pyridoxal 5'- phosphate co-enzyme.[15]
Main article: GABA transaminase inhibitor
Phenylethylidenehydrazine (PEH)
Rosmarinic acid[16]
^ "4-aminobutyrate aminotransferase - Identical Protein Groups - NCBI". www.ncbi.nlm.nih.gov. Retrieved 2020-09-29.
^ a b Iftikhar H, Batool S, Deep A, Narasimhan B, Sharma PC, Malhotra M (February 2017). "In silico analysis of the inhibitory activities of GABA derivatives on 4-aminobutyrate transaminase". Arabian Journal of Chemistry. 10: S1267–75. doi:10.1016/j.arabjc.2013.03.007.
^ "BRENDA - Information on EC 2.6.1.19 - 4-aminobutyrate-2-oxoglutarate transaminase". www.brenda-enzymes.org. Retrieved 2020-09-24.
^ a b Tunnicliff G (1986). "4-Aminobutyrate Transaminase". In Boulton AA, Baker GB, Yu PH (eds.). Neurotransmitter Enzymes. Vol. 5. pp. 389–420. doi:10.1385/0-89603-079-2:389. ISBN 0-89603-079-2.
^ a b c Shelp BJ, Bown AW, Zarei A (2017). "4-Aminobutyrate (GABA): a metabolite and signal with practical significance". Botany. 95 (11): 1015–32. doi:10.1139/cjb-2017-0135. hdl:1807/79639.
^ Cao J, Barbosa JM, Singh N, Locy RD (July 2013). "GABA transaminases from Saccharomyces cerevisiae and Arabidopsis thaliana complement function in cytosol and mitochondria". Yeast. 30 (7): 279–89. doi:10.1002/yea.2962. PMID 23740823. S2CID 1303165.
^ Fait A, Fromm H, Walter D, Galili G, Fernie AR (January 2008). "Highway or byway: the metabolic role of the GABA shunt in plants". Trends in Plant Science. 13 (1): 14–9. doi:10.1016/j.tplants.2007.10.005. PMID 18155636.
^ a b Bown AW, Shelp BJ (September 1997). "The Metabolism and Functions of [gamma]-Aminobutyric Acid". Plant Physiology. 115 (1): 1–5. doi:10.1104/pp.115.1.1. PMC 158453. PMID 12223787.
^ a b Ricci L, Frosini M, Gaggelli N, Valensin G, Machetti F, Sgaragli G, Valoti M (May 2006). "Inhibition of rabbit brain 4-aminobutyrate transaminase by some taurine analogues: a kinetic analysis". Biochemical Pharmacology. 71 (10): 1510–9. doi:10.1016/j.bcp.2006.02.007. PMID 16540097.
^ Sherif FM, Ahmed SS (April 1995). "Basic aspects of GABA-transaminase in neuropsychiatric disorders". Clinical Biochemistry. 28 (2): 145–54. doi:10.1016/0009-9120(94)00074-6. PMID 7628073.
^ "GABA-TRANSAMINASE DEFICIENCY". www.omim.org. Retrieved 2020-10-18.
^ a b Bönnighausen J, Gebhard D, Kröger C, Hadeler B, Tumforde T, Lieberei R, et al. (December 2015). "Disruption of the GABA shunt affects mitochondrial respiration and virulence in the cereal pathogen Fusarium graminearum". Molecular Microbiology. 98 (6): 1115–32. doi:10.1111/mmi.13203. PMID 26305050. S2CID 45755014.
^ Liu W, Peterson PE, Carter RJ, Zhou X, Langston JA, Fisher AJ, Toney MD (August 2004). "Crystal structures of unbound and aminooxyacetate-bound Escherichia coli gamma-aminobutyrate aminotransferase". Biochemistry. 43 (34): 10896–905. doi:10.1021/bi049218e. PMID 15323550.
^ a b Storici P, De Biase D, Bossa F, Bruno S, Mozzarelli A, Peneff C, et al. (January 2004). "Structures of gamma-aminobutyric acid (GABA) aminotransferase, a pyridoxal 5'-phosphate, and [2Fe-2S] cluster-containing enzyme, complexed with gamma-ethynyl-GABA and with the antiepilepsy drug vigabatrin". The Journal of Biological Chemistry. 279 (1): 363–73. doi:10.1074/jbc.M305884200. PMID 14534310. S2CID 42918710.
