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http://mathhelpforum.com/advanced-math-topics/280495-formula-area-object-bounded-lines-radii-cartesian-plane.html | # Thread: Formula For Area of Object Bounded By Lines and Radii on Cartesian Plane
1. ## Formula For Area of Object Bounded By Lines and Radii on Cartesian Plane
At one time, I had a formula used in a FORTRAN program that provided the area of an object comprised of lines and radii. I provided an NC input file that contained x-y coordinates along with x-y offsets for a radius and it also indicated CW or CCW circular interpolation. The areas under the lines and radii would be summed when travelling left to right and subtracted when going right to left. I know that I can calculate the areas manually. Can someone point me to where I could find that formula?
I know I can calculate the area under a line using the trapezoidal formula.
I can calculate:
Area under a radius: Area inside arc & below= area of segment + area under line x increasing clockwise or x decreasing counterclockwise motion
Area outside arc & below = area under line - area of segment x increasing clockwise or x decreasing counterclockwise.
When in Relative Quadrants ! & II sum areas. When in Relative Quadrants III & IV subtract areas. Circular motion will be calculated by quadrant so note about theta </= 90 should be expanded to state area calculated by quadrant.
I found the formula in the appendix of a calculus book published ~1970-1973. I only remember that it was blue and looked similar to the jpg file.
I know it's a stretch to find the formula but I do not know how to re-create it.
Thanks,
Barry
Attached Thumbnails
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https://www.bartleby.com/questions-and-answers/abuate3-dar.-valuate-vt-iv.-ftc-part-one-use-the-fundamental-theorem-of-calculus-to-write-an-express/be155cd8-d9c9-40c7-9545-ccb7b6a61709 | # abuate3 dar.valuateVTIV.FTC (part one)Use the fundamental theorem of calculus to write an expression for the anti-derivative of the function y = f(x) = sin(13).If F(x)12 V t+ldt, find the derivative F'(x).Use the fundamental theorem of calculus to find the derivative of the functiony = g(x)-j" cos(t2) dt.Exercise 11 in section 4.4
Question
1 views
I am studying for a quiz and do not understand a few of the concepts that will be covered on it. Can you help me with the problem under "IV. FTC (part one)" found in the attached document? Thank you.
check_circle
Step 1
The fundamental theorem of calculus states that for a continuous real-valued function, f on a closed interval [a, b], let the function F be defined, for all x in [a, b] as
Step 2
Here, the function is given as y=sin(x3) which is continuous on any closed interval, hence consider a closed interval [a,b]. Use the fundamental theorem of calculus and obtain the antiderivative F as follows.
Step 3
The antiderivative F which is uniformly c...
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http://mathhelpforum.com/advanced-math-topics/279434-total-relation.html | ## Total relation
Hey,
Let A be a finite set and R is an order relation on A.
Prove:
If R is not total,then there exist two different total order relations containing R. | {"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": 0.8404300212860107, "perplexity": 952.7724124002694}, "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-2019-18/segments/1555578553595.61/warc/CC-MAIN-20190422115451-20190422141451-00516.warc.gz"} |
https://tex.stackexchange.com/questions/547579/alignment-in-tabularx-using-exp-siunitx | # Alignment in tabularx using exp. siunitx
It's never ending story about that -> previous post. Simply, I just want to align to 'pm' sign my number columns. I've already tried replacing headers {l | l *{2}{>{\centering\arraybackslash}X}} with {l | l l l *{2}{>{\centering\arraybackslash}X} and than to check one column with siunitx: {l | l S[table-align-uncertainty, separate-uncertainty=true] l l *{2}{>{\centering\arraybackslash}X} but it doesn't work at all.
MWE:
\documentclass{article}
\usepackage{float, enumitem, amsmath}
\usepackage{rotating}
\usepackage{booktabs, makecell, multirow, tabularx, threeparttable}
\usepackage{colortbl, color} %I preffer that instead of xcolor because xcolor give me errors with beamer
\usepackage{caption}
\captionsetup{belowskip=-5pt}
\captionsetup[table]{justification=raggedright,singlelinecheck=off}
\begin{document}
\begin{table}
\centering
\renewcommand\arraystretch{1.1}
\caption{Simple caption.}
\label{tab:setTag}
\begin{tabularx}{\linewidth}{l | l *{2}{>{\centering\arraybackslash}X}}
\Xhline{1pt}
& \thead{Method 1} & \thead{Method 2} \\
\Xhline{0.7pt}
& & \multicolumn{2}{c}{First section} \\
\Xcline{3-4}{0.6pt}
&Very long name & $11,5 \pm 0,5$ & $444,34 \pm 71,9$ \\
& Very long name & $109,2 \pm 75,3$ & $2,8 \pm 664,36$ \\
& Very long name& $9438 \pm 8$ & $256,0 \pm 98,1$ \\
& Very long name & $11,5 \pm 0,5$ & $444,34 \pm 71,9$ \\
& Very long name & $109,2 \pm 75,3$ & $2,8 \pm 664,36$ \\
& Very long name & $9438 \pm 8$ & $256,0 \pm 98,1$ \\
\end{tabularx}
\end{table}
\end{document}
I've changed header to \begin{tabularx}{\linewidth}{l | X S[table-align-uncertainty, separate-uncertainty=true] S[table-align-uncertainty, separate-uncertainty=true]*{2}{>{\centering\arraybackslash}}} and data from first row to 11,5(5) & 444,34(719 and I get that result (strange Methods headers):
I've now had errors with array package '>{}' at wrong position. Can someone please tell me which one is the correct one? And another one with 'missing # inserted in alignment preamble' and the third one 'missing inserted' but I don't know why because it shows line with \end{tabularx}. New part of the code: \begin{tabularx}{\linewidth}{l | X S[table-align-uncertainty, separate-uncertainty=true] S[table-align-uncertainty, separate-uncertainty=true]*{2}{>{\centering\arraybackslash}}} \Xhline{1pt} & \thead[l]{Results} & {\thead{Method 1}} & {\thead{Method 2}} \\ \Xhline{0.7pt} & & \multicolumn{2}{c}{First section} \\ \Xcline{3-4}{0.6pt} \multirow{3}{*}{\rothead{Description 1}} &Very long name &11,5(5)$&$444,34(719)$\\ & Very long name &$109,2(753)$&$2,8(63)$\\ & Very long name&$9438(8)$&$256,0(981)$\\ &Very long name &$11,5(5)$&$444,34(719)$\\ & Very long name &$109,2(753)$&$2,8(63)$\\ & Very long name&$9438(8)$&$256,0(981)\$ \\
\end{tabularx}
It shows (which is obviously not align):
• If you want 11,5 \pm 0,5 as the output in an S type columns of the table, you should use 11,5(5) as the input in the S type column. – leandriis Jun 2 '20 at 18:49
• Why do you use tabularx here? Your table is already narrow enough to fit into the textwidth. By using tabularx here you only stretch the table introducing unneccessary white space. – leandriis Jun 2 '20 at 18:53
• @leandriis my whole table is much wider (the name of the columns etc.) but for the MWE and just align operation it is not necessary to provide the whole one (in the previous post I showed how huge the table is) :) – Dominika Jun 2 '20 at 18:55
• I see. Thanks for the explanation. I was a bit confused since the dummy text you included in your previous example is something entirely different than numbers with their uncertainties which you showed in this example code. – leandriis Jun 2 '20 at 19:00
• I know, so sorry - just learning how to quick describe problem with as much information as I can gave. So, my 'real table' has very long 'Results' section and in some of the 'First section'/'Second section' etc. a lot of numbers data. – Dominika Jun 2 '20 at 19:03
The following should work:
\documentclass{article}
\usepackage{float, enumitem, amsmath}
\usepackage{rotating}
\usepackage{booktabs, makecell, multirow, tabularx, threeparttable}
\usepackage{colortbl, color} %I preffer that instead of xcolor because xcolor give me errors with beamer
\usepackage{caption}
\captionsetup{belowskip=-5pt}
\captionsetup[table]{justification=raggedright,singlelinecheck=off}
\usepackage{siunitx}
\begin{document}
\sisetup{table-align-uncertainty, separate-uncertainty=true}
\begin{tabularx}{\linewidth}{l | X S[table-format=4.1(3)] S[table-format=3.2(5)]}
\Xhline{1pt}
& {\thead{Method 1}} & {\thead{Method 2}} \\
\Xhline{0.7pt}
& & \multicolumn{2}{c}{First section} \\
\Xcline{3-4}{0.6pt}
& Very long name & 11,5(5) & 444,34(7190) \\
& Very long name & 109,2(753) & 2,80(66436) \\
& Very long name & 9438(8) & 256,0(981) \\
& Very long name & 11,5(5) & 444,34(7190) \\
& Very long name & 109,2(753) & 2,80(66436) \\
& Very long name & 9438(8) & 256,0(981) \\
\end{tabularx}
\end{document}
• yes, aligment works pretty well but rotated column looking very bad now (I've changed yours \multirow{3}{*}{\rothead{Description 1}} to \multirow{6}{*}{\rothead{Description 1}}, but it doesn't help – Dominika Jun 2 '20 at 21:23
• besides rotating text issue I also putted in description example how some of the numbers looks like. The +/- symbol is too close to them - please take a look (last image). – Dominika Jun 2 '20 at 21:35
• thank you so much, you also help me a lot with better understanding tables! – Dominika Jun 2 '20 at 21:47
• If you want to add numbers with more than 1 decimal place into the first column, you will have to adjust the table-format option of this specific column accordingly. In order to add a short horizontal line in a S type column, use {--} instead of - -. – leandriis Jun 2 '20 at 22:12
• Got it, but when I put S[table-format=4.10(5)] or S[table-format=4.1(100)] or S[table-format=400.1(5)] it's just ridiculous and nasty. How it works? – Dominika Jun 2 '20 at 22:15
just answering your comment under leandriis answer, you can simply rotate the text directly there is no need for a multirow, also as I commented previously in chat I would never use tabularx for a table of data like this, tabularx is all about tables of text and line breaking in columns.
\documentclass{article}
\usepackage{float, enumitem, amsmath}
\usepackage{rotating}
\usepackage{booktabs, makecell, multirow, tabularx, threeparttable}
\usepackage{colortbl, color} %I preffer that instead of xcolor because xcolor give me errors with beamer
\usepackage{caption}
\captionsetup{belowskip=-5pt}
\captionsetup[table]{justification=raggedright,singlelinecheck=off}
\usepackage{siunitx}
\begin{document}
\sisetup{table-align-uncertainty, separate-uncertainty=true}
\centering
\begin{tabular}{l | l S[table-format=4.1(3)] S[table-format=3.2(5)]}
\Xhline{1pt}
& {\thead{Method 1}} & {\thead{Method 2}} \\
\Xhline{0.7pt}
\smash{\rotatebox[origin=r]{90}{\bfseries Description 1}}
& & \multicolumn{2}{c}{First section} \\
\Xcline{3-4}{0.6pt}
& Very long name & 11,5(5) & 444,34(7190) \\
& Very long name & 109,2(753) & 2,80(66436) \\
& Very long name & 9438(8) & 256,0(981) \\
& Very long name & 11,5(5) & 444,34(7190) \\
& Very long name & 109,2(753) & 2,80(66436) \\
& Very long name & 9438(8) & 256,0(981) \\
\end{tabular}
\end{document}
• I will be remember that, thank you for your time but I put Description 1 in multirow because I want to forced this text to be centered. In original table I have 16 rows (like my first post here, linked in description) so I think centered looks better – Dominika Jun 2 '20 at 21:39
• @Dominika you can easily move it down, just put it in a later row or add an \hspace after the 1 multirow desn't force vertical centering – David Carlisle Jun 2 '20 at 21:40
• Of course - that's not a problem with odd number, but I will try. Once again thank you so much for your time and advices. – Dominika Jun 2 '20 at 21:42
• @Dominika actually you should accept lenadriis answer that had better siunitx use than the answer I was going to give, which was your main question, I just stole the entire code from that answer to add this extra comment, I'll feel a bit bad if I steal the points as well:-) – David Carlisle Jun 2 '20 at 21:44
• Okay, right - I apologise. I've already changed, but thank you for your both :) Now I will try to use it in the much bigger table, hope it will work. Fingers crossed! – Dominika Jun 2 '20 at 21:49 | {"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": 0.8174256086349487, "perplexity": 4242.602433554468}, "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-2021-21/segments/1620243989819.92/warc/CC-MAIN-20210518094809-20210518124809-00203.warc.gz"} |
https://causal-fermion-system.com/theory/physics/principles/ | # The Theory of Causal Fermion Systems
## Underlying Physical Principles
### Prerequisites
Causal fermion systems evolved from an attempt to combine several physical principles in a coherent mathematical framework. As a result, these principles appear in a specific way:
• The principle of causalityA causal fermion system gives rise to a causal structure and a time direction. The causal action principle is compatible with this notion of causality in the sense that the pairs of points with spacelike separation do not enter the Euler-Lagrange equations. In simple terms, points with spacelike separation do not interact.
• The local gauge principle. Local gauge freedom becomes apparent when representing the physical wave functions in bases of the spin spaces. More precisely, choosing a pseudo-orthonormal basis $(\mathfrak{e}_\alpha(x))_{\alpha=1,\ldots, \text{dim}(S_x)}$ of each spin space $(S_x, \prec .|. \succ_x)$, a physical wave function $\psi^u$ can be represented as
$\displaystyle \psi^u(x) = \sum_{\alpha=1}^{\text{dim} S_x} \psi^\alpha(x)\: \mathfrak{e}_\alpha(x)$
with component functions $\psi^1, \ldots, \psi^{\text{dim} S_x}$. The freedom in choosing the basis $(\mathfrak{e}_\alpha)$ is described by the group of unitary transformations with respect to the indefinite spin inner product. This gives rise to the transformations
$\displaystyle \mathfrak{e}_\alpha(x) \rightarrow \sum_\beta U^{-1}(x)^\beta_\alpha\; \mathfrak{e}_\beta(x)$ and $\displaystyle \psi^\alpha(x) \rightarrow \sum_\beta U(x)^\alpha_\beta\: \psi^\beta(x)$ .
As the basis $(\mathfrak{e}_\alpha)$ can be chosen independently at each spacetime point, one obtains local gauge transformations of the wave functions, where the gauge group is determined to be the isometry group of the spin scalar product. The causal action is gauge invariant in the sense that it does not depend on the choice of spinor bases.
• The Pauli exclusion principle. This can be seen in various ways. One formulation of the Pauli exclusion principle states that every fermionic one-particle state can be occupied by at most one particle. In this formulation, the Pauli exclusion principle is respected because every wave function can either be represented in the form $\psi^u$ (the state is occupied) with $u \in \H$ or it cannot be represented as a physical wave function (the state is not occupied). Via these two conditions, the fermionic projector encodes for every state the occupation numbers $1$ and $0$, respectively, but it is impossible to describe higher occupation numbers.
More technically, one may obtain the connection to the fermionic Fock space formalism by choosing an orthonormal basis $u_1, \ldots, u_f$ of ${\mathcal{H}}$ and forming the $f$-particle Hartree-Fock state
$\Psi := \psi^{u_1} \wedge \cdots \wedge \psi^{u_f}$ .
Clearly, the choice of the orthonormal basis is unique only up to the unitary transformations
$\displaystyle u_i \rightarrow \tilde{u}_i = \sum_{j=1}^f U_{ij} \,u_j$ with $U \in \text{U}(f)$ .
Due to the anti-symmetrization, this transformation changes the corresponding Hartree-Fock state only by an irrelevant phase factor,
$\psi^{\tilde{u}_1} \wedge \cdots \wedge \psi^{\tilde{u}_f} = \det U \: \psi^{u_1} \wedge \cdots \wedge \psi^{u_f}$ .
