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https://www.neetprep.com/questions/126-Physics/684-Mechanical-Properties-Solids?courseId=18 | A gas undergoes a process in which its pressure P and volume V are related as ${\mathrm{VP}}^{\mathrm{n}}=\mathrm{constant}$. The bulk modulus for the gas in the process is
[Only for Dropper and XII batch]
1. $\frac{P}{n}$
2. ${P}^{1/n}$
3. nP
4. ${P}^{n}$
Concept Questions :-
Stress-strain
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
The young's modulus of a wire of length 'L' and radius 'r' is 'Y'. If length is reduced to L/2 and radius r/2 then young's modulus will be
1. Y/2
2. Y
3. 2Y
4. 4Y
Concept Questions :-
Hooke's law
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
Three wires A,B,C made of the same material and radius have different lengths. The graphs in the figure show the elongation-load variation. The longest wire is
1. A
2. B
3. C
4. All
Concept Questions :-
Stress-strain curve
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
The breaking stress of a wire depends upon
1. material of the wire
2. length of the wire
4. shape of the cross section
Concept Questions :-
Elasticity
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
The elastic energy stored in a wire of Young's Modulus Y is -
1. $\mathrm{Y}×\frac{{\mathrm{strain}}^{2}}{\mathrm{volume}}$
2. $\mathrm{stress}×\mathrm{strain}×\mathrm{volume}$
3. $\frac{{\mathrm{strain}}^{2}×\mathrm{volume}}{2\mathrm{Y}}$
4. $\frac{1}{2}×stress×strain×volume$
Concept Questions :-
Stress-strain
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
If Young modulus (Y) equal to bulk modulus (B). Then the Poisson ratio is :
1. $\frac{1}{3}$
2. $\frac{2}{3}$
3. $\frac{1}{2}$
4. $\frac{1}{4}$
Concept Questions :-
Poisson ratio
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
The bulk modulus of a spherical object is B. If it is subjected to uniform pressure p, the fractional decrease in radius is
(a)$\frac{P}{B}$
(b) $\frac{B}{3p}$
(c) $\frac{3p}{B}$
(d)$\frac{p}{3B}$
Concept Questions :-
Stress-strain
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
The Young's modulus of steel is twice that of brass. Two wires of same length and of same area of cross-section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weight added to the steel and brass wires must be in the ratio of
(a)1:2
(b)2:1
(c)4:1
(d)1:1
Concept Questions :-
Stress-strain
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
Copper of fixed volume V is drawn into wire of length l. When this wire is subjected to a constant force F, the extension produced in the wire is Δl. Which of the following graphs is a straight line?
(a)Δl versues 1/l
(b)Δl versus l2
(c)Δl versus 1/l2
(d) Δl versus l
Concept Questions :-
Hooke's law
High Yielding Test Series + Question Bank - NEET 2020
Difficulty Level:
The following four wires are made of the same material. Which of then will have the largest extension when the same tension is applied?
(a) Length=50cm, diameter=0.5mm
(b) Length=100cm, diameter=1mm
(c) Length=200cm, diameter=2mm
(d) Length=300cm, diameter=3mm
Concept Questions :-
Hooke's law | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 16, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9087586998939514, "perplexity": 4287.214787278061}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347407001.36/warc/CC-MAIN-20200530005804-20200530035804-00594.warc.gz"} |
https://rd.springer.com/article/10.1007/s12220-020-00561-5?error=cookies_not_supported&code=adb7e3d8-487c-4cf7-82d0-d196be172567 | # Commutators of Cauchy–Fantappiè Type Integrals on Generalized Morrey Spaces on Complex Ellipsoids
## Abstract
Let $$\Omega$$ be a domain which belongs to a class of bounded weakly pseudoconvex domains of finite type in $${\mathbb {C}}^n$$, let $$d\lambda$$ be the Monge–Ampère boundary measure on $$b\Omega$$ and $$\varrho \ge 0$$ be a non-decreasing function. The aim of this paper is to establish the characterizations of boundedness and compactness for the commutator operators of Cauchy–Fantappiè type integrals with $$L^1(b\Omega ,d\lambda )$$ functions on the generalized Morrey spaces $$L^{p}_\varrho (b\Omega ,d\lambda )$$, with $$p\in (1, \infty )$$.
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Arai, H., Mizuhara, T.: Morrey spaces on spaces of homogeneous type and estimates for $$\Box _b$$ and the Cauchy–Szegö projection. Math. Nachr. 186, 5–20 (1997)
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Duong, X.T., Lacey, M., Li, J., Wick, B.D., Wu, Q.: Commutators of Cauchy type integrals for domains in $${\mathbb{C}}^n$$ with minimal smoothness, to appear on Indiana Univ. Maths. J. (2019)
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Morrey, C.B.: Some recent developments in the theory of partial differential equations. Bull. Am. Math. Soc. 68, 279–297 (1962)
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Pérez, C.: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128, 163–185 (1995)
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Stein, E.M.: Three projection operators in complex analysis. In: Colloquium De Giorgi 2010–2012, pp. 49–59
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Tao, J., Yang, D., Yang, D.: Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces. Math. Methods Appl. Sci. 42, 1631–1651 (2019)
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Uchiyama, A.: On the compactness of operators of Hankel type. Tohoku Math. J. 30, 163–171 (1978)
## Acknowledgements
The authors would like to thank the referee(s) for valuable suggestions and comments that led to the improvement of the paper.
## Author information
Authors
### Corresponding author
Correspondence to Xuan Thinh Duong.
### Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A part of the paper was completed during a scientific stay of the authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM), whose hospitality is gratefully appreciated. Xuan Thinh Duong was supported by the Australian Research Council through the Discovery Project 190100970. Ly Kim Ha was funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under Grant Number B2019-18-01.
## Rights and permissions
Reprints and Permissions
Dao, N.A., Duong, X.T. & Ha, L.K. Commutators of Cauchy–Fantappiè Type Integrals on Generalized Morrey Spaces on Complex Ellipsoids. J Geom Anal (2021). https://doi.org/10.1007/s12220-020-00561-5
• Accepted:
• Published:
### Keywords
• BMO
• VMO
• Commutators
• Singular integral operators
• Convex domains of finite type
• 47B40
• 47B70
• 32A55
• 32A26
• 32A37
• 32A50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9187425971031189, "perplexity": 4766.280742523142}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178381230.99/warc/CC-MAIN-20210307231028-20210308021028-00500.warc.gz"} |
https://www.physicsforums.com/threads/simple-tension-problem.68012/ | # Simple Tension Problem
1. Mar 20, 2005
### wetcarpet
There are two wires connected to a ceiling with a light fixture hanging at their ends. The light fixture exerts a force of 80N, while the wires make a 40 degree angle with the ceiling. What is the tension in the two strings? The diagram roughly looks like this:
---------------
- 40 40 -
- -
- -
-
-
80N
I assumed that this was a simple trigonometry problem and set about solving it like so:
{a} I assumed that since the object was not moving downward, there had to be an 80N force exterted upward. Hence, I split the large triangle into two parts using a positive 80N force, making a 90 degree angle with the ceiling, and a 50 degree angle at the base of the larger triangle.
{b} From there I used trigonometry:
cos(50) = 80N/Hyp.
Hyp. = 80N/cos(50)
Hyp. = 124.5N
Yet, the software (Webassign.com) is telling me that the tension is not 124.5N. What am I doing wrong?
P.S.- I also tried assuming that there is not an 80N positive force exerted upward, and thereby tried isolating each X and Y vector for each string seperately. I still got the same answer of 124.5N.
2. Mar 20, 2005
### wetcarpet
3. Mar 20, 2005
### Nylex
If they want the value for T in each string, you need to divide that value by 2. Resolve the forces perpendicular to the string and you get 2Tsin 40 = 80.
4. Mar 20, 2005
### wetcarpet
I thank you kindly, dividing 124.5N by two did work, and I appreciate your explanation as well.
Last edited: Mar 20, 2005
Similar Discussions: Simple Tension Problem | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8708094954490662, "perplexity": 1045.5625753090153}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891817437.99/warc/CC-MAIN-20180225205820-20180225225820-00085.warc.gz"} |
http://tex.stackexchange.com/questions/60536/how-to-remove-parenthesis-in-an-equation-number-created-by-tag | # How to remove parenthesis in an equation number created by \tag?
This is probably simple, but I can't figure it out. How can I remove the parentheses around a \tag in an align equation? I would like the second line (and only that) to contain [2] instead of ([2]). A workaround would be to give it no label, to see the label [2] as part of the equation, and move it to the right text margin by hand. But how? \hfill does not insert space here.
\documentclass{scrartcl}
\usepackage[T1]{fontenc}
\usepackage[american,ngerman]{babel}
\usepackage{amsmath}
\begin{document}
\begin{align}
a &= b \\
c &= d \tag{[2]}
\end{align}
\end{document}
-
Use \tag*, see p. 3 of amsmath documentation. – egreg Jun 20 '12 at 11:27
Ahh... I even used that before some time ago... Thanks, egreg! – Marius Hofert Jun 20 '12 at 11:31
@egreg: I made a CW anser so Marius can accept it an this question won’t stay unanswered. Hope you don’t mind :-) – Tobi Jun 20 '12 at 11:38
Thanks a lot, Tobi :-) – Marius Hofert Jun 20 '12 at 12:43
As egreg said, use the starred version \tag* which is the same as \tag except that it does not automatically enclose the tag in parentheses.
\documentclass{article}
$$a + b = c \tag*{[2]}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9432695508003235, "perplexity": 2202.8130746832676}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042986423.95/warc/CC-MAIN-20150728002306-00037-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://xcorr.net/2012/02/07/using-a-particle-filter-to-decode-place-cells/ | ### Using a particle filter to decode place cells
In the last post, I discussed using an extended Kalman filter to decode place cells, based on the algorithm published in Brown et al. (1998). The results looked pretty good. EKFs are certainly better than population vector approaches that don’t consider the sequential nature of the decoding task. The fact that the path of the rat must be continuous should be certainly be used in the decoding process.
However, extended Kalman filters are rather inflexible. They only deal with systems which are quasi-linear, and they are a bit ad hoc. Place cells are not especially good system to use EKFs, since the rate of the cells is a highly nonlinear function of the position.
A flexible alternative to the EKF is the particle filter, which is a sequential Monte Carlo method. The idea behind the vanilla version of the particle filter called sequential importance resampling (SIR) is very simple. As in other Monte Carlo methods, the probability distribution of the current state (at time t – 1) is represented by a set of samples. These samples are called particles. To obtain the probability distribution of the state at the following time point, it suffices to propagate the probability through these equations:
$p(x_t|y^n_1...y^n_t) = K p(y^n_t|x_t) p(x_t|y^n_1...y^n_{t-1})$
$p(x_t|y^n_1...y^n_{t-1}) = \int p(x_{t-1}|y^n_1...y^n_{t-1}) p(x_t|x_{t-1}) dx_{t-1}$
The second equation is cake; since the probability distribution is represented by samples, one only needs to propagate the samples through the transition distribution. This typically involves nudging the position of the particles by random amounts determined by the prior. Then:
$p(x_t|y^n_1...y^n_t) \approx K \sum_i p(y^n_t|x_t^i) \delta(x_t - x_t^i)$
An estimate of the current state is then given by the expected value of this probability distribution. Now eventually the weights for most particles will go to zero, hence a resampling step is required. If a distribution is given by a weighted sum of delta functions, then it can be sampled by taking multinomial draws corresponding to the weights. If you decide to do this at every time point, then you get the SIR particle filter with unconditional resampling, which is the form of particle filter used in Brockwell, Rojas and Kass (2004; Recursive Bayesian decoding of motor cortical signals by particle filtering). There’s fancier forms of particle filters available, which involve using better proposal distributions, but I’m waiting for my ReBEL license (a Matlab toolkit) to try them out.
I tried both the EKF and particle filter using the same simulation method as in the previous post. Here’s an example decoded path using the EKF:
And the same with the particle filter:
You’ll notice that the reconstruction is much more reliable for the particle filter (MSE of .11 and .07 in the x and y direction versus .19 and .16 for the EKF). Repeating these simulations 100 times, I found a mean MSE of 0.095 in both directions for the particle filter (with 1000 particles) and .13 for the EKF (the median MSE was 10-20% better). So the effect is certainly large enough to matter in practice, especially if the errors are not random but last for a large number of time samples.
The EKF seems to get stuck for long periods of time after rapid movements. My intuition here is that in some circumstances the posterior of the state has some weird shape with multiple local minima. The EKF, which performs a local search, gets stuck in a local minimum for some number of time steps, until by chance a path is created towards the global minimum. Indeed, you can plot the state distribution given by the particle filter, and frequently you’ll see bimodal distributions like this one:
Now Brown et al. mention that sometimes their algorithm doesn’t track sudden movements very well. This could be due to this issue of the decoder getting stuck in a local minimum, but it could also be due to a failure of the model to capture the super-Gaussian changes in position from frame to frame. Indeed, they show in Figure 3 that the density of position changes is not well captured by a Gaussian, and suggest a Laplacian distribution might be more appropriate. I simulated paths where the velocity was given by $v=(r\cos(\theta),r\sin(\theta)$ where r had an exponential distribution with mean .03 and $\theta$ had a uniform distribution.
I attempted to decode the place cells with three models: EKF with incorrect transition probability, particle filter with incorrect transition probability, and particle filter with correct transition probability. Here the EKF error increased marginally from .13 to .14, while the particle filter with both correct and incorrect transition probability got MSEs of .095. From this simulation, it appears that changing the prior barely has an effect on the ability to reconstruct the stimulus. This is counter-intuitive, and would suggest that the reconstruction is mostly driven by the likelihood term rather than the prior term. I don’t really buy this result, and I’d like to try the same with real data. It does point towards the idea that the problem with the EKF is not so much its inability to capture the super-Gaussian nature of the transitions but really the dumber problem that the posterior has a weird shape and it just can’t deal with that well.
In any case, the particle filter is quite a bit simpler to program than the IEKF, it’s faster, and it’s more flexible. I’d like to try it on the CRCNS hippocampal dataset by Buzsáki whenever I have the chance.
Here’s the script I used to run the particle filter:
function [xs,Ws,particles] = pfDecoder(Y,params,W)
%Here I implement the method of Brockwell, Rojas and Kass (2004)
nparticles = 1000;
%Draw from the initial state distribution
xtildes = randn(nparticles,2);
xs = zeros(size(Y,1),2);
Ws = zeros(size(Y,1),2,2);
particles = zeros(size(Y,1),nparticles,2);
Wsqrt = W^(1/2);
%Main loop
for ii = 1:size(Y,1)
%Step 2: compute weights according to w = p(y|v_t=xtilde)
ws = computeLogLikelihoods(xtildes,Y(ii,:)',params);
%Normalize weights
ws = exp(ws-max(ws));
ws = ws/sum(ws);
%Step 3: importance sampling
idx = mnrnd(nparticles,ws);
S = bsxfun(@le,bsxfun(@plus,zeros(nparticles,1),1:max(idx)),idx');
[idxs,~] = find(S);
xtildes = xtildes(idxs,:);
particles(ii,:,:) = xtildes;
%Step 3.5
xs(ii,:) = mean(xtildes);
Ws(ii,:,:) = cov(xtildes);
%Step 4: Propagate each particle through the state equation
xtildes = xtildes + (Wsqrt*randn(2,nparticles))';
if mod(ii,100) == 0
fprintf('Iteration %d\n',ii);
end
end
end
function lls = computeLogLikelihoods(xtildes,y,params)
%Predicted rate for each neuron given Gaussian RF models
xdelta = bsxfun(@times,bsxfun(@minus,xtildes(:,1),params(:,1)'),1./(params(:,3)')).^2;
ydelta = bsxfun(@times,bsxfun(@minus,xtildes(:,2),params(:,2)'),1./(params(:,4)')).^2;
loglambdas = bsxfun(@plus,-.5*(xdelta+ydelta),params(:,5)');
lambdas = exp(loglambdas);
%Compute negative log-posterior
%First part is due to likelihood of data given position, second part is
%the prior prob. of positions
lls = -(-loglambdas*y + sum(lambdas,2));
end | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 3, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8709129095077515, "perplexity": 784.9462462784855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187821088.8/warc/CC-MAIN-20171017110249-20171017130249-00463.warc.gz"} |
https://www.physicsforums.com/threads/percentage-error-analysis.286360/ | # Percentage Error Analysis
1. Jan 21, 2009
### Air
1. The problem statement, all variables and given/known data
I have a resistor which has value written as $5K\Omega$ and I measured the value and it is $4.8K\Omega$. I want to work out the error.
2. The attempt at a solution
I can use simple percentage calculation to get $\frac{4800-5000}{5000} \times 100 = -4$% error but I've been told that derivative can also be used to calculate error, how would I go about doing that? Thanks in advance.
2. Jan 21, 2009
### chrisk
Determining % error using a derivative requires a continuous function that is derivable. Since the data is discrete and only two points are given a derivative in this case cannot be taken.
3. Jan 21, 2009
### turin
I think you may be confusing percent error with expected error. As chrisk suggests, you need to have a theoretical function that depends continuously on various physical parameters, and then you take the derivative w.r.t. those parameters, and then multiply by the precision of those parameters (basically). This error estimation using derivatives is not a percent error, because it makes no assumption regarding what the result of the measurement should be.
Similar Discussions: Percentage Error Analysis | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9785764217376709, "perplexity": 406.090141218013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320539.37/warc/CC-MAIN-20170625152316-20170625172316-00552.warc.gz"} |
http://mathhelpforum.com/calculus/210109-converge-uniformly.html | # Math Help - converge uniformly?
1. ## converge uniformly?
On $[a,b]$, the function sequence $\{f_n\},\{g_n\}$ converge uniformly to $f,g$ respectively. Suppose there exists positive sequence $M_n$ such that $f_n(x)\leq M_n, g_n(x)\leq M_n,\ \forall\ x\in [a,b]$. Prove that $f_ng_n$ converge unformly to $fg$ on $[a,b]$
PS: If $M_n=M$, I know how to prove. But this?....Would you help me?
2. ## Re: converge uniformly?
To show that $f_ng_n$ converges uniformly to $fg$ on $[a,b]$, we have to show there exists an N such that for all x in the interval $[a,b]$ and $n \geq N$,
$|f_ng_n - fg|\leq \epsilon$
I'm sorry I'm a bit rusty, but perhaps it would help to note that you can consider an upper bound for your positive sequence $M_n$. If the least upper bound for $M_n$ is a real number, say M, then you can say $f_n \leq M$ and $g_n \leq M$ for all n. If there is no least upper bound for your positive sequence, I'm not sure how to continue.
3. ## Re: converge uniformly?
So what you're saying is that $f_n$ and $g_n$ are sequences of bounded functions that converge uniformly to $f$ and $g$ respectively on $[a,b]$, and you want to prove that $f_ng_n$ converges uniformly to $fg$.
You said you know how to do it if $f_n$ and $g_n$ are uniformly bounded - that is, there are $M_f$ and $M_g$ such that $|f_n(x)| on $[a,b]$ and $|g_n(x)| on $[a,b]$. So prove that first:
Let $N$ be such that $|f_n(x)-f(x)|<\frac{1}{2}$ for all $x\in[a,b]$. Then for all $n>N$:
$|f_n(x)|\le |f_n(x)-f(x)|+|f(x)-f_N(x)|+|f_N(x)|\le \frac{1}{2}+\frac{1}{2}+M_N=M_N+1$
and I think you can probably fill in the rest of the proof.
- Hollywood
4. ## Re: converge uniformly?
Thank you very much indeed. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 39, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9970564246177673, "perplexity": 112.46083346237195}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042989507.42/warc/CC-MAIN-20150728002309-00011-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://kea-monad.blogspot.com/2007/11/connes-onical-time.html | # Arcadian Functor
occasional meanderings in physics' brave new world
Name:
Location: New Zealand
Marni D. Sheppeard
## Saturday, November 03, 2007
### Connes-onical Time
In his latest post, Alain Connes comments on the canonical nature of time evolution for noncommutative spaces. In M Theory, the analogue should be a whole heirarchy of Planck constants. For example, the Weyl relations of the quantum torus depend on the parameter \$\hbar\$. These spaces are studied in a very nice paper from 1993 by Alan Weinstein, where \$\hbar\$ is associated to time evolution for a particle on an ordinary torus. The particle is initially concentrated at a point, for \$\hbar = 0\$, but quickly becomes non-localised.
Louise Riofrio also has many fascinating posts on an emergent thermodynamic Time associated to Planck's number. In M theory we may view this as an approximation to a 3-Time picture, which brings to mind the twistor triality of Sparling, or the 2-Time theory of Itzhak Bars. It seems that wherever we look, the canonical Arrow raises its head.
#### 4 Comments:
Matti Pitkanen said...
There is a nice discussion about the canonical time evolution for factors of type III, which could teach a lot about what is involved to a motivated layman (such as me).
Type III time evolution is unique apart from the time parameter defining the duration of evolution and inner automorphism, the presence of which could be interpreted in terms of universal gauge invariance.
One can imagine two kinds of almost unique dynamics both having kind of universal gauge invariances.
1. For factors of type III canonical time evolution with inner automorphisms as gauge symmetries. I have tried to understand whether this kind of dynamics could emerge in TGD as a kind of fusion of HFF II_1 dynamics: the differences in the realizations of these factors are rather delicate: same infinite braid system seems to allow both IIs and IIIs. What is problematic with factors of type III is that the trace of unit is infinite: this could lead to difficulties and the divergences of QFT could relate very closely to this.
2. For HFFs of type II_1 the M-matrix defined by Connes tensor product and with measurement resolution defined by inclusion N subset M. This codes to the statement that Hermitian elements of N are symmetries of M-matrix and thus define gauge invariance like symmetry but due to measurement resolution. The old flavor SU(n) of hadron physics, almost forgotten now, might be a physical example of this kind of symmetry.
These symmetries are indeed very much like local gauge symmetries acting on a finite number of tensor factors in the tensor product representation of HFF. In the representation as a group algebra of S_infty or braid algebra B_infty these transformations act on finite number of braid strands only. Gauge symmetries in discrete real line: this is the interpretation.
The counterparts of global gauge transformations -outer automorphisms - correspond to transformations acting on the closure that is on infinite number of braid strands/tensor factors. A typical example is representation of S_n as diagonal group with elements gxgxg.... with g element of S_n.
The physical beauty here is that the periodicity of the imbedding of S_n (or of any finite group G, or compact gauge group actually) allows to represent the infinite system situation by a finite braid at space-time level: no need to have infinite number of replicas of same basic situation.
Inner-outer<---->local-global: this translation should reduce considerably the frustrations of poor physicist trying to understand what these mathematicians are saying.
There are also other questions (in TGD context). What about HFF of type II_infty with trace having all possible values. Could bosonic and fermionic degrees of freedom combine to form this kind of factor as a tensor product. I feel that this is not a good option and superconformal symmetry would suggest direct sum of II_1:s in both degrees of freedom.
Most naturally one would have a direct infinite direct sum of HFFs of II_1 with M-matrix in each summand in both F and B degrees of freedom. This would bring naturally p-adic thermodynamics (each value of conformal weight defining one summand) and other kinds of thermodynamics. But I could be wrong;-)!
The main message would be that one could start from a hierarchy of universal M-matrices with different measurement resolutions and work out their physical representations. Just the opposite for the usual approach. One should show many things. If one believes in TGD one should show that the universal gauge invariance provides a representation of various superconformal symmetries of light-like 3-surfaces; that number theoretic braids emerge naturally; etc... Of course, also other representations of this universal dynamics could exist looking totally different (dualities).
November 03, 2007 7:30 PM
Matti Pitkanen said...
A comment about spectrum of hbar, which Kea assigns to M theory.
In my own approach the spectrum of hbars emerges naturally from the generalization of the imbedding space. There are close connections to Jones inclusions but when I try to express this connection with two sentences I hear only strange humming in my head;-). Perhaps, a correlate for quantum entanglement to my adviser which I am unable to reduce;-).
In any case, the hierarchy of Planck constant is an essential element of quantum TGD proper now and realizes the notion quantum criticality. For instance, the understanding of Higgs expectation led to the identification of number theoretic braids and to a direct construction of the space-time sheet associated with given light-like 3-surfaces and this is a real victory.
Time evolutions (M-matrix) involve Planck constant but I cannot see how these time evolutions alone could imply the spectrum of Planck constants. I wish I would understand this better.
November 03, 2007 8:54 PM
Matti Pitkanen said...
Still a comment about the posting of Alain Connes. Connes mentions 3-D foliations V which give rise to type III factors. Foliation property requires a slicing of V one-form v to which slices are orthogonal. v satisfies thus integrability conditions. If so called Godbillon-Vey invariant is non-vanishing, factor of type III is obtained using Schrodinger amplitudes for which the flow lines of foliation define the time evolution.
In TGD light-like 3-surfaces are natural candidates for V and it is interesting to concretize the situation in this context. The one-form v defined by the induced Kahler gauge potential A defining also a braiding is a unique identification for v. The foliation property requires that v multiplied by suitable scalar is gradient. This gives dA= w\wedge A, A=-dpsi/psi =-dlog(psi). Something proportional to log(psi) can be taken as a third coordinate varying along flow lines of A: the braiding flow defines a continuous sequence of maps of partonic 2-surface to itself.
Physically this means the possibility of a super-conducting phase with order parameter, essentially phase depending only on psi only. This would describe supra current flowing along flow lines of A. Integrability in general is not true and one cannot assign Schrodinger time evolution with the flow lines of v (except possibly trivial time evolution for which energy vanishes!). One could not speak about supra flow along lines of A since the flow would be mixing.
Connes tells that a factor of type III results if Godbillon-Vey invariant defined as the integral of dw\wedge w over 3-manifold is non-vanishing. The operators of the algebra in question are transversal operators acting on Schrodinger amplitudes in each slice. Essentially Schrodinger equation in 3-D space-time would be in question with factor of type III resulting from the exotic choice of the time coordinate defining the slicing.
In TGD Schroedinger amplitudes are replaced by second quantized induced spinor fields. Hence one does not face the problem whether it makes sense to speak about Schroedinger time evolution along the time-lines of foliation or not. Also the fact that "time evolution" for the modified Dirac operator corresponds to single position dependent generalized eigenvalue identified as Higgs expectation same for all transversal modes (essentially z^n labelled by conformal weight) is crucial since it saves from the problems caused by the possible non-existence of Schroedinger evolution.
November 03, 2007 9:04 PM
L. Riofrio said...
Thanbks again for many interesting links to researchers like Connes. The most interesting science seems to be happening outside the mainstream.
November 04, 2007 4:28 AM | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8983915448188782, "perplexity": 1196.40823940908}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936462099.15/warc/CC-MAIN-20150226074102-00287-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://scholarlypublishingcollective.org/psup/biblical-research/article-abstract/5/1/87/300579/The-Seventh-Johannine-Sign-A-Study-in-John-s?redirectedFrom=fulltext | ## Abstract
A proper identification of the Johannine signs significantly sharpens one's grasp of John's Christology. While there is substantial agreement regarding six signs, no consensus has yet been achieved regarding possible further signs in John. After an exploration of the "signs" concept in the OT, three criteria for Johannine signs are established, resulting in a working definition. These preliminary considerations then function as parameters in the search for potential additional signs in the Fourth Gospel. The essay concludes with a brief discussion of the implications of the study's findings for the structure of John's Gospel. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8245772123336792, "perplexity": 2483.116207560448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948620.60/warc/CC-MAIN-20230327092225-20230327122225-00241.warc.gz"} |
https://math.stackexchange.com/questions/839964/soft-question-about-the-square-root | # Soft question about the square root [duplicate]
I got to thinking about the square root the other day, and there's this thing that bugs me in the back of my mind. As far as I know, $\sqrt{4}$ is unambiguously $2$, and nothing else, as the square root of a number is defined as the positive root of that number. Yet, when solving algebraic equations, people (myself included) seem to follow this logic:
Solve: $x^2 = 9$
Solution: $\sqrt{x^2} = \sqrt{9} \Rightarrow x = \pm3$
All of a sudden, people love the minus sign! But this is obviously notationally incorrect, even though $x$ really is $\pm 3$. For myself, I made a deal with myself a long time ago: square roots of numbers are always positive, square roots of unknowns always have 2 roots (at least in $\mathbb C$ (counted with multiplicity)).
This itch really needs to be scratched, driving me crazy! :) Thanks in advance.
• I am not able to see a question in your post. What is your question? – MJD Jun 19 '14 at 20:16
Consider the function $f(x) = \sqrt{x}$. This is a function that gives the positive square root of a number. As you have noted a positive real number has two square roots; one is positive and the other is negative. It is by convention the let $\sqrt{x}$ denote the positive square root. If I want the negative square root I explicitly write $g(x) = -\sqrt{x}$. Note $f(x)$ and $g(x)$ are respectively the top and bottom half of the parabola given by $y^2 = x$. Note we cannot achieve this parabola by a single function of $x$ (as I would teach my college algebra students this parabola fails the vertical line test). Thus is summary $\sqrt{x}$ can only denote the positive square root because we want $\sqrt{x}$ to be a well-defined function. Otherwise $\sqrt{x}$ is a multivalued function. For those you you who have been exposed to complex analysis you know that $\log(z)$ is a multivalued function and we often pick a branch to work with. Something similar is going on here.
Now consider solving equations. When solving equations we typically want all solutions. Thus we will say $x = \pm \sqrt{9} = \pm 3$ when solving $x^2 = 9$. Here we are showing that there are two solutions. If you want to leave $\sqrt{x}$ as a multivalued function it would not be necessary to write it this way with the $\pm$. In most cases people like a "principle" (positive) square root so that $\sqrt{x}$ is really a function. This makes it necessary to explictly denote $-\sqrt{x}$ when we want it.
• THIS!!! Never thought about thinking thinking about it this way. If I reflect back on my thin understanding of complex analysis, I want to remember that for for $z= re^i\theta$, $log(z)$ is not uniquely defined, and so we pick the principal branch because $log(z)$ cant possibly be welldefined, as it fails to be injective, it repeats for all $2i\pi n$ where n is an integer. Now I want to think about $f(x) = \sqrt{x}$ like we're choosing the a branch of "square root", is this pretty close? Feel free to use mature mathematical language, I'm more knowledgeable than the question lets on. – JuliusL33t Jun 19 '14 at 20:51
• PS: the positivity of the square root is just convention, defining $\sqrt{x}$ as positive is just convention, math would work just as fine with the negative definition, it would just be very cumbersome! It this true? – JuliusL33t Jun 19 '14 at 20:55
• Right, $\log(re^{i\theta}) = \log(r) + i\theta'$ for $\theta' = \theta + 2i\pi n$ for any integer $n$. Where on the right side we use the regular real logarithm. Then we make it well-defined by picking a branch. For example pick the unique $\theta'$ so that $0 \leq \theta' < 2\pi$. Or we could just as well pick $-\pi < \theta' \leq \pi$ or any other half open interval of length $2\pi$. – John Machacek Jun 19 '14 at 20:59
• Yes, you could by convention let $\sqrt{x}$ denote the negative square root. Then you would explicitly right $+\sqrt{x}$ for the positive square root. Like you say math would still work, but this convention would be odd having to put in the $+$ sign. But I think you got it. The main thing is $\sqrt{x}$ can mean one of two things, we have to pick which one we want it to mean. – John Machacek Jun 19 '14 at 21:03
• Awesome! Thanks! :) – JuliusL33t Jun 19 '14 at 21:04
How about this? \eqalign{\rm x^2=9&\iff\rm x^2-9=0 \\&\rm\iff x^2-3^2=0\\&\rm\iff (x-3)(x+3)=0\\&\rm\iff x=3\color{grey}{\text{ or }}x=-3\\&\rm\iff x=\pm3.} You got a negative solution because if you square it, you will get a positive number, whose square root is also a positive number. It's all because $\color{white}{\overline{\color{black}{\rm\sqrt{(x)^2}=\sqrt{(-x)^2}}}}.$
If $x^2 = 9$ then $\sqrt{x^2} = \sqrt{9}$. What you seem to be missing is that $\sqrt{x^2}$ is not $x$ but $|x|$ so the equation simplifies to $|x| = 3$. The only two values of $x$ which have absolute value $3$ are $x = 3$ and $x = -3$, i.e. $x = \pm 3$.
Square roots of real numbers are always positive. However, when solving an equation, you're not actually "taking the square root of both sides of the equation". When you say $x^2 = 9 \implies x = \pm 3$, the steps behind it are: \begin{align} x^2 &= 9 \\ x^2 - 9 &= 0 \\ (x-3)(x+3) &= 0 \\ \implies x = 3 \quad \mbox{or} \quad x &= -3 \end{align} Do not have doubts, $\sqrt{9} = 3$, always. In general, $\sqrt{x^2} = |x|$.
Your mistake is assuming that the inverse operation of the function $x^2$ is $\sqrt{x}$, when, in fact, it is $\pm \sqrt{x}$.
e.g. if we're solving $x^2=9$, the operation that undoes the $x^2$ is $\pm \sqrt{...}$, so we get: $$\pm \sqrt{x^2}=\pm9 \iff \pm x= \pm 9,$$ the only solutions to which are $x\in \{-3,3\}.$
Here's one way to think about it. Square root is really something to the $\frac 12$ power. So $\sqrt x = x^{\frac 12}$.
Exponents are all about multiplication. So if $x>1$ then we can say:$$x^a > x^b \implies a>b$$ With that in mind, $$x^0 < x^{\frac 12} < x^1$$ So $$1 < x^{\frac 12} < x$$
The problem is that $\sqrt{x^2}$ is not equal to $x$. Rather, $\sqrt{x^2} = |x|$. So the solution to the problem is: $$x^2 = 9$$ $$\sqrt{x^2} = \sqrt{9}$$ $$|x| = 3$$ $$x = \pm 3$$
This error (of thinking that $\sqrt{x^2} = x$) comes up a lot, in a lot of contexts. See my answer to this question. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9826290607452393, "perplexity": 187.27368023310873}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178374686.69/warc/CC-MAIN-20210306100836-20210306130836-00293.warc.gz"} |
https://dantopology.wordpress.com/2010/07/ | # Sequential spaces, V
In the previous post Sequential spaces, IV, we show that the uncountable product of sequential spaces is not sequential (e.g. the product $2^{\omega_1}$ is not sequential). What is more remarkable is that the product of two sequential spaces needs not be sequential. We present an example of a first countable space and a Frechet space whose product is not a k-space (thus not sequential). For the previous discussion on this blog on sequential spaces and k-spaces, see the links at the end of this post.
Let $\mathbb{R}$ be the real line and let $\mathbb{N}$ be the set of all positive integers. Let $X$ be the space $\mathbb{R}-\left\{1,\frac{1}{2},\frac{1}{3},\cdots\right\}$ with the topology inherited from the usual topology on the real line. Let $Y=\mathbb{R}$ with the positive integers identified as one point (call this point $p$). We claim that $X \times Y$ is not a k-space and thus not a sequential space. To this end, we define a non-closed $A \subset X \times Y$ such that $K \cap A$ is closed in $K$ for all compact $K \subset X \times Y$.
Let $A=\bigcup \limits_{i=1}^\infty A_i$ where for each $i \in \mathbb{N}$, the set $A_i$ is defined by the following:
$\displaystyle A_i =\left\{\biggl(\frac{1}{i}+\frac{a_i}{j},i+\frac{0.5}{j} \biggr) \in X \times Y:j \in \mathbb{N}\right\}$
where $\displaystyle a_i=\biggl(\frac{1}{i}-\frac{1}{i+1} \biggr) 10^{-i}$.
Clearly $A$ is not closed as $(0,p) \in \overline{A}-A$. In fact in the product space $X \times Y$, the point $(0,p)$ is the only limit point of the set $A$. Another observation is that for each $n \in \mathbb{N}$, $(0,p)$ is not a limit point of $\bigcup \limits_{i=1}^n A_i$. Furthermore, if $z_i \in A_i$ for each $i \in S$ where $S$ is an infinite subset of $\mathbb{N}$, then $(0,p)$ is not a limit point of $\left\{z_i:i \in S\right\}$. It follows that no infinite subset of $A$ is compact. Consequently, $K \cap A$ is finite for each compact $K \subset X \times Y$. Thus $X \times Y$ is not a k-space. To see that $X \times Y$ is not sequential directly, observe that $A$ is sequentially closed.
Previous posts on sequential spaces and k-spaces
# Sequentially compact spaces, I
All spaces under consideration are Hausdorff. Countably compactness and sequentially compactness are notions related to compactness. A countably compact space is one in which every counable open cover has a finite subcover, or equivalently, every countably infinite subset has a limit point. The limit points contemplated here are from the topological point of view, i.e. the point $p \in X$ is a limit point of $A \subset X$ if every open subset of $X$ containing $p$ contains a point of $A$ distinct from $p$. On the other hand, a space $X$ is sequentially compact if every sequence $\left\{x_n:n=1,2,3,\cdots\right\}$ of points of $X$ has a subsequence that converges. We present examples showing that the notion of sequentially compactness is different from compactness.
Let $\omega_1$ be the first uncountable ordinal. The space of all countable ordinals $W=[0,\omega_1)$ with the ordered topology is sequentially compact and not compact. Let $\left\{w_n\right\}$ be a sequence of points in $W$. Let $A=\left\{w_n:n=1,2,3,\cdots\right\}$. If $A$ is finite, then the sequence $\left\{w_n\right\}$ is eventually constant and thus has a convergent subsequence. So assume $A$ is an infinite set. Then we can choose an increasing sequence of integers $n(1),n(2),n(3),\cdots$ such that $w_{n(1)}. Let $\alpha<\omega_1$ be the least upper bound of all $w_{n(j)}$. Then subsequence $w_{n(j)}$ converges to $\alpha$.
The notion of sequentially compactness is not to be confused with the notion of being a sequential space. The space $[0,\omega_1]=\omega_1+1$, the space of countable ordinals with one additional point $\omega_1$ at the end, is a sequentially compact space for the same reason that $[0,\omega_1)$ is sequentially compact. However, $[0,\omega_1]$ is not sequential. Note that $[0,\omega_1)$ is a sequentially closed set but not closed in $[0,\omega_1]$. On the other hand, being a sequential space does not imply compactness or sequentially compactness, e.g. the real line $\mathbb{R}$.
For discussion of sequential spaces, see Sequential spaces, I, Sequential spaces, II and Sequential spaces, III.
We now present an example of a compact space that is not sequentially compact. Let $I=[0,1]$ be the unit interval. Let $2=\left\{0,1\right\}$, the two-point discrete space. Let $X=2^{I}$ be the product space of uncountably many copies of $2=\left\{0,1\right\}$ indexed by $I$. We show that $X$ is not sequentially compact. To this end, we define a sequence $\left\{f_n\right\}$ that has not convergent subsequence.
For any $y \in \mathbb{R}$, let $[y]$ be the greatest integer less than or equal to $y$. For each $t \in I$ and for each $n=1,2,3,\cdots$, let $t_n$ be:
$t_n=10^n t-[10^n t]$.
For example, if $q=\frac{1}{\sqrt{2}}=0.7071067811 \cdots$, then $q_1=0.071067811 \cdots$, $q_2=0.71067811 \cdots$ and $q_3=0.1067811 \cdots$. For each $n=1,2,3,\cdots$, define $f_n:I \mapsto 2$ by the following:
$\displaystyle f_n(t)=\left\{\begin{matrix}0&\thinspace t_n <0.5 \\{1}&\thinspace t_n \ge 0.5 \end{matrix}\right.$
With the above example, $f_1(q)=0$, $f_2(q)=1$, $f_3(q)=0$, $f_4(q)=0$ and so on. In general, if the $(n+1)^{st}$ decimal place of the number $t$ is less then $5$, then $f_n(t)=0$. Otherwise $f_n(t)=1$.
Let’s observe that if $g_n \in X=2^{I}$ converges to $g \in X$, then $g_n(t) \in X=2^{I}$ converges to $g(t) \in X$ for each $t \in I$ (hence the product topology is called the topology of pointwise convergence). We claim that the sequence $\left\{f_n\right\}$ has no convergence subsequence. To this end, we show that each subsequence of $\left\{f_n\right\}$ does not converge at some $t \in I$.
Let $n(1) be any increasing sequence of positive integers. We define $t \in I$ such that $f_{n(1)}(t),f_{n(2)}(t),f_{n(3)}(t),\cdots$ is an alternating sequence of zeros and ones. Consider $t \in I$ satisfying the following:
For each $j \le n(1)$, the $j^{th}$ decimal place of $t$ is $9$,
For each $n(1), the $j^{th}$ decimal place of $t$ is $1$,
For each $n(2), the $j^{th}$ decimal place of $t$ is $9$ and so on.
For example, if $n(1)=2$, $n(2)=5$ and $n(3)=9$, then let $t=0.9911199991 \cdots$. With this in mind, $f_{n(1)}(t),f_{n(2)}(t),f_{n(3)}(t),\cdots$ is an alternating sequence of zeros and ones.
Reference
1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.
# Sequential spaces, IV
Uncountable product of first countable spaces can never be first countable (see The product of first countable spaces). It turns out that uncountable products of first countable spaces cannot even be sequential. In this post we show that the product of uncountably many Hausdorff spaces, each of which has at least two points, can never be sequential. This follows from the fact that the product space $2^S$ is not sequential where $2=\left\{0,1\right\}$ is the two-point discrete space and $S$ is any uncountable set. The space $2^S$ can be embedded as a closed subspace of a product of uncountably many Hausdorff spaces, each of which has at least two points.
For discussion on this blog about sequential spaces, see Sequential spaces, I, Sequential spaces, II and Sequential spaces, III.
Let $S$ be an uncountable set. Let $X=2^S$ be the product of uncountably many copies of $2=\left\{0,1\right\}$ indexed by the set $S$. Let $Y$ be all points $x \in X$ such that $x(\alpha)=x_\alpha=0$ for all but countably many $\alpha \in S$. In other words, $Y$ is the $\Sigma$-product of $2=\left\{0,1\right\}$. Clearly $Y$ is not closed in $X$. We claim that $Y$ is sequentially closed in $X$.
For each $z \in Y$, let $W_z=\left\{\alpha \in S:z_\alpha \ne 0\right\}$. Note that each $W_z$ is countable. Suppose that $\left\{y_n\right\}$ is a sequence of points of $Y$ such that $y_n \rightarrow y \in X$. We show that $y \in Y$. Let $W$ be the union of all $W_{y_n}$, which is a countable set. For all $\alpha \in S-W$, $y_n(\alpha)=0$ and thus $y(\alpha)=y_\alpha=0$. Thus $y \in Y$. This shows that $Y$ is sequentially closed in $X$.It follows that $2^S$ is not sequential.
As stated at the beginning of the post, any uncountable product of Hausdorff spaces, each of which has at least two points, can never be sequential. Consequently, the property of being a sequential spaces is not preserved in uncountable products.
# Sequential spaces, III
This is a continuation of the discussion on sequential spaces started with the post Sequential spaces, I and Sequential spaces, II and k-spaces, I. The topology in a sequential space is generated by the convergent sequences. The convergence we are interested in is from a topological view point and not necessarily from a metric (i.e. distance) standpoint. In our discussion, a sequence $\left\{x_n\right\}_{n=1}^\infty$ converges to $x$ simply means for each open set $O$ containing $x$, $O$ contains $x_n$ for all but finitely many $n$. In any topological space, there are always trivial convergent sequences. These are sequences of points that are eventually constant, i.e. the sequences $\left\{x_n\right\}$ where for some $n$, $x_n=x_j$ for $j \ge n$. Any convergent sequence that is not eventually constant is called a non-trivial convergent sequence. We present an example of a space where there are no non-trivial convergent sequences of points. This space is derived from the Euclidean topology on the real line. This space has no isolated point in this space and yet has no non-trivial convergent sequences and has no infinite compact sets. From this example, we make some observations about sequential spaces and k-spaces.
The space we define here is obtained by modifying the Euclidean topology on the real line. Let $\mathbb{R}$ be the real line. Let $\tau_e$ be the Euclidean topology on the real line. Consider the following collection of subsets of the real line:
$\mathcal{B}=\left\{U-C:U \in \tau_e \text{ and } \lvert C \lvert \le \omega\right\}$
It can be verified that $\mathcal{B}$ is a base for a topology $\tau$ on $\mathbb{R}$. In fact this topology is finer than the Euclidean topology. Denote $\mathbb{R}$ with this finer topology by $X$. Clearly $X$ is Hausdorff since the Euclidean topology is. Any countable subset of $X$ is closed. Thus $X$ is not separable (no countable set can be dense). This space is a handy example of a hereditarily Lindelof space that is not separable. The following lists some properties of $X$:
1. $X$ is herditarily Lindelof.
2. There are no non-trivial convergent sequences in $X$.
3. All compact subsets of $X$ are finite.
4. $X$ is not a k-space and is thus not sequential.
Discussion of 1. This follows from the fact that the real line with the Euclidean topology is hereditarily Lindelof and the fact that each open set in $X$ is an Euclidean open set minus a countable set.
Discussion of 2 This follows from the fact that every countable subset of $X$ is closed. If a non-trivial sequence $\left\{x_n\right\}$ were to converge to $x \in X$, then $\left\{x_n:n=1,2,3,\cdots\right\}$ would be a countable subset of $X$ that is not closed.
Discussion of 3. Let $A \subset X$ be an infinite set. If $A$ is bounded in the Euclidean topology, then there would be a non-trivial convergent sequence of points of $A$ in the Euclidean topology, say, $x_n \mapsto x$. Let $U_0=X-\left\{x_n:n=1,2,3,\cdots\right\}$, which is open in $X$. For $n \ge 1$, let $U_n$ be Euclidean open such that $x_n \in U_n$. We also require that all $U_n$ are pairwise disjoint and not contain $x$. Then $U_0,U_1,U_2,\cdots$ form an open cover of $A$ (in the topology of $X$) that has no finite subcover. So any bounded infinite $A$ is not compact in $X$.
Suppose $A$ is unbounded in the Euclidean topology. Then $A$ contains a closed and discrete subset $\left\{x_1,x_2,x_3,\cdots\right\}$ in the Euclidean topology. We can find Euclidean open sets $U_n$ that are pairwise disjoint such that $x_n \in U_n$ for each $n$. Let $U_0=X-\left\{x_n:n=1,2,3,\cdots\right\}$, which is open in $X$. Then $U_0,U_1,U_2,\cdots$ form an open cover of $A$ (in the topology of $X$) that has no finite subcover. So any unbounded infinite $A$ is not compact in $X$.
Discussion of 4. Note that every point of $X$ is a non-isolated point. Just pick any $x \in X$. Then $X-\left\{x\right\}$ is not closed in $X$. However, according to 3, $K \cap (X-\left\{x\right\})$ is finite and is thus closed in $K$ for every compact $K \subset X$. Thus $X$ is not a k-space.
General Discussion
Suppose $\tau$ is the topology for the space $Y$. Let $\tau_s$ be the set of all sequentially open sets with respect to $\tau$ (see Sequential spaces, II). Let $\tau_k$ be the set of all compactly generated open sets with respect to $\tau$ (see k-spaces, I). The space $Y$ is a sequential space (a k-space ) if $\tau=\tau_s$ ($\tau=\tau_k$). Both $\tau_s$ and $\tau_k$ are finer than $\tau$, i.e. $\tau \subset \tau_s$ and $\tau \subset \tau_k$. When are $\tau_s$ and $\tau_k$ discrete? We discuss sequential spaces and k-spaces separately.
Observations on Sequential Spaces
With respect to the space $(Y,\tau)$, we discuss the following four properties:
• A. $\$ No non-trivial convergent sequences.
• B. $\$ $\tau_s$ is a discrete topology.
• C. $\$ $\tau$ is a discrete topology.
• D. $\$ Sequential, i.e., $\tau=\tau_s$.
Observation 1
The topology $\tau_s$ is discrete if and only if $Y$ has no non-trivial convergent sequences, i.e. $A \Longleftrightarrow B$.
If $\tau_s$ is a discrete topology, then every subset of $Y$ is sequentially open and every subset is sequentially closed. Hence there can be no non-trivial convergent sequences. If there are no non-trivial convergent sequences, every subset of the space is sequentially closed (thus every subset is sequentially open).
Observation 2
Given that $Y$ has no non-trivial convergent sequences, $Y$ is not discrete if and only if $Y$ is not sequential. Equivalently, given property A, $C \Longleftrightarrow D$.
Given that there are no non-trivial convergent sequences in $Y$, $\tau_s$ is discrete. For $(Y,\tau)$ to be sequential, $\tau=\tau_s$. Thus for a space $Y$ that has no non-trivial convergent sequences, the only way for $Y$ to be sequential is that it is a discrete space.
Observation 3
Given $Y$ is not discrete, $Y$ has no non-trivial convergent sequences implies that $Y$ is not sequential, i.e. given $\text{not }C$, $A \Longrightarrow \text{not }D$. The converse does not hold.
Observation 3 is a rewording of observation 2. To see that the converse of observation 3 does not hold, consider $Y=[0,\omega_1]=\omega_1+1$, the successor ordinal to the first uncountable ordinal with the order topology. It is not sequential as the singleton set $\left\{\omega_1\right\}$ is sequentially open and not open.
Observations on k-spaces
The discussion on k-spaces mirrors the one on sequential spaces. With respect to the space $(Y,\tau)$, we discuss the following four properties:
• E. $\$ No infinite compact sets.
• F. $\$ $\tau_k$ is a discrete topology.
• G. $\$ $\tau$ is a discrete topology.
• H. $\$ k-space, i.e., $\tau=\tau_k$.
Observation 4
The topology $\tau_k$ is discrete if and only if $Y$ has no infinite compact sets, i.e. $E \Longleftrightarrow F$.
If $\tau_k$ is a discrete topology, then every subset of $Y$ is a compactly generated open set. In particular, for every compact $K \subset Y$, every subset of $K$ is open in $K$. This means $K$ is discrete and thus must be finite. Hence there can be no infinite compact sets if $\tau_k$ is discrete. If there are no infinite compact sets, every subset of the space is a compactly generated closed set (thus every subset is a compactly generated open set).
Observation 5
Given that $Y$ has no infinite compact sets, $Y$ is not discrete if and only if $Y$ is not a k-space. Equivalently, given property E, $G \Longleftrightarrow H$.
Given that there are no infinite compact sets in $Y$, $\tau_k$ is discrete. For $(Y,\tau)$ to be a k-space, $\tau=\tau_k$. Thus for a space $Y$ that has no infinite compact sets, the only way for $Y$ to be a k-space is that it is a discrete space.
Observation 6
Given $Y$ is not discrete, $Y$ has no infinite compact sets implies that $Y$ is not a k-space, i.e. given $\text{not }G$, $E \Longrightarrow \text{not }H$. The converse does not hold.
Observation 6 is a rewording of observation 5. To see that the converse of observation 6 does not hold, consider the topological sum of a non-k-space and an infinite compact space.
Remark
In the space $X$ defined above by removing countable sets from Euclidean open subsets of the real line, there are no infinite compact sets and no non-trivial convergent sequences. Yet the space is not discrete. Thus it can neither be a sequential space nor a k-space. Another observation we would like to make is that no infinite compact sets implies no non-trivial convergent sequences ($E \Longrightarrow A$). However, the converse is not true. Consider $\beta(\omega)$, the Stone-Cech compactification of $\omega$, the set of all nonnegative integers.
Reference
1. Engelking, R. General Topology, Revised and Completed edition, 1989, Heldermann Verlag, Berlin.
2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 314, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.994901180267334, "perplexity": 102.86570021474265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987756350.80/warc/CC-MAIN-20191021043233-20191021070733-00151.warc.gz"} |
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An incremental method to compute (Abel) matrix inverses bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 07/09/2010, 06:31 AM (This post was last modified: 07/09/2010, 06:43 AM by bo198214.) I fiddled a bit around with Gottfried's suggestion of LU decomposition of the Abel matrix (though in the end the formula is independent of the LU decomposition). The annoying thing about calculating the intuitive Abel function (by solving the equation Ax=b where A is the Abel matrix, x the powerseries development of the Abel function and b=(1,0,...)) that if you want to increas the matrix size you have to solve the complete equation again without being able to use your previous solution. Now I found a way how you can compute the inverse of the $A_n$ matrix by using the $A_{n-1}$ Abel matrix. I dissect the matrix as follows, for brevity I set $A=A_{n-1}$: $ A_n=\left(\begin{array}{ccc|c} \phantom{1}& &\phantom{1} & \\ &A& &\acute{a}\\ & & & \\\hline &\grave{a}& &a_n \end{array}\right)$ $\acute{a}$ means column vector and $\grave{a}$ means row vector. The final incremental formula is then: ${A_n}^{-1} =(A^{-1})_{+0} + \frac{(-A^{-1}\acute{a}\oplus 1)(-\grave{a}A^{-1}\oplus 1)}{a_n-\grave{a}A^{-1}\acute{a}}$ Where $\oplus 1$ means adding the entry 1 to the vector and $(A^{-1})_{+0}$ is $A^{-1}$ extended to a nxn matrix by filling with 0's. The deriviation is perhaps too uninteresting and cumbersome to put, but I can post it if inquired. tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 07/09/2010, 12:02 PM (This post was last modified: 07/09/2010, 12:05 PM by tommy1729.) what is an abel matrix ? i know carleman and bell matrix ... whats the difference ? maybe its the matrix equivalent of the infinite linear equations of andrews slog ... bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 07/10/2010, 04:46 AM (This post was last modified: 07/10/2010, 04:49 AM by bo198214.) (07/09/2010, 12:02 PM)tommy1729 Wrote: what is an abel matrix ? i know carleman and bell matrix ... whats the difference ? maybe its the matrix equivalent of the infinite linear equations of andrews slog ... Oh the term was introduced by Andrew and indeed is the matrix equivalent of the infinite equation system of Andrew's slog. Its C-I transposed (C is the infinite Carleman matrix, I the infinite identity matrix) and first column removed (which consists of only 0's) and then square truncated according to your needs of precision. But the above inversion formula does not depend on A being an Abel matrix. It works for every invertible matrix. But currently I see that the internal inversion algorithms of Sage are even faster to recompute the whole inverse than doing an incremtal inverse with the above formula *sigh* Gottfried Ultimate Fellow Posts: 767 Threads: 119 Joined: Aug 2007 07/20/2010, 12:13 PM (This post was last modified: 07/20/2010, 12:14 PM by Gottfried.) (07/09/2010, 06:31 AM)bo198214 Wrote: I fiddled a bit around with Gottfried's suggestion of LU decomposition of the Abel matrix (though in the end the formula is independent of the LU decomposition). (...)Hi Henryk - I've also tried a similar thing: to compute the new column/row of the LU-factors by the old ones - optimally by reference to the previous row/column only... without success so far. Triggered by your msg I looked at a symbolic representation, using the symbol "u" for the log of the base (which can be substituted by 1 if the base is e = exp(1)). So B is the Bell-matrix for x->exp(u*x), B1 is truncate(B - I) , Then L (lower) , D (diagonal), U (upper) the inverses of the LU-factors of B1, such if it converges, B1^-1 = U*D*L , where for the slog we need only the first column of L. Looking at the last column of U with the idea to compose it by the previous column only (or by some composition of some earlier columns) I notice, that the entries are polynomials in u and I didn't see yet a simple possibility to determine that polynomials based on that of the previous colums. The same is true analoguously for the n'th entry in D and for the n'th row in L. If I determine the coefficients for the powerseries for the slog by U*D*L[,0], then I see, that the degree of the polynomials, by which the coefficients are defined, increase binomially (n,2) and in our context of tetration I can't remember any recursionformula for such a situation. (In my treatize about the symbolic description of the coefficients for the dxp-Bell-matrix I came across the same growthrate of the order of the polynomials and did not find a formula how to compute one coefficient only by its index and possibly some known constants) It would be nice to find such a formula for "the last" column in U and "last row" in L by a short recursion - this would then also allow to dismiss that matrix-inversion(s) completely... but I think, your derivation is comparably complex (so that the inversion is even faster)? Gottfried ah - ps: I tried also to express it in terms of (u-1) or (u+1) instead of u - but with little progress. Gottfried Helms, Kassel « Next Oldest | Next Newest »
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https://worldwidescience.org/topicpages/a/approximate+analytical+solutions.html | #### Sample records for approximate analytical solutions
1. Approximate analytic solutions to the NPDD: Short exposure approximations
Science.gov (United States)
Close, Ciara E.; Sheridan, John T.
2014-04-01
There have been many attempts to accurately describe the photochemical processes that take places in photopolymer materials. As the models have become more accurate, solving them has become more numerically intensive and more 'opaque'. Recent models incorporate the major photochemical reactions taking place as well as the diffusion effects resulting from the photo-polymerisation process, and have accurately described these processes in a number of different materials. It is our aim to develop accessible mathematical expressions which provide physical insights and simple quantitative predictions of practical value to material designers and users. In this paper, starting with the Non-Local Photo-Polymerisation Driven Diffusion (NPDD) model coupled integro-differential equations, we first simplify these equations and validate the accuracy of the resulting approximate model. This new set of governing equations are then used to produce accurate analytic solutions (polynomials) describing the evolution of the monomer and polymer concentrations, and the grating refractive index modulation, in the case of short low intensity sinusoidal exposures. The physical significance of the results and their consequences for holographic data storage (HDS) are then discussed.
2. Approximated analytical solution to an Ebola optimal control problem
Science.gov (United States)
Hincapié-Palacio, Doracelly; Ospina, Juan; Torres, Delfim F. M.
2016-11-01
An analytical expression for the optimal control of an Ebola problem is obtained. The analytical solution is found as a first-order approximation to the Pontryagin Maximum Principle via the Euler-Lagrange equation. An implementation of the method is given using the computer algebra system Maple. Our analytical solutions confirm the results recently reported in the literature using numerical methods.
3. The Analytical Approximate Solution of the 2D Thermal Displacement
Institute of Scientific and Technical Information of China (English)
Chu－QuanGuan; Zeng－YuanGuo; 等
1996-01-01
The 2D plane gas flow under heating (with nonentity boundary condition)has been discussed by the analytical approach in this paper.The approximate analytical solutions have been obtained for the flow passing various kinds of heat sources.Solutions demonstrate the thermal displacement phenomena are strongly depend on the heating intensity.
4. Approximate analytical solution for the isothermal Lane Emden equation in a spherical geometry
Science.gov (United States)
Soliman, Moustafa Aly; Al-Zeghayer, Yousef
2015-10-01
This paper obtains an approximate analytical solution for the isothermal Lane-Emden equation that models a self-gravitating isothermal sphere. The approximate solution is obtained by perturbation methods in terms of small and large distance parameters. The approximate solution is compared with the numerical solution. The approximate solution obtained is valid for all values of the distance parameter.
5. Approximate analytical solutions and approximate value of skin friction coefficient for boundary layer of power law fluids
Institute of Scientific and Technical Information of China (English)
SU Xiao-hong; ZHENG Lian-cun; JIANG Feng
2008-01-01
This paper presents a theoretical analysis for laminar boundary layer flow in a power law non-Newtonian fluids.The Adomian analytical decomposition technique is presented and an approximate analytical solution is obtained.The approximate analytical solution can be expressed in terms of a rapid convergent power series with easily computable terms.Reliability and efficiency of the approximate solution are verified by comparing with numerical solutions in the literature.Moreover,the approximate solution can be successfully applied to provide values for the skin friction coefficient of the laminar boundary layer flow in power law non-Newtonian fluids.
6. Approximate analytic solutions of stagnation point flow in a porous medium
Science.gov (United States)
Kumaran, V.; Tamizharasi, R.; Vajravelu, K.
2009-06-01
An efficient and new implicit perturbation technique is used to obtain approximate analytical series solution of Brinkmann equation governing the two-dimensional stagnation point flow in a porous medium. Analytical approximate solution of the classical two-dimensional stagnation point flow is obtained as a limiting case. Also, it is shown that the obtained higher order series solutions agree well with the computed numerical solutions.
7. Approximate analytical solutions for excitation and propagation in cardiac tissue
Science.gov (United States)
Greene, D'Artagnan; Shiferaw, Yohannes
2015-04-01
It is well known that a variety of cardiac arrhythmias are initiated by a focal excitation in heart tissue. At the single cell level these currents are typically induced by intracellular processes such as spontaneous calcium release (SCR). However, it is not understood how the size and morphology of these focal excitations are related to the electrophysiological properties of cardiac cells. In this paper a detailed physiologically based ionic model is analyzed by projecting the excitation dynamics to a reduced one-dimensional parameter space. Based on this analysis we show that the inward current required for an excitation to occur is largely dictated by the voltage dependence of the inward rectifier potassium current (IK 1) , and is insensitive to the detailed properties of the sodium current. We derive an analytical expression relating the size of a stimulus and the critical current required to induce a propagating action potential (AP), and argue that this relationship determines the necessary number of cells that must undergo SCR in order to induce ectopic activity in cardiac tissue. Finally, we show that, once a focal excitation begins to propagate, its propagation characteristics, such as the conduction velocity and the critical radius for propagation, are largely determined by the sodium and gap junction currents with a substantially lesser effect due to repolarizing potassium currents. These results reveal the relationship between ion channel properties and important tissue scale processes such as excitation and propagation.
8. Approximate Analytic Solutions for the Two-Phase Stefan Problem Using the Adomian Decomposition Method
Directory of Open Access Journals (Sweden)
Xiao-Ying Qin
2014-01-01
Full Text Available An Adomian decomposition method (ADM is applied to solve a two-phase Stefan problem that describes the pure metal solidification process. In contrast to traditional analytical methods, ADM avoids complex mathematical derivations and does not require coordinate transformation for elimination of the unknown moving boundary. Based on polynomial approximations for some known and unknown boundary functions, approximate analytic solutions for the model with undetermined coefficients are obtained using ADM. Substitution of these expressions into other equations and boundary conditions of the model generates some function identities with the undetermined coefficients. By determining these coefficients, approximate analytic solutions for the model are obtained. A concrete example of the solution shows that this method can easily be implemented in MATLAB and has a fast convergence rate. This is an efficient method for finding approximate analytic solutions for the Stefan and the inverse Stefan problems.
9. Finding Approximate Analytic Solutions to Differential Equations by Seed Selection Genetic Programming
Institute of Scientific and Technical Information of China (English)
侯进军
2007-01-01
@@ 1 Seed Selection Genetic Programming In Genetic Programming, each tree in population shows an algebraic or surmounting expression, and each algebraic or surmounting expression shows an approximate analytic solution to differential equations.
10. Approximation analytical solutions for a unified plasma sheath model by double decomposition method
Institute of Scientific and Technical Information of China (English)
FangJin－Qing
1998-01-01
A unified plasma sheath model and its potential equation are proposed.Any higher-order approximation analytical solutions for the unified plasma sheath potential equation are derived by double decomposition method.
11. Approximate Analytical Solutions for a Class of Laminar Boundary-Layer Equations
Institute of Scientific and Technical Information of China (English)
Seripah Awang Kechil; Ishak Hashim; Sim Siaw Jiet
2007-01-01
A simple and efficient approximate analytical technique is presented to obtain solutions to a class of two-point boundary value similarity problems in fluid mechanics. This technique is based on the decomposition method which yields a general analytic solution in the form of a convergent infinite series with easily computable terms. Comparative study is carried out to show the accuracy and effectiveness of the technique.
12. Analytical approximate solution of the cooling problem by Adomian decomposition method
Science.gov (United States)
2009-02-01
The Adomian decomposition method (ADM) can provide analytical approximation or approximated solution to a rather wide class of nonlinear (and stochastic) equations without linearization, perturbation, closure approximation, or discretization methods. In the present work, ADM is employed to solve the momentum and energy equations for laminar boundary layer flow over flat plate at zero incidences with neglecting the frictional heating. A trial and error strategy has been used to obtain the constant coefficient in the approximated solution. ADM provides an analytical solution in the form of an infinite power series. The effect of Adomian polynomial terms is considered and shows that the accuracy of results is increased with the increasing of Adomian polynomial terms. The velocity and thermal profiles on the boundary layer are calculated. Also the effect of the Prandtl number on the thermal boundary layer is obtained. Results show ADM can solve the nonlinear differential equations with negligible error compared to the exact solution.
13. Approximate analytical solution of MHD flow of an Oldroyd 8-constant fluid in a porous medium
Directory of Open Access Journals (Sweden)
Faisal Salah
2014-12-01
Full Text Available The steady flow in an incompressible, magnetohydrodynamic (MHD Oldroyd 8-constant fluid in a porous medium with the motion of an infinite plate is investigated. Using modified Darcy’s law of an Oldroyd 8-constant fluid, the equations governing the flow are modelled. The resulting nonlinear boundary value problem is solved using the homotopy analysis method (HAM. The obtained approximate analytical solutions clearly satisfy the governing nonlinear equations and all the imposed initial and boundary conditions. The convergence of the HAM solutions for different orders of approximation is demonstrated. For the Newtonian case, the approximate analytical solution via HAM is shown to be in close agreement with the exact solution. Finally, the variations of velocity field with respect to the magnetic field, porosity and non-Newtonian fluid parameters are graphically shown and discussed.
14. A New Homotopy Analysis Method for Approximating the Analytic Solution of KdV Equation
Directory of Open Access Journals (Sweden)
Vahid Barati
2014-01-01
Full Text Available In this study a new technique of the Homotopy Analysis Method (nHAM is applied to obtain an approximate analytic solution of the well-known Korteweg-de Vries (KdV equation. This method removes the extra terms and decreases the time taken in the original HAM by converting the KdV equation to a system of first order differential equations. The resulted nHAM solution at third order approximation is then compared with that of the exact soliton solution of the KdV equation and found to be in excellent agreement.
15. Approximate analytical solution of diffusion equation with fractional time derivative using optimal homotopy analysis method
Directory of Open Access Journals (Sweden)
S. Das
2013-12-01
Full Text Available In this article, optimal homotopy-analysis method is used to obtain approximate analytic solution of the time-fractional diffusion equation with a given initial condition. The fractional derivatives are considered in the Caputo sense. Unlike usual Homotopy analysis method, this method contains at the most three convergence control parameters which describe the faster convergence of the solution. Effects of parameters on the convergence of the approximate series solution by minimizing the averaged residual error with the proper choices of parameters are calculated numerically and presented through graphs and tables for different particular cases.
16. Approximate Analytic and Numerical Solutions to Lane-Emden Equation via Fuzzy Modeling Method
Directory of Open Access Journals (Sweden)
De-Gang Wang
2012-01-01
Full Text Available A novel algorithm, called variable weight fuzzy marginal linearization (VWFML method, is proposed. This method can supply approximate analytic and numerical solutions to Lane-Emden equations. And it is easy to be implemented and extended for solving other nonlinear differential equations. Numerical examples are included to demonstrate the validity and applicability of the developed technique.
17. Symbolic computation of analytic approximate solutions for nonlinear fractional differential equations
Science.gov (United States)
Lin, Yezhi; Liu, Yinping; Li, Zhibin
2013-01-01
The Adomian decomposition method (ADM) is one of the most effective methods to construct analytic approximate solutions for nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, Rach (2008) [22], the Adomian decomposition method and the Padé approximants technique, a new algorithm is proposed to construct analytic approximate solutions for nonlinear fractional differential equations with initial or boundary conditions. Furthermore, a MAPLE software package is developed to implement this new algorithm, which is user-friendly and efficient. One only needs to input the system equation, initial or boundary conditions and several necessary parameters, then our package will automatically deliver the analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the scope and demonstrate the validity of our package, especially for non-smooth initial value problems. Our package provides a helpful and easy-to-use tool in science and engineering simulations. Program summaryProgram title: ADMP Catalogue identifier: AENE_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENE_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 12011 No. of bytes in distributed program, including test data, etc.: 575551 Distribution format: tar.gz Programming language: MAPLE R15. Computer: PCs. Operating system: Windows XP/7. RAM: 2 Gbytes Classification: 4.3. Nature of problem: Constructing analytic approximate solutions of nonlinear fractional differential equations with initial or boundary conditions. Non-smooth initial value problems can be solved by this program. Solution method: Based on the new definition of the Adomian polynomials [1], the Adomian decomposition method and the Pad
18. Approximate analytical solution to the Boussinesq equation with a sloping water-land boundary
Science.gov (United States)
Tang, Yuehao; Jiang, Qinghui; Zhou, Chuangbing
2016-04-01
An approximate solution is presented to the 1-D Boussinesq equation (BEQ) characterizing transient groundwater flow in an unconfined aquifer subject to a constant water variation at the sloping water-land boundary. The flow equation is decomposed to a linearized BEQ and a head correction equation. The linearized BEQ is solved using a Laplace transform. By means of the frozen-coefficient technique and Gauss function method, the approximate solution for the head correction equation can be obtained, which is further simplified to a closed-form expression under the condition of local energy equilibrium. The solutions of the linearized and head correction equations are discussed from physical concepts. Especially for the head correction equation, the well posedness of the approximate solution obtained by the frozen-coefficient method is verified to demonstrate its boundedness, which can be further embodied as the upper and lower error bounds to the exact solution of the head correction by statistical analysis. The advantage of this approximate solution is in its simplicity while preserving the inherent nonlinearity of the physical phenomenon. Comparisons between the analytical and numerical solutions of the BEQ validate that the approximation method can achieve desirable precisions, even in the cases with strong nonlinearity. The proposed approximate solution is applied to various hydrological problems, in which the algebraic expressions that quantify the water flow processes are derived from its basic solutions. The results are useful for the quantification of stream-aquifer exchange flow rates, aquifer response due to the sudden reservoir release, bank storage and depletion, and front position and propagation speed.
19. Approximate Analytical Solutions of the Regularized Long Wave Equation Using the Optimal Homotopy Perturbation Method
Directory of Open Access Journals (Sweden)
Constantin Bota
2014-01-01
Full Text Available The paper presents the optimal homotopy perturbation method, which is a new method to find approximate analytical solutions for nonlinear partial differential equations. Based on the well-known homotopy perturbation method, the optimal homotopy perturbation method presents an accelerated convergence compared to the regular homotopy perturbation method. The applications presented emphasize the high accuracy of the method by means of a comparison with previous results.
20. Interaction and charge transfer between dielectric spheres: exact and approximate analytical solutions
CERN Document Server
Lindén, Fredrik; Zettergren, Henning
2016-01-01
We present exact analytical solutions for charge transfer reactions between two arbitrarily charged hard dielectric spheres. These solutions, and the corresponding exact ones for sphere-sphere interaction energies, include sums that describe polarization effects to infinite orders in the inverse of the distance between the sphere centers. In addition, we show that these exact solutions may be approximated by much simpler analytical expressions that are useful for many practical applications. This is exemplified through calculations of Langevin type cross sections for forming a compound system of two colliding spheres and through calculations of electron transfer cross sections. We find that it is important to account for dielectric properties and finite sphere sizes in such calculations, which for example may be useful for describing the evolution, growth, and dynamics of nanometer sized dielectric objects such as molecular clusters or dust grains in different environments including astrophysical ones.
1. Approximate semi-analytical solutions for the steady-state expansion of a contactor plasma
CERN Document Server
Camporeale, E; MacDonald, E A
2015-01-01
We study the steady-state expansion of a collisionless, electrostatic, quasi-neutral plasma plume into vacuum, with a fluid model. We analyze approximate semi-analytical solutions, that can be used in lieu of much more expensive numerical solutions. In particular, we focus on the earlier studies presented in Parks and Katz (1979), Korsun and Tverdokhlebova (1997), and Ashkenazy and Fruchtman (2001). By calculating the error with respect to the numerical solution, we can judge the range of validity for each solution. Moreover, we introduce a generalization of earlier models that has a wider range of applicability, in terms of plasma injection profiles. We conclude by showing a straightforward way to extend the discussed solutions to the case of a plasma plume injected with non-null azimuthal velocity.
2. Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: A new iterative algorithm
Science.gov (United States)
El-Ajou, Ahmad; Arqub, Omar Abu; Momani, Shaher
2015-07-01
In this paper, explicit and approximate solutions of the nonlinear fractional KdV-Burgers equation with time-space-fractional derivatives are presented and discussed. The solutions of our equation are calculated in the form of rabidly convergent series with easily computable components. The utilized method is a numerical technique based on the generalized Taylor series formula which constructs an analytical solution in the form of a convergent series. Five illustrative applications are given to demonstrate the effectiveness and the leverage of the present method. Graphical results and series formulas are utilized and discussed quantitatively to illustrate the solution. The results reveal that the method is very effective and simple in determination of solution of the fractional KdV-Burgers equation.
3. Editorial: Special Issue on Analytical and Approximate Solutions for Numerical Problems
Directory of Open Access Journals (Sweden)
Walailak Journal of Science and Technology
2014-08-01
Full Text Available Though methods and algorithms in numerical analysis are not new, they have become increasingly popular with the development of high speed computing capabilities. Indeed, the ready availability of high speed modern digital computers and easy-to-employ powerful software packages has had a major impact on science, engineering education and practice in the recent past. Researchers in the past had to depend on analytical skills to solve significant engineering problems but, nowadays, researchers have access to tremendous amount of computation power under their fingertips, and they mostly require understanding the physical nature of the problem and interpreting the results. For some problems, several approximate analytical solutions already exist for simple cases but finding new solution to complex problems by designing and developing novel techniques and algorithms are indeed a great challenging task to give approximate solutions and sufficient accuracy especially for engineering purposes. In particular, it is frequently assumed that deriving an analytical solution for any problem is simpler than obtaining a numerical solution for the same problem. But in most of the cases relationships between numerical and analytical solutions complexities are exactly opposite to each other. In addition, analytical solutions are limited to relatively simple problems while numerical ones can be obtained for complex realistic situations. Indeed, analytical solutions are very useful for testing (benchmarking numerical codes and for understanding principal physical controls of complex processes that are modeled numerically. During the recent past, in order to overcome some numerical difficulties a variety of numerical approaches were introduced, such as the finite difference methods (FDM, the finite element methods (FEM, and other alternative methods. Numerical methods typically include material on such topics as computer precision, root finding techniques, solving
4. Higher order analytical approximate solutions to the nonlinear pendulum by He's homotopy method
Energy Technology Data Exchange (ETDEWEB)
Belendez, A; Pascual, C; Alvarez, M L; Mendez, D I; Yebra, M S; Hernandez, A [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain)], E-mail: [email protected]
2009-01-15
A modified He's homotopy perturbation method is used to calculate the periodic solutions of a nonlinear pendulum. The method has been modified by truncating the infinite series corresponding to the first-order approximate solution and substituting a finite number of terms in the second-order linear differential equation. As can be seen, the modified homotopy perturbation method works very well for high values of the initial amplitude. Excellent agreement of the analytical approximate period with the exact period has been demonstrated not only for small but also for large amplitudes A (the relative error is less than 1% for A < 152 deg.). Comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient.
5. A nonlinear model arising in the buckling analysis and its new analytic approximate solution
Energy Technology Data Exchange (ETDEWEB)
Khan, Yasir [Zhejiang Univ., Hangzhou, ZJ (China). Dept. of Mathematics; Al-Hayani, Waleed [Univ. Carlos III de Madrid, Leganes (Spain). Dept. de Matematicas; Mosul Univ. (Iraq). Dept. of Mathematics
2013-05-15
An analytical nonlinear buckling model where the rod is assumed to be an inextensible column and prismatic is studied. The dimensionless parameters reduce the constitutive equation to a nonlinear ordinary differential equation which is solved using the Adomian decomposition method (ADM) through Green's function technique. The nonlinear terms can be easily handled by the use of Adomian polynomials. The ADM technique allows us to obtain an approximate solution in a series form. Results are presented graphically to study the efficiency and accuracy of the method. To the author's knowledge, the current paper represents a new approach to the solution of the buckling of the rod problem. The fact that ADM solves nonlinear problems without using perturbations and small parameters can be judged as a lucid benefit of this technique over the other methods. (orig.)
6. Approximate analytical solutions to the condensation-coagulation equation of aerosols
CERN Document Server
Smith, Naftali; Svensmark, Henrik
2015-01-01
We present analytical solutions to the steady state injection-condensation-coagulation equation of aerosols in the atmosphere. These solutions are appropriate under different limits but more general than previously derived analytical solutions. For example, we provide an analytic solution to the coagulation limit plus a condensation correction. Our solutions are then compared with numerical results. We show that the solutions can be used to estimate the sensitivity of the cloud condensation nuclei number density to the nucleation rate of small condensation nuclei and to changes in the formation rate of sulfuric acid.
7. An analytical approximation method for the stabilizing solution of the Hamilton-Jacobi equation based on stable manifold theory
NARCIS (Netherlands)
Sakamoto, Noboru; Schaft, Arjan J. van der
2007-01-01
In this paper, an analytical approximation approach for the stabilizing solution of the Hamilton-Jacobi equation using stable manifold theory is proposed. The proposed method gives approximated flows on the stable manifold of the associated Hamiltonian system and provides approximations of the stabl
8. Super stellar clusters with a bimodal hydrodynamic solution: an Approximate Analytic Approach
CERN Document Server
Wünsch, R; Palous, J; Tenorio-Tagle, G
2007-01-01
We look for a simple analytic model to distinguish between stellar clusters undergoing a bimodal hydrodynamic solution from those able to drive only a stationary wind. Clusters in the bimodal regime undergo strong radiative cooling within their densest inner regions, which results in the accumulation of the matter injected by supernovae and stellar winds and eventually in the formation of further stellar generations, while their outer regions sustain a stationary wind. The analytic formulae are derived from the basic hydrodynamic equations. Our main assumption, that the density at the star cluster surface scales almost linearly with that at the stagnation radius, is based on results from semi-analytic and full numerical calculations. The analytic formulation allows for the determination of the threshold mechanical luminosity that separates clusters evolving in either of the two solutions. It is possible to fix the stagnation radius by simple analytic expressions and thus to determine the fractions of the depo...
9. Analytical Approximate Solutions for the Cubic-Quintic Duffing Oscillator in Terms of Elementary Functions
Directory of Open Access Journals (Sweden)
A. Beléndez
2012-01-01
Full Text Available Accurate approximate closed-form solutions for the cubic-quintic Duffing oscillator are obtained in terms of elementary functions. To do this, we use the previous results obtained using a cubication method in which the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation. Explicit approximate solutions are then expressed as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn. Then we obtain other approximate expressions for these solutions, which are expressed in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean is used and the rational harmonic balance method is applied to obtain the periodic solution of the original nonlinear oscillator.
10. An approximate and an analytical solution to the carousel-pendulum problem
Energy Technology Data Exchange (ETDEWEB)
Vial, Alexandre [Pole Physique, Mecanique, Materiaux et Nanotechnologies, Universite de technologie de Troyes, 12, rue Marie Curie BP-2060, F-10010 Troyes Cedex (France)], E-mail: [email protected]
2009-09-15
We show that an improved solution to the carousel-pendulum problem can be easily obtained through a first-order Taylor expansion, and its accuracy is determined after the obtention of an unusable analytical exact solution, advantageously replaced by a numerical one. It is shown that the accuracy is unexpectedly high, even when the ratio length of the pendulum to carousel radius approaches unity. (letters and comments)
11. Approximate Analytical Solution for Nonlinear System of Fractional Differential Equations by BPs Operational Matrices
Directory of Open Access Journals (Sweden)
Mohsen Alipour
2013-01-01
Full Text Available We present two methods for solving a nonlinear system of fractional differential equations within Caputo derivative. Firstly, we derive operational matrices for Caputo fractional derivative and for Riemann-Liouville fractional integral by using the Bernstein polynomials (BPs. In the first method, we use the operational matrix of Caputo fractional derivative (OMCFD, and in the second one, we apply the operational matrix of Riemann-Liouville fractional integral (OMRLFI. The obtained results are in good agreement with each other as well as with the analytical solutions. We show that the solutions approach to classical solutions as the order of the fractional derivatives approaches 1.
12. On Approximate Analytical Solutions of Nonlinear Vibrations of Inextensible Beams using Parameter-Expansion Method
DEFF Research Database (Denmark)
Kimiaeifar, Amin; Lund, Erik; Thomsen, Ole Thybo;
2010-01-01
In this work, an analytical method, which is referred to as Parameter-expansion Method is used to obtain the exact solution for the problem of nonlinear vibrations of an inextensible beam. It is shown that one term in the series expansion is sufficient to obtain a highly accurate solution, which ...... is valid for the whole domain of the problem. A comparison of the obtained the numerical solution demonstrates that PEM is effective and convenient for solving such problems. After validation of the obtained results, the system response and stability are also discussed....
13. Analytic Approximate Solutions to the Boundary Layer Flow Equation over a Stretching Wall with Partial Slip at the Boundary.
Science.gov (United States)
Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan
2016-01-01
Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations.
14. Analytical approximate solution for nonlinear space-time fractional Klein-Gordon equation
Institute of Scientific and Technical Information of China (English)
Khaled A.Gepreel; Mohamed S.Mohamed
2013-01-01
The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation.The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space-time fractional derivatives KleinGordon equation.This method introduces a promising tool for solving many space-time fractional partial differential equations.This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.
15. Analytical solutions for the surface response to small amplitude perturbations in boundary data in the shallow-ice-stream approximation
Directory of Open Access Journals (Sweden)
G. H. Gudmundsson
2008-07-01
Full Text Available New analytical solutions describing the effects of small-amplitude perturbations in boundary data on flow in the shallow-ice-stream approximation are presented. These solutions are valid for a non-linear Weertman-type sliding law and for Newtonian ice rheology. Comparison is made with corresponding solutions of the shallow-ice-sheet approximation, and with solutions of the full Stokes equations. The shallow-ice-stream approximation is commonly used to describe large-scale ice stream flow over a weak bed, while the shallow-ice-sheet approximation forms the basis of most current large-scale ice sheet models. It is found that the shallow-ice-stream approximation overestimates the effects of bed topography perturbations on surface profile for wavelengths less than about 5 to 10 ice thicknesses, the exact number depending on values of surface slope and slip ratio. For high slip ratios, the shallow-ice-stream approximation gives a very simple description of the relationship between bed and surface topography, with the corresponding transfer amplitudes being close to unity for any given wavelength. The shallow-ice-stream estimates for the timescales that govern the transient response of ice streams to external perturbations are considerably more accurate than those based on the shallow-ice-sheet approximation. In particular, in contrast to the shallow-ice-sheet approximation, the shallow-ice-stream approximation correctly reproduces the short-wavelength limit of the kinematic phase speed given by solving a linearised version of the full Stokes system. In accordance with the full Stokes solutions, the shallow-ice-sheet approximation predicts surface fields to react weakly to spatial variations in basal slipperiness with wavelengths less than about 10 to 20 ice thicknesses.
16. Analytic solutions for Baxter's model of sticky hard sphere fluids within closures different from the Percus-Yevick approximation.
Science.gov (United States)
Gazzillo, Domenico; Giacometti, Achille
2004-03-08
We discuss structural and thermodynamical properties of Baxter's adhesive hard sphere model within a class of closures which includes the Percus-Yevick (PY) one. The common feature of all these closures is to have a direct correlation function vanishing beyond a certain range, each closure being identified by a different approximation within the original square-well region. This allows a common analytical solution of the Ornstein-Zernike integral equation, with the cavity function playing a privileged role. A careful analytical treatment of the equation of state is reported. Numerical comparison with Monte Carlo simulations shows that the PY approximation lies between simpler closures, which may yield less accurate predictions but are easily extensible to multicomponent fluids, and more sophisticate closures which give more precise predictions but can hardly be extended to mixtures. In regimes typical for colloidal and protein solutions, however, it is found that the perturbative closures, even when limited to first order, produce satisfactory results.
17. An approximate analytical solution of free convection problem for vertical isothermal plate via transverse coordinate Taylor expansion
CERN Document Server
Leble, Sergey
2013-01-01
The model under consideration is based on approximate analytical solution of two dimensional stationary Navier-Stokes and Fourier-Kirchhoff equations. Approximations are based on the typical for natural convection assumptions: the fluid noncompressibility and Bousinesq approximation. We also assume that ortogonal to the plate component (x) of velocity is neglectible small. The solution of the boundary problem is represented as a Taylor Series in $x$ coordinate for velocity and temperature which introduces functions of vertical coordinate (y), as coefficients of the expansion. The correspondent boundary problem formulation depends on parameters specific for the problem: Grashoff number, the plate height (L) and gravity constant. The main result of the paper is the set of equations for the coefficient functions for example choice of expansion terms number. The nonzero velocity at the starting point of a flow appears in such approach as a development of convecntional boundary layer theory formulation.
18. Quantum dynamics of incoherently driven V-type system: Analytic solutions beyond the secular approximation
CERN Document Server
Dodin, Amro; Brumer, Paul
2015-01-01
We present closed-form analytic solutions to non-secular Bloch-Redfield master equations for quantum dynamics of a V-type system driven by weak coupling to a thermal bath. We focus on noise-induced Fano coherences among the excited states induced by incoherent driving of the V-system initially in the ground state. For suddenly turned-on incoherent driving, the time evolution of the coherences is determined by the damping parameter $\\zeta=\\frac{1}{2}(\\gamma_1+\\gamma_2)/\\Delta_p$, where $\\gamma_i$ are the radiative decay rates of the excited levels $i=1,2$, and $\\Delta_p=\\sqrt{\\Delta^2 + (1-p^2)\\gamma_1\\gamma_2}$ depends on the excited-state level splitting $\\Delta>0$ and the angle between the transition dipole moments in the energy basis. The coherences oscillate as a function of time in the underdamped limit ($\\zeta\\gg1$), approach a long-lived quasi-steady state in the overdamped limit ($\\zeta\\ll 1$), and display an intermediate behavior at critical damping ($\\zeta= 1$). The sudden incoherent turn-on generat...
19. Approximate Analytical Solutions for Mathematical Model of Tumour Invasion and Metastasis Using Modified Adomian Decomposition and Homotopy Perturbation Methods
Directory of Open Access Journals (Sweden)
Norhasimah Mahiddin
2014-01-01
Full Text Available The modified decomposition method (MDM and homotopy perturbation method (HPM are applied to obtain the approximate solution of the nonlinear model of tumour invasion and metastasis. The study highlights the significant features of the employed methods and their ability to handle nonlinear partial differential equations. The methods do not need linearization and weak nonlinearity assumptions. Although the main difference between MDM and Adomian decomposition method (ADM is a slight variation in the definition of the initial condition, modification eliminates massive computation work. The approximate analytical solution obtained by MDM logically contains the solution obtained by HPM. It shows that HPM does not involve the Adomian polynomials when dealing with nonlinear problems.
20. Analytical approximations for spiral waves
Energy Technology Data Exchange (ETDEWEB)
Löber, Jakob, E-mail: [email protected]; Engel, Harald [Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, EW 7-1, 10623 Berlin (Germany)
2013-12-15
We propose a non-perturbative attempt to solve the kinematic equations for spiral waves in excitable media. From the eikonal equation for the wave front we derive an implicit analytical relation between rotation frequency Ω and core radius R{sub 0}. For free, rigidly rotating spiral waves our analytical prediction is in good agreement with numerical solutions of the linear eikonal equation not only for very large but also for intermediate and small values of the core radius. An equivalent Ω(R{sub +}) dependence improves the result by Keener and Tyson for spiral waves pinned to a circular defect of radius R{sub +} with Neumann boundaries at the periphery. Simultaneously, analytical approximations for the shape of free and pinned spirals are given. We discuss the reasons why the ansatz fails to correctly describe the dependence of the rotation frequency on the excitability of the medium.
1. Approximate Analytical Solution to the Fractional Lane-Emden Equation of the Polytropic Gas Sphere
CERN Document Server
Nouh, Mohamed I
2016-01-01
Lane-Emden equation could be used to model stellar interiors, star clusters and many configurations in astrophysics. Unfortunately, there is an exact solution only for the polytropic index n=0,1 and 5. In the present paper, a series solution for the fractional lane-Emden equation is presented. The solution is performed in the frame of modified Rienmann liouville derivatives. The results indicate that the series converges for the polytropic index range 0<=n <= 4.99 with fractional parameter \\alpha spreads over all range 0<\\alpha <= 1. Comparison with the numerical solution revealed a good agreement with a maximum relative error 0.001. The obtained results recover the well-known series solutions when \\alpha=1.