^ Awad R, Muhammad A, Durst T, Trudeau VL, Arnason JT (August 2009). "Bioassay-guided fractionation of lemon balm (Melissa officinalis L.) using an in vitro measure of GABA transaminase activity". Phytotherapy Research. 23 (8): 1075–81. doi:10.1002/ptr.2712. PMID 19165747. S2CID 23127112.
Scott EM, Jakoby WB (April 1959). "Soluble gamma-aminobutyric-glutamic transaminase from Pseudomonas fluorescens". The Journal of Biological Chemistry. 234 (4): 932–6. PMID 13654294.
Aurich H (October 1961). "[On the beta-alanine-alpha-ketoglutarate transaminase from Neurospora crassa]" [On the beta-alanine-alpha-ketoglutarate transaminase from Neurospora crassa]. Hoppe-Seyler's Zeitschrift für Physiologische Chemie (in German). 326: 25–33. doi:10.1515/bchm2.1961.326.1.25. PMID 13863304.
Schousboe A, Wu JY, Roberts E (July 1973). "Purification and characterization of the 4-aminobutyrate--2,ketoglutarate transaminase from mouse brain". Biochemistry. 12 (15): 2868–73. doi:10.1021/bi00739a015. PMID 4719123.
Parviz M, Vogel K, Gibson KM, Pearl PL (November 2014). "Disorders of GABA metabolism: SSADH and GABA-transaminase deficiencies" (PDF). Journal of Pediatric Epilepsy. 3 (4): 217–227. doi:10.3233/PEP-14097. PMC 4256671. PMID 25485164.
Wikimedia Commons has media related to 4-aminobutyrate transaminase.
4-Aminobutyrate+Transaminase at the US National Library of Medicine Medical Subject Headings (MeSH)
Pearl PL, Parviz M, Hodgeman R, Gibson KM, Reimschisel T (2015). "GABA-transaminase deficiency". MedLink Neurology.
Retrieved from "https://en.wikipedia.org/w/index.php?title=4-aminobutyrate_transaminase&oldid=1077021112"
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వికీపీడియా:Alternative text for images - వికీపీడియా
వికీపీడియా:Alternative text for images
"WP:ALT" redirects here. You may also be looking for Wikipedia:Alternative outlets, Wikipedia:Main Page alternatives or WikiProject Alternative music.
Every image with more than decorative value, including math-mode equations, should specify alternative text (alt text). Alternative text is used as a replacement for an image, whenever the image cannot be seen. This can happen, for example, when someone:
uses a screen reader (e.g. a visually impaired person)
uses a text-only browser (e.g. browsing from a mobile phone)
uses a graphical browser with images turned off
has not yet downloaded the image
browses results from a Web search
fails to download the image because of a network problem
copies an extract from a Web page into a word processor.
Alternative text is also used for other purposes. For example, Google's image search uses it to help return appropriate images. Finally, good choice of alternative text and captions makes life easier for people who are viewing the source of an article, either when editing it, or in a diff, or in Wikipedia's internal search.
To add alt text to an image, specify an "alt=" parameter. Here is an example image specified with alt text, next to what its output looks like:
National flag of France.
[[Image:Flag of France.svg|thumb|alt=Vertical tricolor flag (blue, white, red).|National flag of France.]]
In equations formatted in Wikipedia's math mode, the alt text is specified within the initial math tag. For example, this:
<math alt="3/5 = 0.6">\tfrac{3}{5} = 0.6</math>
{\displaystyle {\tfrac {3}{5}}=0.6}
Unlike images, alt text in the math tag takes quotation marks.
Crafting appropriate alternative text can be difficult since it must try to meet several criteria. Above all, it should convey a correct description to visually impaired Wikipedians using screen readers such as JAWS. However, alt text should also make sense in a graphical browser with images turned off, and it should fit with the surrounding text when viewed with a text-only browser. Trying to accommodate all these criteria can be daunting! This article aims to make the whole process easier.
Alt text is not the same as a caption. The alt text is meant for those who cannot see the image, whereas the caption is intended for all readers. In general, the alt text summarizes the image, whereas the caption helps all readers to interpret the image, to focus on its most essential elements and to connect it with the article text.
1 Captions and alt text serve different functions
2 Choosing good alternative text
3 Alternative text is always necessary on Wikipedia
3.1 Alternative text for icons
3.2 Suppressing the link
4 Describe the image, not its file
5.2 Result in your browser
5.3 What people see with images turned off
Captions and alt text serve different functions[మార్చు]
A 3rd century BC coin depicts the co-rulers of Ptolemaic Egypt: Ptolemy II Philadelphus (left), patron and ex-pupil of Philitas; and Philadelphus' sister and wife Arsinoe II, possibly also an ex-pupil.