Thus the configuration of the physical wave functions can be described by a fermionic multi-particle wave function. The Pauli exclusion principle becomes apparent in the total anti-symmetrization of this wave function.
Clearly, the above Hartree-Fock state $\Psi$ does not account for quantum entanglement. Indeed, the description of entanglement requires more general Fock space constructions (→ connection to quantum field theory in the mathematics section).
• The equivalence principleStarting from a causal fermion system $({\mathcal{H}}, {\mathcal{F}}, \rho)$, spacetime $M:= \text{supp} \rho$ is given as the support of the universal measure. Thus spacetime is a topological space (with the topology on $M$ induced by sup-norm on $\text{L}({\mathcal{H}}$). In situations when spacetime has a smooth manifold structure, one can describe spacetime by choosing coordinates. However, there is no distinguished coordinate systems, giving rise to the freedom of performing general coordinate transformations. The causal action as well as all the constraints are invariant under such transformations. In this sense, the equivalence principle is implemented in the Theory of Causal Fermion Systems.
However, other physical principles are missing. For example, the principle of locality is not included. Indeed, the causal action principle is non-local, and locality is recovered only in the continuum limit. Moreover, our concept of causality is quite different from causation (in the sense that the past determines the future) or microlocality (stating that the observables of spacelike separated regions must commute).
Author | {"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": 0.988990306854248, "perplexity": 334.50825106439686}, "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/1679296946584.94/warc/CC-MAIN-20230326235016-20230327025016-00318.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/216439-cosets-form-partition-x.html | # Math Help - Cosets form a partition of X
1. ## Cosets form a partition of X
Hello guys, I hope I'm posting in the right place.
I'm having problems solve the following problem: Let Y be a subspace of a vector space X. Show that the distinct cosets x + Y (x in X) form a partition of X.
I don't quite understand how these cosets work so I couldn't think of any way to approach this problem.
2. ## Re: Cosets form a partition of X
Odd, I thought I was pretty good at linear algebra but I also thought that "cosets" was a topic from group theory! But, of course, the set of vectors, with the single operation of addition is a group so the 'coset' x+ Y, for a given x, is the set of all vectors of the form x+ y where y is any vector in Y. To show that x+ Y is a "partition" of X, we need to show that every vector in one and only one of those sets.
Let v be any vector in X. Choose any vector y in Y, let x= z- y.
Now, suppose v is in both x+ Y and x'+ Y, with $x\ne x'$. That is, $v= x+ y_1$ and $v= x'+ y_2$ with both $y_1$ and $y_2$ in Y. So we have $x+ y_1= x'+ y_2$ from which it follows that $x- x'= y_2- y_1$. What does that tell 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": 7, "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": 0.9757245182991028, "perplexity": 218.82745573423267}, "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-2015-48/segments/1448398445142.9/warc/CC-MAIN-20151124205405-00036-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/definition-clarification-for-fourier-transform.833551/ | # Definition clarification for Fourier transform
1. Sep 20, 2015
### space-time
I have been studying Fourier transforms lately. Specifically, I have been studying the form of the formula that uses the square root of 2π in the definition. Now here is the problem:
In some sources, I see the forward and inverse transforms defined as such:
F(k) = [1/(√2π)] ∫-∞ f(x)eikx dx
f(x) = [1/(√2π)] ∫-∞ f(k)eikx dk
In other cases, I've seen:
F(k) = [1/(√2π)] ∫-∞ f(x)e-ikx dx
f(x) = [1/(√2π)] ∫-∞ f(k)eikx dk
Notice that in the first version of the forward transform (the one that solves for F(k)), the exponential in the integrand has a positive sign in the exponent ikx, while in the 2nd version it has a negative ikx.
Which version is correct? Are they both correct and it is a matter of convention? Are neither correct?
Also, is there some way to do a multiple dimensional Fourier transform using volume integrals? If so, what is the formula for that (preferably including (√2π))?
2. Sep 20, 2015
### MisterX
Only the 2nd pair is correct. There are a couple conventional issues, but no matter what the sign on the exponent has to change for the inverse transform relative to the forward transform.
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# Solution: Based on molecular structure, arrange the following oxyacids in order of increasing acid strength:HClO3, HIO3, HBrO3
###### Problem
Based on molecular structure, arrange the following oxyacids in order of increasing acid strength:
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https://www.physicsforums.com/threads/confusion-over-hydraulic-gradient-l-parameter.835218/ | # I Confusion Over Hydraulic Gradient, L parameter
Tags:
1. Sep 30, 2015
### Typhon4ever
I've come across two different approaches to quantifying what l is in the equation for hydraulic gradient Δh/L. In this first picture L is the parallel distance along the datum across the reference plane
But in this second picture L is the length along the pipe
Why are the two L's different? I'm asking because there's a picture in a book of a sloping sand layer sandwiched between clay layers and L is taken to be like in the first image but the idea of a permeable sand layer between two effectively impermeable clay layers looks like the 2nd pipe image.
2. Oct 5, 2015
### Greg Bernhardt
Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
3. Mar 2, 2016
### Claudio Meier
The correct interpretation is that of the second figure: the length to compute the gradient is that "travelled" by the water. After all, the hydraulic gradient is the spatial rate at which head (energy per unit weight of water) is lost or dissipated; basically: how many meters of head are lost per meter of distance travelled?
The first figure shows an unconfined aquifer in which the vertical scale is distorted or exaggerated; in most cases, the slope of unconfined aquifers is very flat, so that if one measures L in the horizontal, the difference with the actual distance travelled is negligible (because cosine of a small angle tends to 1, so that the horizontal distance will be almost equal to the length of the hypotenuse of the triangle).
Indeed, the Dupuit-Forchheimer assumption used to solve many groundwater and well problems assumes that flow is horizontal in unconfined aquifers, neglecting the small vertical component of the flow (by the way, note the irony: Dupuit means "of the well" in French).
Note that I wrote "travelled" between quotation marks above, because the actual distance takes the tortuosity of the flow paths into account, and we are not doing that here: our distances are measured assuming that there are no solid particles continuously deflecting the flow at the small scale.
Hope this helps,
Claudio Meier
Draft saved Draft deleted
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https://www.physicsforums.com/threads/vector-subspace.288856/ | # Vector subspace
1. Jan 31, 2009
### fk378
1. The problem statement, all variables and given/known data
Let F be the field of all real numbers and let V be the set of all sequences (a1,a2,....a_n,...), a_i in F, where equality, addition, and scalar multiplication are defined component-wise.
(a) Prove that V is a vector space over F
(b) Let W={(a1, a2,....,a_n,...) in V | lim a_n = 0 as n-->inf}. Prove that W is a subspace of V.
(c) Let U={(a1,...,a_n,...) in V | summation of (a_i)^2 is finite, i evaluated from 1 to inf}. Prove that U is a subspace of V and is contained in W.
3. The attempt at a solution
I know that in order for W to be a subspace of V, W must form a vector space over F under the operations of V. I've already proved (a). Do I need to know the limit of a_n to prove (b) or is that just for (c)? It seems like proving (b) is pretty similar to (a), right? Any tips on proving (c)?
2. Jan 31, 2009
### descendency
In part b, the lim a_n = 0 as n --> inf just means that only those a_n that are contained in V that meet the criteria lim a_n = 0 as n --> inf are contained in the set.
So to prove part b, you just need to show that addition and scalar multiplication are closed in the subspace.
Sorry, I'm not too sure about part C (I don't want to guess and tell you wrong either).
3. Jan 31, 2009
### slider142
Suppose {a_i} is a sequence in U such that the a_i's do not tend to 0 as i increases without bound. Consider the infinite series Sum[(a_i)^2] = Sum[b_i] where each b_i is positive and does not tend to 0. Suppose the sum converges to L, which means for each r > 0, there exists a number N so that | L - Sum[b_i] | < r where i ranges from 0 to n for all n > N. If the terms b_i are never negative and never tend to 0, can this condition be satisfied? (Note that the condition that the b_i's tend to 0 is that for all d > 0, there is some N so that |b_i| < d for all i > N.)
4. Feb 1, 2009
### fk378
I don't really understand, but I would say that the condition cannot be satisfied because if the terms b_i never tend to 0 then the summation must diverge.
5. Feb 1, 2009
### slider142
If you can show that rigorously, then you have shown that each element of U is necessarily an element of W, and is thus contained in W. The only thing left is to show that sums of elements in U remain in U, and so do scalar multiples, which is the easy part.
6. Feb 1, 2009
### fk378
How would I show that the sum of any multiples of elements of U is still in U? Wouldn't I have to show that the sum of every sequence^2 is finite?
7. Feb 2, 2009
### slider142
As an example, suppose {a_i} is an element of U. Then the series (a_1)^2 + (a_2)^2 + ... converges. Consider the element s*{a_i} defined to be {s*a_i}, so we now consider the series (s*a_1)^2 + (s*a_2)^2 + ... = s^2*(a_1)^2 + s^2*(a_2)^2 + ... = s^2*((a_1)^2 + (a_2)^2 + ...) = s^2*A where A is the number that the original series converges to, showing constructively that the component-wise defined multiple also converges. Now you just have to consider what happens to a component-wise sum.
As for the latter question, you would already have shown that by showing that U is contained in W (by contradiction). (Need more of a hint?)
Have something to add?
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## Derivation of Planck’s radiation law – part 4 (final part)
In part 3 of this blog series I explained how Max Planck found a mathematical formula to fit the observed Blackbody spectrum, but that when he presented it to the German Physics Society on the 19th of October 1900 he had no physical explanation for his formula. Remember, the formula he found was
$E_{\lambda} \; d \lambda = \frac{ A }{ \lambda^{5} } \frac{ 1 }{ (e^{a/\lambda T} -1) } \; d\lambda$
if we express it in terms of wavelength intervals. If we express it in terms of frequency intervals it is
$E_{\nu} \; d \nu = A^{\prime} \nu^{3} \frac{ 1 }{ (e^{ a^{\prime} \nu / T } - 1) } \; d\nu$
Planck would spend six weeks trying to find a physical explanation for this equation. He struggled with the problem, and in the process was forced to abandon many aspects of 19th Century physics in both the fields of thermodynamics and electromagnetism which he had long cherished. I will recount his derivation – it is not the only one and maybe in coming blog posts I can show how his formula can be derived from other arguments, but this is the method Planck himself used.
## Radiation in a cavity
As we saw in the derivation of the Rayleigh-Jeans law (see part 3 here, and links in that to parts 1 and 2), blackbody radiation can be modelled as an idealised cavity which radiates through a small hole. Importantly, the system is given enough time for the radiation and the material from which the cavity is made to come into thermal equilibrium with each other. This means that the walls of the cavity are giving energy to the radiation at the same rate that the radiation is giving energy to the walls.
Using classical physics, as we did in the derivation of the Rayleigh-Jeans law, we saw that the energy density (the energy per unit volume) is
$\frac{du}{d\nu} = \left( \frac{ 8 \pi kT }{ c^{3} } \right) \nu^{2}$
After trying to derive his equation based on standard thermodynamic arguments, which failed, Planck developed a model which, he found, was able to produce his equation. How did he do this?
### Harmonic Oscillators
First, he knew from classical electromagnetic theory that an oscillating electron radiates (as it is accelerating), and he reasoned that when the cavity was in thermal equilibrium with the radiation in the cavity, the electrons in the walls of the cavity would oscillate and it was they that produced the radiation.
After much trial and error, he decided upon a model where the electrons were attached to massless springs. He could model the radiation of the electrons by modelling them as a whole series of harmonic oscillators, but with different spring stiffnesses to produce the different frequencies observed in the spectrum.
As we have seen (I derived it here), in classical physics the energy of a harmonic oscillator depends on both its amplitude of oscillation squared ($E \propto A^{2}$); and it also depends on its frequency of oscillation squared ($E \propto \nu^{2}$). The act of heating the cavity to a particular temperature is what, in Planck’s model, set the electrons oscillating; but whether a particular frequency oscillator was set in motion or not would depend on the temperature.
If it were oscillating, it would emit radiation into the cavity and absorb it from the cavity. He knew from the shape of the blackbody curve (and, by now, his equation which fitted it), that the energy density $E d\nu$ at any particular frequency started off at zero for high frequencies (UV), then rose to a peak, and then dropped off again at low frequencies (in the infrared).
So, Planck imagined that the number of oscillators with a particular resonant frequency would determine how much energy came out in that frequency interval. He imagined that there were more oscillators with a frequency which corresponded to the maximum in the blackbody curve, and fewer oscillators at higher and lower frequencies. He then had to figure out how the total energy being radiated by the blackbody would be shared amongst all these oscillators, with different numbers oscillating at different frequencies.
He found that he could not derive his formula using the physics that he had long accepted as correct. If he assumed that the energy of each oscillator went as the square of the amplitude, as it does in classical physics, his formula was not reproduced. Instead, he could derive his formula for the blackbody radiation spectrum only if the oscillators absorbed and emitted packets of energy which were proportional to their frequency of oscillation, not to the square of the frequency as classical physics argued. In addition, he found that the energy could only come in certain sized chunks, so for an oscillator at frequency $\nu, \; E = nh\nu$, where $n$ is an integer, and $h$ is now known as Planck’s constant.
What does this mean? Well, in classical physics, an oscillator can have any energy, which for a particular oscillator vibrating at a particular frequency can be altered by changing the amplitude. Suppose we have an oscillator vibrating with an amplitude of 1 (in abitrary units), then because the energy goes as the square of the amplitude its energy is $E=1^{2} =1$. If we increase the amplitude to 2, the energy will now be $E=2^{2} = 4$. But, if we wanted an energy of 2, we would need an amplitude of $\sqrt{2} = 1.414$, and if we wanted an energy of $3$ we would need an amplitude of $\sqrt{3} = 1.73$.
In classical physics, there is nothing to stop us having an amplitude of 1.74, which would give us an energy of 3.0276 (not 3), or an amplitude of 1.72 whichg would give us an energy of 2.9584 (not 3). But, what Planck found is that this was not allowed for his oscillators, they did not seem to obey the classical laws of physics. The energy could only be integers of $h\nu$, so $E=0h\nu, 1h\nu, 2h\nu, 3h\nu, 4h\nu$ etc.
Then, as I said above, he further assumed that the total energy at a particular frequency was given by the energy of each oscillator at that frequency multiplied by the number of oscillators at that frequency. The frequency of a particular oscillator was, he imagined, determined by its stiffness (Hooke’s constant). The energy of a particular oscillator at a particular frequency could be varied by the amplitude of its oscillations.
Let us assume, just to illustrate the idea, that the value of h is 2. If the total energy in the blackbody at a particular frequency of, say, 10 (in arbitrary units) were 800 (also in arbitrary units), this would mean that the energy of each chunk ($E=h \nu$) was $E = 2 \times 10 = 20$. So, the number of chunks at that frequency would then be $800/20 = 40$. 40 oscillators, each with an energy of 20, would be oscillating to give us our total energy of 800 at that frequency.
Because of this quantised energy, we can write that $E_{n} = nh \nu$, where $n=0,1,2,3, \cdots$.