2. Deriving analytic solutions for compact binary inspirals without recourse to adiabatic approximations
CERN Document Server
2016-01-01
We utilize the dynamical renormalization group formalism to calculate the real space trajectory of a compact binary inspiral for long times via a systematic resummation of secularly growing terms. This method generates closed form solutions without orbit averaging, and the accuracy can be systematically improved. The expansion parameter is v^5 \ 3. Linear analytical solution to the phase diversity problem for extended objects based on the Born approximation NARCIS (Netherlands) Andrei, R.M.; Smith, C.S.; Fraanje, P.R.; Verhaegen, M.; Korkiakoski, V.A.; Keller, C.U.; Doelman, N.J. 2012-01-01 In this paper we give a new wavefront estimation technique that overcomes the main disadvantages of the phase diversity (PD) algorithms, namely the large computational complexity and the fact that the solutions can get stuck in a local minima. Our approach gives a good starting point for an iterativ 4. Approximation of the observed motion of bolides by the analytical solution of the equations of meteor physics Science.gov (United States) Gritsevich, M. I. 2007-12-01 A great volume of data has been accumulated thus far related to the photoregistration of the paths of meteor bodies in the terrestrial atmosphere. Most images have been obtained by four bolide networks, which operate in the USA, Canada, Europe, and Spain in different time periods. The approximation of the actual data using theoretical models makes it possible to achieve additional estimates, which do not directly follow from the observations. In the present study, we suggest an algorithm to find such parameters of the theoretical relationship between the height and the velocity of the bolide motion that help to fit observations along the luminous part of the trajectories in the best way. The main difference from previous studies is that the given observations are approximated using the analytical solution of the equations of meteor physics. The model presented in this study was applied to a number of bright meteors observed by the Canadian camera network and by the US Prairie network and to the Benésov bolide, which is one of the largest fireballs registered by the European network. The correct mathematical modeling of meteor events in the atmosphere is necessary for further estimates of the key parameters, including the extra-atmospheric mass, the ablation coefficient, and the effective enthalpy of evaporation of entering bodies. In turn, this information is needed by some applications, namely, those aimed at studying the problems of asteroid and comet security, to develop measures of planetary defense, and to determine the bodies that can reach Earth’s surface. 5. Existence and Analytic Approximation of Solutions of Duffing Type Nonlinear Integro-Differential Equation with Integral Boundary Conditions Directory of Open Access Journals (Sweden) Alsaedi Ahmed 2009-01-01 Full Text Available A generalized quasilinearization technique is developed to obtain a sequence of approximate solutions converging monotonically and quadratically to a unique solution of a boundary value problem involving Duffing type nonlinear integro-differential equation with integral boundary conditions. The convergence of order for the sequence of iterates is also established. It is found that the work presented in this paper not only produces new results but also yields several old results in certain limits. 6. An approximate analytical solution to the ultra-filtration profile in a hemodialysis process between parallel porous plates Institute of Scientific and Technical Information of China (English) LU JunFeng; LU WenQiang 2008-01-01 In a hemodialysis process, the blood that runs through straight channels exchanges substances with the dialysate through a semi-permeable membrane. The waste products, such as urea and creatinine, are therefore removed from the plasma by the membrane. In the analysis of this process, determination of the ultra-filtration profile along the porous membrane surface remains a difficult problem. In this work, an analytical solution to the derivation of such a profile was detailed, and the feasibility of this solution was discussed. The ultra-filtration profile was found to be in a cosine shape. 7. Approximate Analytical Solution for the 2nd Order Moments of a SDOF Hysteretic Oscillator with Low Yield Levels Excited by Stationary Gaussian White Noise DEFF Research Database (Denmark) Micaletti, R. C.; Cakmak, A. S.; Nielsen, Søren R. K.; and using analytically-available information, physical reasoning, and approximations supported by empirical observation, an equation for the probability of the oscillator being in the plastic state is derived. Upon numerical solution of this equation, analytical approximations to the response moments can...... will be achieved, i.e., excluding the case of an elastic-perfectly-plastic oscillator, algebraic equations for the response moments are found. By the nature of the problem, these moments depend on the probability of the oscillator being in the plastic state. Upon considering oscillators with low yield levels... 8. Rigorous analytical approximation of tritronquée solution to Painlevé-I and the first singularity Science.gov (United States) Adali, A.; Tanveer, S. 2016-10-01 We use a recently developed method [1,2] to determine approximate expression for tritronquée solution for P-1: y″ + 6y2 - x = 0 in a domain D with rigorous bounds. In particular we rigorously confirm the location of the closest singularity from the origin to be at x = -770766/323285 = - 2.3841687675 ⋯ to within 5 ×10-6 accuracy, in agreement with previous numerical calculations [6]. 9. Finite volume approximation of the three-dimensional flow equation in axisymmetric, heterogeneous porous media based on local analytical solution KAUST Repository Salama, Amgad 2013-09-01 In this work the problem of flow in three-dimensional, axisymmetric, heterogeneous porous medium domain is investigated numerically. For this system, it is natural to use cylindrical coordinate system, which is useful in describing phenomena that have some rotational symmetry about the longitudinal axis. This can happen in porous media, for example, in the vicinity of production/injection wells. The basic feature of this system is the fact that the flux component (volume flow rate per unit area) in the radial direction is changing because of the continuous change of the area. In this case, variables change rapidly closer to the axis of symmetry and this requires the mesh to be denser. In this work, we generalize a methodology that allows coarser mesh to be used and yet yields accurate results. This method is based on constructing local analytical solution in each cell in the radial direction and moves the derivatives in the other directions to the source term. A new expression for the harmonic mean of the hydraulic conductivity in the radial direction is developed. Apparently, this approach conforms to the analytical solution for uni-directional flows in radial direction in homogeneous porous media. For the case when the porous medium is heterogeneous or the boundary conditions is more complex, comparing with the mesh-independent solution, this approach requires only coarser mesh to arrive at this solution while the traditional methods require more denser mesh. Comparisons for different hydraulic conductivity scenarios and boundary conditions have also been introduced. © 2013 Elsevier B.V. 10. Differentially rotating force-free magnetosphere of an aligned rotator: analytical solutions in split-monopole approximation CERN Document Server Timokhin, Andrey 2007-01-01 In this paper we consider stationary force-free magnetosphere of an aligned rotator when plasma in the open field line region rotates differentially due to presence of a zone with the accelerating electric field in the polar cap of pulsar. We study the impact of differential rotation on the current density distribution in the magnetosphere. Using split-monopole approximation we obtain analytical expressions for physical parameters of differentially rotating magnetosphere. We find the range of admitted current density distributions under the requirement that the potential drop in the polar cap is less than the vacuum potential drop. We show that the current density distribution could deviate significantly from the classical'' Michel distribution and could be made almost constant over the polar cap even when the potential drop in the accelerating zone is of the order of 10 per cents of the vacuum potential drop. We argue that differential rotation of the open magnetic field lines could play an important role ... 11. Analytical Approximation to the (e)-Wave Solutions of the Hulthén Potential in Tridiagonal Representation Institute of Scientific and Technical Information of China (English) ZHANG Min-Cang; HUANG-FU Guo-Qing 2011-01-01 @@ The Schr(o)dinger equation with the Hulthén potential is studied by working in a complete square integrable basis that carries a tridiagonal matrix representation of the wave operator.The arbitrary e-wave solutions are obtained by using an approximation of the centrifugal term.The resulting three-term recursion relation for the expansion coefficients of the wavefunction is presented and the wavefunctions are expressed in terms of the Jocobi polynomial.The discrete spectrum of the bound states is obtained by the diagonalization of the recursion relation.%The Schr(o)dinger equation with the Hulthén potential is studied by working in a complete square integrable basis that carries a tridiagonal matrix representation of the wave operator. The arbitrary e-wave solutions are obtained by using an approximation of the centrifugal term. The resulting three-term recursion relation for the expansion coefficients of the wavefunction is presented and the wavefunctions are expressed in terms of the Jocobi polynomial. The discrete spectrum of the bound states is obtained by the diagonalization of the recursion relation. 12. Frankenstein's glue: transition functions for approximate solutions Science.gov (United States) Yunes, Nicolás 2007-09-01 Approximations are commonly employed to find approximate solutions to the Einstein equations. These solutions, however, are usually only valid in some specific spacetime region. A global solution can be constructed by gluing approximate solutions together, but this procedure is difficult because discontinuities can arise, leading to large violations of the Einstein equations. In this paper, we provide an attempt to formalize this gluing scheme by studying transition functions that join approximate analytic solutions together. In particular, we propose certain sufficient conditions on these functions and prove that these conditions guarantee that the joined solution still satisfies the Einstein equations analytically to the same order as the approximate ones. An example is also provided for a binary system of non-spinning black holes, where the approximate solutions are taken to be given by a post-Newtonian expansion and a perturbed Schwarzschild solution. For this specific case, we show that if the transition functions satisfy the proposed conditions, then the joined solution does not contain any violations to the Einstein equations larger than those already inherent in the approximations. We further show that if these functions violate the proposed conditions, then the matter content of the spacetime is modified by the introduction of a matter shell, whose stress energy tensor depends on derivatives of these functions. 13. Strongly nonlinear oscillators analytical solutions CERN Document Server Cveticanin, Livija 2014-01-01 This book provides the presentation of the motion of pure nonlinear oscillatory systems and various solution procedures which give the approximate solutions of the strong nonlinear oscillator equations. The book presents the original author’s method for the analytical solution procedure of the pure nonlinear oscillator system. After an introduction, the physical explanation of the pure nonlinearity and of the pure nonlinear oscillator is given. The analytical solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameter is considered. Special attention is given to the one and two mass oscillatory systems with two-degrees-of-freedom. The criteria for the deterministic chaos in ideal and non-ideal pure nonlinear oscillators are derived analytically. The method for suppressing chaos is developed. Important problems are discussed in didactic exercises. The book is self-consistent and suitable as a textbook for students and also for profess... 14. An approximate analytical approach to resampling averages DEFF Research Database (Denmark) Malzahn, Dorthe; Opper, M. 2004-01-01 Using a novel reformulation, we develop a framework to compute approximate resampling data averages analytically. The method avoids multiple retraining of statistical models on the samples. Our approach uses a combination of the replica "trick" of statistical physics and the TAP approach for appr......Using a novel reformulation, we develop a framework to compute approximate resampling data averages analytically. The method avoids multiple retraining of statistical models on the samples. Our approach uses a combination of the replica "trick" of statistical physics and the TAP approach... 15. Numerical and approximate solutions for plume rise Science.gov (United States) Krishnamurthy, Ramesh; Gordon Hall, J. Numerical and approximate analytical solutions are compared for turbulent plume rise in a crosswind. The numerical solutions were calculated using the plume rise model of Hoult, Fay and Forney (1969, J. Air Pollut. Control Ass.19, 585-590), over a wide range of pertinent parameters. Some wind shear and elevated inversion effects are included. The numerical solutions are seen to agree with the approximate solutions over a fairly wide range of the parameters. For the conditions considered in the study, wind shear effects are seen to be quite small. A limited study was made of the penetration of elevated inversions by plumes. The results indicate the adequacy of a simple criterion proposed by Briggs (1969, AEC Critical Review Series, USAEC Division of Technical Information extension, Oak Ridge, Tennesse). 16. Approximate Analytical Solution for One-Dimensional Solidification Problem of a Finite Superheating Phase Change Material Including the Effects of Wall and Thermal Contact Resistances Directory of Open Access Journals (Sweden) Hamid El Qarnia 2012-01-01 Full Text Available This work reports an analytical solution for the solidification of a superheating phase change material (PCM contained in a rectangular enclosure with a finite height. The analytical solution has been obtained by solving nondimensional energy equations by using the perturbation method for a small perturbation parameter: the Stefan number, ε. This analytical solution, which takes into account the effects of the superheating of PCM, finite height of the enclosure, thickness of the wall, and wall-solid shell interfacial thermal resistances, was expressed in terms of nondimensional temperature distributions of the bottom wall of the enclosure and both PCM phases, and the dimensionless solid-liquid interface position and its dimensionless speed. The developed solution was firstly compared with that existing in the literature for the case of nonsuperheating PCM. The predicted results agreed well with those published in the literature. Next, a parametric study was carried out in order to study the impacts of the dimensionless control parameters on the dimensionless temperature distributions of the wall, the solid shell, and liquid phase of the PCM, as well as the solid-liquid interface position and its dimensionless speed. 17. Nonlinear ordinary differential equations analytical approximation and numerical methods CERN Document Server Hermann, Martin 2016-01-01 The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march... 18. Analytical Ballistic Trajectories with Approximately Linear Drag Directory of Open Access Journals (Sweden) Giliam J. P. de Carpentier 2014-01-01 Full Text Available This paper introduces a practical analytical approximation of projectile trajectories in 2D and 3D roughly based on a linear drag model and explores a variety of different planning algorithms for these trajectories. Although the trajectories are only approximate, they still capture many of the characteristics of a real projectile in free fall under the influence of an invariant wind, gravitational pull, and terminal velocity, while the required math for these trajectories and planners is still simple enough to efficiently run on almost all modern hardware devices. Together, these properties make the proposed approach particularly useful for real-time applications where accuracy and performance need to be carefully balanced, such as in computer games. 19. Analytical Special Solutions of the Bohr Hamiltonian CERN Document Server Bonatsos, D; Petrellis, D; Terziev, P A; Yigitoglu, I 2005-01-01 The following special solutions of the Bohr Hamiltonian are briefly described: 1) Z(5) (approximately separable solution in five dimensions with gamma close to 30 degrees), 2) Z(4) (exactly separable gamma-rigid solution in four dimensions with gamma = 30 degrees), 3) X(3) (exactly separable gamma-rigid solution in three dimensions with gamma =0). The analytical solutions obtained using Davidson potentials in the E(5), X(5), Z(5), and Z(4) frameworks are also mentioned. 20. Approximate Analytic Solutions of Time-Fractional Hirota-Satsuma Coupled KdV Equation and Coupled MKdV Equation Directory of Open Access Journals (Sweden) Jincun Liu 2013-01-01 Full Text Available By introducing the fractional derivative in the sense of Caputo and combining the pretreatment technique to deal with long nonlinear items, the generalized two-dimensional differential transform method is proposed for solving the time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation. The presented method is a numerical method based on the generalized Taylor series expansion which constructs an analytical solution in the form of a polynomial. The numerical results show that the generalized two-dimensional differential transform method is very effective for the fractional coupled equations. 1. Wave system and its approximate similarity solutions Institute of Scientific and Technical Information of China (English) Liu Ping; Fu Pei-Kai 2011-01-01 Recently,a new (2+1)-dimensional shallow water wave system,the (2+1)-dimensional displacement shallow water wave system (2DDSWWS),was constructed by applying the variational principle of the analytic mechanics in the Lagrange coordinates. The disadvantage is that fluid viscidity is not considered in the 2DDSWWS,which is the same as the famous Kadomtsev-Petviashvili equation and Korteweg-de Vries equation. Applying dimensional analysis,we modify the 2DDSWWS and add the term related to the fluid viscidity to the 2DDSWWS. The approximate similarity solutions of the modified 2DDSWWS (M2DDSWWS) is studied and four similarity solutions are obtained. For the perfect fluids,the coefficient of kinematic viscosity is zero,then the M2DDSWWS will degenerate to the 2DDSWWS. 2. A Newton-Krylov method with an approximate analytical Jacobian for implicit solution of Navier-Stokes equations on staggered overset-curvilinear grids with immersed boundaries Science.gov (United States) Asgharzadeh, Hafez; Borazjani, Iman 2017-02-01 diagonal of the Jacobian further improves the performance by 42-74% compared to the full Jacobian. The NKM with an analytical Jacobian showed better performance than the fixed point Runge-Kutta because it converged with higher time steps and in approximately 30% less iterations even when the grid was stretched and the Reynold number was increased. In fact, stretching the grid decreased the performance of all methods, but the fixed-point Runge-Kutta performance decreased 4.57 and 2.26 times more than NKM with a diagonal and full Jacobian, respectivley, when the stretching factor was increased. The NKM with a diagonal analytical Jacobian and matrix-free method with an analytical preconditioner are the fastest methods and the superiority of one to another depends on the flow problem. Furthermore, the implemented methods are fully parallelized with parallel efficiency of 80-90% on the problems tested. The NKM with the analytical Jacobian can guide building preconditioners for other techniques to improve their performance in the future. 3. Comparing numerical and analytic approximate gravitational waveforms Science.gov (United States) Afshari, Nousha; Lovelace, Geoffrey; SXS Collaboration 2016-03-01 A direct observation of gravitational waves will test Einstein's theory of general relativity under the most extreme conditions. The Laser Interferometer Gravitational-Wave Observatory, or LIGO, began searching for gravitational waves in September 2015 with three times the sensitivity of initial LIGO. To help Advanced LIGO detect as many gravitational waves as possible, a major research effort is underway to accurately predict the expected waves. In this poster, I will explore how the gravitational waveform produced by a long binary-black-hole inspiral, merger, and ringdown is affected by how fast the larger black hole spins. In particular, I will present results from simulations of merging black holes, completed using the Spectral Einstein Code (black-holes.org/SpEC.html), including some new, long simulations designed to mimic black hole-neutron star mergers. I will present comparisons of the numerical waveforms with analytic approximations. 4. Analytic Approximate Solutions for MHD Boundary-Layer Viscoelastic Fluid Flow over Continuously Moving Stretching Surface by Homotopy Analysis Method with Two Auxiliary Parameters Directory of Open Access Journals (Sweden) M. M. Rashidi 2012-01-01 Full Text Available In this study, a steady, incompressible, and laminar-free convective flow of a two-dimensional electrically conducting viscoelastic fluid over a moving stretching surface through a porous medium is considered. The boundary-layer equations are derived by considering Boussinesq and boundary-layer approximations. The nonlinear ordinary differential equations for the momentum and energy equations are obtained and solved analytically by using homotopy analysis method (HAM with two auxiliary parameters for two classes of visco-elastic fluid (Walters’ liquid B and second-grade fluid. It is clear that by the use of second auxiliary parameter, the straight line region in ℏ-curve increases and the convergence accelerates. This research is performed by considering two different boundary conditions: (a prescribed surface temperature (PST and (b prescribed heat flux (PHF. The effect of involved parameters on velocity and temperature is investigated. 5. Analytic approximate radiation effects due to Bremsstrahlung Energy Technology Data Exchange (ETDEWEB) Ben-Zvi I. 2012-02-01 The purpose of this note is to provide analytic approximate expressions that can provide quick estimates of the various effects of the Bremsstrahlung radiation produced relatively low energy electrons, such as the dumping of the beam into the beam stop at the ERL or field emission in superconducting cavities. The purpose of this work is not to replace a dependable calculation or, better yet, a measurement under real conditions, but to provide a quick but approximate estimate for guidance purposes only. These effects include dose to personnel, ozone generation in the air volume exposed to the radiation, hydrogen generation in the beam dump water cooling system and radiation damage to near-by magnets. These expressions can be used for other purposes, but one should note that the electron beam energy range is limited. In these calculations the good range is from about 0.5 MeV to 10 MeV. To help in the application of this note, calculations are presented as a worked out example for the beam dump of the R&D Energy Recovery Linac. 6. Frankenstein's Glue: Transition functions for approximate solutions CERN Document Server Yunes, N 2006-01-01 Approximations are commonly employed to find approximate solutions to the Einstein equations. These solutions, however, are usually only valid in some specific spacetime region. A global solution can be constructed by gluing approximate solutions together, but this procedure is difficult because discontinuities can arise, leading to large violations of the Einstein equations. In this paper, we provide an attempt to formalize this gluing scheme by studying transition functions that join approximate solutions together. In particular, we propose certain sufficient conditions on these functions and proof that these conditions guarantee that the joined solution still satisfies the Einstein equations to the same order as the approximate ones. An example is also provided for a binary system of non-spinning black holes, where the approximate solutions are taken to be given by a post-Newtonian expansion and a perturbed Schwarzschild solution. For this specific case, we show that if the transition functions satisfy the... 7. Radiative Transfer in spheres I. Analytical Solutions CERN Document Server Aboughantous, C 2001-01-01 A nonsingular analytical solution for the transfer equation in a pure absorber is obtained in central symmetry and in a monochromatic radiation field. The native regular singularity of the equation is removed by applying a linear transformation to the frame of reference. Two different ap-proaches are used to carry out the solution. In the first approach the angular derivative is interpreted in an original way that made it possible to discard this derivative from the equation for all black body media without upsetting the conservation of energy. In this approach the analytic solution is expressible in terms of exponential integrals without approximations but for practical considerations the solution is presented in the form of Gauss-Legendre quadrature for quantitative evaluation of the solutions. In the second approach the angular derivative is approximated by a new set of discrete ordinates that guarantees the closer of the set of equations and the conservation of energy. The solutions from the two approache... 8. Exact analytical solutions for ADAFs CERN Document Server Habibi, Asiyeh; Shadmehri, Mohsen 2016-01-01 We obtain two-dimensional exact analytic solutions for the structure of the hot accretion flows without wind. We assume that the only non-zero component of the stress tensor isT_{r\\varphi}$. Furthermore we assume that the value of viscosity coefficient$\\alpha$varies with$\\theta$. We find radially self-similar solutions and compare them with the numerical and the analytical solutions already studied in the literature. The no-wind solution obtained in this paper may be applied to the nuclei of some cool-core clusters. 9. Analytic solutions of a class of nonlinearly dynamic systems Energy Technology Data Exchange (ETDEWEB) Wang, M-C [System Engineering Institute of Tianjin University, Tianjin, 300072 (China); Zhao, X-S; Liu, X [Tianjin University of Technology and Education, Tianjin, 300222 (China)], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] 2008-02-15 In this paper, the homotopy perturbation method (HPM) is applied to solve a coupled system of two nonlinear differential with first-order similar model of Lotka-Volterra and a Bratus equation with a source term. The analytic approximate solutions are derived. Furthermore, the analytic approximate solutions obtained by the HPM with the exact solutions reveals that the present method works efficiently. 10. Frankenstein's glue: transition functions for approximate solutions Energy Technology Data Exchange (ETDEWEB) Yunes, Nicolas [Center for Gravitational Wave Physics, Institute for Gravitational Physics and Geometry, Department of Physics, Pennsylvania State University, University Park, PA 16802-6300 (United States) 2007-09-07 Approximations are commonly employed to find approximate solutions to the Einstein equations. These solutions, however, are usually only valid in some specific spacetime region. A global solution can be constructed by gluing approximate solutions together, but this procedure is difficult because discontinuities can arise, leading to large violations of the Einstein equations. In this paper, we provide an attempt to formalize this gluing scheme by studying transition functions that join approximate analytic solutions together. In particular, we propose certain sufficient conditions on these functions and prove that these conditions guarantee that the joined solution still satisfies the Einstein equations analytically to the same order as the approximate ones. An example is also provided for a binary system of non-spinning black holes, where the approximate solutions are taken to be given by a post-Newtonian expansion and a perturbed Schwarzschild solution. For this specific case, we show that if the transition functions satisfy the proposed conditions, then the joined solution does not contain any violations to the Einstein equations larger than those already inherent in the approximations. We further show that if these functions violate the proposed conditions, then the matter content of the spacetime is modified by the introduction of a matter shell, whose stress-energy tensor depends on derivatives of these functions. 11. Approximate solution of a nonlinear partial differential equation NARCIS (Netherlands) Vajta, M. 2007-01-01 Nonlinear partial differential equations (PDE) are notorious to solve. In only a limited number of cases can we find an analytic solution. In most cases, we can only apply some numerical scheme to simulate the process described by a nonlinear PDE. Therefore, approximate solutions are important for t 12. Analytical solution for the Feynman ratchet. Science.gov (United States) Pesz, Karol; Gabryś, Barbara J; Bartkiewicz, Stanisław J 2002-12-01 A search for an analytical, closed form solution of the Fokker-Planck equation with periodic, asymmetric potentials (ratchets) is presented. It is found that logarithmic-type potential functions (related to "entropic" ratchets) allow for an approximate solution within a certain range of parameters. An expression for the net current is calculated and it is shown that the efficiency of the rocked entropic ratchet is always low. 13. Approximations of solutions to retarded integrodifferential equations Directory of Open Access Journals (Sweden) Dhirendra Bahuguna 2004-11-01 Full Text Available In this paper we consider a retarded integrodifferential equation and prove existence, uniqueness and convergence of approximate solutions. We also give some examples to illustrate the applications of the abstract results. 14. The approximate solutions of nonlinear Boussinesq equation Science.gov (United States) Lu, Dianhen; Shen, Jie; Cheng, Yueling 2016-04-01 The homotopy analysis method (HAM) is introduced to solve the generalized Boussinesq equation. In this work, we establish the new analytical solution of the exponential function form. Applying the homotopy perturbation method to solve the variable coefficient Boussinesq equation. The results indicate that this method is efficient for the nonlinear models with variable coefficients. 15. Approximate solutions for fractured wells producing layered reservoirs Energy Technology Data Exchange (ETDEWEB) Bennett, C.O.; Camacho-V., R; Raghavan, R.; Reynolds, A.C. 1985-10-01 New analytical solutions for the response at a well intercepting a layered reservoir are derived. The well is assumed to produce at a constant rate or a constant pressure. We examine reservoir systems without interlayer communication and document the usefulness of these solutions, which enable us to obtain increased physical understanding of the performance of fractured wells in layered reservoirs. The influence of vertical variations in fracture conductivity is also considered. Example applications of the approximations derived here are also presented. 16. Analytic solution for a quartic electron mirror Energy Technology Data Exchange (ETDEWEB) Straton, Jack C., E-mail: [email protected] 2015-01-15 A converging electron mirror can be used to compensate for spherical and chromatic aberrations in an electron microscope. This paper presents an analytical solution to a diode (two-electrode) electrostatic mirror including the next term beyond the known hyperbolic shape. The latter is a solution of the Laplace equation to second order in the variables perpendicular to and along the mirror's radius (z{sup 2}−r{sup 2}/2) to which we add a quartic term (kλz{sup 4}). The analytical solution is found in terms of Jacobi cosine-amplitude functions. We find that a mirror less concave than the hyperbolic profile is more sensitive to changes in mirror voltages and the contrary holds for the mirror more concave than the hyperbolic profile. - Highlights: • We find the analytical solution for electron mirrors whose curvature has z4 dependence added to the usual z{sup 2} – r{sup 2}/2 terms. • The resulting Jacobi cosine-amplitude function reduces to the well-known cosh solution in the limit where the new term is 0. • This quartic term gives a mirror designer additional flexibility for eliminating spherical and chromatic aberrations. • The possibility of using these analytical results to approximately model spherical tetrode mirrors close to axis is noted. 17. Approximate Series Solutions for Nonlinear Free Vibration of Suspended Cables Directory of Open Access Journals (Sweden) Yaobing Zhao 2014-01-01 Full Text Available This paper presents approximate series solutions for nonlinear free vibration of suspended cables via the Lindstedt-Poincare method and homotopy analysis method, respectively. Firstly, taking into account the geometric nonlinearity of the suspended cable as well as the quasi-static assumption, a mathematical model is presented. Secondly, two analytical methods are introduced to obtain the approximate series solutions in the case of nonlinear free vibration. Moreover, small and large sag-to-span ratios and initial conditions are chosen to study the nonlinear dynamic responses by these two analytical methods. The numerical results indicate that frequency amplitude relationships obtained with different analytical approaches exhibit some quantitative and qualitative differences in the cases of motions, mode shapes, and particular sag-to-span ratios. Finally, a detailed comparison of the differences in the displacement fields and cable axial total tensions is made. 18. Analytical solutions of moisture flow equations and their numerical evaluation Energy Technology Data Exchange (ETDEWEB) Gibbs, A.G. 1981-04-01 The role of analytical solutions of idealized moisture flow problems is discussed. Some different formulations of the moisture flow problem are reviewed. A number of different analytical solutions are summarized, including the case of idealized coupled moisture and heat flow. The evaluation of special functions which commonly arise in analytical solutions is discussed, including some pitfalls in the evaluation of expressions involving combinations of special functions. Finally, perturbation theory methods are summarized which can be used to obtain good approximate analytical solutions to problems which are too complicated to solve exactly, but which are close to an analytically solvable problem. 19. Approximate solutions of theWei Hua oscillator using the Pekeris approximation and Nikiforov–Uvarov method Indian Academy of Sciences (India) P K Bera 2012-01-01 The approximate analytical bound-state solutions of the Schrödinger equation for the Wei Hua oscillator are carried out in N-dimensional space by taking Pekeris approximation scheme to the orbital centrifugal term. Solutions of the corresponding hyper-radial equation are obtained using the conventional Nikiforov–Uvarov (NU) method. 20. Analytical solutions of the lattice Boltzmann BGK model CERN Document Server Zou, Q; Doolen, G D; Zou, Qisu; Hou, Shuling; Doolen, Gary D. 1995-01-01 Abstract: Analytical solutions of the two dimensional triangular and square lattice Boltzmann BGK models have been obtained for the plain Poiseuille flow and the plain Couette flow. The analytical solutions are written in terms of the characteristic velocity of the flow, the single relaxation time representation of these two flows without any approximation. 1. On approximative solutions of multistopping problems CERN Document Server Faller, Andreas; 10.1214/10-AAP747 2012-01-01 In this paper, we consider multistopping problems for finite discrete time sequences$X_1,...,X_n$.$m$-stops are allowed and the aim is to maximize the expected value of the best of these$m$stops. The random variables are neither assumed to be independent not to be identically distributed. The basic assumption is convergence of a related imbedded point process to a continuous time Poisson process in the plane, which serves as a limiting model for the stopping problem. The optimal$m$-stopping curves for this limiting model are determined by differential equations of first order. A general approximation result is established which ensures convergence of the finite discrete time$m$-stopping problem to that in the limit model. This allows the construction of approximative solutions of the discrete time$m$-stopping problem. In detail, the case of i.i.d. sequences with discount and observation costs is discussed and explicit results are obtained. 2. Rough Sets in Approximate Solution Space Institute of Scientific and Technical Information of China (English) Hui Sun; Wei Tian; Qing Liu 2006-01-01 As a new mathematical theory, Rough sets have been applied to processing imprecise, uncertain and in complete data. It has been fruitful in finite and non-empty set. Rough sets, however, are only served as the theoretic tool to discretize the real function. As far as the real function research is concerned, the research to define rough sets in the real function is infrequent. In this paper, we exploit a new method to extend the rough set in normed linear space, in which we establish a rough set,put forward an upper and lower approximation definition, and make a preliminary research on the property of the rough set. A new tool is provided to study the approximation solutions of differential equation and functional variation in normed linear space. This research is significant in that it extends the application of rough sets to a new field. 3. Analytical approximations to the spectra of quark-antiquark potentials Energy Technology Data Exchange (ETDEWEB) Amore, Paolo [Facultad de Ciencias, Universidad de Colima, Bernal DIaz del Castillo 340, Colima (Mexico); De Pace, Arturo [Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via Giuria 1, I-10125 Turin (Italy); Lopez, Jorge [Physics Department, University of Texas at El Paso, El Paso, TX (United States) 2006-07-15 A method recently devised to obtain analytical approximations to certain classes of integrals is used in combination with the WKB expansion to derive accurate analytical expressions for the spectrum of quantum potentials. The accuracy of our results is verified by comparing them both with the literature on the subject and with the numerical results obtained with a Fortran code. As an application of the method that we propose, we consider meson spectroscopy with various phenomenological potentials. 4. Analytical approximations to the spectra of quark antiquark potentials Science.gov (United States) Amore, Paolo; DePace, Arturo; Lopez, Jorge 2006-07-01 A method recently devised to obtain analytical approximations to certain classes of integrals is used in combination with the WKB expansion to derive accurate analytical expressions for the spectrum of quantum potentials. The accuracy of our results is verified by comparing them both with the literature on the subject and with the numerical results obtained with a Fortran code. As an application of the method that we propose, we consider meson spectroscopy with various phenomenological potentials. 5. Approximate solutions of general perturbed KdV-Burgers equations Directory of Open Access Journals (Sweden) Baojian Hong 2014-09-01 Full Text Available In this article, we present some approximate analytical solutions to the general perturbed KdV-Burgers equation with nonlinear terms of any order by applying the homotopy analysis method (HAM. While compared with the Adomain decomposition method (ADM and the homotopy perturbation method (HPM, the HAM contains the auxiliary convergence-control parameter$\\hbar$and the control function$H(x,t$, which provides a useful way to adjust and control the convergence region of solution series. The numerical results reveal that HAM is accurate and effective when it is applied to the perturbed PDEs. 6. A Statistical Mechanics Approach to Approximate Analytical Bootstrap Averages DEFF Research Database (Denmark) Malzahn, Dorthe; Opper, Manfred 2003-01-01 We apply the replica method of Statistical Physics combined with a variational method to the approximate analytical computation of bootstrap averages for estimating the generalization error. We demonstrate our approach on regression with Gaussian processes and compare our results with averages ob...... obtained by Monte-Carlo sampling.......We apply the replica method of Statistical Physics combined with a variational method to the approximate analytical computation of bootstrap averages for estimating the generalization error. We demonstrate our approach on regression with Gaussian processes and compare our results with averages... 7. ANALYTIC SOLUTIONS OF MATRIX RICCATI EQUATIONS WITH ANALYTIC COEFFICIENTS NARCIS (Netherlands) Curtain, Ruth; Rodman, Leiba 2010-01-01 For matrix Riccati equations of platoon-type systems and of systems arising from PDEs, assuming the coefficients are analytic or rational functions in a suitable domain, analyticity of the stabilizing solution is proved under various hypotheses. General results on analytic behavior of stabilizing so 8. Toward making the mean spherical approximation of primitive model electrolytes analytic: an analytic approximation of the MSA screening parameter. Science.gov (United States) Gillespie, Dirk 2011-01-28 The mean spherical approximation (MSA) for the primitive model of electrolytes provides reasonable estimates of thermodynamic quantities such as the excess chemical potential and screening length. It is especially widely used because of its explicit formulas so that numerically solving equations is minimized. As originally formulated, the MSA screening parameter Γ (akin to the reciprocal of the Debye screening length) does not have an explicit analytic formula; an equation for Γ must be solved numerically. Here, an analytic approximation for Γ is presented whose relative error is generally ≲10(-5). If more accuracy is desired, one step of an iterative procedure (which also produces an explicit formula for Γ) is shown to give relative errors within machine precision in many cases. Even when ion diameter ratios are ∼10 and ion valences are ∼10, the relative error for the analytic approximation is still ≲10(-3) and for the single iterative substitution it is ≲10(-9). 9. Analytical approximations for stick-slip vibration amplitudes DEFF Research Database (Denmark) Thomsen, Jon Juel; Fidlin, A. 2003-01-01 The classical "mass-on-moving-belt" model for describing friction-induced vibrations is considered, with a friction law describing friction forces that first decreases and then increases smoothly with relative interface speed. Approximate analytical expressions are derived for the conditions... 10. Analytic Approximation of Energy Resolution in Cascaded Gaseous Detectors Directory of Open Access Journals (Sweden) Dezső Varga 2016-01-01 Full Text Available An approximate formula has been derived for gain fluctuations in cascaded gaseous detectors such as Gas Electron Multipliers (GEMs, based on the assumption that the charge collection, avalanche formation, and extraction steps are independent cascaded processes. In order to test the approximation experimentally, a setup involving a standard GEM layer has been constructed to measure the energy resolution for 5.9 keV gamma particles. The formula reasonably traces both the charge collection and the extraction process dependence of the energy resolution. Such analytic approximation for gain fluctuations can be applied to multi-GEM detectors where it aids the interpretation of measurements as well as simulations. 11. Analytic approximation of energy resolution in cascaded gaseous detectors CERN Document Server Varga, Dezső 2016-01-01 An approximate formula has been derived for gain fluctuations in cascaded gaseous detectors such as GEM-s, based on the assumption that the charge collection, avalanche formation and extraction steps are independent cascaded processes. In order to test the approximation experimentally, a setup involving a standard GEM layer has been constructed to measure the energy resolution for 5.9 keV gamma particles. The formula reasonably traces both the charge collection as well as the extraction process dependence of the energy resolution. Such analytic approximation for gain fluctuations can be applied to multi-GEM detectors where it aids the interpretation of measurements as well as simulations. 12. Comprehensive Analytical Solution for Linkages Approximating A 3-Order Osculating Straight Line%实现三阶密切直线导路机构解析全解 Institute of Scientific and Technical Information of China (English) 钱卫香 2013-01-01 For given frame and one of its points on the prescribed straight line and its orientation, the angle of one side link is selected as design parameter for solving the difficult optimal search problem. Based on Euler-Savary equation and curvature-stagnation point equation, all of straight line guidance mechanisms with Ball point on the coupler curve are obtained by the analytical method. The visualization of related properties of comprehensive solutions are realized to guide designers to search optimization easilly. After imposing kinematic constraints, the feasible mechanism regions can be computed. The method proposed in this paper is applied to choose the optimal design scheme for linkages approximating a 3-order osculating straight line. The results of synthesis examples verify the correctness and effectiveness of the proposed formulas and method.%对于给定机架和欲逼近直线上的点及方向角,为解决寻优难问题,选取一个连架杆方位角为设计参数,基于Euler-Savary方程和曲率-驻点曲线方程,采用解析法求解含鲍尔点直线机构全解,并实现全部机构尺寸和直线性能的图形可视化.施加运动学约束,计算可行解区间,为具有三阶密切直线导路机构的最优运动设计提供一种先进有效的设计方法,最后通过设计示例验证公式与分析方法的正确性和可行性. 13. Analytical Solution of the Time Fractional Fokker-Planck Equation Directory of Open Access Journals (Sweden) Sutradhar T. 2014-05-01 Full Text Available A nonperturbative approximate analytic solution is derived for the time fractional Fokker-Planck (F-P equation by using Adomian’s Decomposition Method (ADM. The solution is expressed in terms of Mittag- Leffler function. The present method performs extremely well in terms of accuracy, efficiency and simplicity. 14. Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading Institute of Scientific and Technical Information of China (English) 2009-01-01 Analytical and semi-analytical solutions are presented for anisotropic functionally graded beams subject to an arbitrary load,which can be expanded in terms of sinusoidal series.For plane stress problems,the stress function is assumed to consist of two parts,one being a product of a trigonometric function of the longitudinal coordinate(x) and an undetermined function of the thickness coordinate(y),and the other a linear polynomial of x with unknown coefficients depending on y.The governing equations satisfied by these y-dependent functions are derived.The expressions for stresses,resultant forces and displacements are then deduced,with integral constants determinable from the boundary conditions.While the analytical solution is derived for the beam with material coefficients varying exponentially or in a power law along the thickness,the semi-analytical solution is sought by making use of the sub-layer approximation for the beam with an arbitrary variation of material parameters along the thickness.The present analysis is applicable to beams with various boundary conditions at the two ends.Three numerical examples are presented for validation of the theory and illustration of the effects of certain parameters. 15. Analytical solution and semi-analytical solution for anisotropic functionally graded beam subject to arbitrary loading Institute of Scientific and Technical Information of China (English) HUANG DeJin; DING Haodiang; CHEN WeiQiu 2009-01-01 Analytical and semi-analytical solutions are presented for anisotropic functionally graded beams sub-ject to an arbitrary load, which can be expanded in terms of sinusoidal series. For plane stress prob-lems, the stress function is assumed to consist of two parts, one being a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (y), and the other a linear polynomial of x with unknown coefficients depending on y. The governing equa-tions satisfied by these y-dependent functions are derived. The expressions for stresses, resultant forces and displacements are then deduced, with integral constants determinable from the boundary conditions. While the analytical solution is derived for the beam with material coefficients varying exponentially or in a power law along the thickness, the semi-analytical solution is sought by making use of the sub-layer approximation for the beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Three numerical examples are presented for validation of the theory and illustration of the effects of certain parameters. 16. Approximate Relativistic Solutions for One-Dimensional Cylindrical Coaxial Diode Institute of Scientific and Technical Information of China (English) 曾正中; 刘国治; 邵浩 2002-01-01 Two approximate analytical relativistic solutions for one-dimensional, space-chargelimited cylindrical coaxial diode are derived and utilized to compose best-fitting approximate solutions. Comparison of the best-fitting solutions with the numerical one demonstrates an error of about 11% for cathode-inside arrangement and 12% in the cathode-outside case for ratios of larger to smaller electrode radius from 1.2 to 10 and a voltage above 0.5 MV up to 5 MV. With these solutions the diode lengths for critical self-magnetic bending and for the condition under which the parapotential model validates are calculated to be longer than 1 cm up to more than 100 cm depending on voltage, radial dimensions and electrode arrangement. The influence of ion flow from the anode on the relativistic electron-only solution is numerically computed, indicating an enhancement factor of total diode current of 1.85 to 4.19 related to voltage, radial dimension and electrode arrangement. 17. Haze of surface random systems: An approximate analytic approach CERN Document Server Simonsen, Ingve; Andreassen, Erik; Ommundsen, Espen; Nord-Varhaug, Katrin 2009-01-01 Approximate analytic expressions for haze (and gloss) of Gaussian randomly rough surfaces for various types of correlation functions are derived within phase-perturbation theory. The approximations depend on the angle of incidence, polarization of the incident light, the surface roughness,$\\sigma$, and the average of the power spectrum taken over a small angular interval about the specular direction. In particular it is demonstrated that haze(gloss) increase(decrease) with$\\sigma/\\lambda$as$\\exp(-A(\\sigma/\\lambda)^2)$and decreases(increase) with$a/\\lambda$, where$a$is the correlation length of the surface roughness, in a way that depends on the specific form of the correlation function being considered. These approximations are compared to what can be obtained from a rigorous Monte Carlo simulation approach, and good agreement is found over large regions of parameter space. Some experimental results for the angular distribution of the transmitted light through polymer films, and their haze, are presen... 18. A new approximate analytical approach for dispersion relation of the nonlinear Klein-Gordon equation. Science.gov (United States) Lim, C. W.; Wu, B. S.; He, L. H. 2001-12-01 A novel approach is presented for obtaining approximate analytical expressions for the dispersion relation of periodic wavetrains in the nonlinear Klein-Gordon equation with even potential function. By coupling linearization of the governing equation with the method of harmonic balance, we establish two general analytical approximate formulas for the dispersion relation, which depends on the amplitude of the periodic wavetrain. These formulas are valid for small as well as large amplitude of the wavetrain. They are also applicable to the large amplitude regime, which the conventional perturbation method fails to provide any solution, of the nonlinear system under study. Three examples are demonstrated to illustrate the excellent approximate solutions of the proposed formulas with respect to the exact solutions of the dispersion relation. (c) 2001 American Institute of Physics. 19. ANALYTIC SOLUTION AND NUMERICAL SOLUTION TO ENDOLYMPH EQUATION USING FRACTIONAL DERIVATIVE Institute of Scientific and Technical Information of China (English) 2008-01-01 In this paper,we study the solution to the endolymph equation using the fractional derivative of arbitrary orderλ(0<λ<1).The exact analytic solution is given by using Laplace transform in terms of Mittag-Leffler functions.We then evaluate the approximate numerical solution using MATLAB. 20. Analytic anisotropic solution for holography CERN Document Server Ren, Jie 2016-01-01 An exact solution to Einstein's equations for holographic models is presented and studied. The IR geometry has a timelike cousin of the Kasner singularity, which is the less generic case of the BKL (Belinski-Khalatnikov-Lifshitz) singularity, and the UV is asymptotically AdS. This solution describes a holographic RG flow between them. The solution's appearance is an interpolation between the planar AdS black hole and the AdS soliton. The causality constraint is always satisfied. The boundary condition for the current-current correlation function and the Laplacian in the IR is examined in detail. There is no infalling wave in the IR, but instead, there is a normalizable solution in the IR. In a special case, a hyperscaling-violating geometry is obtained after a dimension reduction. 1. Analytical Solutions for Beams Passing Apertures with Sharp Boundaries CERN Document Server Luz, Eitam; Malomed, Boris A 2016-01-01 An approximation is elaborated for the paraxial propagation of diffracted beams, with both one- and two-dimensional cross sections, which are released from apertures with sharp boundaries. The approximation applies to any beam under the condition that the thickness of its edges is much smaller than any other length scale in the beam's initial profile. The approximation can be easily generalized for any beam whose initial profile has several sharp features. Therefore, this method can be used as a tool to investigate the diffraction of beams on complex obstacles. The analytical results are compared to numerical solutions and experimental findings, which demonstrates high accuracy of the approximation. For an initially uniform field confined by sharp boundaries, this solution becomes exact for any propagation distance and any sharpness of the edges. Thus, it can be used as an efficient tool to represent the beams, produced by series of slits with a complex structure, by a simple but exact analytical solution. 2. Construction and use of numerical-analytical approximating functions Science.gov (United States) Serazutdinov, M. N. 2016-11-01 The article goes over the methodology of constructing numerical-analytical approximating functions, satisfying the given boundary conditions for the function of its derivatives in the circuit areas of various shapes. The methodology is based on presenting the unknown function as a series in a complete set of functions that do not satisfy the given boundary conditions on the contour of the area, but additionally numerically defined near the contour to satisfy the boundary conditions. The additional definition of the functions near the area contour is performed numerically based on finite-difference relations. The main advantage of the stated method is the ability to build a relatively simple approximating functions satisfying the given boundary conditions on the boundary of complex shaped areas. The examples of applying the described method for solving the boundary value problem of a plate of different shapes. The possibility of using numerical-analytical functions for solving boundary value problems that contain higher derivatives up to fourth order is shown. 3. ANALYTICAL SOLUTION OF NONLINEAR BAROTROPIC VORTICITY EQUATION Institute of Scientific and Technical Information of China (English) WANG Yue-peng; SHI Wei-hui 2008-01-01 The stability of nonlinear barotropic vorticity equation was proved. The necessary and sufficient conditions for the initial value problem to be well-posed were presented. Under the conditions of well-posedness, the corresponding analytical solution was also gained. 4. The exact renormalization group and approximation solutions CERN Document Server Morris, T R 1994-01-01 We investigate the structure of Polchinski's formulation of the flow equations for the continuum Wilson effective action. Reinterpretations in terms of I.R. cutoff greens functions are given. A promising non-perturbative approximation scheme is derived by carefully taking the sharp cutoff limit and expanding in irrelevancy' of operators. We illustrate with two simple models of four dimensional$\\lambda \\varphi^4$theory: the cactus approximation, and a model incorporating the first irrelevant correction to the renormalized coupling. The qualitative and quantitative behaviour give confidence in a fuller use of this method for obtaining accurate results. 5. Fast, Approximate Solutions for 1D Multicomponent Gas Injection Problems DEFF Research Database (Denmark) Jessen, Kristian; Wang, Yun; Ermakov, Pavel 2001-01-01 initial and injection compositions (Riemann problems). For fully self-sharpening systems, in which all key tie lines are connected by shocks, the analytical solutions obtained are rigorously accurate, while for systems in which some key tie lines are connected by spreading waves, the analytical solutions... 6. Analytic solutions of nonlinear Cournot duopoly game Directory of Open Access Journals (Sweden) Akio Matsumoto 2005-01-01 Full Text Available We construct a Cournot duopoly model with production externality in which reaction functions are unimodal. We consider the case of a Cournot model which has a stable equilibrium point. Then we show the existence of analytic solutions of the model. Moreover, we seek general solutions of the model in the form of nonlinear second-order difference equation. 7. A Simple Analytic Solution for Tachyon Condensation CERN Document Server Erler, Theodore 2009-01-01 In this paper we present a new and simple analytic solution for tachyon condensation in open bosonic string field theory. Unlike the B_0 gauge solution, which requires a carefully regulated discrete sum of wedge states subtracted against a mysterious "phantom" counter term, this new solution involves a continuous integral of wedge states, and no regularization or phantom term is necessary. Moreover, we can evaluate the action and prove Sen's conjecture in a mere few lines of calculation. 8. Helical tractor beam: analytical solution of Rayleigh particle dynamics. Science.gov (United States) Carretero, Luis; Acebal, Pablo; Garcia, Celia; Blaya, Salvador 2015-08-10 We analyze particle dynamics in an optical force field generated by helical tractor beams obtained by the interference of a cylindrical beam with a topological charge and a co-propagating temporally de-phased plane wave. We show that, for standard experimental conditions, it is possible to obtain analytical solutions for the trajectories of particles in such force field by using of some approximations. These solutions show that, in contrast to other tractor beams described before, the intensity becomes a key parameter for the control of particle trajectories. Therefore, by tuning the intensity value the particle can describe helical trajectories upstream and downstream, a circular trajectory in a fixed plane, or a linear displacement in the propagation direction. The approximated analytical solutions show good agreement to the corresponding numerical solutions of the exact dynamical differential equations. 9. Analytic continuation by averaging Padé approximants Science.gov (United States) Schött, Johan; Locht, Inka L. M.; Lundin, Elin; Grânäs, Oscar; Eriksson, Olle; Di Marco, Igor 2016-02-01 The ill-posed analytic continuation problem for Green's functions and self-energies is investigated by revisiting the Padé approximants technique. We propose to remedy the well-known problems of the Padé approximants by performing an average of several continuations, obtained by varying the number of fitted input points and Padé coefficients independently. The suggested approach is then applied to several test cases, including Sm and Pr atomic self-energies, the Green's functions of the Hubbard model for a Bethe lattice and of the Haldane model for a nanoribbon, as well as two special test functions. The sensitivity to numerical noise and the dependence on the precision of the numerical libraries are analyzed in detail. The present approach is compared to a number of other techniques, i.e., the nonnegative least-squares method, the nonnegative Tikhonov method, and the maximum entropy method, and is shown to perform well for the chosen test cases. This conclusion holds even when the noise on the input data is increased to reach values typical for quantum Monte Carlo simulations. The ability of the algorithm to resolve fine structures is finally illustrated for two relevant test functions. 10. Approximate analytical solution of the Dirac equation for pseudospin symmetry with modified Po schl-Teller potential and trigonometric Scarf II non-central potential using asymptotic iteration method Science.gov (United States) Pratiwi, B. N.; Suparmi, A.; Cari, C.; Husein, A. S.; Yunianto, M. 2016-08-01 We apllied asymptotic iteration method (AIM) to obtain the analytical solution of the Dirac equation in case exact pseudospin symmetry in the presence of modified Pcischl- Teller potential and trigonometric Scarf II non-central potential. The Dirac equation was solved by variables separation into one dimensional Dirac equation, the radial part and angular part equation. The radial and angular part equation can be reduced into hypergeometric type equation by variable substitution and wavefunction substitution and then transform it into AIM type equation to obtain relativistic energy eigenvalue and wavefunctions. Relativistic energy was calculated numerically by Matlab software. And then relativistic energy spectrum and wavefunctions were visualized by Matlab software. The results show that the increase in the radial quantum number nr causes decrease in the relativistic energy spectrum. The negative value of energy is taken due to the pseudospin symmetry limit. Several quantum wavefunctions were presented in terms of the hypergeometric functions. 11. Analytical solution methods for geodesic motion CERN Document Server Hackmann, Eva 2015-01-01 The observation of the motion of particles and light near a gravitating object is until now the only way to explore and to measure the gravitational field. In the case of exact black hole solutions of the Einstein equations the gravitational field is characterized by a small number of parameters which can be read off from the observables related to the orbits of test particles and light rays. Here we review the state of the art of analytical solutions of geodesic equations in various space--times. In particular we consider the four dimensional black hole space--times of Pleba\\'nski--Demia\\'nski type as far as the geodesic equation separates, as well as solutions in higher dimensions, and also solutions with cosmic strings. The mathematical tools used are elliptic and hyperelliptic functions. We present a list of analytic solutions which can be found in the literature. 12. Analytical solutions of the simplified Mathieu’s equation Directory of Open Access Journals (Sweden) Nicolae MARCOV 2016-03-01 Full Text Available Consider a second order differential linear periodic equation. The periodic coefficient is an approximation of the Mathieu’s coefficient. This equation is recast as a first-order homogeneous system. For this system we obtain analytical solutions in an explicit form. The first solution is a periodic function. The second solution is a sum of two functions, the first is a continuous periodic function, but the second is an oscillating function with monotone linear increasing amplitude. We give a formula to directly compute the slope of this increase, without knowing the second numeric solution. The periodic term of the second solution may be computed directly. The coefficients of fundamental matrix of the system are analytical functions. 13. Maximum likelihood molecular clock comb: analytic solutions. Science.gov (United States) Chor, Benny; Khetan, Amit; Snir, Sagi 2006-04-01 Maximum likelihood (ML) is increasingly used as an optimality criterion for selecting evolutionary trees, but finding the global optimum is a hard computational task. Because no general analytic solution is known, numeric techniques such as hill climbing or expectation maximization (EM), are used in order to find optimal parameters for a given tree. So far, analytic solutions were derived only for the simplest model--three taxa, two state characters, under a molecular clock. Four taxa rooted trees have two topologies--the fork (two subtrees with two leaves each) and the comb (one subtree with three leaves, the other with a single leaf). In a previous work, we devised a closed form analytic solution for the ML molecular clock fork. In this work, we extend the state of the art in the area of analytic solutions ML trees to the family of all four taxa trees under the molecular clock assumption. The change from the fork topology to the comb incurs a major increase in the complexity of the underlying algebraic system and requires novel techniques and approaches. We combine the ultrametric properties of molecular clock trees with the Hadamard conjugation to derive a number of topology dependent identities. Employing these identities, we substantially simplify the system of polynomial equations. We finally use tools from algebraic geometry (e.g., Gröbner bases, ideal saturation, resultants) and employ symbolic algebra software to obtain analytic solutions for the comb. We show that in contrast to the fork, the comb has no closed form solutions (expressed by radicals in the input data). In general, four taxa trees can have multiple ML points. In contrast, we can now prove that under the molecular clock assumption, the comb has a unique (local and global) ML point. (Such uniqueness was previously shown for the fork.). 14. Analytic vortex solutions on compact hyperbolic surfaces CERN Document Server Maldonado, R 2015-01-01 We construct, for the first time, Abelian-Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations. 15. Analytic vortex solutions on compact hyperbolic surfaces Science.gov (United States) Maldonado, Rafael; Manton, Nicholas S. 2015-06-01 We construct, for the first time, abelian Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations. 16. Analytic solutions of an unclassified artifact / Energy Technology Data Exchange (ETDEWEB) Trent, Bruce C. 2012-03-01 This report provides the technical detail for analytic solutions for the inner and outer profiles of the unclassified CMM Test Artifact (LANL Part Number 157Y-700373, 5/03/2001) in terms of radius and polar angle. Furthermore, analytic solutions are derived for the legacy Sheffield measurement hardware, also in terms of radius and polar angle, using part coordinates, i.e., relative to the analytic profile solutions obtained. The purpose of this work is to determine the exact solution for the “cosine correction” term inherent to measurement with the Sheffield hardware. The cosine correction is required in order to interpret the actual measurements taken by the hardware in terms of an actual part definition, or “knot-point spline definition,” that typically accompanies a component drawing. Specifically, there are two portions of the problem: first an analytic solution must be obtained for any point on the part, e.g., given the radii and the straight lines that define the part, it is required to find an exact solution for the inner and outer profile for any arbitrary polar angle. Next, the problem of the inspection of this part must be solved, i.e., given an arbitrary sphere (representing the inspection hardware) that comes in contact with the part (inner and outer profiles) at any arbitrary polar angle, it is required to determine the exact location of that intersection. This is trivial for the case of concentric circles. In the present case, however, the spherical portion of the profiles is offset from the defined center of the part, making the analysis nontrivial. Here, a simultaneous solution of the part profiles and the sphere was obtained. 17. Approximate Solution of Time-Fractional Advection-Dispersion Equation via Fractional Variational Iteration Method Directory of Open Access Journals (Sweden) Birol İbiş 2014-01-01 Full Text Available This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE involving Jumarie’s modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM. FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs. 18. Analytical And Numerical Approximation of Effective Diffusivities in The Cytoplasm of Biological Cells CERN Document Server Hanke, Michael 2010-01-01 The simulation of the metabolism in mammalian cells becomes a severe problem if spatial distributions must be taken into account. Especially the cytoplasm has a very complex geometric structure which cannot be handled by standard discretization techniques. In the present paper we propose a homogenization technique for computing effective diffusion constants. This is accomplished by using a two-step strategy. The first step consists of an analytic homogenization from the smallest to an intermediate scale. The homogenization error is estimated by comparing the analytic diffusion constant with a numerical estimate obtained by using real cell geometries. The second step consists of a random homogenization. Since no analytical solution is known to this homogenization problem, a numerical approximation algorithm is proposed. Although rather expensive this algorithm provides a reasonable estimate of the homogenized diffusion constant. 19. Approximate solutions for fractured wells producing layered reservoirs Energy Technology Data Exchange (ETDEWEB) Bennett, C.O.; Reynolds, A.C.; Raghavan, R. 1983-01-01 New analytical solutions for the response at a well intercepting a layered reservoir are derived. The well is assumed to produce at a constant rate or at a constant pressure. Reservoir systems with and without interlayer communication were examined. The utility of these solutions is documented. An increased physical understanding of fractured wells in layered reservoirs was obtained from these solutions. The influence of vertical variations in fracture conductivity is considered also. 15 references. 20. Approximate solution for frequency synchronization in a finite-size Kuramoto model. Science.gov (United States) Wang, Chengwei; Rubido, Nicolás; Grebogi, Celso; Baptista, Murilo S 2015-12-01 Scientists have been considering the Kuramoto model to understand the mechanism behind the appearance of collective behavior, such as frequency synchronization (FS) as a paradigm, in real-world networks with a finite number of oscillators. A major current challenge is to obtain an analytical solution for the phase angles. Here, we provide an approximate analytical solution for this problem by deriving a master solution for the finite-size Kuramoto model, with arbitrary finite-variance distribution of the natural frequencies of the oscillators. The master solution embodies all particular solutions of the finite-size Kuramoto model for any frequency distribution and coupling strength larger than the critical one. Furthermore, we present a criterion to determine the stability of the FS solution. This allows one to analytically infer the relationship between the physical parameters and the stable behavior of networks. 1. N Level System with RWA and Analytical Solutions Revisited CERN Document Server Fujii, K; Kato, R; Wada, Y; Fujii, Kazuyuki; Higashida, Kyoko; Kato, Ryosuke; Wada, Yukako 2003-01-01 In this paper we consider a model of an atom with n energy levels interacting with n(n-1)/2 external (laser) fields which is a natural extension of two level system, and assume the rotating wave approximation (RWA) from the beginning. We revisit some construction of analytical solutions (which correspond to Rabi oscillations) of the model in the general case and examine it in detail in the case of three level system. 2. Hamilton's Principle and Approximate Solutions to Problems in Classical Mechanics Science.gov (United States) Schlitt, D. W. 1977-01-01 Shows how to use the Ritz method for obtaining approximate solutions to problems expressed in variational form directly from the variational equation. Application of this method to classical mechanics is given. (MLH) 3. Approximate Solutions of Klein-Gordon Equation with Kratzer Potential Directory of Open Access Journals (Sweden) H. Hassanabadi 2011-01-01 Full Text Available Approximate solutions of the D-dimensional Klein-Gordon equation are obtained for the scalar and vector general Kratzer potential for any l by using the ansatz method. The energy behavior is numerically discussed. 4. A New Approximate Fundamental Solution for Orthotropic Plate Institute of Scientific and Technical Information of China (English) WU Pei-liang; L(U) Yan-ping 2002-01-01 A weight double trigonometric series is presented as an approximate fundamental solution for orthotropic plate.Integral equation of orthotropic plate bending is solved by a new method, which only needs one basic boundary integral Eq., puts one fictitious boundary outside plate domain. Examples show that the approximate fundamental solution and solving method proposed in this paper are simple, reliable and quite precise. And they are applicable for various boundary conditions. 5. Analytical approximations for a conservative nonlinear singular oscillator in plasma physics Directory of Open Access Journals (Sweden) A. Mirzabeigy 2012-10-01 Full Text Available A modified variational approach and the coupled homotopy perturbation method with variational formulation are exerted to obtain periodic solutions of a conservative nonlinear singular oscillator in plasma physics. The frequency–amplitude relations for the oscillator which the restoring force is inversely proportional to the dependent variable are achieved analytically. The approximate frequency obtained using the coupled method is more accurate than the modified variational approach and ones obtained using other approximate methods and the discrepancy between the approximate frequency using this coupled method and the exact one is lower than 0.31% for the whole range of values of oscillation amplitude. The coupled method provides a very good accuracy and is a promising technique to a lot of practical engineering and physical problems. 6. An Approximate Solution to the Equation of Motion for Large-Angle Oscillations of the Simple Pendulum with Initial Velocity Science.gov (United States) Johannessen, Kim 2010-01-01 An analytic approximation of the solution to the differential equation describing the oscillations of a simple pendulum at large angles and with initial velocity is discussed. In the derivation, a sinusoidal approximation has been applied, and an analytic formula for the large-angle period of the simple pendulum is obtained, which also includes… 7. Approximated and User Steerable tSNE for Progressive Visual Analytics. Science.gov (United States) Pezzotti, Nicola; Lelieveldt, Boudewijn; van der Maaten, Laurens; Hollt, Thomas; Eisemann, Elmar; Vilanova, Anna 2016-05-19 Progressive Visual Analytics aims at improving the interactivity in existing analytics techniques by means of visualization as well as interaction with intermediate results. One key method for data analysis is dimensionality reduction, for example, to produce 2D embeddings that can be visualized and analyzed efficiently. t-Distributed Stochastic Neighbor Embedding (tSNE) is a well-suited technique for the visualization of high-dimensional data. tSNE can create meaningful intermediate results but suffers from a slow initialization that constrains its application in Progressive Visual Analytics. We introduce a controllable tSNE approximation (A-tSNE), which trades off speed and accuracy, to enable interactive data exploration. We offer real-time visualization techniques, including a density-based solution and a Magic Lens to inspect the degree of approximation. With this feedback, the user can decide on local refinements and steer the approximation level during the analysis. We demonstrate our technique with several datasets, in a real-world research scenario and for the real-time analysis of high-dimensional streams to illustrate its effectiveness for interactive data analysis. 8. Approximating solutions of neutral stochastic evolution equations with jumps Institute of Scientific and Technical Information of China (English) 2009-01-01 In this paper, we establish existence and uniqueness of the mild solutions to a class of neutral stochastic evolution equations driven by Poisson random measures in some Hilbert space. Moreover, we adopt the Faedo-Galerkin scheme to approximate the solutions. 9. Exact, approximate solutions and error bounds for coupled implicit systems of partial differential equations Directory of Open Access Journals (Sweden) Lucas Jódar 1992-01-01 Full Text Available In this paper coupled implicit initial-boundary value systems of second order partial differential equations are considered. Given a finite domain and an admissible error ϵ an analytic approximate solution whose error is upper bounded by ϵ in the given domain is constructed in terms of the data. 10. Approximate polynomial solutions for Riccati differential equations using the squared remains minimization method Science.gov (United States) Bota, C.; Cǎruntu, B.; Bundǎu, O. 2013-10-01 In this paper we applied the Squared Remainder Minimization Method (SRMM) to find analytic approximate polynomial solutions for Riccati differential equations. Two examples are included to demonstrated the validity and applicability of the method. The results are compared to those obtained by other methods. 11. Phase change material solidification in a finned cylindrical shell thermal energy storage: An approximate analytical approach Directory of Open Access Journals (Sweden) Mosaffa Amirhossein 2013-01-01 Full Text Available Results are reported of an investigation of the solidification of a phase change material (PCM in a cylindrical shell thermal energy storage with radial internal fins. An approximate analytical solution is presented for two cases. In case 1, the inner wall is kept at a constant temperature and, in case 2, a constant heat flux is imposed on the inner wall. In both cases, the outer wall is insulated. The results are compared to those for a numerical approach based on an enthalpy method. The results show that the analytical model satisfactory estimates the solid-liquid interface. In addition, a comparative study is reported of the solidified fraction of encapsulated PCM for different geometric configurations of finned storage having the same volume and surface area of heat transfer. 12. Analytic number theory, approximation theory, and special functions in honor of Hari M. Srivastava CERN Document Server Rassias, Michael 2014-01-01 This book, in honor of Hari M. Srivastava, discusses essential developments in mathematical research in a variety of problems. It contains thirty-five articles, written by eminent scientists from the international mathematical community, including both research and survey works. Subjects covered include analytic number theory, combinatorics, special sequences of numbers and polynomials, analytic inequalities and applications, approximation of functions and quadratures, orthogonality, and special and complex functions. The mathematical results and open problems discussed in this book are presented in a simple and self-contained manner. The book contains an overview of old and new results, methods, and theories toward the solution of longstanding problems in a wide scientific field, as well as new results in rapidly progressing areas of research. The book will be useful for researchers and graduate students in the fields of mathematics, physics, and other computational and applied sciences. 13. Analytic interatomic forces in the random phase approximation CERN Document Server Ramberger, Benjamin; Kresse, Georg 2016-01-01 We discuss that in the random phase approximation (RPA) the first derivative of the energy with respect to the Green's function is the self-energy in the GW approximation. This relationship allows to derive compact equations for the RPA interatomic forces. We also show that position dependent overlap operators are elegantly incorporated in the present framework. The RPA force equations have been implemented in the projector augmented wave formalism, and we present illustrative applications, including ab initio molecular dynamics simulations, the calculation of phonon dispersion relations for diamond and graphite, as well as structural relaxations for water on boron nitride. The present derivation establishes a concise framework for forces within perturbative approaches and is also applicable to more involved approximations for the correlation energy. 14. Analytical Evaluation of Beam Deformation Problem Using Approximate Methods DEFF Research Database (Denmark) Barari, Amin; Kimiaeifar, A.; Domairry, G. 2010-01-01 The beam deformation equation has very wide applications in structural engineering. As a differential equation, it has its own problem concerning existence, uniqueness and methods of solutions. Often, original forms of governing differential equations used in engineering problems are simplified... 15. Analytical Solution for Isentropic Flows in Solids Science.gov (United States) Heuzé, Olivier 2009-12-01 In the XIXth century, Riemann gave the equations system and the exact solution for the isentropic flows in the case of the ideal gas. But to our knowledge, nothing has been done to apply it to condensed media. Many materials of practical interest, for instance metals, obey to the linear law D = c+s u, where D is the shock velocity, u the particle velocity, and c and s properties of the material. We notice that s is strongly linked to the fundamental derivative. This means that the assumption of constant fundamental derivative is useful in this case, as it was with the isentropic gamma in the Riemann solution. Then we can apply the exact Riemann solution for these materials. Although the use of the hypergeometric function is complicated in this case, we obtain a very good approximation with the development in power series. 16. Analytic Solution to Nonlinear Dynamical System of Dragon Washbasin Institute of Scientific and Technical Information of China (English) 贾启芬; 李芳; 于雯; 刘习军; 王大钧 2004-01-01 Based on phase-plane orbit analysis, the mathematical model of piecewise-smooth systems of multi-degree-of-freedom under the mode coordinate is established. Approximate analytical solution under the physical coordinate of multi-degree-of-freedom self-excited vibration induced by dry friction of piecewise-smooth nonlinear systems is derived by means of average method, the results of which agree with those of the numerical solution. An effective and reliable analytical method investigating piecewise-smooth nonlinear systems of multi-degree-of-freedom has been given. Furthermore, this paper qualitatively analyses the curves about stationary amplitude versus rubbing velocity of hands and versus natural frequency of hands, and about angular frequency versus rubbing velocity of hands. The results provide a theoretical basis for identifying parameters of the system and the analysis of steady region. 17. Analytical approximations for the amplitude and period of a relaxation oscillator Directory of Open Access Journals (Sweden) Golkhou Vahid 2009-01-01 Full Text Available Abstract Background Analysis and design of complex systems benefit from mathematically tractable models, which are often derived by approximating a nonlinear system with an effective equivalent linear system. Biological oscillators with coupled positive and negative feedback loops, termed hysteresis or relaxation oscillators, are an important class of nonlinear systems and have been the subject of comprehensive computational studies. Analytical approximations have identified criteria for sustained oscillations, but have not linked the observed period and phase to compact formulas involving underlying molecular parameters. Results We present, to our knowledge, the first analytical expressions for the period and amplitude of a classic model for the animal circadian clock oscillator. These compact expressions are in good agreement with numerical solutions of corresponding continuous ODEs and for stochastic simulations executed at literature parameter values. The formulas are shown to be useful by permitting quick comparisons relative to a negative-feedback represillator oscillator for noise (10× less sensitive to protein decay rates, efficiency (2× more efficient, and dynamic range (30 to 60 decibel increase. The dynamic range is enhanced at its lower end by a new concentration scale defined by the crossing point of the activator and repressor, rather than from a steady-state expression level. Conclusion Analytical expressions for oscillator dynamics provide a physical understanding for the observations from numerical simulations and suggest additional properties not readily apparent or as yet unexplored. The methods described here may be applied to other nonlinear oscillator designs and biological circuits. 18. Analytic solutions for unconfined groundwater flow over a stepped base Science.gov (United States) Fitts, Charles R.; Strack, Otto D. L. 1996-03-01 Two new exact solutions are presented for uniform unconfined groundwater flow over a stepped base; one for a step down in the direction of flow, the other for a step up in the direction of flow. These are two-dimensional solutions of Laplace's equation in the vertical plane, and are derived using the hodograph method and conformal mappings on Riemann surfaces. The exact solutions are compared with approximate one-dimensional solutions which neglect the resistance to vertical flow. For small horizontal hydraulic gradients typical of regional groundwater flow, little error is introduced by neglecting the vertical resistance to flow. This conclusion may be extended to two-dimensional analytical models in the horizontal plane, which neglect the vertical resistance to flow and treat the aquifer base as a series of flat steps. 19. Generating exact solutions to Einstein's equation using linearized approximations Science.gov (United States) Harte, Abraham I.; Vines, Justin 2016-10-01 We show that certain solutions to the linearized Einstein equation can—by the application of a particular type of linearized gauge transformation—be straightforwardly transformed into solutions of the exact Einstein equation. In cases with nontrivial matter content, the exact stress-energy tensor of the transformed metric has the same eigenvalues and eigenvectors as the linearized stress-energy tensor of the initial approximation. When our gauge exists, the tensorial structure of transformed metric perturbations identically eliminates all nonlinearities in Einstein's equation. As examples, we derive the exact Kerr and gravitational plane wave metrics from standard harmonic-gauge approximations. 20. Generating exact solutions to Einstein's equation using linearized approximations CERN Document Server Harte, Abraham I 2016-01-01 We show that certain solutions to the linearized Einstein equation can---by the application of a particular type of linearized gauge transformation---be straightforwardly transformed into solutions of the exact Einstein equation. In cases with nontrivial matter content, the exact stress-energy tensor of the transformed metric has the same eigenvalues and eigenvectors as the linearized stress-energy tensor of the initial approximation. When our gauge exists, the tensorial structure of transformed metric perturbations identically eliminates all nonlinearities in Einstein's equation. As examples, we derive the exact Kerr and gravitational plane wave metrics from standard harmonic-gauge approximations. 1. On Analytical Solutions to Beam-Hardening Science.gov (United States) Rigaud, G. 2017-01-01 When polychromatic X-rays propagate through a material, for instance in computerized tomography (CT), low energy photons are more attenuated resulting in a "harder" beam. The beam-hardening phenomenon breaks the monochromatic radiation model based on the Radon transform giving rise to artifacts in CT reconstructions to the detriment of visual inspection and automated segmentation algorithms. We propose first a simplified analytic representation for the beam-hardening. Besides providing a general understanding of the phenomenon, this model proposes to invert the beam-hardening effect for homogeneous objects leading to classical monochromatic data. For heterogeneous objects, no analytical reconstruction of the density can be derived without more prior information. However, the beam-hardening is shown to be a smooth operation on the data and thus to preserve the encoding of the singularities of the attenuation map within the data. A microlocal analysis encourages the use of contour extraction methods to solve partially the beam-hardening effect even for heterogeneous objects. The application of both methods, exact analytical solution for homogeneous objects and feature extraction for heterogeneous ones, on real data demonstrates their relevancy and efficiency. 2. Analytic regularity and collocation approximation for elliptic PDEs with random domain deformations KAUST Repository Castrillon, Julio 2016-03-02 In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by N random variables. The elliptic problem is remapped onto a corresponding PDE with a fixed deterministic domain. We show that the solution can be analytically extended to a well defined region in CN with respect to the random variables. A sparse grid stochastic collocation method is then used to compute the mean and variance of the QoI. Finally, convergence rates for the mean and variance of the QoI are derived and compared to those obtained in numerical experiments. 3. Analytical Models of Exoplanetary Atmospheres. II. Radiative Transfer via the Two-Stream Approximation CERN Document Server Heng, Kevin; Lee, Jaemin 2014-01-01 We present a comprehensive analytical study of radiative transfer using the method of moments and include the effects of non-isotropic scattering in the coherent limit. Within this unified formalism, we derive the governing equations and solutions describing two-stream radiative transfer (which approximates the passage of radiation as a pair of outgoing and incoming fluxes), flux-limited diffusion (which describes radiative transfer in the deep interior) and solutions for the temperature-pressure profiles. Generally, the problem is mathematically under-determined unless a set of closures (Eddington coefficients) is specified. We demonstrate that the hemispheric (or hemi-isotropic) closure naturally derives from the radiative transfer equation if energy conservation is obeyed, while the Eddington closure produces spurious enhancements of both reflected light and thermal emission. We further demonstrate that traditional non-isothermal treatments of each atmospheric layer lead to unphysical contributions to the ... 4. Dataset concerning the analytical approximation of the Ae3 temperature Directory of Open Access Journals (Sweden) B.L. Ennis 2017-02-01 The dataset includes the terms of the function and the values for the polynomial coefficients for major alloying elements in steel. A short description of the approximation method used to derive and validate the coefficients has also been included. For discussion and application of this model, please refer to the full length article entitled “The role of aluminium in chemical and phase segregation in a TRIP-assisted dual phase steel” 10.1016/j.actamat.2016.05.046 (Ennis et al., 2016 [1]. 5. Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition Directory of Open Access Journals (Sweden) Deniz Agirseven 2012-01-01 Full Text Available Finite difference and homotopy analysis methods are used for the approximate solution of the initial-boundary value problem for the delay parabolic partial differential equation with the Dirichlet condition. The convergence estimates for the solution of first and second orders of difference schemes in Hölder norms are obtained. A procedure of modified Gauss elimination method is used for the solution of these difference schemes. Homotopy analysis method is applied. Comparison of finite difference and homotopy analysis methods is given on the problem. 6. An accurate two-phase approximate solution to the acute viral infection model Energy Technology Data Exchange (ETDEWEB) Perelson, Alan S [Los Alamos National Laboratory 2009-01-01 During an acute viral infection, virus levels rise, reach a peak and then decline. Data and numerical solutions suggest the growth and decay phases are linear on a log scale. While viral dynamic models are typically nonlinear with analytical solutions difficult to obtain, the exponential nature of the solutions suggests approximations can be found. We derive a two-phase approximate solution to the target cell limited influenza model and illustrate the accuracy using data and previously established parameter values of six patients infected with influenza A. For one patient, the subsequent fall in virus concentration was not consistent with our predictions during the decay phase and an alternate approximation is derived. We find expressions for the rate and length of initial viral growth in terms of the parameters, the extent each parameter is involved in viral peaks, and the single parameter responsible for virus decay. We discuss applications of this analysis in antiviral treatments and investigating host and virus heterogeneities. 7. Explicit solutions of the radiative transport equation in the P{sub 3} approximation Energy Technology Data Exchange (ETDEWEB) Liemert, André, E-mail: [email protected]; Kienle, Alwin [Institut für Lasertechnologien in der Medizin und Meßtechnik an der Universität Ulm, Helmholtzstr.12, Ulm D-89081 (Germany) 2014-11-01 Purpose: Explicit solutions of the monoenergetic radiative transport equation in the P{sub 3} approximation have been derived which can be evaluated with nearly the same computational effort as needed for solving the standard diffusion equation (DE). In detail, the authors considered the important case of a semi-infinite medium which is illuminated by a collimated beam of light. Methods: A combination of the classic spherical harmonics method and the recently developed method of rotated reference frames is used for solving the P{sub 3} equations in closed form. Results: The derived solutions are illustrated and compared to exact solutions of the radiative transport equation obtained via the Monte Carlo (MC) method as well as with other approximated analytical solutions. It is shown that for the considered cases which are relevant for biomedical optics applications, the P{sub 3} approximation is close to the exact solution of the radiative transport equation. Conclusions: The authors derived exact analytical solutions of the P{sub 3} equations under consideration of boundary conditions for defining a semi-infinite medium. The good agreement to Monte Carlo simulations in the investigated domains, for example, in the steady-state and time domains, as well as the short evaluation time needed suggests that the derived equations can replace the often applied solutions of the diffusion equation for the homogeneous semi-infinite medium. 8. Approximate solutions of non-linear circular orbit relative motion in curvilinear coordinates Science.gov (United States) Bombardelli, Claudio; Gonzalo, Juan Luis; Roa, Javier 2017-01-01 A compact, time-explicit, approximate solution of the highly non-linear relative motion in curvilinear coordinates is provided under the assumption of circular orbit for the chief spacecraft. The rather compact, three-dimensional solution is obtained by algebraic manipulation of the individual Keplerian motions in curvilinear, rather than Cartesian coordinates, and provides analytical expressions for the secular, constant and periodic terms of each coordinate as a function of the initial relative motion conditions or relative orbital elements. Numerical test cases are conducted to show that the approximate solution can be effectively employed to extend the classical linear Clohessy-Wiltshire solution to include non-linear relative motion without significant loss of accuracy up to a limit of 0.4-0.45 in eccentricity and 40-45° in relative inclination for the follower. A very simple, quadratic extension of the classical Clohessy-Wiltshire solution in curvilinear coordinates is also presented. 9. Approximate solutions of non-linear circular orbit relative motion in curvilinear coordinates Science.gov (United States) Bombardelli, Claudio; Gonzalo, Juan Luis; Roa, Javier 2016-07-01 A compact, time-explicit, approximate solution of the highly non-linear relative motion in curvilinear coordinates is provided under the assumption of circular orbit for the chief spacecraft. The rather compact, three-dimensional solution is obtained by algebraic manipulation of the individual Keplerian motions in curvilinear, rather than Cartesian coordinates, and provides analytical expressions for the secular, constant and periodic terms of each coordinate as a function of the initial relative motion conditions or relative orbital elements. Numerical test cases are conducted to show that the approximate solution can be effectively employed to extend the classical linear Clohessy-Wiltshire solution to include non-linear relative motion without significant loss of accuracy up to a limit of 0.4-0.45 in eccentricity and 40-45° in relative inclination for the follower. A very simple, quadratic extension of the classical Clohessy-Wiltshire solution in curvilinear coordinates is also presented. 10. Analytical Solution of Multicompartment Solute Kinetics for Hemodialysis Directory of Open Access Journals (Sweden) Przemysław Korohoda 2013-01-01 Full Text Available Objective. To provide an exact solution for variable-volume multicompartment kinetic models with linear volume change, and to apply this solution to a 4-compartment diffusion-adjusted regional blood flow model for both urea and creatinine kinetics in hemodialysis. Methods. A matrix-based approach applicable to linear models encompassing any number of compartments is presented. The procedure requires the inversion of a square matrix and the computation of its eigenvalues λ, assuming they are all distinct. This novel approach bypasses the evaluation of the definite integral to solve the inhomogeneous ordinary differential equation. Results. For urea two out of four eigenvalues describing the changes of concentrations in time are about 105 times larger than the other eigenvalues indicating that the 4-compartment model essentially reduces to the 2-compartment regional blood flow model. In case of creatinine, however, the distribution of eigenvalues is more balanced (a factor of 102 between the largest and the smallest eigenvalue indicating that all four compartments contribute to creatinine kinetics in hemodialysis. Interpretation. Apart from providing an exact analytic solution for practical applications such as the identification of relevant model and treatment parameters, the matrix-based approach reveals characteristic details on model symmetry and complexity for different solutes. 11. ANALYTIC SOLUTIONS OF SYSTEMS OF FUNCTIONAL EQUATIONS Institute of Scientific and Technical Information of China (English) LiuXinhe 2003-01-01 Let r be a given positive number.Denote by D=D the closed disc in the complex plane C whose center is the origin and radius is r.For any subset K of C and any integer m ≥1,write A(Dm,K)={f|f:Dm→Kis a continuous map,and f|(Dm)*is analytic).For H∈A(Dm,C)(m≥2),f∈A(D,D)and z∈D,write ψH(f)(z)=H(z,f(z)……fm=1(x)).Suppose F,G∈A(D2n+1,C),and Hk,Kk∈A(Dk,C),k=2,……,n.In this paper,the system of functional equations {F(z,f(z),f2(ψHz(f)(z))…,fn(ψk2(g)(x))… gn(ψKn(g)(z)))=0 G(z,f(z),f2(ψH2(f)(z))…fn(ψHn(f)(z)),g(z),g2(ψk2(g)(x))…,gn(ψkn(g)(z)))=0(z∈D)is studied and some conditions for the system of equations to have a solution or a unique solution in A(D,D)×A（D，D）are given. 12. Derivation of Varying Specific Heat Gasdynamic Functions,Normal Shock Analytical Solution and its Improvements Institute of Scientific and Technical Information of China (English) TsuiChih－Ya 1992-01-01 A set of new gasdynamic functions with varying specific heat are deriveo for the first time.An original analytical solution of normal shock waves is owrked out therewith.This solution is thereafter further improved by not involving total temperature,Illustrative examples of comparison are given,including also some approximate solutions to show the orders of their errors. 13. Explicit Analytical Solutions of Coupled Fluid Flow Transfer Equation in Heterogeneous Porous Media Institute of Scientific and Technical Information of China (English) 张娜; 蔡睿贤 2002-01-01 Explicit analytical solutions are presented for the coupled fluid flow transfer equation in heterogeneous porous media. These analytical solutions are useful for their description of actual flow fields and as benchmark solutions to check the rapidly developing numerical calculations and to study various computational methods such as the discrete approximations of the governing equations and grid generation methods. In addition, some novel mathematical methods are used in the analyses. 14. Novel determination of differential-equation solutions: universal approximation method Science.gov (United States) Leephakpreeda, Thananchai 2002-09-01 In a conventional approach to numerical computation, finite difference and finite element methods are usually implemented to determine the solution of a set of differential equations (DEs). This paper presents a novel approach to solve DEs by applying the universal approximation method through an artificial intelligence utility in a simple way. In this proposed method, neural network model (NNM) and fuzzy linguistic model (FLM) are applied as universal approximators for any nonlinear continuous functions. With this outstanding capability, the solutions of DEs can be approximated by the appropriate NNM or FLM within an arbitrary accuracy. The adjustable parameters of such NNM and FLM are determined by implementing the optimization algorithm. This systematic search yields sub-optimal adjustable parameters of NNM and FLM with the satisfactory conditions and with the minimum residual errors of the governing equations subject to the constraints of boundary conditions of DEs. The simulation results are investigated for the viability of efficiently determining the solutions of the ordinary and partial nonlinear DEs. 15. Optimum Approximate Solution of Herschel-Bulkley Fluid Formula Institute of Scientific and Technical Information of China (English) XU Gui-yun; LIN Xue-dong; ZHANG Yong-zhong 2004-01-01 During calculating the fluid resistence with Herschel-Bulkley formula, the deviation is very large in some regions. In order to decrease the deviation, the optimized parameters of approximate solution are obtained through mathematic analysis and 3-D optimization calculation. In the close region of relative radius of the core flow, the continuity of deviation is determined with the limit methods. By analysis, the results indicate that the deviation in the area around the discontinuous nodes is very large; the deviation is the function of fluid parameters, approximate solution parameters and the relative radius of the core flow. While the fluid parameters keep certain, the 3-D figures of the deviation are drawn. The slice plane is used to seek the extremums of multi-peaks surface; The optimized parameters of approximate formula make the approximate formula in the regions of the certain deviation. The available area of relative radius of the core flow increases by 43.2%. It is more valuable for the calculation of flow resistance in pipe transportation. 16. Approximate solutions of common fixed-point problems CERN Document Server Zaslavski, Alexander J 2016-01-01 This book presents results on the convergence behavior of algorithms which are known as vital tools for solving convex feasibility problems and common fixed point problems. The main goal for us in dealing with a known computational error is to find what approximate solution can be obtained and how many iterates one needs to find it. According to know results, these algorithms should converge to a solution. In this exposition, these algorithms are studied, taking into account computational errors which remain consistent in practice. In this case the convergence to a solution does not take place. We show that our algorithms generate a good approximate solution if computational errors are bounded from above by a small positive constant. Beginning with an introduction, this monograph moves on to study: · dynamic string-averaging methods for common fixed point problems in a Hilbert space · dynamic string methods for common fixed point problems in a metric space · dynamic string-averaging version of the proximal... 17. Analytic solution of Hubbell's model of local community dynamics CERN Document Server McKane, A; Sole, R; Kane, Alan Mc; Alonso, David; Sole, Ricard 2003-01-01 Recent theoretical approaches to community structure and dynamics reveal that many large-scale features of community structure (such as species-rank distributions and species-area relations) can be explained by a so-called neutral model. Using this approach, species are taken to be equivalent and trophic relations are not taken into account explicitly. Here we provide a general analytic solution to the local community model of Hubbell's neutral theory of biodiversity by recasting it as an urn model i.e.a Markovian description of states and their transitions. Both stationary and time-dependent distributions are analysed. The stationary distribution -- also called the zero-sum multinomial -- is given in closed form. An approximate form for the time-dependence is obtained by using an expansion of the master equation. The temporal evolution of the approximate distribution is shown to be a good representation for the true temporal evolution for a large range of parameter values. 18. JOVIAN STRATOSPHERE AS A CHEMICAL TRANSPORT SYSTEM: BENCHMARK ANALYTICAL SOLUTIONS Energy Technology Data Exchange (ETDEWEB) Zhang Xi; Shia Runlie; Yung, Yuk L., E-mail: [email protected] [Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125 (United States) 2013-04-20 We systematically investigated the solvable analytical benchmark cases in both one- and two-dimensional (1D and 2D) chemical-advective-diffusive systems. We use the stratosphere of Jupiter as an example but the results can be applied to other planetary atmospheres and exoplanetary atmospheres. In the 1D system, we show that CH{sub 4} and C{sub 2}H{sub 6} are mainly in diffusive equilibrium, and the C{sub 2}H{sub 2} profile can be approximated by modified Bessel functions. In the 2D system in the meridional plane, analytical solutions for two typical circulation patterns are derived. Simple tracer transport modeling demonstrates that the distribution of a short-lived species (such as C{sub 2}H{sub 2}) is dominated by the local chemical sources and sinks, while that of a long-lived species (such as C{sub 2}H{sub 6}) is significantly influenced by the circulation pattern. We find that an equator-to-pole circulation could qualitatively explain the Cassini observations, but a pure diffusive transport process could not. For slowly rotating planets like the close-in extrasolar planets, the interaction between the advection by the zonal wind and chemistry might cause a phase lag between the final tracer distribution and the original source distribution. The numerical simulation results from the 2D Caltech/JPL chemistry-transport model agree well with the analytical solutions for various cases. 19. Approximate analytical scattering phase function dependent on microphysical characteristics of dust particles. Science.gov (United States) Kocifaj, Miroslav 2011-06-10 The approximate bulk-scattering phase function of a polydisperse system of dust particles is derived in an analytical form. In the theoretical solution, the particle size distribution is modeled by a modified gamma function that can satisfy various media differing in modal radii. Unlike the frequently applied power law, the modified gamma distribution shows no singularity when the particle radius approaches zero. The approximate scattering phase function is related to the parameters of the size distribution function. This is an important advantage compared to the empirical Henyey-Greenstein (HG) approximation, which is a simple function of the average cosine. However, any optimized value of average cosine of the HG function cannot provide the information on particle microphysical characteristics, such as the size distribution function. In this paper, the mapping between average cosine and the parameters of size distribution function is given by a semianalytical expression that is applicable in rapid numerical simulations on various dust populations. In particular, the modal radius and half-width can be quickly estimated using the presented formulas. 20. A solution of LIDAR problem in double scattering approximation CERN Document Server Leble, Sergey 2011-01-01 A problem of monoenergetic particles pulse reflection from half-infinite stratified medium is considered in conditions of elastic scattering with absorbtion account. The theory is based on multiple scattering series solution of Kolmogorov equation for one-particle distribution function. The analytical representation for first two terms are given in compact form for a point impulse source and cylindric symmetrical detector. Reading recent articles on the LIDAR sounding of environment (e.g. Atmospheric and Oceanic Optics (2010) 23: 389-395, Kaul, B. V.; Samokhvalov, I. V. http://www.springerlink.com/content/k3p2p3582674xt21/) one recovers standing interest to the related direct and inverse problems. A development of the result fo the case of n-fold scattering and polarization account as well as correspondent convergence series problem solution of the Kolmogorov equation will be published in nearest future. 1. On the equivalence of approximate stationary axially symmetric solutions of Einstein field equations CERN Document Server Boshkayev, Kuantay; Toktarbay, Saken; Zhami, Bakytzhan 2015-01-01 We study stationary axially symmetric solutions of the Einstein vacuum field equations that can be used to describe the gravitational field of astrophysical compact objects in the limiting case of slow rotation and slight deformation. We derive explicitly the exterior Sedrakyan-Chubaryan approximate solution, and express it in analytical form, which makes it practical in the context of astrophysical applications. In the limiting case of vanishing angular momentum, the solution reduces to the well-known Schwarzschild solution in vacuum. We demonstrate that the new solution is equivalent to the exterior Hartle-Thorne solution. We establish the mathematical equivalence between the Sedrakyan-Chubaryan, Fock-Abdildin and Hartle-Thorne formalisms. 2. An Approximate Solution for Flow between Two Disks Rotating about Distinct Axes at Different Speeds Directory of Open Access Journals (Sweden) H. Volkan Ersoy 2007-01-01 Full Text Available The flow of a linearly viscous fluid between two disks rotating about two distinct vertical axes is studied. An approximate analytical solution is obtained by taking into account the case of rotation with a small angular velocity difference. It is shown how the velocity components depend on the position, the Reynolds number, the eccentricity, the ratio of angular speeds of the disks, and the parameters satisfying the conditions u=0 and ν=0 in midplane. 3. Abrupt PN junctions: Analytical solutions under equilibrium and non-equilibrium Science.gov (United States) Khorasani, Sina 2016-08-01 We present an explicit solution of carrier and field distributions in abrupt PN junctions under equilibrium. An accurate logarithmic numerical method is implemented and results are compared to the analytical solutions. Analysis of results shows reasonable agreement with numerical solution as well as the depletion layer approximation. We discuss extensions to the asymmetric junctions. Approximate relations for differential capacitance C-V and current-voltage I-V characteristics are also found under non-zero external bias. 4. Migration of radionuclides through sorbing media analytical solutions--II Energy Technology Data Exchange (ETDEWEB) Pigford, T.H.; Chambre, P.L.; Albert, M. 1980-10-01 This report presents analytical solutions, and the results of such solutions, for the migration of radionuclides in geologic media. Volume 1 contains analytical solutions for one-dimensional equilibrium transport in infinite media and multilayered media. One-dimensional non-equilibrium transport solutions are also included. Volume 2 contains analytical solutions for transport in a one-dimensional field flow with transverse dispersion as well as transport in multi-dimensional flow. A finite element solution of the transport of radionuclides through porous media is discussed. (DMC) 5. Approximate Solution of the Point Reactor Kinetic Equations of Average One-Group of Delayed Neutrons for Step Reactivity Insertion Directory of Open Access Journals (Sweden) S. Yamoah 2012-04-01 Full Text Available The understanding of the time-dependent behaviour of the neutron population in a nuclear reactor in response to either a planned or unplanned change in the reactor conditions is of great importance to the safe and reliable operation of the reactor. In this study two analytical methods have been presented to solve the point kinetic equations of average one-group of delayed neutrons. These methods which are both approximate solution of the point reactor kinetic equations are compared with a numerical solution using the Euler’s first order method. To obtain accurate solution for the Euler method, a relatively small time step was chosen for the numerical solution. These methods are applied to different types of reactivity to check the validity of the analytical method by comparing the analytical results with the numerical results. From the results, it is observed that the analytical solution agrees well with the numerical solution. 6. Analytical solution of basic equations set of atmospheric motion Institute of Scientific and Technical Information of China (English) SHI Wei-hui; SHEN Chun; WANG Yue-peng 2007-01-01 On condition that the basic equations set of atmospheric motion possesses the best stability in the smooth function classes, the structure of solution space for local analytical solution is discussed, by which the third-class initial value problem with typicaiity and application is analyzed. The calculational method and concrete expressions of analytical solution about the well-posed initial value problem of the third kind are given in the analytic function classes. Near an appointed point, the relevant theoretical and computational problems about analytical solution of initial value problem are solved completely in the meaning of local solution. Moreover, for other type of problems for determining solution, the computational method and process of their stable analytical solution can be obtained in a similar way given in this paper. 7. On Approximate Solutions of Functional Equations in Vector Lattices Directory of Open Access Journals (Sweden) Bogdan Batko 2014-01-01 Full Text Available We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra. The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equation F(x+y+F(x+F(y≠0⇒F(x+y=F(x+F(y in Riesz spaces, the Cauchy equation with squares F(x+y2=(F(x+F(y2 in f-algebras, and the quadratic functional equation F(x+y+F(x-y=2F(x+2F(y in Riesz spaces. 8. BV solutions and viscosity approximations of rate-independent systems CERN Document Server Mielke, Alexander; Savare', Giuseppe 2009-01-01 In the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of 'BV solutions' involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting BV solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions ... 9. Analytical Solution for Stellar Density in Globular Clusters Indian Academy of Sciences (India) M. A. Sharaf; A. M. Sendi 2011-09-01 In this paper, four parameters analytical solution will be established for the stellar density function in globular clusters. The solution could be used for any arbitrary order of outward decrease of the cluster’s density. 