The alt text description should not duplicate information already present in the caption or the main text of the article. However, material can be added to help the listener identify the subject of the image. Here is an example adapted from the article Philitas of Cos:
[[Image:Oktadrachmon Ptolemaios II Arsinoe II.jpg|thumb|alt=Gold coin showing heads and shoulders of a well-fed king and queen in ancient Greek clothing; the king is more prominent.|A 3rd century BC coin depicts the co-rulers of [[Ptolemaic Egypt]]: [[Ptolemy II Philadelphus]] (left), patron and ex-pupil of Philitas; and Philadelphus' sister and wife [[Arsinoe II of Egypt|Arsinoe II]], possibly also an ex-pupil.]]
The alt text describes the coin's appearance, whereas the caption focuses on its explanation. Again, the basic rule is that alt text is written for those who can't see the image, whereas captions are written for everyone.
Choosing good alternative text[మార్చు]
Alternative text exists as a substitute for the image, for those who cannot see it. It should not interpret the image, or suggest its meaning, since that role belongs to the caption instead. Original research should be scrupulously avoided when writing alt text. In general, a sighted reader should be able to confirm by eye that the alt text describes the image. On the other hand, the alt text need not summarize irrelevant details of the image.
The alt text should make sense when read aloud and in the absence of the image itself. It should be composed of complete sentences and punctuated correctly, with screen readers in mind. It should sit comfortably in the flow of the article.
A helpful way to think about alternative text is to imagine that the Web page text, including all alternative text, is the script for an audio cassette. Your listeners aren't necessarily blind, so they may be interested in hearing about what something looks like. But they can't possibly see any images on the cassette, so referring to an image itself will sound silly. If you write your alternative text with this in mind, it should work well in all the situations listed above.
Alternative text is always necessary on Wikipedia[మార్చు]
Because the Wikipedia images are also links, the alternative text must always be specified. Otherwise, a screen reader would read the file name by default.[WCAG-TECH 1]
Alternative text for icons[మార్చు]
Since icons does not carry important information, only a short and general alt text is needed. For example, stub icon should have the following alt text: alt=Stub icon.
Icons usually have no thumb parameter. In this case, and if there is a caption, MediaWiki will copy the caption in the alt parameter. For example, the result of [[Image:P Eiffel.png|48x24px|Eiffel tower icon]] will be:
<a href="/wiki/Fichier:P_Eiffel.png" class="image" title="Eiffel tower icon"><img alt="Eiffel tower icon" src="http://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/P_Eiffel.png/26px-P_Eiffel.png" width="26" height="24" /></a>
Suppressing the link[మార్చు]
When the link on the image is not relevant (for example, link on icons are not relevant), you may decide to suppress it. To do so, you have two options:
When you add an empty link= parameter, Mediawiki produces an empty alt text (alt=""). Since there is no link the image becomes accessible.[WCAG-TECH 2]
For example, the result of [[Image:P Eiffel.png|48x24px|link=]] is:
Use css to have the icon in background. Here is an example:
<div class="icon eiffel">Textual content associated with the icon</div>
Add the following code in MediaWiki:Common.css:
.icon.eiffel {
background: url(http://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/P_Eiffel.png/30px-P_Eiffel.png) no-repeat left center;
padding-left:1px 4px 1px 24px;
Describe the image, not its file[మార్చు]
Alternative text describes the image, not its file or format. Thus, you shouldn't mention
the dimensions of the file
instructions on how to view a bigger version of the image.
the fact that it is a substitute for an image:
Good: "The straight chain form consists of four C H O H groups..."
Bad: "This image shows that the straight chain form consists of four C H O H groups..."
Wikipedia code[మార్చు]
[[Image:Antikythera philosopher.JPG|thumb|left|upright|alt=Bronze head of bearded man.|The ''Philosopher'' ({{circa|250–200 BC}}) from the [[Antikythera wreck]] illustrates the style used by Hecataeus in his bronze of Philitas.]]
The phrase "Bronze head of bearded man." is the text that will be rendered in place of the image.
Result in your browser[మార్చు]
The Philosopher (కాలం. 250–200 BC) from the Antikythera wreck illustrates the style used by Hecataeus in his bronze of Philitas.
The above might be displayed in a graphical web browser that is capable of showing images. In this case, the image serves to illustrate the description of a (different) bronze statue.
What people see with images turned off[మార్చు]
Bronze head of bearded man.