### The number of oscillators at each frequency
The next thing Planck needed to do was derive an expression for the number of oscillators at each frequency. Again, after much trial and error he found that he had to borrow an idea first proposed by Austrian physicist Ludwig Boltzmann to describe the most likely distribution of energies of atoms or molecules in a gas in thermal equilibrium. Boltzmann found that the number of atoms or molecules with a particular energy $E$ was given by
$N_{E} \propto e^{-E/kT}$
where $E$ is the energy of that state, $T$ is the temperature of the gas and $k$ is now known as Boltzmann’s constant. The equation is known as the Boltzmann distribution, and Planck used it to give the number of oscillators at each frequency. So, for example, if $N_{0}$ is the number of oscillators with zero energy (in the so-called ground-state), then the numbers in the 1st, 2nd, 3rd etc. levels ($N_{1}, N_{2}, N_{3},\cdots$) are given by
$N_{1} = N_{0} e^{ -E_{1}/kT }, \; N_{2} = N_{0} e^{ -E_{2}/kT }, \; N_{3} = N_{0} e^{ -E_{3}/kT }, \cdots$
But, as $E_{n} = nh \nu$, we can write
$N_{1} = N_{0} e^{ -h \nu /kT }, \; N_{2} = N_{0} e^{ -2h \nu /kT }, \; N_{3} = N_{0} e^{ -3h \nu /kT }, \cdots$
Planck modelled blackbody radiation as a series of harmonic oscillators with equally spaced energy levels
To make it easier to write, we are going to substitute $x = e^{ -h \nu / kT }$, so we have
$N_{1} = N_{0}x, \; N_{2} = N_{0} x^{2}, \; N_{3} = N_{0} x^{3}, \cdots$
The total number of oscillators $N_{tot}$ is given by
$N_{tot} = N_{0} + N_{1} + N_{2} + N_{3} + \cdots = N_{0} ( 1 + x + x^{2} + x^{3} + \cdots)$
Remember, this is the number of oscillators at each frequency, so the energy at each frequency is given by the number at each frequency multiplied by the energy of each oscillator at that frequency. So
$E_{1}=N_{1} h \nu , \; E_{2} = N_{2} 2h \nu , \; E_{3} = N_{3} 3h \nu, \cdots$
which we can now write as
$E_{1} = h \nu N_{0}x, \; E_{2} = 2h \nu N_{0}x^{2}, \; E_{3} = 3h \nu N_{0}x^{3}, \cdots$
The total energy $E_{tot}$ is given by
$E_{tot} = E_{0} + E_{1} + E_{2} + E_{3} + \cdots = N_{0} h \nu (0 + x + 2x^{2} + 3x^{3} + \cdots)$
The average energy $\langle E \rangle$ is given by
$\langle E \rangle = \frac{ E_{tot} }{ N_{tot} } = \frac{ N_{0} h \nu (0 + x + 2x^{2} + 3x^{3} + \cdots) }{ N_{0} ( 1 + x + x^{2} + x^{3} + \cdots ) }$
The two series inside the brackets can be summed. The sum of the series in the numerator, which we will call $S_{1}$ is given by
$S_{1} = \frac{ x - (n+1)x^{n+1} + nx^{n+2} }{ (1-x)^{2} }$
(for the proof of this, see for example here)
The series in the denominator, which we will call $S_{2}$, is just a geometric progression. The sum of such a series is simply
$S_{2} = \frac{ 1 - x^{n} }{ (1-x) }$
Both series are in $x$, but remember $x = e^{-h \nu / kT}$. Also, both series are from a frequency of $\nu = 0 \text{ to } \infty$, and $e^{-h \nu /kT} < 1$, which means the sums converge and can be simplified.
$S_{1} \rightarrow \frac{x}{ (1-x)^{2} } \text{ and } S_{2} \rightarrow \frac{ 1 }{(1-x)}$
which means that $\langle E \rangle = (h \nu S_{1})/S_{2}$ is given by
$\langle E \rangle = \frac{ h \nu x }{ (1-x)^{2} } \times \frac{ (1-x) }{1} = \frac{h \nu x}{ (1-x) }$
and so we can write that the average energy is
$\boxed{ \langle E \rangle = \frac{h \nu}{( 1/x - 1) } = \frac{h \nu}{ (e^{h \nu/kT} - 1) } }$
## The radiance per frequency interval
In our derivation of the Rayleigh-Jeans law (in this blog here), we showed that, using classical physics, the energy density $du$ per frequency interval was given by
$du = \frac{ 8 \pi }{ c^{3} } kT \nu^{2} \, d \nu$
where $kT$ was the energy of each mode of the electromagnetic radiation. We need to replace the $kT$ in this equation with the average energy for the harmonic oscillators that we have just derived above. So, we re-write the energy density as
$du = \frac{ 8 \pi }{ c^{3} } \frac{ h \nu }{ (e^{h\nu/kT} - 1) } \nu^{2} \; d\nu = \frac{ 8 \pi h \nu^{3} }{ c^{3} } \frac{ 1 }{ (e^{h\nu/kT} - 1) } \; d\nu$
$du$ is the energy density per frequency interval (usually measured in Joules per metre cubed per Hertz), and by replacing $kT$ with the average energy that we derived above the radiation curve does not go as $\nu^{2}$ as in the Rayleigh-Jeans law, but rather reaches a maximum and turns over, avoiding the ultraviolet catastrophe.
It is more common to express the Planck radiation law in terms of the radiance per unit frequency, or the radiance per unit wavelength, which are written $B_{\nu}$ and $B_{\lambda}$ respectively. Radiance is the power per unit solid angle per unit area. So, as a first step to go from energy density to radiance we will divide by $4 \pi$, the total solid angle. This gives
$\frac{ 2 h \nu^{3} }{ c^{3} } \frac{ 1 }{ (e^{h\nu/kT} - 1) } \; d\nu$
We want the power per unit area, not the energy per unit volume. To do this we first note that power is energy per unit time, and second that to go from unit volume to unit area we need to multiply by length. But, for EM radiation, length is just $ct$. So, we need to divide by $t$ and multiply by $ct$, giving us that the radiance per frequency interval is
$\boxed{ B_{\nu} = \frac{ 2h \nu^{3} }{ c^{2} } \frac{ 1 }{ (e^{h\nu/kT} - 1) } \; d\nu }$
which is the way the Planck radiation law per frequency interval is usually written.
## Radiance per unit wavelength interval
If you would prefer the radiance per wavelength interval, we note that $\nu = c/\lambda$ and so $d\nu = -c/\lambda^{2} \; d\lambda$. Ignoring the minus sign (which is just telling us that as the frequency increases the wavelength decreases), and substituting for $\nu$ and $d\nu$ in terms of $\lambda$ and $d\lambda$, we can write
$B_{\lambda} = \frac{ 2h }{ c^{2} } \frac{ c^{3} }{ \lambda^{3} } \frac{ 1 }{ ( e^{hc/\lambda kT} - 1 ) } \frac{ c }{ \lambda^{2} } \; d\lambda$
Tidying up, this gives
$\boxed{ B_{\lambda} = \frac{ 2hc^{2} }{ \lambda^{5} } \frac{ 1 }{ ( e^{hc/\lambda kT} - 1 ) } \; d\lambda }$
which is the way the Planck radiation law per wavelength interval is usually written.
## Summary
To summarise, in order to reproduce the formula which he had empirically derived and presented in October 1900, Planck found that he he could only do so if he assumed that the radiation was produced by oscillating electrons, which he modelled as oscillating on a massless spring (so-called “harmonic oscillators”). The total energy at any given frequency would be given by the energy of a single oscillator at that frequency multiplied by the number of oscillators oscillating at that frequency.
However, he had to assume that
1. The energy of each oscillator was not related to either the square of the amplitude of oscillation or the square of the frequency of oscillation (as it would be in classical physics), but rather to the square of the amplitude and the frequency, $E \propto \nu$.
2. The energy of each oscillator could only be a multiple of some fundamental “chunk” of radiation, $h \nu$, so $E_{n} = nh\nu$ where $n=0,1,2,3,4$ etc.
3. The number of oscillators with each energy $E_{n}$ was given by the Boltzmann distribution, so $N_{n} = N_{0} e^{-nh\nu/kT}$ where $N_{0}$ is the number of oscillators in the lowest energy state.
In a way, we can imagine that the oscillators at higher frequencies (to the high-frequency side of the peak of the blackbody) are “frozen out”. The quantum of energy for a particular oscillator, given by $E_{n}=nh\nu$, is just too large to exist at the higher frequencies. This avoids the ultraviolet catastrophe which had stumped physicists up until this point.
By combining these assumptions, Planck was able in November 1900 to reproduce the exact equation which he had derived empirically in October 1900. In doing so he provided, for the first time, a physical explanation for the observed blackbody curve.
• Part 1 of this blogseries is here.
• Part 2 is here.
• Part 3 is here.
Read Full Post »
## Does centrifugal force exist?
For several weeks now I have been planning to write a blog about centrifugal force, mainly prompted by seeing a post by John Gribbin on Facebook of the xkcd cartoon about it. In the cartoon James Bond is threatened with torture on a centrifuge. Here is a link to the original cartoon.
The xkcd cartoon about centrifugal force involves James Bond being tortured on a centrifuge
I have taught mechanics many times to physics undergraduates, and they are often confused about centripetal force and centrifugal force, and what the difference is between them. Some have heard that centrifugal force doesn’t really exist, just as Bond states in this cartoon. What is the real story?
## Rotating frames of reference
Everyone reading this (apart from a few “flat-Earth adherents” maybe) knows that we live on the surface of a planet which is rotating on its axis once a day. This means that we do not live in an inertial frame of reference (an inertial frame is one which is not accelerating), as clearly being on the surface of a spinning planet means that we are experiencing acceleration all the time; as we are not travelling in a straight line. That acceleration is provided by the force of gravity, and it stops us from going off in a straight line into space!
Because we are living in a non-inertial frame of reference we need to modify Newton’s laws of motion to properly describe such a non-inertial frame (which I am going to call a “rotating frame” from now on, although a rotating frame is just one example of a non-inertial frame but it is the one relevant to us on the surface of a rotating Earth).
Let us consider our usual Cartesian coordinate system. The unit vector in the x-direction is usually written as $\hat{\imath}$, the one in the y-direction as $\hat{\jmath}$, and the one in the z-direction as $\hat{k}$. We are going to consider an object rotating about the $\hat{k}$ (z-axis) direction.
We will consider two reference frames, one which stays fixed (the inertial reference frame), denoted by $(\hat{\imath},\hat{\jmath},\hat{k})$, and a second reference frame which rotates with the rotation, denoted by $(\hat{\imath}_{r} ,\hat{\jmath}_{r} ,\hat{k}_{r})$, where the subscript $r$ reminds us that this is the rotating frame of reference.
For the derivation below I am going to assume that we are considering motion with a constant radius $r$. I want to illustrate how centrifugal force arrises in a rotating frame such as being on the surface of our Earth. Our Earth is not spherical, but at any given point the size of the radius does not change, so this is a reasonable simplification.
As I showed in this blog on angular velocity, we can write the linear velocity $\vec{v}$ of an object moving in a circle as
$\vec{v} = \frac{ d \vec{r} }{ dt } = \vec{\omega} \times \vec{r}$
where $\vec{r}$ is the radius vector and $\vec{\omega}$ is the angular velocity.
Writing $\vec{r}$ in terms of its x,y and z-components in our inertial (non-rotating) frame, $\vec{r}=(\hat{\imath},\hat{\jmath},\hat{k})$, so in general we then have
$\vec{v} = \frac{ d \vec{r} }{ dt } \rightarrow \frac{ d \hat{\imath} }{dt} = \vec{\omega} \times \hat{\imath}, \; \; \frac{ d \hat{\jmath} }{dt} = \vec{\omega} \times \hat{\jmath} , \; \; \frac{ d \hat{k} }{dt} = \vec{\omega} \times \hat{k}$
Let us consider the specific case of a small rotation $\delta \theta$ about the $\hat{k}$ axis, as shown in the figure below. As the figure shows, in our inertial (fixed) frame of reference, the new direction of the x-axis is now $\hat{\imath} + \delta \hat{\imath}$, and the new direction of the y-axis is $\hat{\jmath} + \delta \hat{\jmath}$. The direction of the $\hat{k}$ axis is unchanged.
We are going to rotate about the z-axis ($\hat{k}$ direction) by an angle $\delta \theta$
Because we are rotating about the $\hat{k}$ axis, the angular velocity is in this direction, and so we can write (using the right-hand rule for vector products as I blogged about here)
$\vec{\omega} \times \hat{\imath} = \omega \hat{\jmath}, \; \; \vec{\omega} \times \hat{\jmath} = -\omega \hat{\imath}, \; \; \vec{\omega} \times \hat{k} =0$
Let us now consider some vector $\vec{a}$, which we will write in the rotating frame of reference as
$\vec{a} = a_{x} \hat{\imath}_{r} + a_{y} \hat{\jmath}_{r} + a_{z} \hat{k}_{r}$
If we now look at the rate of change of this vector in the rotating frame we have
$\left( \frac{d \vec{a} }{dt} \right)_{r} = \frac{d}{dt}(a_{x}\hat{\imath}_{r}) + \frac{d}{dt}(a_{y}\hat{\jmath}_{r}) + \frac{d}{dt}(a_{z}\hat{k}_{r})$
In the rotating frame of reference, $\hat{\imath}_{r}, \hat{\jmath}_{r}$ and $\hat{k}_{r}$ do not change with time, so we can write
$\left( \frac{d \vec{a} }{dt} \right)_{r} = \frac{ da_{x} }{dt} \hat{\imath}_{r} + \frac{ da_{y} }{dt} \hat{\jmath}_{r} + \frac{ da_{z} }{dt} \hat{k}_{r}$
In the inertial frame of reference $\hat{\imath}_{r}, \hat{\jmath}_{r}$ and $\hat{k}_{r}$ move, so
$\left( \frac{d \vec{a} }{dt} \right)_{i} = \frac{d}{dt} (a_{x} \hat{\imath}_{r}) + \frac{d}{dt} (a_{y} \hat{\jmath}_{r}) + \frac{d}{dt} (a_{z} \hat{k}_{r})$
$\left( \frac{d \vec{a} }{dt} \right)_{i} = \frac{ da_{x} }{dt}\hat{\imath}_{r} + \frac{ da_{y} }{dt}\hat{\jmath}_{r} + \frac{ da_{z} }{dt}\hat{k}_{r} + a_{x} \frac{d \hat{\imath}_{r} }{dt} + a_{y} \frac{d \hat{\jmath}_{r} }{dt} + a_{z} \frac{d \hat{k}_{r} }{dt}$
But, we can write (see above) that
$\frac{d\hat{\imath}_{r} }{dt} = \vec{\omega} \times \hat{\imath}_{r}, \; \; \frac{d\hat{\jmath}_{r} }{dt} = \vec{\omega} \times \hat{\jmath}_{r}, \; \; \frac{d\hat{k}_{r} }{dt} = \vec{\omega} \times \hat{k}_{r}$
and so
$\left( \frac{d \vec{a} }{dt} \right)_{i} = \frac{ da_{x} }{dt}\hat{\imath}_{r} + \frac{ da_{y} }{dt}\hat{\jmath}_{r} + \frac{ da_{z} }{dt}\hat{k}_{r} + a_{x} \vec{\omega} \times \hat{\imath}_{r} + + a_{y} \vec{\omega} \times \hat{\jmath}_{r} + + a_{z} \vec{\omega} \times \hat{k}_{r}$
$\left( \frac{d \vec{a} }{dt} \right)_{i} = \frac{ da_{x} }{dt}\hat{\imath}_{r} + \frac{ da_{y} }{dt}\hat{\jmath}_{r} + \frac{ da_{z} }{dt}\hat{k}_{r} + \vec{\omega} \times \vec{a}$
$\boxed{ \left( \frac{d \vec{a} }{dt} \right)_{i} = \left( \frac{d \vec{a} }{dt} \right)_{r} + (\vec{\omega} \times \vec{a}) }$
## A fixed point on the Earth’s surface
Let us now consider the point $\vec{a} = \vec{r}$, where $\vec{r}$ is a fixed point on the Earth’s surface. We can write
$\left( \frac{d \vec{r} }{dt} \right)_{i} = \left( \frac{d \vec{r} }{dt} \right)_{r} + (\vec{\omega} \times \vec{r})$
But, in the rotating frame of reference this point does not change with time, so
$\left( \frac{d \vec{r} }{dt} \right)_{r} = 0$
and so
$\left( \frac{d \vec{r} }{dt} \right)_{i} = (\vec{\omega} \times \vec{r}) = \omega r \sin(\theta)$
where $\theta$ is the angle between the Earth’s rotation axis and the latitude of the point (so $\theta = 90^{\circ} - \text{ latitude}$).