10. Fast and Analytical EAP Approximation from a 4th-Order Tensor Directory of Open Access Journals (Sweden) Aurobrata Ghosh 2012-01-01 Full Text Available Generalized diffusion tensor imaging (GDTI was developed to model complex apparent diffusivity coefficient (ADC using higher-order tensors (HOTs and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile does not correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the ensemble average propagator (EAP. Though interesting methods for estimating a positive ADC using 4th-order diffusion tensors were developed, GDTI in general was overtaken by other approaches, for example, the orientation distribution function (ODF, since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper, we present a novel closed-form approximation of the EAP using Hermite polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate our approach with 4th-order tensors on synthetic data and in vivo human data. 11. An analytical solution in the complex plane for the luminosity distance in flat cosmology CERN Document Server Zaninetti, L 2016-01-01 We present an analytical solution for the luminosity distance in spatially flat cosmology with pressureless matter and the cosmological constant. The complex analytical solution is made of a real part and a negligible imaginary part. The real part of the luminosity distance allows finding the two parameters$H_0$and$\\om$. A simple expression for the distance modulus for SNs of type Ia is reported in the framework of the minimax approximation. 12. Mathematical analysis of recent analytical approximations to the collapse of an empty spherical bubble. Science.gov (United States) Amore, Paolo; Fernández, Francisco M 2013-02-28 We analyze the Rayleigh equation for the collapse of an empty bubble and provide an explanation for some recent analytical approximations to the model. We derive the form of the singularity at the second boundary point and discuss the convergence of the approximants. We also give a rigorous proof of the asymptotic behavior of the coefficients of the power series that are the basis for the approximate expressions. 13. Analytical Solution of Projectile Motion with Quadratic Resistance and Generalisations CERN Document Server Ray, Shouryya 2013-01-01 The paper considers the motion of a body under the influence of gravity and drag of the surrounding fluid. Depending on the fluid mechanical regime, the drag force can exhibit a linear, quadratic or even more general dependence on the velocity of the body relative to the fluid. The case of quadratic drag is substantially more complex than the linear case, as it nonlinearly couples both components of the momentum equation, and no explicit analytic solution is known for a general trajectory. After a detailed account of the literature, the paper provides such a solution in form of a series expansion. This result is discussed in detail and related to other approaches previously proposed. In particular, it is shown to yield certain approximate solutions proposed in the literature as limiting cases. The solution technique employs a strategy to reduce systems of ordinary differential equations with a triangular dependence of the right-hand side on the vector of unknowns to a single equation in an auxiliary variable.... 14. A compact analytic solution describing optoacoustic phenomenon in absorbing fluid CERN Document Server Cundin, Luisiana; Barsalou, Norman; Voss, Shannon 2012-01-01 Derivation of an analytic, closed-form solution for Q-switched laser induced optoacoustic phenomenon in absorbing fluid media is presented. The solution assumes spherical symmetry as well for the forcing function, which represents heat deposition from Q-switched lasers. The Greens solution provided is a suitable kernel to generate more complex solutions arising in optoacoustics, optoacoustic spectroscopy, photoacoustic and photothermal problems. 15. AN ANALYTICAL SOLUTION FOR AN EXPONENTIAL TYPE DISPERSION PROCESS Institute of Scientific and Technical Information of China (English) 王子亭 2001-01-01 The dispersion process in heterogeneous porous media is distance-dependent,which results from multi-scaling property of heterogeneous structure. An analytical model describing the dispersion with an exponential dispersion function is built, which is transformed into ODE problem with variable coefficients, and obtained analytical solution for two type boundary conditions using hypergeometric function and inversion technique.According to the analytical solution and computing results the difference between the exponential dispersion and constant dispersion process is analyzed. 16. Semi-analytic solution to planar Helmholtz equation Directory of Open Access Journals (Sweden) Tukač M. 2013-06-01 Full Text Available Acoustic solution of interior domains is of great interest. Solving acoustic pressure fields faster with lower computational requirements is demanded. A novel solution technique based on the analytic solution to the Helmholtz equation in rectangular domain is presented. This semi-analytic solution is compared with the finite element method, which is taken as the reference. Results show that presented method is as precise as the finite element method. As the semi-analytic method doesn’t require spatial discretization, it can be used for small and very large acoustic problems with the same computational costs. 17. Surface conductivity in electrokinetic systems with porous and charged interfaces: Analytical approximations and numerical results. Science.gov (United States) Barbati, Alexander C; Kirby, Brian J 2016-07-01 We derive an approximate analytical representation of the conductivity for a 1D system with porous and charged layers grafted onto parallel plates. Our theory improves on prior work by developing approximate analytical expressions applicable over an arbitrary range of potentials, both large and small as compared to the thermal voltage (RTF). Further, we describe these results in a framework of simplifying nondimensional parameters, indicating the relative dominance of various physicochemical processes. We demonstrate the efficacy of our approximate expression with comparisons to numerical representations of the exact analytical conductivity. Finally, we utilize this conductivity expression, in concert with other components of the electrokinetic coupling matrix, to describe the streaming potential and electroviscous effect in systems with porous and charged layers. 18. Analytical solution for the diffusion of a capacitor discharge generated magnetic field pulse in a conductor Directory of Open Access Journals (Sweden) Ilmārs Grants 2016-06-01 Full Text Available Powerful forces arise when a pulse of a magnetic field in the order of a few tesla diffuses into a conductor. Such pulses are used in electromagnetic forming, impact welding of dissimilar materials and grain refinement of solidifying alloys. Strong magnetic field pulses are generated by the discharge current of a capacitor bank. We consider analytically the penetration of such pulse into a conducting half-space. Besides the exact solution we obtain two simple self-similar approximate solutions for two sequential stages of the initial transient. Furthermore, a general solution is provided for the external field given as a power series of time. Each term of this solution represents a self-similar function for which we obtain an explicit expression. The validity range of various approximate analytical solutions is evaluated by comparison to the exact solution. 19. Analytical solution for the diffusion of a capacitor discharge generated magnetic field pulse in a conductor Science.gov (United States) Grants, Ilmārs; Bojarevičs, Andris; Gerbeth, Gunter 2016-06-01 Powerful forces arise when a pulse of a magnetic field in the order of a few tesla diffuses into a conductor. Such pulses are used in electromagnetic forming, impact welding of dissimilar materials and grain refinement of solidifying alloys. Strong magnetic field pulses are generated by the discharge current of a capacitor bank. We consider analytically the penetration of such pulse into a conducting half-space. Besides the exact solution we obtain two simple self-similar approximate solutions for two sequential stages of the initial transient. Furthermore, a general solution is provided for the external field given as a power series of time. Each term of this solution represents a self-similar function for which we obtain an explicit expression. The validity range of various approximate analytical solutions is evaluated by comparison to the exact solution. 20. A Comprehensive Analytical Solution of the Nonlinear Pendulum Science.gov (United States) Ochs, Karlheinz 2011-01-01 In this paper, an analytical solution for the differential equation of the simple but nonlinear pendulum is derived. This solution is valid for any time and is not limited to any special initial instance or initial values. Moreover, this solution holds if the pendulum swings over or not. The method of approach is based on Jacobi elliptic functions… 1. Analytical solutions of coupled-mode equations for microring resonators Indian Academy of Sciences (India) ZHAO C Y 2016-06-01 We present a study on analytical solutions of coupled-mode equations for microring resonators with an emphasis on occurrence of all-optical EIT phenomenon, obtained by using a cofactor. As concrete examples, analytical solutions for a$3 \\times 3$linearly distributed coupler and a circularly distributed coupler are obtained. The former corresponds to a non-degenerate eigenvalue problem and the latter corresponds to a degenerate eigenvalue problem. For comparison and without loss of generality, analytical solution for a$4 \\times 4$linearly distributed coupler is also obtained. This paper may be of interest to optical physics and integrated photonics communities. 2. A general solution and approximation for the diffusion of gas in a spherical coal sample Institute of Scientific and Technical Information of China (English) Wang Yucang; Xue Sheng; Xie Jun 2014-01-01 The square root relationship of gas release in the early stage of desorption is widely used to provide a simple and fast estimation of the lost gas in coal mines. However, questions arise as to how the relation-ship was theoretically derived, what are the assumptions and applicable conditions and how large the error will be. In this paper, the analytical solutions of gas concentration and fractional gas loss for the dif-fusion of gas in a spherical coal sample were given with detailed mathematical derivations based on the diffusion equation. The analytical solutions were approximated in case of small values of time and the error analyses associated with the approximation were also undertaken. The results indicate that the square root relationship of gas release is the first term of the approximation, and care must be taken in using the square root relationship as a significant error might be introduced with increase in the lost time and decrease in effective diameter of a spherical coal sample. 3. Analytic solutions of topologically disjoint systems DEFF Research Database (Denmark) Armstrong, J. R.; Volosniev, A. G.; Fedorov, D. V.; 2015-01-01 We describe a procedure to solve an up to$2N$problem where the particles are separated topologically in$N$groups with at most two particles in each. Arbitrary interactions are allowed between the (two) particles within one group. All other interactions are approximated by harmonic oscillator ... 4. The comparative study on analytical solutions and numerical solutions of displacement in transversely isotropic rock mass Science.gov (United States) Zhang, Zhizeng; Zhao, Zhao; Li, Yongtao 2016-06-01 This paper attempts to verify the correctness of the analytical displacement solution in transversely isotropic rock mass, and to determine the scope of its application. The analytical displacement solution of a circular tunnel in transversely isotropic rock mass was derived firstly. The analytical solution was compared with the numerical solution, which was carried out by FLAC3D software. The results show that the expression of the analytical displacement solution is correct, and the allowable engineering range is that the dip angle is less than 15 degrees. 5. Reactive silica transport in fractured porous media: Analytical solutions for a system of parallel fractures Science.gov (United States) Yang, Jianwen 2012-04-01 A general analytical solution is derived by using the Laplace transformation to describe transient reactive silica transport in a conceptualized 2-D system involving a set of parallel fractures embedded in an impermeable host rock matrix, taking into account of hydrodynamic dispersion and advection of silica transport along the fractures, molecular diffusion from each fracture to the intervening rock matrix, and dissolution of quartz. A special analytical solution is also developed by ignoring the longitudinal hydrodynamic dispersion term but remaining other conditions the same. The general and special solutions are in the form of a double infinite integral and a single infinite integral, respectively, and can be evaluated using Gauss-Legendre quadrature technique. A simple criterion is developed to determine under what conditions the general analytical solution can be approximated by the special analytical solution. It is proved analytically that the general solution always lags behind the special solution, unless a dimensionless parameter is less than a critical value. Several illustrative calculations are undertaken to demonstrate the effect of fracture spacing, fracture aperture and fluid flow rate on silica transport. The analytical solutions developed here can serve as a benchmark to validate numerical models that simulate reactive mass transport in fractured porous media. 6. Analytic Solution of Strongly Coupling Schroedinger Equation CERN Document Server Liao, J Y; Liao, Jinfeng; Zhuang, Pengfei 2002-01-01 The recently developed expansion method for ground states of strongly coupling Schr\\"odinger equations by Friedberg, Lee and Zhao is extended to excited states. The coupling constant dependence of bound states for power-law central forces$V(r) \\propto g^k r^nis particularly studied. With the extended method all the excited states of the Hydrogen atom problem are resolved and the low-lying states for Yukawa potential are approximately obtained. 7. Generalized Analytical Solutions for Nonlinear Positive-Negative Index Couplers Directory of Open Access Journals (Sweden) Zh. Kudyshev 2012-01-01 Full Text Available We find and analyze a generalized analytical solution for nonlinear wave propagation in waveguide couplers with opposite signs of the linear refractive index, nonzero phase mismatch between the channels, and arbitrary nonlinear coefficients. 8. ANALYTICAL SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS Institute of Scientific and Technical Information of China (English) 胡建兰; 张汉林 2003-01-01 The following partial differential equations are studied: generaliz ed fifth-orderKdV equation, water wave equation, Kupershmidt equation, couples KdV equation. Theanalytical solutions to these problems via using various ansaiz es by introducing a second-order ordinary differential equation are found out. 9. Analytic solutions for marginal deformations in open superstring field theory Energy Technology Data Exchange (ETDEWEB) Okawa, Y. 2007-04-15 We extend the calculable analytic approach to marginal deformations recently developed in open bosonic string field theory to open superstring field theory formulated by Berkovits. We construct analytic solutions to all orders in the deformation parameter when operator products made of the marginal operator and the associated superconformal primary field are regular. (orig.) 10. New software solutions for analytical spectroscopists Science.gov (United States) Davies, Antony N. 1999-05-01 Analytical spectroscopists must be computer literate to effectively carry out the tasks assigned to them. This has often been resisted within organizations with insufficient funds to equip their staff properly, a lack of desire to deliver the essential training and a basic resistance amongst staff to learn the new techniques required for computer assisted analysis. In the past these problems were compounded by seriously flawed software which was being sold for spectroscopic applications. Owing to the limited market for such complex products the analytical spectroscopist often was faced with buying incomplete and unstable tools if the price was to remain reasonable. Long product lead times meant spectrometer manufacturers often ended up offering systems running under outdated and sometimes obscure operating systems. Not only did this mean special staff training for each instrument where the knowledge gained on one system could not be transferred to the neighbouring system but these spectrometers were often only capable of running in a stand-alone mode, cut-off from the rest of the laboratory environment. Fortunately a number of developments in recent years have substantially changed this depressing picture. A true multi-tasking operating system with a simple graphical user interface, Microsoft Windows NT4, has now been widely introduced into the spectroscopic computing environment which has provided a desktop operating system which has proved to be more stable and robust as well as requiring better programming techniques of software vendors. The opening up of the Internet has provided an easy way to access new tools for data handling and has forced a substantial re-think about results delivery (for example Chemical MIME types, IUPAC spectroscopic data exchange standards). Improved computing power and cheaper hardware now allows large spectroscopic data sets to be handled without too many problems. This includes the ability to carry out chemometric operations in 11. Zero Viscosity Limit for Analytic Solutions of the Primitive Equations Science.gov (United States) Kukavica, Igor; Lombardo, Maria Carmela; Sammartino, Marco 2016-10-01 The aim of this paper is to prove that the solutions of the primitive equations converge, in the zero viscosity limit, to the solutions of the hydrostatic Euler equations. We construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic Euler equation and boundary layer correctors as the first order term, and an error that we show to be {O(√{ν})}. The main assumption is spatial analyticity of the initial datum. 12. Analytical solutions for the Rabi model CERN Document Server Yu, Lixian; Liang, Qifeng; Chen, Gang; Jia, Suotang 2012-01-01 The Rabi model that describes the fundamental interaction between a two-level system with a quantized harmonic oscillator is one of the simplest and most ubiquitous models in modern physics. However, this model has not been solved exactly because it is hard to find a second conserved quantity besides the energy. Here we present a unitary transformation to map this unsolvable Rabi model into a solvable Jaynes-Cummings-like model by choosing a proper variation parameter. As a result, the analytical energy spectrums and wavefunctions including both the ground and the excited states can be obtained easily. Moreover, these explicit results agree well with the direct numerical simulations in a wide range of the experimental parameters. In addition, based on our obtained energy spectrums, the recent experimental observation of Bloch-Siegert in the circuit quantum electrodynamics with the ultrastrong coupling can be explained perfectly. Our results have the potential application in the solid-state quantum information... 13. Approximate Solution of D-Dimensional Klein-Gordon Equation with Hulthen-Type Potential via SUSYQM Institute of Scientific and Technical Information of China (English) H. Hassanabadi; S. Zarrinkamar; H. Rahimov 2011-01-01 Approximate analytical solutions of the D-dimensional Klein-Gordon equation are obtained for the scalar and vector general Hulthen-type potential and position-dependent mass with any l by using the concept of supersymmetric quantum mechanics （SUSYQM）. The problem is numerically discussed for some cases of parameters. 14. Analytical solutions for the fractional Fisher's equation Directory of Open Access Journals (Sweden) H. Kheiri 2015-06-01 Full Text Available In this paper, we consider the inhomogeneous time-fractional nonlinear Fisher equation with three known boundary conditions. We first apply a modified Homotopy perturbation method for translating the proposed problem to a set of linear problems. Then we use the separation variables method to solve obtained problems. In examples, we illustrate that by right choice of source term in the modified Homotopy perturbation method, it is possible to get an exact solution. 15. Analytical Solution for the Current Distribution in Multistrand Superconducting Cables CERN Document Server Bottura, L; Fabbri, M G 2002-01-01 Current distribution in multistrand superconducting cables can be a major concern for stability in superconducting magnets and for field quality in particle accelerator magnets. In this paper we describe multistrand superconducting cables by means of a distributed parameters circuit model. We derive a system of partial differential equations governing current distribution in the cable and we give the analytical solution of the general system. We then specialize the general solution to the particular case of uniform cable properties. In the particular case of a two-strand cable, we show that the analytical solution presented here is identical to the one already available in the literature. For a cable made of N equal strands we give a closed form solution that to our knowledge was never presented before. We finally validate the analytical solution by comparison to numerical results in the case of a step-like spatial distribution of the magnetic field over a short Rutherford cable, both in transient and steady ... 16. Big Data Security Analytic Solution using Splunk Directory of Open Access Journals (Sweden) P.Charishma, 2015-04-01 Full Text Available Over the past decade, usage of online applications is experiencing remarkable growth. One of the main reasons for the success of web application is its “Ease of Access” and availability on internet. The simplicity of the HTTP protocol makes it easy to steal and spoof identity. The business liability associated with protecting online information has increased significantly and this is an issue that must be addressed. According to SANSTop20, 2013 list the number one targeted server side vulnerability are Web Applications. So, this has made detecting and preventing attacks on web applications a top priority for IT companies. In this paper, a rational solution is brought to detect events on web application and provides Security intelligence, log management and extensible reporting by analyzing web server logs. 17. Analytic solution of simplified Cardan's shaft model Directory of Open Access Journals (Sweden) Zajíček M. 2014-12-01 Full Text Available Torsional oscillations and stability assessment of the homokinetic Cardan shaft with a small misalignment angle is described in this paper. The simplified mathematical model of this system leads to the linearized equation of the Mathieu's type. This equation with and without a stationary damping parameter is considered. The solution of the original differential equation is identical with those one of the Fredholm’s integral equation with degenerated kernel assembled by means of a periodic Green's function. The conditions of solvability of such problem enable the identification of the borders between stability and instability regions. These results are presented in the form of stability charts and they are verified using the Floquet theory. The correctness of oscillation results for the system with periodic stiffness is then validated by means of the Runge-Kutta integration method. 18. AN ANALYTICAL SOLUTION FOR CALCULATING THE INITIATION OF SEDIMENT MOTION Institute of Scientific and Technical Information of China (English) Thomas LUCKNER; Ulrich ZANKE 2007-01-01 This paper presents an analytical solution for calculating the initiation of sediment motion and the risk of river bed movement. It thus deals with a fundamental problem in sediment transport, for which no complete analytical solution has yet been found. The analytical solution presented here is based on forces acting on a single grain in state of initiation of sediment motion. The previous procedures for calculating the initiation of sediment motion are complemented by an innovative combination of optical surface measurement technology for determining geometrical parameters and their statistical derivation as well as a novel approach for determining the turbulence effects of velocity fluctuations. This two aspects and the comparison of the solution functions presented here with the well known data and functions of different authors mainly differ the presented solution model for calculating the initiation of sediment motion from previous approaches. The defined values of required geometrical parameters are based on hydraulically laboratory tests with spheres. With this limitations the derivated solution functions permit the calculation of the effective critical transport parameters of a single grain, the calculation of averaged critical parameters for describing the state of initiation of sediment motion on the river bed, the calculation of the probability density of the effective critical velocity as well as the calculation of the risk of river bed movement. The main advantage of the presented model is the closed analytical solution from the equilibrium of forces on a single grain to the solution functions describing the initiation of sediment motion. 19. Homotopic Approximate Solutions for the Perturbed CKdV Equation with Variable Coefficients Directory of Open Access Journals (Sweden) Dianchen Lu 2014-01-01 Full Text Available This work concerns how to find the double periodic form of approximate solutions of the perturbed combined KdV (CKdV equation with variable coefficients by using the homotopic mapping method. The obtained solutions may degenerate into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. Moreover, the first order approximate solutions and the second order approximate solutions of the variable coefficients CKdV equation in perturbation εun are also induced. 20. Stability of small-amplitude torus knot solutions of the localized induction approximation Energy Technology Data Exchange (ETDEWEB) Calini, Annalisa; Ivey, Thomas, E-mail: [email protected] [Department of Mathematics, College of Charleston, Charleston, SC 29424 (United States) 2011-08-19 We study the linear stability of small-amplitude torus knot solutions of the localized induction approximation equation for the motion of a thin vortex filament in an ideal fluid. Such solutions can be constructed analytically through the connection with the focusing nonlinear Schroedinger equation using the method of isoperiodic deformations. We show that these (p, q) torus knots are generically linearly unstable for p < q, while we provide examples of neutrally stable (p, q) torus knots with p > q, in contrast with an earlier linear stability study by Ricca (1993 Chaos 3 83-95; 1995 Chaos 5 346; 1995 Small-scale Structures in Three-dimensional Hydro and Magneto-dynamics Turbulence (Lecture Notes in Physics vol 462) (Berlin: Springer)). We also provide an interpretation of the original perturbative calculation in Ricca (1995), and an explanation of the numerical experiments performed by Ricca et al (1999 J. Fluid Mech. 391 29-44), in light of our results. 1. Stability of small-amplitude torus knot solutions of the localized induction approximation Science.gov (United States) Calini, Annalisa; Ivey, Thomas 2011-08-01 We study the linear stability of small-amplitude torus knot solutions of the localized induction approximation equation for the motion of a thin vortex filament in an ideal fluid. Such solutions can be constructed analytically through the connection with the focusing nonlinear Schrödinger equation using the method of isoperiodic deformations. We show that these (p, q) torus knots are generically linearly unstable for p q, in contrast with an earlier linear stability study by Ricca (1993 Chaos 3 83-95 1995 Chaos 5 346; 1995 Small-scale Structures in Three-dimensional Hydro and Magneto-dynamics Turbulence (Lecture Notes in Physics vol 462) (Berlin: Springer)). We also provide an interpretation of the original perturbative calculation in Ricca (1995), and an explanation of the numerical experiments performed by Ricca et al (1999 J. Fluid Mech. 391 29-44), in light of our results. 2. Analytic solution of an oscillatory migratory alpha^2 stellar dynamo CERN Document Server Brandenburg, Axel 2016-01-01 Analytic solutions of the mean-field induction equation predict a nonoscillatory dynamo for uniform helical turbulence or constant alpha effect in unbounded or periodic domains. Oscillatory dynamos are generally thought impossible for constant alpha. We present an analytic solution for a one-dimensional bounded domain resulting in oscillatory solutions for constant alpha, but different (Dirichlet and von Neumann or perfect conductor and vacuum) boundary conditions on the two ends. We solve a second order complex equation and superimpose two independent solutions to obey both boundary conditions. The solution has time-independent energy density. On one end where the function value vanishes, the second derivative is finite, which would not be correctly reproduced with sine-like expansion functions where a node coincides with an inflection point. The obtained solution may serve as a benchmark for numerical dynamo experiments and as a pedagogical illustration that oscillatory dynamos are possible for dynamos with... 3. Transmission Line Adapted Analytical Power Charts Solution Science.gov (United States) Sakala, Japhet D.; Daka, James S. J.; Setlhaolo, Ditiro; Malichi, Alec Pulu 2016-08-01 The performance of a transmission line has been assessed over the years using power charts. These are graphical representations, drawn to scale, of the equations that describe the performance of transmission lines. Various quantities that describe the performance, such as sending end voltage, sending end power and compensation to give zero voltage regulation, may be deduced from the power charts. Usually required values are read off and then converted using the appropriate scales and known relationships. In this paper, the authors revisit this area of circle diagrams for transmission line performance. The work presented here formulates the mathematical model that analyses the transmission line performance from the power charts relationships and then uses them to calculate the transmission line performance. In this proposed approach, it is not necessary to draw the power charts for the solution. However the power charts may be drawn for the visual presentation. The method is based on applying derived equations and is simple to use since it does not require rigorous derivations. 4. Analytical modeling of bargaining solutions for multicast cellular services Directory of Open Access Journals (Sweden) Giuseppe Araniti 2013-07-01 Full Text Available Nowadays, the growing demand for group-oriented services over mobile devices has lead to the definition of new communication standards and multimedia applications in cellular systems. In this article we study the use of game theoretic solutions for these services to model and perform a trade-off analysis between fairness and efficiency in the resources allocation. More precisely, we model bargaining solutions for the multicast data services provisioning and introduce the analytical resolution for the proposed solutions. 5. Analytical First-Order Molecular Properties and Forces within the Adiabatic Connection Random Phase Approximation. Science.gov (United States) Burow, Asbjörn M; Bates, Jefferson E; Furche, Filipp; Eshuis, Henk 2014-01-14 The random phase approximation (RPA) is an increasingly popular method for computing molecular ground-state correlation energies within the adiabatic connection fluctuation-dissipation theorem framework of density functional theory. We present an efficient analytical implementation of first-order RPA molecular properties and nuclear forces using the resolution-of-the-identity (RI) approximation and imaginary frequency integration. The centerpiece of our approach is a variational RPA energy Lagrangian invariant under unitary transformations of occupied and virtual reference orbitals, respectively. Its construction requires the solution of a single coupled-perturbed Kohn-Sham equation independent of the number of perturbations. Energy gradients with respect to nuclear displacements and other first-order properties such as one-particle densities or dipole moments are obtained from partial derivatives of the Lagrangian. Our RPA energy gradient implementation exhibits the same [Formula: see text] scaling with system size N as a single-point RPA energy calculation. In typical applications, the cost for computing the entire gradient vector with respect to nuclear displacements is ∼5 times that of a single-point RPA energy calculation. Derivatives of the quadrature nodes and weights used for frequency integration are essential for RPA gradients with an accuracy consistent with RPA energies and can be included in our approach. The quality of RPA equilibrium structures is assessed by comparison to accurate theoretical and experimental data for covalent main group compounds, weakly bonded dimers, and transition metal complexes. RPA outperforms semilocal functionals as well as second-order Møller-Plesset (MP2) theory, which fails badly for the transition metal compounds. Dipole moments of polarizable molecules and weakly bound dimers show a similar trend. RPA harmonic vibrational frequencies are nearly of coupled cluster singles, doubles, and perturbative triples quality 6. RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS Institute of Scientific and Technical Information of China (English) Reinhard Hochmuth 2002-01-01 This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1 ] are chosen as a starting point for characterizations of functions in Besov spaces B , (0,1) with 0＜σ＜∞ and (1+σ)-1＜τ＜∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results. 7. An analytical dynamo solution for large-scale magnetic fields of galaxies CERN Document Server Chamandy, Luke 2016-01-01 We present an effectively global analytical asymptotic galactic dynamo solution for the regular magnetic field of an axisymmetric thin disc in the saturated state. This solution is constructed by combining two well-known types of local galactic dynamo solution, parameterized by the disc radius. Namely, the critical (zero growth) solution obtained by treating the dynamo equation as a perturbed diffusion equation is normalized using a non-linear solution that makes use of the no-z$' approximation and the dynamical$\\alpha$-quenching non-linearity. This overall solution is found to be reasonably accurate when compared with detailed numerical solutions. It is thus potentially useful as a tool for predicting observational signatures of magnetic fields of galaxies. In particular, such solutions could be painted onto galaxies in cosmological simulations to enable the construction of synthetic polarized synchrotron and Faraday rotation measure (RM) datasets. Further, we explore the properties of our numerical solut... 8. Corrected Analytical Solution of the Generalized Woods-Saxon Potential for Arbitrary$\\ell$States CERN Document Server Bayrak, O 2015-01-01 The bound state solution of the radial Schr\\"{o}dinger equation with the generalized Woods-Saxon potential is carefully examined by using the Pekeris approximation for arbitrary$\\ell$states. The energy eigenvalues and the corresponding eigenfunctions are analytically obtained for different$n$and$\\ell$quantum numbers. The obtained closed forms are applied to calculate the single particle energy levels of neutron orbiting around$^{56}$Fe nucleus in order to check consistency between the analytical and Gamow code results. The analytical results are in good agreement with the results obtained by Gamow code for$\\ell=0$. 9. A hybrid ICT-solution for smart meter data analytics DEFF Research Database (Denmark) Liu, Xiufeng; Nielsen, Per Sieverts 2016-01-01 Smart meters are increasingly used worldwide. Smart meters are the advanced meters capable of measuring energy consumption at a fine-grained time interval, e.g., every 15 min. Smart meter data are typically bundled with social economic data in analytics, such as meter geographic locations, weather...... conditions and user information, which makes the data sets very sizable and the analytics complex. Data mining and emerging cloud computing technologies make collecting, processing, and analyzing the so-called big data possible. This paper proposes an innovative ICT-solution to streamline smart meter data...... analytics. The proposed solution offers an information integration pipeline for ingesting data from smart meters, a scalable platform for processing and mining big data sets, and a web portal for visualizing analytics results. The implemented system has a hybrid architecture of using Spark or Hive for big... 10. Exact analytical solutions for the Poiseuille and Couette-Poiseuille flow of third grade fluid between parallel plates Science.gov (United States) Danish, Mohammad; Kumar, Shashi; Kumar, Surendra 2012-03-01 Exact analytical solutions for the velocity profiles and flow rates have been obtained in explicit forms for the Poiseuille and Couette-Poiseuille flow of a third grade fluid between two parallel plates. These exact solutions match well with their numerical counter parts and are better than the recently developed approximate analytical solutions. Besides, effects of various parameters on the velocity profile and flow rate have been studied. 11. On the General Analytical Solution of the Kinematic Cosserat Equations KAUST Repository Michels, Dominik L. 2016-09-01 Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness. 12. An analytical solution to the equation of motion for the damped nonlinear pendulum DEFF Research Database (Denmark) Johannessen, Kim 2014-01-01 An analytical approximation of the solution to the differential equation describing the oscillations of the damped nonlinear pendulum at large angles is presented. The solution is expressed in terms of the Jacobi elliptic functions by including a parameter-dependent elliptic modulus. The analytical...... of the damped nonlinear pendulum is presented, and it is shown that the period of oscillation is dependent on time. It is established that, in general, the period is longer than that of a linearized model, asymptotically approaching the period of oscillation of a damped linear pendulum.... 13. AN EXACT ANALYTICAL SOLUTION FOR THE INTERSTELLAR MAGNETIC FIELD IN THE VICINITY OF THE HELIOSPHERE Energy Technology Data Exchange (ETDEWEB) Röken, Christian [Universität Regensburg, Fakultät für Mathematik, Regensburg (Germany); Kleimann, Jens; Fichtner, Horst, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Ruhr-Universität Bochum, Fakultät für Physik und Astronomie, Institut für Theoretische Physik IV, Bochum (Germany) 2015-06-01 An analytical representation of the interstellar magnetic field in the vicinity of the heliosphere is derived. The three-dimensional field structure close to the heliopause is calculated as a solution of the induction equation under the assumption that it is frozen into a prescribed plasma flow resembling the characteristic interaction of the solar wind with the local interstellar medium. The usefulness of this analytical solution as an approximation to self-consistent magnetic field configurations obtained numerically from the full MHD equations is illustrated by quantitative comparisons. 14. An analytical approximation of the growth function in Friedmann-Lema\\^itre universes CERN Document Server Kasai, Masumi 2010-01-01 We present an analytical approximation formula for the growth function in a spatially flat cosmology with dust and a cosmological constant. Our approximate formula is written simply in terms of a rational function. We also show the approximate formula in a dust cosmology without a cosmological constant, directly as a function of the scale factor in terms of a rational function. The single rational function applies for all, open, closed and flat universes. Our results involve no elliptic functions, and have very small relative error of less than 0.2 per cent over the range of the scale factor$1/1000 \\la a \\lid 1$and the density parameter$0.2 \\la \\Omega_{\\rmn{m}} \\lid 1$for a flat cosmology, and less than$0.4$per cent over the range$0.2 \\la \\Omega_{\\rmn{m}} \\la 4for a cosmology without a cosmological constant. 15. Approximation solutions for indifference pricing under general utility functions NARCIS (Netherlands) Chen, An; Pelsser, Antoon; Vellekoop, Michel 2008-01-01 With the aid of Taylor-based approximations, this paper presents results for pricing insurance contracts by using indifference pricing under general utility functions. We discuss the connection between the resulting "theoretical" indifference prices and the pricing rule-of-thumb that practitioners u 16. Approximation solution of Schrodinger equation for Q-deformed Rosen-Morse using supersymmetry quantum mechanics (SUSY QM) Energy Technology Data Exchange (ETDEWEB) Alemgadmi, Khaled I. K., E-mail: [email protected]; Suparmi; Cari [Department of Physics, the State University of Surabaya (Unesa), Jl. Ketintang, Surabaya 60231 (Indonesia); Deta, U. A., E-mail: [email protected] [Departmet of Physics, Sebelas Maret University, Jl. Ir. Sutami 36A Kentingan, Surakarta 57126 (Indonesia) 2015-09-30 The approximate analytical solution of Schrodinger equation for Q-Deformed Rosen-Morse potential was investigated using Supersymmetry Quantum Mechanics (SUSY QM) method. The approximate bound state energy is given in the closed form and the corresponding approximate wave function for arbitrary l-state given for ground state wave function. The first excited state obtained using upper operator and ground state wave function. The special case is given for the ground state in various number of q. The existence of Rosen-Morse potential reduce energy spectra of system. The larger value of q, the smaller energy spectra of system. 17. Approximate Solutions of Interactive Dynamic Influence Diagrams Using Model Clustering DEFF Research Database (Denmark) Zeng, Yifeng; Doshi, Prashant; Qiongyu, Cheng 2007-01-01 Interactive dynamic influence diagrams (I-DIDs) offer a transparent and semantically clear representation for the sequential decision-making problem over multiple time steps in the presence of other interacting agents. Solving I-DIDs exactly involves knowing the solutions of possible models... 18. Classification and Approximate Functional Separable Solutions to the Generalized Diffusion Equations with Perturbation Science.gov (United States) Ji, Fei-Yu; Zhang, Shun-Li 2013-11-01 In this paper, the generalized diffusion equation with perturbation ut = A(u;ux)uII+eB(u;ux) is studied in terms of the approximate functional variable separation approach. A complete classification of these perturbed equations which admit approximate functional separable solutions is presented. Some approximate solutions to the resulting perturbed equations are obtained by examples. 19. Analytical approximation of the neutrino oscillation matter effects at large θ{sub 13} Energy Technology Data Exchange (ETDEWEB) Agarwalla, Sanjib Kumar [Institute of Physics, Sachivalaya Marg, Sainik School Post,Bhubaneswar 751005, Orissa (India); Kao, Yee [Department of Chemistry and Physics, Western Carolina University,Cullowhee, NC 28723 (United States); Takeuchi, Tatsu [Center for Neutrino Physics, Physics Department, Virginia Tech,Blacksburg, VA 24061 (United States); Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo,Kashiwa-shi, Chiba-ken 277-8583 (Japan) 2014-04-07 We argue that the neutrino oscillation probabilities in matter are best understood by allowing the mixing angles and mass-squared differences in the standard parametrization to ‘run’ with the matter effect parameter a=2√2G{sub F}N{sub e}E, where N{sub e} is the electron density in matter and E is the neutrino energy. We present simple analytical approximations to these ‘running’ parameters. We show that for the moderately large value of θ{sub 13}, as discovered by the reactor experiments, the running of the mixing angle θ{sub 23} and the CP violating phase δ can be neglected. It simplifies the analysis of the resulting expressions for the oscillation probabilities considerably. Approaches which attempt to directly provide approximate analytical expressions for the oscillation probabilities in matter suffer in accuracy due to their reliance on expansion in θ{sub 13}, or in simplicity when higher order terms in θ{sub 13} are included. We demonstrate the accuracy of our method by comparing it to the exact numerical result, as well as the direct approximations of Cervera et al., Akhmedov et al., Asano and Minakata, and Freund. We also discuss the utility of our approach in figuring out the required baseline lengths and neutrino energies for the oscillation probabilities to exhibit certain desirable features. 20. An Approximate Solution for Spherical and Cylindrical Piston Problem Indian Academy of Sciences (India) S K Singh; V P Singh 2000-02-01 A new theory of shock dynamics (NTSD) has been derived in the form of a finite number of compatibility conditions along shock rays. It has been used to study the growth and decay of shock strengths for spherical and cylindrical pistons starting from a non-zero velocity. Further a weak shock theory has been derived using a simple perturbation method which admits an exact solution and also agrees with the classical decay laws for weak spherical and cylindrical shocks. 1. Foam for Enhanced Oil Recovery: Modeling and Analytical Solutions NARCIS (Netherlands) Ashoori, E. 2012-01-01 Foam increases sweep in miscible- and immiscible-gas enhanced oil recovery by decreasing the mobility of gas enormously. This thesis is concerned with the simulations and analytical solutions for foam flow for the purpose of modeling foam EOR in a reservoir. For the ultimate goal of upscaling our mo 2. Analytical solutions for geodesics in black hole spacetimes CERN Document Server Hackmann, Eva 2015-01-01 We review the analytical solution methods for the geodesic equations in Kerr-Newman-Taub-NUT-de Sitter spacetimes and its subclasses in terms of elliptic and hyperelliptic functions. A short guide to corresponding literature for general timelike and lightlike motion is also presented. 3. Analytical solution for the convectively-mixed atmospheric boundary layer NARCIS (Netherlands) Ouwersloot, H.G.; Vilà-Guerau de Arellano, J. 2013-01-01 Based on the prognostic equations of mixed-layer theory assuming a zeroth order jump at the entrainment zone, analytical solutions for the boundary-layer height evolution are derived with different degrees of accuracy. First, an exact implicit expression for the boundary-layer height for a situation 4. Decision Exploration Lab : A Visual Analytics Solution for Decision Management NARCIS (Netherlands) Broeksema, Bertjan; Baudel, Thomas; Telea, Alex; Crisafulli, Paolo 2013-01-01 We present a visual analytics solution designed to address prevalent issues in the area of Operational Decision Management (ODM). In ODM, which has its roots in Artificial Intelligence (Expert Systems) and Management Science, it is increasingly important to align business decisions with business goa 5. Linear power spectra in cold+hot dark matter models analytical approximations and applications CERN Document Server Ma Chung Pei 1996-01-01 This paper presents simple analytic approximations to the linear power spectra, linear growth rates, and rms mass fluctuations for both components in a family of cold+hot dark matter (CDM+HDM) models that are of current cosmological interest. The formulas are valid for a wide range of wavenumber, neutrino fraction, redshift, and Hubble constant: k\\lo 10\\,h Mpc^{-1}, 0.05\\lo \\onu\\lo 0.3, 0\\le z\\lo 15, and 0.5\\lo h \\lo 0.8. A new, redshift-dependent shape parameter \\Gamma_\ 6. Calculation of photon attenuation coefficients of elements and compounds from approximate semi-analytical formulae Energy Technology Data Exchange (ETDEWEB) Roteta, M.; Baro, J.; Fernandez-Varea, J. M.; Salvat, F. 1994-07-01 The FORTRAN 77 code PHOTAC to compute photon attenuation coefficients of elements and compounds is described. The code is based on the semi analytical approximate atomic cross sections proposed by Baro et al. (1994). Photoelectric cross sections for coherent and incoherent scattering and for pair production are obtained as integrals of the corresponding differential cross sections. These integrals are evaluated, to a pre-selected accuracy, by using a 20-point Gauss adaptive integration algorithm. Calculated attenuation coefficients agree with recently compiled databases to within - 1%, in the energy range from 1 keV to 1 GeV. The complete source listing of the program PHOTAC is included. (Author) 14 refs. 7. Some analytical approximations to radiative transfer theory and their application for the analysis of reflectance data Science.gov (United States) Kokhanovsky, Alexander; Hopkinson, Ian 2008-03-01 We derive an analytical approximation in the framework of the radiative transfer theory for use in the analysis of diffuse reflectance measurements. This model uses two parameters to describe a material, the transport free path length, l, and the similarity parameter, s. Using a simple algebraic expression, s and l can be applied for the determination of the absorption coefficient Kabs, which can be easily compared to absorption coefficients measured using transmission spectroscopy. l and Kabs can be seen as equivalent to the S and K parameters, respectively, in the Kubelka-Munk formulation. The advantage of our approximation is a clear basis in the complete radiative transfer theory. We demonstrate the application of our model to a range of different paper types and to fabrics treated with known levels of a dye. 8. Analytical solutions for Tokamak equilibria with reversed toroidal current Energy Technology Data Exchange (ETDEWEB) Martins, Caroline G. L.; Roberto, M.; Braga, F. L. [Departamento de Fisica, Instituto Tecnologico de Aeronautica, Sao Jose dos Campos, Sao Paulo 12228-900 (Brazil); Caldas, I. L. [Instituto de Fisica, Universidade de Sao Paulo, 05315-970 Sao Paulo, SP (Brazil) 2011-08-15 In tokamaks, an advanced plasma confinement regime has been investigated with a central hollow electric current with negative density which gives rise to non-nested magnetic surfaces. We present analytical solutions for the magnetohydrodynamic equilibria of this regime in terms of non-orthogonal toroidal polar coordinates. These solutions are obtained for large aspect ratio tokamaks and they are valid for any kind of reversed hollow current density profiles. The zero order solution of the poloidal magnetic flux function describes nested toroidal magnetic surfaces with a magnetic axis displaced due to the toroidal geometry. The first order correction introduces a poloidal field asymmetry and, consequently, magnetic islands arise around the zero order surface with null poloidal magnetic flux gradient. An analytic expression for the magnetic island width is deduced in terms of the equilibrium parameters. We give examples of the equilibrium plasma profiles and islands obtained for a class of current density profile. 9. Nonlinear inertial oscillations of a multilayer eddy: An analytical solution Science.gov (United States) Dotsenko, S. F.; Rubino, A. 2008-06-01 Nonlinear axisymmetric oscillations of a warm baroclinic eddy are considered within the framework of an reduced-gravity model of the dynamics of a multilayer ocean. A class of exact analytical solutions describing pure inertial oscillations of an eddy formation is found. The thicknesses of layers in the eddy vary according to a quadratic law, and the horizontal projections of the velocity in the layers depend linearly on the radial coordinate. Owing to a complicated structure of the eddy, weak limitations on the vertical distribution of density, and an explicit form of the solution, the latter can be treated as a generalization of the exact analytical solutions of this form that were previously obtained for homogeneous and baroclinic eddies in the ocean. 10. ANALYTICAL SOLUTION OF FILLING AND EXHAUSTING PROCESS IN PNEUMATIC SYSTEM Institute of Scientific and Technical Information of China (English) 2006-01-01 The filling and exhausting processes in a pneumatic system are involved with many factors,and numerical solutions of many partial differential equations are always adapted in the study of those processes, which have been proved to be troublesome and less intuitive. Analytical solutions based on loss-less tube model and average friction tube model are found respectively by using fluid net theory,and they fit the experimental results well. The research work shows that: Fluid net theory can be used to solve the analytical solution of filling and exhausting processes of pneumatic system, and the result of loss-less tube model is close to that of average friction model, so loss-less tube model is recommended since it is simpler, and the difference between filling time and exhausting time is determined by initial and final pressures, the volume of container and the section area of tube, and has nothing to do with the length of the tube. 11. Dual coupled radiative transfer equation and diffusion approximation for the solution of the forward problem in fluorescence molecular imaging Science.gov (United States) Gorpas, Dimitris; Andersson-Engels, Stefan 2012-03-01 The solution of the forward problem in fluorescence molecular imaging is among the most important premises for the successful confrontation of the inverse reconstruction problem. To date, the most typical approach has been the application of the diffusion approximation as the forward model. This model is basically a first order angular approximation for the radiative transfer equation, and thus it presents certain limitations. The scope of this manuscript is to present the dual coupled radiative transfer equation and diffusion approximation model for the solution of the forward problem in fluorescence molecular imaging. The integro-differential equations of its weak formalism were solved via the finite elements method. Algorithmic blocks with cubature rules and analytical solutions of the multiple integrals have been constructed for the solution. Furthermore, specialized mapping matrices have been developed to assembly the finite elements matrix. As a radiative transfer equation based model, the integration over the angular discretization was implemented analytically, while quadrature rules were applied whenever required. Finally, this model was evaluated on numerous virtual phantoms and its relative accuracy, with respect to the radiative transfer equation, was over 95%, when the widely applied diffusion approximation presented almost 85% corresponding relative accuracy for the fluorescence emission. 12. Approximate solutions to infinite dimensional LQ problems over infinite time horizon Institute of Scientific and Technical Information of China (English) PAN; Liping; ZHANG; Xu; CHEN; Qihong 2006-01-01 This paper is addressed to develop an approximate method to solve a class of infinite dimensional LQ optimal regulator problems over infinite time horizon. Our algorithm is based on a construction of approximate solutions which solve some finite dimensional LQ optimal regulator problems over finite time horizon, and it is shown that these approximate solutions converge strongly to the desired solution in the double limit sense. 13. THE ANALYTICAL SOLUTION FOR SEDIMENT REACTION AND DIFFUSION EQUATION WITH GENERALIZED INITIAL-BOUNDARY CONDITIONS Institute of Scientific and Technical Information of China (English) 熊岳山; 韦永康 2001-01-01 The sediment reaction and diffusion equation with generalized initial and boundary condition is studied. By using Laplace transform and Jordan lemma , an analytical solution is got, which is an extension of analytical solution provided by Cheng Kwokming James ( only diffusion was considered in analytical solution of Cheng ). Some problems arisen in the computation of analytical solution formula are also analysed. 14. Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach Institute of Scientific and Technical Information of China (English) G. Darmani; S. Setayeshi; H. Ramezanpour 2012-01-01 In this paper an efficient computational method based on extending the sensitivity approach （SA） is proposed to find an analytic exact solution of nonlinear differential difference equations. In this manner we avoid solving the nonlinear problem directly. By extension of sensitivity approach for differential difference equations （DDEs）, the nonlinear original problem is transformed into infinite linear differential difference equations, which should be solved in a recursive manner. Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained. Numerical examples are employed to show the effectiveness of the proposed approach. 15. Nonlinear Whitham-Broer-Kaup Wave Equation in an Analytical Solution Directory of Open Access Journals (Sweden) S. A. Zahedi 2008-01-01 Full Text Available This study presented a new approach for the analysis of a nonlinear Whitham-Broer-Kaup equation dealing with propagation of shallow water waves with different dispersion relations. The analysis was based on a kind of analytical method, called Variational Iteration Method (VIM. To illustrate the capability of the approach, some numerical examples were given and the propagation and the error of solutions were shown in comparison to those of exact solution. In clear conclusion, the approach was efficient and capable to obtain the analytical approximate solution of this set of wave equations while these solutions could straightforwardly show some facts of the described process deeply such as the propagation. This method can be easily extended to other nonlinear wave equations and so can be found widely applicable in this field of science. 16. A composite analytical solution for large break LOCA Energy Technology Data Exchange (ETDEWEB) Purdy, P. [Bruce Power, Tiverton, Ontario (Canada); Girard, R. [Hydro-Quebec, Quebec (Canada); Marczak, J. [Ontario Power Generation, Ontario (Canada); Taylor, D. [New Brunswick Power, Fredericton, New Brunswick (Canada); Zemdegs, R. [Candu Energy Inc., Mississauga, Ontario (Canada); Kapaklili, T. [CANDU Owner' s Group, Toronto, Ontario (Canada); Balog, G. [AMEC NSS, Ontario (Canada); Kozluk, M. [Independent Consultant (Canada); Oliva, A. [Candesco, Ontario (Canada) 2011-07-01 The Canadian CANDU Industry is implementing a composite analytical solution to demonstrate, with high confidence, adequate safety margins for Large Break Loss of Coolant Accidents (LBLOCA) in existing CANDU reactors. The approach involves consolidating a number of individual approaches in a manner that alleviates reliance on any single analytical method or activity. Using a multi-layered approach, the objective of this composite solution is to use a variety of reinforcing analytical approaches such that they complement one another to collectively form a robust solution. The composite approach involves: i) systematic reclassification of LBLOCA to beyond design basis events based on the frequency of the limiting initiating events; ii) more realistic modeling of break opening characteristics; iii) application of Best Estimate and Uncertainty (BEAU) analysis methodology to provide a more realistic representation of the margins; iv) continued application of Limit of Operating Envelope (LOE) methodology to demonstrate the adequacy of margins at the extremes of the operating envelope; v) characterizing the coolant void reactivity, with associated uncertainties; and vi) defining suitable acceptance criteria, accounting for the available experimental database and uncertainties. The approach is expected to confirm the adequacy of existing design provisions and, as such, better characterize the overall safety significance of LBLOCA in CANDU reactors. This paper describes the composite analytical approach and its development, implementation and current status. (author) 17. Analytic Solution of Strongly Coupling Schr(o)dinger Equations Institute of Scientific and Technical Information of China (English) LIAO Jin-Feng; ZHUANG Peng-Fei 2004-01-01 A recently developed expansion method for analytically solving the ground states of strongly coupling Schrodinger equations by Friedberg,Lee,and Zhao is extended to excited states and applied to power-law central forces for which scaling properties are proposed.As examples for application of the extended method,the Hydrogen atom problem is resolved and the low-lying states of Yukawa potential are approximately obtained. 18. Simple and Accurate Analytical Solutions of the Electrostatically Actuated Curled Beam Problem KAUST Repository Younis, Mohammad I. 2014-08-17 We present analytical solutions of the electrostatically actuated initially deformed cantilever beam problem. We use a continuous Euler-Bernoulli beam model combined with a single-mode Galerkin approximation. We derive simple analytical expressions for two commonly observed deformed beams configurations: the curled and tilted configurations. The derived analytical formulas are validated by comparing their results to experimental data in the literature and numerical results of a multi-mode reduced order model. The derived expressions do not involve any complicated integrals or complex terms and can be conveniently used by designers for quick, yet accurate, estimations. The formulas are found to yield accurate results for most commonly encountered microbeams of initial tip deflections of few microns. For largely deformed beams, we found that these formulas yield less accurate results due to the limitations of the single-mode approximations they are based on. In such cases, multi-mode reduced order models need to be utilized. 19. Capacity of the circular plate condenser: analytical solutions for large gaps between the plates Science.gov (United States) Rao, T. V. 2005-11-01 A solution of Love's integral equation (Love E R 1949 Q. J. Mech. Appl. Math. 2 428), which forms the basis for the analysis of the electrostatic field due to two equal circular co-axial parallel conducting plates, is considered for the case when the ratio, τ, of distance of separation to radius of the plates is greater than 2. The kernel of the integral equation is expanded into an infinite series in odd powers of 1/τ and an approximate kernel accurate to {\\cal O}(\\tau^{-(2N+1)}) is deduced therefrom by terminating the series after an arbitrary but finite number of terms, N. The approximate kernel is rearranged into a degenerate form and the integral equation with this kernel is reduced to a system of N linear equations. An explicit analytical solution is obtained for N = 4 and the resulting analytical expression for the capacity of the circular plate condenser is shown to be accurate to {\\cal O}(\\tau^{-9}) . Analytical expressions of lower orders of accuracy with respect to 1/τ are deduced from the four-term (i.e., N = 4) solution and predictions (of capacity) from the expressions of different orders of accuracy (with respect to 1/τ) are compared with very accurate numerical solutions obtained by solving the linear system for large enough N. It is shown that the {\\cal O}(\\tau^{-9}) approximation predicts the capacity extremely well for any τ >= 2 and an {\\cal O}(\\tau^{-3}) approximation gives, for all practical purposes, results of adequate accuracy for τ >= 4. It is further shown that an approximate solution, applicable for the case of large distances of separation between the plates, due to Sneddon (Sneddon I N 1966 Mixed Boundary Value Problems in Potential Theory (Amsterdam: North-Holland) pp 230-46) is accurate to {\\cal O}(\\tau^{-6}) for τ >= 2. 20. Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials CERN Document Server Finster, Felix 2008-01-01 A method is presented for obtaining rigorous error estimates for approximate solutions of the Riccati equation, with real or complex potentials. Our main tool is to derive invariant region estimates for complex solutions of the Riccati equation. We explain the general strategy for applying these estimates and illustrate the method in typical examples, where the approximate solutions are obtained by glueing together WKB and Airy solutions of corresponding one-dimensional Schr"odinger equations. 1. On the Partial Analytical Solution of the Kirchhoff Equation KAUST Repository Michels, Dominik L. 2015-09-01 We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have the disadvantage that the quality of solutions decreases enormously with increasing temporal step sizes, which results from the numerical stiffness of the underlying partial differential equations. To prevent that, we apply a differential Thomas decomposition and a Lie symmetry analysis to derive explicit analytical solutions to specific parts of the Kirchhoff system. These solutions are general and depend on arbitrary functions, which we set up according to the numerical solution of the remaining parts. In contrast to a purely numerical handling, this reduces the numerical solution space and prevents the system from becoming unstable. The differential Kirchhoff equation describes the dynamic equilibrium of one-dimensional continua, i.e. slender structures like fibers. We evaluate the advantage of our method by simulating a cilia carpet. 2. Analytic solution of an oscillatory migratory α2 stellar dynamo Science.gov (United States) Brandenburg, A. 2017-02-01 Context. Analytic solutions of the mean-field induction equation predict a nonoscillatory dynamo for homogeneous helical turbulence or constant α effect in unbounded or periodic domains. Oscillatory dynamos are generally thought impossible for constant α. Aims: We present an analytic solution for a one-dimensional bounded domain resulting in oscillatory solutions for constant α, but different (Dirichlet and von Neumann or perfect conductor and vacuum) boundary conditions on the two boundaries. Methods: We solve a second order complex equation and superimpose two independent solutions to obey both boundary conditions. Results: The solution has time-independent energy density. On one end where the function value vanishes, the second derivative is finite, which would not be correctly reproduced with sine-like expansion functions where a node coincides with an inflection point. The field always migrates away from the perfect conductor boundary toward the vacuum boundary, independently of the sign of α. Conclusions: The obtained solution may serve as a benchmark for numerical dynamo experiments and as a pedagogical illustration that oscillatory migratory dynamos are possible with constant α. 3. Analytical exact solution of the non-linear Schroedinger equation Energy Technology Data Exchange (ETDEWEB) Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da [Universidade de Brasilia (UnB), DF (Brazil). Inst. de Fisica. Grupo de Fisica e Matematica 2011-07-01 Full text: In this work we present how to classify and obtain analytical solutions of the Schroedinger equation with a generic non-linearity in 1+1 dimensions. Our approach is based on the determination of Lie symmetry transformation mapping solutions into solutions, and non-classical symmetry transformations, mapping a given solution into itself. From these symmetries it is then possible to reduce the equation to a system of ordinary differential equations which can then be solved using standard methods. The generic non-linearity is handled by considering it as an additional unknown in the determining equations for the symmetry transformations. This results in an over-determined system of non-linear partial differential equations. Its solution can then be determined in some cases by reducing it to the so called involutive (triangular) form, and then solved. This reduction is very tedious and can only performed using a computer algebra system. Once the determining system is solved, we obtain the explicit form for the non-linearity admitting a Lie or non-classical symmetry. The analytical solutions are then derived by solving the reduced ordinary differential equations. The non-linear determining system for the non-classical symmetry transformations and Lie symmetry generators are obtaining using the computer algebra package SADE (symmetry analysis of differential equations), developed at our group. (author) 4. Error analysis of analytic solutions for self-excited near-symmetric rigid bodies - A numerical study Science.gov (United States) Kia, T.; Longuski, J. M. 1984-01-01 Analytic error bounds are presented for the solutions of approximate models for self-excited near-symmetric rigid bodies. The error bounds are developed for analytic solutions to Euler's equations of motion. The results are applied to obtain a simplified analytic solution for Eulerian rates and angles. The results of a sample application of the range and error bound expressions for the case of the Galileo spacecraft experiencing transverse torques demonstrate the use of the bounds in analyses of rigid body spin change maneuvers. 5. Dynamic characteristics of resonant gyroscopes study based on the Mathieu equation approximate solution Institute of Scientific and Technical Information of China (English) Fan Shang-Chun; Li Yan; Guo Zhan-She; Li Jing; Zhuang Hai-Han 2012-01-01 Dynamic characteristics of the resonant gyroscope are studied based on the Mathieu equation approximate solution in this paper.The Mathieu equation is used to analyze the parametric resonant characteristics and the approximate output of the resonant gyroscope.The method of small parameter perturbation is used to analyze the approximate solution of the Mathieu equation.The theoretical analysis and the numerical simulations show that the approximate solution of the Mathieu equation is close to the dynamic output characteristics of the resonant gyroscope.The experimental analysis shows that the theoretical curve and the experimental data processing results coincide perfectly,which means that the approximate solution of the Mathieu equation can present the dynamic output characteristic of the resonant gyroscope.The theoretical approach and the experimental results of the Mathieu equation approximate solution are obtained,which provides a reference for the robust design of the resonant gyroscope. 6. ANALYTICAL SOLUTIONS TO STRESS CONCENTRATION PROBLEM IN PLATES CONTAINING RECTANGULAR HOLE UNDER BIAXIAL TENSIONS Institute of Scientific and Technical Information of China (English) Yi Yang; Jike Liu; Chengwu Cai 2008-01-01 The stress concentration problem in structures with a circular or elliptic hole can be investigated by analytical methods.For the problem with a rectangular hole,only approximate results are derived.This paper deduces the analytical solutions to the stress concentration problem in plates with a rectangular hole under biaxial tensions.By using the U-transformation technique and the finite element method,the analytical displacement solutions of the finite element equations are derived in the series form.Therefore,the stress concentration can then be discussed easily and conveniently.For plate problem the bilinear rectangular element with four nodes is taken as an example to demonstrate the applicability of the proposed method.The stress concentration factors for various ratios of height to width of the hole are obtained. 7. Analytical correction of an extension of the "MU Fraction Approximation" for Varian enhanced dynamic wedges. Science.gov (United States) Gossman, Michael S; Sharma, Subhash C 2010-04-01 The most common method to determine enhanced dynamic wedge factors begins with the use of segmented treatment tables. These segmental dose delivery values set as a function of upper jaw position are the backbone of a calculation process coined the "MU Fraction Approximation." Analytical and theoretical attempts have been made to extend and alter the mathematics for this approximation for greater accuracy. A set of linear equations in the form of a matrix are introduced here which correct one published extension of the MU Fraction Approximation as it applies to both symmetric and asymmetric photon fields. The matrix results are compared to data collected from a commissioned Varian Eclipse Treatment Planning System and previously published research for Varian linear accelerators. A total enhanced dynamic wedge factor with excellent accuracy was achieved in comparison to the most accurate previous research found. The deviation seen here is only 0.4% and 1.0% for symmetric and asymmetric fields respectively, for both 6MV and 18MV photon beams. 8. Analytical solution of the simplified spherical harmonics equations in spherical turbid media Science.gov (United States) Edjlali, Ehsan; Bérubé-Lauzière, Yves 2016-10-01 We present for the first time an analytical solution for the simplified spherical harmonics equations (so-called SPN equations) in the case of a steady-state isotropic point source inside a spherical homogeneous absorbing and scattering medium. The SPN equations provide a reliable approximation to the radiative transfer equation for describing light transport inside turbid media. The SPN equations consist of a set of coupled partial differential equations and the eigen method is used to obtain a set of decoupled equations, each resembling the heat equation in the Laplace domain. The equations are solved for the realistic partial reflection boundary conditions accounting for the difference in refractive indices between the turbid medium and its environment (air) as occurs in practical cases of interest in biomedical optics. Specifically, we provide the complete solution methodology for the SP3, which is readily applicable to higher orders as well, and also give results for the SP5. This computationally easy to obtain solution is investigated for different optical properties of the turbid medium. For validation, the solution is also compared to the analytical solution of the diffusion equation and to gold standard Monte Carlo simulation results. The SP3 and SP5 analytical solutions prove to be in good agreement with the Monte Carlo results. This work provides an additional tool for validating numerical solutions of the SPN equations for curved geometries. 9. An analytic cosmology solution of Poincaré gauge gravity Science.gov (United States) Lu, Jianbo; Chee, Guoying 2016-06-01 A cosmology of Poincaré gauge theory is developed. An analytic solution is obtained. The calculation results agree with observation data and can be compared with the ΛCDM model. The cosmological constant puzzle is the coincidence and fine tuning problem are solved naturally at the same time. The cosmological constant turns out to be the intrinsic torsion and curvature of the vacuum universe, and is derived from the theory naturally rather than added artificially. The dark energy originates from geometry, includes the cosmological constant but differs from it. The analytic expression of the state equations of the dark energy and the density parameters of the matter and the geometric dark energy are derived. The full equations of linear cosmological perturbations and the solutions are obtained. 10. Analytic solutions for seismic travel time and ray path geometry through simple velocity models. Energy Technology Data Exchange (ETDEWEB) Ballard, Sanford 2007-12-01 The geometry of ray paths through realistic Earth models can be extremely complex due to the vertical and lateral heterogeneity of the velocity distribution within the models. Calculation of high fidelity ray paths and travel times through these models generally involves sophisticated algorithms that require significant assumptions and approximations. To test such algorithms it is desirable to have available analytic solutions for the geometry and travel time of rays through simpler velocity distributions against which the more complex algorithms can be compared. Also, in situations where computational performance requirements prohibit implementation of full 3D algorithms, it may be necessary to accept the accuracy limitations of analytic solutions in order to compute solutions that satisfy those requirements. Analytic solutions are described for the geometry and travel time of infinite frequency rays through radially symmetric 1D Earth models characterized by an inner sphere where the velocity distribution is given by the function V (r) = A-Br{sup 2}, optionally surrounded by some number of spherical shells of constant velocity. The mathematical basis of the calculations is described, sample calculations are presented, and results are compared to the Taup Toolkit of Crotwell et al. (1999). These solutions are useful for evaluating the fidelity of sophisticated 3D travel time calculators and in situations where performance requirements preclude the use of more computationally intensive calculators. It should be noted that most of the solutions presented are only quasi-analytic. Exact, closed form equations are derived but computation of solutions to specific problems generally require application of numerical integration or root finding techniques, which, while approximations, can be calculated to very high accuracy. Tolerances are set in the numerical algorithms such that computed travel time accuracies are better than 1 microsecond. 11. Analytical solution for inviscid flow inside an evaporating sessile drop OpenAIRE Masoud, Hassan; Felske, James D. 2008-01-01 Inviscid flow within an evaporating sessile drop is analyzed. The field equation, E^2(Psi)=0, is solved for the stream function. The exact analytical solution is obtained for arbitrary contact angle and distribution of evaporative flux along the free boundary. Specific results and computations are presented for evaporation corresponding to both uniform flux and purely diffusive gas phase transport into an infinite ambient. Wetting and non-wetting contact angles are considered with flow patter... 12. Analytical Analysis and Numerical Solution of Two Flavours Skyrmion CERN Document Server Hadi, Miftachul; Hermawanto, Denny 2010-01-01 Two flavours Skyrmion will be analyzed analytically, in case of static and rotational Skyrme equations. Numerical solution of a nonlinear scalar field equation, i.e. the Skyrme equation, will be worked with finite difference method. This article is a more comprehensive version of \\textit{SU(2) Skyrme Model for Hadron} which have been published at Journal of Theoretical and Computational Studies, Volume \\textbf{3} (2004) 0407. 13. Molecular clock fork phylogenies: closed form analytic maximum likelihood solutions. Science.gov (United States) Chor, Benny; Snir, Sagi 2004-12-01 Maximum likelihood (ML) is increasingly used as an optimality criterion for selecting evolutionary trees, but finding the global optimum is a hard computational task. Because no general analytic solution is known, numeric techniques such as hill climbing or expectation maximization (EM) are used in order to find optimal parameters for a given tree. So far, analytic solutions were derived only for the simplest model-three-taxa, two-state characters, under a molecular clock. Quoting Ziheng Yang, who initiated the analytic approach,"this seems to be the simplest case, but has many of the conceptual and statistical complexities involved in phylogenetic estimation."In this work, we give general analytic solutions for a family of trees with four-taxa, two-state characters, under a molecular clock. The change from three to four taxa incurs a major increase in the complexity of the underlying algebraic system, and requires novel techniques and approaches. We start by presenting the general maximum likelihood problem on phylogenetic trees as a constrained optimization problem, and the resulting system of polynomial equations. In full generality, it is infeasible to solve this system, therefore specialized tools for the molecular clock case are developed. Four-taxa rooted trees have two topologies-the fork (two subtrees with two leaves each) and the comb (one subtree with three leaves, the other with a single leaf). We combine the ultrametric properties of molecular clock fork trees with the Hadamard conjugation to derive a number of topology dependent identities. Employing these identities, we substantially simplify the system of polynomial equations for the fork. We finally employ symbolic algebra software to obtain closed formanalytic solutions (expressed parametrically in the input data). In general, four-taxa trees can have multiple ML points. In contrast, we can now prove that each fork topology has a unique(local and global) ML point. 14. An analytical dynamo solution for large-scale magnetic fields of galaxies Science.gov (United States) Chamandy, Luke 2016-11-01 We present an effectively global analytical asymptotic galactic dynamo solution for the regular magnetic field of an axisymmetric thin disc in the saturated state. This solution is constructed by combining two well-known types of local galactic dynamo solution, parametrized by the disc radius. Namely, the critical (zero growth) solution obtained by treating the dynamo equation as a perturbed diffusion equation is normalized using a non-linear solution that makes use of the no-z' approximation and the dynamical α-quenching non-linearity. This overall solution is found to be reasonably accurate when compared with detailed numerical solutions. It is thus potentially useful as a tool for predicting observational signatures of magnetic fields of galaxies. In particular, such solutions could be painted on to galaxies in cosmological simulations to enable the construction of synthetic polarized synchrotron and Faraday rotation measure data sets. Further, we explore the properties of our numerical solutions, and their dependence on certain parameter values. We illustrate and assess the degree to which numerical solutions based on various levels of approximation, common in the dynamo literature, agree with one another. 15. Propagation of quasiplane nonlinear waves in tubes and the approximate solutions of the generalized Burgers equation. Science.gov (United States) Bednarik, Michal; Konicek, Petr 2002-07-01 This paper deals with using the generalized Burgers equation for description of nonlinear waves in circular ducts. Two new approximate solutions of the generalized Burgers equation (GBE) are presented. These solutions take into account the boundary layer effects. The first solution is valid for the preshock region and gives more precise results than the Fubini solution, whereas the second one is valid for the postshock (sawtooth) region and provides better results than the Fay solution. The approximate solutions are compared with numerical results of the GBE. Furthermore, the limits of validity of the used model equation are discussed with respect to boundary conditions and radius of a circular duct. 16. Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation Directory of Open Access Journals (Sweden) Berenguer MI 2010-01-01 Full Text Available This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of this type of equation, due to some properties of certain biorthogonal systems for the Banach spaces and . 17. Identification of standing fronts in steady state fluid flows: exact and approximate solutions for propagating MHD modes Science.gov (United States) Pantellini, Filippo; Griton, Léa 2016-10-01 The spatial structure of a steady state plasma flow is shaped by the standing modes with local phase velocity exactly opposite to the flow velocity. The general procedure of finding the wave vectors of all possible standing MHD modes in any given point of a stationary flow requires numerically solving an algebraic equation. We present the graphical procedure (already mentioned by some authors in the 1960's) along with the exact solution for the Alfvén mode and approximate analytic solutions for both fast and slow modes. The technique can be used to identify MHD modes in space and laboratory plasmas as well as in numerical simulations. 18. Communication: An efficient analytic gradient theory for approximate spin projection methods Science.gov (United States) Hratchian, Hrant P. 2013-03-01 Spin polarized and broken symmetry density functional theory are popular approaches for treating the electronic structure of open shell systems. However, spin contamination can significantly affect the quality of predicted geometries and properties. One scheme for addressing this concern in studies involving broken-symmetry states is the approximate projection method developed by Yamaguchi and co-workers. Critical to the exploration of potential energy surfaces and the study of properties using this method will be an efficient analytic gradient theory. This communication introduces such a theory formulated, for the first time, within the framework of general post-self consistent field (SCF) derivative theory. Importantly, the approach taken here avoids the need to explicitly solve for molecular orbital derivatives of each nuclear displacement perturbation, as has been used in a recent implementation. Instead, the well-known z-vector scheme is employed and only one SCF response equation is required. 19. Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport Energy Technology Data Exchange (ETDEWEB) Litvinenko, Yuri E.; Effenberger, Frederic, E-mail: [email protected] [Department of Mathematics, University of Waikato, P.B. 3105 Hamilton (New Zealand) 2014-12-01 Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we analytically solve a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model. 20. A transversely localized light in a waveguide: the analytical solution and its potential application Science.gov (United States) Arslanov, Narkis M.; Moiseev, Sergey A.; Kamli, Ali A. 2017-02-01 Investigation of light in waveguide structures is a topical modern problem that has long-standing historical roots. A parallel-plate waveguide is a basic model in these studies and is intensively used in numerous investigations of nano-optics, integrated circuits and nanoplasmonics. In this letter we have first found an approximate analytical solution which describes the light modes with high accuracy in the subwavelength waveguides. The solution provides a way of obtaining a clear understanding of the light properties within the broadband spectral range in the waveguide with various physical parameters. The potential of the analytical solution for studies of light fields in the waveguides of nano-optics and nanoplasmonics has also been discussed. 1. Iterative Algorithm for Finding Approximate Solutions of a Class of Mixed Variational-like Inequalities Institute of Scientific and Technical Information of China (English) Liu-chuan Zeng 2004-01-01 The purpose of this paper is to investigate the iterative algorithm for finding approximate solutions of a class of mixed variational-like inequalities in a real Hilbert space,where the iterative algorithm is presented by virtue of the auxiliary principle technique.On one hand,the existence of approximate solutions of this class of mixed variational-like inequalities is proven.On the other hand,it is shown that the approximate solutions converge strongly to the exact solution of this class of mixed variational-like inequalities. 2. Approximation of the solution of certain nonlinear ODEs with linear complexity Science.gov (United States) Dratman, Ezequiel 2010-03-01 We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the "continuous" equation. Furthermore, we exhibit an algorithm computing an [epsilon]-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is linear in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required. 3. Analytical solutions of the electrostatically actuated curled beam problem KAUST Repository Younis, Mohammad I. 2014-07-24 This works presents analytical expressions of the electrostatically actuated initially deformed cantilever beam problem. The formulation is based on the continuous Euler-Bernoulli beam model combined with a single-mode Galerkin approximation. We derive simple analytical expressions for two commonly observed deformed beams configurations: the curled and tilted configurations. The derived analytical formulas are validated by comparing their results to experimental data and numerical results of a multi-mode reduced order model. The derived expressions do not involve any complicated integrals or complex terms and can be conveniently used by designers for quick, yet accurate, estimations. The formulas are found to yield accurate results for most commonly encountered microbeams of initial tip deflections of few microns. For largely deformed beams, we found that these formulas yield less accurate results due to the limitations of the single-mode approximation. In such cases, multi-mode reduced order models are shown to yield accurate results. © 2014 Springer-Verlag Berlin Heidelberg. 4. Approximate analytic method for high-apogee twelve-hour orbits of artificial Earth's satellites Science.gov (United States) Vashkovyaka, M. A.; Zaslavskii, G. S. 2016-09-01 We propose an approach to the study of the evolution of high-apogee twelve-hour orbits of artificial Earth's satellites. We describe parameters of the motion model used for the artificial Earth's satellite such that the principal gravitational perturbations of the Moon and Sun, nonsphericity of the Earth, and perturbations from the light pressure force are approximately taken into account. To solve the system of averaged equations describing the evolution of the orbit parameters of an artificial satellite, we use both numeric and analytic methods. To select initial parameters of the twelve-hour orbit, we assume that the path of the satellite along the surface of the Earth is stable. Results obtained by the analytic method and by the numerical integration of the evolving system are compared. For intervals of several years, we obtain estimates of oscillation periods and amplitudes for orbital elements. To verify the results and estimate the precision of the method, we use the numerical integration of rigorous (not averaged) equations of motion of the artificial satellite: they take into account forces acting on the satellite substantially more completely and precisely. The described method can be applied not only to the investigation of orbit evolutions of artificial satellites of the Earth; it can be applied to the investigation of the orbit evolution for other planets of the Solar system provided that the corresponding research problem will arise in the future and the considered special class of resonance orbits of satellites will be used for that purpose. 5. On the analytical solution of Fornberg–Whitham equation with the new fractional derivative Indian Academy of Sciences (India) Olaniyi Samuel Iyiola; Gbenga Olayinka Ojo 2015-10-01 Motivated by the simplicity, natural and efficient nature of the new fractional derivative introduced by R Khalil et al in J. Comput. Appl. Math. 264, 65 (2014), analytical solution of space-time fractional Fornberg–Whitham equation is obtained in series form using the relatively new method called q-homotopy analysis method (q-HAM). The new fractional derivative makes it possible to introduce fractional order in space to the Fornberg–Whitham equation and be able to obtain its solution. This work displays the elegant nature of the application of q-HAM to solve strongly nonlinear fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for nonlinear differential equations. Comparisons are made on the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions. 6. Seismic responses of a hemispherical alluvial valley to SV Waves: a three-dimensional analytical approximation Institute of Scientific and Technical Information of China (English) Chenggang Zhao; Jun Dong; Fuping Gao; D.-S.Jeng 2006-01-01 An analytical solution to the three-dimensional scattering and diffraction of plane SV-waves by a saturated hemispherical alluvial valley in elastic halfspace is obtained by using Fourier-Bessel series expansion technique.The hemispherical alluvial valley with saturated soil deposits is simulated with Biot's dynamic theory for saturated porous media.The following conclusions based on numerical results can be drawn:(1) there are a significant differences in the seismic response simulation between the previous single-phase models and the present two-phase model;(2)the normalized displacements on the free surface of the alluvial valley depend mainly on the incident wave angles,the dimensionless frequency of the incident SV waves and the porosity of sediments;(3)with the increase of the incident angle,the displacement distributions become more complicated,and the displacements on the free surface of the alluvial valley increase as the porosity of sediments increases. 7. Effects of dynamic contact angle on liquid infiltration into horizontal capillary tubes: (semi)-analytical solutions. Science.gov (United States) Hilpert, Markus 2009-09-01 We generalize Washburn's analytical solution for capillary flow in a horizontally oriented tube by accounting for a dynamic contact angle. We consider two general models for dynamic contact angle: the uncompensated Young force on the contact line depends on the capillary number in the form of either (1) a power law with exponent beta or (2) a power series. By considering the ordinary differential equation (ODE) for the velocity of the gas-liquid interface instead of the ODE for the interface position, we are able to derive new analytical solutions. For both dynamic contact angle models, we derive analytical solutions for the travel time of the gas-liquid interface as a function of interface velocity. The interface position as a function of time can be obtained through numerical integration. For the power law and beta=1 (an approximation of Cox's model for dynamic contact angle), we obtain an analytical solution for both interface position and velocity as a function of time. For the power law and beta=3, we can express the interface velocity as a function of time. 8. A new analytic solution for 2nd-order Fermi acceleration CERN Document Server Mertsch, Philipp 2011-01-01 A new analytic solution for 2nd-order Fermi acceleration is presented. In particular, we consider time-dependent rates for stochastic acceleration, diffusive and convective escape as well as adiabatic losses. The power law index q of the turbulence spectrum is unconstrained and can therefore account for Kolmogorov (q = 5/3) and Kraichnan (q = 3/2) turbulence, Bohm diffusion (q = 1) as well as the hard-sphere approximation (q = 2). This considerably improves beyond solutions known to date and will prove a useful tool for more realistic modelling of 2nd-order Fermi acceleration in a variety of astrophysical environments. 9. Functions of diffraction correction and analytical solutions in nonlinear acoustic measurement CERN Document Server Alliès, Laurent; Nadi, M 2008-01-01 This paper presents an analytical formulation for correcting the diffraction associated to the second harmonic of an acoustic wave, more compact than that usually used. This new formulation, resulting from an approximation of the correction applied to fundamental, makes it possible to obtain simple solutions for the second harmonic of the average acoustic pressure, but sufficiently precise for measuring the parameter of nonlinearity B/A in a finite amplitude method. Comparison with other expressions requiring numerical integration, show the solutions are precise in the nearfield. 10. Mathematic Model and Analytic Solution for a Cylinder Subject to Exponential Function Institute of Scientific and Technical Information of China (English) LIU Wen; SHAN Rui 2009-01-01 Hollow cylinders are widely used in spacecraft, rockets, weapons, metallurgy, materials, and mechanical manufacturing industries, and so on, hydraulic bulging roll cylinder and hydraulic press work all belong to hollow cylinders. However, up till now, the solution of the cylinder subjected to the pressures in the three-dimensional space is still at the stage of the analytical solution to the normal pressure or the approximate solution to the variable pressure by numerical method. The analytical solution to the variable pressure of the cylinder has not yet made any breakthrough in theory and can not meet accurate theoretical analysis and calculation requirements of the cylindrical in Engineering. In view of their importance, the precision calculation and theoretical analysis are required to investigate on engineering. A stress function which meets both the biharmonic equations and boundary conditions is constructed in the three-dimensional space. Furthermore, the analytic solution of a hollow cylinder subjected to exponential function distributed variable pressure on its inner and outer surfaces is deduced. By controlling the pressure subject to exponential function distributed variable pressure in the hydraulic bulging roller without any rolling load, using a static tester to record the strain supported hydraulic bulging roll, and comparing with the theoretical calculation, the experimental test result has a higher degree of agreement with the theoretical calculation. Simultaneously, the famous Lamè solution can be deduced when given the unlimited length of cylinder along the axis. The analytic solution paves the way for the mathematic building and solution of hollow cylinder with randomly uneven pressure. 11. Analytic energy gradients for the coupled-cluster singles and doubles method with the density-fitting approximation. Science.gov (United States) Bozkaya, Uğur; Sherrill, C David 2016-05-07 An efficient implementation is presented for analytic gradients of the coupled-cluster singles and doubles (CCSD) method with the density-fitting approximation, denoted DF-CCSD. Frozen core terms are also included. When applied to a set of alkanes, the DF-CCSD analytic gradients are significantly accelerated compared to conventional CCSD for larger molecules. The efficiency of our DF-CCSD algorithm arises from the acceleration of several different terms, which are designated as the "gradient terms": computation of particle density matrices (PDMs), generalized Fock-matrix (GFM), solution of the Z-vector equation, formation of the relaxed PDMs and GFM, back-transformation of PDMs and GFM to the atomic orbital (AO) basis, and evaluation of gradients in the AO basis. For the largest member of the alkane set (C10H22), the computational times for the gradient terms (with the cc-pVTZ basis set) are 2582.6 (CCSD) and 310.7 (DF-CCSD) min, respectively, a speed up of more than 8-folds. For gradient related terms, the DF approach avoids the usage of four-index electron repulsion integrals. Based on our previous study [U. Bozkaya, J. Chem. Phys. 141, 124108 (2014)], our formalism completely avoids construction or storage of the 4-index two-particle density matrix (TPDM), using instead 2- and 3-index TPDMs. The DF approach introduces negligible errors for equilibrium bond lengths and harmonic vibrational frequencies. 12. Residence time distributions for hydrologic systems: Mechanistic foundations and steady-state analytical solutions Science.gov (United States) Leray, Sarah; Engdahl, Nicholas B.; Massoudieh, Arash; Bresciani, Etienne; McCallum, James 2016-12-01 This review presents the physical mechanisms generating residence time distributions (RTDs) in hydrologic systems with a focus on steady-state analytical solutions. Steady-state approximations of the RTD in hydrologic systems have seen widespread use over the last half-century because they provide a convenient, simplified modeling framework for a wide range of problems. The concept of an RTD is useful anytime that characterization of the timescales of flow and transport in hydrologic systems is important, which includes topics like water quality, water resource management, contaminant transport, and ecosystem preservation. Analytical solutions are often adopted as a model of the RTD and a broad spectrum of models from many disciplines has been applied. Although these solutions are typically reduced in dimensionality and limited in complexity, their ease of use makes them preferred tools, specifically for the interpretation of tracer data. Our review begins with the mechanistic basis for the governing equations, highlighting the physics for generating a RTD, and a catalog of analytical solutions follows. This catalog explains the geometry, boundary conditions and physical aspects of the hydrologic systems, as well as the sampling conditions, that altogether give rise to specific RTDs. The similarities between models are noted, as are the appropriate conditions for their applicability. The presentation of simple solutions is followed by a presentation of more complicated analytical models for RTDs, including serial and parallel combinations, lagged systems, and non-Fickian models. The conditions for the appropriate use of analytical solutions are discussed, and we close with some thoughts on potential applications, alternative approaches, and future directions for modeling hydrologic residence time. 13. Analytical solutions of seawater intrusion in sloping confined and unconfined coastal aquifers Science.gov (United States) Lu, Chunhui; Xin, Pei; Kong, Jun; Li, Ling; Luo, Jian 2016-09-01 Sloping coastal aquifers in reality are ubiquitous and well documented. Steady state sharp-interface analytical solutions for describing seawater intrusion in sloping confined and unconfined coastal aquifers are developed based on the Dupuit-Forchheimer approximation. Specifically, analytical solutions based on the constant-flux inland boundary condition are derived by solving the discharge equation for the interface zone with the continuity conditions of the head and flux applied at the interface between the freshwater zone and the interface zone. Analytical solutions for the constant-head inland boundary are then obtained by developing the relationship between the inland freshwater flux and hydraulic head and combining this relationship with the solutions of the constant-flux inland boundary. It is found that for the constant-flux inland boundary, the shape of the saltwater interface is independent of the geometry of the bottom confining layer for both aquifer types, despite that the geometry of the bottom confining layer determines the location of the interface tip. This is attributed to that the hydraulic head at the interface is identical to that of the coastal boundary, so the shape of the bed below the interface is irrelevant to the interface position. Moreover, developed analytical solutions with an empirical factor on the density factor are in good agreement with the results of variable-density flow numerical modeling. Analytical solutions developed in this study provide a powerful tool for assessment of seawater intrusion in sloping coastal aquifers as well as in coastal aquifers with a known freshwater flux but an arbitrary geometry of the bottom confining layer. 14. An Approximate Solution for the Circumsolar Flow Field of a Sun Moving Through the Local Interstellar Medium (LISM) Science.gov (United States) Ratkiewicz, Romana E.; Scherer, Klaus; Fahr, Hans J.; Cuzzi, Jeffrey N. (Technical Monitor) 1994-01-01 The solar system is in relative motion with respect to the ambient interstellar medium. The supersonic solar wind is expected to pass through the termination shock, thus the solar wind plasma eventually has to enter into an asymptotic outflow geometry appropriately adopted to this counterflow situation. Many attempts have been done to simulate the interaction between the solar wind and the LISM numerically. In this paper we generalize a Parker type analytical solution of the counterflow. The idea is to introduce a special kind of compressibility of the solar wind flow. With the assumption that only a transversal component of the density gradient normal to the flow lines exists we are able to calculate a full set of hydrodynamical quantities describing the circumsolar flow field of a Sun moving through the LISM. The equations governing the velocity and density fields lead to analytical solutions which can be taken as good approximations to the more general case of compressible plasma flows. 15. Approximate solution for nonlinear model of the second and half order reactions in porous catalyst by decomposition method Institute of Scientific and Technical Information of China (English) 2003-01-01 The problem of the process of coupled diffusion and reaction in catalyst pellets is considered for the case of second and half order reactions. The Adomian decomposition method is used to solve the non-linear model. For the second, half and first order reactions, analytical approximate solutions are obtained. The variation of reactant concentration in the catalyst pellet and the effectiveness factors at φ＜10 are determined and compared with those by the BAND's finite difference numerical method developed by Newman. At lower values of φ, the decomposition solution with 3 terms gives satisfactory agreement with the numerical solution; at higher values of φ, as the term number in the decomposition method is increased, an acceptable agreement between the two methods is achieved. In general, the solution with 6 terms gives a satisfactory agreement. 16. On the evolution of the snow line in protoplanetary discs II: Analytic approximations CERN Document Server Martin, Rebecca G 2013-01-01 We examine the evolution of the snow line in a protoplanetary disc that contains a dead zone (a region of zero or low turbulence). The snow line is within a self-gravitating part of the dead zone, and we obtain a fully analytic solution for its radius. Our formula could prove useful for future observational attempts to characterise the demographics of planets outside the snow line. External sources such as comic rays or X-rays from the central star can ionise the disc surface layers and allow the magneto-rotational instability to drive turbulence there. We show that provided that the surface density in this layer is less than about 50 g/cm^2, the dead zone solution exists, after an initial outbursting phase, until the disc is dispersed by photoevaporation. We demonstrate that the snow line radius is significantly larger than that predicted by a fully turbulent disc model, and that in our own solar system it remains outside of the orbital radius of the Earth. Thus, the inclusion of a dead zone into a protoplan... 17. ANALYTICAL SOLUTION OF GROUNDWATER FLUCTUATIONS IN ESTUARINE AQUIFER Institute of Scientific and Technical Information of China (English) CHEN Jing; ZHOU Zhi-fang; JIA Suo-bao 2005-01-01 As a basic factor in the environment of estuary, tidal effects in the coastal aquifer have recently attracted much attention because tidal dynamic also greatly influences the solute transport in the coastal aquifer. Previous studies on tidal dynamic of coastal aquifers have focused on the inland propagation of oceanic tides in the cross-shore direction, a configuration that is essentially one-dimensional. Two-dimensional analytical solutions for groundwater level fluctuation in recent papers are localized in presenting the effect of both oceanic tides and estuarine tides in quadrantal aquifer. A two-dimensional model of groundwater fluctuations in estuarine zone in proposed in this paper. Using complex transform, the two-dimensional flow equation subject to periodic boundary condition is changed into time-independent elliptic problem. Based on Green function method, an analytical solution for groundwater fluctuations in fan-shaped aquifer is derived. The response to of groundwater tidal loading in an estuary and ocean is discussed. The result show that its more extensive application than recent studies. 18. Comparison between analytical and numerical solution of mathematical drying model Science.gov (United States) Shahari, N.; Rasmani, K.; Jamil, N. 2016-02-01 Drying is often related to the food industry as a process of shifting heat and mass inside food, which helps in preserving food. Previous research using a mass transfer equation showed that the results were mostly concerned with the comparison between the simulation model and the experimental data. In this paper, the finite difference method was used to solve a mass equation during drying using different kinds of boundary condition, which are equilibrium and convective boundary conditions. The results of these two models provide a comparison between the analytical and the numerical solution. The result shows a close match between the two solution curves. It is concluded that the two proposed models produce an accurate solution to describe the moisture distribution content during the drying process. This analysis indicates that we have confidence in the behaviour of moisture in the numerical simulation. This result demonstrated that a combined analytical and numerical approach prove that the system is behaving physically. Based on this assumption, the model of mass transfer was extended to include the temperature transfer, and the result shows a similar trend to those presented in the simpler case. 19. Approximate Damped Oscillatory Solutions for Compound KdV-Burgers Equation and Their Error Estimates Institute of Scientific and Technical Information of China (English) Wei-guo ZHANG; Yan ZHAO; Xiao-yan TENG 2012-01-01 In this paper,we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates.We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation.We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions.Furthermore,we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation.We obtain two critical values of r,and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value,while it appears as a damped oscillatory wave if |r| is less than some critical value.By means of analysis and the undetermined coefficients method,we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation.Based on the above discussions and according to the evolution relations of orbits in the global phase portraits,we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method.Finally,using the homogenization principle,we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions.Moreover,we also give the error estimates for these approximate solutions. 20. Cubic autocatalysis in a reaction-diffusion annulus: semi-analytical solutions Science.gov (United States) Alharthi, M. R.; Marchant, T. R.; Nelson, M. I. 2016-06-01 Semi-analytical solutions for cubic autocatalytic reactions are considered in a circularly symmetric reaction-diffusion annulus. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for this novel geometry. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found using a degenerate Hopf bifurcation analysis. A novel feature of this geometry is the effect, of varying the width of the annulus, on the static and dynamic multiplicity. The results show that for a thicker annulus, Hopf bifurcations and multiple steady-state solutions occur in a larger portion of parameter space. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations. 1. The passage of an infinite swept airfoil through an oblique gust. [approximate solution for aerodynamic response Science.gov (United States) Adamczyk, J. L. 1974-01-01 An approximate solution is reported for the unsteady aerodynamic response of an infinite swept wing encountering a vertical oblique gust in a compressible stream. The approximate expressions are of closed form and do not require excessive computer storage or computation time, and further, they are in good agreement with the results of exact theory. This analysis is used to predict the unsteady aerodynamic response of a helicopter rotor blade encountering the trailing vortex from a previous blade. Significant effects of three dimensionality and compressibility are evident in the results obtained. In addition, an approximate solution for the unsteady aerodynamic forces associated with the pitching or plunging motion of a two dimensional airfoil in a subsonic stream is presented. The mathematical form of this solution approaches the incompressible solution as the Mach number vanishes, the linear transonic solution as the Mach number approaches one, and the solution predicted by piston theory as the reduced frequency becomes large. 2. Analytic solution to a class of integro-differential equations Directory of Open Access Journals (Sweden) Xuming Xie 2003-03-01 Full Text Available In this paper, we consider the integro-differential equation $$epsilon^2 y''(x+L(xmathcal{H}(y=N(epsilon,x,y,mathcal{H}(y,$$ wheremathcal{H}(y[x]=frac{1}{pi}(Pint_{-infty}^{infty} frac{y(t}{t-x}dt$is the Hilbert transform. The existence and uniqueness of analytic solution in appropriately chosen space is proved. Our method consists of extending the equation to an appropriately chosen region in the complex plane, then use the Contraction Mapping Theorem. 3. Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials Science.gov (United States) Finster, Felix; Smoller, Joel 2010-09-01 A method is presented for obtaining rigorous error estimates for approximate solutions of the Riccati equation, with real or complex potentials. Our main tool is to derive invariant region estimates for complex solutions of the Riccati equation. We explain the general strategy for applying these estimates and illustrate the method in typical examples, where the approximate solutions are obtained by gluing together WKB and Airy solutions of corresponding one-dimensional Schrödinger equations. Our method is motivated by, and has applications to, the analysis of linear wave equations in the geometry of a rotating black hole. 4. Analytic solution of differential equation for gyroscope's motions Science.gov (United States) Tyurekhodjaev, Abibulla N.; Mamatova, Gulnar U. 2016-08-01 Problems of motion of a rigid body with a fixed point are one of the urgent problems in classical mechanics. A feature of this problem is that, despite the important results achieved by outstanding mathematicians in the last two centuries, there is still no complete solution. This paper obtains an analytical solution of the problem of motion of an axisymmetric rigid body with variable inertia moments in resistant environment described by the system of nonlinear differential equations of L. Euler, involving the partial discretization method for nonlinear differential equations, which was built by A. N. Tyurekhodjaev based on the theory of generalized functions. To such problems belong gyroscopic instruments, in particular, and especially gyroscopes. 5. An Analytical Solution for Lateral Buckling Critical Load Calculation of Leaning-Type Arch Bridge Directory of Open Access Journals (Sweden) Ai-rong Liu 2014-01-01 Full Text Available An analytical solution for lateral buckling critical load of leaning-type arch bridge was presented in this paper. New tangential and radial buckling models of the transverse brace between the main and stable arch ribs are established. Based on the Ritz method, the analytical solution for lateral buckling critical load of the leaning-type arch bridge with different central angles of main arch ribs and leaning arch ribs under different boundary conditions is derived for the first time. Comparison between the analytical results and the FEM calculated results shows that the analytical solution presented in this paper is sufficiently accurate. The parametric analysis results show that the lateral buckling critical load of the arch bridge with fixed boundary conditions is about 1.14 to 1.16 times as large as that of the arch bridge with hinged boundary condition. The lateral buckling critical load increases by approximately 31.5% to 41.2% when stable arch ribs are added, and the critical load increases as the inclined angle of stable arch rib increases. The differences in the center angles of the main arch rib and the stable arch rib have little effect on the lateral buckling critical load. 6. Comparison of input parameters regarding rock mass in analytical solution and numerical modelling Science.gov (United States) Yasitli, N. E. 2016-12-01 Characteristics of stress redistribution around a tunnel excavated in rock are of prime importance for an efficient tunnelling operation and maintaining stability. As it is a well known fact that rock mass properties are the most important factors affecting stability together with in-situ stress field and tunnel geometry. Induced stresses and resultant deformation around a tunnel can be approximated by means of analytical solutions and application of numerical modelling. However, success of these methods depends on assumptions and input parameters which must be representative for the rock mass. However, mechanical properties of intact rock can be found by laboratory testing. The aim of this paper is to demonstrate the importance of proper representation of rock mass properties as input data for analytical solution and numerical modelling. For this purpose, intact rock data were converted into rock mass data by using the Hoek-Brown failure criterion and empirical relations. Stress-deformation analyses together with yield zone thickness determination have been carried out by using analytical solutions and numerical analyses by using FLAC3D programme. Analyses results have indicated that incomplete and incorrect design causes stability and economic problems in the tunnel. For this reason during the tunnel design analytical data and rock mass data should be used together. In addition, this study was carried out to prove theoretically that numerical modelling results should be applied to the tunnel design for the stability and for the economy of the support. 7. Analytical Solution for the Size of the Minimum Dominating Set in Complex Networks CERN Document Server Nacher, Jose C 2016-01-01 Domination is the fastest-growing field within graph theory with a profound diversity and impact in real-world applications, such as the recent breakthrough approach that identifies optimized subsets of proteins enriched with cancer-related genes. Despite its conceptual simplicity, domination is a classical NP-complete decision problem which makes analytical solutions elusive and poses difficulties to design optimization algorithms for finding a dominating set of minimum cardinality in a large network. Here we derive for the first time an approximate analytical solution for the density of the minimum dominating set (MDS) by using a combination of cavity method and Ultra-Discretization (UD) procedure. The derived equation allows us to compute the size of MDS by only using as an input the information of the degree distribution of a given network. 