This example shows what might be displayed in a web browser with images turned off. Many modern browsers allow users to turn off images (for example, if they are using a low-bandwidth connection). The contents of the alt text are often rendered in place of the image. Here, the phrase serves as a useful replacement for the image.
Caveat: Though this is not technically considered an appropriate usage for the alt text contents, many browsers nevertheless display it in this manner. If possible, choose alt text which makes sense in this format.
Until 2008, the alt text of an image was automatically the same as its caption, an approach with two drawbacks. First, this meant that the image caption was read twice by the screen reader to the listener, once for the alt text and once for the caption itself. Second, captions do not always describe their image, but rather describe its meaning for the reader.
A new system was introduced in October 2008. It allows Wikipedians to specify the image alt text independently of its caption. The caption is always the last parameter in the Image link, whereas the alt text is specified by an "alt=" parameter. In this new system, the alt text serves to describe the image to those who cannot see it, whereas the caption conveys essential information beyond what can be seen.
Need more help?[మార్చు]
If you're still unsure about the best alternative text for an image, leave a note on this article's talk page, and someone will help you out.
The following references link to the techniques for WCAG2.0:
↑ Failure of 2.4.4, 2.4.9 and 4.1.2 due to using null alt on an image where the image is the only content in a link, A accessibility level
↑ Using null alt text and no title attribute on img elements for images that AT should ignore, A accessibility level
Use the alt attribute - World Wide Web Consortium
W3C Web Content Accessibility Guidelines 1.0, section on ALT text
"anybrowser.org" web site
another "Best Viewed With Any Browser" web page
"https://te.wikipedia.org/w/index.php?title=వికీపీడియా:Alternative_text_for_images&oldid=959947" నుండి వెలికితీశారు
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This workshop was a successful and happy meeting of 23 researchers working on various aspects of partial differential equations on singular spaces and noncompact manifolds. This field encompasses such diverse topics as
L^2
cohomology, harmonic analysis and spectral analysis on locally symmetric spaces, spectral geometry, and constructions of metrics with special geometry. All of these topics, and several others, were covered extensively at this meeting. One noteworthy feature was a series of four expository survey talks by leading experts. These served to stimulate extensive discussion which carried over into all the other sessions. Lizhen Ji presented a view of the extensive compactification theory for locally symmetric spaces; Werner M\"uller gave a detailed survey of harmonic analysis on locally symmetric spaces, leading up to a presentation of his important new work in this area; Thomas Schick discussed the topological and
C^*
algebraic techniques now used extensively in index theory; finally, Michael Singer explained some of the analytic aspects of the burgeoning field of extremal K\"ahler metrics. The rest of the talks complemented these surveys. For example, Leslie Saper presented an overview of his recent and ongoing work on the
L^2
cohomology of locally symmetric spaces and the algebraic machinery of
{\mathcal L}
-modules he has developed for this goal. Gilles Carron talked about the
L^2
cohomology of QALE spaces and Eugenie Hunsicker's talk explored further relationships between
L^2
and intersection cohomology theory on manifolds with fibred boundaries, while Ulrich Bunke discussed his recent work on twisted K-theory. There were several talks on more traditional problems in spectral geometry, including one by Iosif Polterovich on new methods to obtain lower bounds for the spectral function of the Laplacian, one on estimates for the first positive eigenvalue of the Dirac operator by Bernd Ammann, Daniel Grieser's talk on the `max flow min cut' method adapted from graph theory applied to the estimation of the Cheeger constant, and Sergiu Moroianu's talk on the disappearance of the essential spectrum for certain magnetic Laplacians. Talks focusing on metrics with special geometry included the one by Robin Graham on the Dirichlet-to-Neumann operator for Poincare-Einstein metrics, Hartmut Weiss' results on deformations of three-dimensional constant curvature cone metrics, and Frank Pacard's presentation of his construction, with Arezzo, of new constant scalar curvature K\"ahler metrics. Finally, Nader Yeganefar discussed his work on topological restrictions associated with quadratic curvature decay. One of the things that stood out about this meeting was the spirited participation by a number of young researchers, many of whom are clearly poised to take their places amongst the next generation of leaders in this field. The quality of the talks was uniformly high, and their clarity certainly furthered the main goal, which was to further stimulate the discussion between the various groups of researchers at this meeting. Several major results were announced and discussed here. These illustrate that this field is now reaching a certain maturity. A number of these problems have been outstanding for many years, but new ideas have finally allowed for more serious attacks on problems of a truly global nature, and on problems involving iterated singularities. In any case, the meeting certainly demonstrated the current vitality of this field.