Let us now calculate the acceleration in an inertial frame in terms of acceleration in a rotating frame. Writing $\vec{a}$ as $\vec{r}$ as above, we now have
$\left( \frac{d \vec{r} }{dt} \right)_{i} = \left( \frac{d \vec{r} }{dt} \right)_{r} + (\vec{\omega} \times \vec{r})$
To make things easier to write, we will re-write
$\left( \frac{d \vec{r} }{dt} \right)_{i} = \frac{d \vec{r}_{i} }{dt} \text{ and } \left( \frac{d \vec{r} }{dt} \right)_{r} = \frac{d \vec{r}_{r} }{dt}$
so
$\frac{d \vec{r}_{i} }{dt} = \frac{d \vec{r}_{r} }{dt} + (\vec{\omega} \times \vec{r})$
$\vec{v}_{i} = \vec{v}_{r} + (\vec{\omega} \times \vec{r})$
If we now differentiate $\vec{v}_{i}$ with respect to time, we will have the acceleration in the inertial frame
$\left( \frac{d \vec{v}_{i} }{dt} \right)_{i} = \left( \frac{d \vec{v}_{i} }{dt} \right)_{r} + (\vec{\omega} \times \vec{v}_{i})$
But, $\vec{v}_{i} = \vec{v}_{r} + (\vec{\omega} \times \vec{r})$
so
$\left( \frac{d \vec{v}_{i} }{dt} \right)_{i} = \frac{d}{dt}(\vec{v}_{r} + \vec{\omega} \times \vec{r})_{r} + \vec{\omega} \times (\vec{v}_{r} + \vec{\omega} \times \vec{r})$
Expanding this out we get
$\left( \frac{d \vec{v}_{i} }{dt} \right)_{i} = \left( \frac{d \vec{v}_{r} }{dt} \right)_{r} + \frac{d}{dt}(\vec{\omega} \times \vec{r}_{r}) + \vec{\omega} \times \vec{v}_{r} + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{r})$
$\vec{a}_{i} = \vec{a}_{r} + 2\vec{\omega} \times \vec{v}_{r} + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{r})$
Multiplying the acceleration by the mass $m$ to get a force
$m\vec{a}_{i} = m\vec{a}_{r} + 2m\vec{\omega} \times \vec{v}_{r} + m\vec{\omega} \times (\vec{\omega} \times \vec{r}_{r})$
So, writing the force in the rotating frame in terms of the force in the inertial frame, we have
$\boxed{ m\vec{a}_{r} = m\vec{a}_{i} - 2m\vec{\omega} \times \vec{v}_{r} - m\vec{\omega} \times (\vec{\omega} \times \vec{r}_{r}) }$
So,
$\boxed{\vec{F}_{r} = \vec{F}_{i} - 2m\vec{\omega} \times \vec{v}_{r} - m\vec{\omega} \times (\vec{\omega} \times \vec{r}_{r}) }$
Notice that there are two extra terms (Term A and Term B) in the equation on the right, I have highlighted them below.
If we compare the force in a rotating frame to an inertial frame, two extra terms (Term A and Term B) arise. Term A is the Coriolis force, Term B is the centrifugal force
Term A is what we call the Coriolis force, which depends on the velocity in the rotating frame $v_{r}$. It is the force which causes water going down a plughole to rotate about the hole and to move anti-clockwise in the northern hemisphere and clockwise in the southern hemisphere. It is also the force which determines the direction of rotation of low pressure systems in the atmosphere. I will discuss the coriolis force more in a future blog.
Term B is the centrifugal force, the force we were aiming to derive in this blogpost. The strength of the centrifugal force depends on the position of the object in the rotating frame – $r_{r}$.
## What is the direction of the centrifugal force
The direction of $(\vec{\omega} \times \vec{r}_{r})$ can be found using the right-hand rule for the vector product, which I blogged about here. Remembering that the direction of $\vec{r}_{r}$\$ is radially outwards from the centre of the Earth, and the direction of $\vec{\omega}$ is the direction of the Earth’s axis (pointing north), then the direction of $\vec{\omega} \times \vec{r}_{r}$ is towards the east (right if looking at the Earth with the North pole up).
We now need to take the vector produce of $\vec{\omega}$ with a vector in this eastwards direction, and again using the right-hand rule gives us that the direction of $(\vec{\omega} \times \vec{r}_{r})$ is outwards (not radially from the centre of the Earth, but at right angles to the axis of the Earth). But, notice the centrifugal force has a minus sign in front of it, so the direction of the centrifugal force is outwards, away and at right angles to the Earth’s axis.
The direction of the centrifugal force is away from the axis of rotation, as shown in this diagram
This means that it acts to reduce the force of gravity which keeps us on the Earth’s surface. It also depends on the angle between where you are and the Earth’s axis, so is greatest at the equator and goes to zero at the pole. It means that you will weight slightly less than if the Earth were not rotating, but the effect is quite small and you would not notice such a difference going from the pole to the equator.
## What is the strength of the centrifugal acceleration due to Earth’s rotation?
Let us calculate the centrifugal force at the Earth’s equator, where it is at its greatest.
At the equator, we can write that the centrifugal acceleration has a value of
$\omega^{2} r \text{ as } \theta = 90^{\circ}$
We can calculate $\omega$ for the Earth by remembering that it takes 24 hours to rotate once, and $\omega$ is related to the period $T$ of rotation via
$\omega = \frac{2 \pi }{ T}$
We need to convert the period $T$ to seconds, so $T = 24 \times 60 \times 60 = 86400 \; s$. This gives that
$\omega = 7.272 \times 10^{-5} \text{ rad/s }$
If we take the Earth’s radius to be 6,378.1 km (this is the radius at the equator), then we have that
$\omega^{2} r = 0.0337 \text{ m/s/s}$
Compare this to the acceleration due to gravity which pulls us towards the Earth’s surface, which is 9.81 m/s/s and we can see that the centrifugal force at its greatest is only $0.34 \%$ of the acceleration due to gravity. Tiny.
It is, however, noticeable when you are on a roundabout, and is used on fairground rides where you spin inside a drum and the floor moves away leaving you pinned to the wall of the drum. The force you feel pushing against this wall is the centrifugal force, and it is very real for you in that rotating frame!
So, there we have it, centrifugal force does exist in a rotating frame of reference, but does not exist from the perspective of someone in an inertial frame of reference.
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## Einstein’s general relativity centenary
There has been quite a bit of mention in the media this last week or so that it is 100 years since Albert Einstein published his ground-breaking theory of gravity – the general theory of relativity. Yet, there seems to be some confusion as to when this theory was first published, in some places you will see 1915, in others 1916. So, I thought I would try and clear up this confusion by explaining why both dates appear.
Albert Einstein in Berlin circa 1915/16 when his General Theory of Relativity was first published
## From equivalence to the field equations
Everyone knew that Einstein was working on a new theory of gravity. As I blogged about here, he had his insight into the equivalence between acceleration and gravity in 1907, and ever since then he had been developing his ideas to create a new theory of gravity.
He had come up with his principle of equivalence when he was asked in the autumn of 1907 to write a review article of his special theory of relativity (his 1905 theory) for Jahrbuch der Radioaktivitätthe (the Yearbook of Electronics and Radioactivity). That paper appeared in 1908 as Relativitätsprinzip und die aus demselben gezogenen Folgerungen (On the Relativity Principle and the Conclusions Drawn from It) (Jahrbuch der Radioaktivität, 4, 411–462).
In 1908 he got his first academic appointment, and did not return to thinking about a generalisation of special relativity until 1911. In 1911 he published a paper Einfluss der Schwerkraft auf die Ausbreitung des Lichtes (On the Influence of Gravitation on the Propagation of Light) (Annalen der Physik (ser. 4), 35, 898–908), in which he calculated for the first time the deflection of light produced by massive bodies. But, he also realised that, to properly develop his ideas of a new theory of gravity, he would need to learn some mathematics which was new to him. In 1912, he moved to Zurich to work at the ETH, his alma mater. He asked his friend Marcel Grossmann to help him learn this new mathematics, saying “You’ve got to help me or I’ll go crazy.”
Grossmann gave Einstein a book on non-Euclidean geometry. Euclidean geometry, the geometry of flat surfaces, is the geometry we learn in school. The geometry of curved surfaces, so-called Riemann geometry, had first been developed in the 1820s by German mathematician Carl Friedrich Gauss. By the 1850s another German mathematician, Bernhard Riemann developed this geometry of curved surfaces even further, and this was the Riemann geometry textbook which Grossmann gave to Einstein in 1912. Mastering this new mathematics proved very difficult for Einstein, but he knew that he needed to master it to be able to develop the equations for general relativity.
These equations were not ready until late 1915. Everyone knew Einstein was working on them, and in fact he was offered and accepted a job in Berlin in 1914 as Berlin wanted him on their staff when the new theory was published. The equations of general relativity were first presented on the 25th of November 1915, to the Prussian Academy of Sciences. The lecture Feldgleichungen der Gravitation (The Field Equations of Gravitation) was the fourth and last lecture that Einstein gave to the Prussian Academy on his new theory (Preussische Akademie der Wissenschaften, Sitzungsberichte, 1915 (part 2), 844–847), the previous three lectures, given on the 4th, 11th and 18th of November, had been leading up to this. But, in fact, Einstein did not have the field equations ready until the last few days before the fourth lecture!
The peer-reviewed paper of the theory (which also contains the field equations) did not appear until 1916 in volume 49 of Annalen der PhysikGrundlage der allgemeinen Relativitätstheorie (The Foundation of the General Theory of Relativity) Annalen der Physik (ser. 4), 49, 769–822. The paper was submitted by Einstein on the 20th of March 1916.
The beginning of Einstein’s first peer-reviewed paper on general relativity, which was received by Annalen der Physik on the 20th of March 1916
In a future blog, I will discuss Einstein’s field equations, but hopefully I have cleared up the confusion as to why some people refer to 1915 as the year of publication of the General Theory of Relativity, and some people choose 1916. Both are correct, which allows us to celebrate the centenary twice!
You can read more about Einstein’s development of the general theory of relativity in our book 10 Physicists Who Transformed Our Understanding of Reality. Order your copy here
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## Derivation of the moment of inertia of an annulus
Following on from my derivation of the moment of inertia of a disk, in this blog I will derive the moment of inertia of an annulus. By an annulus, I mean a disk which has the inner part missing, as shown below.
An annulus is a disk of small thickness $t$ with the inner part missing. The annulus goes from some inner radius $r_{1}$ to an outer radius $r$.
To derive its moment of inertia, we return to our definition of the moment of inertia, which for a volume element $dV$ is given by
$dI = r_{\perp}^{2} dm$
where $dm$ is the mass of the volume element $dV$. We are going to initially consider the moment of inertia about the z-axis, and so for this annulus it will be
$I_{zz} = \int _{r_{1}} ^{r} r_{\perp}^{2} dm$
where $r_{1}$ and $r$ are the inner radius and outer radius of the annulus respectively. As with the disk, the mass $dm$ of the volume element $dV$ is related to its volume and density via
$dm = \rho dV$
(assuming that the annulus has a uniform density). The volume element $dV$ can be found as before by considering a ring at a radius of $r$ which a width $dr$ and a thickness $t$. The volume of this will be
$dV = (2 \pi r dr) t$
and so we can write the mass $dm$ as
$dm = (2 \pi \rho t)rdr$
Thus we can write the moment of inertia $I_{zz}$ as
$I_{zz} = \int _{r_{1}} ^{r} r_{\perp}^{2} dm = 2 \pi \rho t \int _{r_{1}} ^{r} r_{\perp}^{3} dr$
Integrating this between $r_{1}$ and $r$ we get
$I_{zz} = 2 \pi \rho t [ \frac{ r^{4} - r_{1}^{4} }{4} ] = \frac{1}{2} \pi \rho t (r^4 - r_{1}^{4}) \text{ (Equ. 1)}$
But, we can re-write $(r^{4} - r_{1}^{4})$ as $(r^{2} + r_{1}^{2})(r^{2} - r_{1}^{2})$ (remember $x^{2} - y^{2}$ can be written as $(x+y)(x-y)$). So, wen can write Eq. (1) as
$I_{zz} = \frac{1}{2} \pi \rho t (r^{2} + r_{1}^{2})(r^{2} - r_{1}^{2}) \text{ (Equ. 2)}$
The total mass $M_{a}$ of the annulus can be found by considering the total mass of a disk of radius $r$ (which we will call $M_{2}$) and then subtracting the mass of the inner part, a disk of radius $r_{1}$ (which we will call $M_{1}$). The mass of a disk is just its density multiplied by its area multiplied by its thickness.
$M_{2} = \pi \rho t r^{2} \text{ and } M_{1} = \pi \rho t r_{1}^{2}$
so the mass $M_{a}$ of the annulus is
$M_{a} = M_{2} - M_{1} = \pi \rho t r^{2}- \pi \rho t r_{1}^{2} = \pi \rho t (r^{2} - r_{1}^{2})$
Substituting this expression for $M_{a}$ into equation (2) above, we can write that the moment of inertia for an annulus, which goes from an inner radius of $r_{1}$ to an outer radis of $r$, about the z-axis is
$\boxed{ I_{zz} = \frac{1}{2} M_{a} (r^{2} + r_{1}^{2}) }$
## Comparison to the moment of inertia of a disk
As we saw in this blog, the moment of inertia of a disk is $I_{zz} = \frac{1}{2} Mr^{2}$. It may therefore seem, at first sight, that the moment of inertia of an annulus is more than that of a disk. This would be true if they have the same mass, but if they have the same thickness and density the mass of an annulus will be much less.
Let us compare the moment of inertia of a disk and an annulus for the 4 following cases.
The same density and thickness, $r_{1} = 0.5 r$
The same density and thickness, $r_{1} = 0.9 r$
The same mass, $r_{1} = 0.5 r$
The same mass, $r_{1} = 0.9 r$
## The same density and thickness, $r_{1}=0.5r$
We are first going to compare the moment of inertia of a disk of mass $M$ with that of an annulus which goes from half the radius of the disk to the radius of the disk (i.e. $r_{1} \text{ the inner radius of the annulus, is } = 0.5 r$.
For the disk, its mass will be
$M = \rho t (\pi r^{2}) = \pi \rho t r^{2}$
The mass of the annulus, $M_{a}$, will be this mass less the mass of the missing part $M_{1}$, so
$M_{a} = M - M_{1} = M - \pi \rho t (r_{1})^{2} = \pi \rho t (r^{2} - (0.5r)^{2})= \pi \rho t (1-0.25)r^{2}$
$M_{a} = \pi \rho t (0.75)r^{2} = 0.75 M$
The moment of inertia of the disk will be
$I_{d} = \frac{1}{2} M r^{2}$
The moment of inertia of the annulus will be
$I_{a} = \frac{1}{2} M_{a} (r^{2} + r_{1}^{2}) = \frac{1}{2} (0.75M)(r^{2} + (0.5r)^{2}) = \frac{1}{2} (0.75M)(1.25r^{2}) = \frac{1}{2} (0.9375) M r^{2}$
So, for this case, $I_{a} = 0.9375 I_{d}$, i.e. slightly less than the disk.
## The same density and thickness, $r_{1}=0.9r$
Let us now consider the second case, with an annulus of the same density and thickness as the disk, and its inner radius being 90% of the outer radius, $r_{1} = 0.9r$. Now, the mass of the missing part of the disk, $M_{1}$ will be
$M_{1} = \rho t (\pi r_{1}^{2}) = \rho t \pi (0.9r)^{2} = 0.81 \rho t \pi r^{2} = 0.81M$
which means that the mass of the annulus, $M_{a}$ is
$M_{a} = M - M_{1} = M-0.81M=0.19M$
The moment of inertia of the annulus will then be
$I_{a} = \frac{1}{2}M_{a}(r^{2}+r_{1}^{2}) = \frac{1}{2}(0.19M)(r^{2}+(0.9r)^{2})=\frac{1}{2}(0.19M)((1.81)r^{2} = \frac{1}{2}(0.1539)Mr^{2}$
and so in this case
$I_{a} = 0.1539 I_{d}$
which is much less than the moment of inertia of the disk.