8. The Analytic Solution of the s-Process for Heavy Element Institute of Scientific and Technical Information of China (English) 2002-01-01 In this paper, we investigate the net-work equation of s-process. After divide the s-process into twostandard forms, we get the analytic solution of the net-work equation. With our analytic solution, we 9. An approximate global solution of Einstein's equations for a finite body CERN Document Server Cabezas, J A; Molina, A; Ruiz, E 2006-01-01 We obtain an approximate global stationary and axisymmetric solution of Einstein's equations which can be considered as a simple star model: a self-gravitating perfect fluid ball with constant mass density rotating in rigid motion. Using the post-Minkowskian formalism (weak-field approximation) and considering rotation as a perturbation (slow-rotation approximation), we find approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic and quo-harmonic coordinates. In both cases, interior and exterior solutions are matched, in the sense of Lichnerowicz, on the surface of zero pressure to obtain a global solution. The resulting metric depends on three arbitrary constants: mass density, rotational velocity and the star radius at the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined by these three parameters. It is easy to show that this type of fluid cannot be a source of the Kerr metric 10. Approximation of solutions to retarded differential equations with applications to population dynamics Directory of Open Access Journals (Sweden) D. Bahuguna 2005-01-01 Full Text Available We consider a retarded differential equation with applications to population dynamics. We establish the convergence of a finite-dimensional approximations of a unique solution, the existence and uniqueness of which are also proved in the process. 11. A transformed analytical model for thermal noise of FinFET based on fringing field approximation Science.gov (United States) Madhulika Sharma, Savitesh; Dasgupta, S.; Kartikeyant, M. V. 2016-09-01 This paper delineates the effect of nonplanar structure of FinFETs on noise performance. We demonstrate the thermal noise analytical model that has been inferred by taking into account the presence of an additional inverted region in the extended (underlap) S/D region due to finite gate electrode thickness. Noise investigation includes the effects of source drain resistances which become significant as channel length becomes shorter. In this paper, we evaluate the additional noise caused by three dimensional (3-D) structure of the single fin device and then extended analysis of the multi-fin and multi-fingers structure. The addition of fringe field increases its minimum noise figure and noise resistance of approximately 1 dB and 100 Ω respectively and optimum admittance increases to 5.45 mƱ at 20 GHz for a device operating under saturation region. Hence, our transformed model plays a significant function in evaluation of accurate noise performance at circuit level. Project supported in part by the All India Council for Technical Education (AICTE). 12. Looking into Analytical Approximations for Three-flavor Neutrino Oscillation Probabilities in Matter CERN Document Server Li, Yu-Feng; Zhou, Shun; Zhu, Jing-yu 2016-01-01 Motivated by tremendous progress in neutrino oscillation experiments, we derive a new set of simple and compact formulas for three-flavor neutrino oscillation probabilities in matter of a constant density. A useful definition of the$\\eta$-gauge neutrino mass-squared difference$\\Delta^{}_* \\equiv \\eta \\Delta^{}_{31} + (1-\\eta) \\Delta^{}_{32}$is introduced, where$\\Delta^{}_{ji} \\equiv m^2_j - m^2_i$for$ji = 21, 31, 32$are the ordinary neutrino mass-squared differences and$0 \\leq \\eta \\leq 1$is a real and positive parameter. Expanding neutrino oscillation probabilities in terms of$\\alpha \\equiv \\Delta^{}_{21}/\\Delta^{}_*$, we demonstrate that the analytical formulas can be remarkably simplified for$\\eta = \\cos^2 \\theta^{}_{12}$, with$\\theta_{12}^{}$being the solar mixing angle. As a by-product, the mapping from neutrino oscillation parameters in vacuum to their counterparts in matter is obtained at the order of${\\cal O}(\\alpha^2)$. Finally, we show that our approximate formulas are not only valid f... 13. Looking into analytical approximations for three-flavor neutrino oscillation probabilities in matter Science.gov (United States) Li, Yu-Feng; Zhang, Jue; Zhou, Shun; Zhu, Jing-yu 2016-12-01 Motivated by tremendous progress in neutrino oscillation experiments, we derive a new set of simple and compact formulas for three-flavor neutrino oscillation probabilities in matter of a constant density. A useful definition of the η-gauge neutrino mass-squared difference Δ∗ ≡ ηΔ31 + (1 - η)Δ32 is introduced, where Δ ji ≡ m j 2 - m i 2 for ji = 21 , 31 , 32 are the ordinary neutrino mass-squared differences and 0 ≤ η ≤ 1 is a real and positive parameter. Expanding neutrino oscillation probabilities in terms of α ≡ Δ21 /Δ∗, we demonstrate that the analytical formulas can be remarkably simplified for η = cos2 θ 12, with θ 12 being the solar mixing angle. As a by-product, the mapping from neutrino oscillation parameters in vacuum to their counterparts in matter is obtained at the order of O({α}^2) . Finally, we show that our approximate formulas are not only valid for an arbitrary neutrino energy and any baseline length, but also still maintaining a high level of accuracy. 14. All-coupling polaron optical response: Analytic approaches beyond the adiabatic approximation Science.gov (United States) Klimin, S. N.; Tempere, J.; Devreese, J. T. 2016-09-01 In the present work, the problem of an all-coupling analytic description for the optical conductivity of the Fröhlich polaron is treated, with the goal being to bridge the gap in the validity range that exists between two complementary methods: on the one hand, the memory-function formalism and, on the other hand, the strong-coupling expansion based on the Franck-Condon picture for the polaron response. At intermediate coupling, both methods were found to fail as they do not reproduce diagrammatic quantum Monte Carlo results. To resolve this, we modify the memory-function formalism with respect to the Feynman-Hellwarth-Iddings-Platzman approach in order to take into account a nonquadratic interaction in a model system for the polaron. The strong-coupling expansion is extended beyond the adiabatic approximation by including in the treatment nonadiabatic transitions between excited polaron states. The polaron optical conductivity that we obtain at T =0 by combining the two extended methods agrees well, both qualitatively and quantitatively, with the diagrammatic quantum Monte Carlo results in the whole available range of the electron-phonon coupling strength. 15. New chemical evolution analytical solutions including environment effects CERN Document Server Spitoni, E 2015-01-01 In the last years, more and more interest has been devoted to analytical solutions, including inflow and outflow, to study the metallicity enrichment in galaxies. In this framework, we assume a star formation rate which follows a linear Schmidt law, and we present new analytical solutions for the evolution of the metallicity (Z) in galaxies. In particular, we take into account environmental effects including primordial and enriched gas infall, outflow, different star formation efficiencies, and galactic fountains. The enriched infall is included to take into account galaxy-galaxy interactions. Our main results can be summarized as: i) when a linear Schmidt law of star formation is assumed, the resulting time evolution of the metallicity Z is the same either for a closed-box model or for an outflow model. ii) The mass-metallicity relation for galaxies which suffer a chemically enriched infall, originating from another evolved galaxy with no pre-enriched gas, is shifted down in parallel at lower Z values, if co... 16. Analytic solutions of tunneling time through smooth barriers Science.gov (United States) Xiao, Zhi; Huang, Hai 2016-03-01 In the discussion of temporary behaviors of quantum tunneling, people usually like to focus their attention on rectangular barrier with steep edges, or to deal with smooth barrier with semi-classical or even numerical calculations. Very few discussions on analytic solutions of tunneling through smooth barrier appear in the literature. In this paper, we provide two such examples, a semi-infinite long barrier V ( x ) = /A 2 [ 1 + tanh ( x / a ) ] and a finite barrier V(x) = A sech2(x/a). To each barrier, we calculate the associated phase time and dwell time after obtaining the analytic solution. The results show that, different from rectangular barrier, phase time or dwell time does increase with the length parameter a controlling the effective extension of the barrier. More interestingly, for the finite barrier, phase time or dwell time exhibits a peak in k-space. A detailed analysis shows that this interesting behavior can be attributed to the strange tunneling probability Ts(k), i.e., Ts(k) displays a unit step function-like profile Θ(k - k0), especially when a is large, say, a ≫ 1/κ, 1/k. And k 0 ≡ √{ m A } / ħ is exactly where the peak appears in phase or dwell time k-spectrum. Thus only those particles with k in a very narrow interval around k0 are capable to dwell in the central region of the barrier sufficiently long. 17. Logical gaps in the approximate solutions of the social learning game and an exact solution. Science.gov (United States) Dai, Wenjie; Wang, Xin; Di, Zengru; Wu, Jinshan 2014-01-01 After the social learning models were proposed, finding solutions to the games becomes a well-defined mathematical question. However, almost all papers on the games and their applications are based on solutions built either upon an ad-hoc argument or a twisted Bayesian analysis of the games. Here, we present logical gaps in those solutions and offer an exact solution of our own. We also introduce a minor extension to the original game so that not only logical differences but also differences in action outcomes among those solutions become visible. 18. Approximate homotopy symmetry method:Homotopy series solutions to the sixth-order Boussinesq equation Institute of Scientific and Technical Information of China (English) 2009-01-01 An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations.The auxiliary parameter has an effect on the convergence of homotopy series solutions.Series solutions and similarity reduction equations from the approximate symmetry method can be retrieved from the approximate homotopy symmetry method. 19. A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force Directory of Open Access Journals (Sweden) Md. Alal Hosen 2015-01-01 Full Text Available In the present paper, a complicated strongly nonlinear oscillator with cubic and harmonic restoring force, has been analysed and solved completely by harmonic balance method (HBM. Investigating analytically such kinds of oscillator is very difficult task and cumbersome. In this study, the offered technique gives desired results and to avoid numerical complexity. An excellent agreement was found between approximate and numerical solutions, which prove that HBM is very efficient and produces high accuracy results. It is remarkably important that, second-order approximate results are almost same with exact solutions. The advantage of this method is its simple procedure and applicable for many other oscillatory problems arising in science and engineering. 20. Analytic Solutions of Three-Level Dressed-Atom Model Institute of Scientific and Technical Information of China (English) WANG Zheng-Ling; YIN Jian-Ping 2004-01-01 On the basis of the dressed-atom model, the general analytic expressions for the eigenenergies, eigenstates and their optical potentials of the A-configuration three-level atom system are derived and analysed. From the calculation of dipole matrix element of different dressed states, we obtain the spontaneous-emission rates in the dressed-atom picture. We find that our general expressions of optical potentials for the three-level dressed atom can be reduced to the same as ones in previous references under the approximation of a small saturation parameter. We also analyse the dependences of the optical potentials of a three-level 85Rb atom on the laser detuning and the dependences of spontaneous-emission rates on the radial position in the dark hollow beam, and discuss the probability (population) evolutions of dressed-atomic eigenstates in three levels in the hollow beam. 1. Refinement of approximated solution of nonlinear differential equation of second order Energy Technology Data Exchange (ETDEWEB) Zhidkov, E.P.; Sidorova, O.V. 1982-01-01 The boundary problem for nonlinear differential equation of the second order is considered. The problem is assumed to have a unique solution, stable over the right part. It was proved that if the step of the net is small, then the corresponding difference value problem has a unique solution, stable over the right part. Expansion over degrees of discrediting step for approximate solutions is established. The expansion allows one to apply the Richardson type extrapolation. Efficiency of extrapolation is illustrated by numerical example. 2. Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems Directory of Open Access Journals (Sweden) Daniel Olvera 2014-01-01 Full Text Available We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM that is based on the homotopy perturbation method (HPM and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameter h by following the homotopy analysis method (HAM. At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves. 3. Comment on 'Application of the homotopy method for analytical solution of non-Newtonian channel flows' Energy Technology Data Exchange (ETDEWEB) Lipscombe, T C [Johns Hopkins University, 2715 North Charles Street, Baltimore, MD 21218 (United States)], E-mail: [email protected] 2010-03-15 We solve exactly the Poiseuille and Couette flows of a non-Newtonian fluid discussed by Roohi et al (2009 Phys. Scr. 79 065009) and thereby show that the approximate analytical solutions provided by the homotopy method must be used with caution. 4. Analytical solutions for the distribution of pressure in permeable formations with ellipsoidal inclusions Energy Technology Data Exchange (ETDEWEB) Sharif, Ahmed 1997-12-31 The reservoir up-scaling problem has been receiving increased attention in recent years. Over the past decade or so, there has been increasing interest in development of computationally efficient methods to determine effective properties or permeability. Those properties were traditionally computed from detailed numerical solutions of the actual reservoir realization. This is an indirect approach requiring substantial computer resources particularly in 3D problems in which the number of grid-blocks often become impractically large. A contrasting strategy is the direct approach in which the effective properties are computed directly from the statistical description of the medium without the aid of an actual reservoir realization. This method will be particularly important for multiphase problems. Among the direct methods, a particularly promising one which motivated this study, is the self-consistent approximation for determining the electric conductivity of heterogeneous media and multiphase materials. In reservoir engineering context, the self-consistent approximation has been recently applied to determine effective permeabilities. This approximation needs analytical solutions for the fluctuation of pressure created in an otherwise homogeneous matrix of infinite dimensions by the submersion of inclusions. The existing solutions are based on models which have limitations on the orientation of permeability tensors and perhaps largely in the geometry of the inclusions. Mathematical models have been developed which strongly generalize the existing inclusion models serving as a basis for the self-consistent approximation. 21 refs., 9 figs., 2 tabs. 5. FORECAST OF WATER TEMPERATURE IN RESERVOIR BASED ON ANALYTICAL SOLUTION Institute of Scientific and Technical Information of China (English) JI Shun-wen; ZHU Yue-ming; QIANG Sheng; ZENG Deng-feng 2008-01-01 The water temperature in reservoirs is difficult to be predicted by numerical simulations. In this article, a statistical model of forecasting the water temperature was proposed. In this model, the 3-D thermal conduction-diffusion equations were converted into a system consisting of 2-D equations with the Fourier expansion and some hypotheses. Then the statistical model of forecasting the water temperature was developed based on the analytical solution to the 2-D thermal equations. The simplified statistical model can elucidate the main physical mechanism of the temperature variation much more clearly than the numerical simulation with the Navier-Stokes equations. Finally, with the presented statistical model, the distribution of water temperature in the Shangyoujiang reservoir was determined. 6. Pseudo analytical solution to time periodic stiffness systems Institute of Scientific and Technical Information of China (English) Wang Yan-Zhong; Zhou Yuan-Zi 2011-01-01 An analytical form of state transition matrix for a system of equations with time periodic stiffness is derived in order to solve the free response and also allow for the determination of system stability and bifurcation. A pseudoclosed form complete solution for parametrically excited systems subjected to inhomogeneous generalized forcing is developed, based on the Fourier expansion of periodic matrices and the substitution of matrix exponential terms via Lagrange-Sylvester theorem. A Mathieu type of equation with large amplitude is presented to demonstrate the method of formulating state transition matrix and Floquet multipliers. A two-degree-of-freedom system with irregular time periodic stiffness characterized by spiral bevel gear mesh vibration is presented to find forced response in stability and instability. The obtained results are presented and discussed. 7. Analytical solution for inviscid flow inside an evaporating sessile drop. Science.gov (United States) Masoud, Hassan; Felske, James D 2009-01-01 Inviscid flow within an evaporating sessile drop is analyzed. The field equation E;{2}psi=0 is solved for the stream function. The exact analytical solution is obtained for arbitrary contact angle and distribution of evaporative flux along the free boundary. Specific results and computations are presented for evaporation corresponding to both uniform flux and purely diffusive gas phase transport into an infinite ambient. Wetting and nonwetting contact angles are considered, with flow patterns in each case being illustrated. The limiting behaviors of small contact angle and droplets of hemispherical shape are treated. All of the above categories are considered for the cases of droplets whose contact lines are either pinned or free to move during evaporation. 8. An analytic solution to asymmetrical bending problem of diaphragm coupling Institute of Scientific and Technical Information of China (English) 2008-01-01 Because rigidity of either hub or rim of diaphragm coupling is much greater than that of the disk, and asymmetrical bending is under the condition of high speed revolution, an assumption is made that each circle in the middle plane before deforma-tion keeps its radius unchanged after deformation, but the plane on which the circle lies has a varying deflecting angle. Based on this assumption, and according to the principle of energy variation, the corresponding Euler's equation can be obtained, which has the primary integral. By neglecting some subsidiary factors, an analytic solution is obtained. Applying these formulas to a hyperbolic model of diaphragm, the results show that the octahedral shear stress varies less along either radial or thickness direction, but fluctu-ates greatly and periodically along circumferential direction. Thus asymmetrical bending significantly affects the material's fatigue. 9. An analytical solution for determination of small contact angles from sessile drops of arbitrary size. Science.gov (United States) Allen, Jeffrey S 2003-05-15 An analytical solution to the capillary equation of Young and Laplace is derived that allows determination of the static contact angle based on the volume of a sessile drop and the wetted area of the substrate. This solution does not require numerical integration to determine the drop profile and accounts for surface deformation due to gravitational effects. Calculation of the static contact angle by this method is remarkably simple and accurate when the contact angle is less than 30 degrees. A natural scaling arises in the solution, which provides indication of when a drop is small enough so as to neglect gravitational influences on the surface shape which, for small contact angles, is generally less than 1 microl. The technique described has the simplicity of the spherical cap approximation but remains accurate for any size of sessile drop. 10. Measurement of Actinides in Molybdenum-99 Solution Analytical Procedure Energy Technology Data Exchange (ETDEWEB) Soderquist, Chuck Z. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States); Weaver, Jamie L. [Pacific Northwest National Lab. (PNNL), Richland, WA (United States) 2015-11-01 This document is a companion report to a previous report, PNNL 24519, Measurement of Actinides in Molybdenum-99 Solution, A Brief Review of the Literature, August 2015. In this companion report, we report a fast, accurate, newly developed analytical method for measurement of trace alpha-emitting actinide elements in commercial high-activity molybdenum-99 solution. Molybdenum-99 is widely used to produce 99mTc for medical imaging. Because it is used as a radiopharmaceutical, its purity must be proven to be extremely high, particularly for the alpha emitting actinides. The sample of 99Mo solution is measured into a vessel (such as a polyethylene centrifuge tube) and acidified with dilute nitric acid. A gadolinium carrier is added (50 µg). Tracers and spikes are added as necessary. Then the solution is made strongly basic with ammonium hydroxide, which causes the gadolinium carrier to precipitate as hydrous Gd(OH)3. The precipitate of Gd(OH)3 carries all of the actinide elements. The suspension of gadolinium hydroxide is then passed through a membrane filter to make a counting mount suitable for direct alpha spectrometry. The high-activity 99Mo and 99mTc pass through the membrane filter and are separated from the alpha emitters. The gadolinium hydroxide, carrying any trace actinide elements that might be present in the sample, forms a thin, uniform cake on the surface of the membrane filter. The filter cake is first washed with dilute ammonium hydroxide to push the last traces of molybdate through, then with water. The filter is then mounted on a stainless steel counting disk. Finally, the alpha emitting actinide elements are measured by alpha spectrometry. 11. Analytical solution of the Klein Gordon equation for a quadratic exponential-type potential Science.gov (United States) Ezzatpour, Somayyeh; Akbarieh, Amin Rezaei 2016-07-01 In this research study, analytical solutions of the Klein Gordon equation by considering the potential as a quadratic exponential will be presented. However, the potential is assumed to be within the framework of an approximation for the centrifugal potential in any state. The Nikiforov-Uvarov method is used to calculate the wave function, as well as corresponding exact energy equation, in bound states. We finally concluded that the quadratic exponential-type potential under which the results were deduced, led to outcomes that were comparable to the results obtained from the well-known potentials in some special cases. 12. Analytical Solutions of Time Periodic Electroosmotic Flow in a Semicircular Microchannel Directory of Open Access Journals (Sweden) Shaowei Wang 2015-01-01 Full Text Available The time periodic electroosmotic flow of Newtonian fluids through a semicircular microchannel is studied under the Debye–Hückel approximation. Analytical series of solutions are found, and they consist of a time-dependent oscillating part and a time-dependent generating or transient part. Some new physical phenomena are found. The electroosmotic flow driven by an alternating electric field is not periodic in time, but quasi-periodic. There is a phase shift between voltage and flow, which is only dependent on the frequency of external electric field. 13. The Analytical Solution of the Schr\\"odinger Particle in Multiparameter Potential CERN Document Server Taş, Ahmet 2016-01-01 In this study, we present analytical solutions of the Schr\\"odinger equation with the Multiparameter potential containing the different types of physical potential via the asymptotic iteration method (AIM) by applying a Pekeris-type approximation to the centrifugal potential. For any n and l (states) quantum numbers, we get the bound state energy eigenvalues numerically and the corresponding eigenfunctions.Furthermore, we compare our results with the ones obtained in previous works and it is seen that our numerical results are in good agreement with the literature. 14. Variability of the vertical profile of wind speed: characterization at various time scales and analytical approximation Science.gov (United States) Jourdier, Bénédicte; Plougonven, Riwal; Drobinski, Philippe; Dupont, Jean-Charles 2014-05-01 Wind measurements are key for the wind resource assessment. But as wind turbines get higher, wind measurement masts are often lower than the future wind turbine hub height. Therefore one of the first steps in the energy yield assessment is the vertical extrapolation of wind measurements. Such extrapolation is often done by approximating the vertical profile of wind speed with an analytical expression: either a logarithmic law which has a theoretical basis in Monin-Obukhov similarity theory; or a power law which is empirical. The present study analyzes the variability of the wind profile and how this variability affects the results of the vertical extrapolation methods. The study is conducted with data from the SIRTA observatory, 20km south of Paris (France). A large set of instrumentation is available, including sonic anemometers at 10 and 30 meters, a LIDAR measuring wind speeds from 40 to 200 meters and a SODAR measuring wind speeds starting from 100m up to 1km. The comparison between the instruments enables to characterize the measurements uncertainties. The observations show that close to the ground the wind is stronger during daytime and weaker at night while higher, around 150 m, the wind is weaker during daytime and stronger at night. Indeed the wind shear has a pronounced diurnal cycle. The vertical extrapolation methods currently used in the industry do not usually take into account the strong variability of the wind profile. The often fit the parameters of the extrapolation law, not on each time step, but on time-averaged profiles. The averaging period may be the whole measurement period or some part of it: there may be one constant parameter computed on the wind profile that was averaged on the whole year of measures, or the year of measures may be divided into a small number of cases (for example into night or daytime data, or into 4 seasons) and the parameter is adjusted for each case. The study analyzes thoroughly the errors generated by both 15. A Method for Generating Approximate Similarity Solutions of Nonlinear Partial Differential Equations Directory of Open Access Journals (Sweden) Mazhar Iqbal 2014-01-01 Full Text Available Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium. 16. Approximate Solutions of Nonlinear Partial Differential Equations by Modified q-Homotopy Analysis Method Directory of Open Access Journals (Sweden) Shaheed N. Huseen 2013-01-01 Full Text Available A modified q-homotopy analysis method (mq-HAM was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012. The proposed method provides an approximate solution by rewriting the nth-order nonlinear differential equation in the form of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed. 17. Analytical solutions for elastic binary nanotubes of arbitrary chirality Science.gov (United States) Jiang, Lai; Guo, Wanlin 2016-09-01 Analytical solutions for the elastic properties of a variety of binary nanotubes with arbitrary chirality are obtained through the study of systematic molecular mechanics. This molecular mechanics model is first extended to chiral binary nanotubes by introducing an additional out-of-plane inversion term into the so-called stick-spiral model, which results from the polar bonds and the buckling of binary graphitic crystals. The closed-form expressions for the longitudinal and circumferential Young's modulus and Poisson's ratio of chiral binary nanotubes are derived as functions of the tube diameter. The obtained inversion force constants are negative for all types of binary nanotubes, and the predicted tube stiffness is lower than that by the former stick-spiral model without consideration of the inversion term, reflecting the softening effect of the buckling on the elastic properties of binary nanotubes. The obtained properties are shown to be comparable to available density functional theory calculated results and to be chirality and size sensitive. The developed model and explicit solutions provide a systematic understanding of the mechanical performance of binary nanotubes consisting of III-V and II-VI group elements. 18. Analytical solutions for elastic binary nanotubes of arbitrary chirality Science.gov (United States) Jiang, Lai; Guo, Wanlin 2016-12-01 Analytical solutions for the elastic properties of a variety of binary nanotubes with arbitrary chirality are obtained through the study of systematic molecular mechanics. This molecular mechanics model is first extended to chiral binary nanotubes by introducing an additional out-of-plane inversion term into the so-called stick-spiral model, which results from the polar bonds and the buckling of binary graphitic crystals. The closed-form expressions for the longitudinal and circumferential Young's modulus and Poisson's ratio of chiral binary nanotubes are derived as functions of the tube diameter. The obtained inversion force constants are negative for all types of binary nanotubes, and the predicted tube stiffness is lower than that by the former stick-spiral model without consideration of the inversion term, reflecting the softening effect of the buckling on the elastic properties of binary nanotubes. The obtained properties are shown to be comparable to available density functional theory calculated results and to be chirality and size sensitive. The developed model and explicit solutions provide a systematic understanding of the mechanical performance of binary nanotubes consisting of III-V and II-VI group elements. 19. General analytical solutions for DC/AC circuit network analysis CERN Document Server Rubido, Nicolás; Baptista, Murilo S 2014-01-01 In this work, we present novel general analytical solutions for the currents that are developed in the edges of network-like circuits when some nodes of the network act as sources/sinks of DC or AC current. We assume that Ohm's law is valid at every edge and that charge at every node is conserved (with the exception of the source/sink nodes). The resistive, capacitive, and/or inductive properties of the lines in the circuit define a complex network structure with given impedances for each edge. Our solution for the currents at each edge is derived in terms of the eigenvalues and eigenvectors of the Laplacian matrix of the network defined from the impedances. This derivation also allows us to compute the equivalent impedance between any two nodes of the circuit and relate it to currents in a closed circuit which has a single voltage generator instead of many input/output source/sink nodes. Contrary to solving Kirchhoff's equations, our derivation allows to easily calculate the redistribution of currents that o... 20. POLYNOMIAL SOLUTIONS TO PIEZOELECTRIC BEAMS(Ⅱ)--ANALYTICAL SOLUTIONS TO TYPICAL PROBLEMS Institute of Scientific and Technical Information of China (English) DING Hao-jiang; JIANG Ai-min 2005-01-01 For the orthotropic piezoelectric plane problem, a series of piezoelectric beams is solved and the corresponding analytical solutions are obtained with the trialand-error method on the basis of the general solution in the case of three distinct eigenvalues, in which all displacements, electrical potential, stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials. These problems are cantilever beam with cross force and point charge at free end, cantilever beam and simply-supported beam subjected to uniform loads on the upper and lower surfaces, and cantilever beam subjected to linear electrical potential. 1. Finite-element solution of the coupled-channel Schrödinger equation using high-order accuracy approximations Science.gov (United States) Abrashkevich, A. G.; Abrashkevich, D. G.; Kaschiev, M. S.; Puzynin, I. V. 1995-01-01 The finite element method (FEM) is applied to solve the bound state (Sturm-Liouville) problem for systems of ordinary linear second-order differential equations. The convergence, accuracy and the range of applicability of the high-order FEM approximations (up to tenth order) are studied systematically on the basis of numerical experiments for a wide set of quantum-mechanical problems. The analytical and tabular forms of giving the coefficients of differential equations are considered. The Dirichlet and Neumann boundary conditions are discussed. It is shown that the use of the FEM high-order accuracy approximations considerably increases the accuracy of the FE solutions with substantial reduction of the requirements on the computational resources. The results of the FEM calculations for various quantum-mechanical problems dealing with different types of potentials used in atomic and molecular calculations (including the hydrogen atom in a homogeneous magnetic field) are shown to be well converged and highly accurate. 2. Unsteady fluid flow in a slightly curved pipe: A comparative study of a matched asymptotic expansions solution with a single analytical solution Science.gov (United States) Messaris, Gerasimos A. T.; Hadjinicolaou, Maria; Karahalios, George T. 2016-08-01 The present work is motivated by the fact that blood flow in the aorta and the main arteries is governed by large finite values of the Womersley number α and for such values of α there is not any analytical solution in the literature. The existing numerical solutions, although accurate, give limited information about the factors that affect the flow, whereas an analytical approach has an advantage in that it can provide physical insight to the flow mechanism. Having this in mind, we seek analytical solution to the equations of the fluid flow driven by a sinusoidal pressure gradient in a slightly curved pipe of circular cross section when the Womersley number varies from small finite to infinite values. Initially the equations of motion are expanded in terms of the curvature ratio δ and the resulting linearized equations are solved analytically in two ways. In the first, we match the solution for the main core to that for the Stokes boundary layer. This solution is valid for very large values of α. In the second, we derive a straightforward single solution valid to the entire flow region and for 8 ≤ α stresses and is compared to the analogous forms presented in other studies. The two solutions give identical results to each other regarding the axial flow but differ in the secondary flow and the circumferential wall stress, due to the approximations employed in the matched asymptotic expansion process. The results on the stream function from the second solution are in agreement with analogous results from other numerical solutions. The second solution predicts that the atherosclerotic plaques may develop in any location around the cross section of the aortic wall unlike to the prescribed locations predicted by the first solution. In addition, it gives circumferential wall stresses augmented by approximately 100% with respect to the matched asymptotic expansions, a factor that may contribute jointly with other pathological factors to the faster aging of the 3. On the gravitational signature of zonal flows in Jupiter-like planets: An analytical solution and its numerical validation Science.gov (United States) Kong, Dali; Zhang, Keke; Schubert, Gerald 2017-02-01 It is expected that the Juno spacecraft will provide an accurate spectrum of the Jovian zonal gravitational coefficients that would be affected by both the deep zonal flow, if it exists, and the basic rotational distortion. We derive the first analytical solution, under the spheroidal-shape approximation, for the density anomaly induced by an internal zonal flow in rapidly rotating Jupiter-like planets. We compare the density anomaly of the analytical solution to that obtained from a fully numerical solution based on a three-dimensional finite element method; the two show excellent agreement. We apply the analytical solution to a rapidly rotating Jupiter-like planet and show that there exists a close relationship between the spatial structure of the zonal flow and the spectrum of zonal gravitational coefficients. We check the accuracy of the spheroidal-shape approximation by computing both the spheroidal and non-spheroidal solutions with exactly the same physical parameters. We also discuss implications of the new analytical solution for interpreting the future high-precision gravitational measurements of the Juno spacecraft. 4. Higher accurate approximate solutions for the simple pendulum in terms of elementary functions Energy Technology Data Exchange (ETDEWEB) Belendez, Augusto; Frances, Jorge; Ortuno, Manuel; Gallego, Sergi; Guillermo Bernabeu, Jose, E-mail: [email protected] [Departamento de Fisica, IngenierIa de Sistemas y TeorIa de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain) 2010-05-15 A closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions is obtained. To do this, the exact expression for the maximum tension of the string of the pendulum is first considered and a trial approximate solution depending on some parameters is used, which is substituted in the tension equation. We obtain the parameters for the approximate by means of a term-by-term comparison of the power series expansion for the approximate maximum tension with the corresponding series for the exact one. We believe that this letter may be a suitable and fruitful exercise for teaching and better understanding nonlinear oscillations of a simple pendulum in undergraduate courses on classical mechanics. (letters and comments) 5. Concerning an analytical solution of some families of Kepler’s transcendental equation Directory of Open Access Journals (Sweden) Slavica M. Perovich 2016-03-01 Full Text Available The problem of finding an analytical solution of some families of Kepler transcendental equation is studied in some detail, by the Special Trans Functions Theory – STFT. Thus, the STFT mathematical approach in the form of STFT iterative methods with a novel analytical solutions are presented. Structure of the STFT solutions, numerical results and graphical simulations confirm the validity of the basic principle of the STFT. In addition, the obtained analytical results are compared with the calculated values of other analytical methods for alternative proving its significance. Undoubtedly, the proposed novel analytical approach implies qualitative improvement in comparison with conventional numerical and analytical methods. 6. Analytic self-similar solutions of the Oberbeck-Boussinesq equations Science.gov (United States) Barna, I. F.; Mátyás, L. 2015-09-01 In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtoniain Navier-Stokes --- with Boussinesq approximation --- and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are all analytic and can be expressed with the help of the error functions. The temperature field has a strongly damped oscillating behavior which is an interesting feature. 7. Analytic self-similar solutions of the Oberbeck-Boussinesq equations CERN Document Server Barna, I F 2015-01-01 In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtoniain Navier-Stokes --- with Boussinesq approximation --- and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are all analytic and can be expressed with the help of the error functions. The temperature field has a strongly damped oscillating behavior which is an interesting feature. 8. AN EFFECTIVE CONTINUOUS ALGORITHM FOR APPROXIMATE SOLUTIONS OF LARGE SCALE MAX-CUT PROBLEMS Institute of Scientific and Technical Information of China (English) Cheng-xian Xu; Xiao-liang He; Feng-min Xu 2006-01-01 An effective continuous algorithm is proposed to find approximate solutions of NP-hard max-cut problems. The algorithm relaxes the max-cut problem into a continuous nonlinear programming problem by replacing n discrete constraints in the original problem with one single continuous constraint. A feasible direction method is designed to solve the resulting nonlinear programming problem. The method employs only the gradient evaluations of the objective function, and no any matrix calculations and no line searches are required.This greatly reduces the calculation cost of the method, and is suitable for the solution of large size max-cut problems. The convergence properties of the proposed method to KKT points of the nonlinear programming are analyzed. If the solution obtained by the proposed method is a global solution of the nonlinear programming problem, the solution will provide an upper bound on the max-cut value. Then an approximate solution to the max-cut problem is generated from the solution of the nonlinear programming and provides a lower bound on the max-cut value. Numerical experiments and comparisons on some max-cut test problems (small and large size) show that the proposed algorithm is efficient to get the exact solutions for all small test problems and well satisfied solutions for most of the large size test problems with less calculation costs. 9. Homogenous Balance Method and Exact Analytical Solutions for Whitham-Broer-Kaup Equations in Shallow Water Institute of Scientific and Technical Information of China (English) XIAZhi 2004-01-01 Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and other exact analytical solution for Whitham-Broer-Kaup equations(WBK) are derived. These solutions contain Fan's, Xie's and Yan's results and other new types of analytical solutions, such as rational function solutions and periodic solutions. The method can also be applied to solve more nonlinear differential equations. 10. Approximate solution of a model of biological immune responses incorporating delay. Science.gov (United States) Fowler, A C 1981-01-01 A model of the humoral immune response, proposed by Dibrov, Livshits and Volkenstein (1977b), in which the antibody production by a constant target cell population depends on the antigenic stimulation at earlier times, is considered from an analytic standpoint. A method of approximation based on a consideration of the asymptotic limit of "large" delay in the antibody response is shown to be applicable, and to give results similar to those obtained numerically by the above authors. The relevance of this type of approximation to other systems exhibiting "outbreak" phenomena is discussed. 11. On the bound-state solutions of the Manning-Rosen potential including improved approximation to the orbital centrifugal term CERN Document Server Ikhdair, Sameer M 2011-01-01 The approximate analytical bound state solution of the Schr\\"odinger equation for the Manning-Rosen potential is carried out by taking a new approximation scheme to the orbital centrifugal term. The Nikiforov-Uvarov method is used in the calculations. We obtain analytic forms for the energy eigenvalues and the corresponding normalized wave functions in terms of the Jacobi polynomials or hypergeometric functions for different screening parameters 1/b. The rotational-vibrational energy states for a few diatomic molecules are calculated for arbitrary quantum numbers n and l with different values of the potential parameter {\\alpha}. The present numerical results agree within five decimal digits with the previously reported results for different 1/b values. A few special cases of the s-wave (l=0) Manning-Rosen potential and the Hulth\\'en potential are also studied. Keywords: Energy eigenvalues; Manning-Rosen potential; Nikiforov-Uvarov method, Approximation schemes. 03.65.-w; 02.30.Gp; 03.65.Ge; 34.20.Cf 12. An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method Energy Technology Data Exchange (ETDEWEB) Belendez, A., E-mail: [email protected] [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Mendez, D.I. [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Fernandez, E. [Departamento de Optica, Farmacologia y Anatomia, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Marini, S. [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Pascual, I. [Departamento de Optica, Farmacologia y Anatomia, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain) 2009-08-03 The nonlinear oscillations of a Duffing-harmonic oscillator are investigated by an approximated method based on the 'cubication' of the initial nonlinear differential equation. In this cubication method the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude, A. The replacement of the original nonlinear equation by an approximate Duffing equation allows us to obtain explicit approximate formulas for the frequency and the solution as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function, respectively. These explicit formulas are valid for all values of the initial amplitude and we conclude this cubication method works very well for the whole range of initial amplitudes. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones is demonstrated and discussed and the relative error for the approximate frequency is as low as 0.071%. Unlike other approximate methods applied to this oscillator, which are not capable to reproduce exactly the behaviour of the approximate frequency when A tends to zero, the cubication method used in this Letter predicts exactly the behaviour of the approximate frequency not only when A tends to infinity, but also when A tends to zero. Finally, a closed-form expression for the approximate frequency is obtained in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean as well as Legendre's formula to approximately obtain this mean are used. 13. New approximate solutions per unit of time for periodically checked systems with different lifetime distributions Directory of Open Access Journals (Sweden) J. Rodrigues Dias 2006-11-01 Full Text Available Systems with different lifetime distributions, associated with increasing, decreasing, constant, and bathtub-shaped hazard rates, are examined in this paper. It is assumed that a failure is only detected if systems are inspected. New approximate solutions for the inspection period and for the expected duration of hidden faults are presented, on the basis of the assumption that only periodic and perfect inspections are carried out. By minimizing total expected cost per unit of time, on the basis of numerical results and a range of comparisons, the conclusion is drawn that these new approximate solutions are extremely useful and simple to put into practice. 14. An Approximate Analytical Method for the Evaluation of the Concentrations and Current for Hybrid Enzyme Biosensor OpenAIRE 2013-01-01 Mathematical modeling of amperometric biosensor with cyclic reaction is discussed. Analytical expressions pertaining to the concentration of substrate, cosubstrate, reducing agent and medial product and current for hybrid enzyme biosensor are obtained in terms of Thiele module and saturation parameters. In this paper, a powerful analytical method, called homotopy analysis method (HAM) is used to solve the system of nonlinear differential equations. Furthermore, in this work the numerical simu... 15. Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method Energy Technology Data Exchange (ETDEWEB) Belendez, A. [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain)], E-mail: [email protected]; Hernandez, A.; Belendez, T.; Neipp, C.; Marquez, A. [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain) 2008-03-17 He's homotopy perturbation method is used to calculate higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(x). We find He's homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 1.56% for all values of oscillation amplitude, while this relative error is 0.30% for the second iteration and as low as 0.057% when the third-order approximation is considered. Comparison of the result obtained using this method with those obtained by different harmonic balance methods reveals that He's homotopy perturbation method is very effective and convenient. 16. Ermod: fast and versatile computation software for solvation free energy with approximate theory of solutions. Science.gov (United States) Sakuraba, Shun; Matubayasi, Nobuyuki 2014-08-05 ERmod is a software package to efficiently and approximately compute the solvation free energy using the method of energy representation. Molecular simulation is to be conducted at two condensed-phase systems of the solution of interest and the reference solvent with test-particle insertion of the solute. The subprogram ermod in ERmod then provides a set of energy distribution functions from the simulation trajectories, and another subprogram slvfe determines the solvation free energy from the distribution functions through an approximate functional. This article describes the design and implementation of ERmod, and illustrates its performance in solvent water for two organic solutes and two protein solutes. Actually, the free-energy computation with ERmod is not restricted to the solvation in homogeneous medium such as fluid and polymer and can treat the binding into weakly ordered system with nano-inhomogeneity such as micelle and lipid membrane. ERmod is available on web at http://sourceforge.net/projects/ermod. 17. Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods Energy Technology Data Exchange (ETDEWEB) Belendez, A [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Pascual, C [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Fernandez, E [Departamento de Optica, FarmacologIa y AnatomIa, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Neipp, C [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Belendez, T [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain) 2008-02-15 A modified He's homotopy perturbation method is used to calculate higher-order analytical approximate solutions to the relativistic and Duffing-harmonic oscillators. The He's homotopy perturbation method is modified by truncating the infinite series corresponding to the first-order approximate solution before introducing this solution in the second-order linear differential equation, and so on. We find this modified homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. The approximate formulae obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation, including the limiting cases of amplitude approaching zero and infinity. For the relativistic oscillator, only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 1.6% for small and large values of oscillation amplitude, while this relative error is 0.65% for two iterations with two harmonics and as low as 0.18% when three harmonics are considered in the second approximation. For the Duffing-harmonic oscillator the relative error is as low as 0.078% when the second approximation is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance methods reveals that the former is very effective and convenient. 18. Food Adulteration: From Vulnerability Assessment to New Analytical Solutions. Science.gov (United States) Cavin, Christophe; Cottenet, Geoffrey; Blancpain, Carine; Bessaire, Thomas; Frank, Nancy; Zbinden, Pascal 2016-01-01 Crises related to the presence of melamine in milk or horse meat in beef have been a wake-up call to the whole food industry showing that adulteration of food raw materials is a complex issue. By analysing the situation, it became clear that the risk-based approach applied to ensure the safety related to chemical contaminants in food is not adequate for food fraud. Therefore, a specific approach has been developed to evaluate adulteration vulnerabilities within the food chain. Vulnerabilities will require the development of new analytical solutions. Fingerprinting methodologies can be very powerful in determining the status of a raw material without knowing the identity of each constituent. Milk adulterated by addition of adulterants with very different chemical properties could be detected rapidly by Fourier-transformed mid-infrared spectroscopy (FT-mid-IR) fingerprinting technology. In parallel, a fast and simple multi-analytes liquid-chromatography tandem mass-spectrometry (LC/MS-MS) method has been developed to detect either high levels of nitrogen-rich compounds resulting from adulteration or low levels due to accidental contamination either in milk or in other sensitive food matrices. To verify meat species authenticity, DNA-based methods are preferred for both raw ingredients and processed food. DNA macro-array, and more specifically the Meat LCD Array have showed efficient and reliable meat identification, allowing the simultaneous detection of 32 meat species. While the Meat LCD Array is still a targeted approach, DNA sequencing is a significant step towards an untargeted one. 19. OPTIMAL APPROXIMATE SOLUTION OF THE MATRIX EQUATION AXB=C OVER SYMMETRIC MATRICES Institute of Scientific and Technical Information of China (English) Anping Liao; Yuan Lei 2007-01-01 Let SE denote the least-squares symmetric solution set of the matrix equation AXB=C,where A,B and C are given matrices of suitable size.To find the optimal approximate solution in the set SE to a given matrix,we give a new feasible method based on the projection theorem,the generalized SVD and the canonical correction decomposition. 20. SERIES PERTURBATIONS APPROXIMATE SOLUTIONS TO N-S EQUATIONS AND MODIFICATION TO ASYMPTOTIC EXPANSION MATCHED METHOD Institute of Scientific and Technical Information of China (English) 李大鸣; 张红萍; 高永祥 2002-01-01 A method that series perturbations approximate solutions to N-S equations with boundary conditions was discussed and adopted. Then the method was proved in which the asymptotic solutions of viscous fluid flow past a sphere were deducted. By the ameliorative asymptotic expansion matched method, the matched functions are determined easily and the ameliorative curve of drag coefficient is coincident well with measured data in the case that Reynolds number is less than or equal to 40 000. 1. Solutions of random-phase approximation equation for positive-semidefinite stability matrix CERN Document Server Nakada, H 2016-01-01 It is mathematically proven that, if the stability matrix$\\mathsf{S}$is positive-semidefinite, solutions of the random-phase approximation (RPA) equation are all physical or belong to Nambu-Goldstone (NG) modes, and the NG-mode solutions may form Jordan blocks of$\\mathsf{N\\,S}$($\\mathsf{N}$is the norm matrix) but their dimension is not more than two. This guarantees that the NG modes in the RPA can be separated out via canonically conjugate variables. 2. Analysing an Analytical Solution Model for Simultaneous Mobility Directory of Open Access Journals (Sweden) Md. Ibrahim Chowdhury 2013-12-01 Full Text Available Current mobility models for simultaneous mobility h ave their convolution in designing simultaneous movement where mobile nodes (MNs travel randomly f rom the two adjacent cells at the same time and also have their complexity in the measurement of th e occurrences of simultaneous handover. Simultaneou s mobility problem incurs when two of the MNs start h andover approximately at the same time. As Simultaneous mobility is different for the other mo bility pattern, generally occurs less number of tim es in real time; we analyze that a simplified simultaneou s mobility model can be considered by taking only symmetric positions of MNs with random steps. In ad dition to that, we simulated the model using mSCTP and compare the simulation results in different sce narios with customized cell ranges. The analytical results shows that with the bigger the cell sizes, simultaneous handover with random steps occurrences become lees and for the sequential mobility (where initial positions of MNs is predetermined with ran dom steps, simultaneous handover is more frequent. 3. On the Rational Approximation of Analytic Functions Having Generalized Types of Rate of Growth Directory of Open Access Journals (Sweden) Devendra Kumar 2012-01-01 Full Text Available The present paper is concerned with the rational approximation of functions holomorphic on a domain G⊂C, having generalized types of rates of growth. Moreover, we obtain the characterization of the rate of decay of product of the best approximation errors for functions f having fast and slow rates of growth of the maximum modulus. 4. Analytical solution of Boussinesq equations as a model of wave generation Science.gov (United States) Wiryanto, L. H.; Mungkasi, S. 2016-02-01 When a uniform stream on an open channel is disturbed by existing of a bump at the bottom of the channel, the surface boundary forms waves growing splitting and propagating. The model of the wave generation can be a forced Korteweg de Vries (fKdV) equation or Boussinesq-type equations. In case the governing equations are approximated from steady problem, the fKdV equation is obtained. The model gives two solutions representing solitary-like wave, with different amplitude. However, phyically there is only one profile generated from that process. Which solution is occured, we confirm from unsteady model. The Boussinesq equations are proposed to determine the stabil solution of the fKdV equation. From the linear and steady model, its solution is developed to determine the analytical solution of the unsteady equations, so that it can explain the physical phenomena, i.e. the process of the wave generation, wave splitting and wave propagation. The solution can also determine the amplitude and wave speed of the waves. 5. An analytical solution to patient prioritisation in radiotherapy based on utilitarian optimisation. Science.gov (United States) Ebert, M A; Li, W; Jennings, L 2014-03-01 The detrimental impact of a radiotherapy waiting list can in part be compensated by patient prioritisation. Such prioritisation is phrased as an optimisation problem where the probability of local control for the overall population is the objective to be maximised and a simple analytical solution derived. This solution is compared with a simulation of a waiting list for the same population of patients. It is found that the analytical solution can provide an optimal ordering of patients though cannot explicitly constrain optimal waiting times. The simulation-based solution was undertaken using both the analytical solution and a numerical optimisation routine for daily patient ordering. Both solutions provided very similar results with the analytical approach reducing the calculation time of the numerical solution by several orders of magnitude. It is suggested that treatment delays due to resource limitations and resulting waiting lists be incorporated into treatment optimisation and that the derived analytical solution provides a mechanism for this to occur. 6. The adiabatic approximation solutions of cylindrical and spherical dust ion-acoustic solitary waves Institute of Scientific and Technical Information of China (English) 吕克璞; 豆福全; 孙建安; 段文山; 石玉仁 2005-01-01 By using the equivalent particle theory, the adiabatic approximation solutions of the Korteweg-de Vries type equation (including KdV equation, cylindrical KdV equation and spherical KdV equation) in dust ion-acoustic solitary waves were obtained. The method can be extended to other nonlinear evolution equations. 7. An approximate global solution to the gravitational field of a perfect fluid in slow rotation CERN Document Server Cabezas, J A 2006-01-01 Using the Post-Minkowskian formalism and considering rotation as a perturbation, we compute an approximate interior solution for a stationary perfect fluid with constant density and axial symmetry. A suitable change of coordinates allows this metric to be matched to the exterior metric to a particle with a pole-dipole-quadrupole structure, relating the parameters of both. 8. Solution of Equations of Internal Ballistics for the Composite Charge Using Lagrange Density Approximation Directory of Open Access Journals (Sweden) D. K. Narvilkar 1979-07-01 Full Text Available In the present paper, the equations of internal ballistics of composite charge consisting of N component charge with quadratic form are solved. Largange density approximation and hydrodynamic flow behaviour, have been assumed and the solutions are obtained for the composite charge for these assumptions. 9. Average optimization of the approximate solution of operator equations and its application Institute of Scientific and Technical Information of China (English) WANG; xinghua(王兴华); MA; Wan(马万) 2002-01-01 In this paper, a definition of the optimization of operator equations in the average case setting is given. And the general result (Theorem 1) about the relevant optimization problem is obtained. This result is applied to the optimization of approximate solution of some classes of integral equations. 10. Electromechanics: An analytic solution for graded biological cell. Science.gov (United States) Chan, Kin Lok; Yu, K. W. 2007-03-01 Electromechanics of graded material has been established recently to study the effective response of inhomogeneous graded spherical particles under an external ac electric field.[1, 2]Such particles having a complex dielectric profile varies along the radius of the particles. The gradation in the colloidal particles is modeled by assuming both the dielectric and conductivity vary along the radius. More precisely, both the dielectric and conductivity function are assumed to be a isotopic linear function dependence on the radius variable r, namely, ɛ(r)=ɛ(0)+A1r, σ(r)=σ(0)+A2r.In this talk, we will present the exact analytical solutions of the dipole moment of such particle in terms of the hypergeometric functions, and the effective electric response in dilute limit. Moreover, we applied the dielectric dispersion spectral representation (DDSR) to study the Debye Behavior of the cell. Our exact results may be applied to graded biological cell suspensions, as their interior must be inhomogeneous in nature. [1] En-Bo Wei, L. Dong, K. W. Yu, Journal of Applied Physics 99, 054101(2006) [2] L. Dong, Mikko Karttunen, K. W. Yu, Phys. Rev. E, Vol. 72, art. no. 016613 (2005) 11. New analytic solutions for modeling vertical gravity gradient anomalies Science.gov (United States) Kim, Seung-Sep; Wessel, Paul 2016-05-01 Modern processing of satellite altimetry for use in marine gravimetry involves computing the along-track slopes of observed sea-surface heights, projecting them into east-west and north-south deflection of the vertical grids, and using Laplace's equation to algebraically obtain a grid of the vertical gravity gradient (VGG). The VGG grid is then integrated via overlapping, flat Earth Fourier transforms to yield a free-air anomaly grid. Because of this integration and associated edge effects, the VGG grid retains more short-wavelength information (e.g., fracture zone and seamount signatures) that is of particular importance for plate tectonic investigations. While modeling of gravity anomalies over arbitrary bodies has long been a standard undertaking, similar modeling of VGG anomalies over oceanic features is not commonplace yet. Here we derive analytic solutions for VGG anomalies over simple bodies and arbitrary 2-D and 3-D sources. We demonstrate their usability in determining mass excess and deficiency across the Mendocino fracture zone (a 2-D feature) and find the best bulk density estimate for Jasper seamount (a 3-D feature). The methodologies used herein are implemented in the Generic Mapping Tools, available from gmt.soest.hawaii.edu. 12. Electronic states of graphene nanoribbons and analytical solutions Directory of Open Access Journals (Sweden) Katsunori Wakabayashi, Ken-ichi Sasaki, Takeshi Nakanishi and Toshiaki Enoki 2010-01-01 Full Text Available Graphene is a one-atom-thick layer of graphite, where low-energy electronic states are described by the massless Dirac fermion. The orientation of the graphene edge determines the energy spectrum of π-electrons. For example, zigzag edges possess localized edge states with energies close to the Fermi level. In this review, we investigate nanoscale effects on the physical properties of graphene nanoribbons and clarify the role of edge boundaries. We also provide analytical solutions for electronic dispersion and the corresponding wavefunction in graphene nanoribbons with their detailed derivation using wave mechanics based on the tight-binding model. The energy band structures of armchair nanoribbons can be obtained by making the transverse wavenumber discrete, in accordance with the edge boundary condition, as in the case of carbon nanotubes. However, zigzag nanoribbons are not analogous to carbon nanotubes, because in zigzag nanoribbons the transverse wavenumber depends not only on the ribbon width but also on the longitudinal wavenumber. The quantization rule of electronic conductance as well as the magnetic instability of edge states due to the electron–electron interaction are briefly discussed. 13. Deriving Coarse-Grained Charges from All-Atom Systems: An Analytic Solution. Science.gov (United States) McCullagh, Peter; Lake, Peter T; McCullagh, Martin 2016-09-13 An analytic method to assign optimal coarse-grained charges based on electrostatic potential matching is presented. This solution is the infinite size and density limit of grid-integration charge-fitting and is computationally more efficient by several orders of magnitude. The solution is also minimized with respect to coarse-grained positions which proves to be an extremely important step in reproducing the all-atom electrostatic potential. The joint optimal-charge optimal-position coarse-graining procedure is applied to a number of aggregating proteins using single-site per amino acid resolution. These models provide a good estimate of both the vacuum and Debye-Hückel screened all-atom electrostatic potentials in the vicinity and in the far-field of the protein. Additionally, these coarse-grained models are shown to approximate the all-atom dimerization electrostatic potential energy of 10 aggregating proteins with good accuracy. 14. SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations Science.gov (United States) Gusev, Sergei V.; Shiriaev, Anton S.; Freidovich, Leonid B. 2016-07-01 Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies. 15. STUDY ON EXACT ANALYTICAL SOLUTIONS FOR TWO SYSTEMS OF NONLINEAR EVOLUTION EQUATIONS Institute of Scientific and Technical Information of China (English) 闫振亚; 张鸿庆 2001-01-01 The homogeneous balance method was improved and applied to two systems of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions contain Wang's and Zhang's results and other new types of analytical solutions, such as rational fraction solutions and periodic solutions. The way can also be applied to solve more nonlinear partial differential equations. 16. Analytic Solutions of Some Self-Adjoint Equations by Using Variable Change Method and Its Applications OpenAIRE Mehdi Delkhosh; Mohammad Delkhosh 2012-01-01 Many applications of various self-adjoint differential equations, whose solutions are complex, are produced (Arfken, 1985; Gandarias, 2011; and Delkhosh, 2011). In this work we propose a method for the solving some self-adjoint equations with variable change in problem, and then we obtain a analytical solutions. Because this solution, an exact analytical solution can be provided to us, we benefited from the solution of numerical Self-adjoint equations (Mohynl-Din, 2009; Allame and Azal, 2011;... 17. A model for reactive nonadiabatic transitions: Comparison between exact numerical and approximate analytical results Science.gov (United States) Child, M. S.; Baer, M. 1981-03-01 Exact diabatic/adiabatic branching ratios and final state distributions are presented for a reactive model for nonadiabatic transitions, applicable to situations where the coupling term is approximately constant over the region where the interpotential seam crosses the two valleys. Comparison is made with the Bauer-Fischer-Gilmore (BFG) and Franck-Condon (FC) models for a variety of situations. A new index γ=(vRΔGR/vrΔGR), where subscripts R and r denote translational and vibrational variables, respectively, is introduced as a measure of the validity of the two approximations. The FC approximation is shown to become exact for γ≳≳1, while the BFG approximation is preferred for γ<<1. 18. Polynomial-based approximate solutions to the Boussinesq equation near a well Science.gov (United States) Telyakovskiy, Aleksey S.; Kurita, Satoko; Allen, Myron B. 2016-10-01 This paper presents a method for constructing polynomial-based approximate solutions to the Boussinesq equation with cylindrical symmetry. This equation models water injection at a single well in an unconfined aquifer; as a sample problem we examine recharge of an initially empty aquifer. For certain injection regimes it is possible to introduce similarity variables, reducing the original problem to a boundary-value problem for an ordinary differential equation. The approximate solutions introduced here incorporate both a singular part to model the behavior near the well and a polynomial part to model the behavior in the far field. Although the nonlinearity of the problem prevents decoupling of the singular and polynomial parts, the paper presents an approach for calculating the solution based on its spatial moments. This approach yields closed-form expressions for the position of the wetting front and for the form of the phreatic surface. Comparison with a highly accurate numerical solution verifies the accuracy of the newly derived approximate solutions. 19. Analytical approximations of diving-wave imaging in constant-gradient medium KAUST Repository Stovas, Alexey 2014-06-24 Full-waveform inversion (FWI) in practical applications is currently used to invert the direct arrivals (diving waves, no reflections) using relatively long offsets. This is driven mainly by the high nonlinearity introduced to the inversion problem when reflection data are included, which in some cases require extremely low frequency for convergence. However, analytical insights into diving waves have lagged behind this sudden interest. We use analytical formulas that describe the diving wave’s behavior and traveltime in a constant-gradient medium to develop insights into the traveltime moveout of diving waves and the image (model) point dispersal (residual) when the wrong velocity is used. The explicit formulations that describe these phenomena reveal the high dependence of diving-wave imaging on the gradient and the initial velocity. The analytical image point residual equation can be further used to scan for the best-fit linear velocity model, which is now becoming a common sight as an initial velocity model for FWI. We determined the accuracy and versatility of these analytical formulas through numerical tests. 20. Communication: Analytic continuation of the virial series through the critical point using parametric approximants. Science.gov (United States) Barlow, Nathaniel S; Schultz, Andrew J; Weinstein, Steven J; Kofke, David A 2015-08-21 The mathematical structure imposed by the thermodynamic critical point motivates an approximant that synthesizes two theoretically sound equations of state: the parametric and the virial. The former is constructed to describe the critical region, incorporating all scaling laws; the latter is an expansion about zero density, developed from molecular considerations. The approximant is shown to yield an equation of state capable of accurately describing properties over a large portion of the thermodynamic parameter space, far greater than that covered by each treatment alone. 1. Effects of dynamic contact angle on liquid infiltration into inclined capillary tubes: (semi)-analytical solutions. Science.gov (United States) Hilpert, Markus 2009-09-01 In a recent paper, we generalized Washburn's analytical solution for capillary flow in a horizontally oriented tube by accounting for a dynamic contact angle. In this paper, we derive solutions for flow in inclined tubes that account for gravity. We again consider two general models for dynamic contact angle: the uncompensated Young force on the contact line depends on the capillary number in the form of (1) a power law with exponent beta, or (2) a polynomial. A dimensional analysis shows that, aside from the parameters for the model for the uncompensated Young force, the problem is defined through four nondimensional parameters: (1) the advancing equilibrium contact angle, (2) the initial contact angle, (3) a Bond number, and (4) nondimensional liquid pressure at the tube inlet relative to the constant gas pressure. For both contact angle models, we derive analytical solutions for the travel time of the gas-liquid interface as a function of interface velocity. The interface position as a function of travel time can be obtained through numerical integration. For the power law and beta=1 (an approximation of Cox's model for dynamic contact angle), we obtain an analytical solution for travel time as a function of interface position, as Washburn did for constant contact angle. Four different flow scenarios may occur: the interface moves (1) upward and approaches the height of capillary rise, (2) downward with the steady-state velocity, (3) downward while approaching the steady-state velocity from an initially higher velocity, or (4) downward while approaching the steady-state velocity from an initially smaller velocity. 2. An analytical solution for light field modes in waveguides with nonideal cladding CERN Document Server Arslanov, N M; Moiseev, S A 2015-01-01 We have obtained an analytical solution for the dispersion relation of the light field modes in the nanowaveguide structure. The solution has been analyzed for the planar waveguide with metamaterial claddings and dielectric core. The analytical solution is valid within the broadband spectral range and is confirmed by existing numerical calculations. The developed theoretical approach opens vast possibilities for the analytical investigations of the light fields in the various waveguides. 3. ANALYTICAL SOLUTION FOR FIXED-FIXED ANISOTROPIC BEAM SUBJECTED TO UNIFORM LOAD Institute of Scientific and Technical Information of China (English) DING Hao-jiang; HUANG De-jin; WANG Hui-ming 2006-01-01 The analytical solutions of the stresses and displacements were obtained for fixed-fixed anisotropic beams subjected to uniform load. A stress function involving unknown coefficients was constructed, and the general expressions of stress and displacement were obtained by means of Airy stress function method. Two types of the description for the fixed end boundary condition were considered. The introduced unknown coefficients in stress function were determined by using the boundary conditions. The analytical solutions for stresses and displacements were finally obtained. Numerical tests show that the analytical solutions agree with the FEM results. The analytical solution supplies a classical example for the elasticity theory. 4. Approximate analytical descriptions of the stationary single-vortex Marangoni convection inside an evaporating sessile droplet of capillary size CERN Document Server Barash, L Yu 2013-01-01 Three versions of an approximate analytical description of the stationary single vortex Marangoni convection in an axially symmetrical sessile drop of capillary size are studied for arbitrary contact angle and compared with the results of numerical simulations. The first approach is heuristic extension of the well-known lubrication approximation. Two other descriptions are developed here and named n\\tau- and rz-description. They are free from most of restrictive assumptions of the lubrication approach. For droplets with large contact angles they result in better accuracy compared to the heuristic extension of the lubrication approach, which still gives reasonable results within the accuracy 10-30 per cent. For droplets with small contact angles all three analytical descriptions well agree with the numerical data. 5. Distribution of Steps with Finite-Range Interactions: Analytic Approximations and Numerical Results Science.gov (United States) GonzáLez, Diego Luis; Jaramillo, Diego Felipe; TéLlez, Gabriel; Einstein, T. L. 2013-03-01 While most Monte Carlo simulations assume only nearest-neighbor steps interact elastically, most analytic frameworks (especially the generalized Wigner distribution) posit that each step elastically repels all others. In addition to the elastic repulsions, we allow for possible surface-state-mediated interactions. We investigate analytically and numerically how next-nearest neighbor (NNN) interactions and, more generally, interactions out to q'th nearest neighbor alter the form of the terrace-width distribution and of pair correlation functions (i.e. the sum over n'th neighbor distribution functions, which we investigated recently.[2] For physically plausible interactions, we find modest changes when NNN interactions are included and generally negligible changes when more distant interactions are allowed. We discuss methods for extracting from simulated experimental data the characteristic scale-setting terms in assumed potential forms. 6. Analytical approximation of the InGaZnO thin-film transistors surface potential Science.gov (United States) Colalongo, Luigi 2016-10-01 Surface-potential-based mathematical models are among the most accurate and physically based compact models of thin-film transistors, and in turn of indium gallium zinc oxide TFTs, available today. However, the need of iterative computations of the surface potential limits their computational efficiency and diffusion in CAD applications. The existing closed-form approximations of the surface potential are based on regional approximations and empirical smoothing functions that could result not accurate enough in particular to model transconductances and transcapacitances. In this work we present an extremely accurate (in the range of nV) and computationally efficient non-iterative approximation of the surface potential that can serve as a basis for advanced surface-potential-based indium gallium zinc oxide TFTs models. 7. Approximate solutions for Forchheimer flow during water injection and water production in an unconfined aquifer Science.gov (United States) Mathias, Simon A.; Moutsopoulos, Konstantinos N. 2016-07-01 Understanding the hydraulics around injection and production wells in unconfined aquifers associated with rainwater and reclaimed water aquifer storage schemes is an issue of increasing importance. Much work has been done previously to understand the mathematics associated with Darcy's law in this context. However, groundwater flow velocities around injection and production wells are likely to be sufficiently large such as to induce significant non-Darcy effects. This article presents a mathematical analysis to look at Forchheimer's equation in the context of water injection and water production in unconfined aquifers. Three different approximate solutions are derived using quasi-steady-state assumptions and the method of matched asymptotic expansion. The resulting approximate solutions are shown to be accurate for a wide range of practical scenarios by comparison with a finite difference solution to the full problem of concern. The approximate solutions have led to an improved understanding of the flow dynamics. They can also be used as verification tools for future numerical models in this context. 8. Approximate solutions to the quantum problem of two opposite charges in a constant magnetic field Energy Technology Data Exchange (ETDEWEB) Ardenghi, J.S., E-mail: [email protected] [IFISUR, Departamento de Física (UNS-CONICET), Avenida Alem 1253, Bahía Blanca, Buenos Aires (Argentina); Gadella, M., E-mail: [email protected] [Department of Theoretical, Atomic Physics and Optics and IMUVA, University of Valladolid, 47011 Valladolid (Spain); Grinnell College, Department of Physics, Grinnell, 50112 IA (United States); Negro, J., E-mail: [email protected] [Department of Theoretical, Atomic Physics and Optics and IMUVA, University of Valladolid, 47011 Valladolid (Spain) 2016-05-06 We consider two particles of equal mass and opposite charge in a plane subject to a perpendicular constant magnetic field. This system is integrable but not superintegrable. From the quantum point of view, the solution is given by two fourth degree Hill differential equations which involve the energy as well as a second constant of motion. There are two solvable approximations in relation to the value of a parameter. Starting from each of these approximations, a consistent perturbation theory can be applied to get approximate values of the energy levels and of the second constant of motion. - Highlights: • We have studied the quantum model of two charged particles on a plane with opposite charges and a perpendicular constant magnetic field. • This model is integrable, although not superintegrable. • The model under study is described by two fourth degree Hill equations, one trigonometric and the other hyperbolic. • We have considered two distinct approximations that have exact solution. • We have applied a perturbative method to improve the approximation. 9. A simple analytic approximation to the Rayleigh-Bénard stability threshold NARCIS (Netherlands) Prosperetti, Andrea 2011-01-01 The Rayleigh-Bénard linear stability problem is solved by means of a Fourier series expansion. It is found that truncating the series to just the first term gives an excellent explicit approximation to the marginal stability relation between the Rayleigh number and the wave number of the perturbatio 10. Lanthanide salts solutions: representation of osmotic coefficients within the binding mean spherical approximation. Science.gov (United States) Ruas, Alexandre; Moisy, Philippe; Simonin, Jean-Pierre; Bernard, Olivier; Dufrêche, Jean-François; Turq, Pierre 2005-03-24 Osmotic coefficients of aqueous solutions of lanthanide salts are described using the binding mean spherical approximation (BIMSA) model based on the Wertheim formalism for association. The lanthanide(III) cation and the co-ion are allowed to form a 1-1 ion pair. Hydration is taken into account by introducing concentration-dependent cation size and solution permittivity. An expression for the osmotic coefficient, derived within the BIMSA, is used to fit data for a wide variety of lanthanide pure salt aqueous solutions at 25 degrees C. A total of 38 lanthanide salts have been treated, including perchlorates, nitrates, and chlorides. For most solutions, good fits could be obtained up to high ionic strengths. The relevance of the fitted parameters has been discussed, and a comparison with literature values has been made (especially the association constants) when available. 11. Approximate solutions to the deep bed filtration problem; Solucoes aproximadas para o problema de deposicao profunda Energy Technology Data Exchange (ETDEWEB) Silva, Julio M.; Marchesin, Dan [Instituto de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, RJ (Brazil) 2008-07-01 The deep bed filtration problem is closely related to secondary oil recovery. In this work we derive explicit solutions to two filtration problems. The filtration function varies non-linearly with the Darcy speed and linearly with the deposition, but very little. The first solution is built by the method of perturbations and although it is only an approximation it is available in multiple symmetries, including the radial geometry used in the field. The main motivation is the validation of numerical methods. The second solution is exact but it is only available in the linear symmetry, i.e., laboratory geometry. We use it to verify the accuracy of the first solution, but it can also be used to simulate the deposition in experiments. (author) 12. Analytic Solution for Magnetohydrodynamic Stagnation Point Flow towards a Stretching Sheet Institute of Scientific and Technical Information of China (English) DING Qi; ZHANG Hong-Qing 2009-01-01 A steady two-dimensional magnetohydrodynamic stagnation point flow towards a stretching sheet with variable surface temperature is investigated. The analytic solution is obtained by homotopy analysis method. Theconvergence region is computed and the feature of the solution is discussed. 13. Analytical solutions for reactive transport under an infiltration-redistribution cycle. Science.gov (United States) Severino, Gerardo; Indelman, Peter 2004-05-01 Transport of reactive solute in unsaturated soils under an infiltration-redistribution cycle is investigated. The study is based on the model of vertical flow and transport in the unsaturated zone proposed by Indelman et al. [J. Contam. Hydrol. 32 (1998) 77], and generalizes it by accounting for linear nonequilibrium kinetics. An exact analytical solution is derived for an irreversible desorption reaction. The transport of solute obeying linear kinetics is modeled by assuming equilibrium during the redistribution stage. The model which accounts for nonequilibrium during the infiltration and assumes equilibrium at the redistribution stage is termed partial equilibrium infiltration-redistribution model (PEIRM). It allows to derive approximate closed form solutions for transport in one-dimensional homogeneous soils. These solutions are further applied to computing the field-scale concentration by adopting the Dagan and Bresler [Soil Sci. Soc. Am. J. 43 (1979) 461] column model. The effect of soil heterogeneity on the solute spread is investigated by modeling the hydraulic saturated conductivity as a random function of horizontal coordinates. The quality of the PEIRM is illustrated by calculating the critical values of the Damköhler number which provide the achievable accuracy in estimating the solute mass in the mobile phase. The distinguishing feature of transport during the infiltration-redistribution cycle as compared to that of infiltration only is the finite depth of solute penetration. For irreversible desorption, the maximum solute penetration W/theta(r) is determined by the amount of applied water W and the residual water content theta(r). For sorption-desorption kinetics, the maximum depth of penetration z(r)(e, infinity ) also depends on the ratio between the rate of application and the column-saturated conductivity. It is shown that z(r)(e, infinity ) is bounded between the depths W/(theta(r)+K(d)) and W/theta(r) corresponding to the maximum solute 14. Analytical approximation for AC losses in thin power-law superconductors Energy Technology Data Exchange (ETDEWEB) Sokolovsky, V; Meerovich, V [Physics Department, Ben-Gurion University of the Negev, PO Box 653, Beer-Sheva, 84105 (Israel) 2007-08-15 AC losses in the superconducting parts of tapes and multifilamentary coated conductors that are subjected to AC magnetic fields are an important component of the total losses in such composites. The analytical expression for AC losses in a thin superconducting strip with a power-law voltage-current characteristic and critical current depending on a magnetic field is obtained for the case of asymptotically high perpendicular magnetic fields. The losses caused by closure currents are estimated. The results show that the Bean model gives significantly understated values for coated conductors. The applicability of the obtained expressions is analyzed. 15. Analytical solutions for benchmarking cold regions subsurface water flow and energy transport models: one-dimensional soil thaw with conduction and advection Science.gov (United States) Kurylyk, Barret L.; McKenzie, Jeffrey M; MacQuarrie, Kerry T. B.; Voss, Clifford I. 2014-01-01 Numerous cold regions water flow and energy transport models have emerged in recent years. Dissimilarities often exist in their mathematical formulations and/or numerical solution techniques, but few analytical solutions exist for benchmarking flow and energy transport models that include pore water phase change. This paper presents a detailed derivation of the Lunardini solution, an approximate analytical solution for predicting soil thawing subject to conduction, advection, and phase change. Fifteen thawing scenarios are examined by considering differences in porosity, surface temperature, Darcy velocity, and initial temperature. The accuracy of the Lunardini solution is shown to be proportional to the Stefan number. The analytical solution results obtained for soil thawing scenarios with water flow and advection are compared to those obtained from the finite element model SUTRA. Three problems, two involving the Lunardini solution and one involving the classic Neumann solution, are recommended as standard benchmarks for future model development and testing. 16. Compressible flow in front of an axisymmetric blunt object: analytic approximation and astrophysical implications CERN Document Server Keshet, Uri 2016-01-01 Compressible flows around blunt objects have diverse applications, but current analytic treatments are inaccurate and limited to narrow parameter regimes. We show that the gas-dynamic flow in front of an axisymmetric blunt body is accurately derived analytically using a low order expansion of the perpendicular gradients in terms of the parallel velocity. This reproduces both subsonic and supersonic flows measured and simulated for a sphere, including the transonic regime and the bow shock properties. Some astrophysical implications are outlined, in particular for planets in the solar wind and for clumps and bubbles in the intergalactic medium. The bow shock standoff distance normalized by the obstacle curvature is$\\sim 2/(3g)$in the strong shock limit, where$g$is the compression ratio. For a subsonic Mach number$M$approaching unity, the thickness$\\delta$of an initially weak, draped magnetic layer is a few times larger than in the incompressible limit, with amplification$\\sim ({1+1.3M^{2.6}})/({3\\delt...
17. Analytical approximation to the dynamics of a binary stars system with time depending mass variation
CERN Document Server
López, Gustavo V
2016-01-01
We study the classical dynamics of a binary stars when there is an interchange of mass between them. Assuming that one of the star is more massive than the other, the dynamics of the lighter one is analyzed as a function of its time depending mass variation. Within our approximations and models for mass transference, we obtain a general result which establishes that if the lightest star looses mass, its period increases. If the lightest star win mass, its period decreases.
18. Analytical Approximation Method for the Center Manifold in the Nonlinear Output Regulation Problem
Science.gov (United States)
Suzuki, Hidetoshi; Sakamoto, Noboru; Celikovský, Sergej
In nonlinear output regulation problems, it is necessary to solve the so-called regulator equations consisting of a partial differential equation and an algebraic equation. It is known that, for the hyperbolic zero dynamics case, solving the regulator equations is equivalent to calculating a center manifold for zero dynamics of the system. The present paper proposes a successive approximation method for obtaining center manifolds and shows its effectiveness by applying it for an inverted pendulum example.
19. Approximate Ad Hoc Parametric Solutions for Nonlinear First-Order PDEs Governing Two-Dimensional Steady Vector Fields
Directory of Open Access Journals (Sweden)
M. P. Markakis
2010-01-01
Full Text Available Through a suitable ad hoc assumption, a nonlinear PDE governing a three-dimensional weak, irrotational, steady vector field is reduced to a system of two nonlinear ODEs: the first of which corresponds to the two-dimensional case, while the second involves also the third field component. By using several analytical tools as well as linear approximations based on the weakness of the field, the first equation is transformed to an Abel differential equation which is solved parametrically. Thus, we obtain the two components of the field as explicit functions of a parameter. The derived solution is applied to the two-dimensional small perturbation frictionless flow past solid surfaces with either sinusoidal or parabolic geometry, where the plane velocities are evaluated over the body's surface in the case of a subsonic flow.
20. Comparison of exact solution with Eikonal approximation for elastic heavy ion scattering
Science.gov (United States)
Dubey, Rajendra R.; Khandelwal, Govind S.; Cucinotta, Francis A.; Maung, Khin Maung
1995-01-01
A first-order optical potential is used to calculate the total and absorption cross sections for nucleus-nucleus scattering. The differential cross section is calculated by using a partial-wave expansion of the Lippmann-Schwinger equation in momentum space. The results are compared with solutions in the Eikonal approximation for the equivalent potential and with experimental data in the energy range from 25A to 1000A MeV.
1. MHD FLOW OF A NEWTONIAN FLUID OVER A STRETCHING SHEET: AN APPROXIMATE SOLUTION
Institute of Scientific and Technical Information of China (English)
Chakraborty, B.K; Mazumdar, H.P.
2000-01-01
An approximate solution to the problem of steady laminar flow of a viscous incompressible electrically con ducting fluid over a stretching sheet is presented. The approach is based on the idea of stretching the variables of the flow problem and then using least squares method to minimize the residual of a differential equation. The effects of the magnetic field on the flow characteristics are demonstrated through numerical computations with di f ferent values of the Hartman monber.
2. Solución aproximada de sistemas diferenciales mixtos Approximated solution of differentials mixed systems
Directory of Open Access Journals (Sweden)
Jorge I. Castaño–Bedoya
2009-12-01
Full Text Available En este artículo se propone encontrar una solución aproximada para problemas de valor en la frontera y problemas de valor inicial de un sistema diferencial utilizando el método de los desarrollos de Fer.In this paper we propose to find an approximate solution to boundary value problems and initial value differential system problems using the method of Fer developments.
3. Computing a Finite Size Representation of the Set of Approximate Solutions of an MOP
CERN Document Server
Schuetze, Oliver; Tantar, Emilia; Talbi, El-Ghazali
2008-01-01
Recently, a framework for the approximation of the entire set of $\\epsilon$-efficient solutions (denote by $E_\\epsilon$) of a multi-objective optimization problem with stochastic search algorithms has been proposed. It was proven that such an algorithm produces -- under mild assumptions on the process to generate new candidate solutions --a sequence of archives which converges to $E_{\\epsilon}$ in the limit and in the probabilistic sense. The result, though satisfactory for most discrete MOPs, is at least from the practical viewpoint not sufficient for continuous models: in this case, the set of approximate solutions typically forms an $n$-dimensional object, where $n$ denotes the dimension of the parameter space, and thus, it may come to perfomance problems since in practise one has to cope with a finite archive. Here we focus on obtaining finite and tight approximations of $E_\\epsilon$, the latter measured by the Hausdorff distance. We propose and investigate a novel archiving strategy theoretically and emp...
4. Exact Analytical Solutions to the Two-Mode Mean-Field Model Describing Dynamics of a Split Bose-Einstein Condensate
Institute of Scientific and Technical Information of China (English)
WU Ying; YANG Xiao-Xue
2002-01-01
We present the analytical solutions to the two-mode mean-field model for a split Bose Einstein condensate.These explicit solutions completely determine the system's dynamics under the two-mode mean-field approximation for all possible initial conditions.
5. Exact and approximate Fourier rebinning algorithms for the solution of the data truncation problem in 3-D PET.
Science.gov (United States)
Bouallègue, Fayçal Ben; Crouzet, Jean-François; Comtat, Claude; Fourcade, Marjolaine; Mohammadi, Bijan; Mariano-Goulart, Denis
2007-07-01
This paper presents an extended 3-D exact rebinning formula in the Fourier space that leads to an iterative reprojection algorithm (iterative FOREPROJ), which enables the estimation of unmeasured oblique projection data on the basis of the whole set of measured data. In first approximation, this analytical formula also leads to an extended Fourier rebinning equation that is the basis for an approximate reprojection algorithm (extended FORE). These algorithms were evaluated on numerically simulated 3-D positron emission tomography (PET) data for the solution of the truncation problem, i.e., the estimation of the missing portions in the oblique projection data, before the application of algorithms that require complete projection data such as some rebinning methods (FOREX) or 3-D reconstruction algorithms (3DRP or direct Fourier methods). By taking advantage of all the 3-D data statistics, the iterative FOREPROJ reprojection provides a reliable alternative to the classical FOREPROJ method, which only exploits the low-statistics nonoblique data. It significantly improves the quality of the external reconstructed slices without loss of spatial resolution. As for the approximate extended FORE algorithm, it clearly exhibits limitations due to axial interpolations, but will require clinical studies with more realistic measured data in order to decide on its pertinence.
6. Interacting steps with finite-range interactions: Analytical approximation and numerical results
Science.gov (United States)
Jaramillo, Diego Felipe; Téllez, Gabriel; González, Diego Luis; Einstein, T. L.
2013-05-01
We calculate an analytical expression for the terrace-width distribution P(s) for an interacting step system with nearest- and next-nearest-neighbor interactions. Our model is derived by mapping the step system onto a statistically equivalent one-dimensional system of classical particles. The validity of the model is tested with several numerical simulations and experimental results. We explore the effect of the range of interactions q on the functional form of the terrace-width distribution and pair correlation functions. For physically plausible interactions, we find modest changes when next-nearest neighbor interactions are included and generally negligible changes when more distant interactions are allowed. We discuss methods for extracting from simulated experimental data the characteristic scale-setting terms in assumed potential forms.
7. Exact analytical solution of shear-induced flexural vibration of functionally graded piezoelectric beam
Science.gov (United States)
Sharma, Pankaj; Parashar, Sandeep Kumar
2016-05-01
The priority of this paper is to obtain the exact analytical solution for free flexural vibration of FGPM beam actuated using the d15 effect. In piezoelectric actuators, the potential use of d15 effect has been of particular interest for engineering applications since shear piezoelectric coefficient d15 is much higher than the other piezoelectric coupling constants d31 and d33. The applications of shear actuators are to induce and control the flexural vibrations of beams and plates. In this study, a modified Timoshenko beam theory is used where electric potential is assumed to vary sinusoidaly along the thickness direction. The material properties are assumed to be graded across the thickness in accordance with power law distribution. Hamiltons principle is employed to obtain the equations of motion along with the associated boundary conditions for FGPM beams. Exact analytical solution is derived thus obtained equations of motion. Results for clamped-clamped and clamped-free boundary conditions are presented. The presented result and method shell serve as benchmark for comparing the results obtained from the other approximate methods.
8. Application of the homotopy method for analytical solution of non-Newtonian channel flows
Energy Technology Data Exchange (ETDEWEB)
Roohi, Ehsan [Department of Aerospace Engineering, Sharif University of Technology, PO Box 11365-8639, Azadi Avenue, Tehran (Iran, Islamic Republic of); Kharazmi, Shahab [Department of Mechanical Engineering, Sharif University of Technology, PO Box 11365-8639, Azadi Avenue, Tehran (Iran, Islamic Republic of); Farjami, Yaghoub [Department of Computer Engineering, University of Qom, Qom (Iran, Islamic Republic of)], E-mail: [email protected]
2009-06-15
This paper presents the homotopy series solution of the Navier-Stokes and energy equations for non-Newtonian flows. Three different problems, Couette flow, Poiseuille flow and Couette-Poiseuille flow have been investigated. For all three cases, the nonlinear momentum and energy equations have been solved using the homotopy method and analytical approximations for the velocity and the temperature distribution have been obtained. The current results agree well with those obtained by the homotopy perturbation method derived by Siddiqui et al (2008 Chaos Solitons Fractals 36 182-92). In addition to providing analytical solutions, this paper draws attention to interesting physical phenomena observed in non-Newtonian channel flows. For example, it is observed that the velocity profile of non-Newtonian Couette flow is indistinctive from the velocity profile of the Newtonian one. Additionally, we observe flow separation in non-Newtonian Couette-Poiseuille flow even though the pressure gradient is negative (favorable). We provide physical reasoning for these unique phenomena.
9. Analytical Approximation of the Deconvolution of Strongly Overlapping Broad Fluorescence Bands
Science.gov (United States)
Dubrovkin, J. M.; Tomin, V. I.; Ushakou, D. V.
2016-09-01
A method for deconvoluting strongly overlapping spectral bands into separate components that enables the uniqueness of the deconvolution procedure to be monitored was proposed. An asymmetric polynomial-modified function subjected to Fourier filtering (PMGFS) that allowed more accurate and physically reasonable band shapes to be obtained and also improved significantly the deconvolution convergence was used as the band model. The method was applied to the analysis of complexation in solutions of the molecular probe 4'-(diethylamino)-3-hydroxyflavone with added LiCl. Two-band fluorescence of the probe in such solutions was the result of proton transfer in an excited singlet state and overlapped strongly with stronger spontaneous emission of complexes with the ions. Physically correct deconvolutions of overlapping bands could not always be obtained using available software.
10. Thermoelastic analysis of spent fuel and high level radioactive waste repositories in salt. A semi-analytical solution. [JUDITH
Energy Technology Data Exchange (ETDEWEB)
St. John, C.M.
1977-04-01
An underground repository containing heat generating, High Level Waste or Spent Unreprocessed Fuel may be approximated as a finite number of heat sources distributed across the plane of the repository. The resulting temperature, displacement and stress changes may be calculated using analytical solutions, providing linear thermoelasticity is assumed. This report documents a computer program based on this approach and gives results that form the basis for a comparison between the effects of disposing of High Level Waste and Spent Unreprocessed Fuel.
11. Analytical solution of the linearized hillslope-storage Boussinesq equation for exponential hillslope width functions
NARCIS (Netherlands)
Troch, P.A.A.; Loon, van A.H.; Hilberts, A.G.J.
2004-01-01
This technical note presents an analytical solution to the linearized hillslope-storage Boussinesq equation for subsurface flow along complex hillslopes with exponential width functions and discusses the application of analytical solutions to storage-based subsurface flow equations in catchment stud
12. Semiconductor quantum wells with BenDaniel-Duke boundary conditions: approximate analytical results
Science.gov (United States)
Barsan, Victor; Ciornei, Mihaela-Cristina
2017-01-01
The Schrödinger equation for a particle moving in a square well potential with BenDaniel-Duke boundary conditions is solved. Using algebraic approximations for trigonometric functions, the transcendental equations of the bound states energy are transformed into tractable, algebraic equations. For the ground state and the first excited state, they are cubic equations; we obtain simple formulas for their physically interesting roots. The case of higher excited states is also analysed. Our results have direct applications in the physics of type I and type II semiconductor heterostructures.
13. Analytical mechanics solutions to problems in classical physics
CERN Document Server
Merches, Ioan
2014-01-01
Fundamentals of Analytical Mechanics Constraints Classification Criteria for Constraints The Fundamental Dynamical Problem for a Constrained Particle System of Particles Subject to Constraints Lagrange Equations of the First KindElementary Displacements Generalities Real, Possible and Virtual Displacements Virtual Work and Connected Principles Principle of Virtual WorkPrinciple of Virtual Velocities Torricelli's Principle Principles of Analytical Mechanics D'alembert's Principle Configuration Space Generalized Forces Hamilton's Principle The Simple Pendulum Problem Classical (Newtonian) Formal
14. Approximate Solutions of Nonlinear Fractional Kolmogorov—Petrovskii—Piskunov Equations Using an Enhanced Algorithm of the Generalized Two-Dimensional Differential Transform Method
Science.gov (United States)
Song, Li-Na; Wang, Wei-Guo
2012-08-01
By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.
15. Approximate Solutions of Nonlinear Fractional Kolmogorov-Petrovskii-Piskunov Equations Using an Enhanced Algorithm of the Generalized Two-Dimensional Differential Transform Method
Institute of Scientific and Technical Information of China (English)
宋丽娜; 王维国
2012-01-01
By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.
16. Analytical solutions for thermal forcing vortices in boundary layer and its applications
Institute of Scientific and Technical Information of China (English)
LIU Xiao-ran; LI Guo-ping
2007-01-01
Using the Boussinesq approximation, the vortex in the boundary layer is assumed to be axisymmetrical and thermal-wind balanced system forced by diabatic heating and friction, and is solved as an initial-value problem of linearized vortex equation set in cylindrical coordinates. The impacts of thermal forcing on the flow field structure of vortex are analyzed. It is found that thermal forcing has significant impacts on the flow field structure, and the material representative forms of these impacts are closely related to the radial distribution of heating. The discussion for the analytical solutions for the vortex in the boundary layer can explain some main structures of the vortex over the Tibetan Plateau.
17. Analytical approximations for spatial stochastic gene expression in single cells and tissues.
Science.gov (United States)
Smith, Stephen; Cianci, Claudia; Grima, Ramon
2016-05-01
Gene expression occurs in an environment in which both stochastic and diffusive effects are significant. Spatial stochastic simulations are computationally expensive compared with their deterministic counterparts, and hence little is currently known of the significance of intrinsic noise in a spatial setting. Starting from the reaction-diffusion master equation (RDME) describing stochastic reaction-diffusion processes, we here derive expressions for the approximate steady-state mean concentrations which are explicit functions of the dimensionality of space, rate constants and diffusion coefficients. The expressions have a simple closed form when the system consists of one effective species. These formulae show that, even for spatially homogeneous systems, mean concentrations can depend on diffusion coefficients: this contradicts the predictions of deterministic reaction-diffusion processes, thus highlighting the importance of intrinsic noise. We confirm our theory by comparison with stochastic simulations, using the RDME and Brownian dynamics, of two models of stochastic and spatial gene expression in single cells and tissues.
18. A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems
Science.gov (United States)
Ying, Zu-guang; Luo, Yin-miao; Zhu, Wei-qiu; Ni, Yi-qing; Ko, Jan-ming
2012-04-01
A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.
19. Analytical determination of the two-body gravitational interaction potential at the 4th post-Newtonian approximation
CERN Document Server
Bini, Donato
2013-01-01
We complete the analytical determination, at the 4th post-Newtonian approximation, of the main radial potential describing the gravitational interaction of two bodies within the effective one-body formalism. The (non logarithmic) coefficient a_5 (nu) measuring this 4th post-Newtonian interaction potential is found to be linear in the symmetric mass ratio nu. Its nu-independent part a_5 (0) is obtained by an analytical gravitational self-force calculation that unambiguously resolves the formal infrared divergencies which currently impede its direct post-Newtonian calculation. Its nu-linear part a_5 (nu) - a_5 (0) is deduced from recent results of Jaranowski and Sch\\"afer, and is found to be significantly negative.
20. Analytical solution of population balance equation involving aggregation and breakage in terms of auxiliary equation method
Zehra Pinar; Abhishek Dutta; Guido Bény; Turgut Öziş
2015-01-01
This paper presents an effective analytical simulation to solve population balance equation (PBE), involving particulate aggregation and breakage, by making use of appropriate solution(s) of associated complementary equation via auxiliary equation method (AEM). Travelling wave solutions of the complementary equation of a nonlinear PBE with appropriately chosen parameters is taken to be analogous to the description of the dynamic behaviour of the particulate processes. For an initial proof-of-concept, a general case when the number of particles varies with respect to time is chosen. Three cases, i.e. (1) balanced aggregation and breakage, (2) when aggregation can dominate and (3) breakage can dominate, are selected and solved for their corresponding analytical solutions. The results are then compared with the available analytical solution, based on Laplace transform obtained from literature. In this communication, it is shown that the solution approach proposed via AEM is flexible and therefore more efficient than the analytical approach used in the literature.
1. Can We Remove Secular Terms for Analytical Solution of Groundwater Response under Tidal Influence?
CERN Document Server
Munusamy, Selva Balaji
2016-01-01
This paper presents a secular term removal methodology based on the homotopy perturbation method for analytical solutions of nonlinear problems with periodic boundary condition. The analytical solution for groundwater response to tidal fluctuation in a coastal unconfined aquifer system with the vertical beach is provided as an example. The non-linear one-dimensional Boussinesq's equation is considered as the governing equation for the groundwater flow. An analytical solution is provided for non-dimensional Boussinesq's equation with cosine harmonic boundary condition representing tidal boundary condition. The analytical solution is obtained by using homotopy perturbation method with a virtual embedding parameter. The present approach does not require pre-specified perturbation parameter and also facilitates secular terms elimination in the perturbation solution. The solutions starting from zeroth-order up to third-order are obtained. The non-dimensional expression, $A/D_{\\infty}$ emerges as an implicit parame...
2. A Table Lookup Method for Exact Analytical Solutions of Nonlinear Fractional Partial Differential Equations
Directory of Open Access Journals (Sweden)
Ji Juan-Juan
2017-01-01
Full Text Available A table lookup method for solving nonlinear fractional partial differential equations (fPDEs is proposed in this paper. Looking up the corresponding tables, we can quickly obtain the exact analytical solutions of fPDEs by using this method. To illustrate the validity of the method, we apply it to construct the exact analytical solutions of four nonlinear fPDEs, namely, the time fractional simplified MCH equation, the space-time fractional combined KdV-mKdV equation, the (2+1-dimensional time fractional Zoomeron equation, and the space-time fractional ZKBBM equation. As a result, many new types of exact analytical solutions are obtained including triangular periodic solution, hyperbolic function solution, singular solution, multiple solitary wave solution, and Jacobi elliptic function solution.
3. Approximate N-Player Nonzero-Sum Game Solution for an Uncertain Continuous Nonlinear System.
Science.gov (United States)
Johnson, Marcus; Kamalapurkar, Rushikesh; Bhasin, Shubhendu; Dixon, Warren E
2015-08-01
An approximate online equilibrium solution is developed for an N -player nonzero-sum game subject to continuous-time nonlinear unknown dynamics and an infinite horizon quadratic cost. A novel actor-critic-identifier structure is used, wherein a robust dynamic neural network is used to asymptotically identify the uncertain system with additive disturbances, and a set of critic and actor NNs are used to approximate the value functions and equilibrium policies, respectively. The weight update laws for the actor neural networks (NNs) are generated using a gradient-descent method, and the critic NNs are generated by least square regression, which are both based on the modified Bellman error that is independent of the system dynamics. A Lyapunov-based stability analysis shows that uniformly ultimately bounded tracking is achieved, and a convergence analysis demonstrates that the approximate control policies converge to a neighborhood of the optimal solutions. The actor, critic, and identifier structures are implemented in real time continuously and simultaneously. Simulations on two and three player games illustrate the performance of the developed method.
4. An Examination of the Multi-Peaked Analytically Extended Function for Approximation of Lightning Channel-Base Currents
CERN Document Server
Lundengård, Karl; Javor, Vesna; Silvestrov, Sergei
2016-01-01
A multi-peaked version of the analytically extended function (AEF) intended for approximation of multi-peaked lightning current wave-forms will be presented along with some of its basic properties. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks as well as a given charge trans-fer and specific energy will also be described. This framework is used to find parameters for some common single-peak wave-forms and some advantages and disadvantages of the approach will be discussed.
5. Approximate Nonnegative Symmetric Solution of Fully Fuzzy Systems Using Median Interval Defuzzification
Directory of Open Access Journals (Sweden)
R. Ezzati
2014-09-01
Full Text Available We propose an approach for computing an approximate nonnegative symmetric solution of some fully fuzzy linear system of equations, where the components of the coefficient matrix and the right hand side vector are nonnegative fuzzy numbers, considering equality of the median intervals of the left and right hand sides of the system. We convert the m×n fully fuzzy linear system to two m×n real linear systems, one being related to the cores and the other being concerned with spreads of the solution. We propose an approach for solving the real systems using the modified Huang method of the Abaffy-Broyden-Spedicato (ABS class of algorithms. An appropriate constrained least squares problem is solved when the solution does not satisfy nonnegative fuzziness conditions, that is, when the obtained solution vector for the core system includes a negative component, or the solution of the spread system has at least one negative component, or there exists an index for which the component of the spread is greater than the corresponding component of the core. As a special case, we discuss fuzzy systems with the components of the coefficient matrix as real crisp numbers. We finally present two computational algorithms and illustrate their effectiveness by solving some randomly generated consistent as well as inconsistent systems.
6. Analytical solution and meaning of feasible regions in two-component three-way arrays.
Science.gov (United States)
Omidikia, Nematollah; Abdollahi, Hamid; Kompany-Zareh, Mohsen; Rajkó, Róbert
2016-10-01
Although many efforts have been directed to the development of approximation methods for determining the extent of feasible regions in two- and three-way data sets; analytical determination (i.e. using only finite-step direct calculation(s) instead of the less exact numerical ones) of feasible regions in three-way arrays has remained unexplored. In this contribution, an analytical solution of trilinear decomposition is introduced which can be considered as a new direct method for the resolution of three-way two-component systems. The proposed analytical calculation method is applied to the full rank three-way data array and arrays with rank overlap (a type of rank deficiency) loadings in a mode. Close inspections of the analytically calculated feasible regions of rank deficient cases help us to make clearer the information gathered from multi-way problems frequently emerged in physics, chemistry, biology, agricultural, environmental and clinical sciences, etc. These examinations can also help to answer, e.g., the following practical question: "Is two-component three-way data with proportional loading in a mode actually a three-way data array?" By the aid of the additional information resulted from the investigated feasible regions of two-component three-way data arrays with proportional profile in a mode, reasons for the inadequacy of the seemingly trilinear data treatment methods published in the literature (e.g., U-PLS/RBL-LD that was used for extraction of quantitative and qualitative information reported by Olivieri et al. (Anal. Chem. 82 (2010) 4510-4519)) could be completely understood.