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A monotone method for fourth order boundary value problems involving a factorizable linear operator | EMS Press
We consider the nonlinear fourth order beam equation
u^{\iv}=f(t,u,u''),
with boundary conditions corresponding to the periodic or the hinged beam problem. In presence of upper and lower solutions, we consider a monotone method to obtain solutions. The main idea is to write the equation in the form
u^{\iv}-cu''+du=g(t,u,u''),
c
d
are adequate constants, and use maximum principles and a suitable decomposition of the operator appearing in the left-hand side.
P. Habets, Margarita Ramalho, A monotone method for fourth order boundary value problems involving a factorizable linear operator. Port. Math. 64 (2007), no. 3, pp. 255–279
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Polygon - Citizendium
A polygon is a two-dimensional geometric closed figure bounded by a continuous set of line segments. The word derives from the Greek word for angle, γωνία, and the Greek word for many, πολλος (also πολυς), hence literally a polygon is a "many-angle". A polygon, in Euclidean geometry, must have at least three sides. A polygon of three sides is called a triangle, four sides a quadrilateral, five sides a pentagon, six sides a hexagon. Figures with more sides are typically named with the Greek numeral for the number of sides, followed by "-gon". Mathematicians discussing the properties of polygons with large numbers of sides will often use the formulation n-gon, where n is replaced by the number of sides (i.e., a 17-gon or a 100-gon). When discussing the properties of classes of polygons which include polygons of different numbers of sides, mathematicians will sometimes refer to n-gon, without substituting the n.
The line segments bounding a polygon are known as sides, and the points where the sides meet are vertices (singular vertex).
A polygon is known as a simple polygon if none of its sides cross other sides, and each vertex is the meeting point of only two sides. A polygon which does not meet this criterion is a complex polygon. A polygon is called convex when there are no line segments which connect two points within the polygon which pass outside the polygon. Convex polygons have no internal angles between two adjacent sides greater than 180 degrees of arc. A polygon which has all sides of equal length is known as an equilateral polygon. A polygon which has all internal angles equal is known as an equiangular polygon. If the number of sides is greater than three, an equilateral polygon is not necessarily an equiangular polygon. A convex polygon which has all sides and all internal angles equal is known as a regular polygon. A complex polygon which has all sides and all angles equal is known as a regular star polygon.
All polygons have the same number of sides and vertices. The sum of the interior angles of a simple polygon, R, is
{\displaystyle R=\pi *(n-2)}
{\displaystyle R=180*(n-2)}
(in degrees) where n is the number of sides of the polygon. Other than for triangles and quadrilaterals, there are no general formulas for determining the area of a polygon; the area must be determined by dividing the polygon into separate pieces whose areas can be determined (usually triangles), and adding the area of all the parts.
Regular polygons have properties which are more easily determined analytically. For a given regular polygon with side length s and number of sides n, interior angle at each vertex is
{\displaystyle \theta ={\tfrac {\pi (n-2)}{n}}=\left(1-{\tfrac {2}{n}}\right)\pi }
, the perimeter p is:
{\displaystyle p=sn}
, and the area A is:
{\displaystyle A={\tfrac {1}{4}}s^{2}\cot \left({\tfrac {\pi }{n}}\right)}
Many polygons are named, and for 3-gons and 4-gons, there are particular names for special cases. Some of these are listed below.
right triangle One internal angle is a right angle (90 degrees)
isoceles triangle Two sides of the same length
equilateral triangle All three sides of the same length, three angles equal (60 degrees).
4 quadrilateral, quadrangle, tetragon quadrangle and tetragon are not common usages
trapezoid two sides parallel
parallelogram both pairs of non-adjacent sides are parallel
rhombus equilateral (also a parallelogram)
rectangle equiangular (also a parallelogram)
n>10 usually n-gon there are rules for using Greek numbers or constructing polygon names, but these are not frequently used.