## The same mass, $r_{1}=0.5r$
In this third case, the mass of the annulus is the same as the mass of the disk, and its inner radius is 50% of the radius of the disk. This would, of course, require the annulus to either have a greater density than the disk, or to be thicker (or both). So, $M_{a} = M$. The moment of inertia of the annulus will be
$I_{a} = \frac{1}{2} M(r^{2} + r_{1}^{2}) = \frac{1}{2} M(r^{2} + (0.5r)^{2}) = \frac{1}{2} M(r^{2} + 0.25r^{2}) = \frac{1}{2} M(1.25)r^{2}$
$I_{a}= 1.25 I_{d}$
## The same mass, $r_{1}=0.9r$
The last case we will consider is an annulus with its inner radius being 90% of the outer radius, but its mass the same. So, $M_{a} = M$. The moment of inertia of the annulus will be
$I_{a} = \frac{1}{2} M(r^{2} + r_{1}^{2}) = \frac{1}{2} M(r^{2} + (0.9r)^{2}) = \frac{1}{2} M(r^{2} + 0.81r^{2}) = \frac{1}{2} M(1.81)r^{2}$
$I_{a}= 1.81 I_{d}$
## Summary
To summarise, we have
The same density and thickness, $r_{1} = 0.5 r, \; \; I_{a}=0.9375 I_{d}$
The same density and thickness, $r_{1} = 0.9 r, \; \; I_{a}=0.1539 I_{d}$
The same mass, $r_{1} = 0.5 r, \; \; I_{a}=1.25 I_{d}$
The same mass, $r_{1} = 0.9 r, \; \; I_{a}=1.81 I_{d}$
So, as these calculations show, if keeping the mass of a flywheel down is important, then a larger moment of inertia will be achieved by concentrating most of that mass in the outer parts of the flywheel, as this photograph below shows.
If keeping mass down is important, a flywheel’s moment of inertia can be increased by concentrating most of the mass in its outer parts
In the next blogpost in this series I will calculate the moment of inertia of a solid sphere.
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## Derivation of the moment of inertia of a disk
In physics, the rotational equivalent of mass is something called the moment of inertia. The definition of the moment of inertia of a volume element $dV$ which has a mass $dm$ is given by
$dI = r_{\perp}^{2} dm$
where $r_{\perp}$ is the perpendicular distance from the axis of rotation to the volume element. To find the total moment of inertia of an object, we need to sum the moment of inertia of all the volume elements in the object over all values of distance from the axis of rotation. Normally we consider the moment of inertia about the vertical (z-axis), and we tend to denote this by $I_{zz}$. We can write
$I_{zz} = \int _{r_{1}} ^{r_{2}} r_{\perp}^{2} dm$
The moment of inertia about the other two cardinal axes are denoted by $I_{xx}$ and $I_{yy}$, but we can consider the moment of inertia about any convenient axis.
## Derivation of the moment of inertia of a disk
In this blog, I will derive the moment of inertia of a disk. In upcoming blogs I will derive other moments of inertia, e.g. for an annulus, a solid sphere, a spherical shell and a hollow sphere with a very thin shell.
For our purposes, a disk is a solid circle with a small thickness $t$ ($t \ll r$, small in comparison to the radius of the disk). If it has a thickness which is comparable to its radius, it becomes a cylinder, which we will discuss in a future blog. So, our disk looks something like this.
A disk of small thickness $t$, with a radius of $r$
To calculate the moment of inertia of this disk about the z-axis, we sum the moment of inertia of a volume element $dV$ from the centre (where $r=0$) to the outer radius $r$.
$I_{zz} = \int_{r=0} ^{r=r} r_{\perp} ^{2} dm \text{ (Equ. 1)}$
The mass element $dm$ is related to the volume element $dV$ via the equation
$dm = \rho dV$ (where $\rho$ is the density of the volume element). We will assume in this example that the density $\rho(r)$ of the disk is uniform; but in principle if we know its dependence on $r, \; \rho (r) = f(r)$, this would not be a problem.
The volume element $dV$ can be calculated by considering a ring at a radius $r$ with a width $dr$ and a thickness $t$. The volume of this ring is just this rings circumference multiplied by its width multiplied by its thickness.
$dV = (2 \pi r dr) t$
so we can write
$dm = \rho (2 \pi r dr) t$
and hence we can write equation (1) as
$I_{zz} = \int_{r=0} ^{r=r} r_{\perp} ^{2} \rho (2 \pi r dr) t = 2 \pi \rho t \int_{r=0} ^{r=r} r_{\perp} ^{3} dr$
Integrating between a radius of $r=0$ and $r$, we get
$I_{zz} = 2 \pi \rho t [ \frac{ r^{4} }{ 4 } -0 ] = \frac{1}{2} \pi \rho t r^{4} \text{ (Equ. 2)}$
If we now define the total mass of the disk as $M$, where
$M = \rho V$
and $V$ is the total volume of the disk. The total volume of the disk is just its area multiplied by its thickness,
$V = \pi r^{2} t$
and so the total mass is
$M = \rho \pi r^{2} t$
Using this, we can re-write equation (2) as
$\boxed{ I_{zz} = \frac{1}{2} \pi \rho t r^{4} = \frac{1}{2} Mr^{2} }$
## What are the moments of inertia about the x and y-axes?
To find the moment of inertia about the x or the y-axis we use the perpendicular axis theorem. This states that, for objects which lie within a plane, the moment of inertia about the axis parallel to this plane is given by
$I_{zz} = I_{xx} + I_{yy}$
where $I_{xx}$ and $I_{yy}$ are the two moments of inertia in the plane and perpendicular to each other.
We can see from the symmetry of the disk that the moment of inertia about the x and y-axes will be the same, so $I_{zz} = 2I_{xx}$. Therefore we can write
$\boxed{ I_{xx} = I_{yy} = \frac{1}{2}I_{zz} = \frac{1}{4} Mr^{2} }$
## Flywheels
Flywheels are used to store rotational energy. This is useful when the source of energy is not continuous, as they can help provide a continuous source of energy. They are used in many types of motors including modern cars.
It is because of an disk’s moment of inertia that it can store rotational energy in this way. Just as with mass in the linear case, it requires a force to change the rotational speed (angular velocity) of an object. The larger the moment of inertia, the larger the force required to change its angular velocity. As we can see above from the equation for the moment of inertia of a disk, for two flywheels of the same mass a thinner larger one will store more energy than a thicker smaller one because its moment of inertia increases as the square of the radius of the disk.
Sometimes mass is a critical factor, and next time I will consider the case of an annulus, where the inner part of the disk is removed.
Read Full Post »
## Harmonic Oscillators
Today I was planning to post the fourth and final part of my series of blogs about the derivation of Planck’s radiation law. But, I realised on Sunday that it would not be ready, so I’m postponing it until next Thursday (17th). Parts 1, 2 and 3 are here, here and here respectively).
One of the reasons for this is that my time is being consumed by writing articles for 30-second Einstein which I talked about on Tuesday. Another reason is that I am scrambling to finish a slew of things by the 15th (next Tuesday!!), as I have been asked to go on another cruise to give astronomy lectures. More about that next week 🙂
There is a third reason, I have realised that I have not yet done a blog about harmonic oscillators, which is a necessary part of understanding Planck’s derivation. So, that is the subject of today’s blogpost.
Harmonic oscillators is another term for something which is exhibiting simple harmonic motion, and I did blog here about how a pendulum exhibits simple harmonic motion (SHM), and how this relates to circular motion. Another example of SHM is a spring oscillating back and forth. Whether the spring is vertical or horizontal, if it is displaced from its equilibrium position it will exhibit SHM. So, a spring is a harmonic oscillator.
## The frequency of a harmonic oscillator
The restoring force on a spring when it is displaced from its equilibrium position is given by Hooke’s law, which states
$\vec{F} = - k \vec{x}$
where $\vec{F} \text{ and } \vec{x}$ are the force and displacement respectively (vector quantities), and the minus sign is telling us that the force acts in the opposite direction to the displacement; that is it is a restoring force which is directed back towards the equilibrium position. The term $k$ is known as Hooke’s constant, and is basically the stiffness of the spring.
Because we can also write the force in terms of mass and acceleration (Newton’s 2nd law of motion), and acceleration is the second derivate of displacement, we can write
$m \vec{a} = m \frac{ d^{2}\vec{x} }{ dt^{2} }= - k \vec{x}$
If we divide by $m$ we get an expression for the acceleration, which is
$\boxed{ \vec{a} = - \frac{k}{m} \vec{x} }$
which, if you compare it to the equation for SHM for a pendulum, has the same form. The usual way to write equations of SMH is to write
$\vec{a} = - \omega^{2} \vec{x}$
where $\omega$ is the angular velocity, and as I mentioned in the blog I did on the pendulum, $\omega$ is related to the period of the SHM, via $T = 2 \pi / \omega$.
For our derivation of Planck’s radiation law, the parts which we need to know about are that the frequency of a harmonic oscillator, $\nu$ is given by $\nu = \omega / 2 \pi$, and so depends only on $\omega, \; \boxed{\nu \propto \omega }$. So the frequency of the spring’s oscillations depends only on k/m, we can write $\boxed{ \nu \propto k/m }$. A stiffer spring oscillates with a higher frequency, more mass (in either the spring or what is attached to it) will reduce the frequency of the oscillations.
## The energy of a harmonic oscillator
The other thing we need to know about to understand Planck’s derivation of his blackbody radiation law is the energy of the harmonic oscillator. This is always constant, but is divided between kinetic energy and potential energy. The kinetic energy is at a maximum when the spring is at its equilibrium position, at this moment it actually has zero potential energy.
The velocity of a harmonic oscillator $v$ can be found my differentiating the displacement $x$ with respect to time. The expression for displacement (see my blog here on SHM in a pendulum) is
$x(t) = A sin ( \omega t )$
where $A$ is the maximum displacement (amplitude) of the oscillstions. So
$v = \frac{dx}{dt} = A \omega cos ( \omega t)$
This will be a maximum when $cos( \omega t) = 1$ and so
$v_{max} = A \omega$
which means that the maximum kinetic energy, and hence the total energy of the harmonic oscillator is given by
$\text{Total energy} = E = \frac{1}{2}mv_{max}^{2} = \frac{1}{2} m A^{2} \omega^{2}$
As the frequency $\nu$ is just $\omega / 2 \pi$, this tells us that, for a harmonic oscillator of a given mass, the energy depends on both the square of frequency $\nu$ and the square of the size of the oscillations (larger oscillations mean more energy, double the size of the oscillations and the energy goes up by a factor of four). Mathematically we can write this as $\boxed{ E \propto A^{2} }$ and $\boxed{ E \propto \nu^{2} }$.
As we shall see, the theoretical explanation which Planck concocted to explain his blackbody curve involved assuming the walls of the cavity producing the radiation oscillated in resonance with the radiation; this is why I needed to derive these things on this blog today.
Read Full Post »
## Derivation of Planck’s radiation law – part 3
As I have outlined in parts 1 and 2 of this series (see here and here), in the 1890s, mainly through the work of the Physikalisch-Technische Reichsanstalt (PTR) in Germany, the exact shape of the blackbody spectrum began to be well determined. By mid-1900, with the last remaining observations in the infrared being completed, its shape from the UV through the visible and into the infrared was well determined for blackbodies with a wide range of temperatures.
I also described in part 2 that in 1896 Wilhelm Wien came up with a law, based on a thermodynamical argument, which almost explained the blackbody spectrum. The form of his equation (which we now know as Wien’s distribution law) is
$\boxed{ E_{ \lambda } d \lambda = \frac{ A }{ \lambda ^{5} } e^{ -a / \lambda T } d \lambda }$
Notice I said almost. Below I show two plots which I have done showing the Wien distribution law curve and the actual blackbody curve for a blackbody at a temperature of $T=4000 \text{Kelvin}$. As you can see, they are not an exact match, the Wien distribution law fails on the long-wavelength side of the peak of the blackbody curve.
Comparison of the Wien distribution law and the actual blackbody curve for a blackbody at a temperature of $T=4000 \text{Kelvin}$. Although they agree very well on the short wavelength side of the peak, the Wien law drops away too quickly on the long-wavelength side compared to the observed blackbody spectrum.
A zoomed-in view to highlight the difference between the Wien distribution law and the actual blackbody curve for a blackbody at a temperature of $T=4000 \text{Kelvin}$. Although they agree very well on the short wavelength side of the peak, the Wien law drops away too quickly on the long-wavelength side compared to the observed blackbody spectrum.
## Planck’s “act of desperation”
By October 1900 Max Planck had heard of the latest experimental results from the PTR which showed, beyond any doubt, that Wien’s distribution law did not fit the blackbody spectrum at longer wavelengths. Planck, along with Wien, was hoping that the results from earlier in the year were in error, but when new measurements by a different team at the PTR showed that Wien’s distribution law failed to match the observed curve in the infrared, Planck decided he would try and find a curve that would fit the data, irrespective of what physical explanation may lie behind the mathematics of the curve. In essence, he was prepared to try anything to get a fit.
Planck would later say of this work
Briefly summarised, what I did can be described as simply an act of desperation
What was this “act of desperation”, and why did Planck resort to it? Planck was 42 when he unwittingly started what would become the quantum revolution, and his act of desperation to fit the blackbody curve came after all other options seemed to be exhausted. Before I show the equation that he found to be a perfect fit to the data, let me say a little bit about Planck’s background.
## Who was Max Planck?
Max Karl Ernst Ludwig Planck was born in Kiel in 1858. At the time, Kiel was part of Danish Holstein. He was born into a religious family, both his paternal great-grandfather and grandfather had been distiguished theologians, and his father became professor of constitutional law at Munich University. So he came from a long line of men who venerated the laws of God and Man, and Planck himself very much followed in this tradition.
He attended the most renowned secondary school in Munich, the Maximilian Gymnasium, always finishing near the top of his class (but not quite top). He excelled through hard work and self discipline, although he may not have had quite the inherent natural ability of the few who finished above him. At 16 it was not the famous taverns of Munich which attracted him, but rather the opera houses and concert halls; he was always a serious person, even in his youth.
In 1874, aged 16, he enrolled at Munich University and decided to study physics. He spent three years studying at Munich, where he was told by one of his professors ‘it is hardly worth entering physics anymore’; at the time it was felt by many that there was nothing major left to discover in the subject.
In 1877 Planck moved from Munich to the top university in the German-speaking world – Berlin. The university enticed Germany’s best-known physicist, Herman von Helmholtz, from his position at Heidelberg to lead the creation of what would become the best physics department in the world. As part of creating this new utopia, Helmholtz demanded the building of a magnificient physics institute, and when Planck arrived in 1877 it was still being built. Gustav Kirchhoff, the first person to systematically study the nature of blackbody radiation in the 1850s, was also enticed from Heidelberg and made professor of theoretical physics.
Planck found both Helmholtz and Kirchhoff to be uninspring lecturers, and was on the verge of losing interest in physics when he came across the work of Rudolf Clausius, a professor of physics at Bonn University. Clausius’ main research was in thermodynamics, and it is he who first formulated the concept of entropy, the idea that things naturally go from order to disorder and which, possibly more than any other idea in physics, gives an arrow to the direction of time.
Planck spent only one year in Berlin, before he returned to Munich to work on his doctoral thesis, choosing to explore the concept of irreversibility, which was at the heart of Claussius’ idea of entropy. Planck found very little interest in his chosen topic from his professors in Berlin, and not even Claussius answered his letters. Planck would later say ‘The effect of my dissertation on the physicists of those days was nil.’