7. On analytical solutions of the generalized Boussinesq equation
Science.gov (United States)
Kudryashov, Nikolay A.; Volkov, Alexandr K.
2016-06-01
Extended Boussinesq equation for the description of the Fermi-Pasta-Ulam problem is studied. It is analysed with the Painlevé test. It is shown, that the equation does not pass the Painlevé test, although necessary conditions for existence of the meromorphic solution are carried out. Method of the logistic function is introduced for Solitary wave solutions of the considered equation. Elliptic solutions for studied equation are constructed and discussed.
8. Exact Markov chain and approximate diffusion solution for haploid genetic drift with one-way mutation.
Science.gov (United States)
Hössjer, Ola; Tyvand, Peder A; Miloh, Touvia
2016-02-01
The classical Kimura solution of the diffusion equation is investigated for a haploid random mating (Wright-Fisher) model, with one-way mutations and initial-value specified by the founder population. The validity of the transient diffusion solution is checked by exact Markov chain computations, using a Jordan decomposition of the transition matrix. The conclusion is that the one-way diffusion model mostly works well, although the rate of convergence depends on the initial allele frequency and the mutation rate. The diffusion approximation is poor for mutation rates so low that the non-fixation boundary is regular. When this happens we perturb the diffusion solution around the non-fixation boundary and obtain a more accurate approximation that takes quasi-fixation of the mutant allele into account. The main application is to quantify how fast a specific genetic variant of the infinite alleles model is lost. We also discuss extensions of the quasi-fixation approach to other models with small mutation rates.
9. Analytic continuation of solutions of some nonlinear convolution partial differential equations
Directory of Open Access Journals (Sweden)
Hidetoshi Tahara
2015-01-01
Full Text Available The paper considers a problem of analytic continuation of solutions of some nonlinear convolution partial differential equations which naturally appear in the summability theory of formal solutions of nonlinear partial differential equations. Under a suitable assumption it is proved that any local holomorphic solution has an analytic extension to a certain sector and its extension has exponential growth when the variable goes to infinity in the sector.
10. Analytical Applications of Electrified Interfaces Between Two Immiscible Solutions
Science.gov (United States)
1993-04-07
electrode potentiostate needed for the iwork is described. Experimental techniques involving potentiometry , polarography with dropping electrode...convert any potentiostat to the 4-electrode potentiostat needed for the work is described. Experimental techniques involving potentiometry , polarography...potentiostat input. Analytical applications Potentiometry Potentiometric measurements on ITIES are related to the principle of ionl selective electrodes (ISE
11. Exact Solution Versus Gaussian Approximation for a Non-Ideal Bose Gas in One-Dimension
CERN Document Server
Tommasini, P; Natti, P L
1997-01-01
We investigate ground-state and excitation spectrum of a system of non-relativistic bosons in one-dimension interacting through repulsive, two-body contact interactions in a self-consistent Gaussian mean-field approximation which consists in writing the variationally determined density operator as the most general Gaussian functional of the quantized field operators. There are mainly two advantages in working with one-dimension. First, the existence of an exact solution for the ground-state and excitation energies. Second, neither in the perturbative results nor in the Gaussian approximation itself we do not have to deal with the three-dimensional patologies of the contact interaction . So that this scheme provides a clear comparison between these three different results. PACS numbers : 05.30.-d, 05.30.Jp, 67.40.Db
12. ANALYTICAL SOLUTION FOR WAVES IN PLANETS WITH ATMOSPHERIC SUPERROTATION. II. LAMB, SURFACE, AND CENTRIFUGAL WAVES
Energy Technology Data Exchange (ETDEWEB)
Peralta, J.; López-Valverde, M. A. [Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, 18008 Granada (Spain); Imamura, T. [Institute of Space and Astronautical Science-Japan Aerospace Exploration Agency 3-1-1, Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210 (Japan); Read, P. L. [Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford (United Kingdom); Luz, D. [Centro de Astronomia e Astrofísica da Universidade de Lisboa (CAAUL), Observatório Astronómico de Lisboa, Tapada da Ajuda, 1349-018 Lisboa (Portugal); Piccialli, A., E-mail: [email protected] [LATMOS, UVSQ, 11 bd dAlembert, 78280 Guyancourt (France)
2014-07-01
This paper is the second in a two-part study devoted to developing tools for a systematic classification of the wide variety of atmospheric waves expected on slowly rotating planets with atmospheric superrotation. Starting with the primitive equations for a cyclostrophic regime, we have deduced the analytical solution for the possible waves, simultaneously including the effect of the metric terms for the centrifugal force and the meridional shear of the background wind. In those cases where the conditions for the method of the multiple scales in height are met, these wave solutions are also valid when vertical shear of the background wind is present. A total of six types of waves have been found and their properties were characterized in terms of the corresponding dispersion relations and wave structures. In this second part, we study the waves' solutions when several atmospheric approximations are applied: Lamb, surface, and centrifugal waves. Lamb and surface waves are found to be quite similar to those in a geostrophic regime. By contrast, centrifugal waves turn out to be a special case of Rossby waves that arise in atmospheres in cyclostrophic balance. Finally, we use our results to identify the nature of the waves behind atmospheric periodicities found in polar and lower latitudes of Venus's atmosphere.
13. Numerical solution of nonlinear partial differential equations of mixed type. [finite difference approximation
Science.gov (United States)
Jameson, A.
1976-01-01
A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
14. An approximate solution to the stress and deformation states of functionally graded rotating disks
Science.gov (United States)
Sondhi, Lakshman; Sanyal, Shubhashis; Saha, Kashi Nath; Bhowmick, Shubhankar
2016-07-01
The present work employs variational principle to investigate the stress and deformation states and estimate the limit angular speed of functionally graded high-speed rotating annular disks of constant thickness. Assuming a series approximation following Galerkin's principle, the solution of the governing equation is obtained. In the present study, elasticity modulus and density of the disk material are taken as power function of radius with the gradient parameter ranging between 0.0 and 1.0. Results obtained from numerical solutions are validated with benchmark results and are found to be in good agreement. The results are reported in dimensional form and presented graphically. The results provide a substantial insight in understanding the behavior of FGM rotating disks with constant thickness and different gradient parameter. Furthermore, the stress and deformation state of the disk at constant angular speed and limit angular speed is investigated to explain the existence of optimum gradient parameters.
15. Explicit analytical wave solutions of unsteady 1D ideal gas flow with friction and heat transfer
Institute of Scientific and Technical Information of China (English)
2001-01-01
Several families of algebraically explicit analytical wavesolutions are derived for the unsteady 1D ideal gas flow with friction and heat-transfer, which include one family of travelling wave solutions, three families of standing wave solutions and one standing wave solution. \\{Among\\} them, the former four solution families contain arbitrary functions, so actually there are infinite analytical wave solutions having been derived. Besides their very important theoretical meaning, such analytical wave solutions can guide the development of some new equipment, and can be the benchmark solutions to promote the development of computational fluid dynamics. For example, we can use them to check the accuracy, convergence and effectiveness of various numerical computational methods and to improve the numerical computation skills such as differential schemes, grid generation ways and so on.
16. Analytic Solutions of Some Self-Adjoint Equations by Using Variable Change Method and Its Applications
Directory of Open Access Journals (Sweden)
Mehdi Delkhosh
2012-01-01
Full Text Available Many applications of various self-adjoint differential equations, whose solutions are complex, are produced (Arfken, 1985; Gandarias, 2011; and Delkhosh, 2011. In this work we propose a method for the solving some self-adjoint equations with variable change in problem, and then we obtain a analytical solutions. Because this solution, an exact analytical solution can be provided to us, we benefited from the solution of numerical Self-adjoint equations (Mohynl-Din, 2009; Allame and Azal, 2011; Borhanifar et al. 2011; Sweilam and Nagy, 2011; Gülsu et al. 2011; Mohyud-Din et al. 2010; and Li et al. 1996.
17. Analytic solutions of transcendental equations with application to automatics
Directory of Open Access Journals (Sweden)
Górecki Henryk
2016-12-01
Full Text Available In the paper the extremal dynamic error x(τ and the moment of time τ are considered. The extremal value of dynamic error gives information about accuracy of the system. The time τ gives information about velocity of transient. The analytical formulae enable design of the system with prescribed properties. These formulae are calculated due to the assumption that x(τ is a function of the roots s1, ..., sn of the characteristic equation.
18. Approximate solutions to a nonintegrable problem of propagation of elliptically polarised waves in an isotropic gyrotropic nonlinear medium, and periodic analogues of multisoliton complexes
Energy Technology Data Exchange (ETDEWEB)
Makarov, V A; Petnikova, V M; Potravkin, N N; Shuvalov, V V [International Laser Center, M. V. Lomonosov Moscow State University, Moscow (Russian Federation)
2014-02-28
Using the linearization method, we obtain approximate solutions to a one-dimensional nonintegrable problem of propagation of elliptically polarised light waves in an isotropic gyrotropic medium with local and nonlocal components of the Kerr nonlinearity and group-velocity dispersion. The consistent evolution of two orthogonal circularly polarised components of the field is described analytically in the case when their phases vary linearly during propagation. The conditions are determined for the excitation of waves with a regular and 'chaotic' change in the polarisation state. The character of the corresponding nonlinear solutions, i.e., periodic analogues of multisoliton complexes, is analysed. (nonlinear optical phenomena)
19. Analytical solutions of simply supported magnetoelectroelastic circular plate under uniform loads
Institute of Scientific and Technical Information of China (English)
陈江英; 丁皓江; 侯鹏飞
2003-01-01
In this paper, the axisymmetric general solutions of transversely isotropic magnetoelectroelastic media are expressed with four harmonic displacement functions at first. Then, based on the solutions, the analytical three-dimensional solutions are provided for a simply supported magnetoelectroelastic circular plate subjected to uniform loads. Finally, the example of circular plate is presented.
20. Analytical Solution for Transient Water Table Heights and Outflows from Inclined Ditch-Drained Terrains
NARCIS (Netherlands)
Verhoest, N.E.C.; Pauwels, V.R.N.; Troch, P.A.; Troch, De F.P.
2002-01-01
This paper presents two analytical solutions of the linearized Boussinesq equation for an inclined aquifer, drained by ditches, subjected to a constant recharge rate. These solutions are based on different initial conditions. First, the transient solution is obtained for an initially fully saturated
1. Analytical solution for electromagnetic scattering from a sphere of uniaxial left-handed material
Institute of Scientific and Technical Information of China (English)
GENG You-lin; HE Sai-ling
2006-01-01
Based on the analytical solution of electromagnetic scattering by a uniaxial anisotropic sphere in the spectral domain,an analytical solution to the electromagnetic scattering by a uniaxial left-handed materials (LHMs) sphere is obtained in terms of spherical vector wave functions in a uniaxial anisotropic LHM medium. The expression of the analytical solution contains only some one-dimensional integral which can be calculated easily. Numerical results show that Mie series of plane wave scattering by an isotropic LHM sphere is a special case of the present method. Some numerical results of electromagnetic scattering ofa uniaxial anisotropic sphere by a plane wave are given.
2. Analytical theories of transport in concentrated electrolyte solutions from the MSA.
Science.gov (United States)
Dufrêche, J-F; Bernard, O; Durand-Vidal, S; Turq, P
2005-05-26
Ion transport coefficients in electrolyte solutions (e.g., diffusion coefficients or electric conductivity) have been a subject of extensive studies for a long time. Whereas in the pioneering works of Debye, Hückel, and Onsager the ions were entirely characterized by their charge, recent theories allow specific effects of the ions (such as the ion size dependence or the pair association) to be obtained, both from simulation and from analytical theories. Such an approach, based on a combination of dynamic theories (Smoluchowski equation and mode-coupling theory) and of the mean spherical approximation (MSA) for the equilibrium pair correlation, is presented here. The various predicted equilibrium (osmotic pressure and activity coefficients) and transport coefficients (mutual diffusion, electric conductivity, self-diffusion, and transport numbers) are in good agreement with the experimental values up to high concentrations (1-2 mol L(-1)). Simple analytical expressions are obtained, and for practical use, the formula are given explicitly. We discuss the validity of such an approach which is nothing but a coarse-graining procedure.
3. Weak and Strong Solutions for the Stokes Approximation of Non-homogeneous Incompressible Navier-Stokes Equations
Institute of Scientific and Technical Information of China (English)
2007-01-01
In this paper, the Dirichlet problem of Stokes approximate of non-homogeneous incompressible Navier-Stokes equations is studied. It is shown that there exist global weak solutions as well as global and unique strong solution for this problem, under the assumption that initial density ρ0(x) is bounded away from 0 and other appropriate assumptions (see Theorem 1 and Theorem 2). The semi-Galerkin method is applied to construct the approximate solutions and a prior estimates are made to elaborate upon the compactness of the approximate solutions.
4. Efficient approximation of the solution of certain nonlinear reaction--diffusion equation I: the case of small absorption
CERN Document Server
Dratman, Ezequiel
2011-01-01
We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that, if \\emph{the absorption is small enough}, compared with the flux in the boundary, there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the "continuous" equation. Furthermore, we exhibit an algorithm computing an $\\epsilon$-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is {\\em linear} in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.
5. Efficient approximation of the solution of certain nonlinear reaction--diffusion equation II: the case of large absorption
CERN Document Server
Dratman, Ezequiel
2011-01-01
We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that, if the absorption is large enough, compared with the flux in the boundary, there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the "continuous" equation. Furthermore, we exhibit an algorithm computing an $\\epsilon$-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is {\\em linear} in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.
6. Analyticity of solutions of analytic non-linear general elliptic boundary value problems,and some results about linear problems
Institute of Scientific and Technical Information of China (English)
WANG Rouhuai
2006-01-01
The main aim of this paper is to discuss the problem concerning the analyticity of the solutions of analytic non-linear elliptic boundary value problems.It is proved that if the corresponding first variation is regular in Lopatinski(i) sense,then the solution is analytic up to the boundary.The method of proof really covers the case that the corresponding first variation is regularly elliptic in the sense of Douglis-Nirenberg-Volevich,and hence completely generalize the previous result of C.B.Morrey.The author also discusses linear elliptic boundary value problems for systems of ellip tic partial differential equations where the boundary operators are allowed to have singular integral operators as their coefficients.Combining the standard Fourier transform technique with analytic continuation argument,the author constructs the Poisson and Green's kernel matrices related to the problems discussed and hence obtain some representation formulae to the solutions.Some a priori estimates of Schauder type and Lp type are obtained.
7. ORTHOGONAL POLYNOMIALS AND DETERMINANT FORMULAS OF FUNCTION-VALUED PAD(E)-TYPE APPROXIMATION USING FOR SOLUTION OF INTEGRAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
GU Chuan-qing; PAN Bao-zhen; WU Bei-bei
2006-01-01
To solve Fredholm integral equations of the second kind, a generalized linear functional is introduced and a new function-valued Padé-type approximation is defined.By means of the power series expansion of the solution, this method can construct an approximate solution to solve the given integral equation. On the basis of the orthogonal polynomials, two useful determinant expressions of the numerator polynomial and the denominator polynomial for padé-type approximation are explicitly given.
8. Analytical Solutions to Non-linear Mechanical Oscillation Problems
DEFF Research Database (Denmark)
Kaliji, H. D.; Ghadimi, M.; Barari, Amin
2011-01-01
In this paper, the Max-Min Method is utilized for solving the nonlinear oscillation problems. The proposed approach is applied to three systems with complex nonlinear terms in their motion equations. By means of this method, the dynamic behavior of oscillation systems can be easily approximated u...
9. General Analytical Solutions of Scalar Field Cosmology with Arbitrary Potential
CERN Document Server
Dimakis, N; Zampeli, Adamantia; Paliathanasis, Andronikos; Christodoulakis, T; Terzis, Petros A
2016-01-01
We present the solution space for the case of a minimally coupled scalar field with arbitrary potential in a FLRW metric. This is made possible due to the existence of a nonlocal integral of motion corresponding to the conformal Killing field of the two-dimensional minisuperspace metric. The case for both spatially flat and non flat are studied first in the presence of only the scalar field and subsequently with the addition of non interacting perfect fluids. It is verified that this addition does not change the general form of the solution, but only the particular expressions of the scalar field and the potential. The results are applied in the case of parametric dark energy models where we derive the scalar field equivalence solution for some proposed models in the literature.
10. Analytical solution for multilayer plates using general layerwise plate theory
Directory of Open Access Journals (Sweden)
Vuksanović Đorđe M.
2005-01-01
Full Text Available This paper deals with closed-form solution for static analysis of simply supported composite plate, based on generalized laminate plate theory (GLPT. The mathematical model assumes piece-wise linear variation of in-plane displacement components and a constant transverse displacement through the thickness. It also include discrete transverse shear effect into the assumed displacement field, thus providing accurate prediction of transverse shear stresses. Namely, transverse stresses satisfy Hook's law, 3D equilibrium equations and traction free boundary conditions. With assumed displacement field, linear strain-displacement relation, and constitutive equations of the lamina, equilibrium equations are derived using principle of virtual displacements. Navier-type closed form solution of GLPT, is derived for simply supported plate, made of orthotropic laminae, loaded by harmonic and uniform distribution of transverse pressure. Results are compared with 3D elasticity solutions and excellent agreement is found.
11. A Quantum Dot with Spin-Orbit Interaction--Analytical Solution
Science.gov (United States)
Basu, B.; Roy, B.
2009-01-01
The practical applicability of a semiconductor quantum dot with spin-orbit interaction gives an impetus to study analytical solutions to one- and two-electron quantum dots with or without a magnetic field.
12. The analyticity of solutions to a class of degenerate elliptic equations
Institute of Scientific and Technical Information of China (English)
2010-01-01
In the present paper,the analyticity of solutions to a class of degenerate elliptic equations is obtained.A kind of weighted norms are introduced and under such norms some degenerate elliptic operators are of weak coerciveness.
13. Analytical solutions for transport processes fluid mechanics, heat and mass transfer
CERN Document Server
Brenn, Günter
2017-01-01
This book provides analytical solutions to a number of classical problems in transport processes, i.e. in fluid mechanics, heat and mass transfer. Expanding computing power and more efficient numerical methods have increased the importance of computational tools. However, the interpretation of these results is often difficult and the computational results need to be tested against the analytical results, making analytical solutions a valuable commodity. Furthermore, analytical solutions for transport processes provide a much deeper understanding of the physical phenomena involved in a given process than do corresponding numerical solutions. Though this book primarily addresses the needs of researchers and practitioners, it may also be beneficial for graduate students just entering the field. .
14. Analytical Solutions for Steady Heat Transfer in Longitudinal Fins with Temperature-Dependent Properties
Directory of Open Access Journals (Sweden)
Partner L. Ndlovu
2013-01-01
Full Text Available Explicit analytical expressions for the temperature profile, fin efficiency, and heat flux in a longitudinal fin are derived. Here, thermal conductivity and heat transfer coefficient depend on the temperature. The differential transform method (DTM is employed to construct the analytical (series solutions. Thermal conductivity is considered to be given by the power law in one case and by the linear function of temperature in the other, whereas heat transfer coefficient is only given by the power law. The analytical solutions constructed by the DTM agree very well with the exact solutions even when both the thermal conductivity and the heat transfer coefficient are given by the power law. The analytical solutions are obtained for the problems which cannot be solved exactly. The effects of some physical parameters such as the thermogeometric fin parameter and thermal conductivity gradient on temperature distribution are illustrated and explained.
15. Application of Analytic Solution in Relative Motion to Spacecraft Formation Flying in Elliptic Orbit
Science.gov (United States)
Cho, Hancheol; Park, Sang-Young; Choi, Kyu-Hong
2008-09-01
The current paper presents application of a new analytic solution in general relative motion to spacecraft formation flying in an elliptic orbit. The calculus of variations is used to analytically find optimal trajectories and controls for the given problem. The inverse of the fundamental matrix associated with the dynamic equations is not required for the solution in the current study. It is verified that the optimal thrust vector is a function of the fundamental matrix of the given state equations. The cost function and the state vector during the reconfiguration can be analytically obtained as well. The results predict the form of optimal solutions in advance without having to solve the problem. Numerical simulation shows the brevity and the accuracy of the general analytic solutions developed in the current paper.
16. Analytic method for solitary solutions of some partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Ugurlu, Yavuz [Firat University, Department of Mathematics, 23119 Elazig (Turkey); Kaya, Dogan [Firat University, Department of Mathematics, 23119 Elazig (Turkey)], E-mail: [email protected]
2007-10-22
In this Letter by considering an improved tanh function method, we found some exact solutions of the clannish random walker's parabolic equation, the modified Korteweg-de Vries (KdV) equation, and the Sharma-Tasso-Olver (STO) equation with its fission and fusion, the Jaulent-Miodek equation.
17. Analytical solutions of weakly coupled map lattices using recurrence relations
Energy Technology Data Exchange (ETDEWEB)
Sotelo Herrera, Dolores, E-mail: [email protected] [Applied Maths, EUITI, UPM, Ronda de Valencia, 3-28012 Madrid (Spain); San Martin, Jesus [Applied Maths, EUITI, UPM, Ronda de Valencia, 3-28012 Madrid (Spain); Dep. Fisica Matematica y de Fluidos, UNED, Senda del Rey 9-28040 Madrid (Spain)
2009-07-20
By using asymptotic methods recurrence relations are found that rule weakly CML evolution, with both global and diffusive coupling. The solutions obtained from these relations are very general because they do not hold restrictions about boundary conditions, initial conditions and number of oscilators in the CML. Furthermore, oscillators are ruled by an arbitraty C{sup 2} function.
18. General scalar-tensor cosmology: analytical solutions via noether symmetry
Science.gov (United States)
Massaeli, Erfan; Motaharfar, Meysam; Sepangi, Hamid Reza
2017-02-01
We analyze the cosmology of a general scalar-tensor theory which encompasses generalized Brans-Dicke theory, Gauss-Bonnet gravity, non-minimal derivative gravity, generalized Galilean gravity and also the general k-essence type models. Instead of taking into account phenomenological considerations we adopt a Noether symmetry approach, as a physical criterion, to single out the form of undetermined functions in the action. These specified functions symmetrize equations of motion in the simplest possible form which result in exact solutions. Demanding de Sitter, power-law and bouncing universe solutions in the absence and presence of matter density leads to exploring new as well as well-investigated models. We show that there are models for which the dynamics of the system allows a transition from a decelerating phase (matter dominated era) to an accelerating phase (dark energy epoch) and could also lead to general Brans-Dicke with string correction without a self-interaction potential. Furthermore, we classify the models based on a phantom or quintessence dark energy point of view. Finally, we obtain the condition for stability of a de Sitter solution for which the solution is an attractor of the system.
19. Least-Squares Solutions of the Matrix Equation ATXA = B Over Bisymmetric Matrices and its Optimal Approximation
Institute of Scientific and Technical Information of China (English)
2007-01-01
A real n × n symmetric matrix X = (xij)n×n is called a bisymmetric matrix if xij = xn+1-j,n+1-i. Based on the projection theorem, the canonical correlation decomposition and the generalized singular value decomposition, a method useful for finding the least-squares solutions of the matrix equation ATXA = B over bisymmetric matrices is proposed. The expression of the least-squares solutions is given. Moreover, in the corresponding solution set, the optimal approximate solution to a given matrix is also derived. A numerical algorithm for finding the optimal approximate solution is also described.
20. An Analytical Solution for Acoustic Emission Source Location for Known P Wave Velocity System
Directory of Open Access Journals (Sweden)
Longjun Dong
2014-01-01
Full Text Available This paper presents a three-dimensional analytical solution for acoustic emission source location using time difference of arrival (TDOA measurements from N receivers, N⩾5. The nonlinear location equations for TDOA are simplified to linear equations, and the direct analytical solution is obtained by solving the linear equations. There are not calculations of square roots in solution equations. The method solved the problems of the existence and multiplicity of solutions induced by the calculations of square roots in existed close-form methods. Simulations are included to study the algorithms' performance and compare with the existing technique.
1. Analytical solution based on stream-aquifer interactions in partially penetrating streams
Institute of Scientific and Technical Information of China (English)
Yong HUANG; Zhi-fang ZHOU; Zhong-bo YU
2010-01-01
An analytical solution of drawdown caused by pumping was developed for an aquifer partially penetrated by two streams.The proposed analytical solution modifies Hunt's analytical solution and considers the effects of stream width and the interaction of two streams on drawdown.Advantages of the solution include its simple structure,consisting of the Theis well function and parameters of aquifer and streambed semipervious material.The calculated results show that the proposed analytical solution agrees with a previously developed acceptable solution and the errors between the two solutions are equal to zero without consideration of the effect of stream width.Also,deviations between the two analytical solutions incrcase with stream width.Four cases were studied to examine the effect of two streams on drawdown,assuming that some parameters were changeable,and other parameters were constant,such as the stream width,the distance between the stream and the pumping well,the stream recharge rate,and the leakage coefficient of streambed semipervious material.
2. Analytic study of solutions for the Born-Infeld equation in nonlinear electrodynamics
Science.gov (United States)
Gao, Hui; Xu, Tianzhou; Fan, Tianyou; Wang, Gangwei
2017-03-01
The Born-Infeld equation is an important nonlinear partial differential equation in theoretical and mathematical physics. The Lie group method is used for simplifying the nonlinear partial differential equation, which is partly solved, in which there are some difficulties; to overcome the difficulties, we develop a power series method, and find the solutions in analytic form. In the mean time, a wave propagation (traveling wave) method is developed for solving the equation, and analytic solutions are also constructed.
3. Analytical solutions for the slow neutron capture process of heavy element nucleosynthesis
Institute of Scientific and Technical Information of China (English)
Wu Kai-Su
2009-01-01
In this paper,the network equation for the slow neutron capture process (s-process) of heavy element nucleosynthesis is investigated. Dividing the s-process network reaction chains into two standard forms and using the technique of matrix decomposition,a group of analytical solutions for the network equation are obtained. With the analytical solutions,a calculation for heavy element abundance of the solar system is carried out and the results are in good agreement with the astrophysical measurements.
4. Method of the Logistic Function for Finding Analytical Solutions of Nonlinear Differential Equations
OpenAIRE
Kudryashov, N. A.
2015-01-01
The method of the logistic function is presented for finding exact solutions of nonlinear differential equations. The application of the method is illustrated by using the nonlinear ordinary differential equation of the fourth order. Analytical solutions obtained by this method are presented. These solutions are expressed via exponential functions.logistic function, nonlinear wave, nonlinear ordinary differential equation, Painlev´e test, exact solution
5. APPROXIMATE ANALYTIC SOLUTIONS FOR THE IONIZATION STRUCTURE OF A DUSTY STRÖMGREN SPHERE
Directory of Open Access Journals (Sweden)
A. C. Raga
2015-01-01
Full Text Available Presentamos un modelo de balance global de “esfera de Str ̈om gren” para el caso de regiones HII polvorientas. De este modelo, obtenemo s prescripciones para el radio exterior de las nebulosas en funci ́on del radio de St r ̈omgren R S (de la nebulosa correspondiente libre de polvo y del espesor ́opt ico del polvo. Tambien obtenemos una nueva soluci ́on anal ́ıtica aproximada para e l problema de transporte radiativo, dando formas anal ́ıticas para la fracci ́on de io nizaci ́on en funci ́on del radio. Estas soluciones se comparan con los resultados obte nidos del an ́alisis de esfera de Str ̈omgren. Nuestros resultados pueden ser usado s para evaluar bajo qu ́e condiciones la presencia de polvo puede tener un efecto importante sobre las estructuras de regiones HII
6. Intertemporal Asset Allocation with Habit Formation in Preferences: An Approximate Analytical Solution
DEFF Research Database (Denmark)
Pedersen, Thomas Quistgaard
assets: a risk free asset with constant return and a risky asset with a time-varying premium. We extend the ap- proach proposed by Campbell and Viceira (1999), which builds on log-linearizations of the Euler equation, intertemporal budget constraint, and portfolio return, to also contain the log...
7. APPROXIMATE ANALYTICAL SOLUTION FOR THE ISOTHERMAL LANE EMDEN EQUATION IN A SPHERICAL GEOMETRY
Directory of Open Access Journals (Sweden)
Moustafa Aly Soliman
2015-01-01
Full Text Available Este trabajo obtiene una soluci ́on anal ́ıtica aproximada p ara la ecuaci ́on isoterma de Lane-Emden que modela una esfera isot ́ermica au togravitante. La soluci ́on aproximada se obtiene en t ́erminos de par ́ametro s de distancias peque ̃nos y grandes por el m ́etodo de perturbaciones. La soluci ́on apr oximada se compara con la soluci ́on n ́umerica. La soluci ́on aproximada obteni da es v ́alida para todos los valores del par ́ametro de distancia.
8. An approximate analytical solution for non-Darcy flow toward awell infractured media
Energy Technology Data Exchange (ETDEWEB)
Wu, Yu-Shu
2001-06-08
Estuarine suspended sediment is transported in a mixed nonuniform way under unsteady flows. Sediment of different grain sizes has different characteristics and transport behavior and has a different effect on the ecological system. Therefore classification and fractionization of the mixed sediment are required before the flux is estimated. A fuzzy clustering approach is applied to the classification of suspended fine-grained sediment in the Changjiang Estuary. Two populations are objectively found by considering the standard grain-size distribution statistics of each cluster. The critical grain size of {approx}10??m in diameter is the size limit for cohesive sediments. A grid with equal cell areas is used to estimate fractional sediment fluxes through an estuarine cross section since this type of grid introduces less statistical error in the flux calculation. The sediment transport mechanism is analyzed.
9. Methods for estimating uncertainty in factor analytic solutions
Directory of Open Access Journals (Sweden)
P. Paatero
2013-08-01
Full Text Available EPA PMF version 5.0 and the underlying multilinear engine executable ME-2 contain three methods for estimating uncertainty in factor analytic models: classical bootstrap (BS, displacement of factor elements (DISP, and bootstrap enhanced by displacement of factor elements (BS-DISP. The goal of these methods is to capture the uncertainty of PMF analyses due to random errors and rotational ambiguity. It is shown that the three methods complement each other: depending on characteristics of the data set, one method may provide better results than the other two. Results are presented using synthetic data sets, including interpretation of diagnostics, and recommendations are given for parameters to report when documenting uncertainty estimates from EPA PMF or ME-2 applications.
10. Analytical solutions for space charge fields in TPC drift volumes
CERN Document Server
Rossegger, S; Schnizer, B
2011-01-01
At high particle rates and high multiplicities, Time Projection Chambers can suffer from field distortions due to slow moving ions that accumulate within the drift volume. These variations modify the electron trajectory along the drift path, affecting the tracking performance of the detector. In order to calculate the track distortions due to an arbitrary space charge distribution in a TPC, novel representations of the Green's function for a TPC-like geometry were worked out. This analytical approach permits accurate predictions of track distortions due to an arbitrary space charge distribution (by solving the Langevin equation) as well as the possibility to benchmark common numerical methods to calculate such space charge fields. (C) 2011 Elsevier B.V. All rights reserved.
11. Approximate solution of the multiple watchman routes problem with restricted visibility range.
Science.gov (United States)
Faigl, Jan
2010-10-01
In this paper, a new self-organizing map (SOM) based adaptation procedure is proposed to address the multiple watchman route problem with the restricted visibility range in the polygonal domain W. A watchman route is represented by a ring of connected neuron weights that evolves in W, while obstacles are considered by approximation of the shortest path. The adaptation procedure considers a coverage of W by the ring in order to attract nodes toward uncovered parts of W. The proposed procedure is experimentally verified in a set of environments and several visibility ranges. Performance of the procedure is compared with the decoupled approach based on solutions of the art gallery problem and the consecutive traveling salesman problem. The experimental results show the suitability of the proposed procedure based on relatively simple supporting geometrical structures, enabling application of the SOM principles to watchman route problems in W.
12. APPROXIMATION OF FIXED POINTS AND VARIATIONAL SOLUTIONS FOR PSEUDO-CONTRACTIVE MAPPINGS IN BANACH SPACES
Institute of Scientific and Technical Information of China (English)
Yekini SHEHU
2014-01-01
Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gˆateaux differentiable norm. Assume that every nonempty closed con-vex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive map-pings which is also a unique solution to variational inequality problem involving φ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, con-vex optimization problems, and split feasibility problems. Our result extends many recent important results.
13. Analytical solution for wave-induced response of isotropic poro-elastic seabed
Institute of Scientific and Technical Information of China (English)
2010-01-01
By use of separation of variables,the governing equations describing the Biot consolidation model is firstly transformed into a complex coefficient linear homogeneous ordinary differential equation,and the general solution of the horizontal displacement of seabed is constructed by employing a complex wave number,thus,all the explicit analytical solutions of the Biot consolidation model are determined. By comparing with the experimental results and analytical solution of Yamamoto etc. and the analytical solution of Hsu and Jeng,the validity and superiority of the suggested solution are verified. After investigating the influence of seabed depth on the wave-induced response of isotropic poro-elastic seabed based on the present theory,it can be concluded that the influence depth of wave-induced hydrodynamic pressure in the seabed is equal to the wave length.
14. Analytical solutions for Dirac and Klein-Gordon equations using Backlund transformations
Energy Technology Data Exchange (ETDEWEB)
Zabadal, Jorge R.; Borges, Volnei, E-mail: [email protected], E-mail: [email protected] [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Dept. de Engenharia Mecanica; Ribeiro, Vinicius G., E-mail: [email protected] [Centro Universitario Ritter dos Reis (UNIRITTER), Porto Alegre, RS (Brazil); Santos, Marcio, E-mail: [email protected] [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Centro de Estudos Interdisciplinares
2015-07-01
This work presents a new analytical method for solving Klein-Gordon type equations via Backlund transformations. The method consists in mapping the Klein-Gordon model into a first order system of partial differential equations, which contains a generalized velocity field instead of the Dirac matrices. This system is a tensor model for quantum field theory whose space solution is wider than the Dirac model in the original form. Thus, after finding analytical expressions for the wave functions, the Maxwell field can be readily obtained from the Dirac equations, furnishing a self-consistent field solution for the Maxwell-Dirac system. Analytical and numerical results are reported. (author)
15. Inhomogeneous Poisson point process nucleation: comparison of analytical solution with cellular automata simulation
Directory of Open Access Journals (Sweden)
Paulo Rangel Rios
2009-06-01
Full Text Available Microstructural evolution in three dimensions of nucleation and growth transformations is simulated by means of cellular automata (CA. In the simulation, nuclei are located in space according to a heterogeneous Poisson point processes. The simulation is compared with exact analytical solution recently obtained by Rios and Villa supposing that the intensity is a harmonic function of the spatial coordinate. The simulated data gives very good agreement with the analytical solution provided that the correct shape factor for the growing CA grains is used. This good agreement is auspicious because the analytical expressions were derived and thus are exact only if the shape of the growing regions is spherical.
16. On the Poynting-Robertson Effect and Analytical Solutions
CERN Document Server
Klacka, J
2000-01-01
Solutions of the two-body problem with the simultaneous action of the solar electromagnetic radiation in the form of the Poynting-Robertson effect are discussed. Special attention is devoted to pseudo-circular orbits and terminal values of osculating elements. The obtained results complete those of Klacka and Kaufmannova (1992) and Breiter and Jackson (1998). Terminal values of osculating elements presented in Breiter and Jackson (1998) are of no physical sense due to the fact that relativistic equation of motion containing only first order of $\\vec{v}/c$ was used in the paper.
17. Analytic solutions for degenerate Raman-coupled model
Institute of Scientific and Technical Information of China (English)
Zhang Zhi-Ming; Yu Ya-Fei
2008-01-01
The Raman-coupled interaction between an atom and a single mode of a cavity field is studied. For the cases in which a light field is initially in a coherent state and in a thermal state separately, we have derived the analytic expressions for the time evolutions of atomic population difference W, modulus B of the Bloch vector, and entropy E. We find that the time evolutions of these quantities are periodic with a period of e. The maxima of W and B appear at the scaled interaction time points (τ) = κπ(κ =0, 1, 2,...). At these time points, E = 0, which shows that the atom and the field are not entangled. Between these time points, E ≠ 0, which means that the atom and the field are entangled. When the field is initially in a coherent state, near the maxima, the envelope of W is a Gaussian function with a variance of 1/(4(-n)) ((-n) is the mean number of photons). Under the envelope, W oscillates at a frequency of (-n)/e.When the field is initially in a thermal state, near the maxima, W is a Lorentz function with a width of 1/(-n).
18. An Analytical Solution for Cylindrical Concrete Tank on Deformable Soil
Directory of Open Access Journals (Sweden)
Shirish Vichare
2010-07-01
Full Text Available Cylindrical concrete tanks are commonly used in wastewater treatment plants. These are usually clarifier tanks. Design codes of practice provide methods to calculate design forces in the wall and raft of such tanks. These methods neglect self-weight of tank material and assume extreme, namely ‘fixed’ and ‘hinged’ conditions for the wall bottom. However, when founded on deformable soil, the actual condition at the wall bottom is neither fixed nor hinged. Further, the self-weight of the tank wall does affect the design forces. Thus, it is required to offer better insight of the combined effect of deformable soil and bottom raft stiffness on the design forces induced in such cylindrical concrete tanks. A systematic analytical method based on fundamental equations of shells is presented in this paper. Important observations on variation of design forces across the wall and the raft with different soil conditions are given. Set of commonly used tanks, are analysed using equations developed in the paper and are appended at the end.
19. General Scalar-Tensor cosmology: Analytical solutions via Noether symmetry
CERN Document Server
Masaeli, Erfan; Sepangi, Hamid Reza
2016-01-01
We analyze the cosmology of a general Scalar-Tensor theory which encompasses generalized Brans-Dicke theory, Gauss-Bonnet gravity, non-minimal derivative gravity, generalized Galileon gravity and also the general k-essence type models. Instead of taking into account phenomenological considerations we adopt a Noether symmetry approach, as a physical criterion, to single out the form of undetermined functions in the action. These specified functions symmetrize equations of motion in the simplest possible form which result in exact solutions. Demanding de Sitter, power-law and bouncing universe solutions in the absence and presence of matter density leads to exploring new as well as well-investigated models. We show that there are models for which dynamics of the system allow transition from a decelerating phase (matter dominated era) to an accelerating phase (dark energy epoch) and could also lead to general Brans-Dicke with string correction without a self-interaction potential. Furthermore, we classify the mo...
20. Analytic crack solutions for tilt fields around hydraulic fractures
Energy Technology Data Exchange (ETDEWEB)
Warpinski, N.R.
2000-01-05
The recent development of downhole tiltmeter arrays for monitoring hydraulic fractures has provided new information on fracture growth and geometry. These downhole arrays offer the significant advantages of being close to the fracture (large signal) and being unaffected by the free surface. As with surface tiltmeter data, analysis of these measurements requires the inversion of a crack or dislocation model. To supplement the dislocation models of Davis [1983], Okada [1992] and others, this work has extended several elastic crack solutions to provide tilt calculations. The solutions include constant-pressure 2D, penny-shaped, and 3D-elliptic cracks and a 2D-variable-pressure crack. Equations are developed for an arbitrary inclined fracture in an infinite elastic space. Effects of fracture height, fracture length, fracture dip, fracture azimuth, fracture width and monitoring distance on the tilt distribution are given, as well as comparisons with the dislocation model. The results show that the tilt measurements are very sensitive to the fracture dimensions, but also that it is difficult to separate the competing effects of the various parameters.
1. Continuum theory of critical phenomena in polymer solutions: Formalism and mean field approximation
Science.gov (United States)
Goldstein, Raymond E.; Cherayil, Binny J.
1989-06-01
A theoretical description of the critical point of a polymer solution is formulated directly from the Edwards continuum model of polymers with two- and three-body excluded-volume interactions. A Hubbard-Stratonovich transformation analogous to that used in recent work on the liquid-vapor critical point of simple fluids is used to recast the grand partition function of the polymer solution as a functional integral over continuous fields. The resulting Landau-Ginzburg-Wilson (LGW) Hamiltonian is of the form of a generalized nonsymmetric n=1 component vector model, with operators directly related to certain connected correlation functions of a reference system. The latter is taken to be an ensemble of Gaussian chains with three-body excluded-volume repulsions, and the operators are computed in three dimensions by means of a perturbation theory that is rapidly convergent for long chains. A mean field theory of the functional integral yields a description of the critical point in which the power-law variations of the critical polymer volume fraction φc, critical temperature Tc, and critical amplitudes on polymerization index N are essentially identical to those found in the Flory-Huggins theory. In particular, we find φc ˜N-1/2, Tθ-Tc˜N-1/2 with (Tθ the theta temperature), and that the composition difference between coexisting phases varies with reduced temperature t as N-1/4t1/2. The mean field theory of the interfacial tension σ between coexisting phases near the critical point, developed by considering the LGW Hamiltonian for a weakly inhomogeneous solution, yields σ˜N-1/4t3/2, with the correlation length diverging as ξ˜N1/4t-1/2 within the same approximation, consistent with the mean field limit of de Gennes' scaling form. Generalizations to polydisperse systems are discussed.
2. A new method for deriving analytical solutions of partial differential equations-Algebraically explicit analytical solutions of two-buoyancy natural convection in porous media
Institute of Scientific and Technical Information of China (English)
2008-01-01
Analytical solutions of governing equations of various phenomena have their irre-placeable theoretical meanings. In addition, they can also be the benchmark solu-tions to verify the outcomes and codes of numerical solutions, and even to develop various numerical methods such as their differencing schemes and grid generation skills as well. A hybrid method of separating variables for simultaneous partial differential equation sets is presented. It is proposed that different methods of separating variables for different independent variables in the simultaneous equa-tion set may be used to improve the solution derivation procedure, for example, using the ordinary separating method for some variables and using extraordinary methods of separating variables, such as the separating variables with addition promoted by the first author, for some other variables. In order to prove the ability of the above-mentioned hybrid method, a lot of analytical exact solutions of two-buoyancy convection in porous media are successfully derived with such a method. The physical features of these solutions are given.
3. Cutting solid figures by plane - analytical solution and spreadsheet implementation
Science.gov (United States)
Benacka, Jan
2012-07-01
In some secondary mathematics curricula, there is a topic called Stereometry that deals with investigating the position and finding the intersection, angle, and distance of lines and planes defined within a prism or pyramid. Coordinate system is not used. The metric tasks are solved using Pythagoras' theorem, trigonometric functions, and sine and cosine rules. The basic problem is to find the section of the figure by a plane that is defined by three points related to the figure. In this article, a formula is derived that gives the positions of the intersection points of such a plane and the figure edges, that is, the vertices of the section polygon. Spreadsheet implementations of the formula for cuboid and right rectangular pyramids are presented. The user can check his/her graphical solution, or proceed if he/she is not able to complete the section.
4. Nonlinear Helicons ---an analytical solution elucidating multi-scale structure
CERN Document Server
Abdelhamid, Hamdi M
2016-01-01
The helicon waves exhibit varying characters depending on plasma parameters, geometry, and wave numbers. Here we elucidate an intrinsic multi-scale property embodied by the combination of dispersive effect and nonlinearity. The extended magnetohydrodynamics model (exMHD) is capable of describing wide range of parameter space. By using the underlying Hamiltonian structure of exMHD, we construct an exact nonlinear solution which turns out to be a combination of two distinct modes, the helicon and Trivelpiece-Gould (TG) waves. In the regime of relatively low frequency or high density, however, the combination is made of the TG mode and an ion cyclotron wave (slow wave). The energy partition between these modes is determined by the helicities carried by the wave fields.
5. Manufactured analytical solutions for isothermal full-Stokes ice sheet models
Directory of Open Access Journals (Sweden)
A. Sargent
2010-08-01
Full Text Available We present the detailed construction of a manufactured analytical solution to time-dependent and steady-state isothermal full-Stokes ice sheet problems. The solutions are constructed for two-dimensional flowline and three-dimensional full-Stokes ice sheet models with variable viscosity. The construction is done by choosing for the specified ice surface and bed a velocity distribution that satisfies both mass conservation and the kinematic boundary conditions. Then a compensatory stress term in the conservation of momentum equations and their boundary conditions is calculated to make the chosen velocity distributions as well as the chosen pressure field into exact solutions. By substituting different ice surface and bed geometry formulas into the derived solution formulas, analytical solutions for different geometries can be constructed.
The boundary conditions can be specified as essential Dirichlet conditions or as periodic boundary conditions. By changing a parameter value, the analytical solutions allow investigation of algorithms for a different range of aspect ratios as well as for different, frozen or sliding, basal conditions. The analytical solutions can also be used to estimate the numerical error of the method in the case when the effects of the boundary conditions are eliminated, that is, when the exact solution values are specified as inflow and outflow boundary conditions.
6. Some analytical properties of solutions of differential equations of noninteger order
Directory of Open Access Journals (Sweden)
S. M. Momani
2004-01-01
Full Text Available The analytical properties of solutions of the nonlinear differential equations x(α(t=f(t,x, α∈ℝ, 0<α≤1 of noninteger order have been investigated. We obtained two results concerning the frame curves of solutions. Moreover, we proved a result on differential inequality with fractional derivatives.
7. Analytical solutions for spin response functions in model storage rings with Siberian Snakes
Energy Technology Data Exchange (ETDEWEB)
Mane, S.R. [Convergent Computing Inc., P.O. Box 561, Shoreham, NY 11786 (United States)], E-mail: [email protected]
2009-03-01
I present analytical solutions for the spin response functions for radial field rf dipole spin flippers in models of storage rings with one Siberian Snake or two diametrically opposed orthogonal Siberian Snakes. The solutions can serve as benchmarks tests for computer programs. The spin response functions can be used to calculate the resonance strengths for radial field rf dipole spin flippers in storage rings.
8. Plasma flow structures as analytical solution of a magneto-hydro-dynamic model with pressure
Science.gov (United States)
Paccagnella, R.
2012-03-01
In this work starting from a set of magnetohydrodynamic (MHD) equations that describe the dynamical evolution for the pressure driven resistive/interchange modes in a magnetic confinement system, global solutions for the plasma flow relevant for toroidal pinches like tokamaks and reversed field pinches (RFPs) are derived. Analytical solutions for the flow stream function associated with the dominant modes are presented.
9. ANALYTICAL SOLUTION FOR WAVES IN PLANETS WITH ATMOSPHERIC SUPERROTATION. I. ACOUSTIC AND INERTIA-GRAVITY WAVES
Energy Technology Data Exchange (ETDEWEB)
Peralta, J.; López-Valverde, M. A. [Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, 18008 Granada (Spain); Imamura, T. [Institute of Space and Astronautical Science-Japan Aerospace Exploration Agency 3-1-1, Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210 (Japan); Read, P. L. [Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford (United Kingdom); Luz, D. [Centro de Astronomia e Astrofísica da Universidade de Lisboa (CAAUL), Observatório Astronómico de Lisboa, Tapada da Ajuda, 1349-018 Lisboa (Portugal); Piccialli, A., E-mail: [email protected] [LATMOS, UVSQ, 11 bd dAlembert, 78280 Guyancourt (France)
2014-07-01
This paper is the first of a two-part study devoted to developing tools for a systematic classification of the wide variety of atmospheric waves expected on slowly rotating planets with atmospheric superrotation. Starting with the primitive equations for a cyclostrophic regime, we have deduced the analytical solution for the possible waves, simultaneously including the effect of the metric terms for the centrifugal force and the meridional shear of the background wind. In those cases when the conditions for the method of the multiple scales in height are met, these wave solutions are also valid when vertical shear of the background wind is present. A total of six types of waves have been found and their properties were characterized in terms of the corresponding dispersion relations and wave structures. In this first part, only waves that are direct solutions of the generic dispersion relation are studied—acoustic and inertia-gravity waves. Concerning inertia-gravity waves, we found that in the cases of short horizontal wavelengths, null background wind, or propagation in the equatorial region, only pure gravity waves are possible, while for the limit of large horizontal wavelengths and/or null static stability, the waves are inertial. The correspondence between classical atmospheric approximations and wave filtering has been examined too, and we carried out a classification of the mesoscale waves found in the clouds of Venus at different vertical levels of its atmosphere. Finally, the classification of waves in exoplanets is discussed and we provide a list of possible candidates with cyclostrophic regimes.
10. Analytical Structuring of Periodic and Regular Cascading Solutions in Self-Pulsing Lasers
Directory of Open Access Journals (Sweden)
Belkacem Meziane
2008-01-01
Full Text Available A newly proposed strong harmonic-expansion method is applied to the laser-Lorenz equations to analytically construct a few typical solutions, including the first few expansions of the well-known period-doubling cascade that characterizes the system in its self-pulsing regime of operation. These solutions are shown to evolve in accordance with the driving frequency of the permanent solution that we recently reported to illustrate the system. The procedure amounts to analytically construct the signal Fourier transform by applying an iterative algorithm that reconstitutes the first few terms of its development.
11. Nonlinear analytical solution for one-dimensional consolidation of soft soil under cyclic loading
Institute of Scientific and Technical Information of China (English)
XIE Kang-he; QI Tian; DONG Ya-qin
2006-01-01
This paper presents an analytical solution for one-dimensional consolidation of soft soil under some common types of cyclic loading such as trapezoidal cyclic loading, based on the assumptions proposed by Davis and Raymond (1965) that the decrease in permeability is proportional to the decrease in compressibility during the consolidation process of the soil and that the distribution of initial effective stress is constant with depth. It is verified by the existing analytical solutions in special cases. Using the solution obtained, some diagrams are prepared and the relevant consolidation behavior is investigated.
12. Explicit analytical solutions of the anisotropic Brinkman model for the natural convection in porous media
Institute of Scientific and Technical Information of China (English)
CAI; Ruixian(蔡睿贤); ZHANG; Na(张娜)
2002-01-01
Some algebraically explicit analytical solutions are derived for the anisotropic Brinkman model an improved Darcy model describing the natural convection in porous media. Besides their important theoretical meaning (for example, to analyze the non-Darcy and anisotropic effects on the convection), such analytical solutions can be the benchmark solutions to promoting the develop ment of computational heat and mass transfer. For instance, we can use them to check the accuracy,convergence and effectiveness of various numerical computational methods and to improve numerical calculation skills such as differential schemes and grid generation ways.
13. Matching of analytical and numerical solutions for neutron stars of arbitrary rotation
Energy Technology Data Exchange (ETDEWEB)
Pappas, George, E-mail: [email protected] [Section of Astrophysics, Astronomy, and Mechanics, Department of Physics, University of Athens, Panepistimiopolis Zografos GR15783, Athens (Greece)
2009-10-01
We demonstrate the results of an attempt to match the two-soliton analytical solution with the numerically produced solutions of the Einstein field equations, that describe the spacetime exterior of rotating neutron stars, for arbitrary rotation. The matching procedure is performed by equating the first four multipole moments of the analytical solution to the multipole moments of the numerical one. We then argue that in order to check the effectiveness of the matching of the analytical with the numerical solution we should compare the metric components, the radius of the innermost stable circular orbit (R{sub ISCO}), the rotation frequency and the epicyclic frequencies {Omega}{sub {rho}}, {Omega}{sub z}. Finally we present some results of the comparison.
14. On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem
Directory of Open Access Journals (Sweden)
Soheil Salahshour
2015-02-01
Full Text Available In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville or a solution with increasing length of their support (Hukuhara difference. Then, in order to solve the FFDE analytically, we introduce the fuzzy Laplace transform of the Caputo H-derivative. To the best of our knowledge, there is limited research devoted to the analytical methods to solve the FFDE under the fuzzy Caputo fractional differentiability. An analytical solution is presented to confirm the capability of the proposed method.
15. Approximate axially symmetric solution of the Weyl-Dirac theory of gravitation and the spiral galactic rotation problem
CERN Document Server
Babourova, O V; Kudlaev, P E
2016-01-01
On the basis of the Poincare-Weyl gauge theory of gravitation, a new conformal Weyl-Dirac theory of gravitation is proposed, which is a gravitational theory in Cartan-Weyl spacetime with the Dirac scalar field representing the dark matter model. A static approximate axially symmetric solution of the field equations in vacuum is obtained. On the base of this solution in the Newtonian approximation one considers the problem of rotation velocities in spiral components of galaxies.
16. Algebraically explicit analytical solutions for the unsteady non-Newtonian swirling flow in an annular pipe
Institute of Scientific and Technical Information of China (English)
CAI; Ruixian; GOU; Chenhua
2006-01-01
This paper presents two algebraically explicit analytical solutions for the incompressible unsteady rotational flow of Oldroyd-B type in an annular pipe. The first solution is derived with the common method of separation of variables. The second one is deduced with the method of separation of variables with addition developed in recent years. The first analytical solution is of clear physical meaning and both of them are fairly simple and valuable for the newly developing computational fluid dynamics. They can be used as the benchmark solutions to verify the applicability of the existing numerical computational methods and to inspire new differencing schemes, grid generation ways, etc. Moreover, a steady solution for the generalized second grade rheologic fluid flow is also presented. The correctness of these solutions can be easily proven by substituting them into the original governing equation.
17. Analytic solution of Riccati equations occurring in open-loop Nash multiplayer differential games
Directory of Open Access Journals (Sweden)
L. Jódar
1992-01-01
Full Text Available In this paper we present explicit analytic solutions of coupled Riccati matrix differential systems appearing in open-loop Nash games. Two different cases are considered. Firstly, by means of appropriate algebraic transformations the problem is decoupled so that an explicit solution of the problem is available. The second is based on the existence of a solution of a rectangular Riccati type algebraic matrix equation associated with the problem.
18. Analytical solitary wave solutions of the nonlinear Kronig-Penney model in photonic structures.
Science.gov (United States)
Kominis, Y
2006-06-01
A phase space method is employed for the construction of analytical solitary wave solutions of the nonlinear Kronig-Penney model in a photonic structure. This class of solutions is obtained under quite generic conditions, while the method is applicable to a large variety of systems. The location of the solutions on the spectral band gap structure as well as on the low dimensional space of system's conserved quantities is studied, and robust solitary wave propagation is shown.
19. Analytical solution of two-phase spherical Stefan problem by heat polynomials and integral error functions
Science.gov (United States)
Kharin, Stanislav N.; Sarsengeldin, Merey M.; Nouri, Hassan
2016-08-01
On the base of the Holm model, we represent two phase spherical Stefan problem and its analytical solution, which can serve as a mathematical model for diverse thermo-physical phenomena in electrical contacts. Suggested solution is obtained from integral error function and its properties which are represented in the form of series whose coefficients have to be determined. Convergence of solution series is proved.
20. Analytical Solution for the SU(2)Hedgehog Skyrmion and Static Properties of Nucleons
Institute of Scientific and Technical Information of China (English)
JIA Duo-Jie; WANG Xiao-Wei; LIU Feng
2010-01-01
@@ An analytical solution for symmetric Skyrmion is proposed for the SU(2)Skyrme model,which takes the form of the hybrid form of a kink-like solution,given by the instanton method.The static properties of nucleons is then computed within the framework of collective quantization of the Skyrme model,in a good agreement with that given by the exact numeric solution.The comparisons with the previous results as well as the experimental values are also presented.
1. Analytical solutions to dissolved contaminant plume evolution with source depletion during carbon dioxide storage.
Science.gov (United States)
Yang, Yong; Liu, Yongzhong; Yu, Bo; Ding, Tian
2016-06-01
Volatile contaminants may migrate with carbon dioxide (CO2) injection or leakage in subsurface formations, which leads to the risk of the CO2 storage and the ecological environment. This study aims to develop an analytical model that could predict the contaminant migration process induced by CO2 storage. The analytical model with two moving boundaries is obtained through the simplification of the fully coupled model for the CO2-aqueous phase -stagnant phase displacement system. The analytical solutions are confirmed and assessed through the comparison with the numerical simulations of the fully coupled model. Then, some key variables in the analytical solutions, including the critical time, the locations of the dual moving boundaries and the advance velocity, are discussed to present the characteristics of contaminant migration in the multi-phase displacement system. The results show that these key variables are determined by four dimensionless numbers, Pe, RD, Sh and RF, which represent the effects of the convection, the dispersion, the interphase mass transfer and the retention factor of contaminant, respectively. The proposed analytical solutions could be used for tracking the migration of the injected CO2 and the contaminants in subsurface formations, and also provide an analytical tool for other solute transport in multi-phase displacement system.
2. Analytical solution and computer program (FAST) to estimate fluid fluxes from subsurface temperature profiles
Science.gov (United States)
Kurylyk, Barret L.; Irvine, Dylan J.
2016-02-01
This study details the derivation and application of a new analytical solution to the one-dimensional, transient conduction-advection equation that is applied to trace vertical subsurface fluid fluxes. The solution employs a flexible initial condition that allows for nonlinear temperature-depth profiles, providing a key improvement over most previous solutions. The boundary condition is composed of any number of superimposed step changes in surface temperature, and thus it accommodates intermittent warming and cooling periods due to long-term changes in climate or land cover. The solution is verified using an established numerical model of coupled groundwater flow and heat transport. A new computer program FAST (Flexible Analytical Solution using Temperature) is also presented to facilitate the inversion of this analytical solution to estimate vertical groundwater flow. The program requires surface temperature history (which can be estimated from historic climate data), subsurface thermal properties, a present-day temperature-depth profile, and reasonable initial conditions. FAST is written in the Python computing language and can be run using a free graphical user interface. Herein, we demonstrate the utility of the analytical solution and FAST using measured subsurface temperature and climate data from the Sendia Plain, Japan. Results from these illustrative examples highlight the influence of the chosen initial and boundary conditions on estimated vertical flow rates.
3. An Approximate Solution and Master Curves for Buckling of Symmetrically Laminated Composite Cylinders
Science.gov (United States)
Nemeth, Michael P.
2013-01-01
Nondimensional linear-bifurcation buckling equations for balanced, symmetrically laminated cylinders with negligible shell-wall anisotropies and subjected to uniform axial compression loads are presented. These equations are solved exactly for the practical case of simply supported ends. Nondimensional quantities are used to characterize the buckling behavior that consist of a stiffness-weighted length-to-radius parameter, a stiffness-weighted shell-thinness parameter, a shell-wall nonhomogeneity parameter, two orthotropy parameters, and a nondimensional buckling load. Ranges for the nondimensional parameters are established that encompass a wide range of laminated-wall constructions and numerous generic plots of nondimensional buckling load versus a stiffness-weighted length-to-radius ratio are presented for various combinations of the other parameters. These plots are expected to include many practical cases of interest to designers. Additionally, these plots show how the parameter values affect the distribution and size of the festoons forming each response curve and how they affect the attenuation of each response curve to the corresponding solution for an infinitely long cylinder. To aid in preliminary design studies, approximate formulas for the nondimensional buckling load are derived, and validated against the corresponding exact solution, that give the attenuated buckling response of an infinitely long cylinder in terms of the nondimensional parameters presented herein. A relatively small number of "master curves" are identified that give a nondimensional measure of the buckling load of an infinitely long cylinder as a function of the orthotropy and wall inhomogeneity parameters. These curves reduce greatly the complexity of the design-variable space as compared to representations that use dimensional quantities as design variables. As a result of their inherent simplicity, these master curves are anticipated to be useful in the ongoing development of
4. Analytical solution of electrohydrodynamic flow and transport in rectangular channels: inclusion of double layer effects
KAUST Repository
Joekar-Niasar, V.
2013-01-25
Upscaling electroosmosis in porous media is a challenge due to the complexity and scale-dependent nonlinearities of this coupled phenomenon. "Pore-network modeling" for upscaling electroosmosis from pore scale to Darcy scale can be considered as a promising approach. However, this method requires analytical solutions for flow and transport at pore scale. This study concentrates on the development of analytical solutions of flow and transport in a single rectangular channel under combined effects of electrohydrodynamic forces. These relations will be used in future works for pore-network modeling. The analytical solutions are valid for all regimes of overlapping electrical double layers and have the potential to be extended to nonlinear Boltzmann distribution. The innovative aspects of this study are (a) contribution of overlapping of electrical double layers to the Stokes flow as well as Nernst-Planck transport has been carefully included in the analytical solutions. (b) All important transport mechanisms including advection, diffusion, and electromigration have been included in the analytical solutions. (c) Fully algebraic relations developed in this study can be easily employed to upscale electroosmosis to Darcy scale using pore-network modeling. © 2013 Springer Science+Business Media Dordrecht.
5. Approximate solutions to a weighted mixed-sensitivity H-infinity-control design for irrational transfer matrices
NARCIS (Netherlands)
Curtain, RF; Weiss, M; Zhou, Y
1996-01-01
Approximate solutions to a weighted mixed-sensitivity H-infinity-control problem for an irrational transfer matrix are obtained by solving the same problem for a reduced-order (rational) transfer matrix. Upper and lower bounds are given in terms of the solution to the reduced-order problem and the a
6. Transient radiative transfer in participating media with pulse-laser irradiation-an approximate Galerkin solution
Energy Technology Data Exchange (ETDEWEB)
Okutucu, Tuba [Mechanical and Industrial Engineering Department, Northeastern University, Boston, MA 02115 (United States); Yener, Yaman [Mechanical and Industrial Engineering Department, Northeastern University, Boston, MA 02115 (United States)]. E-mail: [email protected]; Busnaina, Ahmed A. [Mechanical and Industrial Engineering Department, Northeastern University, Boston, MA 02115 (United States)
2007-01-15
An assessment is made of the Galerkin technique as an effective method of solution for transient radiative transfer problems in participating media. A one-dimensional absorbing and isotropically scattering plane-parallel gray medium irradiated with a short-pulse laser on one of its boundaries is considered for the application of the method. The medium is non-emitting and the boundaries are non-reflecting and non-refracting. In the integral formulation of the problem for the source function, the time-wise variation of the radiation intensity at any point and in any direction in the medium is assumed to be the same as the time-wise variation of the average intensity at the same point as an approximation for the application of the method. The transient transmittance and reflectance of the medium are evaluated for various values of the optical thickness, scattering albedo and pulse duration. The results are in agreement with those available in the literature. It is demonstrated that the method is relatively simple to implement and yields accurate results.
7. Transient radiative transfer in participating media with pulse-laser irradiation—an approximate Galerkin solution
Science.gov (United States)
Okutucu, Tuba; Yener, Yaman; Busnaina, Ahmed A.
2007-01-01
An assessment is made of the Galerkin technique as an effective method of solution for transient radiative transfer problems in participating media. A one-dimensional absorbing and isotropically scattering plane-parallel gray medium irradiated with a short-pulse laser on one of its boundaries is considered for the application of the method. The medium is non-emitting and the boundaries are non-reflecting and non-refracting. In the integral formulation of the problem for the source function, the time-wise variation of the radiation intensity at any point and in any direction in the medium is assumed to be the same as the time-wise variation of the average intensity at the same point as an approximation for the application of the method. The transient transmittance and reflectance of the medium are evaluated for various values of the optical thickness, scattering albedo and pulse duration. The results are in agreement with those available in the literature. It is demonstrated that the method is relatively simple to implement and yields accurate results.
8. Algebraically explicit analytical solutions of two-buoyancy natural convection in porous media
Institute of Scientific and Technical Information of China (English)
CAI Ruixian; ZHANG Na; LIU Weiwei
2003-01-01
Analytical solutions of governing equations of various physical phenomena have their own irreplaceable theoretical meaning. In addition, they can also be the benchmark solutions to verify the outcomes and codes of numerical solution, and to develop various numerical methods such as their differencing schemes and grid generation skills as well. In order to promote the development of the discipline of natural convection, three simple algebraically explicit analytical solution sets are derived for a non-linear simultaneous partial differential equation set with five dependent unknown variables, which represents the natural convection in porous media with both temperature and concentration gradients. An extraordinary method separating variables with addition is applied in this paper to deduce solutions.
9. Manufactured analytical solutions for isothermal full-Stokes ice sheet models
Directory of Open Access Journals (Sweden)
A. Sargent
2010-04-01
Full Text Available We present the detailed construction of an exact solution to time-dependent and steady-state isothermal full-Stokes ice sheet problems. The solutions are constructed for two-dimensional flowline and three-dimensional full-Stokes ice sheet models with variable viscosity. The construction is done by choosing for the specified ice surface and bed a velocity distribution that satisfies both mass conservation and the kinematic boundary conditions. Then a compensatory stress term in the conservation of momentum equations and their boundary conditions is calculated to make the chosen velocity distributions as well as the chosen pressure field into exact solutions. By substituting different ice surface and bed geometry formulas into the derived solution formulas, analytical solutions for different geometries can be constructed.
The boundary conditions can be specified as essential Dirichlet conditions or as periodic boundary conditions. By changing a parameter value, the analytical solutions allow investigation of algorithms for a different range of aspect ratios as well as for different, frozen or sliding, basal conditions. The analytical solutions can also be used to estimate the numerical error of the method in the case when the effects of the boundary conditions are eliminated, that is, when the exact solution values are specified as inflow and outflow boundary conditions.
10. Analytical solutions of ac electrokinetics in interdigitated electrode arrays: Electric field, dielectrophoretic and traveling-wave dielectrophoretic forces
Science.gov (United States)
Sun, Tao; Morgan, Hywel; Green, Nicolas G.
2007-10-01
Analysis of the movement of particles in a nonuniform field requires accurate knowledge of the electric field distribution in the system. This paper describes a method for analytically solving the electric field distribution above interdigitated electrode arrays used for dielectrophoresis (DEP) and traveling wave dielectrophoresis (twDEP), using the Schwarz-Christoffel mapping method. The electric field solutions are used to calculate the dielectrophoretic force in both cases, and the traveling wave dielectrophoretic force and the electrorotational torque for the twDEP case. This method requires no approximations and can take into account the Neumann boundary condition used to represent an insulating lid and lower substrate. The analytical results of the electric field distributions are validated for different geometries by comparison with numerical simulations using the finite element method.
11. Material dependence of Casimir interaction between a sphere and a plate: First analytic correction beyond proximity force approximation
CERN Document Server
Teo, L P
2013-01-01
We derive analytically the asymptotic behavior of the Casimir interaction between a sphere and a plate when the distance between them, $d$, is much smaller than the radius of the sphere, $R$. The leading order and next-to-leading order terms are derived from the exact formula for the Casimir interaction energy. They are found to depend nontrivially on the dielectric functions of the objects. As expected, the leading order term coincides with that derived using the proximity force approximation. The result on the next-to-leading order term complements that found by Bimonte, Emig and Kardar [Appl. Phys. Lett. \\textbf{100}, 074110 (2012)] using derivative expansion. Numerical results are presented when the dielectric functions are given by the plasma model or the Drude model, with the plasma frequency (for plasma and Drude models) and relaxation frequency (for Drude model) given respectively by 9eV and 0.035eV, the conventional values used for gold metal. It is found that if plasma model is used instead of Drude...
12. Analytical solution of the Gross-Neveu model at finite density
CERN Document Server
Thies, M
2003-01-01
Recent numerical calculations have shown that the ground state of the Gross-Neveu model at finite density is a crystal. Guided by these results, we can now present the analytical solution to this problem in terms of elliptic functions. The scalar potential is the superpotential of the non-relativistic Lame Hamiltonian. This model can also serve as analytically solvable toy model for a relativistic superconductor in the Larkin-Ovchinnikov-Fulde-Ferrell phase.
13. On Direct Transformation Approach to Asymptotical Analytical Solutions of Perturbed Partial Differential Equation
Institute of Scientific and Technical Information of China (English)
LIU Hong-Zhun; PAN Zu-Liang; LI Peng
2006-01-01
In this article, we will derive an equality, where the Taylor series expansion around ε = 0for any asymptotical analytical solution of the perturbed partial differential equation (PDE) with perturbing parameter ε must be admitted.By making use of the equality, we may obtain a transformation, which directly map the analytical solutions of a given unperturbed PDE to the asymptotical analytical solutions of the corresponding perturbed one. The notion of Lie-B(a)cklund symmetries is introduced in order to obtain more transformations. Hence, we can directly create more transformations in virtue of known Lie-B(a)cklund symmetries and recursion operators of corresponding unperturbed equation. The perturbed Burgers equation and the perturbed Korteweg-de Vries (KdV) equation are used as examples.
14. Analytic Solutions of a Polynomial-Like Iterative Functional Equation near Resonance
Institute of Scientific and Technical Information of China (English)
LIU Ling Xia; SI Jian Guo
2009-01-01
In this paper existence of local analytic solutions of a polynomial-like iterative functional equation is studied. As well as in previous work, we reduce this problem with the Schroder transformation to finding analytic solutions of a functional equation without iteration of the unknown function f. For technical reasons, in previous work the constant α given in the Schr(o)der transformation, i.e., the eigenvalue of the linearized f at its fixed point O, is required to fulfill that α is off the unit circle S1 or lies on the circle with the Diophantine condition. In this paper,we obtain results of analytic solutions in the case of α at resonance, i.e., at a root of the unity and the case of α near resonance under the Brjuno condition.
15. Analytical solutions of steady vibration of free rectangular plate on semi-infinite elastic foundation
Institute of Scientific and Technical Information of China (English)
WANG Chun-ling; HUANG Yi; JIA Ji-hong
2007-01-01
The method of double Fourier transform was employed in the analysis of the semi-infinite elastic foundation with vertical load. And an integral representations for the displacements of the semi-infinite elastic foundation was presented. The analytical solution of steady vibration of an elastic rectangle plate with four free edges on the semi-infinite elastic foundation was also given by combining the analytical solution of the elastic rectangle plate with the integral representation for displacements of the semiinfinite elastic foundation. Some computational results and the analysis on the influence of parameters were presented.
16. Generalization of the analytical solution of neutron point kinetics equations with time-dependent external source
Science.gov (United States)
Seidi, M.; Behnia, S.; Khodabakhsh, R.
2014-09-01
Point reactor kinetics equations with one group of delayed neutrons in the presence of the time-dependent external neutron source are solved analytically during the start-up of a nuclear reactor. Our model incorporates the random nature of the source and linear reactivity variation. We establish a general relationship between the expectation values of source intensity and the expectation values of neutron density of the sub-critical reactor by ignoring the term of the second derivative for neutron density in neutron point kinetics equations. The results of the analytical solution are in good agreement with the results obtained with numerical solution.
17. ANALYTICAL SOLUTION OF BENDING-COMPRESSION COLUMN USING DIFFERENT TENSION-COMPRESSION MODULUS
Institute of Scientific and Technical Information of China (English)
姚文娟; 叶志明
2004-01-01
Based on elastic theory of different tension-compression modulus, the analytical solution was deduced for bending-compression column subject to combined loadings by the flowing coordinate system and phased integration method. The formulations for the neutral axis, stress, strain and displacement were developed, the finite element program was compiled for calculation, and the comparison between the result of finite element and analytical solution were given too. Finally, compare and analyze the result of different modulus and the same modulus, obtain the difference of two theories in result, and propose the reasonable suggestion for the calculation of this structure.
18. An Analytic and Optimal Inverse Kinematic Solution for a 7-DOF Space Manipulator
Institute of Scientific and Technical Information of China (English)
WANG Yingshi; SUN Lei; YAN Wenbin; LIU Jingtai
2014-01-01
An analytic inverse kinematic solution is presented for a 7-DOF (degree of freedom) redundant space manipu-lator. The proposed method can obtain all the feasible solutions in the global joint space, which are denoted by a joint angle parameter. Meanwhile, both the singularity problem and the joint limits are considered in detail. Besides, an optimization approach is provided to get one near optimal inverse kinematic solution from all the feasible solutions. The proposed method can reduce effectively the computational complexity, so that it can be applied online. Finally, the method’s validity is shown by kinematic simulations.
19. Algebraically explicit analytical solutions of unsteady conduction with variable thermal properties in cylindrical coordinate
Institute of Scientific and Technical Information of China (English)
CAI Ruixian; ZHANG Na
2004-01-01
The analytical solutions of unsteady heat conduction with variable thermal properties(thermal conductivity,density and specific heat are functions of temperature or coordinates)are meaningful in theory.In addition,they are very useful to the computational heat conduction to check the numerical solutions and to develop numerical schemes,grid generation methods and so forth.Such solutions in rectangular coordinates have been derived by the authors.Some other solutions for 1-D and 2-D axisymmetrical heat conduction in cylin drical coordinates are given in this paper to promote the heat conduction theory and to develop the relative computational heat conduction.
20. Analytical self-dual solutions in a nonstandard Yang-Mills-Higgs scenario
CERN Document Server
Casana, R; da Hora, E; Santos, C dos
2013-01-01
We have found analytical self-dual solutions within the generalized Yang-Mills-Higgs model introduced in Phys. Rev. D 86, 085034 (2012). Such solutions are magnetic monopoles satisfying Bogomol'nyi-Prasad-Sommerfield (BPS) equations and usual finite energy boundary conditions. Moreover, the new solutions are classified in two different types according to their capability of recovering (or not) the usual 't Hooft--Polyakov monopole. Finally, we compare the profiles of the solutions we found with the standard ones, from which we comment about the main features exhibited by the new configurations. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9214339852333069, "perplexity": 1186.4188590902631}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187828134.97/warc/CC-MAIN-20171024033919-20171024053919-00697.warc.gz"} |
https://www.bionicturtle.com/forum/tags/pseudo-random-number-generation/ | What's new
# pseudo-random-number-generation
1. ### P1.T2.21.6 Bootstrapping and antithetic/control variates
Learning objectives: Explain the use of antithetic and control variates in reducing Monte Carlo sampling error. Describe the bootstrapping method and its advantage over Monte Carlo simulation. Describe pseudo-random number generation. Describe situations where the bootstrapping method is... | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8772563338279724, "perplexity": 4999.010532535507}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358702.43/warc/CC-MAIN-20211129074202-20211129104202-00605.warc.gz"} |
http://www.aimsciences.org/journal/1930-5311/2013/7/3 | # American Institute of Mathematical Sciences
ISSN:
1930-5311
eISSN:
1930-532X
All Issues
## Journal of Modern Dynamics
2013 , Volume 7 , Issue 3
Select all articles
Export/Reference:
2013, 7(3): 329-367 doi: 10.3934/jmd.2013.7.329 +[Abstract](358) +[PDF](583.5KB)
Abstract:
Let $M$ be a smooth compact connected manifold of dimension greater than two, on which there exists a free (modulo zero) smooth circle action that preserves a positive smooth volume. In this article, we construct volume-preserving diffeomorphisms on $M$ that are metrically isomorphic to ergodic translations on the torus of dimension greater than two, where one given coordinate of the translation is an arbitrary Liouville number. To obtain this result, we determine sufficient conditions on translation vectors of the torus that allow us to explicitly construct the sequence of successive conjugacies in Anosov--Katok's method, with suitable estimates of their norm.
2013, 7(3): 369-394 doi: 10.3934/jmd.2013.7.369 +[Abstract](376) +[PDF](278.0KB)
Abstract:
We show that if $M$ is a compact oriented surface of genus $0$ and $G$ is a subgroup of Symp$^\omega_\mu(M)$ that has an infinite normal solvable subgroup, then $G$ is virtually abelian. In particular the centralizer of an infinite order $f \in$ Symp$^\omega_\mu(M)$ is virtually abelian. Another immediate corollary is that if $G$ is a solvable subgroup of Symp$^\omega_\mu(M)$ then $G$ is virtually abelian. We also prove a special case of the Tits Alternative for subgroups of Symp$^\omega_\mu(M)$.
2013, 7(3): 395-427 doi: 10.3934/jmd.2013.7.395 +[Abstract](462) +[PDF](304.6KB)
Abstract:
We prove that the set of bounded geodesics in Teichmüller space is a winning set for Schmidt's game. This is a notion of largeness in a metric space that can apply to measure $0$ and meager sets. We prove analogous closely related results on any Riemann surface, in any stratum of quadratic differentials, on any Teichmüller disk and for intervals exchanges with any fixed irreducible permutation.
2013, 7(3): 429-460 doi: 10.3934/jmd.2013.7.429 +[Abstract](434) +[PDF](323.2KB)
Abstract:
We prove a conjecture of G.A. Margulis on the abundance of certain exceptional orbits of partially hyperbolic flows on homogeneous spaces by utilizing a theory of modified Schmidt games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in mid-1960s.
2013, 7(3): 461-488 doi: 10.3934/jmd.2013.7.461 +[Abstract](402) +[PDF](801.6KB)
Abstract:
Let $K/\mathbf{Q}$ be a degree-$d$ extension. Inside the ring of integers $\mathscr O_K$ we define the set of $k$-free integers $\mathscr F_k$ and a natural $\mathscr O_K$-action on the space of binary $\mathscr O_K$-indexed sequences, equipped with an $\mathscr O_K$-invariant probability measure associated to $\mathscr F_k$. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the work of Cellarosi and Sinai [J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1343--1374] that considered the case $K=\mathbf{Q}$ and $k=2$.
2017 Impact Factor: 0.425 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9152188897132874, "perplexity": 343.13889020227066}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221213666.61/warc/CC-MAIN-20180818114957-20180818134957-00410.warc.gz"} |
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=HNSHCY_2012_v34n3_403 | A NEW LOWER BOUND FOR THE VOLUME PRODUCT OF A CONVEX BODY WITH CONSTANT WIDTH AND POLAR DUAL OF ITS p-CENTROID BODY
• Journal title : Honam Mathematical Journal
• Volume 34, Issue 3, 2012, pp.403-408
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2012.34.3.403
Title & Authors
A NEW LOWER BOUND FOR THE VOLUME PRODUCT OF A CONVEX BODY WITH CONSTANT WIDTH AND POLAR DUAL OF ITS p-CENTROID BODY
Chai, Y.D.; Lee, Young-Soo;
Abstract
In this paper, we prove that if K is a convex body in $\small{E^n}$ and $\small{E_i}$ and $\small{E_o}$ are inscribed ellipsoid and circumscribed ellipsoid of K respectively with $\small{{\alpha}E_i=E_o}$, then $\small{$({\alpha})^{\frac{n}{p}+1}$^n{\omega}^2_n{\geq}V(K)V({\Gamma}^{\ast}_pK){\geq}$(\frac{1}{\alpha})^{\frac{n}{p}+1}$^n{\omega}^2_n}$. Lutwak and Zhang[6] proved that if K is a convex body, $\small{{\omega}^2_n=V(K)V({\Gamma}_pK)}$ if and only if K is an ellipsoid. Our inequality provides very elementary proof for their result and this in turn gives a lower bound of the volume product for the sets of constant width.
Keywords
Convex body;constant width;polar body;volume product;p-centroid body;
Language
English
Cited by
References
1.
G. D. Chakerian and H. Groemer, Convex Bodies of Constant Width, In: Convexity and Its Applications, ed. by P. M. Gruber and J. M. Wills. Birkhauser, Basel, 1983.
2.
H.G. Eggleston, Convexity, Cambridge Univ. Press, 1958.
3.
R. J. Gardner, Geometric Tomography, Cambridge Univ. Press, Cambridge, 1995.
4.
H. Groemer, Stability Theorem for convex domains of constant width, Canad. Math. Bull. 31(1988), 328-337.
5.
R. Howard, Convex bodies of constant width and constant brightness, Adv. Math. 204 (2006), no. 1, 241-261.
6.
E. Lutwak and G. Zhang, Blaschke-Santalo inequality, J. Differential Geom., Vol.47(1997), 1-16.
7.
Z. A.Melzak, A note on sets of constant width. Proc. Amer. Math. Soc. 11 (1960) 493-497. Cambridge Univ. Press, Cambridge, 1993.
8.
I.M.Yaglom and V.G. Boltyanskii, Convex Figures, Hol, Rinehart and Winston, New York, 1961. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 6, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9779415726661682, "perplexity": 1546.5858440632524}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320049.84/warc/CC-MAIN-20170623100455-20170623120455-00338.warc.gz"} |
https://openstax.org/books/elementary-algebra/pages/1-5-visualize-fractions | Elementary Algebra
# 1.5Visualize Fractions
Elementary Algebra1.5 Visualize Fractions
### Learning Objectives
By the end of this section, you will be able to:
• Find equivalent fractions
• Simplify fractions
• Multiply fractions
• Divide fractions
• Simplify expressions written with a fraction bar
• Translate phrases to expressions with fractions
Be Prepared 1.5
A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Fractions.
### Find Equivalent Fractions
Fractions are a way to represent parts of a whole. The fraction $1313$ means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See Figure 1.11. The fraction $2323$ represents two of three equal parts. In the fraction $23,23,$ the 2 is called the numerator and the 3 is called the denominator.
Figure 1.11 The circle on the left has been divided into 3 equal parts. Each part is $1313$ of the 3 equal parts. In the circle on the right, $2323$ of the circle is shaded (2 of the 3 equal parts).
### Manipulative Mathematics
Doing the Manipulative Mathematics activity “Model Fractions” will help you develop a better understanding of fractions, their numerators and denominators.
### Fraction
A fraction is written $ab,ab,$ where $b≠0b≠0$ and
• a is the numerator and b is the denominator.
A fraction represents parts of a whole. The denominator b is the number of equal parts the whole has been divided into, and the numerator a indicates how many parts are included.
If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate $6666$ pieces, or, in other words, one whole pie.
So $66=1.66=1.$ This leads us to the property of one that tells us that any number, except zero, divided by itself is 1.
### Property of One
$aa=1(a≠0)aa=1(a≠0)$
Any number, except zero, divided by itself is one.
### Manipulative Mathematics
Doing the Manipulative Mathematics activity “Fractions Equivalent to One” will help you develop a better understanding of fractions that are equivalent to one.
If a pie was cut in $66$ pieces and we ate all 6, we ate $6666$ pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate $8888$ pieces, or one whole pie. We ate the same amount—one whole pie.
The fractions $6666$ and $8888$ have the same value, 1, and so they are called equivalent fractions. Equivalent fractions are fractions that have the same value.
Let’s think of pizzas this time. Figure 1.12 shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that $1212$ is equivalent to $48.48.$ In other words, they are equivalent fractions.
Figure 1.12 Since the same amount is of each pizza is shaded, we see that $1212$ is equivalent to $48.48.$ They are equivalent fractions.
### Equivalent Fractions
Equivalent fractions are fractions that have the same value.
How can we use mathematics to change $1212$ into $48?48?$ How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into $88$ pieces instead of just 2. Mathematically, what we’ve described could be written like this as $1·42·4=48.1·42·4=48.$ See Figure 1.13.
Figure 1.13 Cutting each half of the pizza into $44$ pieces, gives us pizza cut into 8 pieces: $1·42·4=48.1·42·4=48.$
This model leads to the following property:
### Equivalent Fractions Property
If $a,b,ca,b,c$ are numbers where $b≠0,c≠0,b≠0,c≠0,$ then
$ab=a·cb·cab=a·cb·c$
If we had cut the pizza differently, we could get
So, we say $12,24,36,and102012,24,36,and1020$ are equivalent fractions.
### Manipulative Mathematics
Doing the Manipulative Mathematics activity “Equivalent Fractions” will help you develop a better understanding of what it means when two fractions are equivalent.
### Example 1.64
Find three fractions equivalent to $25.25.$
Try It 1.127
Find three fractions equivalent to $35.35.$
Try It 1.128
Find three fractions equivalent to $45.45.$
### Simplify Fractions
A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.
For example,
• $2323$ is simplified because there are no common factors of 2 and 3.
• $10151015$ is not simplified because $55$ is a common factor of 10 and 15.
### Simplified Fraction
A fraction is considered simplified if there are no common factors in its numerator and denominator.
The phrase reduce a fraction means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.
In Example 1.64, we used the equivalent fractions property to find equivalent fractions. Now we’ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.
### Equivalent Fractions Property
If $a,b,ca,b,c$ are numbers where $b≠0,c≠0,b≠0,c≠0,$
$thenab=a·cb·canda·cb·c=abthenab=a·cb·canda·cb·c=ab$
### Example 1.65
Simplify: $−3256.−3256.$
Try It 1.129
Simplify: $−4254.−4254.$
Try It 1.130
Simplify: $−4581.−4581.$
Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the equivalent fractions property.
### Example 1.66
#### How to Simplify a Fraction
Simplify: $−210385.−210385.$
Try It 1.131
Simplify: $−69120.−69120.$
Try It 1.132
Simplify: $−120192.−120192.$
We now summarize the steps you should follow to simplify fractions.
### How To
#### Simplify a Fraction.
1. Step 1. Rewrite the numerator and denominator to show the common factors.
If needed, factor the numerator and denominator into prime numbers first.
2. Step 2. Simplify using the equivalent fractions property by dividing out common factors.
3. Step 3. Multiply any remaining factors, if needed.
### Example 1.67
Simplify: $5x5y.5x5y.$
Try It 1.133
Simplify: $7x7y.7x7y.$
Try It 1.134
Simplify: $3a3b.3a3b.$
### Multiply Fractions
Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.
### Manipulative Mathematics
Doing the Manipulative Mathematics activity “Model Fraction Multiplication” will help you develop a better understanding of multiplying fractions.
We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with $34.34.$
Now we’ll take $1212$ of $34.34.$
Notice that now, the whole is divided into 8 equal parts. So $12·34=38.12·34=38.$
To multiply fractions, we multiply the numerators and multiply the denominators.
### Fraction Multiplication
If $a,b,candda,b,candd$ are numbers where $b≠0andd≠0,b≠0andd≠0,$ then
$ab·cd=acbdab·cd=acbd$
To multiply fractions, multiply the numerators and multiply the denominators.
When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In Example 1.68, we will multiply negative and a positive, so the product will be negative.
### Example 1.68
Multiply: $−1112·57.−1112·57.$
Try It 1.135
Multiply: $−1028·815.−1028·815.$
Try It 1.136
Multiply: $−920·512.−920·512.$
When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as $a1.a1.$ So, for example, $3=31.3=31.$
### Example 1.69
Multiply: $−125(−20x).−125(−20x).$
Try It 1.137
Multiply: $113(−9a).113(−9a).$
Try It 1.138
Multiply: $137(−14b).137(−14b).$
### Divide Fractions
Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.
The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of $2323$ is $32.32.$
Notice that $23·32=1.23·32=1.$ A number and its reciprocal multiply to 1.
To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.
The reciprocal of $−107−107$ is $−710,−710,$ since $−107(−710)=1.−107(−710)=1.$
### Reciprocal
The reciprocal of $abab$ is $ba.ba.$
A number and its reciprocal multiply to one $ab·ba=1.ab·ba=1.$
### Manipulative Mathematics
Doing the Manipulative Mathematics activity “Model Fraction Division” will help you develop a better understanding of dividing fractions.
To divide fractions, we multiply the first fraction by the reciprocal of the second.
### Fraction Division
If $a,b,candda,b,candd$ are numbers where $b≠0,c≠0andd≠0,b≠0,c≠0andd≠0,$ then
$ab÷cd=ab·dcab÷cd=ab·dc$
To divide fractions, we multiply the first fraction by the reciprocal of the second.
We need to say $b≠0,c≠0andd≠0b≠0,c≠0andd≠0$ to be sure we don’t divide by zero!
### Example 1.70
Divide: $−23÷n5.−23÷n5.$
Try It 1.139
Divide: $−35÷p7.−35÷p7.$
Try It 1.140
Divide: $−58÷q3.−58÷q3.$
### Example 1.71
Find the quotient: $−78÷(−1427).−78÷(−1427).$
Try It 1.141
Find the quotient: $−727÷(−3536).−727÷(−3536).$
Try It 1.142
Find the quotient: $−514÷(−1528).−514÷(−1528).$
There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.
• “To multiply fractions, multiply the numerators and multiply the denominators.”
• “To divide fractions, multiply the first fraction by the reciprocal of the second.”
Another way is to keep two examples in mind:
The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction.
### Complex Fraction
A complex fraction is a fraction in which the numerator or the denominator contains a fraction.
Some examples of complex fractions are:
$6733458x2566733458x256$
To simplify a complex fraction, we remember that the fraction bar means division. For example, the complex fraction $34583458$ means $34÷58.34÷58.$
### Example 1.72
Simplify: $3458.3458.$
Try It 1.143
Simplify: $2356.2356.$
Try It 1.144
Simplify: $37611.37611.$
### Example 1.73
Simplify: $x2xy6.x2xy6.$
Try It 1.145
Simplify: $a8ab6.a8ab6.$
Try It 1.146
Simplify: $p2pq8.p2pq8.$
### Simplify Expressions with a Fraction Bar
The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.
To simplify the expression $5−37+1,5−37+1,$ we first simplify the numerator and the denominator separately. Then we divide.
$5−37+15−37+1$
$2828$
$1414$
### How To
#### Simplify an Expression with a Fraction Bar.
1. Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator.
2. Step 2. Simplify the fraction.
### Example 1.74
Simplify: $4−2(3)22+2.4−2(3)22+2.$
Try It 1.147
Simplify: $6−3(5)32+3.6−3(5)32+3.$
Try It 1.148
Simplify: $4−4(6)32+3.4−4(6)32+3.$
Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.
$−13=−13negativepositive=negative1−3=−13positivenegative=negative−13=−13negativepositive=negative1−3=−13positivenegative=negative$
For any positive numbers a and b,
$−ab=a−b=−ab−ab=a−b=−ab$
### Example 1.75
Simplify: $4(−3)+6(−2)−3(2)−2.4(−3)+6(−2)−3(2)−2.$
Try It 1.149
Simplify: $8(−2)+4(−3)−5(2)+3.8(−2)+4(−3)−5(2)+3.$
Try It 1.150
Simplify: $7(−1)+9(−3)−5(3)−2.7(−1)+9(−3)−5(3)−2.$
### Translate Phrases to Expressions with Fractions
Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.
The English words quotient and ratio are often used to describe fractions. Remember that “quotient” means division. The quotient of $aa$ and $bb$ is the result we get from dividing $aa$ by $b,b,$ or $ab.ab.$
### Example 1.76
Translate the English phrase into an algebraic expression: the quotient of the difference of m and n, and p.
Try It 1.151
Translate the English phrase into an algebraic expression: the quotient of the difference of a and b, and cd.
Try It 1.152
Translate the English phrase into an algebraic expression: the quotient of the sum of $pp$ and $q,q,$ and $rr$
### Section 1.5 Exercises
#### Practice Makes Perfect
Find Equivalent Fractions
In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
343.
$3838$
344.
$5858$
345.
$5959$
346.
$1818$
Simplify Fractions
In the following exercises, simplify.
347.
$−4088−4088$
348.
$−6399−6399$
349.
$−10863−10863$
350.
$−10448−10448$
351.
$120252120252$
352.
$182294182294$
353.
$−3x12y−3x12y$
354.
$−4x32y−4x32y$
355.
$14x221y14x221y$
356.
$24a32b224a32b2$
Multiply Fractions
In the following exercises, multiply.
357.
$34·91034·910$
358.
$45·2745·27$
359.
$−23(−38)−23(−38)$
360.
$−34(−49)−34(−49)$
361.
$−59·310−59·310$
362.
$−38·415−38·415$
363.
$(−1415)(920)(−1415)(920)$
364.
$(−910)(2533)(−910)(2533)$
365.
$(−6384)(−4490)(−6384)(−4490)$
366.
$(−6360)(−4088)(−6360)(−4088)$
367.
$4·5114·511$
368.
$5·835·83$
369.
$37·21n37·21n$
370.
$56·30m56·30m$
371.
$−8(174)−8(174)$
372.
$(−1)(−67)(−1)(−67)$
Divide Fractions
In the following exercises, divide.
373.
$34÷2334÷23$
374.
$45÷3445÷34$
375.
$−79÷(−74)−79÷(−74)$
376.
$−56÷(−56)−56÷(−56)$
377.
$34÷x1134÷x11$
378.
$25÷y925÷y9$
379.
$518÷(−1524)518÷(−1524)$
380.
$718÷(−1427)718÷(−1427)$
381.
$8u15÷12v258u15÷12v25$
382.
$12r25÷18s3512r25÷18s35$
383.
$−5÷12−5÷12$
384.
$−3÷14−3÷14$
385.
$34÷(−12)34÷(−12)$
386.
$−15÷(−53)−15÷(−53)$
In the following exercises, simplify.
387.
$−8211235−8211235$
388.
$−9163340−9163340$
389.
$−452−452$
390.
$53105310$
391.
$m3n2m3n2$
392.
$−38−y12−38−y12$
Simplify Expressions Written with a Fraction Bar
In the following exercises, simplify.
393.
$22+31022+310$
394.
$19−4619−46$
395.
$4824−154824−15$
396.
$464+4464+4$
397.
$−6+68+4−6+68+4$
398.
$−6+317−8−6+317−8$
399.
$4·36·64·36·6$
400.
$6·69·26·69·2$
401.
$42−12542−125$
402.
$72+16072+160$
403.
$8·3+2·914+38·3+2·914+3$
404.