Retrieved from "https://citizendium.org/wiki/index.php?title=Polygon&oldid=25012"
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§ Proving block matmul using program analysis
It's a somewhat well-known fact that given matrix multiplication:
O = AB
O \in \mathbb R^{2n \times 2m}
O
for output),
A \in \mathbb R^{2n \times r}, B \in \mathbb R^{r \times 2m}
are matrices. We can also write this as follows:
\begin{bmatrix} o_{11} & o_{12} \\ o_{21} & o_{22} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} b_{11} + a_{12} b_{21} & a_{11} b_{12} + a_{12} b_{22} \\ a_{21} b_{11}+ a_{22} b_{21} & a_{21} b_{12} + a_{22} b_{22} \end{bmatrix}
When written as code, the original matrix multiplication is:
// a:[2N][2Z] b:[2Z][2M] -> out:[2N][2M]
int matmul(int N, int Z, int M, int a[N][Z], int b[Z][M], int out[N][M]) {
for(int i = 0; i < 2*N; ++i) {
for(int j = 0; j < 2*M; ++j) {
for(int k = 0; k < 2Z; ++k) out[i][j] += a[i][k] * b[k][j]
and the block-based matrix multiplication is:
int matmulBlock(int N, int Z, int M, int a[N][Z], int b[Z][M], int out[N][M]) {
for (int BI = 0; BI < 2; ++BI) {
for (int BJ = 0; BJ < 2; ++BJ) {
for(int i = BI*N; i < BI*N+N; ++i) {
for(int j = BJ*M; j < BJ*M+M; ++j) {
for(int k = 0; k < 2Z; ++k) { out[i][j] += a[i][k] * b[k][j] }
we wish to show that both of these programs have the same semantics . We will do this by appealing to ideas from program analysis.
§ The key idea
We will consider the statement:
out[i][j] += a[i][k] * b[k][j]
as occuring at an abstract "point in time"
(i, j, k)
in the matmul function. I also occurs at an abstract "point in time"
(BI, BJ, i', j', k')
in the matmulBlock function. We will then show that the loops for(i...) for(j...) for(k...) are fully parallel, and hence we can reorder the loops any way we want. Then, we will show that the ordering imposed by
(BI, BJ, i', j', k')
is a reordering of the original
(i, j, k)
ordering. We do this by showing that there is a bijection:
(i=i_0, j=j_0, k=k_0) \rightarrow (BI=i_0/N, BJ=j_0/N, i=i_0\%N, j=j_0\%N, k=k_0)
Thus, this bijection executes all loops, and does so without affecting the program semantics.
§ Schedules
We'll zoom out a little, to consider some simple programs and understan how to represent parallelism.
void eg1(int N, int M, int out[N][M]) {
out[i][j] = out[i][j-1];
Notice that this program is equivalent to the program with the
i
loop reversed:
void eg1rev(int N, int M, int out[N][M]) {
for(int i = N-1; i >=0; --i) {
What's actually stopping us from reversing the loop for(j...)? it's the fact that the value of, say, out[i=0][j=1] depends on out[i=0][j=0]. We can see that in general, out[i=i_0][j=j_0] depends on out[i=i_0][j=j_0-1]. We can represent this by considering a dependence set :
\{ \texttt{write}:(i_0, j_0-1) \rightarrow \texttt{write}:(i_0, j_0) \}
in general, we can reorder statements as long as we do not change the directions of the arrows in the dependence set. We can imagine the scenario as follows:
| (1, 0)->(1, 1)->(1, 2)->(1, 3) ...
| (0, 0)->(0, 1)->(0, 2)->(0, 3) ....
(i, j) -->
§ Dependence structure of matmul.
§ Fully parallel, reordering
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Sebaran kondisional - Wikipédia Sunda, énsiklopédi bébas
Sebaran kondisional
Diberekeun dua sebaran gabungan variabel random X jeung Y, sebaran kondisional probabiliti of Y given X (written "Y | X") is the probability distribution of Y when X is known to be a particular value.
For discrete random variables, the conditional probability mass function can be written as P(Y = y | X = x). From the definition of conditional probability, this is
{\displaystyle P(Y=y|X=x)={\frac {P(X=x,Y=y)}{P(X=x)}}={\frac {P(X=x|Y=y)P(Y=y)}{P(X=x)}}}
Similarly for continuous random variables, the conditional probability density function can be written as pY|X(y | x) and this is
{\displaystyle p_{Y|X}(y|x)={\frac {p_{X,Y}(x,y)}{p_{X}(x)}}={\frac {p_{X|Y}(x|y)p_{Y}(y)}{p_{X}(x)}}}
where pX,Y(x, y) gives the joint distribution of X and Y, while pX(x) gives the marginal distribution for X.
If for discrete random variables P(Y = y | X = x) = P(Y = y) for all x and y, or for continuous random variables pY|X(y | x) = pY(y) for all x and y, then Y is said to be independent of X (and this implies that X is also independent of Y).