Undeterred, as he began his academic career, thermodynamics and, in particular, the second law (the law of entropy) became the focus of his research. In 1880 Planck became Privatdozent, an unpaid lecturer, at Munich University. He spent five years as a Privatdozent, and it looked like he was never going to get a paid academic position. But in 1885 Gottingen University announced that the subject of its prestigoius essay competition was ‘The Nature of Energy’, right up Planck’s alley. As he was working on his essay for this competition, he was offered an Extraordinary (assistant) professorship at the University of Kiel.
Gottingen took two years to come to a decision about their 1885 essay competition, even though they had only received three entries. They decided that no-one should receive first prize, but Planck was awarded second prize. It later transpired that he was denied first prize because he had supported Helmholtz in a scientific dispute with a member of the Gottingen faculty. This brought him to the attention of Helmholtz, and in November 1888 Planck was asked by Helmholtz to succeed Kirchhoff as professor of theoretical physics in Berlin (he was chosen after Ludwig Boltzmann turned the position down).
And so Planck returned to Berlin in the spring of 1889, eleven years after he had spent a year there, but this time not as a graduate student but as an Extraordinary Professor. In 1892 Planck was promoted to Ordinary (full) Professor. In 1894 both Helmholtz and August Kundt, the head of the department, died within months of each other; leaving Planck at just 36 as the most senior physicist in Germany’s foremost physics department.
Max Planck who, in 1900 at the age of 42, found a mathematical equation which fitted the entire blackbody spectrum correctly.
As part of his new position as the most senior physicist in the Berlin department, he took over the duties of being adviser for the foremost physics journal of the day – Annalen der Physik (the journal in which Einstein would publish in 1905). It was in this role of adviser that he became aware of the work being done at PTR on determining the true spectrum of a blackbody.
Planck regarded the search for a theoretical explanation of the blackbody spectrum as nothing less than the search for the absolute, and as he later stated
Since I had always regarded the search for the absolute as the loftiest goal of all scientific activity, I eagerly set to work
When Wien published his distribution law in 1896, Planck tried to put the law on a solid theoretical foundation by deriving it from first principles. By 1899 he thought he had succeeded, basing his argument on the second law of thermodynamics.
## Planck finds a curve which fits
But, all of this fell apart when it was shown conclusively on the 2nd of February 1900, by Lummer and Pringsheim of the PTR, that Wien’s distribribution law was wrong. Wien’s law failed at high temperatures and long wavelengths (the infrared); a replacement which would fit the experimental curve needed to be found. So, on Sunday the 7th of October, Planck set about trying to find a formula which would reproduce the observed blackbody curve.
He was not quite shooting in the dark, he had three pieces of information to help him. Firstly, Wien’s law worked for the intensity of radiation at short wavelengths. Secondly, it was in the infrared that Wien’s law broke down, at these longer wavelengths it was found that the intensity was directly propotional to the temperature. Thirdly, Wien’s displacement law, which gave the relationship between the wavelength of the peak of the curve and the blackbody’s temperature worked for all observed blackbodies.
After working all night of the 7th of October 1900, Planck found an equation which fitted the observed data. He presented this work to the German Physical Society a few weeks later on Friday the 19th of October, and this was the first time others saw the equation which has now become known as Planck’s law.
The equation he found for the energy in the wavelength interval $d \lambda$ had the form
$\boxed{ E_{\lambda} \; d \lambda = \frac{ A }{ \lambda^{5} } \frac{ 1 }{ (e^{a/\lambda T} - 1) } \; d\lambda }$
(compare this to the Wien distribution law above).
After presenting his equation he sat down; he had no explanation for why this equation worked, no physical understanding of what was going on. That understanding would dawn on him over the next few weeks, as he worked tirelessly to explain the equation on a physical basis. It took him six weeks, and in the process he had to abandon some of the ideas in physics which he held most dear. He found that he had to abandon accepted ideas in both thermodynamics and electromagnetism, two of the cornerstones of 19th Century physics. Next week, in the fourth and final part of this blog-series, I will explain what physical theory Planck used to explain his equation; the theory which would usher in the quantum age.
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http://www.zora.uzh.ch/id/eprint/111518/ | # Mod-$\phi$ convergence
Delbaen, Freddy; Kowalski, Emmanuel; Nikeghbali, Ashkan (2015). Mod-$\phi$ convergence. International Mathematics Research Notices, 2015(11):3445-3485.
## Abstract
Using Fourier analysis, we study local limit theorems in weak-convergence problems. Among many applications, we discuss random matrix theory, some probabilistic models in number theory, the winding number of complex Brownian motion and the classical situation of the central limit theorem, and a conjecture concerning the distribution of values of the Riemann zeta function on the critical line.
## Abstract
Using Fourier analysis, we study local limit theorems in weak-convergence problems. Among many applications, we discuss random matrix theory, some probabilistic models in number theory, the winding number of complex Brownian motion and the classical situation of the central limit theorem, and a conjecture concerning the distribution of values of the Riemann zeta function on the critical line.
## Statistics
### Citations
1 citation in Web of Science®
1 citation in Scopus®
### Altmetrics
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https://arxiv.org/abs/1601.08167 | astro-ph.SR
(what is this?)
# Title: Turbulent reconnection of magnetic bipoles in stratified turbulence
Abstract: We consider strongly stratified forced turbulence in a plane-parallel layer with helicity and corresponding large-scale dynamo action in the lower part and non-helical turbulence in the upper. The magnetic field is found to develop strongly concentrated bipolar structures near the surface. They form elongated bands with a sharp interface between opposite polarities. Unlike earlier experiments with imposed magnetic field, the inclusion of rotation does not strongly suppress the formation of these structures. We perform a systematic numerical study of this phenomenon by varying magnetic Reynolds number, scale separation ratio, and Coriolis number. We focus on the formation of a current sheet between bipolar regions where reconnection of oppositely oriented field lines occurs. We determine the reconnection rate by measuring either the inflow velocity in the vicinity of the current sheet or by measuring the electric field in the reconnection region. We demonstrate that for large Lundquist numbers, S>10^3, the reconnection rate is nearly independent of S in agreement with results of recent numerical simulations performed by other groups in simpler settings.
Comments: 11 pages, 14 figures Subjects: Solar and Stellar Astrophysics (astro-ph.SR) Journal reference: Mon. Not. Roy. Astron. Soc. 459, 4046-4056 (2016) DOI: 10.1093/mnras/stw888 Report number: NORDITA-2016-7 Cite as: arXiv:1601.08167 [astro-ph.SR] (or arXiv:1601.08167v2 [astro-ph.SR] for this version)
## Submission history
From: Sarah Jabbari [view email]
[v1] Fri, 29 Jan 2016 16:00:35 GMT (4048kb)
[v2] Fri, 18 Mar 2016 19:46:58 GMT (4009kb) | {"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": 0.8586885929107666, "perplexity": 2891.431740514644}, "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-2017-43/segments/1508187827853.86/warc/CC-MAIN-20171024014937-20171024034937-00592.warc.gz"} |
https://yutsumura.com/top-10-popular-math-problems-in-2016-2017/top10mathproblems2017/ | top10mathproblems2017
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http://tex.stackexchange.com/questions/83196/setting-texinputs-for-pdflatex-in-ubuntu | # Setting TEXINPUTS for pdflatex in Ubuntu
I have a set of files that are commonly used as inputs in my tex files, so I put them in a directory and set the path in the TEXINPUTS environment variable (in Ubuntu). This works fine when compiling files with the latex command, but if I compile them with pdftex, it gives the following error message:
``````! I can't find file `{header.tex}'.
``````
As far as I can tell, pdftex is completely ignoring the TEXINPUTS environment variable. Does it use a different environment variable? How can I get it to recognise a path?
-
Are you using `pdftex` or `pdflatex`. The latter will read `\input{header.tex}` as a braced argument for file `header.tex`, but `pdftex` is Knuth's plain format with the pdfTeX engine: there, `\input` has primitive syntax only, and the file name searched is `{input.tex}`. – Joseph Wright Nov 18 '12 at 12:38
Aha! pdflatex works, but pdftex doesn't. Thanks for explaining. – thornate Nov 18 '12 at 12:52
I'll write that up in a slightly modified form as an answer, then :-) – Joseph Wright Nov 18 '12 at 13:06
Start your documents with `\NeedsTeXFormat{LaTeX2e}` to get a meaningful error message if you use the wrong program. – Martin Schröder Nov 19 '12 at 7:41
The `latex` command runs LaTeX in DVI-output mode. LaTeX's definition of `\input` allows for a syntax
``````\input{<file-name>}
``````
where the `<file-name>` is read as a braced argument. The `pdflatex` command will do exactly the same but with direct PDF output.
On the other hand, `pdftex` runs plain TeX with direct PDF output. The plain TeX definition for `\input` uses a 'primitive' syntax, in which the name is read as any tokens at all up to the first space. Thus with `pdftex`
``````\input{<file-name>}
``````
ends up looking for a file called `{<file-name>}`, including the braces.
-
It may be worth noting that `\input{filename}` works also in LuaTeX (as opposed to LuaLaTeX). – egreg Nov 18 '12 at 13:46
@egreg One of the 'features' of LuaTeX which gives me headaches :-) – Joseph Wright Nov 18 '12 at 14:00 | {"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": 0.9991143345832825, "perplexity": 2741.520971223771}, "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/1422120842874.46/warc/CC-MAIN-20150124173402-00040-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://laurentlessard.com/bookproofs/finding-the-doctored-coin/ | # Finding the doctored coin
This Riddler puzzle is about repeatedly flipping coins!
On the table in front of you are two coins. They look and feel identical, but you know one of them has been doctored. The fair coin comes up heads half the time while the doctored coin comes up heads 60 percent of the time. How many flips — you must flip both coins at once, one with each hand — would you need to give yourself a 95 percent chance of correctly identifying the doctored coin?
Extra credit: What if, instead of 60 percent, the doctored coin came up heads some P percent of the time? How does that affect the speed with which you can correctly detect it?
Here is my solution.
[Show Solution]
## 4 thoughts on “Finding the doctored coin”
1. Hi Laurent,
I am trying to understand how your curve can fall all the way to 1 when the doctored probability is 100%. In one flip, there is a 50% chance that they are both heads and I learn nothing. So there is a 1/2 chance I need a second flip, and a 1/4 chance I need a third, etc; the geometric series sums to 2. Also I found an odd “stairstep” feature where the average number of guesses jumps discontinuously at each probability p obeying p^m/(1-p)^m=19 (19 = 95% / 5% and m an integer). I put my solution at
As an aside, a Monte-Carlo is a kind of inefficient way to find the distribution of lengths, you can solve the Fokker-Planck equation for the distribution of possible outcomes and eliminate fluctuations…
guy
1. Hi Laurent,
Reading through again, I think what you computed is the number of flips n such that it is more likely than not that one knows with 95% certainty. Rather than the average number of flips needed to reach 95% certainty. Or the number of flips such that a guess after that many flips is 95% certain to be right. Curiously, these each give a different answer, though the first two appear to be close.
1. Hi Guy,
Thanks for the comments! I believe that what I calculated is the test statistic (the equation in the blue box that says n > …) that, when satisfied, implies you are at least 95% confident that you know which coin is the doctored one.
My mistake was in approximating the expected value of n by substituting the true probabilities in place of the empirical ones. This turns out to be a good approximation when $p_2=0.6$, but leads to a nonsensical solution when $p_2=1$, as you pointed out.
If I can find some free time next week, I’ll look over my solution and see if I can patch it up. I think that if I calculate expectations exactly, I will obtain what you found in the second part of your solution (“stop when you know”). In the meantime, do you have any good references that explain the Fokker-Planck approach you mentioned?
1. Jason Weisman says:
I think the question is to determine before starting, how many flips are needed to give yourself a 95 percent chance of correctly identifying the doctored coin. I believe the correct answer to this question with p2=0.6, p1=0.5 is 134. This is different from the published solution which does not take into consideration 50% probability of guessing correctly for cases where there are the same number of heads up flips. | {"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": 0.9321072697639465, "perplexity": 355.6175365534924}, "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/1627046153899.14/warc/CC-MAIN-20210729234313-20210730024313-00507.warc.gz"} |
https://www.cheenta.com/set-theory-isi-b-stat-entrance-objective/ | Set theory | TOMATO ISI B.stat Objective | Problem 53
Try this beautiful problem Based on Set Theory useful for ISI B.Stat Entrance.
Set Theory| ISI B.Stat Entrance | Problem 53
There were 41 candidates in an examination and each candidate was examined in algebra, geometry and calculus. It was found that 12 candidates failed in algebra, 7 failed in geometry and 8 failed in calculus , 2 in geometry and calculus , 3 in calculus and algebra , 6 in algebra and geometry, whereas only 1 failed in one of the three subjects. Then, find the number of candidates who passed in all three subjects?
• $24$
• $26$
• $28$
Key Concepts
SET
Algebra
Cardinal number
Answer: $24$
TOMATO, Problem 53
Challenges and Thrills in Pre College Mathematics
Try with Hints
use set theory concept
Can you now finish the problem ..........
we assume that A ,G,C be the sets of the students who failed on algebra ,geometry and calculus respectively.
Find the complement of $N(A \cup G \cup C)$
can you finish the problem........
Let S be the set of total students i.e N(s) = 41
we assume that A ,G,C be the sets of the students who failed on algebra ,geometry and calculus respectively.
Therefore
N(A)=12
N(G)=7
N(C)=8
and
$N(A \cap G \cap C)$=1
$N(G \cap C)$ =2 ,
$N(C\cap A)$ =3
$N(A\cap G)$ =6
$N(A \cup G \cup C)$=$N(A) + N(G) +N(C) -N(G \cap C) -N (C \cap A) -N(C \cap A) + N(A \cap G \cap C)$=12 + 7+8-2-6-3+1=17
Therefore complement of $N(A \cup G \cup C)$ =41-17=24 | {"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": 0.9420720338821411, "perplexity": 4841.097138955617}, "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-2021-25/segments/1623488519735.70/warc/CC-MAIN-20210622190124-20210622220124-00381.warc.gz"} |
https://www.physicsforums.com/threads/radioactive-decay-given-ax-ay-and-t-for-y-get-t-for-x-not-possible.196312/ | # Radioactive decay. Given Ax, Ay and t½ for Y, get t½ for X. Not possible?
1. Nov 5, 2007
### catkin
[SOLVED] Radioactive decay. Given Ax, Ay and t½ for Y, get t½ for X. Not possible?
1. The problem statement, all variables and given/known data
This is from Advanced Physics by Adams & Allday, spread 8.13 Question 1.
The activity of 20 g of element X is four times the activity of 10 g of element Y. Element Y has a half-life of 20,000 y. What is the half-life of X?
2. Relevant equations
$$A = \lambda N$$
$$\lambda t_{0.5} = 0.69$$
3. The attempt at a solution
Rewriting the first relevant equation in $t_{0.5}$, rather than λ, using the proportionality from the second relevant equation
$$A = 0.69 N / t_{0.5}$$
Considering 10g of both elements
$$A_{X} = 2A_{Y}$$
Expressing these activities in terms of the number of atoms in 10 g and half life
$$0.69 N_{X} / t_{0.5X} = 2 \times 0.69 N_{Y} / t_{0.5Y}$$
$$N_{X} / t_{0.5X} = 2N_{Y} / t_{0.5Y}$$
$$t_{0.5X} = (N_{X} / 2 N_{Y}) t_{0.5Y}$$
Substituting, using years as time units
$$t_{0.5X} = (N_{X} / 2 N_{Y}) {20000}$$
If the number of atoms in 10 g of element X were the same as the number of atoms in 10 g of element Y (there is no reason why it should be) then $t_{0.5X}$ would be 10,000 years (the answer the book gives).