$9·6−4·722+39·6−4·722+3$
405.
$5·6−3·44·5−2·35·6−3·44·5−2·3$
406.
$8·9−7·65·6−9·28·9−7·65·6−9·2$
407.
$52−323−552−323−5$
408.
$62−424−662−424−6$
409.
$7·4−2(8−5)9·3−3·57·4−2(8−5)9·3−3·5$
410.
$9·7−3(12−8)8·7−6·69·7−3(12−8)8·7−6·6$
411.
$9(8−2)−3(15−7)6(7−1)−3(17−9)9(8−2)−3(15−7)6(7−1)−3(17−9)$
412.
$8(9−2)−4(14−9)7(8−3)−3(16−9)8(9−2)−4(14−9)7(8−3)−3(16−9)$
Translate Phrases to Expressions with Fractions
In the following exercises, translate each English phrase into an algebraic expression.
413.
the quotient of r and the sum of s and 10
414.
the quotient of A and the difference of 3 and B
415.
the quotient of the difference of $xandy,and−3xandy,and−3$
416.
the quotient of the sum of $mandn,and4qmandn,and4q$
#### Everyday Math
417.
Baking. A recipe for chocolate chip cookies calls for $3434$ cup brown sugar. Imelda wants to double the recipe. How much brown sugar will Imelda need? Show your calculation. Measuring cups usually come in sets of $14,13,12,and114,13,12,and1$ cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.
418.
Baking. Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs $2323$ cup of condensed milk. How much condensed milk will Nina need? Show your calculation. Measuring cups usually come in sets of $14,13,12,and114,13,12,and1$ cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for $44$ pans of fudge.
419.
Portions Don purchased a bulk package of candy that weighs $55$ pounds. He wants to sell the candy in little bags that hold $1414$ pound. How many little bags of candy can he fill from the bulk package?
420.
Portions Kristen has $3434$ yards of ribbon that she wants to cut into $66$ equal parts to make hair ribbons for her daughter’s 6 dolls. How long will each doll’s hair ribbon be?
#### Writing Exercises
421.
Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn’t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael’s reasoning.
422.
Give an example from everyday life that demonstrates how $12·23is13.12·23is13.$
423.
Explain how you find the reciprocal of a fraction.
424.
Explain how you find the reciprocal of a negative number.
#### Self Check
After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
After looking at the checklist, do you think you are well prepared for the next section? Why or why not?
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As an Amazon Associate we earn from qualifying purchases. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 235, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9435864686965942, "perplexity": 1102.5909085517658}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178360293.33/warc/CC-MAIN-20210228054509-20210228084509-00573.warc.gz"} |
https://quant.stackexchange.com/questions/38114/accrual-in-default-derivation-of-credit-cds-curve/38116 | # Accrual in Default Derivation of Credit CDS Curve
In Trading Credit Curves Part I by JP Morgan we have that each point on a credit (CDS) curve represents:
$$PV(\text{Fee Leg}) = PV(\text{Contingent Leg})$$
which is
$$S_n \sum_{i=1}^{n}\Delta_i PS_i DF_i + \text{Accrual on Default} = (1-R)\sum_{i=1}^{n}(Ps(i-1)-Psi)DF_i$$
where the accrual on Default is $S_n \sum_{i=1}^{n}\frac{\Delta i}{2}(Ps(i-1)-Psi)DF_i$
where $S_n$ is the spread for protection to period n, $\Delta_i$ is the length of time period i in years, $PSi$ is the probability of survival to time t, $DFi$ is the risk free discount factor to time i, $R$ is the recovery rate on default
I cannot understand why the accrual on default bit is there and i cannot see how it has been derived and the reasoning behind it. I really dont see why you dont just sum to time n when there is a default and discount that? I dont understand why we need the $\Delta_i$ in the first term on the LHS as it seems superfluous.
I suppose really I dont understand the LHS of the equation derivation at all.
The formula for the accrual on default $$S_n \sum_{i=1}^n \frac{\Delta_i}{2}(Ps(i-1)-Ps(i))DF_i$$ is just an approximation that says conditional on default occurring within period $i$ (probability of $Ps(i-1)-Ps(i)$), defaults occurs on average in the middle of the period, thus the $\frac{\Delta_i}{2}$ average accrual time from beginning of period to default.
• Ok thanks, yeah i understand this now, but I dont see why we need the $\Delta_i$ in the LHS??? – Permian Feb 8 '18 at 9:24
• $S_n \Delta_i$ is the fixed leg coupon paid on a full period ($S_n$ is a rate, not an amount). – Antoine Conze Feb 8 '18 at 9:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9322752356529236, "perplexity": 619.9021984674329}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314638.49/warc/CC-MAIN-20190819011034-20190819033034-00074.warc.gz"} |
https://hpaulkeeler.com/poisson-point-process-simulation/ | # Simulating a homogeneous Poisson point process on a rectangle
This is the first of a series of posts about simulating Poisson point processes. I’ll start with arguably the simplest Poisson point process on two-dimensional space, which is the homogeneous one defined on a rectangle. Let’s say that we we want to simulate a Poisson point process with intensity $$\lambda>0$$ on a (bounded) rectangular region, for example, the rectangle $$[0,w]\times[0,h]$$ with dimensions $$w>0$$ and $$h>0$$ and area $$A=wh$$. We assume for now that the bottom left corner of the rectangle is at the origin.
## Steps
##### Number of points
The number of points in the rectangle $$[0,w]\times[0,h]$$ is a Poisson random variable with mean $$\lambda A$$. In other words, this random variable is distributed according to the Poisson distribution with parameter $$\lambda A$$, and not just $$\lambda$$, because the number of points depends on the size of the simulation region.
This is the most complicated part of the simulation procedure. As long as your preferred programming language can produce (pseudo-)random numbers according to a Poisson distribution, you can simulate a homogeneous Poisson point process. There’s a couple of different ways used to simulate Poisson random variables, but we will skip the details. In MATLAB, it is done by using the poissrnd function with the argument $$\lambda A$$. In R, it is done similarly with the standard function rpois . In Python, we can use either the scipy.stats.poisson or numpy.random.poisson function from the SciPy or NumPy libraries.
##### Location of points
The points now need to be positioned randomly, which is done by using Cartesian coordinates. For a homogeneous Poisson point process, the $$x$$ and $$y$$ coordinates of each point are independent uniform points, which is also the case for the binomial point process, covered in a previous post. For the rectangle $$[0,w]\times[0,h]$$, the $$x$$ coordinates are uniformly sampled on the interval $$[0,w]$$, and similarly for the $$y$$ coordinates. If the bottom left corner of rectangle is located at the point $$(x_0,y_0)$$, then we just have to shift the random $$x$$ and $$y$$ coordinates by respectively adding $$x_0$$ and $$y_0$$.
Every scientific programming language has a random uniform number generator because it is the default random number generator. In MATLAB, R and SciPy, it is respectively rand, runif and scipy.stats.uniform.
## Code
Here is some code that I wrote for simulating a homogeneous Poisson point process on a rectangle. You will notice that in all the code samples the part that simulates the Poisson point process requires only three lines of code: one line for the number of points and two lines lines for the $$x$$ and $$y$$ coordinates of the points.
##### MATLAB
%Simulation window parameters
xMin=0;xMax=1;
yMin=0;yMax=1;
xDelta=xMax-xMin;yDelta=yMax-yMin; %rectangle dimensions
areaTotal=xDelta*yDelta;
%Point process parameters
lambda=100; %intensity (ie mean density) of the Poisson process
%Simulate Poisson point process
numbPoints=poissrnd(areaTotal*lambda);%Poisson number of points
xx=xDelta*(rand(numbPoints,1))+xMin;%x coordinates of Poisson points
yy=xDelta*(rand(numbPoints,1))+yMin;%y coordinates of Poisson points
%Plotting
scatter(xx,yy);
xlabel('x');ylabel('y');
##### R
Note: it is a bit tricky to write “<-” in the R code (as it automatically changes to the html equivalent in the HTML editor I am using), so I have usually used “=” instead of the usual “<-”.
#Simulation window parameters
xMin=0;xMax=1;
yMin=0;yMax=1;
xDelta=xMax-xMin;yDelta=yMax-yMin; #rectangle dimensions
areaTotal=xDelta*yDelta;
#Point process parameters
lambda=100; #intensity (ie mean density) of the Poisson process
#Simulate Poisson point process
numbPoints=rpois(1,areaTotal*lambda);#Poisson number of points
xx=xDelta*runif(numbPoints)+xMin;#x coordinates of Poisson points
yy=xDelta*runif(numbPoints)+yMin;#y coordinates of Poisson points
#Plotting
plot(xx,yy,'p',xlab='x',ylab='y',col='blue');
##### Python
Note: “lambda” is a reserved word in Python (and other languages), so I have used “lambda0” instead.
import numpy as np
import scipy.stats
import matplotlib.pyplot as plt
#Simulation window parameters
xMin=0;xMax=1;
yMin=0;yMax=1;
xDelta=xMax-xMin;yDelta=yMax-yMin; #rectangle dimensions
areaTotal=xDelta*yDelta;
#Point process parameters
lambda0=100; #intensity (ie mean density) of the Poisson process
#Simulate Poisson point process
numbPoints = scipy.stats.poisson( lambda0*areaTotal ).rvs()#Poisson number of points
xx = xDelta*scipy.stats.uniform.rvs(0,1,((numbPoints,1)))+xMin#x coordinates of Poisson points
yy = yDelta*scipy.stats.uniform.rvs(0,1,((numbPoints,1)))+yMin#y coordinates of Poisson points
#Plotting
plt.scatter(xx,yy, edgecolor='b', facecolor='none', alpha=0.5 )
plt.xlabel("x"); plt.ylabel("y")
##### Julia
After writing this post, I later wrote the code in Julia. The code is here and my thoughts about Julia are here.
## Higher dimensions
If you want to simulate a Poisson point process in a three-dimensional box (typically called a cuboid or rectangular prism), you just need two modifications.
For a box $$[0,w]\times[0,h]\times[0,\ell]$$, the number of points now a Poisson random variable with mean $$\lambda V$$, where $$V= wh\ell$$ is the volume of the box. (For higher dimensions, you need to use $n$-dimensional volume.)
To position the points in the box, you just need an additional uniform variable for the extra coordinate. In other words, the $$x$$, $$y$$ and $$z$$ coordinates are uniformly and independently sampled on the respective intervals $$[0,w]$$, $$[0,h]$$, $$[0,\ell]$$. The more dimensions, the more uniform random variables.
And that’s it.
## Further reading
For simulation of point processes, see, for example, the books Statistical Inference and Simulation for Spatial Point Processes by Møller and Waagepetersen, or Stochastic Geometry and its Applications by Chiu, Stoyan, Kendall and Mecke.
There are books written by spatial statistics experts such as Stochastic Simulation by Ripley and Spatial Point Patterns: Methodology and Applications with R by Baddeley, Rubak and Turner, where the second book covers the spatial statistics R-package spatstat. Kroese and Botev also have a good introduction in the edited collection Stochastic Geometry, Spatial Statistics and Random Fields : Models and Algorithms by Schmidt, where the relevant chapter (number 12) is also freely available online.
More general stochastic simulation books that cover relevant material include Uniform Random Variate Generation by Devroye and Stochastic Simulation: Algorithms and Analysis by Asmussen and Glynn.
## Author: Paul Keeler
I am a researcher with interests in mathematical models involving randomness, particularly models with some element of geometry. Much of my work studies wireless networks with a focus on using tools from probability theory such as point processes. I come from Australia, where I call Melbourne home, but I have lived several years in Europe. I grew up in country Queensland and New South Wales.
## 2 thoughts on “Simulating a homogeneous Poisson point process on a rectangle”
1. Poulomi Mukherjee says:
It is nice.Please help me to generate the same points in ns2.
1. Paul Keeler says:
I don’t know ns2. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8158236145973206, "perplexity": 936.1780404387866}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703538226.66/warc/CC-MAIN-20210123160717-20210123190717-00571.warc.gz"} |
https://arxiv.org/abs/1902.10189v1 | stat.ML
(what is this?)
# Title:Variational Inference to Measure Model Uncertainty in Deep Neural Networks
Abstract: We present a novel approach for training deep neural networks in a Bayesian way. Classical, i.e. non-Bayesian, deep learning has two major drawbacks both originating from the fact that network parameters are considered to be deterministic. First, model uncertainty cannot be measured thus limiting the use of deep learning in many fields of application and second, training of deep neural networks is often hampered by overfitting. The proposed approach uses variational inference to approximate the intractable a posteriori distribution on basis of a normal prior. The variational density is designed in such a way that the a posteriori uncertainty of the network parameters is represented per network layer and depending on the estimated parameter expectation values. This way, only a few additional parameters need to be optimized compared to a non-Bayesian network. We apply this Bayesian approach to train and test the LeNet architecture on the MNIST dataset. Compared to classical deep learning, the test error is reduced by 15%. In addition, the trained model contains information about the parameter uncertainty in each layer. We show that this information can be used to calculate credible intervals for the prediction and to optimize the network architecture for a given training data set.
Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG) Cite as: arXiv:1902.10189 [stat.ML] (or arXiv:1902.10189v1 [stat.ML] for this version)
## Submission history
From: Jan Steinbrener [view email]
[v1] Tue, 26 Feb 2019 20:04:15 UTC (453 KB)
[v2] Fri, 8 Mar 2019 10:55:19 UTC (459 KB) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8568425178527832, "perplexity": 724.0474431244972}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578596541.52/warc/CC-MAIN-20190423074936-20190423100936-00380.warc.gz"} |
http://mathhelpforum.com/calculus/71418-volume-solid-washer-print.html | # Volume of solid/washer
Printable View
• Feb 2nd 2009, 03:13 PM
silencecloak
Volume of solid/washer
$y=2/x$
$y=0$
$x=1$
$x=3$
Rotate about $y = -1$
Outside radius = $(-1-2/x)^2$
Inside radius = 1
$\int_1^3 ((-1-2/x)^2-1)dx = 22.1831495912$
Is this correct?
Thanks in advance
• Feb 2nd 2009, 03:28 PM
Jester
Quote:
Originally Posted by silencecloak
$y=2/x$
$y=0$
$x=1$
$x=3$
Rotate about $y = -1$
Outside radius = $(-1-2/x)^2$
Inside radius = 1
$\int_1^3 ((-1-2/x)^2-1)dx = 22.1831495912$
Is this correct?
Thanks in advance
Outside radius = $(2/x + 1)$ although the square will fix that. Just need a pie
$\pi \int_1^3 ((2/x+1)^2-1)dx = 22.1831495912$
although judging from the answer you put it in.
• Feb 2nd 2009, 03:32 PM
silencecloak
So I did this problem correct? Or that number is just the correct number for the integral i set up?
• Feb 2nd 2009, 03:39 PM
Jester
Quote:
Originally Posted by silencecloak
So I did this problem correct? Or that number is just the correct number for the integral i set up?
Yes, just make sure you have the outer radius
$
r_o = f(x) - L$
(for $f(x)>L$) where $y = f(x)$ the curve you're rotating and $y = L$ the line you're rotating about. Switch f and L if f is under $y= L$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 21, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9851073026657104, "perplexity": 3030.2475844517517}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823997.21/warc/CC-MAIN-20171020082720-20171020102641-00052.warc.gz"} |
http://umj-old.imath.kiev.ua/article/?lang=en&article=5199 | 2019
Том 71
№ 11
Application of a separating transformation to estimates of inner radii of open sets
Abstract
We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets.
English version (Springer): Ukrainian Mathematical Journal 59 (2007), no. 10, pp 1472-1481.
Citation Example: Bakhtin A. K., V'yun V. E. Application of a separating transformation to estimates of inner radii of open sets // Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1313–1321.
Full text | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9360445737838745, "perplexity": 1203.4000992914594}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600401614309.85/warc/CC-MAIN-20200928202758-20200928232758-00387.warc.gz"} |
https://proofwiki.org/wiki/399_is_not_Expressible_as_Sum_of_Fewer_than_19_Fourth_Powers | # 399 is not Expressible as Sum of Fewer than 19 Fourth Powers
## Theorem
$399$ cannot be expressed as the sum of fewer than $19$ fourth powers:
$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$
or:
$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$
## Proof
First note that $5^4 = 625 > 399$.
Then note that $2 \times 4^4 = 512 > 399$.
Hence any expression of $399$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.
For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances of $1^4$ does.
Now we have:
$\displaystyle 399$ $=$ $\displaystyle 4^4 + 3^4 + 3 \times 2^4 + 14 \times 1^4$ $\displaystyle$ $=$ $\displaystyle 4^4 + 8 \times 2^4 + 15 \times 1^4$ $\displaystyle$ $=$ $\displaystyle 4 \times 3^4 + 4 \times 2^4 + 11 \times 1^4$ $\displaystyle$ $=$ $\displaystyle 3 \times 3^4 + 9 \times 2^4 + 12 \times 1^4$ $\displaystyle$ $=$ $\displaystyle 2 \times 3^4 + 14 \times 2^4 + 13 \times 1^4$ $\displaystyle$ $=$ $\displaystyle 1 \times 3^4 + 19 \times 2^4 + 14 \times 1^4$ $\displaystyle$ $=$ $\displaystyle 0 \times 3^4 + 24 \times 2^4 + 15 \times 1^4$
and it can be seen that the first and the third uses the least number of fourth powers, at $19$.
$\blacksquare$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8990417718887329, "perplexity": 87.1175207733545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107897022.61/warc/CC-MAIN-20201028073614-20201028103614-00673.warc.gz"} |
https://encyclopediaofmath.org/wiki/Self-injective_ring | # Self-injective ring
left
A ring that, as a left module over itself, is injective (cf. Injective module). A right self-injective ring is defined in a symmetric way. The classical semi-simple rings and all rings of residues of integers $\mathbf Z /( n)$ are self-injective rings. If $R$ is a self-injective ring with Jacobson radical $J$, then the quotient ring $R/J$ is a regular ring (in the sense of von Neumann). A regular self-injective ring is continuous. Every countable self-injective ring is quasi-Frobenius (cf. Quasi-Frobenius ring). A left self-injective ring is not necessarily right self-injective. The ring of matrices over a self-injective ring and the complete ring of linear transformations of a vector space over a field are self-injective. The rings of endomorphisms of all free left $R$- modules are self-injective rings if and only if $R$ is quasi-Frobenius. If $M$ is the cogenerator of the category of left $R$- modules, then $\mathop{\rm End} _ {R} M$ is a self-injective ring. If the singular ideal of a ring $R$ is zero, then its injective hull can be made into a self-injective ring in a natural way. A group ring $RG$ is left self-injective if and only if $R$ is a self-injective ring and $G$ is a finite group. The direct product of self-injective rings is self-injective. A ring $R$ is isomorphic to the direct product of complete rings of linear transformations over fields if and only if $R$ is a left self-injective ring without nilpotent ideals for which every non-zero left ideal contains a minimal left ideal.
#### References
[1] L.A. Skornyaka, A.V. Mikhalev, "Modules" Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 14 (1976) pp. 57–190 (In Russian) [2] C. Faith, "Algebra" , 1–2 , Springer (1973–1976) [3] J. Lawrence, "A countable self-injective ring is quasi-Frobenius" Proc. Amer. Math. Soc. , 65 : 2 (1977) pp. 217–220
An essential right ideal of a ring $R$ is an ideal $E$ such that $E \cap I \neq 0$ for all non-zero right ideals $I$ of $R$. In a right Ore domain (cf. below) every non-zero right ideal is essential. Let ${\mathcal E} ( R)$ be the set of essential right ideals of $R$;
$$\zeta ( R) = \{ {a \in R } : {a E = 0 \textrm{ for some } E \in {\mathcal E} ( R) } \}$$
is an ideal, called the right singular ideal of $R$.
Let $S$ be the multiplicatively closed subset of regular elements of $R$( i.e. non-zero-divisors of $R$). If $S$ satisfies the right Ore condition (cf. Associative rings and algebras), $R$ is called a right Ore ring. A right Ore domain is an integral domain that is a right Ore ring.
#### References
[a1] J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) pp. Part I, Chapt. 2
How to Cite This Entry:
Self-injective ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Self-injective_ring&oldid=48650
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8893547058105469, "perplexity": 422.10616716552727}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944996.49/warc/CC-MAIN-20230323034459-20230323064459-00052.warc.gz"} |
https://math.stackexchange.com/questions/2409690/dynamical-system-output-differential-equations | # Dynamical system output - differential equations
I am applying at a Faculty for Electrical Engineering, and I have an entrance exam in two days. I have few exams for exercise, from previous years and I keep getting stuck in one particular type of problem:
"A dynamicaI system is described by a differential equation of the following form: $$y''(t) + 5y'(t)+4y(t) = u(t)$$ where t is the independent variable of time, $u(t)$ is the input signal into the system, and $y(t)$ is the system response."
I need to prove that if the system is excited with stepped amplitude signal of $1$, the output of the system in the steady state will reach a constant value of $0.25$.
Can anyone please suggest how can I solve this problem. I think I should use Laplace Transformation to solve it, but I do not know to how transform the above differential equation in some type that can be solved with Laplace. Also, i do not get where does the amplitude signal of $1$ is used.
Any help or hints are welcomed and appreciated, I need all the help I can get. Thank you.
• Maybe this would suit ELECTRICAL ENGINEERING.SE better? – Mr. Xcoder Aug 29 '17 at 9:21
• Thank you for the suggestion, I did not know there is a page for Electrical Engineering. I will try there. – mchingoska Aug 29 '17 at 9:33
• The question definitely belongs to math.SE. To solve this, why not use that every solution of the differential equation is $$y(t)=ae^{-t}+be^{-4t}+\frac13e^{-t}\int_0^tu(s)e^sds-\frac13e^{-4t}\int_0^tu(s)e^{4s}ds$$ for some constants $(a,b)$ depending on the initial conditions $(y(0),y'(0))$, and plug in the function $u(t)$ you are interested in, whatever it is? For instance, if $u(t)=1$ for every $t>0$, one gets a fully explicit formula for $y(t)$, from which the limit when $t\to\infty$ is direct... – Did Aug 29 '17 at 9:36
• An electrical engineer would probably do Laplace tranformation $$(s^2+5s+4)Y=\frac{1}{s}$$ then do the partial fractions, solve the system and ensure that the limit is as expected. Or maybe use the final value theorem instead if allowed. – A.Γ. Aug 29 '17 at 10:05
• I started solving it like that and ended with $$Y=\frac{1}{4}-\frac{1}{3}e^{-t}+\frac{1}{12}e^{-4t}$$ , and I am stuck again. – mchingoska Aug 29 '17 at 10:17
To answer your question you do not actually need to solve the differential equation. Indeed, in steady state conditions both $y´´$ and $y´$ will vanish: hence you are (asymptotically) left with the equation $$4 y(t) = 1$$ (as $u(t)$ is where the step input function comes into play) from which one concludes $$y = 1/4 = 0.25$$
• @KirylPesotski If you feed in the harmonic signal $e^{i\omega t}$ you will get your output $y_p(t)$ as a harmonic oscillation with no existing limit as $t\to\infty$. Your $y_s$ is not a steady state, but an amplitude (frequency response at $\omega$). I do not see your point. – A.Γ. Aug 29 '17 at 11:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8601511120796204, "perplexity": 224.48336361991747}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107880014.26/warc/CC-MAIN-20201022170349-20201022200349-00236.warc.gz"} |
https://scholarworks.iu.edu/dspace/browse?value=noncommutative+probability&type=subject | mirage
# Browsing by Subject "noncommutative probability"
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• ([Bloomington, Ind.] : Indiana University, 2010-05-24)
We study convolutions that arise from noncommutative probability theory. In the case of the free convolutions, we prove that the absolutely continuous part, with respect to the Lebesgue measure, of the free convolution of ... | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9547460675239563, "perplexity": 1816.3849745696318}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860125897.19/warc/CC-MAIN-20160428161525-00185-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://stacks.math.columbia.edu/tag/03T3 | ## 62.6 Derived categories
To set up notation, let $\mathcal{A}$ be an abelian category. Let $\text{Comp}(\mathcal{A})$ be the abelian category of complexes in $\mathcal{A}$. Let $K(\mathcal{A})$ be the category of complexes up to homotopy, with objects equal to complexes in $\mathcal{A}$ and morphisms equal to homotopy classes of morphisms of complexes. This is not an abelian category. Loosely speaking, $D(A)$ is defined to be the category obtained by inverting all quasi-isomorphisms in $\text{Comp}(\mathcal{A})$ or, equivalently, in $K(\mathcal{A})$. Moreover, we can define $\text{Comp}^+(\mathcal{A}), K^+(\mathcal{A}), D^+(\mathcal{A})$ analogously using only bounded below complexes. Similarly, we can define $\text{Comp}^-(\mathcal{A}), K^-(\mathcal{A}), D^-(\mathcal{A})$ using bounded above complexes, and we can define $\text{Comp}^ b(\mathcal{A}), K^ b(\mathcal{A}), D^ b(\mathcal{A})$ using bounded complexes.
Remark 62.6.1. Notes on derived categories.
1. There are some set-theoretical problems when $\mathcal{A}$ is somewhat arbitrary, which we will happily disregard.
2. The categories $K(A)$ and $D(A)$ are endowed with the structure of a triangulated category.
3. The categories $\text{Comp}(\mathcal{A})$ and $K(\mathcal{A})$ can also be defined when $\mathcal{A}$ is an additive category.
The homology functor $H^ i : \text{Comp}(\mathcal{A}) \to \mathcal{A}$ taking a complex $K^\bullet \mapsto H^ i(K^\bullet )$ extends to functors $H^ i : K(\mathcal{A}) \to \mathcal{A}$ and $H^ i : D(\mathcal{A}) \to \mathcal{A}$.
Lemma 62.6.2. An object $E$ of $D(\mathcal{A})$ is contained in $D^+(\mathcal{A})$ if and only if $H^ i(E) =0$ for all $i \ll 0$. Similar statements hold for $D^-$ and $D^+$.
Proof. Hint: use truncation functors. See Derived Categories, Lemma 13.11.5. $\square$
Lemma 62.6.3. Morphisms between objects in the derived category.
1. Let $I^\bullet \in \text{Comp}^+(\mathcal{A})$ with $I^ n$ injective for all $n \in \mathbf{Z}$. Then
$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(K^\bullet , I^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(K^\bullet , I^\bullet ).$
2. Let $P^\bullet \in \text{Comp}^-(\mathcal{A})$ with $P^ n$ is projective for all $n \in \mathbf{Z}$. Then
$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(P^\bullet , K^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(P^\bullet , K^\bullet ).$
3. If $\mathcal{A}$ has enough injectives and $\mathcal{I} \subset \mathcal{A}$ is the additive subcategory of injectives, then $D^+(\mathcal{A})\cong K^+(\mathcal{I})$ (as triangulated categories).
4. If $\mathcal{A}$ has enough projectives and $\mathcal{P} \subset \mathcal{A}$ is the additive subcategory of projectives, then $D^-(\mathcal{A}) \cong K^-(\mathcal{P}).$
Proof. Omitted. $\square$
Definition 62.6.4. Let $F: \mathcal{A} \to \mathcal{B}$ be a left exact functor and assume that $\mathcal{A}$ has enough injectives. We define the total right derived functor of $F$ as the functor $RF: D^+(\mathcal{A}) \to D^+(\mathcal{B})$ fitting into the diagram
$\xymatrix{ D^+(\mathcal{A}) \ar[r]^{RF} & D^+(\mathcal{B}) \\ K^+(\mathcal I) \ar[u] \ar[r]^ F & K^+(\mathcal{B}). \ar[u] }$
This is possible since the left vertical arrow is invertible by the previous lemma. Similarly, let $G: \mathcal{A} \to \mathcal{B}$ be a right exact functor and assume that $\mathcal{A}$ has enough projectives. We define the total left derived functor of $G$ as the functor $LG: D^-(\mathcal{A}) \to D^-(\mathcal{B})$ fitting into the diagram
$\xymatrix{ D^-(\mathcal{A}) \ar[r]^{LG} & D^-(\mathcal{B}) \\ K^-(\mathcal{P}) \ar[u] \ar[r]^ G & K^-(\mathcal{B}). \ar[u] }$
This is possible since the left vertical arrow is invertible by the previous lemma.
Remark 62.6.5. In these cases, it is true that $R^ iF(K^\bullet ) = H^ i(RF(K^\bullet ))$, where the left hand side is defined to be $i$th homology of the complex $F(K^\bullet )$.
Comment #14 by Emmanuel Kowalski on
The short "Notes on derived categories" (remarks-derived-categories) is duplicated in the next Tag 03T4.
Comment #21 by Johan on
That is because we have tags for sections and lemmas, remarks, etc. And lemmas and remarks, etc are items inside sections. So there is some duplication in the material.
Comment #2167 by Alex on
typo: In the definition of $K(\mathcal{A})$ "objects equal to homotopy classes..." should say "morphisms equal to..."
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | {"extraction_info": {"found_math": true, "script_math_tex": 1, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9965048432350159, "perplexity": 269.92651193668803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703515235.25/warc/CC-MAIN-20210118185230-20210118215230-00489.warc.gz"} |
http://mathoverflow.net/questions/533/largest-hyperbolic-disk-embeddable-in-euclidean-3-space | # Largest hyperbolic disk embeddable in Euclidean 3-space?
Hilbert proved that there's no complete regular (C^k for sufficiently large k) isometric embedding of the hyperbolic plane into R^3. On the other hand, the pseudosphere is locally isometric to the hyperbolic plane up to its cusps (though it has the topology of a cylinder). What's the largest hyperbolic disk (with Gaussian curvature -1) that can be smoothly (or C^2, say) isometrically embedded in R^3?
edit: This doesn't seem to be getting many views, so I'll bump this by adding in a rather easy lower bound from the pseudosphere. First, the pseudosphere is parametrized by the region PS={z | Im z ≥ 1, -π < Re z ≤ π} on the upper half plane model of H^2. Let z=x+iy, so that ordered pairs (x,y)∈ H^2 when y>0.
Next, Euclidean circles drawn on in the upper-half plane model with center (x,y\cosh r) and radius y\sinh r correspond to hyperbolic circles with center (x,y) and radius r. I can fit a Euclidean circle of radius π centered at (0,1+π) into the region PS. This corresponds to a hyperbolic disk of radius arctanh(π/(1+π)) ~ 0.993.
Surely one can do better?
edit2: fixed mistakes in formulas above (didn't affect the bound). Here're some pictures:
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I didn't see the exact answer to your question in the Borisenko paper, since section 2.4 only seems to address immersions of subsets of ℍ2 into ℝ3. However, a perturbation of the pseudosphere, Dini's surface, which is an isometrically embedded horodisk, seems to do the trick since it contains arbitrarily large disks in the hyperbolic plane. See Dini's Surface at the Geometry Center.
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Thank you. I believe you mean that Dini's surface is an isometrically embedded horodisk, though? (The horocycle ought to be the cusp of Dini's surface.) I had been meaning to come back to update this question after following up on some of the references in Borisenko's paper, where I found that the immersions weren't particularly close to what I had in mind. – j.c. Nov 2 '09 at 3:12
yes, thanks, I changed it to horodisk. – Ian Agol Nov 2 '09 at 3:32
Robert Bryant shows that Dini's surface is not a horodisk in his answer here mathoverflow.net/questions/149842/… . Instead it is the region between a geodesic and a curve of constant geodesic curvature. – j.c. Nov 25 '13 at 15:29
@j.c.: thanks for the correction - I didn't realize this, but now that I read Robert's answer, I should have realized it wasn't a horodisk. I still think the answer works, by varying the parameter (in the limit, it should approach the immersed horodisk of the pseudosphere in the appropriate sense). – Ian Agol Nov 25 '13 at 23:51
Several russian geometers have addressed this question. I suggest you a survey on Isometric immersions by A. Borisenko (2001, I think) in Russian Mathematical Surveys (it is in English)
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Thanks! If you mean iop.org/EJ/abstract/0036-0279/56/3/R01, then it looks like section 2.4 has exactly what I've been looking for! – j.c. Oct 21 '09 at 21:43
Could you also post an answer here, please? – Ilya Nikokoshev Nov 2 '09 at 0:06
I will edit the question above with an answer (and hopefully some graphics) at some point in the future. – j.c. Nov 2 '09 at 3:13
I can't comment on Noah's answer, so:
The reason for the C^1 condition is that Nash's C1 embedding theorem says that any Riemannian k-manifold with a short embedding into Rn has an isometric C1 embedding into Rn for any n > k. In particular, there is an isometric C1 embedding of the hyperbolic plane into R3. The very beautiful proof proceeds by "pleating" the embedding to be closer and closer to isometric, while keeping it contained in a not-much-larger region.
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Thanks, I didn't know that about the pleating. Like I said, I'm more interested in the smooth cases, but I guess I'll look into the proof of Nash-Kuiper at some point as well. – j.c. Oct 17 '09 at 21:24
for n chosen sufficiently larger than k. – Narasimham Oct 26 '15 at 21:19
I think that you can get an arbitrarily large disk. The proof is by crochet. Since there's a pattern for crocheting constant negative curvature disks where you increase the radius as you go and since we live in 3-space, it follows that you can get arbitrarily large disks.
See this TED talk for some cool applications of hyperbolic crochet to biology, or this article for a more rigorous explanation.
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It's my reading of the second link that these crotcheted surfaces are not even C^1 though. Thanks for the links though. – j.c. Oct 15 '09 at 0:47
I should point out that the crochet article linked above has references to research literature. In particular, it mentions that there are no C^2 embeddings of the whole plane, but there is a C^1 embedding. – S. Carnahan Oct 15 '09 at 0:50
That's actually the reason I put in C^2 in my question. There's a recent book on isometric embeddings, by Han and Hong, which tackles related problems from an analysis viewpoint. There seem to be plenty of results on local isometric embedding without any explicit bounds or constructions, unless there are general ways to back out bounds from such proofs that I'm not aware of. – j.c. Oct 15 '09 at 0:56
I'm confused, my reading of that link is that they don't give a C^1 embedding of the whole hyperbolic plane. But I can't see how they wouldn't be C^1 for arbitrarily large (but finite) radius. – Noah Snyder Oct 15 '09 at 1:14
Yeah, I was just looking at the journal version of that article to see if there were more details, and there the sentence reads "The finite surfaces described here can apparently be extended indefinitely, but they appear always not to be differentiably embedded (see Figure 12)." Figure 12 is a picture of one of their crotcheted surfaces with the center pulled up so that it kind of looks like a pseudosphere, with the caption "Crotcheted pseudosphere". – j.c. Oct 15 '09 at 1:20 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8850571513175964, "perplexity": 625.8602228570836}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860112228.39/warc/CC-MAIN-20160428161512-00103-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/88529/infinite-products-of-representations-of-the-additive-group/88602 | # Infinite products of representations of the additive group
Fix a $\mathbb{Q}$-algebra $R$. Let's call an endomorphism $f : M \to M$ of an $R$-module $M$ locally nilpotent if for every $m \in M$ there is some $n \in \mathbb{N}$ such that $f^n(m)=0$. Equivalently, for every finitely generated submodule $N \subseteq M$ there is some $n \in \mathbb{N}$ with $f^n |_N = 0$. In particular, for a finitely generated $M$ we don't get anything new.
Now consider the following category $C$: Objects are pairs $(M,f)$, consisting of an $R$-module and a locally nilpotent endomorphism $f$ of $M$. A morphism is just a commutative diagram.
Question A. Is there a reference in the literature for the following observation: $C$ is isomorphic to the category of representations of the additive group scheme $\mathbb{G}_{a}$ over $R$.
This is not hard to prove. In fact, one can use the same method as in Example I.8.1 in Milne's script on algebraic groups; but there $R$ is a field and $M$ is supposed to be finite dimensional, so that locally nilpotent = nilpotent. But the same works in general, the $R[T]$-comodule structure on $M$ corresponding to $(M,f)$ is given by $M \to M[T]$, $m \mapsto \sum\limits_{n \geq 0} \frac{f^n(m) T^n}{n!}$.
Question B. How do infinite products look like in $C$?
Note that the description of $C$ above shows that $C$ is cocomplete as as well as complete. The forgetful functor $C \to \mathrm{Mod}(R)$ creates colimits, so they are easy to describe. The same is true for finite limits, so the only limits missing are infinite products. The forgetful functor doesn't create them: For a familiy of objects $(M_i,f_i)$, the endomorphism $\prod_i f_i$ of $\prod_i M_i$ does not have to be locally nilpotent. Thus they will be more complicated.
Question C. What are interesting examples of locally nilpotent endomorphisms which are not nilpotent?
Of course there are many examples: If $M$ is an $\mathbb{N}$-graded module and $f$ is of negative degree, then $f$ is locally nilpotent, but usually $f$ is not nilpotent. A specific example is the derivative $\partial : R[X] \to R[X]$. Are there more interesting examples which don't arise from gradings?
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I don't know a precise reference, but it's a very well known fact. I doubt that the category of representations of $\mathbb{G}_{\mathrm a}$ has infinite products: (algebraic) representations of algebraic groups are unions of finite-dimensional subrepresentations, and this property is not preserved under infinite products. – Angelo Feb 15 '12 at 16:51
This property is not preserved under the "naive product" which is no product at all (see details in Question B). The category has products by some abstract result (Todd, Theo or Mike probably can explain this?). – Martin Brandenburg Feb 15 '12 at 17:02
Question A: Demazure-Gabriel, Groupes Algébriques, II, §2, 2.6
Question B: It is easily checked that the maximal submodule of $\prod M_i$ on which $\prod f_i$ acts locally nilpotent is the categorical product. This may also be described as $\bigcup_n \prod_i \ker f_i^n\subseteq\prod_i M_i$. Actually, since the set-valued functor $(M,f)\mapsto\ker f^n$ is represented by $(R[X]/(X^n),X)$, so commutes with products, and since the equality $M=\bigcup_n \ker f^n$ is equivalent to $f$ acting locally nilpotent, it is clear that the underlying set of the categorical product must be $\bigcup_n \prod_i \ker f_i^n$.
Question C: An example where the filtration $\ker f^n$ does not split is $R=k[[T]]$, $M=k((T))/k[[T]]$, $f=T$.
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Thanks for this answer! Another view on B: Actually $C$ is a coreflective subcategory of the category of $R$-modules equipped with some endomorphism, the coreflector is $(M,f) \mapsto (M,\cup_n \mathrm{ker}(f^n))$. Thus limits in $C$ may be computed by taking the coreflectors of the limits in the larger category. – Martin Brandenburg Feb 16 '12 at 8:00
Right. I should have thought of that. – Angelo Feb 17 '12 at 8:05
## I address mainly Question C in the simplest special case where $R$ is $\mathbb Q$:
In this case you are looking at locally nilpotent endomorphisms of a vector space. Similarity classes of such endomoephisms correspond to isomorphism classes of torsion $\mathbb Q[[t]]$-modules. If you assume that the modules are of countable dimension and reduced, then a classification of isomorphism classes of such modules is given by Ulm's theorem: the isomorphism classes are in bijective correspondence with certain transfinite sequences of non-negative integers (the Ulm invariants). This should allow you to construct many interesting examples.
For the details, I can do no better than referring you to Kaplansky's book Infinite Abelian Groups where Ulm's theorem and its relation to the classification of locally nilpotent endomorphisms are beautifully explained. It is based on Kaplansky's paper with Mackey A generalization of Ulm's theorem, Sum. Bras. Math., 1951 (available in Kaplansky's selected papers).
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This is interesting, thank you. – Martin Brandenburg Feb 16 '12 at 8:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9484403133392334, "perplexity": 161.15064497939096}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049277313.92/warc/CC-MAIN-20160524002117-00122-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/107159/pochhammer-symbol-of-a-differential-and-hypergeometric-polynomials?sort=votes | Pochhammer symbol of a differential, and hypergeometric polynomials
I have a minor result which I'm sure has come up somewhere before but I can't seem to find it.
Consider a confluent hypergeometric function of the form $$\newcommand{\ff}{{}_1F_1} \ff(b+k;b;z)\textrm{, for }k\in\mathbb{N}.$$ Numerical tests suggest that this is always a polynomial of degree $k$ multiplied by an exponential. One can prove this in a dull fashion by using $\ff(b;b;z)=e^z$ and then applying recurrence relations, but I found a cleaner way using the series definition,
$$\ff(b+k;b;z) =\sum_{n=0}^\infty\frac{(b+k)_n}{(b)_n}\frac{z^n}{n!}$$
where $(b)_k=b(b+1)\cdots(b+k-1)$ is the Pochhammer symbol. By exploiting the identities $$\frac{(b+n)_k}{(b)_k}=\frac{\Gamma(b+k+n)\Gamma(b)}{\Gamma(b+k)\Gamma(b+n)}=\frac{(b+n)_k}{(b)_k} \textrm{ and }nz^n=z\frac{d}{dz}z^n,$$ one can easily prove that $$\ff(b+k;b;z)=\frac{\left(b+z\frac{d}{dz}\right)_k}{(b)_k}e^z,$$
by being somewhat liberal with the meaning of the Pochhammer symbol. This is clearly the desired polynomial-times-exponential, and provides an explicit expression for the polynomial that looks kind of like a Rodrigues formula.
Even better, if you put this together with Kummer's first transformation, $$\ff\left(a;b;z\right)=e^{z}\ff\left(b-a;b;-z\right),$$ and the expression for Laguerre polynomials in terms of hypergeometric functions, $L^{(\alpha)}_{n}\left(x\right)=\frac{\left(\alpha+1\right)_{n}}{n!}\ff\left(-n,\alpha+1,x\right)$, you get an analogous result for Laguerre polynomials,
$$L^{(b-1)}_{k}\left(x\right)=\frac{1+k/b}{k!}e^z\left(b+z\frac{d}{dz}\right)_ke^{-z}.$$
Are these results familiar to anyone? Do they fit inside a larger framework? They are not the best thing since sliced bread but they do have a nice simplicity to them, and particularly I would like to cite the appropriate reference if they have appeared before.
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Your last formula is not correct for k=b=1. – Tom Copeland Sep 15 '12 at 9:37
@TomCopeland so it is. Let me check the details. – Emilio Pisanty Sep 15 '12 at 17:03
Formally using the inverse Mellin transform for x>0:
$$e^x f(x\tfrac{d}{dx})e^{-x}=e^x f(x\tfrac{d}{dx}) \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\pi}{\sin(\pi s)} \frac{x^{-s}}{(-s)!} ds$$
$$=e^x \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\pi}{\sin(\pi s)} f(-s) \frac{x^{-s}}{(-s)!} ds.$$
Let $$f(x)=\binom{x+\alpha+\beta}{\beta},$$
then
$$e^x \binom{x\tfrac{d}{dx}+\alpha+\beta}{\beta}e^{-x}=L_{\beta}^{\alpha}(x)=\binom{\alpha+\beta}{\beta} K(-\beta,\alpha+1,x)$$
where $L_{\beta}^{\alpha}(x)$ is the generalized Laguerre function and $K(-\beta,\alpha+1,x),$ Kummer's confluent hypergeometric function.
For an elaboration, see the notes The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions.
See also Rodriguez-like formula on pg. 59 of Bateman's (et al.) Higher Transcendental Functions Vol. I:
$$(x\tfrac{d}{dx}+\alpha)_{n} h(x) = x^{1-\alpha}D^{n}[x^{n+\alpha-1}h(x)],$$
with $D=\tfrac{d}{dx}$, leading to
$$e^x \binom{x\tfrac{d}{dx}+\alpha+n}{n}e^{-x}=e^x x^{-\alpha}\tfrac{D^{n}}{n!}[x^{n+\alpha}e^{-x}]=L_{n}^{\alpha}(x).$$
This can be generalized by using the fractional integro-derivative representation of $K(a,b,x)$ (see Eqn. 13.2.1 on pg. 505 of Abramowitz and Stegun):
$$K(a,b,x)= e^x \tfrac{(b-1)!}{x^{b-1}}\int_{0}^{x} e^{-t}\tfrac{(x-t)^{a-1}}{(a-1)!} \tfrac{t^{b-a-1}}{(b-a-1)!} dt=e^x \tfrac{(b-1)!}{x^{b-1}}D^{-a}[e^{-x}\tfrac{x^{b-a-1}}{(b-a-1)!}],$$
$$e^x {x^{-\alpha}}\tfrac{D^{\beta}}{\beta!}[x^{\beta+\alpha}e^{-x}]=L_{\beta}^{\alpha}(x)=\binom{\alpha+\beta}{\beta} K(-\beta,\alpha+1,x).$$
As an aside, the operator $(xDx)^n=x^nD^nx^n=x^n n! L_n(-\widehat{xD})$ where $\widehat{xD}^n=x^nD^n$, has a long and interesting history. – Tom Copeland Sep 19 '12 at 2:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9115251302719116, "perplexity": 181.9332016627967}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678695499/warc/CC-MAIN-20140313024455-00047-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/75505-vector-difficulties.html | # Math Help - Vector difficulties
1. ## Vector difficulties
I am having some difficulties doing some basic vector questions on an assignment.
a line L1 passes through (1,-1,4) and is parallel to vector (1,-1,1). Another line L2 passes through (2,4,7) and (4,5,10). obtain teh vector lines equations for L1 & L2. Do the two lines intersect, if so at what point?
I think I have correctly worked out the line equations as:
for L1: r = (1,-1,4)+
λ(1,-1,1) Edit: thanks to plato
for L2: r = (2,4,7)+
λ(2,1,3)
But I am unsure how to work out if they intersect, and if so where?
Do i need to calculate their cartisan equations, and if so how do I then determine the intersect point.
2. Originally Posted by nickskely
[I]a line L1 passes through (1,-1,4) and is parallel to vector (1,-1,1). Another line L2 passes through (2,4,7) and (4,5,10). obtain teh vector lines equations for L1 & L2. Do the two lines intersect, if so at what point?
for L2: r = (2,4,7)+[/B]λ(2,1,3)
The first line is $\ell _{1`} :\left\langle {1, - 1,4} \right\rangle + \lambda \left\langle {1, - 1,1} \right\rangle$
3. Thanks for that Plato, but I am still not sure how to proceed to work out if and where the lines intersect?
4. Originally Posted by nickskely
I am still not sure how to proceed to work out if and where the lines intersect?
Write each line in parametric form using a different parameter:
$\ell _1 :\left\{ \begin{gathered}
x = 1 + \lambda \hfill \\
y = - 1 - \lambda \hfill \\
z = 4 + \lambda \hfill \\
\end{gathered} \right.\;\& \,\ell _2 :\left\{ \begin{gathered}
x = 2 + 2\mu \hfill \\
y = 4 + \mu \hfill \\
z = 7 + 3\mu \hfill \\
\end{gathered} \right.$
Now pick two of the coordinates and solve the equations.
(Be sure you check the solution in the coordinate that was not used.)
5. Thanks, but I am still unclear...
are you saying I need to find the value of λ and μ that will make X (if chosen) equal for both coordiantes?
6. Solve the system: $\begin{gathered} 1 + \lambda = 2 + 2\mu \hfill \\ - 1 - \lambda = 4 + \mu \hfill \\ \end{gathered}$.
But make sure that it also works in $4 + \lambda = z = 7 + 3\mu$.
7. ok, I see now
So I got:
λ = -3
μ = -2
and it also satisfies Z
So is -3,-2 my intersection point?
and what would the Z value be?
-3,-2,0?
thanks again | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9289859533309937, "perplexity": 1187.9769466535993}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122122092.80/warc/CC-MAIN-20150124175522-00028-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/4738/are-the-eigenvalues-of-an-observable-always-1-and-1 | # Are the eigenvalues of an observable always -1 and 1?
What are the necessary & sufficient conditions for a matrix to be an observable, and what is the proof that any such matrix has eigenvalues -1 and 1 (if indeed that is the case)? I ask because in the standard Bell experimental setup the measurement outputs are always -1 or 1.
Possibly related: in a previous question I asked whether the squared absolute values of the eigenvalues of a unitary matrix are always 1 (they are).
(In the case of observables on a qubit, having a repeated eigenvalue makes the observable rather uninteresting, because absolutely all pure states are eigenstates in that case; I'd be tempted to call such an observable 'degenerate' in an informal sense as well in that case — though it is on occasion useful to include $$\mathbf 1$$ in an analysis of things to do with single-qubit observables.)
In the analysis of Bell's Theorem, the reason why the observables are taken to be ones with eigenvalues $$\pm1$$ are conventional. It makes them analogous to Pauli spin operators in particular, and it makes a perfectly mixed state have expectation value $$0$$. Having eigenvalues of $$\pm1$$ also allows the expectation values of the operators to describe a bias towards one of two outcomes, and for expectation values of tensor products to be straightforwardly interpreted as a correlation coefficient of outcomes. You could prove versions of Bell's Theorem for observables with other eigenvalues, but those versions could be derived from Bell's Theorem as its usually stated.
• I'm not talking about projectors, I'm talking about projective measurement operators like $\sigma_z$ and $\sigma_x$. Nov 16 '18 at 20:47
• This is a use of the term 'projective measurement operator' that I'm unfamiliar with. (Which reference are you using?) These are certainly observables, and in measuring an observable one may consider the state to be 'projected' onto an eigenstate, but (a) there is nothing in that description that requires that the eigenvalues be $\pm1$, and (b) one usually describes the action of 'projecting' onto an eigenstate terms of a projector. Nov 16 '18 at 20:57
• No, they don't have to be. (Why should they?) Examples of non-unitary observables are the spin-component operators (eigenvalues $\pm\tfrac12$ for spin-$\tfrac12$ particles; eigenvalues $-1,0,+1$ for spin-$1$ particles; etc.) and usually Hamiltonian operators (whose eigenvalues represent the energy levels of a physical system). Nov 16 '18 at 21:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8874306678771973, "perplexity": 403.5592219056649}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587926.9/warc/CC-MAIN-20211026200738-20211026230738-00526.warc.gz"} |
https://www.physicsforums.com/threads/collision-problem.167788/ | Collision problem
1. Apr 27, 2007
neelakash
1. The problem statement, all variables and given/known data
A ball moving parallel to the y axis undergoes an elastic collision with a parabolic mirror y^2=2px.Prove that no matter where the prompt of impact lies it will arrive at the mirror's focus.
2. Relevant equations
3. The attempt at a solution
I believe this can be done in the same way we prove a result for reflection of light in a parabolic mirror.Please guide me if I am wrong.
2. Apr 27, 2007
Dick
It can be done the same way. Assuming the mass of ball is negligible compared with the mirror.
3. Apr 27, 2007
neelakash
Ok,thank you.
4. Apr 27, 2007
neelakash
Sorry,I need to talk more.In that equiavalent optics problem,we used Fermat's law of least path.Here it is not available.So,how should we proceed?
5. Apr 28, 2007
Dick
The 'least path' law still basically applies in the form of a least action principle. But what you really need is just angle of incidence equals angle of reflection. To see this you have to assume there are no frictional forces involved in the collision. This means the wall can only exert a normal force. So the parallel component of the momentum can't change. Conservation of energy now tells you that the normal component of the momentum must just reverse.
6. Apr 28, 2007
neelakash
That's right.But I cannot see anything using only "law" of reflection.How does it mean that the reflected ball passes through focus(P)?One possibility is that the undeviated ball strikes the directrics and from the definition of parabola,e=1...so,SP=SN.That is (d/dt)SN=(d/dt)SPThis means magnitude of velocity remains unchanged iff the ball reflects along SP...Because,for other points the relation is not true.
I understand the way is hanging around this.But,I cannot develop logically...
by the way,I hope in the same way one can prove for two focii in an ellipse?
7. Apr 28, 2007
HallsofIvy
Staff Emeritus
This can be done by calculating the "slope of the tangent line" to the curve of the mirror (i.e. the derivative).
However, I have a problem with the statement. You give the equation of the parabolic mirror as y2= 2px, a parabola "opening" to the right but say the ball is coming "parallel to the y-axis", vertically. The ball is going to hit the back of the mirror and will bounce AWAY from the focus!
8. Apr 28, 2007
Dick
Magnitude of velocity is always unchanged. You have to look at the angles.
9. Apr 28, 2007
neelakash
HallsofIvy,you are right.Actually,I posted the question directly from a book...When I tried,I started with a the geometrical optics analogy in another...so,I overlooked it.
However,how does the differentiation help?
dy/dx=(2p)^(1/2)*(-1/2)[1/{(x)^(1/2)}]
dick,I could not get anything out of ONLY ANGLES.What about the way suggested?
10. Apr 29, 2007
neelakash
I think I got a way.I have to show GEOMETRICALLY the reflected ray passes through (p/2,0)
11. Apr 30, 2007
neelakash
*First you find the co-ordinates of the point of incidence.
*differentiate y^2 and 2px and equate them and find the slope of the normal in terms of co-ordinates of the point of incidence.
*Now you have incident slope 0,normal's slope known,and reflected ray's slope m,say.
*use standard co-ordinate formula that i=r.This gives the value of m.
*The reflected ray's equation can be found.and it is seen that the line passes through (p/2,0) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8434092402458191, "perplexity": 1065.548433098186}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542414.34/warc/CC-MAIN-20161202170902-00016-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://dimacs.rutgers.edu/TechnicalReports/abstracts/1998/98-33.html | Thin Complete Subsequence
Author: Norbert Hegyvári
ABSTRACT
$A$ is said to be {\em complete} if every sufficiently large integer belongs to the sumset of $A$. $A'$ is {\em thin comlete subsequence} of $A$ if $A'$ is complete and $A'(x)=(1+o(1))\log_2x$.
It is proved that $\lim_{n\to \infty}a_{n+1}/a_n=1$ implies the existence of thin complete subsequence.
Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1998/98-33.ps.gz | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.996614933013916, "perplexity": 623.0974432735455}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187825464.60/warc/CC-MAIN-20171022203758-20171022223758-00763.warc.gz"} |
https://groupprops.subwiki.org/w/index.php?title=Conjugacy_class_size_formulas_for_linear_groups_of_degree_two&oldid=38876 | # Conjugacy class size formulas for linear groups of degree two
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
This article gives formulas for the number of conjugacy classes as well as the sizes of individual conjugacy classes for the general linear group of degree two and some other related groups, both for a finite field of size $q$ and for some related rings.
## For a finite field of size $q$
In the formulas below, the field size is $q$. The characteristic of the field is a prime number $p$. $q$ is a prime power with underlying prime $p$. We let $r = \log_pq$, so $q = p^r$ and $r$ is a nonnegative integer.
### Formulas for number of conjugacy classes
Group Symbolic notation Formula for number of conjugacy classes in case of characteristic two (i.e., $p = 2$)
Examples are $q = 2,4,8,16,\dots$
Formula for number of conjugacy classes in case of odd characteristic (equivalent to $q$ being odd).
Example are $q = 3,5,7,9,11,13,17,\dots$
Degree of polynomial in $q$ More information
general linear group of degree two $GL(2,q)$ or $GL(2,\mathbb{F}_q)$ $q^2 - 1$ $q^2 - 1$ 2 element structure of general linear group of degree two over a finite field
projective general linear group of degree two $PGL(2,q)$ or $PGL(2,\mathbb{F}_q)$ $q + 1$ $q + 2$ 1 element structure of projective general linear group of degree two over a finite field
special linear group of degree two $SL(2,q)$ or $SL(2,\mathbb{F}_q)$ $q + 1$ $q + 4$ 1 element structure of special linear group of degree two over a finite field
projective special linear group of degree two $PSL(2,q)$ or $PSL(2,\mathbb{F}_q)$ $q + 1$ $(q + 5)/2$ 1 element structure of projective special linear group of degree two over a finite field
general semilinear group of degree two $\Gamma L(2,q)$ or $\Gamma L(2,\mathbb{F}_q)$ ? ? 2 (essentially, if we treat $r$ as a constant) element structure of general semilinear group of degree two over a finite field
projective semilinear group of degree two $P\Gamma L(2,q)$ or $P\Gamma L(2,\mathbb{F}_q)$ ? ? 2 element structure of projective semilinear group of degree two over a finite field
general affine group of degree two $GA(2,q)$ or $AGL(2,q)$ or $GA(2,\mathbb{F}_q)$ or $AGL(2,\mathbb{F}_q)$ $q^2 + q - 1$ $q^2 + q - 1$ 2 element structure of general affine group of degree two over a finite field
special affine group of degree two $SA(2,q)$ or $ASL(2,q)$ or $SA(2,\mathbb{F}_q)$ or $ASL(2,\mathbb{F}_q)$ ? $2q + 4$ 1 element structure of special affine group of degree two over a finite field | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 46, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8861660957336426, "perplexity": 263.35643748265954}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314696.33/warc/CC-MAIN-20190819073232-20190819095232-00520.warc.gz"} |
https://www.answers.com/general-science/What_is_basic_unit_of_length_mass_and_volume_in_the_metric_system | 0
What is basic unit of length mass and volume in the metric system?
Wiki User
2015-03-08 17:44:42
basic unit of length mass and volume in the metric system are as follows .
basic unit of length in the metric system is meter .
basic unit of mass in the metric system is kg .
basic unit of volume in the metric system is L.
Wiki User
2015-03-08 17:44:42
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Wiki User
2014-12-19 15:07:31
The unit of length is meter.
The unit of mass is kilogram.
The unit of volume is liter. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9987334609031677, "perplexity": 4385.77145552363}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662531762.30/warc/CC-MAIN-20220520061824-20220520091824-00555.warc.gz"} |
http://math.stackexchange.com/questions/242572/scale-and-ratio-try-to-find-x-y-width-height/242574 | # scale and ratio : try to find x,y,width,height
http://i.stack.imgur.com/LbQSu.png
I have to boxes with same ratio.
How can I find position x,y and width,height in the second box ?
-
add comment
## 1 Answer
Multiplying the small x-value (10) with the ratio between the two heights should do the trick. And likewise for everything else. So x in the larger box would be twice 10*100/50 = 20.
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if another box w=300 , height = 156 ? – l2aelba Nov 22 '12 at 11:39
now i think i know now, just basic math :P , my bad – l2aelba Nov 22 '12 at 11:41
add comment | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8165984749794006, "perplexity": 1971.9193793664288}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999675992/warc/CC-MAIN-20140305060755-00052-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.bbc.co.uk/bitesize/guides/zyt4y4j/revision/3 | # Chemical changes and physical changes
Chemical changes happen when chemical reactions occur. They involve the formation of new chemical elements or .
Physical changes do not lead to new chemical substances forming. In a physical change, a substance simply changes physical , eg from a solid to a liquid.
For example:
Liquid water becoming steam (when water boils) is a physical reaction:
Whereas:
Liquid water into hydrogen and oxygen, eg when an electric current is passed through water, is a chemical reaction:
## Collisions and reactions
For a chemical reaction to happen, must collide with each other. However, not all collisions between particles lead to a chemical reaction.
This may be because of the of the substances involved. For example, gold does not react with oxygen (because gold is so unreactive) even though oxygen particles in the air collide with gold in jewellery.
Iron is a more reactive metal than gold so iron reacts with oxygen. However, the reaction of iron with oxygen in the air (rusting) is slow. This shows that not all collisions between oxygen and iron particles lead to a reaction taking place. The reason for this is that collisions may not happen with enough energy for the reaction to take place. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9682881832122803, "perplexity": 982.7897111313397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670597.74/warc/CC-MAIN-20191120162215-20191120190215-00521.warc.gz"} |
http://math.stackexchange.com/questions/295532/how-to-show-that-for-any-meager-set-a-in-baire-space-there-is-a-nowhere-dense | # How to show that for any meager set $A$ in Baire Space, there is a nowhere dense set $C$, such that $A \subseteq C^*$?
Let $X$ be a subset of $\omega^{\omega}$, $X^{*}$ is defined as:$$\{y:(\exists x\in X,\exists N <\omega)(\forall n >N x(n)=y(n))\}$$ which consists of all sequences in $\omega^{\omega}$ that eventually coincide with some $x$ in $X$. Show that for any set $A \in \omega^{\omega}$ that is meager(countable union of nowhere dense set), there is a nowhere dense set $C$ that $A \subseteq C^*$ .
Here's how far I understand the problem. For any meager set $A$,$A^*$ is also meager,which can be written as a union of non-decreasing sequence of nowhere dense sets $\{C_n:n<\omega, C_n \subseteq C_{n+1}\}$. But I got stuck on how to show that the induced sequence $\{C^{*}_n:n<\omega\}$ is eventually constant. In other words, there exists $N$ such that $C^*_N = A^*$.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9835996031761169, "perplexity": 67.89165665974662}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00207-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://environmath.org/2018/07/30/from-17000-kwh-year-to-less-than-zero-my-experience-with-energy-efficiency-and-rooftop-solar-in-mesa-az/ | From 17,000 kWh/year to less than zero: my experience with energy-efficiency and rooftop solar in Mesa, AZ
I wrote several hundred pages of a book that amounted to exhorting people to alter their own habits of residential energy consumption (turn down the heat, you rogues!), as well as upgrade their built environment (e.g. insulate the attic) and appliances, or even, *gasp*, add solar to their roofs. All this because the numbers, at least in the abstract, showed that such banal acts of conservation (not counting solar) can reduce the carbon footprint of residential energy use by at least 30-50%, from a baseline average of about 12 metric tonnes (1 tonne = 1 metric ton) of CO$_2$-equivalent (CO$_2$e). But a demonstration of how, over the course of roughly 5 years, the net energy use in my house actually fell progressively from around 17,000 kWh of electricity per year down to less than zero (net) seems in order.
I will use this post to emphasize that two-thirds of the total energy reduction, and hence carbon reduction, came from inexpensive or free efficiency/behavioral changes. The last third was then fully offset by a relatively cheap 4.48 kW (<$7,000 after tax incentives) solar system. Thus, efficiency and solar should always be pursued in tandem (in my view), as this reduces overall material and energy use far more effectively than solar alone, and minimizes expenses as well. Electricity Consumption Figure 1. Monthly electricity consumption from January, 2011 through July, 2018, with projected use through 2018 (based upon average energy use in 2017 and relative use in 2018 year-to-date). Figure 2. Yearly electricity consumption and approximate timeline of major accompanying efficiency/behavioral changes. Note that solar (added August, 2017) does not affect energy consumption, but is rather a source of energy generation. Figure 1 shows monthly electricity consumption from January, 2011 through July, 2018 (and projections through the end of 2018), as summarized from previous monthly bills, for my single family house of around 1,500 square feet, with an in-ground swimming pool (the house is all-electric, so no gas, etc. need be added to the monthly energy total). Note that this consumption represents all on-site residential energy use before incorporating any solar generation into our calculations. Given the setting in Mesa, AZ (part of the Phoenix metropolis), it is not at all shocking that there is a cyclical summer peak, with a much smaller winter peak. Now, what accounts for the dramatic drop (about 66%) in energy use over the years (note that both the peaks and baseline consumption fall progressively)? The answer is rather banal, amounting to the following collection of changes (along with approximate time of implementation and my estimate of energy savings): • High efficiency lighting turnover, i.e. CFLs and LEDs (2012/2013, saving 1,000-1,500 kWh/year) • Roughly four degree ($^{\circ}$F) change in the thermostat settings: from 78 to 82$^{\circ}$F in the summer months, and from 70-71 to about 67$^{\circ}$F in the winter, plus setting the thermostat back when absent (Starting around 2014, becoming more ambitious in 2016; saving 2,000-2,500 kWh/year) • Blown-in attic insulation (~December 2014, saving additional 500-1,000 kWh/year) • Single-speed pool pump run-time decreased from 6 hours/day to 3 hours/day (started 2016, saving 1,500 kWh/year) • Replaced single-speed pool pump with variable speed pump (started 2017, saving an additional 1,000 kWh/year) • Ultra-low flow shower-heads (1.5 gpm), general reduction in hot water use (started 2015/2016, saving ~1,500-2,000 kWh/year) • General increase in conscientiousness – turning off TVs, unused appliances, etc. (started 2014/2015, saving 500-1,000 kWh/year) • New AC/Heat pump in 2017 (saving additional 750-1,000 kWh) Figure 2 relates yearly energy consumption with the changes above marked, and we see that, in toto, they result in dramatic energy savings, and theory is validated empirically! In fact, such behavioral and efficiency changes were much more important, overall, than solar. Indeed, simply changing the thermostat (free), replacing the lightbulbs (\$100?, \$150?), installing 1.5 gpm showerheads (\$30), and replacing the pool pump with a variable speed pump (\$750) together saved as much as or more energy than the solar system (discussed in a moment). As a brief aside, residential pools are extremely common in the Phoenix area, even in lower-income housing, and a traditional single-speed pool pump usually represents a massive energy drain, sometimes rivaling even the AC. Reducing run-time, or even better, replacing with a variable-speed pump, can reduce this drain by an astounding 90% (see, e.g., this DOE/NREL report) Solar System and Energy Production Figure 3. Daily net energy for three year-long intervals. After the installation of solar in August, 2017, a net negative value implies more electricity was produced than was consumed. Figure 4. Daily total solar generation and total electricity use since commissioning of the rooftop array almost one year ago (in August, and so the winter months are towards the center of the graph). Most days generation exceeds demand, with peak summer and winter exceptions. Entering into service on August 3, 2017, our 14-panel, 4.48 kW grid-tie solar system has generated 7,576 kWh at this point, just a few days shy of a single year of operation, and about 1,700 kWh more than the 5,870 kWh consumed in the meantime. Note that this is a grid-tie system, meaning that, during the night and when household demand exceeds solar generation, electricity is drawn from the grid, while when local solar generation exceeds demand, the excess energy is diverted to the grid to be used by other electricity customers. Figure 3 gives daily net energy for yearly intervals before significant conservation (2011/2012, as approximated from later daily energy use data), the year interval before solar (2016/2017), and after solar (2017/2018). Figure 4 shows daily solar generation and total household consumption superimposed on each other. Now, while the system is net-negative, I still am interested in how much electricity is drawn from the grid, as solar detractors like to point to the fact that grid-tie systems still rely on the grid. Indeed, some go far as to claim that this amounts to solar customers bilking the system and getting a “free ride.” I find this claim absurd on its face, as I still pay a monthly flat fee (one that is appreciably inflated relative to non-solar customers) for the privilege of being connected to the grid, and pay the retail rate for any energy I use. Nevertheless… Using hourly consumption and generation data available from SRP, in Figure 5 I give an example of a week in late May, showing how during most of the day the local load is covered and excess energy goes to the grid, while at other times there is a load on the grid. Summing hours to days, Figure 6 shows the daily solar energy put on the grid, the daily solar energy consumed locally (i.e. by the house itself), and daily grid energy used. As can be seen, even if all excess solar energy was lost, the solar system covers roughly half of the local load, and so, even giving zero credit for extra energy production, rooftop solar is highly effective at reducing one’s energy/carbon footprint. In my case, the entire house now only draws about 3,000 kWh from the grid per year, an over 80% reduction from the 2011/2012 baseline. Figure 5. Hourly energy generation and consumption for an example week in late May. The red bars give solar energy that is used locally (i.e. within the house), while the green bars are the extra solar generation that is put onto the grid. Energy drawn from the grid is represented by blue bars (grid energy is mainly used in the evening after the sun is down, but the AC is still needed). Figure 6. The top series gives daily solar energy used locally, and below is the solar energy put onto the grid. On the bottom is daily grid electricity consumed. Comparing to Figure 3, note how grid energy used at the summer peak has fallen from almost 100 kWh (in 2011/2012) to under 50 kWh with conservation measures, and finally to less than 20 kWh with rooftop solar added to the mix. Carbon Offset Finally, I ask how many tonnes of CO$_2$e emissions are avoided via these changes. We could look at grid-average emissions, i.e. average the total CO$_2$e generated from electricity production over all kWh generated, and thus include near-zero carbon sources such as nuclear and hydro. Such a grid-average electricity emissions factor (EF) can be expressed with units kgCO$_2$/kWh, as can all other EFs in this discussion. However, generation from non-fossil sources generally does not change with changes in demand, whereas fossil generation, almost exclusively coal and natural gas, does change seasonally and daily to meet demand. Thus, any marginal change in electricity consumption affects marginal generation, which again, derives almost exclusively from coal and gas. As we shall see, changes in load preferentially affect gas generation over coal generation, and so, on average in Arizona and using 2017 data, any change in generation has an associated (lifecycle) marginal emissions factor of 0.818 kgCO$_2$/kWh (per kWh delivered, i.e. including transmission and distribution losses), which is less than the EF for all fossil generation, at about 0.961 kgCO$_2$/kWh delivered, but appreciably greater than the (again, lifecycle) grid-average EF of 0.627 kgCO$_2$e/kWh in the AZNM eGRID subregion (for 2016). It follows that my residence, at negative 1,700 kWh for the year-to-date (just shy of one complete year), has net negative CO$_2$e emissions, at about -1.4 metric tonnes of CO$_2$, based upon the marginal emissions factor. Even using the grid-average, we are net negative by slightly over one tonne CO$_2$e. This is down from around 12 metric tonnes of CO$_2$e in 2012 (using the 2012 edition of eGRID and adjustments as detailed below) on a grid-average basis, or at least 14 metric tonnes of CO$_2$e on a marginal basis (and likely more, given that gas has partially displaced coal in recent years). The derivation of these numbers follows. Methodology for a marginal emissions factor We can derive a marginal emissions factor for Arizona, in terms of CO$_2$e/kWh, using hourly power generation and CO$_2$emissions data from the EPA’s Air Markets Program Data, as previously done by Siler-Evans and colleagues [1]. In a not at all tedious exercise, I have downloaded and processed this data for the AZNM eGRID subregion (the regional grid interconnect expected to provide power for my home), and for Arizona state in particular, using hourly data for all generating units. Note that SRP, the utility company in my area, is much more reliant on coal than both the AZNM subregion and AZ state average, and derived only about 1.2% of its actual retail electricity from solar in 2017, while 59% of came from coal and 11% from natural gas (note that these figures are higher than the 53% coal and 10% gas reported by SRP, as SRP’s reports included 9.49% of retail electricity demand being met by energy efficiency programs; factoring this out to determine actual generation gives my numbers). Since most SRP generating resources are located in Arizona, and the few out-of-state units are mainly coal-fired (e.g. in Colorado), I consider the Arizona state average marginal emissions rates for power generation to be a minimum for the SRP territory. Finally, to determine the fossil electricity emissions rates, I factor in upstream natural gas emissions from methane leakage, assumed at 2.4% of gas by mass (the kgCO$_e$/kWh factor for gas generation is then determined for each generating unit using the hourly thermal efficiency), and I also add 13.2 gCO$_2$/kWh(t) for extraction operations. Similarly, coal mine methane and CO$_2$e from mining operations and coal transport are factored into coal emissions based upon the thermal efficiency of each generating unit (details for these calculations are given in the appendix below). From all this, I arrive at a fossil generation emissions rate for each hour of the year (based on year 2017 data). However, this is still not the marginal emissions factor, which represents the emissions from marginal generating sources that respond to small changes in load. Siler-evans et al. [1] derived the marginal emission factor at time$t$(MEF$_t$) as$\text{MEF}_t = \frac{E_{t+1} – E_t}{G_{t+1} – G_t},$where$E_t$is the emissions total at time$t$and$G_t$is the energy generation total. However, attempting to apply this method to 2017 Arizona data has been problematic for me, as the MEF$_t$can be very sensitive when generation changes are very small (i.e. dividing anything by a very small number gives a large number). Therefore, I use an alternative “min-max” method, whereby I find the minimum and maximum generation for each 8-hour period of the year, and then determine the marginal emissions rate from the emissions and generation difference at these points. I also calculate the fraction of marginal generation provided by gas versus coal within each 8-hour period. Arizona fossil generation and emissions rates Figure 7. Daily fossil (coal and gas) electricity generation in AZ is plotted on the top panel, while the associated lifecycle emissions factor is given at the bottom (note the bottom plot is scaled to facilitate comparison to the somewhat lower marginal EFs in Figure 8). Figure 8. The top panel gives the estimated fractions of marginal electricity generation from coal and gas (note that the fractions must by definition sum to 1). The bottom gives daily marginal EFs for Arizona electricity in 2017. While 54% of all fossil generation was coal in 2017, coal could expected to be the marginal generating source only about 38% of the time (more in winter, less in summer), and so marginal EFs are slightly lower than the average fossil EFs. Looking at Arizona fossil generation, we see a clear seasonal trend, with increased total energy generation in the summer (for fairly obvious reasons), with most of this increase covered by gas generation. Thus, coal is a smaller fraction of the generating mix in summer (total coal generation is still high in summer). Total fossil generation divided by source, and the lifecycle fossil emissions factor for each day of 2017 is given in Figure 7. Using the method above, I also derive 8-hour marginal emissions factors, and marginal generating mixes. This is shown in Figure 8 on a daily basis. Note that these emissions factors are at the generation level, while there is a small loss of energy involved in power transmission and distribution. Therefore, at the consumption level, I adjust for a 4.23% power loss (average T&D loss in AZNM subregion in 2016, per EPA eGRID). Now, given hourly solar generation, hourly grid energy consumption, and (approximated from daily) hourly marginal kgCO$_2$e/kWh emissions rates, we can calculate the marginal emissions over the course of the year attributable to my humble, solar-equipped abode. In summary, I calculate: • Averaged over the year, the lifecycle MEF is 0.784 kgCO$_2$e/kWh at the production level, and 0.818 kgCO$_2$e/kWh at the consumption level (factoring in 4.23% T&D losses). • The net marginal effect, using hourly solar production and grid-energy consumption data (year-to-date), of my residence is a net negative 1.4 tonnes CO$_2$e, from a net negative 1,700 kWh. • If we used grid-average EFs for the AZNM subregion, the carbon impact is still net negative by over 1 tonne CO$_2$e. Note: All figures were created by the author using MATLAB, and may be freely reused with proper attribution to Steffen Eikenberry and a link to this post. Appendix: Upstream/lifecycle emissions from coal and gas Following the methodology presented in Chapter 4 of A Fair Share (by the author), I systematically adjust emissions as reported in the 2016 edition of eGIRD for all non-fossil electricity generation for all EPA eGRID subregions to arrive at a grid-average EF of 0.627 kgCO$_2$e/kWh for the AZNM subregion (including 4.23% T&D loss). Chapter 4 is now available for download as supporting material (note that the 2014 edition of eGRID was used in this work: I have updated all numbers to the 2016 edition used here). Natural gas. Also following Chapter 4, I assume a 2.4% mass leak rate from natural gas systems, appreciably higher than EPA estimates but based upon a large body of literature indicating this is too low (as reviewed in that document), and very similar to the 2.3% leak rate estimated in a new comprehensive assessment just published in Science [2]. I assume 13.2 gCO$_2$/kWh(t) for gas extraction. Calculations also assume a gas density of 0.05 lb/ft$^3$, a heat content of 1,034 Btu/ft$^3$(per the EIA), and combustion emissions of 0.05444 kgCO$_2$/ft$^3$(per EPA). I further assume that methane has a 100-year global warming potential of 36 (IPCC AR5 value with climate-carbon feedback). Together, in terms of thermal energy, this yields upstream emissions of 79.5 gCO$_2$e/kWh(t). From EPA AMPD data the thermal efficiency of any operating unit is calculated, and thus total upstream emissions per kWh of electrical energy output are calculated. As a yearly average, natural gas generation in AZ in 2017 had a thermal efficiency of 44.8%, a direct combustion emission factor of 0.4529 kgCO$_2$/kWh(e), and a lifecycle emissions factor of 0.6305 kgCO$_2$/kWh(e). Coal. Upstream emissions stem mainly from coal mine methane, mining operations, and coal transport. From the 2016 EIA Coal Report, we have 728.4 million short tons of coal mined in the US (660.8 million metric tons), and 19.78 million Btu / short ton coal in 2017 (EIA). The most recent EPA greenhouse gas inventory gives 2,153 Gg coal mine methane in 2016, and all together these figures yield upstream coal mine emissions of 18.4 gCO$_2$e/kWh(t). Jamarillo et al. [3] gave coal mining energy for 1997 as 9.081 million barrels of fuel oil, 1.2 billion cubic feet of gas, 34 million gallons of gasoline, and 49.597 billion kWh electricity. I suppose EFs of 12.504 kgCO2e/gallon fuel oil, a gasoline EF of 11.146 kgCO2e/gallon (from Chapter 3 of A Fair Share), and natural gas density, energy content, and upstream emissions as above. For electricity, I set upstream coal emissions to zero as a conservative measure to avoid double-counting, and the method outlined above for grid-average electricity using 2016 EPA eGRID data to give 0.577 kgCO2e/kWh electricity. Dividing among 9.888 billion tonnes of coal mined in 1997 and using a heat content of 23.193 mmBtu/kg finally yields 5 gCO$_2$e/kWh(t). Also using data reported in Jamarillo et al. suggests an additional 5 gCO$_2$e/kWh(t) for coal transportation. Thus, in sum we get 28.4 gCO$_2$e/kW(t) of upstream coal emissions. Averaged over the year 2017, in AZ, coal generation had a thermal efficiency of 33.6%, a direct combustion emissions factor of 1.0482 kgCO$_2$e/kW(e), and a lifecycle emissions factor of 1.1328 kgCO$_2\$e/kW(e).
[1] Siler-Evans, K., Azevedo, I. L., & Morgan, M. G. (2012). Marginal emissions factors for the US electricity system. Environmental science & technology, 46(9), 4742-4748.
[2] Alvarez, R. A., Zavala-Araiza, D., Lyon, D. R., Allen, D. T., Barkley, Z. R., Brandt, A. R., … & Kort, E. A (2018). Assessment of methane emissions from the US oil and gas supply chain. Science, 361(6398), 186-188.
[3] Jaramillo, P., Griffin, W. M., & Matthews, H. S. (2007). Comparative life-cycle air emissions of coal, domestic natural gas, LNG, and SNG for electricity generation. Environmental science & technology, 41(17), 6290-6296. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8360036015510559, "perplexity": 3426.198951893669}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337631.84/warc/CC-MAIN-20221005140739-20221005170739-00160.warc.gz"} |
https://www.physicsforums.com/threads/angular-frequency-time-and-angle.48500/ | # Angular Frequency, Time, and Angle
1. Oct 18, 2004
### Larry717
pi = 3.14159...
angular frequency = 2(pi)f
theta [in radians] = 2(pi)f t
t = theta / 2(pi)f
For theta = 0, t = 0
For theta = 2(pi), t = 1/f
If 0 =< theta <= 2(pi)
Then 0 =< t <= 1/f
[I'm not sure if the notation above
is correct.]
Is the foregoing true for any frequency?
Larry
2. Oct 18, 2004
### kenhcm
When $$\theta=2\pi$$, one whole cycle has been completed. By definition, the time taken for a whole cycle is the period $$T$$. The relationship between the period and the frequency is $$T=1/f$$.
Note that $$\theta$$ denotes angular displacement. There is no need to constraint the value of $$\theta$$ to be less than $$2\pi$$. When it is larger than $$2\pi$$, it just simply means that it has completed more than one cycle. For instance, when $$\theta=4\pi$$, it has completed two cycles and certainly the time taken will be twice of the period, i.e. $$2T$$.
Best regards,
Kenneth
3. Oct 19, 2004
### Larry717
Taking this a step further
Ken,
Thanks for reviewing the basics about frequency, period, and angle.
I still need some help with angles larger than 2pi.
Take the equation for a sinusoidally varying
electric field:
E = Eo cos(2pift)
In the above, t doesn't have to be the period.
Now, for a hypothetical example, let:
and t = 1s
What does cos(2pift) equal?
My calculator gives an error message. Doesn't
the angle have to be be between 0 and 2pi for
the cosine function to work?
Or, doesn't the product of f and t have to vary
between zero and 1? (same thing as above)
Larry
4. Oct 19, 2004
### arildno
The reason why your calculator gives you an error message, is that the argument you've given it is larger than the maximum argument of the cosine function your calculator has been built to handle.
Basically, you're giving it a major headache, and it responds with a grumpy error message.
5. Oct 19, 2004
### Larry717
Getting Closer!
If I understand you correctly, you are saying that an argument of any size
is ok.
Isn't there an algorithm that can be done by hand or by computer program that can take an agument of any size (an angle of any size) and reduce it to an angle between 0 and 2pi?
Recall that for cos(2pift) the argument (the angle) will always be positive. And, that the cosine function must return a value between -1 and 1.
Given the angle 10^10 radians. Given that there is another angle between
0 and 2pi that will return the same value for the cosine function, is the
smaller angle equivalent to the larger angle?
Larry
6. Oct 19, 2004
### kenhcm
Recall that $$\cos(\theta)$$ is a periodic function of $$\theta$$. The period in this case is $$2\pi$$. This function will repeat itself in the range of $$[0,2\pi], [2\pi,4\pi], [4\pi,6\pi]$$, etc. Therefore, whatever value of $$\theta$$ you have, you will always find a value of $$\theta$$ within $$2\pi$$ so that the value of the function is the same.
Hope that this clarifies your doubt.
Best regards,
Kenneth
7. Oct 20, 2004
### Larry717
Clarification
To illustrate more clearly what I'm after I'd like to switch
from units in radians to units in degrees.
Given theta = 1085 deg.
n = 1085/360 = 3.0138889
phi = 360(n-3) = 5 deg.
cos(phi) = cos(theta)
Now, are the angles 5 deg. and 1085 deg. equivalent?
Larry
8. Oct 20, 2004
### Pyrrhus
Yes, they are equivalent. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8821731805801392, "perplexity": 1148.727651015275}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719784.62/warc/CC-MAIN-20161020183839-00001-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.wiredfaculty.com/board/question/CBSE/11/The+s-Block+Elements?question_id=CBSEENCH11007038&q_type=all&page=42 | The s-Block Elements
• Question 165
## Explain why alkaline earth metals are poor reducing agents as compared to alkali metals.
Solution
The ionisation enthalpies of alkaline earth metals are higher and their electrode potentials are less negative than the corresponding alkali metals, therefore alkaline earth metals are weaker reducing agents than alkali metals.
Question 166
## Explain the trend of solubility of carbonate, sulphates and hydroxides of alkaline earth metals ?
Solution
The solubility of carbonates and sulphates of these metals decreases downward in the group. This is because the magnitude of the lattice enthalpy remains almost constant as the carbonate or sulphate is so big that small increase in the size of the cations from Be to Ba does not make any difference. However, the hydration enthalpy decreases from Be2+ to Ba2+ sufficiently with the increase in their size resulting in the decrease of solubility of their carbonates or sulphates.
The solubility of hydroxides of these metals in water increases downward in the group. This is due to the fact that the lattice enthalpy decreases down the group due to increase in the size of the cation of the alkaline earth metal. On the other hand, the hydration enthalpy of the cations of alkaline earth metals decreases as we go down the group. As a result ∆Hsolution (∆Hlattice – ∆Hhydration) becomes more negative and solubility increases.
Question 167
## Account for the following:(i) Be(OH)2 is amphoteric while Mg(OH)2 is basic.(ii) Be(OH)2 is insoluble but Ba(OH)2 is fairly soluble in water.
Solution
(i) This is because I.E. of Mg < I.E. of Be. So bond M – OH can break more easily in Mg(OH)2 than in Be(OH)2.
(ii) This is because with the increase in size (from Be to Ba), the lattice enthalpy decreases significantly but hydration enthalpy remains almost constant.
Question 168
## The hydroxides and carbonates of sodium and potassium are easily soluble in water while the corresponding salts of magnesium and calcium are sparingly soluble in water. Explain ?
Solution
This is due to the larger size of Na and K. as compared to that of Mg and Ca. As a result, the lattice energies of hydroxides and carbonates of sodium and potassium are much lower than most of the corresponding salts of magnesium and calcium. Consequently, the hydroxides and carbonates of sodium and potassium are easily soluble in water while the corresponding salts of magnesium and calcium are sparingly soluble in water. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8985499739646912, "perplexity": 2282.9068217411777}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154089.68/warc/CC-MAIN-20210731141123-20210731171123-00125.warc.gz"} |
http://www.sciforums.com/threads/is-cern-a-waste-of-money.160980/page-3 | # Is CERN a waste of money ?
Discussion in 'Physics & Math' started by Lostinspace, Jul 5, 2018.
Messages:
35
3. ### billvonValued Senior Member
Messages:
14,523
They detect anything that reflects RF.
5. ### James RJust this guy, you know?Staff Member
Messages:
31,639
Sorry. My statement could be read two ways. I meant what you said just there. I'm not sure about the word "primary" there, but the rest is okay.
No, I don't think so. That gives the impression that there are two kinds of space. Only one is needed.
7. ### Write4UValued Senior Member
Messages:
8,192
I believe that most theories try to show being background independent in order to avoid conflict with QM or GR
8. ### James RJust this guy, you know?Staff Member
Messages:
31,639
That's right. GR has no absolute standard of rest. That's why it's called Relativity.
9. ### Write4UValued Senior Member
Messages:
8,192
Moreover, as I understand it, GR deals with a continuous field (a wave function), whereas QM deals with quanta which are not be suitable in describing a smooth field.
Can't have a quantum field..
?
This is where the LHC at Cern comes into play. It actually is able to generate quantum packets from the underlying smooth fields. Pretty neat trick.
https://en.wikipedia.org/wiki/Lepton
Last edited: Jul 9, 2018
10. ### HaydenRegistered Senior Member
Messages:
110
Thoughts could get into speculations so let us wait for a decade or so to see if aLIGO revolutionize the related science.
11. ### LostinspaceRegistered Member
Messages:
35
I disagree , it gives the impression of a spatial whole that can have ''resident occupant'' fields. I do not think Einstein was considering space at all when he was considering space-time. I interpret space-time is a spatial field that can curve comparative to the background space. If we consider the Dirac sea, Maxwell, the Higg's field , Tesla's electrical Universe and space-time, these theories are seemingly trying to interpret the same thing, the ''stuff'' between masses that has no visual representation in respect to colour. Transparent fields leaving an amount of guess work to do about these 'invisible' fields . Could we maybe consider that space-time is the occupying field content of a BH ?
This then would fit together nice to explain space-time is expanding and the BH volume is increasing over time by maybe enphalpic inflation. Could we also consider that different BH's ''inherit'' the same physics as our own observable BH ''interior''?
Last edited: Jul 9, 2018
12. ### sweetpeaValued Senior Member
Messages:
1,056
Robert Watson Watt got the idea for radar when asked by the British Air Ministry about the possibility of making a '' Death Ray'' .
Late edit: It was Skip Wilkins and Robert Watson Watt.
Last edited: Jul 9, 2018
13. ### billvonValued Senior Member
Messages:
14,523
That's like saying Maxwell was not considering magnetism when he came up with the basic theories of electromagnetism. (Of course, he was considering it.)
It can curve, and it is not separate from background space.
Right - but almost nothing has color. So none of them are unique in that respect. Indeed, the only place we can perceive color is a very narrow section of the EM band.
The fact that all fields are not visible to human eyes (outside of EM 400 to 800 nanometers) does not make them more likely to be "guesswork."
14. ### LostinspaceRegistered Member
Messages:
35
Nonsense, that is suggestive to saying space itself is a ''solid'' with a ''rubber'' like structure. You are mistaken, space by definition is just a vast expanse.
Again nonsense, spatial fields are not observable so it is guesswork.
15. ### billvonValued Senior Member
Messages:
14,523
No it's not. Space is not a solid, nor does it have a rubber-like structure. The rubber sheet model is just a way of visualizing spacetime.
I can observe them with simple tools. You can find these tools in any high school science lab, so it's not hard for anyone to do so.
16. ### LostinspaceRegistered Member
Messages:
35
Firstly you claimed :
After talking nonsense you now decide to contradict yourself, which answer is it?
17. ### originIn a democracy you deserve the leaders you elect.Valued Senior Member
Messages:
10,578
It sounds like you are trying to equate a gravitational field to a field like an electric field, in that an electric field is 'in' space, it is not space curving. Is my interpretation correct?
18. ### LostinspaceRegistered Member
Messages:
35
Field curvature would be the correct interpretation as ''simulated'' in the rubber sheet example, you have the correct understanding of what I meant.
19. ### originIn a democracy you deserve the leaders you elect.Valued Senior Member
Messages:
10,578
That is not really telling if my interpretation of your idea is what you are trying to say. In the rubber sheet analogy that was developed, the rubber sheet IS space-time, there is no 'background' space.
20. ### LostinspaceRegistered Member
Messages:
35
I said ''you have the correct understanding''in the other prior post.
I agree the rubber sheet represents space-time and shows space-time curvature, but if you remove the sheet , that does not remove the space that ''underlays'' the ''overlay'' sheet, the sheet curves comparable to the ''background'' space.
21. ### billvonValued Senior Member
Messages:
14,523
Are you really having this much trouble with a relatively simple concept? Space can curve. It is not a solid. It is not like rubber. It is like space. It's the dimensional framework upon which we locate objects. That framework can be warped by gravitational fields. Fields (like the four fields we know about) can propagage through it. There is no one "zero reference frame" - all frames in spacetime are relative.
22. ### LostinspaceRegistered Member
Messages:
35
Nonsense, you seem to be struggling to comprehend the difference between space and a spatial field that occupies the space. Again contradicting yourself, again claiming space can curve but is not made of anything to curve. Please explain your nonsense I have put in bold.
23. ### billvonValued Senior Member
Messages:
14,523
Space is not "made of anything." There is such a thing as empty space, which is the absence of anything in that space. Space, of course, has characteristics, like the Planck constant, a permittivity, a permeability etc. This does not mean that there's "something there."
If you cannot understand that, we are done here. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8030124306678772, "perplexity": 2169.1330939821896}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743011.30/warc/CC-MAIN-20181116111645-20181116133645-00404.warc.gz"} |
https://astarmathsandphysics.com/index.php?option=com_content&view=article&id=3821:the-mann-whitney-u-test&catid=209&Itemid=1738 | ## The Mann Whitney U Test
The Mann Whitney U test allows the comparison of two independent random samples (1 and 2).
The Mann Whitney U statistic is defined as:
- where samples of sizeandare pooled and Ri are the ranks of sample 1.
U can be resolved as the number of times observations in one sample precede observations in the other sample in the ranking.
In most circumstances a two sided test is required; here the alternative hypothesis is that 1 values tend to be distributed differently to 2 values. For a lower side test the alternative hypothesis is that 1 values tend to be smaller than 2 values. For an upper side test the alternative hypothesis is that 1 values tend to be larger than 2 values.
Assumptions of the Mann-Whitney test:
• random samples from populations
• independence within samples and mutual independence between samples
• measurement scale is at least ordinal
A confidence interval for the difference between two measures of location is provided with the sample medians. The assumptions of this method are slightly different from the assumptions of the Mann-Whitney test:
• random samples from populations
• independence within samples and mutual independence between samples
• two population distribution functions are identical apart from a possible difference in location parameters
Example
The following data represent fitness scores from two groups of boys of the same age, those from homes in the town and those from farm homes.
Farm Boys Town Boys 14.811.1 12.718.9 12.2 17.4 14.2
Pooling and sorting gives 11.1, 12.2, 12.7, 14.2, 14.8, 17.4, 18.9.
The Farm Boys have ranks1, 2 and 5.andso
Comparison with values in statistical tables causes us not to reject the null hypothesis of no difference at the 10% level. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8216168880462646, "perplexity": 1120.7865256014209}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046152129.33/warc/CC-MAIN-20210726120442-20210726150442-00187.warc.gz"} |
https://chenfuture.wordpress.com/2011/04/27/bochner%E2%80%99s-formula/ | ### Bocher’s formula
A colleague of mine mentioned to me today the Bocher’s formula for computing the coefficients of the characteristic polynomial of a matrix. It seems that this formula does not appear too often in textbooks or literature. I’ll just write down the formula and the idea of a simple proof here.
Let the characteristic polynomial of a matrix $A$ be
$\displaystyle{p(\lambda)=\lambda^n+a_1\lambda^{n-1}+\cdots+a_n.}$
Then the coefficients can be computed by
$a_1=-tr(A),$
$a_2=-\frac{1}{2}\left(a_1tr(A)+tr(A^2)\right),$
$a_3=-\frac{1}{3}\left(a_2tr(A)+a_1tr(A^2)+tr(A^3)\right),$
$\vdots$
$a_n=-\frac{1}{n}\left(a_{n-1}tr(A)+\cdots+a_1tr(A^{n-1})+tr(A^n)\right).$
To prove the formula, note that the coefficient $a_j$ is the summation of all possible products of j eigenvalues, i.e.,
$\displaystyle{a_j=(-1)^j\sum_{\{t_1\cdots t_j\}\in C_n^j}\lambda_{t_1}\cdots\lambda_{t_j},}$
where $C_n^j$ denotes the j-combination of numbers from 1 to n, and the trace of $A^i$ is the sum of the $j$th power of the eigenvalues, i.e.,
$\displaystyle{tr(A^i)=\sum_{t=1}^n\lambda_t^i.}$
In addition, we have
$\displaystyle{\left(\sum_{\{t_1\cdots t_j\}\in C_n^j}\lambda_{t_1}\cdots\lambda_{t_j}\right)\left(\sum_{t_{j+1}=1}^n\lambda_{t_{j+1}}^i\right)=\sum_{\{t_1\cdots t_{j+1}\}\in C_n^{j+1}}\lambda_{t_1}\cdots\lambda_{t_j}\lambda_{t_{j+1}}^i+\sum_{\{t_1\cdots t_j\}\in C_n^j}\lambda_{t_1}\cdots\lambda_{t_{j-1}}\lambda_{t_j}^{i+1}.}$
The above indicates that the first part of $(-1)^ja_jtr(A^i)$ cancels the second part of $(-1)^{j-1}a_{j-1}tr(A^{i+1})$, whereas the second part of $(-1)^ja_jtr(A^i)$ cancels the first part of $(-1)^{j+1}a_{j+1}tr(A^{i-1}).$ The rest of proof becomes obvious now.