Seen as a function of y for given x, P(Y = y | X = x) is a probability and so the sum over all y (or integral if it is a density) is 1. Seen as a function of x for given y, it is a likelihood, so that the sum over all x need not be 1.
Dicomot ti "https://su.wikipedia.org/w/index.php?title=Sebaran_kondisional&oldid=496041"
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Kind (type theory) - Wikipedia
In the area of mathematical logic and computer science known as type theory, a kind is the type of a type constructor or, less commonly, the type of a higher-order type operator. A kind system is essentially a simply typed lambda calculus "one level up", endowed with a primitive type, denoted
{\displaystyle *}
and called "type", which is the kind of any data type which does not need any type parameters.
A kind is sometimes confusingly described as the "type of a (data) type", but it is actually more of an arity specifier. Syntactically, it is natural to consider polymorphic types to be type constructors, thus non-polymorphic types to be nullary type constructors. But all nullary constructors, thus all monomorphic types, have the same, simplest kind; namely
{\displaystyle *}
Since higher-order type operators are uncommon in programming languages, in most programming practice, kinds are used to distinguish between data types and the types of constructors which are used to implement parametric polymorphism. Kinds appear, either explicitly or implicitly, in languages whose type systems account for parametric polymorphism in a programatically accessible way, such as C++,[1] Haskell and Scala.[2]
2 Kinds in Haskell
{\displaystyle *}
, pronounced "type", is the kind of all data types seen as nullary type constructors, and also called proper types in this context. This normally includes function types in functional programming languages.
{\displaystyle *\rightarrow *}
is the kind of a unary type constructor, e.g. of a list type constructor.
{\displaystyle *\rightarrow *\rightarrow *}
is the kind of a binary type constructor (via currying), e.g. of a pair type constructor, and also that of a function type constructor (not to be confused with the result of its application, which itself is a function type, thus of kind
{\displaystyle *}
{\displaystyle (*\rightarrow *)\rightarrow *}
is the kind of a higher-order type operator from unary type constructors to proper types.[3]
Kinds in Haskell[edit]
(Note: Haskell documentation uses the same arrow for both function types and kinds.)
The kind system of Haskell 98[4] includes exactly two kinds:
{\displaystyle *}
, pronounced "type" is the kind of all data types.
{\displaystyle k_{1}\rightarrow k_{2}}
is the kind of a unary type constructor, which takes a type of kind
{\displaystyle k_{1}}
and produces a type of kind
{\displaystyle k_{2}}
An inhabited type (as proper types are called in Haskell) is a type which has values. For instance, ignoring type classes which complicate the picture, 4 is a value of type Int, while [1, 2, 3] is a value of type [Int] (list of Ints). Therefore, Int and [Int] have kind
{\displaystyle *}
, but so does any function type, for instance Int -> Bool or even Int -> Int -> Bool.
A type constructor takes one or more type arguments, and produces a data type when enough arguments are supplied, i.e. it supports partial application thanks to currying.[5][6] This is how Haskell achieves parametric types. For instance, the type [] (list) is a type constructor - it takes a single argument to specify the type of the elements of the list. Hence, [Int] (list of Ints), [Float] (list of Floats) and even [[Int]] (list of lists of Ints) are valid applications of the [] type constructor. Therefore, [] is a type of kind
{\displaystyle *\rightarrow *}
. Because Int has kind
{\displaystyle *}
, applying [] to it results in [Int], of kind
{\displaystyle *}
. The 2-tuple constructor (,) has kind
{\displaystyle *\rightarrow *\rightarrow *}
, the 3-tuple constructor (,,) has kind
{\displaystyle *\rightarrow *\rightarrow *\rightarrow *}
Kind inference[edit]
Standard Haskell does not allow polymorphic kinds. This is in contrast to parametric polymorphism on types, which is supported in Haskell. For instance, in the following example:
data Tree z = Leaf | Fork (Tree z) (Tree z)
the kind of z could be anything, including
{\displaystyle *}
{\displaystyle *\rightarrow *}
etc. Haskell by default will always infer kinds to be
{\displaystyle *}
, unless the type explicitly indicates otherwise (see below). Therefore the type checker will reject the following use of Tree:
type FunnyTree = Tree [] -- invalid
because the kind of [],
{\displaystyle *\rightarrow *}
does not match the expected kind for z, which is always
{\displaystyle *}
Higher-order type operators are allowed however. For instance:
data App unt z = Z (unt z)
has kind
{\displaystyle (*\rightarrow *)\rightarrow *\rightarrow *}
, i.e. unt is expected to be a unary data constructor, which gets applied to its argument, which must be a type, and returns another type.