4. Question Am I right in thinking there is not enough information in the question to answer it?
2. Nov 6, 2007
### rl.bhat
10 g of x and y cannot have same number of atoms. Nx and Ny depend on their molicular weights.
3. Nov 7, 2007
### catkin
Thanks rl.bhat
It doesn't answer my question though; is the problem soluble?
[Separate issue: what if the elements had the same atomic number? Say Th-234 and Pa-234? Wouldn't the number of atoms in 10 g be the same, at least to the number of significant figures the question implies?]
4. Nov 7, 2007
### rl.bhat
In case of Th- 234 and Pa- 234 the problem is soluble.
5. Nov 7, 2007
### catkin
Thanks again. That makes sense. Is the problem soluable as it is posed in the original question?
6. Nov 7, 2007
### Staff: Mentor
Certainly if X -> Y (or Y -> X) by beta decay, then the same mass would have approximately the same number of atoms within 1% or less.
The question becomes - "does X -> Y, or vice versa, i.e. do they represent sequential steps in a decay chain?"
The approach seems correct. The problem hinges on the assumption of Nx and Ny, which would be determined by the atomic masses.
7. Nov 7, 2007
### catkin
Thanks, astronuc
That helps understanding.
There's nothing in the question to indicate either any decay relationship or the atomic mass relationship between X and Y, though. Decay chains are introduced in a later "spread" in the textbook so should not be necessary for the solution.
I'm increasingly coming to think that the question as set is not soluable.
8. Nov 7, 2007
### Staff: Mentor
Raise this concern with the professor.
With mass and activity, one can get the specific activity, but one needs to know the atomic mass to obtain the number of atoms.
Since you obtained the answer given in the book with the assumption that the atomic mass of X and Y are roughly equal, that would seem to indicate an implicit assumption on the part of the author. If beta decay was involved (e.g. X -> Y), then that is a reasonable assumption.
9. Nov 7, 2007
### catkin
Thanks Astronuc
That's enough to consider this one SOLVED
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Similar Discussions: Radioactive decay. Given Ax, Ay and t½ for Y, get t½ for X. Not possible? | {"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": 0.8871065974235535, "perplexity": 1359.6783184122776}, "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-2017-47/segments/1510934805977.0/warc/CC-MAIN-20171120090419-20171120110419-00788.warc.gz"} |
http://math.stackexchange.com/questions/324517/trace-of-the-inverse-of-a-matrix-times-another-matrix | trace of the inverse of a matrix times another matrix
I do know generally $\text{trace}(A^{-1}B)\not= \sum_i \lambda_{B_i}/\lambda_{A_i}$,
where $\lambda_{A_i}$ and $\lambda_{B_i}$ are the corresponding eigenvalues of matrix $A$ and $B$ respectively,
but is there any cases when this equality can be statified?
-
@sbr for $A=B$ you need to know which is the first eigenvalue and which is the second and so on, else the equality is not true in general – Dominic Michaelis Mar 8 '13 at 8:30
If $A$ and $B$ are simultaneously diagonalisable (or trigonalisable) and you are combining eigenvalues for the same (generalised) eigenvectors then this obviously holds. In other cases (i.e., almost always), all bets are off.
There are many cases: let $A$ be $\gamma \cdot I$ where $I$ is the identiy matrix and $\gamma$ is an arbitrary scalar $\neq 0$, or let $B=0$ (the matrix with every entry zero). | {"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": 0.988586962223053, "perplexity": 154.46812420586036}, "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-2015-27/segments/1435375099361.57/warc/CC-MAIN-20150627031819-00114-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Triangular_distribution | # Triangular distribution
Parameters Probability density function Cumulative distribution function $a:~a\in (-\infty,\infty)$ $b:~a $c:~a\le c\le b\,$ $a \le x \le b \!$ $\begin{cases} 0 & \text{for } x < a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \text{for } a \le x < c, \\[4pt] \frac{2}{b-a} & \text{for } x = c, \\[4pt] \frac{2(b-x)}{(b-a)(b-c)} & \text{for } c < x \le b, \\[4pt] 0 & \text{for } b < x. \end{cases}$ $\begin{cases} 0 & \text{for } x \leq a, \\[2pt] \frac{(x-a)^2}{(b-a)(c-a)} & \text{for } a < x \leq c, \\[4pt] 1-\frac{(b-x)^2}{(b-a)(b-c)} & \text{for } c < x < b, \\[4pt] 1 & \text{for } b \leq x. \end{cases}$ $\frac{a+b+c}{3}$ $\begin{cases} a+\frac{\sqrt{(b-a)(c-a)}}{\sqrt{2}} & \text{for } c \ge \frac{a+b}{2}, \\[6pt] b-\frac{\sqrt{(b-a)(b-c)}}{\sqrt{2}} & \text{for } c \le \frac{a+b}{2}. \end{cases}$ $c\,$ $\frac{a^2+b^2+c^2-ab-ac-bc}{18}$ $\frac{\sqrt 2 (a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^2\!+\!b^2\!+\!c^2\!-\!ab\!-\!ac\!-\!bc)^\frac{3}{2}}$ $-\frac{3}{5}$ $\frac{1}{2}+\ln\left(\frac{b-a}{2}\right)$ $2\frac{(b\!-\!c)e^{at}\!-\!(b\!-\!a)e^{ct}\!+\!(c\!-\!a)e^{bt}} {(b-a)(c-a)(b-c)t^2}$ $-2\frac{(b\!-\!c)e^{iat}\!-\!(b\!-\!a)e^{ict}\!+\!(c\!-\!a)e^{ibt}} {(b-a)(c-a)(b-c)t^2}$
In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b.
## Special cases
### Two points known
The distribution simplifies when c = a or c = b. For example, if a = 0, b = 1 and c = 1, then the PDF and CDF become:
$\left.\begin{matrix}f(x) &=& 2x \\[8pt] F(x) &=& x^2 \end{matrix}\right\} \text{ for } 0 \le x \le 1$
\begin{align} E(X) & = \frac{2}{3} \\[8pt] \mathrm{Var}(X) &= \frac{1}{18} \end{align}
### Distribution of mean of two standard uniform variables
This distribution for a = 0, b = 1 and c = 0.5 is distribution of X = (X1 + X2)/2, where X1, X2 are two independent random variables with standard uniform distribution.
$f(x) = \begin{cases} 4x & \text{for }0 \le x < \frac{1}{2} \\ 4-4x & \text{for }\frac{1}{2} \le x \le 1 \end{cases}$
$F(x) = \begin{cases} 2x^2 & \text{for }0 \le x < \frac{1}{2} \\ 1-2(1-x)^2 & \text{for }\frac{1}{2} \le x \le 1 \end{cases}$
\begin{align} E(X) & = \frac{1}{2} \\[6pt] \operatorname{Var}(X) & = \frac{1}{24} \end{align}
### Distribution of the absolute difference of two standard uniform variables
This distribution for a = 0, b = 1 and c = 0 is distribution of X = |X1 − X2|, where X1, X2 are two independent random variables with standard uniform distribution.
\begin{align} f(x) & = 2 - 2x \text{ for } 0 \le x < 1 \\[6pt] F(x) & = 2x - x^2 \text{ for } 0 \le x < 1 \\[6pt] E(X) & = \frac{1}{3} \\[6pt] \operatorname{Var}(X) & = \frac{1}{18} \end{align}
## Generating Triangular-distributed random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
$\begin{matrix} \begin{cases} X = a + \sqrt{U(b-a)(c-a)} & \text{ for } 0 < U < F(c) \\ & \\ X = b - \sqrt{(1-U)(b-a)(b-c)} & \text{ for } F(c) \le U < 1 \end{cases} \end{matrix}$[1]
Where F(c) = (c-a)/(b-a)
has a Triangular distribution with parameters a, b and c. This can be obtained from the cumulative distribution function.
## Use of the distribution
The triangular distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known but data is scarce (possibly because of the high cost of collection). It is based on a knowledge of the minimum and maximum and an "inspired guess" [2] as to the modal value. For these reasons, the triangle distribution has been called a "lack of knowledge" distribution.
The triangular distribution is therefore often used in business decision making, particularly in simulations. Generally, when not much is known about the distribution of an outcome, (say, only its smallest and largest values) it is possible to use the uniform distribution. But if the most likely outcome is also known, then the outcome can be simulated by a triangular distribution. See for example under corporate finance.
### Project management
The triangular distribution, along with the Beta distribution, is also widely used in project management (as an input into PERT and hence critical path method (CPM)) to model events which take place within an interval defined by a minimum and maximum value.
### Audio dithering
The symmetric triangular distribution is commonly used in audio dithering, where it is called TPDF (Triangular Probability Density Function). | {"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": 22, "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": 0.9989724159240723, "perplexity": 727.4325274388366}, "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/1440644064869.18/warc/CC-MAIN-20150827025424-00072-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://aviation.stackexchange.com/questions/70709/why-is-the-nose-cone-of-su-34-white | # Why is the nose cone of SU-34 white?
The title says it all.
Can be clearly seen for example in this image:
(image source: Wikimedia)
The rest of the jet seems to have a camouflage.
• Tips of the vertical stabs are also white... – Ron Beyer Oct 16 '19 at 13:17 | {"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": 0.8882731795310974, "perplexity": 2535.525561423855}, "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-2020-16/segments/1585370493818.32/warc/CC-MAIN-20200329045008-20200329075008-00548.warc.gz"} |
https://brilliant.org/problems/a-mechanics-problem-by-satish-varma/ | A classical mechanics problem by satish varma
Satish(yellow one) has dropped a ball(red one) of unknown mass on an inclined surface with inclination angle 45 degrees. He wants to find the velocity with which it touches ground.he knew horizontal distance between him and where ball touches ground is 20m. help him finding velocity.
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https://www.physicsforums.com/threads/sounds-springs-and-pendelums.467155/ | # Sounds, Springs, and Pendelums
• Start date
• #1
146
0
## Homework Statement
A vibrator moves one end of a rope up and down to generate a wave. The tension in the rope is 63 N. The frequency is then doubled. To what value must the tension be adjusted, so the new wave has the same wavelength as the old one?
(1/2)kx^2
and/or
(1/2)mv^2
## The Attempt at a Solution
I tried using those equations but i am not getting the right answer :/ Please Help. Any assistance would be appreciated. thnk you
Last edited:
• #2
gneill
Mentor
20,925
2,866
How did you try to use those equations? (I ask because they don't appear to be relevant to the problem at hand, so I'm curious).
• #3
146
0
sorry. i edited the question and changed the question but i forgot to change the other 2 parts.
so...
relevant equation:
I know velocity=frequecy*wavelength, but i do not see where tension comes in in this problem. is there another equation i should use?
• #4
gneill
Mentor
20,925
2,866
Do a search on the keywords: wave velocity tension .
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2K | {"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": 0.9421737194061279, "perplexity": 1399.8173447790975}, "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-2021-25/segments/1623488519735.70/warc/CC-MAIN-20210622190124-20210622220124-00024.warc.gz"} |
https://www.physicsforums.com/threads/what-is-reflection-and-refraction-of-light-at-the-microscopic-scale.549197/ | # What is reflection and refraction of light at the microscopic scale?
Tags:
1. Nov 10, 2011
### Aidyan
I'm not asking for what reflection and refraction are or the usual law governing it, but I would like to understand what they represent at the quantum atomic, molecular level? In a mirror is it about photons absorbed and emitted with the same wavelength and same direction through atomic electron transitions? How can that be? And what is microscopically refraction? Why should a photon traveling between atoms of a trasparent medium change not only its speed but also its direction? I'm a bit confused... Can someone indicate some nice links explaining all that?
2. Nov 11, 2011
### Claude Bile
Reflection and refraction are linked to a single property of a solid; the refractive index. The refractive index is commonly defined as the ratio of the speed of light in the solid compared to the speed of light in a vacuum, however this definition has two main drawbacks.
1. The term "speed of light" is ambiguous when multiple frequencies are concerned, as we can define multiple velocities (group velocity, phase velocity etc).
2. It says nothing about the properties of the solid.
A better definition of refractive index is $1/\sqrt{\epsilon \mu}$ where $\epsilon$ is the electric permittivity and $\mu$ is the magnetic permeability. Permittivity and permeability are defined as the dipoles generated per unit volume, per unit of electric and magnetic field respectively. These are the atomic properties that affect reflection and refraction on an atomic scale.
Claude.
3. Nov 11, 2011
### Aidyan
But why does dipole generation in an amorphous material emit photons always in the same direction? I would have expected an isotropic re-emission of photons in every direction. And how do we have to distinguish dipole generation in the case of reflection from refraction? I think things are a bit more complicate than this.
Last edited: Nov 11, 2011
4. Nov 18, 2011
### Claude Bile
Photons are not being absorbed and re-emitted. Only photons that lie in specific frequency bands are absorbed by any given material. Photons that do not lie in these absorption bands (i.e. those that are transmitted) instead cause the positive and negative charges in nearby atoms to separate (by virtue of the E-field that the photon is a quantum of).
The difference in the laws of reflection and refraction is due to the boundary conditions that apply at an interface between two media with different refractive indices.
Claude. | {"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": 0.8760755658149719, "perplexity": 429.52856544190786}, "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-2018-43/segments/1539583514030.89/warc/CC-MAIN-20181021140037-20181021161537-00501.warc.gz"} |
http://mathoverflow.net/users/39339/pepetoro?tab=activity | # PepeToro
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Trying to learn math :)
# 14 Actions
Oct 31 comment “Partition” of a smooth function in $\mathbb R^2$ Well there is not much difference from what I wrote. What @JochenWengenroth wrote above is correct. I hope things are clear now. Oct 31 comment “Partition” of a smooth function in $\mathbb R^2$ By $f_1$ we abbreviate the smooth function of the variables $(xy,x)$ and by $f_2$ the smooth function of the variables $(xy,y)$. Just as shown in the expansion above. Call $w=xy$, then $\hat f_1(w,x)=\sum a_{mn}w^mx^n$ and $\hat f_2(w,y)=\sum a_{mn}w^my^n$, and make the coefficients coincide. Some of them are zero if necessary. Oct 31 comment “Partition” of a smooth function in $\mathbb R^2$ Both $f_i$ are smooth. Both are used at the decomposition. That is why I showed at least the formal level of the proof. Flat means zero Taylor expansion. The result means that a function in two variables may be partitioned as showed. The importance is that it can be done at the level of smooth functions and not only at the level of formal functions. My application is in dynamical systems for example. It may happen that $xy$ is a first integral (a constant along the trajectories of a vector field). Then it is much easier to integrate, say $f$, if we have such a result. Oct 30 revised “Partition” of a smooth function in $\mathbb R^2$ added 571 characters in body Oct 30 comment “Partition” of a smooth function in $\mathbb R^2$ @GHfromMO nope, I really meant what is written. Imagine in the formal series expansion of the form $x^iy^j$, $f_1$ contains monomials where $i\geq j$ and $f_2$ the rest. Oct 30 revised “Partition” of a smooth function in $\mathbb R^2$ edited tags Oct 30 asked “Partition” of a smooth function in $\mathbb R^2$ Sep 24 awarded Autobiographer Aug 5 awarded Editor Aug 5 revised Smooth normal forms of vector fields (the path method) edited body Aug 5 asked Smooth normal forms of vector fields (the path method) Aug 30 revised Stratification of a smooth map edited tags Aug 30 awarded Student Aug 30 asked Stratification of a smooth map | {"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": 0.8275800347328186, "perplexity": 540.8473154554102}, "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-2015-35/segments/1440645371566.90/warc/CC-MAIN-20150827031611-00151-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://guido.vonrudorff.de/2019/resampling-grids-for-integration/ | # Resampling grids for integration
When evaluating an integral on a grid, the finite resolution thereof can affect the accuracy of the result. In the context of atomistic simulation, this is because a slight change in nuclear coordinates will affect the mapping on the grid.