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https://balbhartisolutions.com/class-7-maths-solutions-chapter-5-practice-set-24/ | Balbharti Maharashtra State Board Class 7 Maths Solutions covers the 7th Std Maths Practice Set 24 Answers Solutions Chapter 5 Operations on Rational Numbers.
## Operations on Rational Numbers Class 7 Practice Set 24 Answers Solutions Chapter 5
Question 1.
Write the following rational numbers in decimal form.
i. $$\frac { 13 }{ 4 }$$
ii. $$\frac { -7 }{ 8 }$$
iii. $$7\frac { 3 }{ 5 }$$
iv. $$\frac { 5 }{ 12 }$$
v. $$\frac { 22 }{ 7 }$$
vi. $$\frac { 4 }{ 3 }$$
vii. $$\frac { 7 }{ 9 }$$
Solution:
i. $$\frac { 13 }{ 4 }$$
ii. $$\frac { -7 }{ 8 }$$
iii. $$7\frac { 3 }{ 5 }$$
iv. $$\frac { 5 }{ 12 }$$
v. $$\frac { 22 }{ 7 }$$
vi. $$\frac { 4 }{ 3 }$$
vii. $$\frac { 7 }{ 9 }$$
Maharashtra Board Class 7 Maths Chapter 5 Operations on Rational Numbers Practice Set 24 Intext Questions and Activities
Question 1.
Without using division, can we tell from the denominator of a fraction, whether the decimal form of the fraction will be a terminating decimal? Find out. (Textbook pg. no. 40)
Solution:
If the prime factorization of the denominator of a fraction has only factors as 2 or 5 or a combination of 2 and 5 then the decimal form of that fractional will be a terminating decimal form.
Consider the fractions $$\frac { 17 }{ 20 }$$ and $$\frac { 19 }{ 6 }$$
Now, 20 = 2 x 2 x 5, and 6 = 2 x 3
∴ $$\frac { 17 }{ 20 }$$ is terminating decimal form while $$\frac { 19 }{ 6 }$$ is recurring decimal form. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9242678880691528, "perplexity": 1301.7020543520794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585911.17/warc/CC-MAIN-20211024050128-20211024080128-00003.warc.gz"} |
https://nrich.maths.org/5576/solution | Exploring Wild & Wonderful Number Patterns
EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.
I'm Eight
Find a great variety of ways of asking questions which make 8.
Dice and Spinner Numbers
If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be each time?
What's in the Box?
Stage: 2 Challenge Level:
We had a good number of solutions sent in with explanations as to how the answers were found. Ansh from Monkfield Park Primary School sent in the following:
For question 1) Answer is 8.
First of all I thought that 7 times 8 will be 56. Then I tried 3 times 8 and that also worked. Next I tried 13 times 8 that was 98 but that did not work so I tried 14 times 8 which was 112 and it was correct. Finally I divided 216 by 8 and ended up with 7.
So that’s how I figured out that the multiplier is 8.
For question 2) Answer is 11
First of all I tried 3 and that did not work so I tried 6 and that did not work either. I also tried 7 and 9 but they both were wrong too. I knew it was not an even number so I skipped 2 and 10. Then I tried 11 and that worked.
So the first one was 13 times 11 = 143.
Next I divided 297 by 11 and the answer was 297.
After that I tried 341 divided by 11 and came up with 31.
Finally I did 1221 divided by 11 which gave me 111.
So the multiplier was 11.
Delaney at Mount Vernon School Maine, U.S. said:
Question 1 Possible factors: 2, 4 or 8
Largest common factor: 8
$56\div2=28$, $28\div2=14$, $14\div2=7$
$24\div2=12$, $12\div2=6$, $6\div2=3$
$112\div2=56$, $56\div2=28$, $28\div2=14$
$216\div2=108$, $108\div2=54$, $54\div2=27$
After I got these answers there were no more common factors and I assumed that 7, 3, 14, and 27 were the original numbers. 8 is the largest common factor because $2^3$ or 2x2x2=8.
Question 2 Only possible factor: 11
None of the numbers were even in this question so I knew that no multiples of two would be a common factor.
143=110+33
297=220+77
1221=1100+121
341=330+11
After dividing into portions divisible by 11, I found the original numbers by dividing by 11.
$143\div11=13$
$341\div11=31$
$1221\div11=111$
$297\div11=27$
There were no more common factors, so I assumed those were the original numbers.
We had this submitted from Jerry at Dulwich College, Shanghai:
First I checked which numbers that 143 can be divided by.
I skipped the multiples of 5 because 143, 297, 341 and 1221 clearly wouldn't be able to be divided by multiples of 5. I first found out 143 can be divided by 11 and 13. But 297 cannot be divided by 13. So I kept working on 11 and all four numbers can be divided by 11. So the winner is 11!
Well, altogether these have been very good responses to this challenge. Keep sending solutions to any other activity you have a go at. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8295406699180603, "perplexity": 615.0907474406126}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042989443.69/warc/CC-MAIN-20150728002309-00211-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.dreamincode.net/forums/topic/337739-algebraic-graph-theory-adjacency-matrix-and-spectrum/ | # Algebraic Graph Theory- Adjacency Matrix and Spectrum
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### #1 macosxnerd101
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# Algebraic Graph Theory- Adjacency Matrix and Spectrum
Posted 05 January 2014 - 06:28 PM
Algebraic Graph Theory- Adjacency Matrix and Spectrum
This tutorial will introduce the adjacency matrix, as well as spectral graph theory. For those familiar with Linear Algebra, the spectrum of a matrix denotes its eigenvalues and their algebraic multiplicities. Spectral Graph Theory deals with the eigenvalues and eigenvectors associated with the Adjacency and Laplacian matrices. Note that the Laplacian matrix will not be explored in this tutorial.
Below is a typeset copy of this tutorial:
The Adjacency Matrix describes the adjacency relation between two vertices; that is, whether or not there exists an edge that is incident to the two vertices. Let A be used to denote the adjacency matrix of a graph. If A describes a simple graph, then Aij = 1 if vertices vi and vj are adjacent. Otherwise, Aij = 0. If the simple graph is undirected, then the matrix is symmetric. That is, if vi is adjacent to vj, then vj is adjacent to vi. This is, in fact, the statement of the Handshake Lemma. If the graph is directed, then the matrix is only symmetric if, for each vertex vi, the in-degree of vi and the out-degree of vi are the same.
Weighted graphs can also be described more precisely using cost matrices. The cost matrix is defined the same way as an adjacency matrix. Rather than an indication of an adjacency, a numeric value is included in the cell to indicate the weight. If a cell has a 0, that means there is no edge connecting two vertices.
Let's now consider some examples.
The Adjacency Matrix has application in counting the number of walks between two vertices of a given length. A walk is a sequence alternating between vertices and edges, starting and ending with vertices. The two vertices on either side of an edge are incident to that edge. Consider some examples based on the graph K5:
Some examples of walks in K5 include:
• v1 - v2 - v5, which is a walk of length 2
• v2 - v4 - v2, which is also a walk of length 2
Now let's explore how the adjacency matrix can be used to count the number of walks of length n in a graph. This is done through matrix exponentiation. That is, let G be a Graph and let A be the corresponding adjacency matrix. Then Anij counts the number of walks of length n between vertices vi and vj in G.
Consider again the example of K5, counting the 2-walks. Let A be the adjacency matrix of K5, with A2 shown below.
From A2, it can be seen that for all walks that are not closed, there are three walks of length 2 in K5. There are four closed walks of length 2 in K5.
Let's examine what exactly is happening. Consider the case when the 2-walk is not closed. For the purpose of illustrating the concept, consider the 2-walks between vertices v1 and v2. Since K5 is a simple graph, there are no loops or multiple edges between two vertices. Thus, there are only three vertices that can connect v1 and v2, without violating the length requirement.
Proof that An describes the number of n-walks in a graph:
The proof is by induction on n a Natural number. Consider the base case of n = 1. So A1 = A. By definition, if vertices vi and vj are adjacent, then Aij = 1. Otherwise, Aij = 0. Thus, A counts the number of walks of length 1 in the graph.
The theorem will be assumed true up to an arbitrary integer k, and proven true for the k+1 case. By definition of matrix multiplication, Ak+1 = Ak * A. By the inductive hypothesis, Ak counts the number of k-walks in the graph, and A counts the number of walks of length 1 in the graph. Consider Aijk+1 = \sum_{x=1}^{k} Aixk * Axj. Thus, for each x, if Aixk has a non-zero value, then there exists at least one walk of length k from vertex vi to vx. Similarly, if Axj = 1, then there exists an edge connecting vertices vx and vj. If both values are non-zero, there exists Aixk walks of length k+1 from vi to vj. Thus, the k+1 case has been shown.
Thus, by the principle of mathematical induction, An describes the number of n-walks between two vertices in a graph.
Spectral Graph Theory
This section will briefly introduce the topic of Spectral Graph Theory. At this point, it is assumed that the reader is familiar with Eigentheory and Matrix Diagonalization. The spectrum of the graph provides numerous insights into graph properties, that are otherwise difficult to ascertain and prove. This section will discuss some of those properties.
Trace of a Graph
The first property to explore deals with the trace of the adjacency matrix. The trace of the matrix, denoted tr(M), is defined as the sum of the diagonal entries of a matrix. Consider an adjacency matrix shown above, all of which are simply graphs. It is easy to see that their traces are 0, as the diagonal entries are 0. So tr(A(K5)) = 0 since the diagonal entries are 0.
The trace of a matrix can also be written as the sum of the eigenvalues for the matrix. Consider a diagonalizable matrix A, written as A = QDQ-1 (through diagonalization). The trace has the property that given a product of matrices, their cyclic permutations do not change the result. So tr(A) = tr(QDQ-1) = tr(Q-1QD) = tr(DQ-1Q). It follows that since QQ-1 = Q-1Q = I, that tr(A) = tr(D), which means that tr(A) is equal to the sum of the eigenvalues of A.
This property that tr(A) = \sum_{i=1}^{n} lambda_{i}, where lambda_{i} is an eigenvalue of A, is useful in counting edges and triangles in a graph. Consider the Handshake Lemma, which states that the sum of the vertex degrees is equal to twice the number of edges (\sum_{i} deg(v_{i}) = 2E). The trace-eigenvalue property provides a combinatorial proof for the Handshake Lemma. Consider A2, the square of the adjacency matrix of the given simple graph. The diagonal entries of A2 are the closed 2-walks of the graph. The only way for a 2-walk on a simple graph to be closed is, starting at vi, go to an adjacent vertex vj, then return to vi. Thus, vi is adjacent to vj if and only if vj is adjacent to vi. So a closed 2-walk on vi counts the number of edges incident to vi (ie., the degree of vi). Consequently, the diagonal entry of A2 corresponding to the closed 2-walks of vj also counts the same edge incident to vj and vi, so the edge is counted twice. Thus, tr(A2) = tr(D * D) = \sum_{i=1}^{n} lambda_{i}^{2} = 2E. In other words, if lambda_{i} is an eigenvalue for A, then lambda_{i}^{2} is an eigenvalue for A2 and the sum of all the lambda_{i}^{2} counts the number of edges in the graph.
Through a similar argument, it is possible to count the number of triangles in a graph. Consider A3, the cube of the adjacency matrix. Its diagonal entries describe the closed 3-walks of the graph. On a simple graph, these are restricted to triangles (C3). Thus, tr(A3) = \sum_{i=1}^{n} lambda_{i} = 6C3. Here, each edge in a given triangle is double counted, in a similar manner as the edges. Thus, double counting three edges returns 6C3.
It should be noted that the trace-eigenvalue property does scale, but with a nuance. Certainly tr(An) for n > 3 counts the number of closed n-walks in the graph. However, there are multiple ways to achieve closed n-walks; whereas with closed walks of length 1-3, there is exactly one way to accomplish each on a simple graph.
Characteristic Polynomial
Just as the trace of the adjacency matrix provides tools for examining properties of the graph, the characteristic polynomial in and of itself also provides the same information. Consider the characteristic polynomial of a graph, f(x) = det(A - xI) = xn + \sum_{i=1}^{n-1} ci xn-i. From the calculation on the determinant, the ci coefficients are the sum of the ith principal minors of the matrix A. More exactingly, ci = (-1)i * \sum_{j} pj, where pj is the jth principal minor of A whose corresponding principal matrix is of dimension i x i. A principal minor is the determinant of a principal matrix, which is a square sub-matrix whose top-left element is on the diagonal of the parent matrix. Additional rows and columns are removed from the parent matrix in the formation of a principal matrix.
Let's explore exactly how the principal minors behave with respect to adjacency matrices (of simple graphs). On the surface, it looks like a lot of linear algebra without a lot of graph theory. While the linear algebra is definitely there, the graph theory is underneath the surface. Consider an example with the graph C4. Below is C4, along with its adjacency matrix and distinct 2x2 principal matrices.
It is easy to see the 1x1 principal minors are all 0's, as the diagonal entries are 0's. So clearly c1 = tr(A) = 0 for simple graphs. Now consider the 2x2 principal minors of C4. There are six 2x2 principal matrices, with the distinct ones listed above. The first 2x2 principal matrix is noted to indicate an adjacency. Let's examine why this is. The first time this principal matrix appear is with A11, keeping the second column. Thus, from the image, it is easy to see that there is an edge between vertices v1 and v2. The given principal matrix is also the adjacency matrix of P2, the Path graph on two vertices, or an edge with two vertices. Thus, clearly, if there is no path of length one between two vertices, then there is no adjacency. Hence, the other two 2x2 principal matrices clearly cannot indicate an adjacency, or an edge. Thus, the sum of the 2x2 principal minors counts the edges in the graph, with -c2 = |E(G)|. Using C4 as an example, there are four edges, indicated by the principal matrices starting at A11 keeping the second row and second and third columns; A22 keeping the fourth row and fourth column; and A33 keeping the fourth row and fourth column.
The idea is the same in looking for triangles in a graph. Simply look for the 3x3 principal matrices which are the same as the adjacency matrix for C3, which is shown below. The determinant of the adjacency matrix of C3 is 2. All other 3x3 principal minors have a determinant of 0. Thus, adding up all the 3x3 principal minors produces the twice the count for the number of triangles in the graph. More exactingly, c3 = -2 C3. There are no triangles in C4, so c3 = 0 in the characteristic polynomial of C4.
Conclusion
This tutorial served as a brief introduction into the broad topic of spectral graph theory. Hopefully it has provided a better appreciation and understanding of the various graph properties that can be explored through use of the graph's spectrum.
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https://stats.stackexchange.com/questions/223050/mcmc-for-probit-logit-model-with-some-1s-flipped-to-0s | MCMC for Probit/Logit model with some 1's flipped to 0's
I would like help constructing a sampler for the following model, which is the latent variable interpretation of either logistic or probit glm (doesn't matter which one to me), with a small twist: there is a probability $p$ that "successes" are flipped to failures after the fact.
Model:
Let $X$ represent the design matrix. Let $\vec{y}$ represent a vector of binary responses
$z = X \vec{\beta} + \vec{\epsilon}$
$\epsilon_i \sim Logis(0,s)$ or $\epsilon_i \sim Normal(0,\phi)$
$y_i = 1$ if $z_i > 0$ AND $rbernoulli(p) == 0$, $y_i = 0$ otherwise
$p \sim beta(\alpha,\beta)$
$\beta_j \sim Normal(0,0.001)$
Whether or not an observation is flipped from 1 to 0 is independent of an observation's "ability" (X values), conditional on our other parameters. Observations may not be flipped from 0 to 1.
Toy Example:
At the University of Foo, statistics students are selected to receive the Bar Award based on things such as academic achievement, extra-curricular participation, and community service. Professors convene and decide to give a certain number of students the award, writing the awardees' names on separate slips of paper, and handing them to the department head to give out the awards. However, as they often are, this department head is quite clumsy and, at random, loses some of the deserving students' names.
We need to build a model to predict student success, taking into account that some of the "losers" are deserving of the award. Last year, we found that 3 of the 20 original awardee's names were lost. Using a Jeffreys' prior with binomial likelihood to estimate the probability of droppage, we develop a $Beta(3.5,17.5)$ posterior on $p$.
I would love to see references to papers where something like this is done, or some help in constructing a sampler for this problem. My own efforts have not lead anywhere.
My Math:
As requested, I am putting here my attempt at a solution.
Above, I have the prior for $\beta$ as a normal. It doesn't really matter to me what it is, so long as it is not very informative. My attempts will use marginal Jeffreys' priors for $\beta$ and $\phi$:
$P(\beta) \propto 1$
$P(\phi) \propto 1/\phi$
These are the Jeffreys priors for regular old linear models, I assume I can use them in the latent varaible model as well. Please inform me if I am wrong.
$$(\vec{z} - X \vec{beta}) \sim N(0,\phi I)$$
$$P(\vec{z},\vec{\beta},\phi | X,y) \propto | \phi I| ^{1/2} e^{-1/2 (z - X \beta)' (\phi I)(z - X\beta)}$$
$$\propto | \phi I| ^{1/2} e^{-\phi/2 (z' - \beta' X')(z - X\beta)}$$
$$\propto | \phi I| ^{1/2} e^{-\phi/2 (z'z - \beta'X'z - z'X\beta + \beta'X'X\beta)}$$
$$\propto | \phi I| ^{1/2} e^{-\phi/2(\beta'X'X\beta - 2z'X\beta )}$$
It would seem $\beta$ is normally distributed conditional on $z$ and $\phi$.
I am not sure how to tie all of this to $y$ and $p$.
• Have you tried finding the posterior distribution for this? – Greenparker Jul 12 '16 at 21:11
• @Greenparker are you referring to the conditional posteriors for each parameter? I'm afraid I'm fairly new to bayesianism, and did not get very far at all trying to work the math out, to the point where I felt it was better omitted from the post. – John Madden Jul 12 '16 at 21:21
• I am referring to the posterior, not the conditional posterior. This will help you understand (1) Is the posterior intractable. Because if the posterior turns out to be a known distribution, you don't need MCMC(!). It is likely, the posterior is not a known distribution. (2) It will force you to write down the parameters of the posterior, and understand the quantities you are dealing with. (3) Most often we use Metropolis-Hastings to sample from the posterior, which needs the kernel os the posterior. So you will have to write it down. That would be the first step. – Greenparker Jul 12 '16 at 21:26
• Alright, I'll give her another go and edit the OP with my progress. – John Madden Jul 12 '16 at 21:30
• @Greenparker OK I gave it a go – John Madden Jul 12 '16 at 22:39
There are two steps to this problem: 1) Obtaining an unbiased estimate of $\beta$ from the fully observed model and 2) Estimating the probability model for the error generating mechanism.
In general the approach to handling these types of problems is using the EM algorithm. A few assumptions are necessary to gain any traction. Assumptions might be along the lines of 1) what is the actual probability model for so called random flipping? Is it an unobserved additive model? Is it a mixture model? Does it depend on other observed factors? Is it even an estimable model? 2) what is the expected number of true positives in the $y$ variable? Is this known? Should it have been a fixed amount, or consistent with some target? 3) What is the known conditional distribution of the $X$ variables given $y$?
So the problem as stated is not defined sufficiently, but below an approach is outlined.
If we assume $X$ is normally distributed conditional upon $Y$, and that the label flipping is done completely at random, a clustering based approach is outlined as follows. Using the distribution of $X$ where $y$ is 0, one would look for probabilistic outliers estimated from a normal mixture model, and presume a small cluster of high risk X values here should belong to the $Y=1$ group. After flipping labels, one would re-estimate $\beta$ for predictions of risk, then re-run the algorithm iteratively until convergence.
An example in practice is simulated here:
## example
set.seed(1234)
n <- 1000
p <- 0.3
e.prob <- 0.05 ## error rate, completely at random model
y.orig <- rbinom(1000, 1, p)
x <- numeric(1000)
x[y.orig == 1] <- rnorm(sum(y.orig), 10, 2)
x[y.orig == 0] <- rnorm(sum(1-y.orig), 3, 3)
## true beta
f <- glm(y ~ x, family=binomial)
# int: -9.2, slope: 1.2
err <- rbinom(1000, 1, e.prob)
y.obs <- y.orig
y.obs[err==1] <- 0
hist(x[y.obs==1], col=rgb(1,0,0,.5), breaks = -10:20, ylim=c(0, 120))
hist(x[y.obs==0], col=rgb(0,1,0,.5), add=T, breaks=-10:20, ylim=c(0, 120))
## initialize em:
y.working <- y.obs
mu <- tapply(x, y.working, mean)
sig <- tapply(x, y.working, var)
## loglikelihood
loglik.prev <- sum(dnorm(x[y.working==0], mean=mu[1], sd = sig[1], log=T))+sum(dnorm(x[y.working==1], mean=mu[2], sd = sig[2], log=T))
itr <- 0
repeat({
itr <- itr+1
## classify least likely negative X as a false negative value
minLL.y0 <- which.min(dnorm(x[y.working==0], mean=mu[1], sd=sig[1], log=T))
index<-which(y.working==0)[minLL.y0]
y.working[index] <- 1
## evaluate updated likelihood, stop if it gets worse
mu <- tapply(x, y.working, mean)
sig <- tapply(x, y.working, sd)
loglik.curr <- sum(dnorm(x[y.working==0], mean=mu[1], sd = sig[1], log=T))+sum(dnorm(x[y.working==1], mean=mu[2], sd = sig[2], log=T))
if(loglik.curr < loglik.prev) {
y.working[index] <- 0 ## set the old value back since that was the ML value
break()
} else {
loglik.prev <- loglik.curr
}
})
table(y.working, y.obs)
table(y.working, y.orig)
par(mfrow=c(2,1))
hist(x[y.obs==1], col=rgb(1,0,0,.5), breaks = -10:20, ylim=c(0, 120), main='Observed')
hist(x[y.obs==0], col=rgb(0,1,0,.5), add=T, breaks=-10:20, ylim=c(0, 120))
hist(x[y.working==1], col=rgb(1,0,0,.5), breaks = -10:20, ylim=c(0, 120), main='Corrected')
hist(x[y.working==0], col=rgb(0,1,0,.5), add=T, breaks=-10:20, ylim=c(0, 120))
legend('topright', bty='n', pch=22, pt.bg = c(rgb(1,0,0,.5), rgb(0,1,0,.5)), c('Positives', 'Negatives'))
3 / 12, not very good really
• Very cool work. Playing around with your code, it seems that this "adjustment" gets the parameter estimates closer to the truth. Thanks a bunch! This serves me as a very cool intro to EM type algorithms. – John Madden Jul 13 '16 at 2:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8067798614501953, "perplexity": 1740.7856921077391}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400217623.41/warc/CC-MAIN-20200924100829-20200924130829-00224.warc.gz"} |
https://cs.stackexchange.com/questions/11043/number-of-possible-search-paths-when-searching-in-bst/49523 | # Number of possible search paths when searching in BST
I have the following question, but don't have answer for this. I would appreciate if my method is correct :
Q. When searching for the key value 60 in a binary search tree, nodes containing the key values 10, 20, 40, 50, 70, 80, 90 are traversed, not necessarily in the order given. How many different orders are possible in which these key values can occur on the search path from the root node containing the value 60?
(A) 35 (B) 64 (C) 128 (D) 5040
From the question, I understand that all nodes given have to be included in traversal and ultimately we have to reach the key, 60. For example, one such combination would be :
10, 20, 40, 50, 90, 80, 70, 60.
Since we have to traverse all nodes given above, we have to start either with 10 or 90. If we start with 20, we will not reach 10 (since 60 > 20 and we will traverse right subtree of 20)
Similarly, we cannot start with 80, because we will not be able to reach 90, since 80>60, we will traverse in left sub tree of 80 & thus not reaching 90.
Lets take 10. The remaining nodes are 20, 40, 50, 70, 80, 90. Next node could be either 20 or 90. We cannot take other nodes for same earlier mentioned reason.
If we consider similarly, at each level we are having two choices. Since there are 7 nodes, two choices for first 6 & no choice for last one. So there are totally
$2*2*2*2*2*2*1$ permutations = $2^6$ = $64$
1. Is this a correct answer?
2. If not, whats the better approach?
3. I would like to generalize. If $n$ nodes are given then total possible search paths would be $2^{n-1}$
If looking for the key 60 we reach a number $K$ less than 60, we go right (where the larger numbers are) and we never meet numbers less than $K$. That argument can be repeated, so the numbers 10, 20, 40, 50 must occur along the search in that order.
Similarly, if looking for the key 60 we reach a number $K$ larger than 60, we go leftt (where the smaller numbers are) and we never meet numbers larger than $K$. Hence the numbers 90, 80, 70 must occur along the search in that order.
The sequences 10, 20, 30, 40, 50 and 90, 80, 70 can then be shuffled together, as long as their subsequences keep intact. Thus we can have 10, 20, 40, 50, 90, 80, 70, but also 10, 20, 90, 30, 40, 80, 70, 50.
We now can compute the number, choosing the position of large and small numbers. See the comment by Aryabhata. We have two sequences of 4 and 3 numbers. How many ways can I shuffle them? In the final 7 positions I have to choose 3 positions for the larger numbers (and the remaining 4 for the smaller numbers). I can choose these in $7 \choose 3$ ways. After fixing these positions we know the full sequence. E.g., my first example has positions S S S S L L L the second has S S L S L L S.
You ask for a generalization. Always the $x$ numbers less than the number found, and the $y$ numbers larger are fixed in their relative order. The smaller numbers must go up, the arger numbers must go down. The number is then $x+y \choose y$.
PS (edited). Thanks to Gilles, who noted that 30 is not in the question.
• I would surely like to try. Since no.s 90,80,70 has to be together, lets consider them as a single no. and it can be placed in among 6 places : _ 10 _ 20 _ 30 _ 40 _ 50 _ So that's $2^6$ If by same analogy, the no.s [10,20,30,40,50] can be placed in 4 places, that's $2^4$ But it has to be divided by common combinations which are occurring (which I am not able to figure out) – avi Apr 5 '13 at 11:19
• @avi No, they do not have to be together, only in that order: 10, 20, 90, 30, 40, 80, 70, 50 is OK. – Hendrik Jan Apr 5 '13 at 11:40
• @avi: Try thinking this way: Big and Small. Now you have 8 spots, with 5 Small and 3 Big. How do you fill them? 8 choose 3. Which comes to 56, and I presume is what Hendrik got too. – Aryabhata Apr 5 '13 at 18:58
• @HendrikJan There was no 30 in the original question, there were only 7 values. And 7 choose 3 is (A). – Gilles Apr 5 '13 at 20:07
• @HendrikJan - can you explain this to me : Always the $x$ numbers less than the number found, and the $y$ numbers larger are fixed in their relative order – avi Apr 6 '13 at 15:41
We will convert Moves to Text. It is given that During Search we have Traversed these nodes
as it can be seen that Red ones are bigger than 60 and blue ones are smaller than 60.
Path to node 60 has involved those nodes. So, one of the possible solution to the problem is $$\{S,S,S,S,L,L,L\}$$ any other solution will contains these moves only. coz at a time on a node we can get directions as S or L on comparison and since its given that those nodes were encountered it means directions were picked from that set.
Hence, total number of possible solutions = all Permutations of that set, which is given by $$\frac{7!}{4! \times 3!} = 35$$ answer = option A | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.827804684638977, "perplexity": 396.6328420520781}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526818.17/warc/CC-MAIN-20190721020230-20190721042230-00504.warc.gz"} |
https://math.stackexchange.com/questions/1801632/references-on-the-moduli-space-of-flat-connections-as-a-symplectic-reduction | # References on the moduli space of flat connections as a symplectic reduction
In their Yang Mills equations over Riemann surfaces paper, Atiyah & Bott famously remark that the moduli space of flat connections on a principal bundle over a compact orientable surface may be obtained as a symplectic reduction on the space of connections (with the curvature as the moment map). In that paper, however, it is only a passing remark. One must be careful about this construction since the spaces involved are infinite-dimensional (e.g. the space of connections). I've only found references which ignore this technical detail and present the ideas in analogy to the finite-dimensional case. While they are helpful to give a simple understanding of what's going on, I was wondering if there is a reference that actually goes through the infinite-dimensional analysis to formalize this process. Any recommendation is appreciated.
• How formal is formal? If you're being really careful you need to start by building a theory of what an infinite-dimensional manifold is. Unless they need to do some actual analysis on this space it's generally less helpful to actually work with Hilbert manifolds of $L^2_k$ connections etc, since usually this Symplectic reduction picture is more valuable as a picture than as a theorem. But if you're really looking for the theorem I can try to write down the details. – user98602 May 27 '16 at 5:17
• @MikeMiller, a sketch of how to formalize it would be helpful. I'm writing a bachelor thesis about this and would like to include some idea of the analysis behind it. I know there are several books that treat infinite-dimensional manifolds (there is one by Palais which I've heard is good), so perhaps they cover the necessary background? – Pedro May 27 '16 at 12:20
Some remarks on infinite-dimensional manifolds. There are two approaches which to me still feel very manifold-ish (there are others yet: Frolicher spaces and diffeological spaces, which feel a bit less so); the approach of Banach manifolds and the much more general approach of Frechet manifolds.
The former is almost precisely the same theory as the theory of finite-dimensional smooth manifolds and is a harmless technical addition. The latter is much more complicated, and I would be hesitant to make any actually technical claims about it, as I am not expert in Kriegl-Michor (the lovely book you cited). The advantages of Frechet manifold theory are that the most obvious and naturally occuring manifolds are Frechet manifolds (spaces of smooth mappings; diffeomorphism groups), and that if you honestly want a theory of infinite-dimensional Lie groups, Banach manifolds will not do (it is a theorem that no Banach Lie group can act transitively and effectively on a finite-dimensional smooth manifold; in particular no large group of diffeomorphisms can be given a Banach Lie group structure).
On the other hand, you lose the technical simplicity of Banach manifolds and a bit of the power of the inverse function theorem. If one wants to do analysis, the inverse function theorem is indispensible, which is why one sees lots of Sobolev completions flying around. There's Nash-Moser but I don't understand it very well and do not think it has the wide applicability of the straight-up inverse function theorem.
Let $\Sigma$ be a surface equipped with a Riemannian metric $g$. Fix a $U(n)$-bundle $E \to \Sigma$, and importantly, fix a smooth connection $A_0$ on $E$. (Whenever I say a connection on $E$, I expect it to preserve the unitary structure; it should be valued in $\mathfrak u(n)$, not just $\mathfrak{gl}(n,\Bbb C)$.) Define the space of $L^2_k$ connections, $\mathcal A_k(E)$, to be the set of connections $A$ on $E$ such that $A-A_0 \in \Omega^1(\text{End}(E))$ is an $L^2_k$ 1-form. (Note that, to make sense of $L^2_k$, I need a connection on $\text{End}(E) \otimes \Lambda^j T^*M$ - but this is provided by my base connection $A_0$ and my metric $g$.) This is topologized precisely so that it is affine over the space of $L^2_k$ sections of $\text{End}(E) \otimes T^*M$ with the $L^2_k$ topology. Henceforth I will suppress the $L^2_k$ in notation. We also have a group $\mathcal G_{k+1}$, of $L^2_{k+1}$ automorphisms of $E$. (These are sections of a principal bundle instead of sections of a bundle, so one needs to be careful about what we mean about $L^2_{k+1}$ section: by Sobolev embedding, $L^2_k$ sections are automatically continuous for $k \geq 0$, though one needs larger $k$ for larger-dimensional base, after which it is easy to make sense of the derivatives being $L^2_k$ and whatnot). This acts on $\mathcal A_k$ in the standard way. We need to increase regularity for the gauge group because its action involves taking a derivative.
Because $\mathcal A_k(E)$ is an affine space over $\Omega^1(\text{End}(E))$, its tangent space at every point $A$ is $\Omega^1(\text{End}(E))$. Slightly more subtle is the tangent space of $\mathcal G_{k+1}(E)$, which is $\Omega^0(\text{End}(E))$. Now as in Atiyah-Bott, we can put a symplectic structure on $\mathcal A_k(E)$, preserved by the action of $\mathcal G_{k+1}(E)$: $\omega(v,w) = \int [v \wedge w]$, where by the inside of the integral I mean the composition of the maps $\wedge: \Omega^1(\text{End}(E)) \to \Omega^2(\text{End}(E) \otimes \text{End}(E))$, and $\Omega^2(\text{End}(E) \otimes \text{End}(E)) \to \Omega^2(B)$ (this is just a fiberwise inner product).
Here is the first infinite-dimensional subtlety. This is nondegenerate in the sense that the induced map $\omega^*: T_x \mathcal A_k \to T_x^* \mathcal A_k$ is injective, but it is not an isomorphism! This is essentially because we're taking the $L^2$ inner product of $L^2_k$ forms, and the dual under the $L^2$ inner product is not $L^2_k$ - it's $L^2_{-k}$. But for the sake of having a symplectic form, this is fine.
Atiyah-Bott gives an argument that the curvature map $F: \mathcal A_k \to \Omega^2_{k-1}(\text{End}(E))$ is the moment map. This argument is perfectly valid. Now in the setting of Hilbert manifolds, the regular value theorem (or transverse intersection theorems etc) is still valid; since you're interested in flat connections, you're interested in particular in $F^{-1}(0)$. Why is $0$ a regular value? Because the derivative of $F$ at $A$ is just the derivative map $d_A$, we're asking why the differential is surjective at any flat connection. Well... it's not. At a flat connection the cokernel of $d_A$ is canonically isomorphic to the $H^2(\text{End}(E);d_A)$, the de Rham cohomology with differential from flat connection $A$. Poincare duality (and the self-duality of $\text{End}(E)$) implies that this is the same as $H^0(\text{End}(E);d_A)$: the dimension of parallel sections of $\text{End}(E)$. How big this dimension is says how reducible the connection $A$ is; as an example, for $G = SU(2)$, there are three possibilities: you could be fully reducible, where $H^0$ is 3-dimensional; you could be a $U(1)$-reducible, where $H^0$ is 1-dimensional; you could be irreducible, where $H^0$ is zero-dimensional. Our notion of transversality needs to have reducibility baked into it.
If $H$ is the centralizer of some subgroup of $G$, $H$ is a possible group to reduce to - in the sense that the holonomy group of a connection can be $H$. We can stratify the space (not linearly, if the structure of these subgroups is complicated) $\mathcal A$ into subsets $\mathcal A_H$ of connections reducible to $H$. On each of these, $F$ is constant rank. So $F^{-1}(0)$ is a stratified space, of which each stratum a manifold.
Unfortunately, we're now in trouble. The quotient by the action of $\mathcal G$ - canonically identified with $\text{Hom}(\pi_1 X, G)/G$ - is not usually a manifold. Consider, for instance, $X = T^2$ and $G = SU(2)$; then this quotient is the "pillowcase", a sphere with four singular points. On the irreducible part of the moduli space, you get a smooth structure and a symplectic structure by the above method (the same way you usually would; we've more or less passed all the technicalities). I think the people who spend a lot of time thinking about this representation variety have an appropriate sort of structure on it even at the singular points, but I'm not really sure what it is. For further reading, you could look at this thesis, which I haven't but looks nice.
• I think a sense of adventure could lead one to work out a lot of the basic ideas about this symplectic quotient even in the singular setting, but I got a bit tired by the end of this, and at the moment am not so adventurous. – user98602 May 27 '16 at 15:34
• Thank you for the detailed answer. I'll try to make as much sense of it as I can. For now, some basic questions: what do you mean by $L^2_k$? When you write $\Omega^1(\text{End}(E))$ you really mean $\Omega^1(M, \text{End}(E))$ (i.e. forms in $M$ with values in $\text{End}(E)$)? I've actually been reading the thesis you've mentioned. It's nice, but doesn't really go into any of the details as above. – Pedro May 27 '16 at 23:32
• @Pedro $L^2_k$ is the notation I like for the Sobolev space $W^{2,k}$ or $H^k$, whichever you prefer: $L^2$ functions with $k$ derivatives that are also $L^2$ functions. You can make sense of this in terms of sections of a bundle over a manifold as long as the manifold has a Riemannian metric and the bundle has a metric and connection; then you define the $L^2_k$ norm on smooth sections to be $$\|\sigma\|^2 = \sum_{i=0}^k \|\nabla^k \sigma\|^2_{L^2},$$ where here $\nabla^k \sigma$ is a section of $E \otimes (T^*M)^k$ (I'm tensoring in $k$ copies of the cotangent bundle). – user98602 May 27 '16 at 23:54
• Then one says that the space of $L^2_k$ sections is the completion of the space of smooth sections under this norm; one could just as well make it an inner product if they like. If you're willing to work with coordinate charts and local trivializations you could get honest $L^2_k$ functions (literal functions) as opposed to elements of a completion, but my taste is to avoid coordinates. Yes, I make a (mild, and I think standard) abuse of notation by ignoring the base manifold in my notation for $\Omega^*(M,E)$. – user98602 May 27 '16 at 23:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9250180125236511, "perplexity": 193.98837319266713}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000367.74/warc/CC-MAIN-20190626154459-20190626180459-00512.warc.gz"} |
http://link.springer.com/article/10.1007%2Fs00422-002-0362-x | Biological Cybernetics
, Volume 87, Issue 5, pp 383-391
First online:
# Converging evidence for a simplified biophysical model of synaptic plasticity
• Harel Z. ShouvalAffiliated withInstitute for Brain and Neural Systems and Department of Physics Brown University, 02912
• , Gastone C. CastellaniAffiliated withPhysics Department and Dimorfipa, Bologna University, Bologna 40121, Italy
• , Brian S. BlaisAffiliated withInstitute for Brain and Neural Systems and Department of Physics Brown University, 02912
• , Luk C. YeungAffiliated withInstitute for Brain and Neural Systems and Department of Physics Brown University, 02912
• , Leon N CooperAffiliated withInstitute for Brain and Neural Systems and Department of Physics Brown University, 02912
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## Abstract.
Different mechanisms that could form the molecular basis for bi-directional synaptic plasticity have been identified experimentally and corresponding biophysical models can be constructed. However, such models are complex and therefore it is hard to deduce their consequences to compare them to existing abstract models of synaptic plasticity. In this paper we examine two such models: a phenomenological one inspired by the phenomena of AMPA receptor insertion, and a more complex biophysical model based on the phenomena of AMPA receptor phosphorylation. We show that under certain approximations both these models can be mapped on to an equivalent, calcium-dependent, differential equation. Intracellular calcium concentration varies locally in each postsynaptic compartment, thus the plasticity rule we extract is a single-synapse rule. We convert this single synapse plasticity equation to a multi-synapse rule by incorporating a model of the NMDA receptor. Finally we suggest a mathematical embodiment of metaplasticity, which is consistent with observations on NMDA receptor properties and dependence on cellular activity. These results, in combination with some of our previous results, produce converging evidence for the calcium control hypothesis including a dependence of synaptic plasticity on the level of intercellular calcium as well as on the temporal pattern of calcium transients. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8104453682899475, "perplexity": 3354.111035153308}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701163729.14/warc/CC-MAIN-20160205193923-00063-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/91141-row-operations-solve-system-equations-print.html | # Row operations to solve a system of equations
• May 30th 2009, 08:42 AM
rpatel
1 Attachment(s)
Row operations to solve a system of equations
Hi
I need help in solving a question involving row operations by converting the equations in a matrix.
i have attached the question
thank you
(Wink)
• May 30th 2009, 12:17 PM
NonCommAlg
Quote:
Originally Posted by rpatel
Hi
I need help in solving a question involving row operations by converting the equations in a matrix.
i have attached the question
thank you
(Wink)
write the augmented matrix of the system and do the row operations until your matrix is in an echelon form, which is: $\begin{bmatrix}1 & 1 & 1 & | & 1 \\ 0 & -1 & -3 & | & -1 \\ 0 & 0 & a-4 & | & 0 \end{bmatrix}.$ you should be able to finish the proof now. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.90667325258255, "perplexity": 536.5689243951391}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720973.64/warc/CC-MAIN-20161020183840-00101-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/sum-of-iid-random-variables-and-mgf-of-normal-distribution.730427/ | Sum of IID random variables and MGF of normal distribution
1. Dec 29, 2013
Luna=Luna
If the distribution of a sum of N iid random variables tends to the normal distribution as n tends to infinity, shouldn't the MGF of all random variables raised to the Nth power tend to the MGF of the normal distribution?
I tried to do this with the sum of bernouli variables and exponential variables and didn'treally get anywhere with either.
Does anyone know if this is even possible and where I can find the proof steps?
2. Dec 29, 2013
Stephen Tashi
You have to specify what type of convergence you're talking about when you say "tends". ( http://en.wikipedia.org/wiki/Convergence_of_random_variables)
The sum of iid random variables doesn't converge (in distribution) to a normal distribution. It's the mean of the sum that converges to a normal distribution. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8653596639633179, "perplexity": 185.10243329044542}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267859923.59/warc/CC-MAIN-20180618012148-20180618032148-00378.warc.gz"} |
http://math.stackexchange.com/questions/199223/solving-normal-distribution-probability | # Solving Normal Distribution Probability
The mean length of 600 stainless steel sticks is 181mm and the standard deviation is 60mm.Assuming that the length is normally distributed,
1) find the probability that a randomly chosen stick is between 150 and 190mm in length.
2)Given that the length of a particular stick is more than 195mm, find the conditional probability that is actual length exceeds 210mm.
I already solve part (1). For part 2 i dont know. Could someone help me out
-
– You Sep 19 '12 at 16:58
not sure. It says conditional probability – David Sep 19 '12 at 16:59
Let $A$ be the event "greater than $195$" and let $B$ be the event "greater than $210$." We want $\Pr(B|A)$. By a standard formula, $$\Pr(B|A)=\frac{\Pr(A\cap B)}{\Pr(A)}.$$
Note that $\Pr(B\cap A)$ is just $\Pr(B)$. If you solved the first problem then you know how to find $\Pr(A)$ and $\Pr(B)$.
Remark: Intuitively, $\Pr(A)$ is the area under a certain "tail" of the normal. Our answer $\dfrac{\Pr(B)}{\Pr(A)}$ is just the ratio of the area past $210$ to the area past $195$.
So this is my steps : P(X>210|X>195)= P(Z>0.483|Z>0.233)=$$\frac{P(Z>0.483)}{P(Z>0.233)}$$.And i got the answer 0.7712. Is my answer right?? – David Sep 19 '12 at 17:26
$\Pr(A)$ is the probability of being more than $14$ above the mean, so it is $\Pr(Z\gt 14/60$. Now use normal tables or software. – André Nicolas Sep 19 '12 at 17:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9511954188346863, "perplexity": 261.3419575084491}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122108378.68/warc/CC-MAIN-20150124175508-00017-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/95327-sequence-problem-print.html | # sequence problem!
• July 16th 2009, 07:54 PM
calculus?!
sequence problem!
(a) Suppose that {an} is a convergent sequence of points all in [0, 1] (ie.
an∈[0, 1] for all n). Prove that limit of an as n-->∞ is also in [0, 1].
(b) Find a convergent sequence {an} of points all in (0, 1) such that limit of an as n-->∞
is not in (0, 1).
• July 16th 2009, 08:09 PM
Jose27
1) Let $\{ a_n \}$ $\subset [0,1]$ such that $a_n \rightarrow a \in \mathbb{R}$ then for all $\epsilon > 0$ there exists an $N$ such that if $n>N \vert a_n -a \vert < \epsilon$ then let $\vert a \vert \leq \vert a-a_n \vert + \vert a_n \vert < \epsilon + \vert a_n \vert \leq \epsilon + 1$ and so $\vert a \vert < 1 + \epsilon$ for all $\epsilon >0$, and so $\vert a \vert \leq 1$. In a similar fashion you prove that $a$ can't be negative.
2) Let $a_n =\frac{1}{n}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.995583176612854, "perplexity": 482.6877356928373}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500815991.16/warc/CC-MAIN-20140820021335-00121-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://libbi.org/papers | # Papers
Please cite the following paper if you use LibBi:
• L. M. Murray, Bayesian state-space modelling on high-performance hardware using LibBi, 2013. [arXiv]
The following papers may be of interest, which detail novel contributions related to the development of LibBi:
• L. M. Murray, S. Singh, P. E. Jacob and A. Lee. Anytime Monte Carlo. 2016. [arXiv].
• Del Moral, P. & Murray, L. M. Sequential Monte Carlo with highly informative observations, 2014. [arXiv]
• L. M. Murray, A. Lee, and P. E. Jacob. Parallel resampling in the particle filter, 2014. [arXiv]
• P. E. Jacob, L. M. Murray, and S. Rubenthaler. Path storage in the particle filter, 2013. Statistics and Computing, to appear. [doi] [arXiv]
• L. M. Murray, GPU acceleration of Runge-Kutta integrators. IEEE Transactions on Parallel and Distributed Systems, vol. 23, pp. 94-101, 2012. [doi]
Papers related to specific applications of LibBi are usually referenced in their respective package, some of which are available in the examples. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8809536695480347, "perplexity": 4802.077845394264}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500095.4/warc/CC-MAIN-20230204075436-20230204105436-00116.warc.gz"} |
https://www.physicsforums.com/threads/reading-through-jackson-gauss-theorem.608306/ | # Reading through Jackson: Gauss Theorem
1. May 23, 2012
### thelonious
1. The problem statement, all variables and given/known data
I'm reading through Jackson and ran into the following:
An application of Gauss's theorem to
∇'$^{2}$G=-4πδ(x-x')
shows that
$\oint$($\partial$G/$\partial$n')da'= -4∏
where G is a Green function given by 1/|x-x'| + F, and F is a function whose Laplacian is zero.
(Sec. 1.10, Formal Solution of Electrostaic Boundary Value Problem)
2. Relevant equations
Divergence theorem?
Gauss's theorem?
3. The attempt at a solution
I don't see how to arrive at the surface integral. This looks a bit like an application of the divergence theorem because of the surface integral term. It also looks something like Gauss's law in differential form. Is this what the author means by applying Gauss's theorem?
2. May 23, 2012
### vela
Staff Emeritus
Nothing too mysterious going on. Gauss's law tells us
$$\int_V \nabla\cdot (\nabla G)\,dv = \oint_S [(\nabla G)\cdot\hat{n}]\,dS$$ The integrand of the surface integral is simply the directional derivative in the $\hat{n}$ direction, which is equal to ∂G/∂n, where n is the coordinate along the direction of $\hat{n}$.
3. May 23, 2012
### thelonious
Thanks -- what was I thinking... G is a 1/r potential...
Similar Discussions: Reading through Jackson: Gauss Theorem | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9317936897277832, "perplexity": 981.853999227514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102307.32/warc/CC-MAIN-20170816144701-20170816164701-00397.warc.gz"} |
http://mathhelpforum.com/algebra/215057-probability-distinct-roots.html | # Math Help - Probability of Distinct roots.
1. ## Probability of Distinct roots.
k is uniformly chosen from the interval (-5,5) . Let p be the probability that the quartic f(x)=kx^4+(k^2+1)x^2+k has 4 distinct real roots such that one of the roots is less than -4 and the other three are greater than -1. Find the value of 1000p.
2. ## Re: Probability of Distinct roots.
let y=x2
Your equation is $ky^2+(k^2+1)y+k$
If you solve this equation for y the value of x will be $\pm y^{1/2}$ So y is greater than or equal to zero if the roots are to be zero.
If the roots are distinct
$y=0$ is not a root and $(k^2+1)^2-4$ is not equal to zero
Find the range of values of k so that one root is below -4 and the others are above -1 and express that range as a proportion of (-5,5) to get a probability. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9405595660209656, "perplexity": 218.6802842085994}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375094662.41/warc/CC-MAIN-20150627031814-00298-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://www.computer.org/csdl/trans/tc/1990/02/t0282-abs.html | Subscribe
Issue No.02 - February (1990 vol.39)
pp: 282-287
ABSTRACT
<p>A universal lower-bound technique for the size and other implementation characteristics of an arbitrary Boolean function as a decision tree and as a two-level AND/OR circuit is derived. The technique is based on the power spectrum coefficients of the n dimensional Fourier transform of the function. The bounds vary from constant to exponential and are tight in many cases. Several examples are presented.</p>
INDEX TERMS
spectral lower bound technique; decision trees; two-level AND/OR circuits; arbitrary Boolean function; power spectrum coefficients; n dimensional Fourier transform; Boolean functions; Fourier transforms; logic circuits; trees (mathematics).
CITATION
Y. Brandman, A. Orlitsky, J. Hennessy, "A Spectral Lower Bound Technique for the Size of Decision Trees and Two-Level AND/OR Circuits", IEEE Transactions on Computers, vol.39, no. 2, pp. 282-287, February 1990, doi:10.1109/12.45216 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8923625349998474, "perplexity": 2958.892445334241}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246640001.64/warc/CC-MAIN-20150417045720-00183-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://www.thefullwiki.org/Transfer_function | Transfer function: Wikis
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
Encyclopedia
A transfer function (also known as the network function) is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a (linear time-invariant) system. With optical imaging devices, for example, it is the Fourier transform of the point spread function (hence a function of spatial frequency) i.e. the intensity distribution caused by a point object in the field of view.
Explanation
The transfer function is commonly used in the analysis of single-input single-output filters, for instance. It is mainly used in signal processing, communication theory, and control theory. The term is often used exclusively to refer to linear, time-invariant systems (LTI), as covered in this article. Most real systems have non-linear input/output characteristics, but many systems, when operated within nominal parameters (not "over-driven") have behavior that is close enough to linear that LTI system theory is an acceptable representation of the input/output behavior.
In its simplest form for continuous-time input signal x(t) and output y(t), the transfer function is the linear mapping of the Laplace transform of the input, X(s), to the output Y(s):
$Y(s) = H(s)\;X(s)$
or
$H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} }$
where H(s) is the transfer function of the LTI system.
In discrete-time systems, the function is similarly written as $H(z) = \frac{Y(z)}{X(z)}$ (see Z transform) and is often referred to as the pulse-transfer function.
Direct derivation from differential equations
Consider a linear differential equation with constant coefficients
$L[u] = \frac{d^nu}{dt^n} + a_1\frac{d^{n-1}u}{dt^{n-1}} + \dots + a_{n-1}\frac{du}{dt} + a_nu = r(t)$
where u and r are suitably smooth functions of t, and L is the operator defined on the relevant function space, that transforms u into r. That kind of equation can be used to constrain the output function u in terms of the forcing function r. The transfer function, written as an operator F[r] = u, is the right inverse of L, since L[F[r]] = r.
Solutions of the homogeneous equation L[u] = 0 can be found by trying u = eλt. That substitution yields the characteristic polynomial
$p_L(\lambda) = \lambda^n + a_1\lambda^{n-1} + \dots + a_{n-1}\lambda + a_n\,$
The inhomogeneous case can be easily solved if the input function r is also of the form r(t) = est. In that case, by substituting u = H(s)est one finds that L[H(s)e st] = est if and only if
$H(s) = \frac{1}{p_L(s)}, \qquad p_L(s) \neq 0.$
Taking that as the definition of the transfer function[1] requires to carefully disambiguate between complex vs. real values, which is traditionally influenced by the interpretation of abs(H(s)) as the gain and -atan(H(s)) as the phase lag.
Signal processing
Let $x(t) \$ be the input to a general linear time-invariant system, and $y(t) \$ be the output, and the bilateral Laplace transform of $x(t) \$ and $y(t) \$ be
$X(s) = \mathcal{L}\left \{ x(t) \right \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} x(t) e^{-st}\, dt$
$Y(s) = \mathcal{L}\left \{ y(t) \right \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} y(t) e^{-st}\, dt$.
Then the output is related to the input by the transfer function $H(s) \$ as
$Y(s) = H(s) X(s) \,$
and the transfer function itself is therefore
$H(s) = \frac{Y(s)} {X(s)}$ .
In particular, if a complex harmonic signal with a sinusoidal component with amplitude $|X| \$, angular frequency $\omega \$ and phase $\arg(X) \$
x(t) = Xejωt = | X | ejt + arg(X))
where X = | X | ejarg(X)
is input to a linear time-invariant system, then the corresponding component in the output is:
$y(t) = Ye^{j\omega t} = |Y|e^{j(\omega t + \arg(Y))}$
and Y = | Y | ejarg(Y).
Note that, in a linear time-invariant system, the input frequency $\omega \$ has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system. The frequency response $H(j \omega) \$ describes this change for every frequency $\omega \$ in terms of gain:
$G(\omega) = \frac{|Y|}{|X|} = |H(j \omega)| \$
and phase shift:
$\phi(\omega) = \arg(Y) - \arg(X) = \arg( H(j \omega))$.
The phase delay (i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is:
$\tau_{\phi}(\omega) = -\begin{matrix}\frac{\phi(\omega)}{\omega}\end{matrix}$.
The group delay (i.e., the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency $\omega \$,
$\tau_{g}(\omega) = -\begin{matrix}\frac{d\phi(\omega)}{d\omega}\end{matrix}$.
The transfer function can also be shown using the Fourier transform which is only a special case of the bilateral Laplace transform for the case where s = jω.
Common transfer function families
While any LTI system can be described by some transfer function or another, there are certain "families" of special transfer functions that are commonly used. Typical infinite impulse response filters are designed to implement one of these special transfer functions.
Some common transfer function families and their particular characteristics are:
Control engineering
In control engineering and control theory the transfer function is derived using the Laplace transform.
The transfer function was the primary tool used in classical control engineering. However, it has proven to be unwieldy for the analysis of multiple-input multiple-output (MIMO) systems, and has been largely supplanted by state space representations for such systems. In spite of this, a transfer matrix can be always obtained for any linear system, in order to analyze its dynamics and other properties: each element of a transfer matrix is a transfer function relating a particular input variable to an output variable.
References
1. ^ The transfer function is defined by 1 / pL(ik) in, e.g., Birkhoff, Garrett; Rota, Gian-Carlo (1978). Ordinary differential equations. New York: John Wiley & Sons. ISBN 0-471-05224-8. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 27, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8917590379714966, "perplexity": 483.8247154204749}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948512121.15/warc/CC-MAIN-20171211033436-20171211053436-00667.warc.gz"} |
http://tex.stackexchange.com/questions/17482/how-can-you-address-the-size-of-surrounding-parentheses-whose-size-is-determined | # How can you address the size of surrounding parentheses whose size is determined by \left and \right? [duplicate]
Possible Duplicate:
How to automatically resize the vertical bar in a set comprehension?
Trying to type the formula for a conditional probability like this
\mathrm{P} \left( whatever \mid whatever \right) = ...,
I'd like to have \mid the same size as the left and right parentheses.
How can I achieve this?
-
## marked as duplicate by Juan A. Navarro, Leo Liu, lockstep, Caramdir, Alan MunnMay 7 '11 at 14:24
@Martin: Thanks for correcting the layout! – j.p. May 5 '11 at 15:05
use \middle| or for a double \middle|\middle|
$\left( \frac{a}{b^2}+1\middle| f(x)\right)$
-
Arg, 12sec to late ... I would have used \middle\vert, but \middle| seems to do the same. – Martin Scharrer May 5 '11 at 15:07
I was 29 seconds too late... – Gonzalo Medina May 5 '11 at 15:11
@Herbert: Thanks, works fine (I just have to adjust the spaces around the | a little)! – j.p. May 5 '11 at 15:13
@Martin: Sorry, but you could have been a little less altruistic and first answer the question before beautifying it. – j.p. May 5 '11 at 15:15
@Herbert: \vert is the same as |, \Vert is the same as \| – egreg May 5 '11 at 16:04
\middle doesn't add spaces around the symbol it produces and treats it as an ordinary symbol, like when we say \bigg| or even \bigg(. On the other hand it can't be put inside \mathrel.
The problem might arise when when \middle is used in a subscript; one can say either of
\left( something \nonscript\;\middle|\nonscript\; something \right)
\left( something \mathrel{}\middle|\mathrel{} something \right)
and the result will be the same as if we used \mid (of course if we decide that this vertical bar must be treated as a relation symbol). This is for writing macros that use the \middle feature, as in direct input one knows what space is necessary.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8979573845863342, "perplexity": 2307.8450754338774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701159654.65/warc/CC-MAIN-20160205193919-00131-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/124761/one-question-about-the-quandle | # One question about the quandle
Given a finite quandle $Q$, for any knot $K$ one can associate an invariant, i.e. the number of proper colorings $p(K)$. Let us consider the inverse $K^{-1}$ and mirror image $K'$ of $K$. My queston is, is there any relation between $p(K^{-1})$ $(p(K'))$ and $p(K)$? In general, with an appropriate finite quandle, can we distinguish a knot from its inverse (mirror image) by counting the number of proper colorings?
It seems that the number of proper colorings is exactly the number of homomorphisms from the knot quandle of $K$ to the quandle $Q$ (trivial homomorphism corresponds to trivial coloring). Hence my question concerns the relation between the knot quandle of $K^{-1}$ $(K')$ and the knot quandle of $K$. I only know that the knot quandle is a complete invariant for unoriented knots.
Any hints are welcome.
-
A stronger question would be whether knot quandles are residually finite (distinguished by homomorphisms onto finite quandles). – Daniel Moskovich Mar 17 '13 at 8:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9388913512229919, "perplexity": 289.5394474078365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645359523.89/warc/CC-MAIN-20150827031559-00074-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.iag.uni-hannover.de/632.html | # Abstract Torelli
Massey products and Fujita decomposition
On the Hodge bundle $f_*\omega_{S/B}$ attached to a fibration $f:S\to B$ over a projective curve $B$ there are two {\em Fujita decompositions}: the first one $f_*\omega_{S/B}=\mathcal{O}^{ \oplus q_f}\oplus E$ splits a trivial bundle $\mathcal{O}^{\oplus q_f}$ of rank equal to the relative irregularity $q_f$, while the second one splits $f_*\omega_{S/B}=U\oplus A$ into a unitary flat bundle $U$ and an ample bundle $A$. The monodromy of $U$ can be infinite in general, as Catanese and Dettweiler show with explicit examples. In this talk we first give a description of the structure of $U$ in terms of "tubular" closed holomorphic forms and then we discuss how a vanishing condition on "Massey products" (also known as adjoint images) forces the monodromy of $U$ to be finite. This is a joint work with Gian Pietro Pirola. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9752964377403259, "perplexity": 343.98670289351514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867140.87/warc/CC-MAIN-20180525160652-20180525180652-00084.warc.gz"} |
http://www.ck12.org/book/CK-12-Chemistry---Intermediate/r25/section/21.3/ | <meta http-equiv="refresh" content="1; url=/nojavascript/"> Acid and Base Strength | CK-12 Foundation
You are reading an older version of this FlexBook® textbook: CK-12 Chemistry - Intermediate Go to the latest version.
# 21.3: Acid and Base Strength
Created by: CK-12
## Lesson Objectives
• Explain the difference between a strong acid or base and a weak acid or base.
• Write equilibrium expressions for the ionizations of weak acids and weak bases. Explain how the value of Ka or Kb relates to the strength of the acid or base.
• Calculate an unknown Ka or Kb from the solution concentration and the pH.
• Calculate the expected pH of a solution containing a given concentration of a weak acid or base that has a known Ka or Kb value.
## Lesson Vocabulary
• acid ionization constant
• base ionization constant
• strong acid
• strong base
• weak acid
• weak base
### Recalling Prior Knowledge
• What is meant by the ionization (or dissociation) of a polar or ionic compound that is dissolved in water?
• How is the pH of a solution containing a strong acid or strong base determined?
Some acids, such as sulfuric acid and nitric acid, are strong acids. Working with strong acids requires special care and protective clothing and eyewear. Other acids, such as acetic acid and citric acid, are encountered in many foods and beverages. These weak acids are considerably less dangerous. In this lesson, you will learn about the differences between strong and weak acids and bases.
## Strong and Weak Acids and Bases
Acids are classified as either strong or weak, based on the extent to which they ionize in water. A strong acid is an acid which is completely ionized in aqueous solution. As we saw earlier, hydrogen chloride (HCl) is a gas as a pure compound, but it ionizes into hydrogen ions and chloride ions upon being dissolved into water.
$\mathrm{HCl}(g) \rightarrow \mathrm{H^+}(aq) + \mathrm{Cl^-}(aq)$
The directional arrow in the equation above indicates that the HCl reactant is converted completely into the product ions. In other words, there is 100% ionization of HCl.
A weak acid is an acid that ionizes only slightly in aqueous solution. Acetic acid, the acid found in vinegar, is a very common weak acid. Its ionization is shown below.
$\mathrm{CH_3COOH}(aq) \rightleftharpoons \mathrm{H^+}(aq) + \mathrm{CH_3COO^-}(aq)$
The ionization of acetic acid is incomplete, and so the equation is shown with a double arrow, indicating equilibrium between the reactant and products. The extent of ionization varies for different weak acids, but it is generally less than 10%. A 0.10 M solution of acetic acid is only about 1.3% ionized, meaning that the equilibrium strongly favors the reactants.
Weak acids, like strong acids, ionize to yield the H+ ion and a conjugate base. In the examples so far, the conjugate bases are the chloride ion (Cl) and the acetate ion (CH3COO). Because HCl is a strong acid, its conjugate base (Cl) is extremely weak. The chloride ion is essentially incapable of taking the H+ ion off of H3O+ and becoming HCl again. In general, the stronger the acid, the weaker its conjugate base. Likewise, the weaker the acid, the stronger its conjugate base. The Table below shows a listing of some common acids and their conjugate bases. The acids are listed from strongest at the top to weakest at the bottom, while the order for the conjugate bases is reversed.
Relative Strengths of Acids and their Conjugate Bases
Acid Conjugate Base
Strong Acids
HCl (hydrochloric acid) Cl (chloride ion)
H2SO4 (sulfuric acid) HSO4 (hydrogen sulfate ion)
HNO3 (nitric acid) NO3 (nitrate ion)
Weak Acids
H3PO4 (phosphoric acid) H2PO4 (dihydrogen phosphate ion)
CH3COOH (acetic acid) CH3COO (acetate ion)
H2CO3 (carbonic acid) HCO3 (hydrogen carbonate ion)
HCN (hydrocyanic acid) CN (cyanide ion)
The top three acids in the table above (Table above) are strong acids and are thus all 100% ionized in solution. The others are weak acids and are only slightly ionized. Phosphoric acid is stronger than acetic acid, so it is ionized to a greater extent. Acetic acid is stronger than carbonic acid, and so on.
### The Acid Ionization Constant, Ka
The ionization for a generic weak acid, HA, can be written in one of two ways.
$\mathrm{HA}(aq) + \mathrm{H_2O}(l) \rightleftharpoons \mathrm{H_3O^+}(aq) + \mathrm{A^-}(aq)$
$\mathrm{HA}(aq) \rightarrow \mathrm{H^+}(aq) + \mathrm{A^-}(aq)$
The water molecule is omitted for simplicity in the second case. This will be the way that acid ionization equations will be written for the remainder of the chapter. Because the acid is weak, an equilibrium expression can be written. An acid ionization constant (Ka) is the equilibrium constant for the ionization of an acid.
$\mathrm{K_a=\dfrac{[H^+][A^-]}{[HA]}}$
The acid ionization represents the fraction of the original acid that has been ionized in solution. Therefore, the numerical value of Ka is a reflection of the strength of the acid. Weak acids with relatively higher Ka values are stronger than acids with relatively lower Ka values. Because strong acids are essentially 100% ionized, the concentration of the acid in the denominator is nearly zero and the Ka value approaches infinity. For this reason, Ka values are generally reported only for weak acids.
The Table below is a listing of acid ionization constants for several acids. Note that polyprotic acids have a distinct ionization constant for each ionization step. Each successive ionization constant for a polyprotic acid is always smaller than the previous one.
Acid Ionization Constants at 25°C
Name of Acid Ionization Equation Ka
Sulfuric acid
$\mathrm{H_2SO_4 \rightleftharpoons H^+ + HSO^-_4}$
$\mathrm{HSO^-_4 \rightleftharpoons H^+ + SO^{2-}_4}$
very large
1.3 × 10−2
Oxalic acid
$\mathrm{H_2C_2O_4 \rightleftharpoons H^+ + HC_2O^-_4}$
$\mathrm{HC_2O^-_4 \rightleftharpoons H^+ + C_2O^{2-}_4}$
6.5 × 10−2
6.1 × 10−5
Phosphoric acid
$\mathrm{H_3PO_4 \rightleftharpoons H^+ + H_2PO^-_4}$
$\mathrm{H_2PO^-_4 \rightleftharpoons H^+ + HPO^{2-}_4}$
$\mathrm{HPO^{2-}_4 \rightleftharpoons H^+ + PO^{3-}_4}$
7.5 × 10−3
6.2 × 10−8
4.8 × 10−13
Hydrofluoric acid $\mathrm{HF \rightleftharpoons H^+ + F^-}$ 7.1 × 10−4
Nitrous acid $\mathrm{HNO_2 \rightleftharpoons H^+ + NO_2^-}$ 4.5 × 10−4
Benzoic acid $\mathrm{C_6H_5COOH \rightleftharpoons H^+ + C_6H_5COO^-}$ 6.5 × 10−5
Acetic acid $\mathrm{CH_3COOH \rightleftharpoons H^+ + CH_3COO^-}$ 1.8 × 10−5
Carbonic acid
$\mathrm{H_2CO_3 \rightleftharpoons H^+ + HCO^-_3}$
$\mathrm{HCO^-_3 \rightleftharpoons H^+ + CO^{2-}_3}$
4.2 × 10−7
4.8 × 10−11
Hydrocyanic acid $\mathrm{HCN \rightleftharpoons H^+ + CN^-}$ 4.9 × 10−10
### The Base Ionization Constant, Kb
As with acids, bases can either be strong or weak, depending on their extent of ionization. A strong base is a base which ionizes completely in aqueous solution. The most common strong bases are soluble metal hydroxide compounds, such as potassium hydroxide. Some metal hydroxides are not as strong because they are not as soluble. For example, calcium hydroxide is only slightly soluble in water. However, the portion that does dissolve completely dissociates into ions.
A weak base is a base that ionizes only slightly in aqueous solution. Recall that a base can be defined as a substance that accepts a hydrogen ion from another substance. When a weak base such as ammonia is dissolved in water, it accepts an H+ ion from water, forming the hydroxide ion and the conjugate acid of the base, the ammonium ion.
$\mathrm{NH_3}(aq) + \mathrm{H_2O}(l) \rightleftharpoons \mathrm{NH^+_4}(aq) + \mathrm{OH^-}(aq)$
The equilibrium greatly favors the reactants, and the extent of ionization of the ammonia molecule is very small.
An equilibrium expression can be written for the reactions of weak bases with water. Again, because the concentration of liquid water is essentially constant, the water is not included in the expression. A base ionization constant (Kb) is the equilibrium constant for the ionization of a base. For ammonia, the expression is:
$\mathrm{K_b=\dfrac{[NH^+_4][OH^-]}{[NH_3]}}$
The numerical value of Kb is a reflection of the strength of the base. Weak bases with relatively high Kb values are stronger than bases with relatively low Kb values. The Table below lists base ionization constants for several weak bases.
Base Ionization Constants at 25°C
Name of Base Ionization Equation Kb
Methylamine $\mathrm{CH_3NH_2 + H_2O \rightleftharpoons CH_3NH^+_3 + OH^-}$ 5.6 × 10−4
Ammonia $\mathrm{NH_3 + H_2O \rightleftharpoons NH^+_4 + OH^-}$ 1.8 × 10−5
Pyridine $\mathrm{C_5H_5N + H_2O \rightleftharpoons C_5H_5NH^+ + OH^-}$ 1.7 × 10−9
Acetate ion $\mathrm{CH_3COO^- + H_2O \rightleftharpoons CH_3COOH + OH^-}$ 5.6 × 10−10
Fluoride ion $\mathrm{F^- + H_2O \rightleftharpoons HF + OH^-}$ 1.4 × 10−11
Urea $\mathrm{H_2NCONH_2 + H_2O \rightleftharpoons H_2CONH^+_3 + OH^-}$ 1.5 × 10−14
Notice that the conjugate base of a weak acid is also a weak base. For example, the acetate ion has a small tendency to accept a hydrogen ion from water to form acetic acid and the hydroxide ion.
## Calculations with Ka and Kb
The numerical value of Ka or Kb can be determined by experiment. A solution of known concentration is prepared, and its pH is measured with an instrument called a pH meter (Figure below).
A pH meter is a laboratory device that provides quick, accurate measurements of the pH of solutions.
Sample Problem 21.4 shows the steps involved to determine the Ka of formic acid (HCOOH).
Sample Problem 21.4: Calculation of an Acid Ionization Constant
A 0.500 M solution of formic acid is prepared, and its pH is measured to be 2.04. Determine the Ka for formic acid.
Step 1: List the known values and plan the problem.
Known
• initial [HCOOH] = 0.500 M
• pH = 2.04
Unknown
• Ka = ?
First, the pH is used to calculate the value of [H+] at equilibrium. An ICE table is set up in order to determine the concentrations of HCOOH and HCOO at equilibrium. All concentrations are then substituted into the Ka expression, and the Ka value is calculated.
Step 2: Solve.
[H+] = 10-pH = 10-2.04 = 9.12 × 10−3
Since each formic acid molecule that ionizes yields one H+ ion and one formate ion (HCOO), the concentrations of H+ and HCOO are equal at equilibrium. We assume that the initial concentrations of each ion are zero, resulting in the following ICE table (Table below). Although the concentration of H+ would technically be 1.0 × 10−7 before any dissociation occurs, we can ignore this very small amount because, in this case, it does not make a noticeable contribution to the final value of [H+].
Concentrations [HCOOH] [H+] [HCOO]
Initial 0.500 0 0
Change −9.12 × 10−3 +9.12 × 10−3 +9.12 × 10−3
Equilibrium 0.491 9.12 × 10−3 9.12 × 10−3
Substituting the equilibrium values into the Ka expression gives the following:
$\mathrm{K_a=\dfrac{[H^+][HCOO^-]}{[HCOOH]}=\dfrac{(9.12 \times 10^{-3})(9.12 \times 10^{-3})}{0.491}=1.7 \times 10^{-4}}$
The value of Ka is consistent with that of a weak acid. Two significant figures are appropriate for the answer, since there are two digits after the decimal point in the reported pH.
Practice Problems
1. Hypochlorous acid (HClO) is a weak acid. What is its Ka if a 0.250 M solution of hypochlorous acid has a pH of 4.07?
Similar steps can be taken to determine the Kb of a base. For example, a 0.750 M solution of the weak base ethylamine (C2H5NH2) has a pH of 12.31.
$\mathrm{C_2H_5NH_2 + H_2O \rightleftharpoons C_2H_5NH^+_3 + OH^-}$
Since one of the products of the ionization reaction is the hydroxide ion, we need to first find the value of [OH] at equilibrium. The pOH is 14 – 12.31 = 1.69. [OH] is then calculated to be 10−1.69 = 2.04 × 10−2 M. The ICE table is then set up as shown below (Table below).
Concentrations [C2H5NH2] [C2H5NH3+] [OH]
Initial 0.750 0 0
Change −2.04 × 10−2 +2.04 × 10−2 +2.04 × 10−2
Equilibrium 0.730 2.04 × 10−2 2.04 × 10−2
Substituting into the Kb expression yields the Kb for ethylamine.
$\mathrm{K_b=\dfrac{[C_2H_5NH^+_3][OH^-]}{[C_2H_5NH_2]}=\dfrac{(2.04 \times 10^{-2})(2.04 \times 10^{-2})}{0.730}=5.7 \times 10^{-4}}$
### Calculating the pH of a Weak Acid or Weak Base
The Ka and Kb values have been determined for a great many acids and bases, as shown in the Acid and Base Ionization Constants Tables (Table above and Table above). These can be used to calculate the pH of any solution of a weak acid or base whose ionization constant is known.
Sample Problem 21.5: Calculating the pH of a Weak Acid
Calculate the pH of a 2.00 M solution of nitrous acid (HNO2). The Ka for nitrous acid can be found in the table above (Table above).
Step 1: List the known values and plan the problem.
Known
• initial [HNO2] = 2.00 M
• Ka = 4.5 × 10−4
Unknown
• pH = ?
First, an ICE table is set up with the variable x used to signify the change in concentration of the substance due to ionization of the acid. Then, the Ka expression is used to solve for x and calculate the pH.
Step 2: Solve.
Concentrations [HNO2] [H+] [NO2]
Initial 2.00 0 0
Change −x +x +x
Equilibrium 2.00 − x x x
The Ka expression and value is used to set up an equation to solve for x.
$\mathrm{K_a=4.5 \times 10^{-4}=\dfrac{(x)(x)}{2.00-x}=\dfrac{x^2}{2.00-x}}$
The quadratic equation is required to exactly solve this equation for x. However, a simplification can be made because of the fact that relatively little of the weak acid will be ionized at equilibrium. Since the reactants are heavily favored, the value of x will be much less than 2.00, so we can say that 2.00 - x is approximately equal to 2.00.
$\mathrm{4.5 \times 10^{-4}=\dfrac{x^2}{2.00-x} \approx \dfrac{x^2}{2.00}}$
$\mathrm{x=\sqrt{4.5 \times 10^{-4}(2.00)}=2.9 \times 10^{-2}\:M=[H^+]}$
Since the variable x represents the hydrogen-ion concentration, the pH of the solution can now be calculated.
pH = -log[H+] = -log[2.9 × 10-2] = 1.54
The pH of a 2.00 M solution of a strong acid would be equal to –log(2.00) = −0.30. The higher pH of the 2.00 M nitrous acid is consistent with it being a weaker acid.
Practice Problem
1. Calculate the pH of a 0.20 M solution of hydrocyanic acid.
The procedure for calculating the pH of a solution of a weak base is similar to that of the weak acid in Sample Problem 21.5. However, the variable x will represent the concentration of the hydroxide ion. The pH is found by taking the negative logarithm to get the pOH, followed by subtracting from 14 to get the pH.
## Lesson Summary
• Strong acids and bases are those that ionize completely in solution. Weak acids and bases ionize only to a very small extent in solution. A stronger acid has a correspondingly weaker conjugate base, and vice-versa.
• The acid ionization constant (Ka) and base ionization constant (Kb) are numerical measures of the strength of acids and bases. The larger the ionization constant, the stronger the acid or base.
• Solutions of known concentration and known pH can be used to find an unknown Ka or Kb.
• For acids or bases with known Ka or Kb, the pH of the solution can be determined as long as the concentration of the solution is also known.
## Lesson Review Questions
### Reviewing Concepts
1. Explain why a strong acid is not the same thing as a concentrated acid.
2. Acid HX is a strong acid, while HY is a weak acid.
1. Compare the relative amounts of HX, H+, and X that are present in solution at equilibrium.
2. Compare the relative amounts of HY, H+, and Y that are present in solution at equilibrium.
3. Write equations for the ionizations of the following acid and base in water.
1. bromous acid, HBrO2
2. bromite ion, BrO2
4. Use the table above (Table above) to list the bases F, CN, and HCO3 in order from weakest to strongest.
5. Write the Ka expressions for the following weak acids.
1. sulfurous acid, H2SO3
2. lactic acid, HC3H5O3
6. Write Kb expressions for the following weak bases.
1. carbonate ion, CO32-
2. aniline, C6H5NH2
### Problems
1. A 1.25 M solution of a certain unknown acid has a pH of 4.73. Calculate the Ka of the acid.
2. A 0.350 M solution of a certain unknown base has a pH of 11.22. Calculate the Kb of the base.
3. Determine the pH of the following solutions.
1. 2.40 M benzoic acid
2. 0.745 M pyridine
4. What concentration of a solution of hydrofluoric acid should be prepared in order to have a pH of 2.00?
## Points to Consider
Acids and bases react with each other in a reaction called a neutralization.
• What are the products of a neutralization reaction?
• How can a neutralization reaction be used to determine the unknown concentration of an acid or base in an aqueous solution?
Mar 29, 2013
Jan 20, 2015 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 33, "texerror": 0, "math_score": 0.8859098553657532, "perplexity": 3274.503206125802}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936462232.5/warc/CC-MAIN-20150226074102-00077-ip-10-28-5-156.ec2.internal.warc.gz"} |
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# The rectangular region above contains two circles and a semi
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The rectangular region above contains two circles and a semi [#permalink]
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The rectangular region above contains two circles and a semicircle, each with a radius of 7. If 22/7 is used as an approximation for π, then the area of the shaded region is approximately
(A) 105
(B) 210
(C) 380
(D) 385
(E) 405
Originally posted by jlgdr on 11 Oct 2013, 17:25.
Last edited by Bunuel on 11 Oct 2013, 17:52, edited 1 time in total.
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Re: The rectangular region above contains two circles and a semi [#permalink]
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11 Oct 2013, 17:56
1
The rectangular region above contains two circles and a semicircle, each with a radius of 7. If 22/7 is used as an approximation for π, then the area of the shaded region is approximately
(A) 105
(B) 210
(C) 380
(D) 385
(E) 405
The dimensions of the rectangle is 5 radii by 2 radii, thus 35 by 14. Therefore its area is 35*14.
The area of the two circles and the semicircle is $$2.5*\pi{r^2}=\frac{5}{2}*\frac{22}{7}*7^2=35*11$$.
Thus the area of the shaded region is approximately 35*14-35*11 = 35(14-11)=35*3=105.
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Re: The rectangular region above contains two circles and a semi [#permalink]
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11 Oct 2013, 20:31
jlgdr wrote:
Attachment:
Untitled.png
The rectangular region above contains two circles and a semicircle, each with a radius of 7. If 22/7 is used as an approximation for π, then the area of the shaded region is approximately
(A) 105
(B) 210
(C) 380
(D) 385
(E) 405
Area of 2 Circles = 2*[22/7 * 7 * 7] = 2 * 154 = 308
Area of semicircle = 1/2 * [ 154 ] = 77.
Total Area of 2 circles + Semicircle = 308 + 77 = 385
Area of Rectangle = 35 * 14 = 490
Area of shaded region = 490 - 385 = 105
Option A
Hope it is clear
Cheers
Qoofi
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Re: The rectangular region above contains two circles and a semi [#permalink]
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14 Apr 2014, 00:34
Hey, can somebody explain here , why the dimensions of the rectangle are 35 by 14, when we are given only the radius is 7.
Help is appreciated.
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Posts: 59671
Re: The rectangular region above contains two circles and a semi [#permalink]
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14 Apr 2014, 01:25
1
dheeraj24 wrote:
Hey, can somebody explain here , why the dimensions of the rectangle are 35 by 14, when we are given only the radius is 7.
Help is appreciated.
Look at the diagram below:
Attachment:
Untitled.png [ 10 KiB | Viewed 4684 times ]
The length of the rectangle is 5r, so 5*7=35 and the width is 2r, so 2*7=14.
Hope it's clear.
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Re: The rectangular region above contains two circles and a semi [#permalink]
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14 Apr 2014, 01:57
Thanks a lot Bunuel.
Bunuel wrote:
dheeraj24 wrote:
Hey, can somebody explain here , why the dimensions of the rectangle are 35 by 14, when we are given only the radius is 7.
Help is appreciated.
Look at the diagram below:
Attachment:
Untitled.png
The length of the rectangle is 5r, so 5*7=35 and the width is 2r, so 2*7=14.
Hope it's clear.
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14 Sep 2018, 14:45
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Re: The rectangular region above contains two circles and a semi [#permalink] 14 Sep 2018, 14:45
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https://www.mathworks.com/help/econ/compare-simulation-smoother-to-smooth.html | # Compare Simulation Smoother to Smoothed States
This example shows how the results of the state-space model simulation smoother (`simsmooth`) compare to the smoothed states (`smooth`).
Suppose that the relationship between the change in the unemployment rate (${x}_{1,t}$) and the nominal gross national product (nGNP) growth rate (${x}_{3,t}$) can be expressed in the following, state-space model form.
`$\left[\begin{array}{c}{x}_{1,t}\\ {x}_{2,t}\\ {x}_{3,t}\\ {x}_{4,t}\end{array}\right]=\left[\begin{array}{cccc}{\varphi }_{1}& {\theta }_{1}& {\gamma }_{1}& 0\\ 0& 0& 0& 0\\ {\gamma }_{2}& 0& {\varphi }_{2}& {\theta }_{2}\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1,t-1}\\ {x}_{2,t-1}\\ {x}_{3,t-1}\\ {x}_{4,t-1}\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 1& 0\\ 0& 1\\ 0& 1\end{array}\right]\left[\begin{array}{c}{u}_{1,t}\\ {u}_{2,t}\end{array}\right]$`
`$\left[\begin{array}{c}{y}_{1,t}\\ {y}_{2,t}\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{c}{x}_{1,t}\\ {x}_{2,t}\\ {x}_{3,t}\\ {x}_{4,t}\end{array}\right]+\left[\begin{array}{cc}{\sigma }_{1}& 0\\ 0& {\sigma }_{2}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{1,t}\\ {\epsilon }_{2,t}\end{array}\right],$`
where:
• ${x}_{1,t}$ is the change in the unemployment rate at time t.
• ${x}_{2,t}$ is a dummy state for the MA(1) effect on ${x}_{1,t}$.
• ${x}_{3,t}$ is the nGNP growth rate at time t.
• ${x}_{4,t}$ is a dummy state for the MA(1) effect on ${x}_{3,t}$.
• ${y}_{1,t}$ is the observed change in the unemployment rate.
• ${y}_{2,t}$ is the observed nGNP growth rate.
• ${u}_{1,t}$ and ${u}_{2,t}$ are Gaussian series of state disturbances having mean 0 and standard deviation 1.
• ${\epsilon }_{1,t}$ is the Gaussian series of observation innovations having mean 0 and standard deviation ${\sigma }_{1}$.
• ${\epsilon }_{2,t}$ is the Gaussian series of observation innovations having mean 0 and standard deviation ${\sigma }_{2}$.
Load the Nelson-Plosser data set, which contains the unemployment rate and nGNP series, among other things.
`load Data_NelsonPlosser`
Preprocess the data by taking the natural logarithm of the nGNP series, and the first difference of each. Also, remove the starting `NaN` values from each series.
```isNaN = any(ismissing(DataTable),2); % Flag periods containing NaNs gnpn = DataTable.GNPN(~isNaN); u = DataTable.UR(~isNaN); T = size(gnpn,1); % Sample size y = zeros(T-1,2); % Preallocate y(:,1) = diff(u); y(:,2) = diff(log(gnpn));```
This example proceeds using series without `NaN` values. However, using the Kalman filter framework, the software can accommodate series containing missing values.
Specify the coefficient matrices.
```A = [NaN NaN NaN 0; 0 0 0 0; NaN 0 NaN NaN; 0 0 0 0]; B = [1 0;1 0 ; 0 1; 0 1]; C = [1 0 0 0; 0 0 1 0]; D = [NaN 0; 0 NaN];```
Specify the state-space model using `ssm`. Verify that the model specification is consistent with the state-space model.
`Mdl = ssm(A,B,C,D)`
```Mdl = State-space model type: ssm State vector length: 4 Observation vector length: 2 State disturbance vector length: 2 Observation innovation vector length: 2 Sample size supported by model: Unlimited Unknown parameters for estimation: 8 State variables: x1, x2,... State disturbances: u1, u2,... Observation series: y1, y2,... Observation innovations: e1, e2,... Unknown parameters: c1, c2,... State equations: x1(t) = (c1)x1(t-1) + (c3)x2(t-1) + (c4)x3(t-1) + u1(t) x2(t) = u1(t) x3(t) = (c2)x1(t-1) + (c5)x3(t-1) + (c6)x4(t-1) + u2(t) x4(t) = u2(t) Observation equations: y1(t) = x1(t) + (c7)e1(t) y2(t) = x3(t) + (c8)e2(t) Initial state distribution: Initial state means are not specified. Initial state covariance matrix is not specified. State types are not specified. ```
Estimate the model parameters, and use a random set of initial parameter values for optimization. Restrict the estimate of ${\sigma }_{1}$ and ${\sigma }_{2}$ to all positive, real numbers using the `'lb'` name-value pair argument. For numerical stability, specify the Hessian when the software computes the parameter covariance matrix, using the `'CovMethod'` name-value pair argument.
```rng(1); params0 = rand(8,1); [EstMdl,estParams] = estimate(Mdl,y,params0,... 'lb',[-Inf -Inf -Inf -Inf -Inf -Inf 0 0],'CovMethod','hessian');```
```Method: Maximum likelihood (fmincon) Sample size: 61 Logarithmic likelihood: -199.397 Akaike info criterion: 414.793 Bayesian info criterion: 431.68 | Coeff Std Err t Stat Prob ---------------------------------------------------- c(1) | 0.03387 0.15213 0.22263 0.82383 c(2) | -0.01258 0.05749 -0.21876 0.82684 c(3) | 2.49855 0.22764 10.97570 0 c(4) | 0.77438 2.58647 0.29939 0.76464 c(5) | 0.13994 2.64359 0.05294 0.95778 c(6) | 0.00367 2.45472 0.00150 0.99881 c(7) | 0.00535 2.11315 0.00253 0.99798 c(8) | 0.00032 0.12685 0.00253 0.99798 | | Final State Std Dev t Stat Prob x(1) | 1.39999 0.00535 261.57973 0 x(2) | 0.21778 0.91641 0.23765 0.81216 x(3) | 0.04730 0.00032 147.40073 0 x(4) | 0.03568 0.00033 107.76114 0 ```
`EstMdl` is an `ssm` model, and you can access its properties using dot notation.
Simulate `1e4` paths of observations from the fitted, state-space model `EstMdl` using the simulation smoother. Specify to simulate observations for each period.
```numPaths = 1e4; SimX = simsmooth(EstMdl,y,'NumPaths',numPaths);```
`SimX` is a `T - 1`-by- `4`-by- `numPaths` matrix containing the simulated states. The rows of `SimX` correspond to periods, the columns correspond to a state in the model, and the pages correspond to paths.
Estimate the smoothed state means, standard deviations, and 95% confidence intervals.
```SmoothBar = mean(SimX,3); SmoothSTD = std(SimX,0,3); SmoothCIL = SmoothBar - 1.96*SmoothSTD; SmoothCIU = SmoothBar + 1.96*SmoothSTD;```
Estimate smooth states using `smooth`.
`SmoothX = smooth(EstMdl,y);`
Plot the smoothed states, and the means of the simulated states and their 95% confidence intervals.
```figure h = plot(dates(2:T),SmoothBar(:,1),'-r',... dates(2:T),SmoothCIL(:,1),':b',... dates(2:T),SmoothCIU(:,1),':b',... dates(2:T),SmoothX(:,1),':k',... 'LineWidth',3); xlabel 'Period'; ylabel 'Unemployment rate'; legend(h([1,2,4]),{'Simulated, smoothed state mean','95% confidence interval',... 'Smoothed states'},'Location','Best'); title 'Smoothed Unemployment Rate'; axis tight```
```figure h = plot(dates(2:T),SmoothBar(:,3),'-r',... dates(2:T),SmoothCIL(:,3),':b',... dates(2:T),SmoothCIU(:,3),':b',... dates(2:T),SmoothX(:,3),':k',... 'LineWidth',3); xlabel 'Period'; ylabel 'nGNP'; legend(h([1,2,4]),{'Simulated, smoothed state mean','95% confidence interval',... 'Smoothed states'},'Location','Best'); title 'Smoothed nGNP'; axis tight```
The simulated state means are practically identical to the smoothed states.
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http://jucourseworkfsdp.musikevents.us/physics-newtons-2nd-law-lab.html | # Physics newtons 2nd law lab
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Physics newtons 2nd law lab
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https://www.physicsforums.com/threads/how-unitary-change-of-basis-related-to-trace.648929/ | # How unitary change of basis related to Trace?
1. Nov 2, 2012
### Shing
1. The problem statement, all variables and given/known data
Shanker 1.7.1
3.)Show that the trace of an operator is unaffected by a unitary change of basis (Equivalently, show $TrΩ=TrU^{\dagger}ΩU$
2. Relevant equations
I can show that via Shanker's hint, but I however can't see how a unitary change of basis links to $TrΩ=TrU^{\dagger}ΩU$, (and it really giving me a headache!) Would anyone be kind enough to explain to me?
Last edited: Nov 2, 2012
2. Nov 2, 2012
### Square
There is some identity which tells you that $\mathrm{Tr}(AB) = \mathrm{Tr}(BA)$ (more generally one could state that the trace is invariant under cyclic permutations). Use it and your problem should be as good as solved.
3. Nov 2, 2012
### Shing
Forgive my ignorance,
But shouldn't "unaffected by a unitary change of basis" be expressed as$TrUΩ=TrΩ$
Last edited: Nov 2, 2012
4. Nov 2, 2012
### Dick
No. A matrix A is a change of basis of a matrix B if A=T^(-1)BT for a nonsingular matrix T.
5. Nov 2, 2012
### Shing
So here is my understanding: we map A to B, A and B represent the same thing in different bases, and their mathematical relation is $A=T^{-1}AT$
am I right?
I have a rough picture now, but Sadly, I still can't see the physical picture, neither the mathematics.
, and shanker surely was telling the truth! One should know a bit more of linear algebra before embrace into it!
Thanks guys, anyway :)
6. Nov 2, 2012
### Shing
Now I got it a bit!
For column vector (1,0) -> (0 ,1)
And T is {(0,1),(1,0)} :)
Did I get it right?
7. Nov 2, 2012
### Dick
If T is unitary then sure that's an option. So T takes (1,0) -> (0,1) and (0,1) -> (1,0). T^(-1) (which happens to be the same as T, but that's usually not the case) does the opposite. So to figure out what A is 'equivalent' (not equal!) to B, you use T to rotate a vector to the basis of B, then let B act on it, then undo the rotation with T^(-1), so A=T^(-1)BT. I know this is vague. But none of this vagueness should stop you from being able to show Tr(B)=Tr(A)=Tr(T^(-1)BT). That change of bases don't change the trace.
Last edited: Nov 2, 2012
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https://www.physicsforums.com/threads/how-much-calc-ii-is-used-for-calc-iii.337083/ | # How much calc II is used for calc III?
1. Sep 13, 2009
### quantum13
Not sure about the forum, sorry.
Are series expands featured prominently in vector calculus?
2. Sep 13, 2009
### VeeEight
Courses vary depending on the school. Infinite series is not covered for most, if not all, "Calc III' courses I have seen (this is usually left for courses in Real Analysis). If you are covering multiple, line, and surface integrals in this class then you should need to know appropriate techniques (integration by parts, trig substitution, etc). Polar coordinates are used frequently to simplify integration.
3. Sep 13, 2009
### Luongo
All of it. Calc III is Calc I and Calc II except in the n-th dimension, integration and differentiation in the n dimensions, calc i and ii are only in the 1st dimension. Vector calculus is basically the same... Calc I and Calc II integration and differentiation. but this time it's in the n-th dimension AND don't just have a magnitude, but also have direction in n-dimensional euclidean vector space. So as you progress in calculus it is the same concept except more variables, and vectors which complicates things alot more than it sounds.
also infinite series are used more for partial differential equations to obtain solutions to complicated equations whose solution isn't compromised entirely of elementary functions and you're introduced to them in calc II
Last edited: Sep 13, 2009
4. Sep 13, 2009
### lurflurf
How many "Calc II' courses have you "seen"? Many courses do one variable series in calc 2 and multiple variable series. Some courses do series in only in one term. Some couses are two or four terms instead of three and are adjusted accordingly. Real analysis is for things like banach spaces, spectral theory, and abstract measure and integration, it would be quite sad if time was wasted introducing series; especially since many students would be third year and would have needed series sooner, like freshman physics.
5. Sep 14, 2009
### quantum13
sorry, my question was horribly put.
i want to study electromagnetism, but I only have calc I (i'm taking calc II in my high school, but we're pretty slow) and I heard that calc III is recommended for electricity and magnetism. my question is, how can I most efficiently study just the math I need while deferring other topics not directly dealt with in E&M for coursework?
6. Sep 14, 2009
### Prologue
Study it out of a 'Physics for scientists and engineers' textbook like Serway or something. They will leave out the heavy calculus stuff but keep the things needed to get a very good grasp on E&M. I self studied out of Serway to learn E&M and watched the ocw mit videos on physics by Walter Lewin. If you do this combo you will be in really good shape. Later on you could read a book like Griffiths to get a more 'mathy' understanding once you have the needed Calc 3 experience.