GHC has the extension PolyKinds, which, together with KindSignatures, allows polymorphic kinds. For example:
data Tree (z :: k) = Leaf | Fork (Tree z) (Tree z)
type FunnyTree = Tree [] -- OK
Since GHC 8.0.1, types and kinds are merged.[7]
System F-omega
Pierce, Benjamin (2002). Types and Programming Languages. MIT Press. ISBN 0-262-16209-1. , chapter 29, "Type Operators and Kinding"
^ "CS 115: Parametric Polymorphism: Template Functions". www2.cs.uregina.ca. Retrieved 2020-08-06.
^ Generics of a Higher Kind
^ Pierce (2002), chapter 32
^ Kinds - The Haskell 98 Report
^ "Chapter 4 Declarations and Binding". Haskell 2010 Language Report. Retrieved 23 July 2012.
^ Miran, Lipovača. "Learn You a Haskell for Great Good!". Making Our Own Types and Typeclasses. Retrieved 23 July 2012.
^ "9.1. Language options — Glasgow Haskell Compiler Users Guide".
Retrieved from "https://en.wikipedia.org/w/index.php?title=Kind_(type_theory)&oldid=1026640571"
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OrbitProblemSolution - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Summation and Difference Equations : LREtools : HypergeometricTerm Subpackage : OrbitProblemSolution
OrbitProblemSolution
solve the sigma-orbit problem
OrbitProblemSolution(
\mathrm{\alpha }
\mathrm{\beta }
, x, r)
\mathrm{\alpha }
first polynomial or an algebraic number
\mathrm{\beta }
second polynomial or an algebraic number
list of equations which gives the tower of hypergeometric extensions
The OrbitProblemSolution(
\mathrm{\alpha }
\mathrm{\beta }
, x, r) command returns the solution of a
\mathrm{\sigma }
-orbit problem, that is, a positive integer n such that
{E}^{n-1}\mathrm{\alpha }\cdot \dots \cdot E\mathrm{\alpha }\cdot \mathrm{\alpha }=\mathrm{\beta }
\mathrm{\alpha }
\mathrm{\beta }
can be algebraic numbers or polynomials in K(r), where K is the ground field and r is the tower of hypergeometric extensions. Each
{r}_{i}
\frac{{\mathrm{Er}}_{i}}{{r}_{i}}
is a rational function over K. E is the shift operator.
\mathrm{\alpha }
\mathrm{\beta }
are algebraic numbers then the procedure solves the classic orbit problem (
{\mathrm{\alpha }}^{n}=\mathrm{\beta }
). Otherwise, it solves the
\mathrm{\sigma }
-orbit problem for polynomials in the tower of hypergeometric extensions. This means that the polynomials can contain hypergeometric terms in their coefficients. These terms are defined in the parameter r. Each hypergeometric term in the list is specified by a name, for example, t. It can be specified directly in the form of an equation, for example,
t=n!
[t,n+1]
. The OrbitProblemSolution function returns
-1
if there is no solution.
If the arguments of the
\mathrm{\sigma }
-orbit problem are algebraic numbers, then the routine directly computes the solution. Otherwise, a hypergeometric dispersion is calculated. For an empty tower of hypergeometric extensions, a simple dispersion is calculated.
\mathrm{with}\left(\mathrm{LREtools}[\mathrm{HypergeometricTerm}]\right):
\mathrm{OrbitProblemSolution}\left(s+1,\left(s+1\right)\left(2s+1\right),x,[s={2}^{x}]\right)
\textcolor[rgb]{0,0,1}{2}
\mathrm{OrbitProblemSolution}\left(t+s,\left(t+s\right)\left(\left(x+1\right)t+2s\right),x,[t=x!,s={2}^{x}]\right)
\textcolor[rgb]{0,0,1}{2}
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§ Whalesong hyperbolic space in detail
We can build a toy model of a space where velocity increases with depth. Let the x-y axis be: left-to-right (→) is positive x, top-to-bottom (↓) is positive y. Now let the velocity at a given location
(x^\star, y^\star)
(y^\star+1, 1)
. That is, velocity along
y
is constant; the velocity along
x
is an increasing function of the current depth. The velocity along
x
increases linearly with depth. Under such a model, our shortest paths will be 'curved' paths.
Very funnily, the wikipedia page on Whale vocalization has a 'selected discograpy' section: Whalesong, selected discography
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