Similar Discussions: How much calc II is used for calc III? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8205172419548035, "perplexity": 1306.156617245846}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948544677.45/warc/CC-MAIN-20171214144324-20171214164324-00119.warc.gz"} |
https://research.kaust.edu.sa/en/publications/scaling-relation-and-regime-map-of-explosive-gasliquid-flow-of-bi | # Scaling relation and regime map of explosive gas–liquid flow of binary Lennard-Jones particle system
Hajime Inaoka, Satoshi Yukawa, Nobuyasu Ito
Research output: Contribution to journalArticlepeer-review
1 Scopus citations
## Abstract
We study explosive gasliquid flows caused by rapid depressurization using a molecular dynamics model of Lennard-Jones particle systems. A unique feature of our model is that it consists of two types of particles: liquid particles, which tend to form liquid droplets, and gas particles, which remain supercritical gaseous states under the depressurization realized by simulations. The system has a pipe-like structure similar to the model of a shock tube. We observed physical quantities and flow regimes in systems with various combinations of initial particle number densities and initial temperatures. It is observed that a physical quantity Q, such as pressure, at position z measured along a pipe-like system at time t follows a scaling relation Q(z,t)=Q(zt) with a scaling function Q(ζ). A similar scaling relation holds for time evolution of flow regimes in a system. These scaling relations lead to a regime map of explosive flows in parameter spaces of local physical quantities. The validity of the scaling relations of physical quantities means that physics of equilibrium systems, such as an equation of state, is applicable to explosive flows in our simulations, though the explosive flows involve highly nonequilibrium processes. In other words, if the breaking of the scaling relations is observed, it means that the explosive flows cannot be fully described by physics of equilibrium systems. We show the possibility of breaking of the scaling relations and discuss its implications in the last section. © 2011 Elsevier B.V. All rights reserved.
Original language English (US) 423-438 16 Physica A: Statistical Mechanics and its Applications 391 3 https://doi.org/10.1016/j.physa.2011.08.018 Published - Feb 2012 Yes | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.899685263633728, "perplexity": 1349.6799541499072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038056325.1/warc/CC-MAIN-20210416100222-20210416130222-00170.warc.gz"} |
http://mathhelpforum.com/algebra/212273-arithmetic-series-problem.html | Math Help - Arithmetic Series problem
1. Arithmetic Series problem
In an arithmetic sequence, the 6th term is half the fourth term and the third term is 15.
i) Find the first term and the common difference
ii) How many terms are needed to give a sum that is less than 65?
I've done part (i) and found a=21 and d=-3, but I'm having trouble with the second part. Here's what I've got so far:
Sn = n/2{2a+(n-1)d}
Sn = n/2{2(21) + (n-1)(-3)}
Sn = n/2{45-3n}
(n/2){45-3n} < 65
n{45-3n} < 130
45n -3n2 < 130
I think I'm on the right track (I think???)...but I don’t think the quadratic inequality isn't giving me whole numbers if I solve for values of n so I’ve gone wrong somewhere. Any help? Thanks
2. Re: Arithmetic Series problem
Are you sure you have written the problem correctly? Yes, a= 21 and d= -3 so the first few terms are 21, 18, 15, .... The "sum" of the first one term (if you call that a sum) is 21, the sum of the first two is 39, the sum of the first three is 54. Those are all less than 65! The sum of the first four terms is 21+ 18+ 15+ 12= 66 so that after the fourth term the sum is NO LONGER less than 65. Of course, $a_9$ and beyond are negative so that the sum begins decreasing and will eventually be less than 65 again. Was that what was meant?
As for "I don’t think the quadratic inequality isn't giving me whole numbers if I solve for values of n", of course it does! The equation 45n- 3n2= 130 has no integer roots but 45n- 3n2< 130 does. The roots of 45n- 3n2= 130 are a little smaller than 4 and a little larger than 11. For all integers between 4 and 11, 5, 6, 7, 8, 9, and 10, 45n- 3n2> 130 so that sum is larger than 65. For all n larger than 11, the sum is again less than 65.
3. Re: Arithmetic Series problem
So now it is enough to solve this quadratic inequality. you may consider the following equivalent inequality $3n^2-45n+130\geq0$
solving the quadratic equation to get the zeros (11.09... and 3.90...) and outside the zeros the quadratic equation is positive
so your series is less than 65 until the third term and from the 12th term
4. Re: Arithmetic Series problem
Ah yes I get it now! Thank you! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8731797337532043, "perplexity": 439.5259395564885}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257831769.86/warc/CC-MAIN-20160723071031-00094-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://wiki.cs.byu.edu/cs-465/homework-2 | Finite Field Arithmetic
For this homework, please submit two pseudo-code samples to answer question 1 and 2 below.
Review sections 4 and 5 in FIPS 197 (see Lab 1) to understand details about finite field arithmetic in AES.
Section 4: Focus on 4.1 and 4.2. 4.3 is less important to study (and more complicated), but it may help you understand section 5.
Section 5: See the high-level pseudo-code for the Cipher algorithm in 5.1. The MixColumns function in 5.1.3 is important to understand. Near the end of that section, it gives you the formulas you need to implement the MixColumns function.
Question 1) Submit pseudo-code for the MixColumns function in 5.1.3. Your submission should demonstrate that you understand how to implement the ideas in section 4. Study these sections and the lecture slides to be able to distinguish between the abstract mathematical ideas (e.g., polynomial representation) and the implementation methodology (i.e., bit shifting, MOD, AND, XOR).
Hint: Start with a Mixcolumns(state) where the state is a 4×4 array of bytes. You can reference each byte with a state(x,y) reference.
Question 2) Submit code for a finite field multiply function that takes two bytes as input and produces a byte as output, the result of multiplying a * b where * is the finite field multiply described in section 4.2. Your code should use an xtime function that is described in 4.2.1.
Hint: Assume a function byte c ffMultiply(byte a, byte b) where the function returns c = a * b.
cs-465/homework-2.txt · Last modified: 2015/08/28 11:07 by seamons | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8227242231369019, "perplexity": 1239.9162971622472}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737225.57/warc/CC-MAIN-20200807202502-20200807232502-00165.warc.gz"} |
http://mathoverflow.net/questions/71716/how-far-will-a-random-walk-on-the-integers-go | # How far will a random walk on the integers go?
Let $S_n$ be the distance walked at time $n$ in a simple symmetric random walk on $\mathbb{Z}$ (with steps $=\pm1$).
It is well known that $\lim_{n\to\infty}(E(|S_n|)/\sqrt{n})=\sqrt{2 /\pi}$.
Can we describe the set $F$ of monotonic "growth" functions with probability 1 of being crossed by $|S_n|$ for $n>1$?
Clearly $f(x)=\sqrt{x}$ is in $F$ and I think I could prove that $g(x)=2f(x/2)$ is in $F$ whenever $f$ is. I have no idea beyond that.
-
Have you heard about the Law of iterated logarithm? en.wikipedia.org/wiki/Law_of_the_iterated_logarithm I think that this may help you. – Leonid Petrov Jul 31 '11 at 8:36
How is $f(x) = \sqrt{x}$ clearly if $F\hspace{.03 in}$? Among other things, $f$ can only be hit by $|S_n|$ at perfect squares. – Ricky Demer Jul 31 '11 at 9:18
Does it makes sense for a limit on $n$ to be a function of $n$? Perhaps you mean the expected value is asymptotic to that function of $n$? – Gerry Myerson Jul 31 '11 at 9:21
Gerry Myerson: yes, I edited accordingly. @Ricky Demer: I replaced "hit" with "crossed" - I hope that clarifies that I'm not looking for equality of two functions, but for $f$'s such that $\limsup |S_n|/f(n) \ge 1$ with probability 1 – Yaakov Baruch Jul 31 '11 at 11:23
@Leonid: I think your comment is the answer to the question (or at least the INTENDED question). – Yaakov Baruch Jul 31 '11 at 11:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8951936364173889, "perplexity": 316.9150474891815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936463708.99/warc/CC-MAIN-20150226074103-00067-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/215158-basis-derivative-transformation-question.html | Math Help - Basis and Derivative Transformation Question
1. Basis and Derivative Transformation Question
True or False: Provide a QUICK proof or a COUNTER-EXAMPLE to each of the following:
1.) If v1,...,vn are any vectors in a vector space V, then the set {v1,...,vn} is a basis for the subspace span {v1,...,vn}.
I believe that this is false because if you had n vectors which were scalar multiples of each other then it wouldn't be a basis because it couldn't span the subspace or wouldn't be linearly independent, which is required. Is that correct/ on the right track?
2.) The derivative transformation d/dx: R[x] ---> R[x] has no eigenvalues or eigenvectors.
I believe that this is false, but I'm less confident on this one. I was thinking if you took the derivative of degree 0 polynomials, i.e. any constant then things wouldn't work? Any help on this one would be appreciated!
2. Re: Basis and Derivative Transformation Question
Hey TimsBobby2.
For the first one: you need to consider the three axioms of the sub-space which are namely a) the zero vector being in the space, b) closure in the sub-space for scalar multiplication and addition.
The basis for a sub-space does not have to be the same as that for the whole space.
For number 2, you should firstly note that the transformation is square and secondly that you have non-zero eigenvalues if the determinant is non-zero.
When does the determinant with respect to the derivative be non-zero? (Hint: Think about when you have a minimum, maximum in a multivariable scenario).
3. Re: Basis and Derivative Transformation Question
Originally Posted by TimsBobby2
True or False: Provide a QUICK proof or a COUNTER-EXAMPLE to each of the following:
1.) If v1,...,vn are any vectors in a vector space V, then the set {v1,...,vn} is a basis for the subspace span {v1,...,vn}.
I believe that this is false because if you had n vectors which were scalar multiples of each other then it wouldn't be a basis because it couldn't span the subspace or wouldn't be linearly independent, which is required. Is that correct/ on the right track?
Yes, that is on the right track. Now, write out what you have said: Let $v_1$ be some vector v, then $v_2= 2v$, etc. A simpler example is just to include the 0 vector in the set!
2.) The derivative transformation d/dx: R[x] ---> R[x] has no eigenvalues or eigenvectors.
I believe that this is false, but I'm less confident on this one. I was thinking if you took the derivative of degree 0 polynomials, i.e. any constant then things wouldn't work? Any help on this one would be appreciated!
I don't understand what you mean by "wouldn't work". It is certainly true that if f(x) is a constant then df/dx= 0. That is df/dx= 0f. What does that tell you about "eigenvalues" and "eigenvectors". | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9598093628883362, "perplexity": 386.9349631605869}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510257966.18/warc/CC-MAIN-20140728011737-00140-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://dsharp.dev/research/tensors | ## Tensor Decompositions
The Singular Value Decomposition (SVD) is a well-trodden subject of many a linear algebra class. Given a matrix $$\mathbf{A}\in\mathbb{R}^{m\times n}$$, we'd like to find $$\mathbf{U}\in\mathbb{R}^{m\times m}$$, $$\mathbf{V}\in\mathbb{R}^{n\times n}$$ and $$\mathbf{\Sigma}\in\mathbb{R}^{m\times n}$$ such that $$\mathbf{A} = \mathbf{U\Sigma V}^T$$, where $$\mathbf{U},\mathbf{V}$$ are orthogonal matrices and $$\mathbf{\Sigma}$$ is a diagonal matrix with positive, decreasing entries on the diagonal. For a more rigorous undertaking, see, e.g., Trefethen and Bau. This SVD can be applied to any number of things, and is frequently used as an optimal way to compress data and exploit structure. For example, if $$\mathbf{U}_p,\mathbf{V}_p$$ are the first $$p$$ columns of $$\mathbf{U},\mathbf{V}$$ respectively, and $$\mathbf{\Sigma}_p$$ corresponds to $$\mathbf{\Sigma}$$ appropriately, then $$\mathbf{A}_p := \mathbf{U}_p\mathbf{\Sigma}_p\mathbf{V}_p^T$$ will minimize $$\|\mathbf{A}-\mathbf{B}\|$$ over all rank-$$p$$ matrices $$B$$. In this sense, we know that the columns of $$\mathbf{U}$$ and $$\mathbf{V}$$ "contain important information" about $$\mathbf{A}$$ in some sense of the phrase, which is exploited in much literature (citations needed).
This motivates important questions on extensions-- can we exploit the same properties in tensors? What information can we recover? The gist of the research I performed on this revolved around the Higher Order SVD (HOSVD) and its applications to parametric model reduction, in some sense. Suppose you had data on some $$N$$ dimensional problem with a $$P$$ dimensional parameter space. Then, if you sample in dimension $$j$$ a number of $$m_j$$ times, and parameter $$j$$ a number of $$q_j$$ times, then you end up with $$m_1\times m_2\times \ldots\times m_N\times q_1\times q_2\times\ldots\times q_P$$ dimensions, which can easily be enormous. The idea in the research, though, is if you can possibly construct this data tensor, what can you do with it? Can you perform something like "looking at the singular vectors"? The answer is yes, and in fact, instead of having only left and right singular vectors, you get singular vectors corresponding to each dimension (i.e. $$m_j$$ and $$q_j$$). Then, intuitively, one can examine the behavior of, say, a discretized PDE along one particular dimension in a way that shows us the important parts. See more here. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8940078616142273, "perplexity": 194.00668651129232}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711390.55/warc/CC-MAIN-20221209043931-20221209073931-00600.warc.gz"} |
http://math.stackexchange.com/questions/120617/validation-of-partial-sum-sum-i-1i-n-dfrac-x-left-left-i-1-right | # Validation of partial sum $\sum _{i=1}^{i=n}\dfrac {x} {\left( \left( i-1\right) x+1\right) \left( ix+1\right) }=1-\dfrac {1} {nx+1}$
I came across the below partial sum where the author did not provide any reasoning and claimed that it is easy to see that the partial sum is so $$\sum _{i=1}^{i=n}\dfrac {x} {\left( \left( i-1\right) x+1\right) \left( ix+1\right) }=1-\dfrac {1} {nx+1}$$ I want to validate this, although i am willing to bet that his claim must be right but, i could n't quite accomplish this. The link does infact confirm it.
Obviously the series converges, my original idea was to get a common denominator and sum up the terms in the numerators, but this ends up being a very laborious process.
Second thought was to try think of a well known partial sum or series and obtain the sum expression based on that, but i can not seem to match this to a pattern of a well known series as it stands atleast.
There seems to be a lot of tools which help with convergence but probably not many when it comes to exact computation. I was hoping if some one could please shed some light on how to validate this and also if we have any other tools in our arsenal to evaluate partial sums.
Any help would be much appreciated.
-
Hint: telescope sum – Blah Mar 15 '12 at 19:36
$$\frac{x}{((i-1)x+1)(ix+1)} = \frac1{(i-1)x+1)} - \frac1{ix+1}$$ – user17762 Mar 15 '12 at 19:38
Thank you guys that was very helpfull. – Hardy Mar 15 '12 at 19:43
@Hardy: Note that the expression that was used is just the partial fractions version of your expression. Partial fractions are not only for integration! Computer Science students end up having to know about partial fractions, because of their usefulness in dealing with common generating functions. – André Nicolas Mar 15 '12 at 20:11
@Hardy: Yes, one gets combinatorial problems. Generating functions are often rational functions, and partial fractions often let us get explicit formulas for the coefficients. – André Nicolas Mar 15 '12 at 22:12
Hint: $$\dfrac {x} {\left( \left( i-1\right) x+1\right) \left( ix+1\right) }={\dfrac {1} {\left( \left( i-1\right) x+1\right)} } - {\dfrac {1} {\left(i) x+1\right)} }$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.881659746170044, "perplexity": 356.2790053329949}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931014329.94/warc/CC-MAIN-20141125155654-00084-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://mathhelpforum.com/geometry/119041-converting-problem-print.html | converting problem
• Dec 7th 2009, 02:12 AM
mohamedsafy
converting problem
first of all i want to know whats mean by gon
2- i need to convert 22g 68c 33cc into degrees minutes seconds and plz explain to me how did u convert it and whats c and whats cc
and at last thx for this forum who helped me alot last year
• Dec 7th 2009, 03:55 AM
mr fantastic
Quote:
Originally Posted by mohamedsafy
first of all i want to know whats mean by gon Mr F says: Read this: Polygons
2- i need to convert 22g 68c 33cc into degrees minutes seconds and plz explain to me how did u convert it and whats c and whats cc
and at last thx for this forum who helped me alot last year
Not sure about the second. What is the context?
• Dec 7th 2009, 11:35 AM
mohamedsafy
Quote:
Originally Posted by mr fantastic
Not sure about the second. What is the context?
no gon is something to measure angles and sometimes it is called grades gon is 1/400 of circle degree is 1/360 of circle i just googled it and now iam ok with converting gon to degrees but i dont know how to convert the "c" and the "cc" into minutes and seconds "c" and "cc" fractions of the gon
• Dec 7th 2009, 02:52 PM
aidan
Quote:
Originally Posted by mohamedsafy
first of all i want to know whats mean by gon
2- i need to convert 22g 68c 33cc into degrees minutes seconds and plz explain to me how did u convert it and whats c and whats cc
and at last thx for this forum who helped me alot last year
Quote:
A unit of angular measure in which the angle of an entire circle is 400 gradians. A right angle is therefore 100 gradians. A gradian is sometimes also called a gon or a grade.
The term "gon" came into general use after World War II. The term "grad" has been around for centuries (well, at least 2). At the time the length of the metre was being established, the use of grads for measuring angles became "metric".
Almost all calculators have a key identified as [DRG].
A grad (or gon) is a decimal form for angular measure.
A right angle, or 1/4 of a circle, is 90 degrees.
90 degress is $\cfrac{\pi}{2}$ radians.
A right angle, or 1/4 of a circle, is 100 grads.
1 grad = 0.01 of a right angle;
1 metric minute = 1c = 0.01g ( 1/100 of a grad )
1 metric second = 1cc = 0.01c ( 1/100 of a metric minute, or 1/10000 of a grad)
Quote:
convert 22g 68c 33cc into degrees minutes seconds
0.226833 of a quarter circle
$0.226833 \times 90\text{degrees}=20.41497 \text{degrees}$
Simply convert the decimal-degrees into the degrees-minutes-seconds required.
Hope that helps.
.
NOTE: c & cc were originally defined as circular measure. It could be stated as 1.57rad for 1.57 radians or 1.57c , meaning circular measure. The c & cc gradually became associated with the 0.01 grad & 0.0001 grad.
• Dec 7th 2009, 07:22 PM
mohamedsafy
Quote:
Originally Posted by aidan
The term "gon" came into general use after World War II. The term "grad" has been around for centuries (well, at least 2). At the time the length of the metre was being established, the use of grads for measuring angles became "metric".
Almost all calculators have a key identified as [DRG].
A grad (or gon) is a decimal form for angular measure.
A right angle, or 1/4 of a circle, is 90 degrees.
90 degress is $\cfrac{\pi}{2}$ radians.
A right angle, or 1/4 of a circle, is 100 grads.
1 grad = 0.01 of a right angle;
1 metric minute = 1c = 0.01g ( 1/100 of a grad )
1 metric second = 1cc = 0.01c ( 1/100 of a metric minute, or 1/10000 of a grad)
$0.226833 \times 90\text{degrees}=20.41497 \text{degrees}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8703060746192932, "perplexity": 3833.513572499591}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824899.43/warc/CC-MAIN-20171021205648-20171021225648-00519.warc.gz"} |
https://repo.scoap3.org/record/32614 | # Enhancing the discovery prospects for SUSY-like decays with a forgotten kinematic variable
Debnath, Dipsikha (0000 0004 1936 8091, Physics Department, University of Florida, Gainesville, FL, 32611, U.S.A.) (NHETC, Department of Physics and Astronomy, Rutgers, The State University of NJ, Piscataway, NJ, 08854, U.S.A.) ; Gainer, James (0000 0001 2188 0957, Department of Physics and Astronomy, University of Hawaii, Honolulu, HI, 96822, U.S.A.) ; Kilic, Can (0000 0004 1936 9924, Theory Group, Department of Physics, The University of Texas at Austin, Austin, TX, 78712, U.S.A.) ; Kim, Doojin (0000 0001 2156 142X, Theoretical Physics Department, CERN, Geneva, Switzerland) (0000 0001 2168 186X, Department of Physics, University of Arizona, Tucson, AZ, 85721, U.S.A.) ; Matchev, Konstantin (0000 0004 1936 8091, Physics Department, University of Florida, Gainesville, FL, 32611, U.S.A.) ; Yang, Yuan-Pao (0000 0004 1936 9924, Theory Group, Department of Physics, The University of Texas at Austin, Austin, TX, 78712, U.S.A.)
07 May 2019
Abstract: The lack of a new physics signal thus far at the Large Hadron Collider motivates us to consider how to look for challenging final states, with large Standard Model backgrounds and subtle kinematic features, such as cascade decays with compressed spectra. Adopting a benchmark SUSY-like decay topology with a four-body final state proceeding through a sequence of two-body decays via intermediate resonances, we focus our attention on the kinematic variable Δ 4 which previously has been used to parameterize the boundary of the allowed four-body phase space. We highlight the advantages of using Δ 4 as a discovery variable, and present an analysis suggesting that the pairing of Δ 4 with another invariant mass variable leads to a significant improvement over more conventional variable choices and techniques.
Published in: JHEP 1905 (2019) 008 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8754657506942749, "perplexity": 1660.347084987941}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256494.24/warc/CC-MAIN-20190521162634-20190521184634-00356.warc.gz"} |
https://zbmath.org/?q=an%3A1101.30010 | ×
# zbMATH — the first resource for mathematics
Conformal mappings between canonical multiply connected domains. (English) Zbl 1101.30010
The authors show that the modified Green’s function of a multiply connected circular domain can be represented by the Schottky-Klein prime function associated with this domain. Using this result, explicit analytic formulae for the conformal mappings from circular domains to domains with parallel, radial or circular slits are constructed.
##### MSC:
30C20 Conformal mappings of special domains 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
Full Text:
##### References:
[1] M. J. Ablowitz and A. S. Fokas, Complex Variables, Cambridge University Press, 1997. · Zbl 0885.30001 [2] H. Baker, Abelian Functions, Cambridge University Press, Cambridge, 1995. · Zbl 0848.14012 [3] A. F. Beardon, A Primer on Riemann Surfaces, London. Math. Soc. Lecture Note Ser. 78, Cambridge University Press, Cambridge, 1984. · Zbl 0546.30001 [4] E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer Verlag, 1994. [5] D. G. Crowdy and J. S. Marshall, Analytical formulae for the Kirchhoff-Routh path function in multiply connected domains, Proc. Roy. Soc. A. 461 (2005), 2477–2501. · Zbl 1186.76630 · doi:10.1098/rspa.2005.1492 [6] D. G. Crowdy and J. S. Marshall, The motion of a point vortex through gaps in walls, to appear in J. Fluid Mech. · Zbl 1085.76014 [7] D. G. Crowdy, Schwarz-Christoffel mappings to multiply connected polygonal domains, Proc. Roy. Soc. A 461 (2005), 2653–2678. · Zbl 1186.30005 · doi:10.1098/rspa.2005.1480 [8] D. G. Crowdy, Genus-N algebraic reductions of the Benney hierarchy within a Schottky model, J. Phys. A: Math. Gen. 38 (2005), 10917–10934. · Zbl 1092.37046 · doi:10.1088/0305-4470/38/50/004 [9] J. Gibbons and S. Tsarev, Conformal mappings and reductions of the Benney equations, Phys Lett. A 258 (1999), 263–271. · Zbl 0936.35184 · doi:10.1016/S0375-9601(99)00389-8 [10] P. Henrici, Applied and Computational Complex Analysis, Wiley Interscience, New York, 1986. · Zbl 0578.30001 [11] G. Julia, Lecons sur la representation conforme des aires multiplement connexes, Gaulthiers-Villars, Paris, 1934. · Zbl 0011.31204 [12] H. Kober, A Dictionary of Conformal Representation, Dover, New York, 1957. [13] P. Koebe, Abhandlungen zur Theorie der konformen Abbildung, Acta Mathematica 41 (1914), 305–344. · JFM 46.0545.02 · doi:10.1007/BF02422949 [14] V. V. Mityushev and S. V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Function Theory, Chapman & Hall/CRC, London, 1999. · Zbl 0957.30002 [15] D. Mumford, C. Series and D. Wright, Indra’s Pearls, Cambridge University Press, 2002. · Zbl 1141.00002 [16] Z. Nehari, Conformal Mapping, au]McGraw-Hill,_ New York, 1952. · Zbl 0048.31503 [17] M. Schiffer, Recent advances in the theory of conformal mapping, appendix to: R. Courant, Dirichlet’s Principle, Conformal Mapping and Minimal Surfaces, 1950. [18] M. Schmies, Computational methods for Riemann surfaces and helicoids with handles, Ph.D. thesis, University of Berlin, 2005. · Zbl 1233.30003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.843720018863678, "perplexity": 2609.887427553829}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703507971.27/warc/CC-MAIN-20210116225820-20210117015820-00409.warc.gz"} |
https://www.physicsforums.com/threads/why-do-stars-show-absorption-lines.745377/ | Why do stars show absorption lines?
1. Mar 26, 2014
zincshow
On an intuitive level, Why do stars show absorption lines? For example, the 121.6nm photon is produced when an electron falls into the lowest level of a hydrogen atom. A 121.6nm photon is absorbed by a hydrogen atom being kicked into an excited state, where it will eventually emit a 121.6nm photon.
As you go farther from the sun (or any star) surface, you would expect far more 121.6nm photons to be emitted then absorbed, but yet you see in the spectrum of a star, a shortage of 121.6nm photons, when logic (mine anyway, which is clearly wrong for some reason) would suggest you should see an excess of 121.6nm photons.
2. Mar 26, 2014
mathman
My guess. The outer layer of a star is cooler than the interior so it absorbs radiation passing through.
3. Mar 27, 2014
Bandersnatch
Imagine a beam of light of continuous spectrum coming at you.
If the photons in the beam encounter an atom of hydrogen on their way to your spectrometre, some of the 121.6nm-frequency photons will interact with the atom by boosting the electron to a higher energy state, and then getting reemitted as the electron falls back to its ground state.
The thing is, the reemitted photons can be moving in any direction. It is extremely unlikely for the reemitted photon to be moving in the exactly same direction that leads to your spectrometre.
So, each of such interactions will deduct photons of certain wavelength from the beam you're observing, giving rise to absorption lines.
4. Apr 5, 2014
snorkack
Many stars do show emission lines instead of absorption lines.
What is the cause of stars showing absorption not emission lines? How are the stars that have absorption lines different from stars that have emission lines?
5. Apr 5, 2014
Bandersnatch
These stars have a layer of hot, low pressure gas that emits them. Since the gas is hot, the energy for emission comes from collisions between atoms, rather than (just)from the incident radiation, so it adds to the background continuous spectrum rather than deducting from it like cool gas does.
6. Apr 6, 2014
Ken G
There are actually (at least) two reasons for stars to show absorption lines, and two reasons to show emission lines. The subtraction of an absorption line can be due to an atmosphere with a dropping temperature as was mentioned, or it can also just be due to scattering back down of light that is trying to escape, even in an isothermal atmosphere, as was also mentioned. Emission lines can be due to a rising atmospheric temperature, as mentioned, but it can also be due to a geometically extended "halo" around the star (often either a disk or a wind). If atoms in the halo create photons in some line, say due to conversion of continuum starlight that is absorbed in the halo, then they can create a geometrically extended circumstellar glow when the star is looked at in the frequency of that line. The star literally looks larger at that frequency! This can create an emission line, even if the circumstellar gas is cooler than the star, and even if some scattering of starlight is occuring in the line.
Scattering can even join with the geometric effect to give both absorption and emission components in the same line-- as happens when a wind creates a "P Cygni" profile. So your question opens up a lot of interesting physics, it is not at all a simple matter to decide why lines have the shapes they do, but that's why the line shapes convey so much potentially important information.
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http://kleine.mat.uniroma3.it/mp_arc-bin/mpa?yn=14-51 | Below is the ascii version of the abstract for 14-51. The html version should be ready soon.
Pavel Exner and Alexander Minakov
Curvature-induced bound states in Robin waveguides and their asymptotical properties
(240K, pdf)
ABSTRACT. We analyze bound states of Robin Laplacian in infinite planar domains with a smooth boundary, in particular, their relations to the geometry of the latter. The domains considered have locally straight boundary being, for instance, locally deformed halfplanes or wedges, or infinite strips, alternatively they are the exterior of a bounded obstacle. In the situation when the Robin condition is strongly attractive, we derive a two-term asymptotic formula in which the next-to-leading term is determined by the extremum of the boundary curvature. We also discuss the non-asymptotic case of attractive boundary interaction and show that the discrete spectrum is nonempty if the domain is a local deformation of a halfplane or a wedge of angle less than $\pi$, and it is void if the domain is concave. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9080051779747009, "perplexity": 584.4534156990293}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805687.20/warc/CC-MAIN-20171119153219-20171119173219-00487.warc.gz"} |
https://brilliant.org/problems/everything-goes-wrong/ | # Everything Goes Wrong
In how many ways a postman can deliver seven letters such that no letter is delivered at correct location?
× | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9268868565559387, "perplexity": 2675.2220852640835}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608954.74/warc/CC-MAIN-20170527133144-20170527153144-00296.warc.gz"} |
http://plantsinaction.science.uq.edu.au/content/525-control-assimilate-transport-source-sink | Back to top
# 5.2.5 - Control of assimilate transport from source to sink
Loading of sugars, potassium and accompanying anions into sieve tubes at sources determines solute concentrations in phloem sap (Table 5.1). The osmotic pressure (Π) of these solutes influences P generated in sieve tubes. Thus, source output determines the total amount of assimilate available for phloem transport as well as the pressure head driving transport along the phloem path to recipient sinks. Withdrawal of assimilates from sieve tubes at the sink end of the phloem path, by the combined activities of phloem unloading and metabolism/compartmentation (Table 5.1), determines Π of phloem sap. Other sink-located membrane transport processes influence Π around sieve tubes. The difference between intra- and extracellular Π of sieve tubes is a characteristic property of each sink and determines P in sink sieve tubes.
## Munns_PlantsInAction_Tab05.1.jpg.png.png
The pressure difference between source and sink ends of the phloem pathway drives sap flow (Equation 5.3) and hence phloem translocation rate (Equation 5.2) from source to sink. The source and sink processes governing the pressure dif-ference (Table 5.1) are metabolically dependent, thus rendering phloem translocation rates susceptible to cellular and environmental influences. The pressure-flow hypothesis predicts that the phloem path contribution to longitudinal transport is determined by the structural properties of sieve tubes (Table 5.1). Variables of particular importance are cross-sectional area (A) of the path (determined by numbers of sieve pores in a sieve plate and sieve-tube numbers) and radius of these pores (sets r in Equation 5.3). These quantities appear in Equations 5.2 and 5.3. Thus, the individual properties of each sink and those of the phloem path connecting that sink to its source will determine the potential rate of assimilate import to the sink (Figure 5.12).
## Fig_p_5.21.png
Figure 5.12. Scheme describing photoassimilate flow from a source leaf linked to two competing sinks, Sink 1 and Sink 2. Assimilate flows through alternative phloem paths (Path 1 and Path 2) each with its own conductance (Kpath) and pressure difference (P) between source and sink. Hence Path 1 is distinguished by Kpath1 and Psink1 and Path 2 by Kpath2and Psink2
The transport rate (R) of assimilate along each phloem path, linking a source with each respective sink, can be predicted from the pressure-flow hypothesis (see Equations 5.2 and 5.3) as:
$R = K_{path} (P_{source} - P_{sink}) C \tag{5.4}$
where path conductance (Kpath) is the product of path hydraulic conductivity (Lp) and cross-sectional area (A). Hence, the relative flows of assimilates between hypothetical sinks (sink 1 and sink 2) shown in Figure 5.12 may be expressed by the following ratio:
$\frac{K_{path1} (P_{source} - P_{sink1}) C} {K_{path2} (P_{source} - P_{sink2}) C} \tag {5.5}$
Partitioning of assimilates between two competing sinks is thus a function of path conductance and P at the sink end of the phloem path (Equation 5.5). Since phloem has spare capacity, any differences in the conductance of the inter-connecting paths (Figure 5.12) would exert little influence on the rate of phloem transport to the competing sinks. Assimilate partitioning between competing sinks would then be determined by the relative capacity of each sink to depress sieve-tube P at the sink end of the respective phloem path. Even when differences in path conductance are experi-mentally imposed, phloem transport rates are sustained by adjustments to the pressure differences between the source and sink ends of the phloem path (Wardlaw 1990).
These conclusions have led to a shift in focus from phloem transport to phloem loading and unloading, which are instrumental in determining the amount of assimilate translocated and its partitioning between competing sinks, respectively. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8335665464401245, "perplexity": 4792.699874079942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439740423.36/warc/CC-MAIN-20200815005453-20200815035453-00206.warc.gz"} |
https://usakairali.com/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1-%E0%B8%AB%E0%B8%A1%E0%B8%B2%E0%B8%A2-%E0%B8%82%E0%B8%AD%E0%B8%87-%E0%B8%9E%E0%B8%A5%E0%B8%B1%E0%B8%87%E0%B8%87%E0%B8%B2%E0%B8%99-%E0%B8%84%E0%B8%A7%E0%B8%B2/ | Energy that is measured by temperature
Thermal radiation in visible light can be seen on this hot metalwork. The term “thermal energy” is used loosely in respective context in physics and mastermind. It can refer to respective different chiseled physical concepts. These include the home energy or heat content of a body of matter and radiation ; heating system, defined as a type of energy transfer ( as is thermodynamic work ) ; and the characteristic energy of a degree of exemption, kelvin B T { \displaystyle k_ { \mathrm { B } } T } , in a system that is described in terms of its microscopic particulate constituents ( where T { \displaystyle T } denotes temperature and thousand B { \displaystyle k_ { \mathrm { B } } } denotes the Boltzmann ceaseless ) .
## relation to heat and inner energy
In thermodynamics, heat is energy transferred to or from a thermodynamic system by mechanisms other than thermodynamic function or transplant of matter. [ 1 ] [ 2 ] [ 3 ] Heat refers to a quantity transferred between systems, not to a property of any one system, or “ contained ” within it. [ 4 ] On the early hand, home energy and heat content are properties of a unmarried arrangement. Heat and knead depend on the way in which an energy transfer occurred, whereas home energy is a property of the country of a system and can therefore be understood without knowing how the energy got there. In a statistical mechanical explanation of an ideal gas, in which the molecules move independently between instantaneous collisions, the inner department of energy is the sum total of the flatulence ‘s freelancer particles ‘ kinetic energies, and it is this kinetic motion that is the source and the impression of the transfer of inflame across a arrangement ‘s boundary. For a gas that does not have particle interactions except for instantaneous collisions, the terminus “ thermal energy ” is effectively synonymous with “ inner energy “. In many statistical physics texts, “ thermal energy ” refers to k T { \displaystyle karat } , the merchandise of Boltzmann ‘s constant and the absolute temperature, besides written as thousand B T { \displaystyle k_ { \text { B } } T } . [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] In a material, specially in condense topic, such as a liquid or a solid, in which the part particles, such as molecules or ions, interact powerfully with one another, the energies of such interactions contribute powerfully to the internal energy of the torso, but are not plainly apparent in the temperature.
The term “ thermal energy ” is besides applied to the energy carried by a inflame flow, [ 10 ] although this can besides just be called heat or quantity of inflame.
## historic context
In an 1847 lecture titled “ On Matter, Living Force, and Heat ”, James Prescott Joule characterised respective terms that are closely related to thermal energy and heat. He identified the terms latent heating system and sensible inflame as forms of heat each affecting clear-cut physical phenomenon, namely the potential and kinetic energy of particles, respectively. [ 11 ] He described latent energy as the energy of interaction in a given shape of particles, i.e. a form of likely energy, and the sensible heat as an energy affecting temperature measured by the thermometer due to the thermal energy, which he called the surviving storm.
## Useless thermal energy
If the minimal temperature of a organization ‘s environment is T east { \displaystyle T_ { \text { e } } } and the system ‘s information is S { \displaystyle S } , then a separate of the system ‘s internal department of energy amounting to S ⋅ T e { \displaystyle S\cdot T_ { \text { e } } } can not be converted into useful make. This is the difference between the home energy and the Helmholtz exempt energy .
## References
source : https://usakairali.com
Category : Nutrition | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.959168553352356, "perplexity": 1110.505299263537}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103341778.23/warc/CC-MAIN-20220627195131-20220627225131-00658.warc.gz"} |
https://proofwiki.org/wiki/False_Statement_implies_Every_Statement/Formulation_2 | # False Statement implies Every Statement/Formulation 2
## Theorem
$\vdash \neg p \implies \left({p \implies q}\right)$
## Proof 1
By the tableau method of natural deduction:
$\vdash \neg p \implies \left({p \implies q}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p$ Assumption (None)
2 2 $p$ Assumption (None)
3 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 2, 1
4 1, 2 $q$ Rule of Explosion: $\bot \mathcal E$ 3
5 1 $p \implies q$ Rule of Implication: $\implies \mathcal I$ 2 – 4 Assumption 2 has been discharged
6 $\neg p \implies \left({p \implies q}\right)$ Rule of Implication: $\implies \mathcal I$ 1 – 5 Assumption 1 has been discharged
$\blacksquare$
## Proof 2
By the tableau method of natural deduction:
$\vdash \neg p \implies \left({p \implies q}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p$ Assumption (None)
2 1 $p \implies q$ Sequent Introduction 1 False Statement implies Every Statement: Formulation 1
3 $\neg p \implies \left({p \implies q}\right)$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged
$\blacksquare$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8911100625991821, "perplexity": 3929.144602413454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987787444.85/warc/CC-MAIN-20191021194506-20191021222006-00235.warc.gz"} |
http://www.maths.manchester.ac.uk/raag/index.php?preprint=0277 | Real Algebraic and Analytic Geometry
Previous Next
277. Jaka Cimprič, Murray Marshall, Tim Netzer:
e-mail: , ,
Submission: 2009, April 9.
Abstract:
We consider the problem of determining the closure $\overline{M}$ of a quadratic module $M$ in a commutative $\Bbb{R}$-algebra with respect to the finest locally convex topology. This is of interest in deciding when the moment problem is solvable \cite{S1} \cite{S2} and in analyzing algorithms for polynomial optimization involving semidefinite programming \cite{L}. The closure of a semiordering is also considered, and it is shown that the space $\mathcal{Y}_M$ consisting of all semiorderings lying over $M$ plays an important role in understanding the closure of $M$. The result of Schm\"udgen for preorderings in \cite{S2} is strengthened and extended to quadratic modules. The extended result is used to construct an example of a non-archimedean quadratic module describing a compact semialgebraic set that has the strong moment property. The same result is used to obtain a recursive description of $\overline{M}$ which is valid in many cases.
Mathematics Subject Classification (2000): 12D15, 14P99, 44A60.
Keywords and Phrases: moment problem, positive polynomials, sums of squares.
Full text, 24p.: dvi 163k, ps.gz 208k, pdf 268k. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9055699110031128, "perplexity": 888.6632343423878}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590295.61/warc/CC-MAIN-20180718154631-20180718174631-00308.warc.gz"} |
https://www.ijert.org/an-improved-adaptive-smo-for-speed-estimation-of-sensorless-dsfoc-induction-motor-drives-and-stability-analysis-using-lyapunov-theorem-at-low-frequencies | # An Improved Adaptive Smo for Speed Estimation of Sensorless Dsfoc Induction Motor Drives and Stability Analysis using Lyapunov Theorem at Low Frequencies
Text Only Version
#### An Improved Adaptive Smo for Speed Estimation of Sensorless Dsfoc Induction Motor Drives and Stability Analysis using Lyapunov Theorem at Low Frequencies
A.Venkatesh
Research Scholar
Electrical and Electronics Engineering, National Institute of Engineering, Mysore, Karnataka, India
Abstract In this paper, An Improved Adaptive Sliding Mode Observer (ASMO) is proposed to a Sensorless DSFOC Induction Motor Drives and their stability is analyzed. ASMO is used to estimate the Rotor Speed, Rotor Resistance, Flux, Stator and Rotor currents and the developed electromagnetic Torques. To improve the robustness and accuracy of an adaptive SMO during very low frequency operation, the sliding mode flux observer(SMFO) uses independent gains as the correction terms. The gains of current and rotor flux SMOs are designed using Lyapunov stability theory to guarantee the stability and fast convergence of the estimated variables. In this paper concentrated on Simulink Blocks and their graphs are analyzed with the help of mathematical approach. Also, comparison of results with the basic conventional controllers are done and the results proved that the proposed ASMO method shows excellent Transient and Steady state speed estimation by the Adaptive Estimators, particularly at low frequencies.
Keywords Induction Motor Drive, ASMO, stabilit,SMFO
1. INTRODUCTION
The induction machine was independently contrived by NIKOLA TESLA in 1888Q. A Polyphase (3 Phase) Induction drive systems have been co-ordinated into full-sized vehicles, Conveyors, neighbourhood electric vehicles(EV), Elevators, golf cars, motorcycles, industrial (or utility) vehicles and even in the amusement park attractions. These Polyphase drives are designed to attain maximum distance, with Low Slip and maximum power and efficiency.
With the use of 3-Phase Induction Motor we can produce direct rotational motion of wide speeds ranging from 0-18000 rpm and uniform power which is not possible with a conventional IC engine which has a restriction in speed limits. With the replacement of Induction motor in the place of Internal combustion engines(IC) which results in superior Torque/Weight ratio.
Induction motor works on the principle of electromagnetic induction and is a singly excited machine.
Figure 1.1 Squirrel Cage Induction Motor
It is evident that electrical energy exhaustion of the appliances can be economized by governing the speed of the motors. Fo that reason these three phase variable speed IM drives are uplifted to be used in the industries today as an a fair solution forever for the Decreasing of electricity generation cost.
With the Advancements of Power electronics and Mechatronics technology, a high speed ultrafast switching devices such as IGBTs and Piezeo MEMS were associated with the mechanical systems and more precise motor control strategies, such as Vector or Field control techniques(like Direct or Feedback and Indirect or Feedforward vector Control techniques) , were developed during the last decade. As a result, today IMs can be used in any kind of variable speed applications (called as Asynchronous Motors), even in servomechanism, where high-speed response with respect to the Loads and on the mark of accuracy is required.
Vector or Field control technique is used for the high performance Adjustable drive systems(Variable frequency drives or V/f Control drives). A complex current is mashed-up from the two quadrature components in the vector or field control scheme, one of which is responsible for the flux level in the motor(called as Flux Vector) and the other component, which controls the torque production(called as Torque Vector) in the motor.
The speed and Rotor resistance sensorless estimation concept along with the implementation of Model Reference Adaptive System (MRAS) schemes was studied[1]. It is a well-known fact that the performance of MRAS based speed estimators is betterwhen compared
with the other speed estimators with regards to its stability approach and design complexity. Although this thesis is all about ASMO based speed estimators, but it is also the aim of this project is to investigate several speed sensorless estimation strategies for IMs. Conceptual Explanations on the different type of control strategies also were briefly discussed. In the view of simulation works is concerned, the MRAS based speed sensorless estimation schemes chosen in this thesis have been implemented in the Direct Stator Field oriented control (DSFOC) to evaluate the Flux and Speed Observers performance.
Present Research efforts have targeted on replacing the positional encoder on the IM ROTOR shaft and to develop the sensorless drives without effecting their static and dynamic performance. Several speed and dynamic Rotor resistance estimation methods of the sensorless drives have been Introduced. They are classified as machine dynamic-model based methods and high- frequency signal injection methods [4]-[6]. The high- frequency signal injection methods are independent of the machine model. So, these schemes are insensitive to parameters variation and they gives an accurate speed and position estimation particularly at very low and zero stator frequencies. However, they cause high frequency noise which leads to system performance detoriation, in addition to that they require a unusual design which was explained in [6]. Machine model-based schemes gives an accurate and robust speed estimation at extreme high and medium speeds. However, their accuracy and robustness depends mainly on the accuracy of the dynamic IM model for the good operation of the particular drive at very low and zero stator frequencies [4]. For operation at very low stator frequency, dynamic machine model-based methods with more precise models and robust state estimators are to be researched to increase their reliability in a particular region of operations [4]-[5]. Different conventional schemes in the literature survey have been widely used such as MRAS observers [7], adaptive flux observer (AFO) [8]-[12], Extend Kalman filter (EKF) [13], and the sliding mode observers (SMO) [14].
SMO, as a variable structure control system(VSCS), is one of these techniques that gained superior affinity because of its simplicity in design [15], robustness, insensitivity to linear and non-linear parameters variation. MRAS speed-estimation methods which are based on SMO have been introduced in [16] to improve the speed- estimation with superior accuracy when compared with the classical MRAS scheme.
Speed and flux estimation schemes along with ASMO have been discussed in [17]-[22]. These schemes utilizes sliding surfaces which involves stator-current errors in flux estimation. Ingeneral, ASMO schemes use a time-variable full order observer/estimator IM model for flux and current estimation. The estimated speed of these adaptive methods, considered as the final stage of the estimation procedure. Therefore, the errors in the estimated speed due to parameters mismatch and noises reflect directly on the rotor flux estimation, which results in decrease in the accuracy of the flux and speed estimators. These unwanted effects declines the drive performance, specifically at very
low and zero stator frequencies, since the fundamental excitation of the motor drive is low. Hence, solution for these issues is helped in concentrating the non adaptive sliding mode observers(Non-ASMO).
SMO(observer) and SMC(controller) for sensorless Direct Torque Controllers(DTC) of IM drives have been discussed in [17-18]. In these research works, the flux estimation was based on Non-ASMO. Specifically in sensorless obervers, without speed adaption to provide increased accuracy in a wide speed range operation, which have been presented and compared with others in [19]- [20].
In general, the sensorless IM drive based ASMO are considered as a Adaptive speed observer. In this adaptive speed observer, SMO employs discontinuous correction terms using the current estimation error. Therefore, the errors in the estimated speed reflect specifically on flux estimation and which degrades theaccuracy of state estimators. Unlike the research works in [21]-[22], this paper takes the help of a current model based ASMO for speed and flux estimation to improve their accuracy of estimations, Which can be realized by designing the flux estimation algorithm with a correction term using current estimation error with separate gains are used for current and flux estimation. These gains are designed based on stability conditions of Lyapunov stability theory. This featured solution will improves accuracy of the rotor flux estimators, and which subsequently, Increases the speed and flux estimation accuracy at very low stator frequency operation. The indirect field oriented control (IFOC) for speed control of a sensorless IM drive using the developed estimation algorithms is built by the help of MATLAB or with help of Simulinks.
2. INSPIRATION FOR FUTURE EXTENSION Most global optimization problems are nonlinear and thus
difficult to solve, and they become even more challenging when uncertainties are present in objective functions and constraints. Efficiency of a Optimization techniques depends upon the search algorithm. Most of the search techniques are single search stage. But in Eagle Strategy there are two searches one is called as Global search and the other is called as the Local Search.
Think Like a Golden Eagle
"The Victorious strategist only seeks battle after the victory has been won, whereas he who is destined to defeat first fights and afterwards looks for victory."- Sun Tzu
The behaviour of Golden Eagles (Aquila chrysaetos) is inspiring. An eagle forages in its own territory by flying freely in a random manner much like then L´evy flights. Once the prey is sighted, the eagle will change its search strategy to an intensive chasing tactics so as to catch the prey as efficiently as possible. There are two important components to an eagles hunting strategy: random search by L´evy flight (or walk) and intensive chase by locking its aim on the target.
For the Future extension for the Stability analysis of a 3- Phase Induction Motor Rotor Speed and Rotor Resistance we can implement ES-PSO(Eagle Strategy with Particle Swarm Optimization) Technique.
3. MATHEMATICAL MODELLING
A typical construction of a squirrel cage IM along with the main parts is as shown in Figure 1.1. Its main advantages are the electrical simplicity and mechanical ruggedness, Self starting and the lack of rotating contacts (brushes or Sliprings
) requires low maintenance and its capability of better speed regulation.
Before going to analyze any motor or generator it is very much important to obtain the machine in terms of its equivalent mathematical equations. Traditional per phase equivalent circuit has been widely used in steady state analysis and design of induction motor, but it is not appreciated to predict the dynamic performance of the motor. The dynamics consider the instantaneous effects of varying voltage/currents, stator frequency, and torque disturbance. The dynamic model of the induction motor is derived by using a two phase motor in direct and quadrature axes. This approach is desirable because of the conceptual simplicity obtained with two sets of windings, one on the rotor and the other in the stator. The equivalence between the three phase and two phase machine models is derived from simple observation, and this approach is suitable for extending it to model an n-phase machine by means of a two phase machine.
REFERENCE FRAMES
The required transformation in voltages, currents, or flux linkages is derived in a generalized way. The reference frames are chosen to be arbitrarily such as stationary, rotor and synchronous reference frames. R.H. Park, in the 1920s,
of Induction Motor is accepted and adopted universally, that is called as Dynamic d q model.
Fig 3.1 to Transformation
Consider a symmetrical 3-phase induction machine with stationary asbscs axis at 120 Degree angle apart. So
we needs to transform the 3-phase stationary reference frame
asbscs variables into 2-phase stationary reference frame
(ds-qs) variables. Assume that ds- qs axes are oriented at angle of 90 Degree(Since it is a 2-Phase) which is as shown in Fig 3.1
proposed a new theory of electrical machine analysis to
V
cos
sin
1 V s
represent the machine in d q model. He transformed the
qs
as
as
stator variables to a synchronously rotating reference frame
V cos(120 ) sin(120 ) 1 V s
(3.1)
fixed in the rotor, which is called Parks transformation.[2]-
bs
ds
[3] V
cos(120 ) sin(120 ) 1 V s
cs
os
He showed that all the time varying inductances that occur due to an electric circuit in relative motion and electric circuits with varying magnetic reluctances could be eliminated. In 1930s, H.C Stanley showed that time varying Inductances in the voltage equations of an induction machine due to electric circuits in relative motion can be eliminated by transforming the rotor variables to a stationary reference frame fixed on the stator. Later, G. Kron proposed a transformation of both stator and rotor variables to a
DYNAMIC EQUATIONS OF INDUCTION MACHINE
Generally, an IM can be described uniquely in an arbitrary rotating frames i.e.,Stationary reference frame or Synchronously rotating frame. Induction Machine modelling equations with ds qs Axes is given by following dynamic equations which are obtained from the KVL equations to the Dynamic Modelled equivalent circuit of an IM.
synchronously rotating reference that moves with the rotating
V s R is
• d s
magnetic field.
AXES TRANSFORMATION
qs s qs dt qs
d
(3.2)
V s R is s
The per phase equivalent circuit of the induction motor is only valid in steady state condition. It doesnt hold good while dealing with the transient response of the motor. In transient response condition the voltages and currents in three phases are not in balance condition. It is too much difficult to study the machine performance by analyzing the three phases. In order to reduce this complexity the
ds s ds dt ds
When above equations (3.2) are converted into d e qe
Axis then above dynamic equations are re-written as follows
v R i d ( )
transformation of axes from 3 to 2 is needed. Another reason for the transformation is to analyze any machine with
qs s qs dt qs e r ds
d
(3.3)
n number of phases. Thus, an equivalent per phase model
vds Rsids dt ds (e r ) qs
For transient studies of a Adjustable/Variable speed drives, it is usually more convenient to simulate an IM when it converts into a stationary reference frame.
Moreover,calculations with stationary reference frame are less complex due to zero sequence frame Component. For small signal stability analysis, a synchronously rotating frame which yields steady-state voltages and currents under balanced conditions.
Fig 3.2 Two-phase equivalent diagram of induction motor
Figure 3.3 shows the qe- de dynamic model equivalent circuit
The IM described in stationary reference frame interms of stator currents and Rotor Fluxes are as follows which was described in Reference[6] is designed as follows in the Figure 3.4
Figure 3.4 Induction Motor Simulation Diagram
Symbol
Parameter
Value
de qe
Synchronously Rotating Reference Frame Direct and Quadrature Axes
d s qs
Stationary Reference Frame Direct and Quadrature Axes (Also known as
Axes)
F
Frequency
[Hz]
I s & I s dr ds
d s Axis Rotor and Stator Currents
[Amp]
Iqr & Iqs
qe Axis Rotor and Stator Currents
[Amp]
e
Angle of Synchronously Rotating Frame
[Degree]
e
Rotor Angle
[Degree]
sl
Slip Angle
[Degree]
Lr & Ls
Rotor & Stator Inductance
[Henry]
Llr & Lls
Rotor & Stator Leakage Inductance
[Henry]
Ldm & Ldm
d e & qe Axis Magnetizing Inductance
[Henry]
Rs & Rr
Stator and Rotor Resistance
[ ]
V s & V s dr ds
Vqr & Vqs
s & s dr ds
qr & qs
d s Axis Rotor and Stator Voltages
qe Axis Rotor and Stator Voltages
d s Axis Rotor and Stator Flux linkage
qe Axis Rotor and Stator Flux linkages
[Volt] [Volt]
Symbol Parameter Value de qe Synchronously Rotating Reference Frame Direct and Quadrature Axes d s qs Stationary Reference Frame Direct and Quadrature Axes (Also known asAxes) F Frequency [Hz] I s & I s dr ds d s Axis Rotor and Stator Currents [Amp] Iqr & Iqs qe Axis Rotor and Stator Currents [Amp] e Angle of Synchronously Rotating Frame [Degree] e Rotor Angle [Degree] sl Slip Angle [Degree] Lr & Ls Rotor & Stator Inductance [Henry] Llr & Lls Rotor & Stator Leakage Inductance [Henry] Ldm & Ldm d e & qe Axis Magnetizing Inductance [Henry] Rs & Rr Stator and Rotor Resistance [ ] V s & V s dr dsVqr & Vqs s & s dr ds qr & qs d s Axis Rotor and Stator Voltagesqe Axis Rotor and Stator Voltagesd s Axis Rotor and Stator Flux linkageqe Axis Rotor and Stator Flux linkages [Volt] [Volt]
of induction motor under synchronously rotating reference
frame, if
vdr vqr 0
and
e 0
then it becomes
stationary reference frame dynamic model.
Figure 3.3 Dynamic de-qe equivalent circuits of machine
4. ADAPTIVE SLIDING MODE OBSERVER (ASMO):
The ability to generate a sliding motion on the error between the measured plant output and the output of the observer ensures that a sliding mode observer produces a set
of states. Estimates that are precisely comparable with the actual output of the plant. Standard Testing function (Plant) along with Adaptive SMO is as shown in the below figure 4.1
5. PROBLEM FORMULATION
This paper defines the Objective Function as reducing the error between the original Induction Machine Angular Rotor Speed and Rotor Resistance and Rotor Flux and Electromagnetic Torque Developed by the Machine along with respect to standard system to make IM as a stabilized system. So, the objective function can be formulated as follows
f (e , e ) 1 e T Pe d 2 aT a bT b
x 2 x x 2 r
(5.1)
Where P and Q are positive definite symmetrical matrices. a and b are the vectors which contains the non-zero elements of A and B matrices of Induction Motor modelling equations. and are the diagonal matrices with positive elements which determines the speed of adaption of the Optimizing Technique.
Where
e e T e T
e T e T
Fig. 4.1 Block diagram of ASMO
x id iq d
q
ASMO TESTING FUNCTION:
This simulation is use to demonstrate the robustness property of sliding mode control. Here, second order stable system is consider, which means that states of the system will reach the equilibrium in infinite time.
The transfer function of system is given
The above objective function is to be tuned by Model Reference Adaptive System(MRAS) technique. The corresponding Block diagram of MRAS is implemented first. Later, ASMO is applied to optimize the above objective function in Eqn 5.1.
G(s)
1
s2 5s 6
CANONICAL FORM FOR THE NOMINAL SYSTEM:
1. First, we will observe the response for system without disturbance using state feedback controller. In order to do so keep the switch 2 in SW 1 position
Canonical Form of representation of a standard Continuous- Time state space system is as follows
X1 (t) A11x1 (t) A12 y(t) B1u(t)
(open plant). Observer the response, you will find that both the state will reach to the zero in infinite time,
2. In next step we will introduce the sinusoidal
Y (t) A21x1 (t) A22 y(t) B2u(t)
(5.2)
disturbance by moving switch from SW1 position to SW 2 position and observer the response of the system; you will find that both the state will oscillate.
Similarly, Adaptive Sliding Mode Observer (ASMO) is represented as follows in State space form is as follows
3. In the last step will apply the SMC controller to the plant with disturbance and observer the response.
The simulation state responses are as shown below
X1 (t) A11x1 (t) A12 y(t) B1u(t) Lv
Y (t) A21x1 (t) A22 y(t) B2u(t) v
(5.3)
Where
(x1, y)
represent the state estimates,
L R(n p) p
is a gain matrix and vi M sgn( yi yi ) where M R
Error of the system along with ASMO is
ex A11ex (t) A12ey (t) Lv
ey A21ex (t) A22ey (t) v
(5.4)
Fig. 4.2 Simulation Results of ASMO Testing Function
A Lyapunov stability function can be examined to define the stability of the Induction Machine along with the state estimators is represented as follows,
V 1 e T Pe d 2 aT a bT b
(5.5)
(5.5)
(5.5)
2 x x 2 r
The Time derivate of above function to guarantee the stability
of the system will be V 0 . Controller gains are adjusted in
such a way that V is a Negative Definite function. The main
aim of ASMO controller parameters adjustements are done to ensure that the state trajectory of the switching function should lies within the specified switching surface. The following state trajectories are used to define the stability of the system.
Fig 5.1. State Trajectories to identify the type of stability
Block Diagram representation of Induction Motor along with the State Estimators( Flux and Speed Estimators) along with PID tuners is as represented as follows in figure 5.2
Fig 5.2 Block Diagram of Induction Motor along ith Estimators
Simulation Diagram of DSFOC Induction Motor without ASMO is as shown in the below figure 5.3
Fig 5.3 Simulation Diagram of DSFOC Induction Motor without ASMO
6. SIMULATION RESULTS
The parameter values are considered for the dynamic Induction Machine modelling are as follows:
Ls 0.1004[Henry], Lr 0.0969 [Henry], Lm 0.0915[Henry], Rr 1.294[Ohm], Rs 1.54 [Ohm],
Fig 6.1: Simulation Results of Direct Axis current Id
DSFOC Induction Motor
Fig 6.2: Simulation Results of Quadrature Axis current Iq
Induction Motor
DSFOC
Fig 6.4 Simulation Results of DSFOC Actual Speed, Measured Speed and
Figure 6.3: Simulation Results of Stator Flux of a DSFOC Induction Motor
Fig 6.4 Simulation Diagram of DSFOC Induction Motor along with ASMO/C
Electromagnetic Torque When Load Torque TL 0
Fig 6.5 Simulation Results of DSFOC Induction Motor with ASMO Actual Speed, Measured Speed and Electromagnetic Torque When Load Torque
TL 0
In this paper, a new ASMO for Rotor speed and flux and rotor resistance estimation of the sensorless speed DSFOC IM has been designed. Lyapunov stability theorem[25] is utilized for the stator current and Rotor flux estimated to determine the tunable observer gains of the current and flux estimators to guarantee the occurance of the stability of the IM within the sliding surfaces.In dynamic state, the estimator speed are calculated and During the steady state, the state estimators(Flux and Speed) error dynamics behaves as a Reduced order state dynamics and subsequently it is controlled only by the error of the Rotor fluxes. So, the Lyapunov function is selected to estimate the speed of the Rotor and its position. Simulation and Experimental results
confirms that the usefulness of the new proposed ASMO for estimating the Rotor Speed of IM at very low and zero stator frequencies. It has been observed that the sensorless drive with the proposed adaptive SMO provides a good performance in comparison with the previous conventional works. But, The SMO algorithm is designed using the dynamic IM mathematical model, Its observability is usually at zero stator frequency. Observability of the IM can be improved by additional stator voltage injection methods and the advanced optimizing techniques like ESPSO[23]-[24] which could be implemented to Increase the effectiveness in reaching the sliding surfaces to attain the better stability which can be considered to be the future extension work.
REFERENCES
1. Y.Narendra Kumar, A.Venkatesh, Ch.Anusha, K.Vani, D.Siddarda Estimation of Speed based on Flux Oriented Control of an Induction Motor drive Model with Reference Adaptive System Scheme in Transactions on Engineering and Sciences Vol. 2, Issue 4, April 2014.
2. Modern Power Electronics and AC drives text book by Bimal K.Bose original Prentice Hall PTR ISBN 0-13-016743-6
3. Electric Motor drives Modelling, Analysis and Control Text book by
R. Krishnan ©2001 Prentice Hall ISBN 0-13-0910147
4. M. Pacas, Sensorless drives in industrial applications, IEEE Ind. Electron.Mag., vol. 5, no. 2, pp. 1623, Jun. 2011.
5. C. S. Staines, C. Caruana, G. M. Asher, and M. Sumner, "Sensorless control of induction machines at zero and low frequency using zero sequence currents, IEEE Trans. Ind. Electron., vol. 53, no. 1, pp.195- 206, Feb. 2006.
6. J. Holtz, "Sensorless control of induction machineswith or without signal injection?," IEEE Trans. Ind. Electron., vol. 53, no. 1, 7-30, Feb. 2006.
7. Y. B. Zbede, S. M. Gadoue, and D. J. Atkinson, "Model predictive MRAS estimator for sensorless induction motor drives," IEEE Trans. Ind. Electron., vol. 63, no. 6, pp. 3511- 3521, Jun. 2016.
8. Z. Yin, Y. Zhang, C. Du, J. Liu, X. Sun, and Y. Zhong, "Research on anti-error performance of speed and flux estimation for induction motors based on robust adaptive state observer," IEEE Trans. Ind. Electron., vol. 63, no. 6, pp. 3499-3510, Jun. 2016.
9. M. Hinkkanen, Analysis and design of full-order flux observers for sensorless induction motors, IEEE Trans. on Ind. Electron., vol. 51, no.5, pp.1033- 1040, 2004.
10. S. Suwankawin, S. Sangwongwanich, Design strategy of an adaptive fullorder observer for speed-sensorless induction-motor drives-tracking performance and stabilization, IEEE Trans. Ind. Electron., vol. 53, no.1, pp. 96119, 2006.
11. L. Harnefors, M. Hinkkanen, Complete stability of reduced-order and fullorder observers for sensorless IM drives, IEEE Trans. Ind. Electron., vol. 55, no.3, pp. 13191329, 2008.
12. B. Chen, T. Wang, W. Yao, K. Lee, Z. Lu, Speed convergence rate- based feedback gains design of adaptive full-order observer in sensorless induction motor drives, IET Electr. Power Appl., vol. 8, no.1, pp. 1322, 2014.
13. M. Barut, S. Bogosyan, and M. Gokasan, Speed-sensorless estimation for induction motors using extended Kalman filters, IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 272280, Feb. 2007.
14. C. Lascu, I. Boldea, and F. Blaabjerg, "Very-Low-Speed Variable- Structure Control of Sensorless Induction Machine Drives Without Signal Injection," IEEE Transactions on Industry Applications, vol. 41, no. 2, pp. 591- 598, March/April 2005.
15. A. Sabanovic, Variable structure systems with sliding modes in motion controlA survey, IEEE Trans. Ind. Informat., vol. 7, no. 2, pp. 212223, May 2011.
16. L. Zhao, J. Huang, H. Liu, B. Li and W. Kong, "Second-order sliding mode observer with online parameter identification for sensorless induction motor drives," IEEE Trans. Ind. Electron., vol. 61, no. 10, pp. 5280- 5289, October 2014.
17. C. Lascu and G. D. Andreescu, "Sliding-mode observer and improved integrator with dc-offset compensation for flux estimation in sensorlesscontrolled induction motors," IEEE Tran on Ind Electronics, vol. 53, no. 3, pp. 785- 794, June 2006.
18. M. Ghanes and G. Zheng, "On sensorless induction motor drives: sliding-mode observer and output feedback controller," IEEE Trans. Ind. Electron., vol. 56, no. 9, pp. 3404-3413, September 2009.
19. C. Lascu, I. Boldea, and F. Blaabjerg, "Comparative study of adaptive and inherently sensorless observers for variable-speed induction-motor drives, " IEEE Trans. Ind. Electron., vol. 53, no. 1, pp. 57-65, Feb. 2006.
20. C. Lascu, I. Boldea, and F. Blaabjerg, "A Class of speed-sensorless slidingmode observers for high-performance induction motor drives," IEEE Trans. Ind. Electron., vol. 56, no. 9, pp. 3394- 3403, Sept. 2009.
21. M. Comanescu, "Single and double compound manifold sliding mode observers for flux and speed estimation of the induction motor drive," IET Electr. Power Appl., vol. 8, Iss. 1, pp. 2938, 2014.
22. M. S. Zaky, M. M. Khater, S. S. Shokralla, and H. A. Yasin, "Wide- speedrange estimation with online parameter identification schemes of sensorless induction motor drives," IEEE Trans on Ind Electronics, vol. 56, no. 5, pp. 1699- 1707, May 2009.
23. 1A.Venkatesh, 2Dr.G.Raja Rao An Improved Adaptive Control System for a Two Wheel Inverted Pendulum-Mobile Robot using Eagle Strategy with a Particle Swarm Optimization © 2017 JETIR September 2017, Volume 4, Issue 6.
24. Hamza Yapici and Nurettin Cetinkaya An Improved Particle Swarm Optimization Algorithm Using Eagle Strategy for Power Loss Minimization Mathematical Problems in Engineering Volume 2017, Article ID 1063045.
25. Katsuhiko Ogata, Modern Control Engineering 4th edition PrenticeHall Publications, Jan 2002 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8533579111099243, "perplexity": 3788.053917124114}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987828425.99/warc/CC-MAIN-20191023015841-20191023043341-00063.warc.gz"} |
http://support.sas.com/documentation/cdl/en/statug/68162/HTML/default/statug_krige2d_details12.htm | # The KRIGE2D Procedure
### Anisotropic Models
Subsections:
In all of the theoretical models considered previously, the lag distance h is entered as a scalar value. This implies that the correlation between the spatial process at two point pairs is dependent only on the separation distance , not on the orientation of the vector . A spatial process described by an SRF {} with this property is called isotropic, as is the associated covariance or semivariogram.
However, real spatial phenomena often show directional effects. Particularly in geologic applications, measurements along a particular direction might be highly correlated, while typically the perpendicular direction shows little or no correlation. Such processes are called anisotropic; see, for example, Journel and Huijbregts (1978, section III.B.4).
When the correlation structure varies across different directions, you need different models for each direction so that you can account correctly for the continuity within the SRF. The following subsections describe how techniques are applied to override the anisotropy effects for computational purposes. First, characteristics of anisotropy are examined.
The semivariogram sill is a measure of the process variability; hence the direction of the highest continuity is perpendicular to the direction where the highest sill occurs. If the sill is the same in all directions, then the direction with the highest range indicates highest continuity. The directions in which the spatial process is most and least correlated are called the major and minor axis of anisotropy, respectively.
In some cases, these directions are known a priori. This can occur in mining applications where the geology of a region is known in advance. In most cases however, nothing is known about possible anisotropy. Depending on the amount of data available, using several directions is usually sufficient to determine the presence of anisotropy and to find the approximate major and minor axis directions; see the discussion in the section Anisotropy in Chapter 122: The VARIOGRAM Procedure, documentation. You can find a detailed example of anisotropy investigation in the section An Anisotropic Case Study with Surface Trend in the Data in Chapter 122: The VARIOGRAM Procedure, documentation.
After you explore an anisotropic process and you identify the minor and major axis directions, you can compute the anisotropy factor parameter R which is defined as
where is the semivariogram range in the direction of the minor axis and is the semivariogram range in the direction of the major axis.
There are two types of anisotropy, depending on which semivariogram characteristics change in different directions. These types are the geometric and the zonal anisotropy, and either or both can be present. Both are examined in detail in the following subsections. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8709991574287415, "perplexity": 582.363725048601}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187828411.81/warc/CC-MAIN-20171024105736-20171024125736-00263.warc.gz"} |
https://www.komal.hu/verseny/1999-02/fiz.e.shtml | English Információ A lap Pontverseny Cikkek Hírek Fórum
Versenykiírás Tudnivalók Nevezési lap Feladatok Eredmények Korábbi évek Arcképcsarnok Munkafüzet
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# Exercises and problems in Physics February 1999
## New experimental problem:
m. 204. Observe the motion of balls started from the side of a bath wiped dry and bunged. Due to the slope of the bottom of the bath, the balls moving to and fro drift towards the bung. How does the average drift velocity in the direction of the slope of the bath depend on the height h of the starting point? Make experiments using balls of different material and size (e.g. steel ball, glass ball, table-tennis ball, etc.), and using different starting points with the same height h as well. (6 points)
## New exercises:
FGy. 3224. 5 cm3 of water is mixed with 5 cm3 of heavy water in one vessel, and 5 g of water is mixed with 5 g of heavy water in another one. Which mixture is of higher density? (3 points)
FGy. 3225. A bus driver is separated from the passengers by a Plexiglas pane mounted behind him. A car moving at a speed of 110 km/h starts to overtake the bus travelling at a speed of 80 km/h. In what direction and at what speed does the reflection of the lights of the car made by the Plexiglas pane move, as seen by a passenger on the bus? (3 points)
FGy. 3226. An inert gas with a volume of 2 dm3 and a pressure of 105 Pa expands to a volume of 6 dm3 in such a way that the process can be represented on a (p,V) diagram by a straight line. At the end of the process, the increase of the internal energy of the gas is twice the work done by the gas.
a) Determine the pressure of the gas in the final state.
b) How much heat does the gas take up during the expansion? (4 points)
FGy. 3227. An astronaut throws a stone on the Moon at an angle of 60o with the horizontal and at an initial speed of 20 m/s. What are the tangential and the normal components of the acceleration of the stone 3 seconds after the throwing? What is the radius of curvature of the trajectory in this point? (5 points)
## New problems:
FF. 3228. A small body of mass m can be fastened to any point of a homogeneous rod of length l and mass M suspended at one end. Where should the small body be fixed so that the period of the rod is minimum? What is this minimum period? (E.g. let m=6M.)
FF. 3229. A body starting from the top of a slope of inclination and height h slides down the slope and rebounds from a wall perpendicular to the slope. It loses some mechanical energy both while sliding and while rebounding. What distance does it cover in how much time before it finally stops? The coefficient of dynamic friction is , the collision number (the ratio of the speeds after to before rebounding) is k. Numerical data: =30o, h=1 m, =, k=.
FF. 3230. The top plate of a charged plane capacitor is fixed, while the bottom one is kept in equilibrium by the gravitational and electrostatic forces. What is the type of the equilibrium when a) the charge, b) the pd. of the capacitor is constant?
FF. 3231. A metal ring with a mass of 1 gram and a charge of 2.10-8 C rotates at high velocity about its symmetry axis perpendicular to its plane. Calculate its magnetic dipole momentum if its momentum is 0.1 kg m2 s-1. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8199160099029541, "perplexity": 588.9942892417797}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689775.56/warc/CC-MAIN-20170923175524-20170923195524-00278.warc.gz"} |
http://math.stackexchange.com/questions/36041/what-is-the-volume-of-x-y-z-in-mathbbr3-geq-0-sqrtx-sqrt/36057 | # What is the volume of $\{ (x,y,z) \in \mathbb{R}^3_{\geq 0} |\; \sqrt{x} + \sqrt{y} + \sqrt{z} \leq 1 \}$?
I have to calculate the volume of the set
$$\{ (x,y,z) \in \mathbb{R}^3_{\geq 0} |\; \sqrt{x} + \sqrt{y} + \sqrt{z} \leq 1 \}$$
and I did this by evaluating the integral
$$\int_0^1 \int_0^{(1-\sqrt{x})^2} \int_0^{(1-\sqrt{x}-\sqrt{y})^2} \mathrm dz \; \mathrm dy \; \mathrm dx = \frac{1}{90}.$$
However, a friend of mine told me that his assistant professor gave him the numerical solutions and it turns out the solution should be $\frac{1}{70}$. Also, I found out that this would be the result of the integral
$$\int_0^1 \int_0^{1-\sqrt{x}} \int_0^{1-\sqrt{x}-\sqrt{y}} \mathrm dz \; \mathrm dy \; \mathrm dx,$$
which is pretty much the same as mine just without squares in the upper bounds. My question is: Is the solution provided by the assistant professor wrong or why do I have to calculate the integral without squared upper bounds?
Also, is there any tool to compute the volume of such sets without knowing how one has to integrate?
-
Because $z \leq (1-\sqrt{x} - \sqrt{x})^2$. Why is the assistant professor correct? – Huy Apr 30 '11 at 16:45
I answered too soon. It looks like you are right. :) – Grumpy Parsnip Apr 30 '11 at 16:48
PS: I of course meant $z \leq (1-\sqrt{x}-\sqrt{y})^2$. Somehow I can't edit my first comment anymore. – Huy Apr 30 '11 at 16:52
An integral $(*)\ \int_B f(x){\rm d}(x)$ over a three-dimensional domain $B$ depends on the exact expression for $f(x)$, $\ x\in{\mathbb R}^n$, and on the exact shape of the domain $B$. The latter is usually defined by a set of inequalities of the form $g_i(x)\leq c_i$. The information about $B$ has to be entered in the course of the reduction of the integral $(*)$ to a sequence of nested integrals. So, as a rule, there is a lot of work involved in the process of reducing everything to the computation and evaluation of primitives.
Now sometimes there is another way of handling such integrals: Maybe we can set up a parametric representation of $B$ with a parameter domain $\tilde B$ which is a standard geometric object like a simplex, a rectangular box or a half sphere. In the case at hand we can use the representation $$g: \quad S\to B,\quad (u,v,w)\mapsto (x,y,z):=(u^2,v^2,w^2)$$ which produces $B$ as an essentially 1-1 image of the standard simplex $$S:=\{(u,v,w)\ |\ u\geq0, v\geq0, w\geq 0, u+v+w\leq1\}\ .$$ In the process we have to compute the Jacobian $J_g(u,v,w)=8uvw$ and obtain the following formula: $${\rm vol}(B)=\int_B 1\ {\rm d}(x)= \int_S 1 \> J_g(u,v,w) \> {\rm d}(u,v,w)=\int_0^1\int_0^{1-u}\int_0^{1-u-v} 8uvw \> dw dv du ={1\over 90}\ .$$ (In this particular example the simplification is only marginal.)
@Willie: I did understand that. :) @Christian: In order to apply this kind of substitution, we needed $g$ to be a diffeomorphism. I'm pretty sure in general such a mapping wouldn't be a diffeomorphism, but is this here the case due to the fact that both $B$ and the standard simplex only contain non-negative components? – Huy May 1 '11 at 11:29
@Christian: As far as I see, even with only non-negative components, this is no diffeomorphism, as the inverse function is not differentiable in $0$. So why is one allowed to do this transformation here? – Huy May 1 '11 at 11:37
@Huy: it's the same this as with polar coordinates: they are not a diffeomorphism on the whole plane, but it is in the plane withouth $x \geq 0$. Here it is a diffeomorphism on the whole plane without the origin; does this fact change your integral? (think about polar coordinates again) – Andy May 1 '11 at 12:00
@Huy: The map $g$ has to be a diffeomorphism $S'\to B'$ where $S\triangle S'$ and $B\triangle B'$ are sets of measure $0$ in their respective ${\mathbb R^d}$'s. – Christian Blatter May 1 '11 at 16:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9060387015342712, "perplexity": 153.49548821214907}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398462709.88/warc/CC-MAIN-20151124205422-00042-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-x-4-x-2-1 | Algebra
Topics
# How do you solve x^4 + x^2 = 1 ?
Mar 14, 2016
Solve as a quadratic in ${x}^{2}$ using the quadratic formula, then take square roots...
#### Explanation:
${x}^{4} + {x}^{2} = 1$
Subtract $1$ from both sides to get:
${x}^{4} + {x}^{2} - 1 = 0$
Writing ${x}^{4} = {\left({x}^{2}\right)}^{2}$ we have:
${\left({x}^{2}\right)}^{2} + \left({x}^{2}\right) - 1 = 0$
This is in the form $a {X}^{2} + b X + c = 0$ with $X = {x}^{2}$, $a = 1$, $b = 1$ and $c = - 1$.
We can use the quadratic formula to find:
${x}^{2} = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
$= \frac{- 1 \pm \sqrt{{1}^{2} - \left(4 \cdot 1 \cdot - 1\right)}}{2 \cdot 1}$
$= \frac{- 1 \pm \sqrt{5}}{2}$
So:
$x = \pm \sqrt{\frac{- 1 + \sqrt{5}}{2}}$
Or:
$x = \pm \sqrt{\frac{- 1 - \sqrt{5}}{2}} = \pm \sqrt{\frac{1 + \sqrt{5}}{2}} i$
##### Impact of this question
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http://physics.stackexchange.com/questions/38548/questions-regarding-standing-waves | # Questions regarding standing waves
I have two questions regarding mechanical waves.
1) We know that standing waves are created when any wave traveling along the medium will reflect back when they reach the end. But in an open organ pipe there is nothing to oppose the wave and reflect it back. So then how are standing waves created in such a case. I did not find the answer in my textbook.
2)In standing waves where is the pressure greater - nodes or anti-nodes?
-
Your first question is answered by this Open University article. To summarise, consider a low pressure region travelling along the tube towards the open end. The air outside is at atmospheric pressure, so when the low pressure region hits the end of the tube air from the atmosphere rushes in and creates a compression wave heading back down the tube. The opposite happens when a high pressure region hits the end of the tube.
The reflection of a sound wave at the end of a tube is an example of an impedance mismatch.
This also bears upon your second question. At an open end the wave inverts i.e. a reflected pressure peak becomes a trough, and a trough becomes a peak. This is in contrast to the closed end where a pressure peak reflects as a peak. This means the pressure changes are lowest at the open end and highest at the closed end.
The reflection at the open end is not 100%: some energy escapes. Actually this is obvious, because if no energy could escape an organ pipe wouldn't make any sound. The sound we hear is the energy that has escaped from the open end.
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In organ pipes,vibrational antinodes will be present at any open end and vibrational nodes will be present at any closed end. When one plays a organ pipe, the air that is pushed through the pipe vibrates and standing waves are formed. If the both ends of the organ are open then the pattern for the fundamental frequency (the lowest frequency and longest wavelength pattern) will have antinodes at the two open ends and a single node in between.
- | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8326899409294128, "perplexity": 206.5882219394664}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375635604.22/warc/CC-MAIN-20150627032715-00187-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://swmath.org/software/8646 | CoReLG
Computing with real Lie algebras: real forms, Cartan decompositions, and Cartan subalgebras. We describe algorithms for performing various tasks related to real simple Lie algebras. These algorithms form the basis of our software package CoReLG, written in the language of the computer algebra system GAP4. First, we describe how to efficiently construct real simple Lie algebras up to isomorphism. Second, we consider a real semisimple Lie algebra 𝔤. We provide an algorithm for constructing a maximally (non-)compact Cartan subalgebra of 𝔤; this is based on the theory of Cayley transforms. We also describe the construction of a Cartan decomposition 𝔤=𝔨⊕𝔭. Using these results, we provide an algorithm to construct all Cartan subalgebras of 𝔤 up to conjugacy; this is a constructive version of a classification theorem due to Sugiura.
Keywords for this software
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References in zbMATH (referenced in 3 articles )
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Sorted by year (citations) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9215674996376038, "perplexity": 699.2626799958266}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542060.60/warc/CC-MAIN-20161202170902-00364-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://jp.mathworks.com/matlabcentral/fileexchange/89719-numerical-differentiation-of-data-derivative?s_tid=prof_contriblnk | ## Numerical Differentiation of Data (derivative)
version 3.0.3 (383 KB) by
Numerical differentiation of data (i.e. arrays) over the domain of the data or at specified points.
Updated 27 Aug 2021
From GitHub
# derivative
Numerical differentiation of data (i.e. arrays).
## Syntax
dy = derivative(x,y)
dy = derivative(x,y,x_star)
## Description
dy = derivative(x,y) returns the derivative of a set of data, vs. (which are stored in y and x). dy stores the derivative of y vs. x at every point in x.
dy = derivative(x,y,x_star) returns the derivative of a set of data, vs. (which are stored in y and x), at the set of points specified by (x_star). x_star can be a scalar or a vector.
• This function is the equivalent of trapz and cumtrapz for numerical differentiation and is especially useful for estimating derivatives where the underlying function is unknown (i.e. we have some data set vs. that describes a function , but this underlying function is unknown and so we use the data set to approximate its derivative).
• If we do know the underlying function , then we can still use derivative to estimate its derivative. We can do this by defining a vector of points x and then populating a vector y with evaluations of at every point in x. However, in this case (where the underlying function is known), iderivative and derivest are better functions to use.
• See "EXAMPLES.mlx" or the "Examples" tab on the File Exchange page for examples.
### Cite As
Tamas Kis (2021). Numerical Differentiation of Data (derivative) (https://github.com/tamaskis/derivative-MATLAB/releases/tag/v3.0.3), GitHub. Retrieved .
##### MATLAB Release Compatibility
Created with R2021a
Compatible with any release
##### Platform Compatibility
Windows macOS Linux | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8856238722801208, "perplexity": 2175.9366575832573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363309.86/warc/CC-MAIN-20211206163944-20211206193944-00300.warc.gz"} |
http://mathonline.wikidot.com/interior-and-boundary-points-of-a-set-in-a-metric-space | Interior and Boundary Points of a Set in a Metric Space
# Interior and Boundary Points of a Set in a Metric Space
Recall from the Interior, Boundary, and Exterior Points in Euclidean Space that if $S \subseteq \mathbb{R}^n$ then a point $\mathbf{a} \in S$ is called an interior point of $S$ if there exists a positive real number $r > 0$ such that the ball centered at $a$ with radius $r$ is a subset of $S$.
Furthermore, a point $\mathbf{a}$ is called a boundary point of $S$ if for every positive real number $r > 0$ we have that there exists points $\mathbf{x}, \mathbf{y} \in B(\mathbf{a}, r)$ such that $\mathbf{x} \in S$ and $\mathbf{y} \in S^c$.
We will now generalize these definitions to metric spaces $(M, d)$.
Definition: Let $(M, d)$ be a metric space and let $S \subseteq M$. A point $a \in S$ is said to be an Interior Point of $S$ if there exists a positive real number $r > 0$ such that the ball centered at $a$ with radius $r$ with respect to the metric $d$ is a subset of $S$, i.e., $B(a, r) \subseteq S$. The set of all interior points of $S$ is called the Interior of $S$ and is denoted $\mathrm{int} (S)$
In shorter terms, a point $a \in S$ is an interior point of $S$ if there exists a ball centered at $a$ that is fully contained in $S$. Note that from the definition above we have that a point can be an interior point of a set only if that point is contained in $S$. Therefore $\mathrm{int} (A) \subseteq A$.
Definition: Let $(M, d)$ be a metric space and let $S \subseteq M$. A point $a \in M$ is said to be a Boundary Point of $S$ if for every positive real number $r > 0$ we have that there exists points $x, y \in B(a, r)$ such that $x \in S$ and $y \in S^c$. The set of all boundary points is called the Boundary of $S$ and is denoted $\partial S$ or $\mathrm{bdry} (S)$.
A point $a \in M$ is said to be a boundary point of $S$ if every ball centered at $a$ contains points in $S$ and points in the complement $S^c$. Notice that from the definition above that a boundary point of a set need not be contained in that set. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9846661686897278, "perplexity": 18.500284920063017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334974.57/warc/CC-MAIN-20220927002241-20220927032241-00261.warc.gz"} |
http://en.wikipedia.org/wiki/Loop_space | # Loop space
Jump to: navigation, search
In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of based maps from the circle S1 to X with the compact-open topology. Two elements of a loop space can be naturally concatenated. With this concatenation operation, a loop space is an A-space. The adjective A describes the manner in which concatenating loops is homotopy coherently associative.
The quotient of the loop space ΩX by the equivalence relation of pointed homotopy is the fundamental group π1(X).
The iterated loop spaces of X are formed by applying Ω a number of times.
An analogous construction of topological spaces without basepoint is the free loop space. The free loop space of a topological space X is the space of maps from S1 to X with the compact-open topology. That is to say, the free loop space of a topological space X is the function space $\mathrm{Map}(S^1,X)$. The free loop space of X is denoted by $\mathcal{L}X$.
The free loop space construction is right adjoint to the cartesian product with the circle, while the loop space construction is right adjoint to the reduced suspension. This adjunction accounts for much of the importance of loop spaces in stable homotopy theory.
## Relation between homotopy groups of a space and those of its loop space
The basic relation between the homotopy groups is $\pi_k(X) \approxeq \pi_{k-1}(\Omega X)$.[1]
More generally,
$[\Sigma Z,X] \approxeq [Z, \Omega X]$
where, $[A,B]$ is the set of homotopy classes of maps $A \rightarrow B$, and $\Sigma A$ is the suspension of A. In general $[A, B]$ does not have a group structure for arbitrary spaces $A$ and $B$. However, it can be shown that $[\Sigma Z,X]$ and $[Z, \Omega X]$ do have natural group structures when $Z$ and $X$ are pointed, and the aforesaid isomorphism is of those groups. [2]
Note that setting $Z = S^{k-1}$ (the $k-1$ sphere) gives the earlier result.
## References
1. ^ http://topospaces.subwiki.org/wiki/Loop_space_of_a_based_topological_space
2. ^ May, J. P. (1999), "8", A Concise Course in Algebraic Topology (PDF), U. Chicago Press, Chicago, retrieved 2008-09-27 (chapter 8, section 2) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 16, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9248206615447998, "perplexity": 319.62057470299965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929899.62/warc/CC-MAIN-20150521113209-00312-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://mathhelpforum.com/number-theory/125796-abstract-algebra-2-a.html | # Math Help - abstract algebra #2
1. ## abstract algebra #2
if p is prime and (a,b)=p, then (a^2,b^2)=?
2. Dear Deepu,
$(a,b)=p$
Take $(a^2,b^2)=k$
Now since, $k\mid{a^2}$ and $k\mid{b^2}$
Therefore, $k\mid{a}$ and $k\mid{b}$
Then since the gcd of "a" and "b" is "p" ; $k\leq{p}$ ---------------A
Also since $p=(a,b)\Rightarrow{p\mid{a}}$ and $p\mid{b}\Rightarrow{p\mid{a^2}}$ and $p\mid{b^2}$
Then since the gcd of $a^2$ and $b^2$ is "k" ; $p\leq{k}$ ----------------B
From A and B ; p=k
Therefore $(a^2,b^2)=p$
Hope this helps.
3. Thank you very much.
4. Originally Posted by Sudharaka
...
Now since, $k\mid{a^2}$ and $k\mid{b^2}$
Therefore, $k\mid{a}$ and $k\mid{b}$
This implication is incorrect.
Consider, for example, $a=6,b=9 \Rightarrow a^2=36, ~b^2=81 \Rightarrow 9 | 36, ~ 9 | 81$ but 9 obviously does not divide 6.
---
Let $(a,b)=p \Rightarrow p|a, ~ p|b \Rightarrow p^2|a^2,~p^2|b^2$
So, we now know that $p^2$ is a common divisor of $a^2,~b^2$. I'll leave it to you to show that it is the greatest common divisor.
5. Suppose $a=p_1^{\alpha_1} p_2^{\alpha_2} \cdot\cdot\cdot p_n^{\alpha_n}$ and
$b=p_1^{\beta_1} p_2^{\beta_2} \cdot\cdot\cdot p_n^{\beta_n}$ where $\alpha_i,\beta_j \geq 0$.
Set $k=(a,b)$. Then $k=p_1^{\min{\alpha_1,\beta_1}} p_2^{\min{\alpha_2,\beta_2}} \cdot\cdot\cdot p_n^{\min{\alpha_n,\beta_n}}$.
Now $a^2=p_1^{2\alpha_1} p_2^{2\alpha_2} \cdot\cdot\cdot p_n^{2\alpha_n}$ and $b^2=p_1^{2\beta_1} p_2^{2\beta_2} \cdot\cdot\cdot p_n^{2\beta_n}$.
So $(a^2,b^2) = p_1^{\min{2\alpha_1,2\beta_1}} p_2^{\min{2\alpha_2,2\beta_2}} \cdot\cdot\cdot p_n^{\min{2\alpha_n,2\beta_n}} = p_1^{2\min{\alpha_1,\beta_1}} p_2^{2\min{\alpha_2,\beta_2}} \cdot\cdot\cdot p_n^{2\min{\alpha_n,\beta_n}} = k^2$.
Hence $(a^2,b^2) = (a,b)^2$.
6. Originally Posted by Defunkt
This implication is incorrect.
Consider, for example, $a=6,b=9 \Rightarrow a^2=36, ~b^2=81 \Rightarrow 9 | 36, ~ 9 | 81$ but 9 obviously does not divide 6.
---
Let $(a,b)=p \Rightarrow p|a, ~ p|b \Rightarrow p^2|a^2,~p^2|b^2$
So, we now know that $p^2$ is a common divisor of $a^2,~b^2$. I'll leave it to you to show that it is the greatest common divisor.
Dear Defunkt,
Thank you for showing my mistake. I tried to solve the problem again using the suggestion you had given, but still haven't had any luck. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 35, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9907042384147644, "perplexity": 426.7892732539533}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375096290.39/warc/CC-MAIN-20150627031816-00151-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://brilliant.org/problems/but-is-the-solution-unique/ | # But is the solution unique?
The answer to this question is a positive integer.
What is the Euler totient function to the answer of this question?
× | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9446477293968201, "perplexity": 338.880111673109}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948583808.69/warc/CC-MAIN-20171216045655-20171216071655-00628.warc.gz"} |
http://math.stackexchange.com/questions/83012/reformulating-a-problem-of-solving-a-system-of-linear-equations | # Reformulating a problem of solving a system of linear equations
Suppose that I have a program that can solve the system of linear equation $Ay=x$ efficiently when $A$ is a Hermitian matrix. I have a system of equations which is $(I+bH)y=x$, where $I$ is the identity matrix and $b$ is a complex scalar and $H$ is the hermitian matrix. Obviously $(I+bH)$ is not hermitian and neither is $(I/b+H)$. Can I reformulate this problem in terms of the Hermitian problem: $Ay=x$ ?
-
Introduce a variable $z=(I+bH)^*x$. Where $(I+bH)^*$ denotes the conjugate transpose of $(I+bH)$. Then solve $(I+bH)^*(I+bH)y=z$ using your method for Hermitian matrices since $(I+bH)^*(I+bH)$ is Hermitian.
$y$ doesn't necessarily solve your original problem, but if it doesn't, then it is a least squares solution to your equation if $(I+bH)^*(I+bH)$ is not singular.
If $(I+bH)^*(I+bH)$ is singular then $y=(I+bH)^+(I+bH)^{+*}z$ is a least squares solution to your original equation, where $A^+$ denotes the Moore-Penrose pseudoinverse of A.
Assuming your algorithm is coded correctly though, it should handle both of these cases already.
-
This does not solve the problem since you have shifted the problem to $z=(I+bH)^*x$ even if $z$ is computed correctly. You recover the original problem when obtaining $x$. – user13838 Dec 2 '11 at 15:08
It was my understanding that y is the variable vector and x is some known vector. This is the usual use of the notation, and since the OP did not specify, then I assumed it was the case. If it is not the case that x represents some known vector, then it is not really clear what the poster is asking. In any case: For any known x we can calculate the corresponding y(provided it exists) using the method I described – Tim Seguine Dec 10 '11 at 20:18
@percusse or to rephrase: I am assuming we are looking for some relation between y and x. I was assuming we are looking for y as a function of x, since the other direction is trivial. We define this relationship then in terms of it's values given a particular x. This is the only reasonable interpretation of the poster's notation that I could think of, and my solution solves this. – Tim Seguine Dec 10 '11 at 20:49
I understand your solution but it is still using a intermediate step where the linear equation $z=(I+bH)^*x$ with a non-hermitian matrix. This is what OP is trying to by-pass, that is solving a linear system with a non-Hermitian A matrix. – user13838 Dec 11 '11 at 20:11
@percusse You apparently don't understand my solution then. We don't have to solve the system $z=(I+bH)^*x$ . x is a given. We only have to apply the transformation to x so that we can obtain z. We never have to find x in terms of z, so there is no solving going on. The only system we have to solve is the Hermitian one. – Tim Seguine Dec 13 '11 at 9:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9742896556854248, "perplexity": 171.13363543710994}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1430452189638.13/warc/CC-MAIN-20150501034949-00023-ip-10-235-10-82.ec2.internal.warc.gz"} |
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