For smooth properties like the electron density, there is a workaround. Akin to Simpson’s rule where polynomials are fitted to the grid points first and then the piecewise defined polynomials are integrated, a refinement of the grid can achieve the similar improvements in integration.
While the smart way would be to not build the grid in memory but rather integrate as you go, the following is a simple and reasonably fast implementation by explicit grid refinement in python.
``````import numpy as np
import scipy.ndimage as snd
import matplotlib.pyplot as plt
# generate data to operate on
data = np.arange(16).reshape(4, 4).astype(np.float)
data[0, 3] = 0
data[0, 1:3] = 5
# visualisation only
f, axs = plt.subplots(1, 2, dpi=300)
ax_2d, ax_scan = axs
ax_2d.imshow(data)
# refine grid: build grid
X, Y = np.mgrid[0:3:50j, 0:3:50j]
positions = np.vstack([X.ravel(), Y.ravel()])
for order in range(6):
# refine grid: fit splines, re-evaluate function on grid
a = snd.map_coordinates(data, positions, mode='nearest', order=order)
ax_scan.plot(np.linspace(0, 3, 50), a[:50], label='Order %d' % order)
ax_scan.legend()``````
The parameter order specifies the interpolation order. You’ll need to set it to match your computational problem. This solution generalizes to arbitrary dimensions, even though the number of gridpoints scales unfavourably. | {"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": 0.882988452911377, "perplexity": 2590.2462138957885}, "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-2020-16/segments/1585370524043.56/warc/CC-MAIN-20200404134723-20200404164723-00176.warc.gz"} |
https://mathhelpboards.com/threads/muffins-gcse-additional-maths-moments-question-from-y-answers.2474/ | ### Welcome to our community
#### CaptainBlack
##### Well-known member
Jan 26, 2012
890
Two painters of mass 50kg and 70 kg stand on a uniform horizontal plank ABCDEFG of mass 40kg and length 8m.
B, C, D and E are respectively 1m, 2m, 5m and 6m from A. The 50kg painter stands at B, the 70kg painter at D and the plank's supported at C and E.
Find the reaction at each support I got the answers to be 1053.5 and 514.5 but my answer book tells me otherwise (although it has been wrong before) can someone please check this and explain where I've gone wrong please?
CB
#### CaptainBlack
##### Well-known member
Jan 26, 2012
890
Two painters of mass 50kg and 70 kg stand on a uniform horizontal plank ABCDEFG of mass 40kg and length 8m.
B, C, D and E are respectively 1m, 2m, 5m and 6m from A. The 50kg painter stands at B, the 70kg painter at D and the plank's supported at C and E.
Find the reaction at each support I got the answers to be 1053.5 and 514.5 but my answer book tells me otherwise (although it has been wrong before) can someone please check this and explain where I've gone wrong please?
CB
Since the reaction forces must sum to the total load (the two painters and the weight of the plank acting at the centre of mass of the plank) which is $$160g = 1569.6 \, {\rm{N}}$$ (taking $$g$$ to be $$9.81\, {\rm{m/s^2}}$$ ) your answer is not impossible so far (using $$g=9.8\, {\rm{m/s^2}}$$ we have exact agreement).
You need to set up a pair of simultaneous equations under the assumption that the system is in equilibrium, the first is that the reaction forces sum to the load forces:
$$R_C+R_E = 160 g$$
The second is obtained by taking moments about some convenient point, as the system is in equilibrium these must sum to zero. A convenient point (as it eliminates one of the unknowns) is either C or E.
$$4 R_C -2 (40 g) -5 (50 g) - 1 (70 g) =0$$
$$R_C = 100 g\, {\rm{N}}$$ ... | {"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": 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": 0.9246149063110352, "perplexity": 730.8624281139811}, "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/1603107874135.2/warc/CC-MAIN-20201020192039-20201020222039-00225.warc.gz"} |
https://en.wikipedia.org/wiki/Talk:Cumulant | # Talk:Cumulant
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## (d/dt) log E(etX) as cumulant generation function
The formula
$E\left(e^{tX}\right)=\exp\left(\sum_{n=1}^\infty\kappa_n t^n/n!\right)\,$
might be written
$\log E\left(e^{tX}\right)=\sum_{n=1}^\infty\kappa_n t^n/n!\,$
The constant term is found by setting t = 0:
$\log E\left(e^0\right)=0$
Zero is not a cumulant, and so the function
$\frac{d}{dt}\log E\left(e^{tX}\right)=\sum_{n=0}^\infty\kappa_{n+1} t^n/n!=\mu+\sigma^2t+\cdots$
better deserves the name 'cumulant generation function'.
Bo Jacoby 12:55, 4 January 2006 (UTC)
I agree that there's no zeroth-order cumulant. But I don't think that's a reason to change the convention to what you've given here. In that version, the coefficient of tn/n! is not the nth cumulant, and that is potentially confusing. Besides, to speak of a zeroth cumulant and say that it's zero regardless of the probability distribution seems harmless at worst. Michael Hardy 00:03, 9 January 2006 (UTC)
I understand your reservations against changing conventions. Note, however, the tempting simplification obtained by differentiation.
The (new cumulant generation function of the) degenerate distribution is 0;
The (..) normal distribution is t.
The (..) bernoulli distribution is (1+(p−1−1)e−t)−1
The (..) binomial distribution is n(1+(p−1−1)e−t)−1
The (..) geometric distribution is (−1+(1−p)−1e−t)−1
The (..) negative binomial distribution is n(−1+(1−p)−1e−t)−1
The (..) poisson distribution is λet
Bo Jacoby 12:21, 31 January 2006 (UTC)
The last point kills your proposal: one wants to be able to speak not only of compositions of cumulant-generating functions, but of compositional inverses in cases where the expected value is not 0. So one wants the graph of the function to pass through (0, 0), with nonzero slope when the expected value is not 0. Michael Hardy 22:54, 31 January 2006 (UTC)
What do you mean? Please explain first and conclude later. The two definitions allow the same operations. The new definition just does not contain a superfluous zero constant term. The graph of the new cumulant-generating function passes through (0, μ) having the slope σ2. Curvature shows departure from normality: μ+σ2t. Bo Jacoby 09:14, 1 February 2006 (UTC)
I don't like that kind of definition for the cumulant generating function. Imagine you have a random variable X and a constant a. If K(t) is the cumulant generating function for X, then what is the cumulant generating function for aX? Using the standard definition it's K(at) whereas using your definition it would be aK(at) which is more complicated. Ossi 22:41, 4 April 2006 (UTC)
I'll comment further on Bo Jacoby's comments some day. But for now, let's note that what is in the article has been the standard convention in books and articles for more than half a century, and Wikipedia is not the place to introduce novel ideas. Michael Hardy 22:55, 4 April 2006 (UTC)
## cumulant (encyclopedia)
I came randomly to see the article : no explanation about a cumulant were in view. A TOC was followed by formulas.
We math people love what we do. Let us try to do more : explain what we do (for this, I need help).
• What class of math object is that
• Who uses it and for what
• Are there plain related concepts to invoke, &c. ? Thanks --DLL 18:47, 9 June 2006 (UTC)
P.S. Wolfram, for example, gives links to : Characteristic Function, Cumulant-Generating Function, Fourier Transform, k-Statistic, Kurtosis, Mean, Moment, Sheppard's Correction, Skewness, Unbiased Estimator, Variance. [Pages Linking Here]. Though I cannot tell if it is pertinent here, maybe a little check might be done ? Thanks again. --DLL 18:56, 9 June 2006 (UTC)
The very first sentence in the article says what cumulants are. Michael Hardy 21:20, 9 June 2006 (UTC)
It only describes what cumulants are, I do not see a formal definition anywhere. As a general rule, shouldn't the first thing in such an article be the definition? Can someone please add the definition? --Innerproduct (talk) 20:45, 2 April 2010 (UTC)
No, that's a very very bad proposed general rule. Sometimes the first sentence should be a definition; more often it should not. One must begin by acquainting the lay reader with the fact that this is a concept in mathematics; sometimes stating the general definition in that same sentence conflicts with that goal. Michael Hardy (talk) 22:27, 2 April 2010 (UTC)
"Innerproduct", I see that you commented nearly four years after the comment you're replying to. It should be perfectly obvious that that comment was about the article as it existed in 2006, and is no longer relevant to the article in its present form.
Your proposed general rule is very bad (even though in some particular cases it makes sense); following it extensively would require people to clean up after you. Michael Hardy (talk) 22:36, 2 April 2010 (UTC)
## joint cumulant
I think that in the formula
$\kappa(X_1,\dots,X_n) =\sum_\pi\prod_{B\in\pi}(|B|-1)!(-1)^{|B|-1}E\left(\prod_{i\in B}X_i\right)$
the number |B| of elements in B should be replaced by the number $|\pi|$ of blocks in $\pi$.
For example, in the given case n=3
$\kappa(X,Y,Z)=E(XYZ)-E(XY)E(Z)-E(XZ)E(Y)-E(YZ)E(X)+2E(X)E(Y)E(Z).\,$
the constant before the term E(XYZ) (which corresponds to $\pi=\{\{1,2,3\}\}$ : only one block of 3 items, i.e. |\pi|=1 and \pi={B} with |B|=3) is $1=(|\pi|-1)!$ and not $2=(|B|-1)!$.
That is certainly correct in this case, and I think probably more generally. I've changed it; I'll come back and look more closely later. Michael Hardy 17:41, 19 September 2006 (UTC)
## Intro
We need an intro --dudzcom 04:52, 24 December 2006 (UTC)
Seconded. I'm a stats n00b and without a basic intro, the encyclopaedic content lacks a basic context. Jddriessen 14:12, 3 March 2007 (UTC)
## k-statistics
I think we should have a section on k-statistics. Could someone knowledgeable write a section describing them and explaining why they are unbiased estimators for the cumulants. Ossi 18:04, 30 December 2006 (UTC)
I have been trying to find information about unbiased and other estimators for ratios of powers of cummulants. In particular, I am interested in estimators for the particular ratio $\frac {{\kappa_{{1}}}^{2}}{\kappa_{{2}}}$. I can use k-statistics to estimate this ratio as simply $\frac {{k_{{1}}}^{2}}{k_{{2}}}$ where $k_{n}$ is the $n^{th}$ k-statistic. This should work, but it is a biased estimator. Are there better, unbiased estimators? User:155.101.22.76 (Talk) 28 Oct 2008
You may find something about this in Kendall&Stuart Vol 1. If there is nothing there, it may not be possible to do anything straightforwardly. Unbiassedness may not necessarily be particularly relevant to you, but if it is then you might try the jackknifing and bootstrapping methods for reducing bias. Melcombe (talk) 09:44, 29 October 2008 (UTC)
The library at my university does not have Kendall&Stuart Vol 1. However, they do have K&S Vol. 2. The first chapter of Vol. 2 is on estimation. Problem 17.10 turns out to be very close to what I need. Assuming normally distributed r.v. and some limits on the allowed values for m and r the Minimum Variance Unbiased Estimator (MVUE) for $\kappa_{1}^{r}\,\kappa_{2}^{m}$ is given by:
$\sum _{i=0}^{1/2\,r}{\frac { \left( -1 \right) ^{i}r!\,\Gamma \left(\frac{\left(n-1\right)}{2}\right) {k_{{1}}}^{r-2\,i} \left( \frac{\left( n-1 \right)}{2} k_{{2}} \right) ^{m+i}}{i!\, \left( r-2\,i \right) !\,\Gamma\left( \frac{\left(n-1\right)}{2}+m+i \right) \left( 2\,n \right) ^{i}}}$
In my case $\kappa_{1}^{2}\,\kappa_{2}^{-1} , r =1, m = -1$, and this reduces to:
$\left(\frac{\left(n-3\right)}{\left(n-1\right)}\frac{k_{1}^2}{k_{2}} - \frac{1}{n}\right)$
Which is precisely the same estimator I derived using other methods also assuming a normal distribution. I did not know that you could reduce bias using jackknifing and bootstrapping methods. Doh! That is great. The distributions I am working with are close to normal. I should be able to use the above MVUE and then reduce any remaining bias using jackknifing or bootstrapping. Thanks. --Stanthomas (talk) 17:34, 30 October 2008 (UTC)
## Cumulant "basis"?
It appears that you can reconstruct a function from its cumulants; that is, it seems like the cumulants define a "basis" of sorts the same way the sin and cos functions define a Fourier basis. Of course, a function isn't a linear combination of its cumulants, so it's not a linear basis, but in some sense it still seems like a basis. Comments? 155.212.242.34 (talk) 22:23, 11 December 2007 (UTC)
If two finite multisets of numbers have the same cumulant generating function, they are equal. The concept of a random variable is somewhat more general than just a multiset of numbers, and complications arise. It is worth while to understand multisets before trying to understand random variables. The derivative of the cumulant distribution function of a continuous random variable should be considered the limiting case of a series of derivatives of cumulant distribution functions of finite multisets of numbers. As probability density functions are nonnegative, they do not make vector spaces, and so the concept of basis does not immediately apply. Bo Jacoby (talk) 16:33, 19 March 2008 (UTC).
## Error in formula
Quote: Some writers prefer to define the cumulant generating function, via the characteristic function, as h(t) where
$h(t)=\log(E (e^{i t X}))=\sum_{n=1}^\infty\kappa_n \cdot\frac{(it)^n}{n!}=\mu\cdot t - \sigma^2\cdot\frac{ t^2}{2} +\cdots\,.$
I suppose the formula should be:
$h(t)=\log(E (e^{i t X}))=\sum_{n=1}^\infty\kappa_n \cdot\frac{(it)^n}{n!}=\mu\cdot i\cdot t - \sigma^2\cdot\frac{ t^2}{2} +\cdots\,.$
Is there a reference? Bo Jacoby (talk) 00:43, 20 March 2008 (UTC).
You're right; the factor of i was missing. I don't think it should be too hard to find references. I wouldn't be surprised if this is in McCullagh's book. Michael Hardy (talk) 18:06, 21 March 2008 (UTC)
References added to article, for this point at least. The Kendall and Stuart ref would be good for many other of the results quoted (but a later edition might be sought out?). Melcombe (talk) 09:18, 17 April 2008 (UTC)
## Improve intro ?
At the end of the intro, the final sentance says: "This characterization of cumulants is valid even for distributions whose higher moments do not exist." This seems to dangle somewhat...
• exactly what is refered to by "this characterisation"?
• it seems to imply there are other characterisations?
• it seems to imply that cumulants might exist even if higher moments do not exist?
Melcombe (talk) 09:30, 17 April 2008 (UTC)
Probably could be improved; I'll think about it. When higher moments do not exist, then neither do higher cumulants. In that case, the characterization of cumulants that says the cumulant-generating function is the logarithm of the moment-generating function is problematic. That is what is meant. As far as other characterizations go, yes of course there are. Michael Hardy (talk) 21:10, 17 April 2008 (UTC)
## Joint cumulants
I want to know more about Joint Cumulants, but this section made no reference to any books or papers. Any suggestions? Thanks! Yongtwang (talk) 13:47, 13 May 2010 (UTC)
You could try the existing Kendall&Stuart reference. It is old but covers the multivariate case, both for theoretical and sample versions of the joint cumulants. Melcombe (talk) 14:52, 14 May 2010 (UTC)
Hey, thanks for the information. I am reading it. Yongtwang (talk) 12:05, 16 May 2010 (UTC)
## Some properties of the cumulant-generating function
The article states that the cumulant-generating function is always convex (not too hard to prove). I wonder if the converse holds: any convex function (+ maybe some regularity conditions) can be a cumulant-generating function of some random variable. // stpasha » 20:03, 2 March 2011 (UTC)
## Applicability to Quantum Mechanics
I was reading this article to get a more broad background on the cumulant expansion, which is useful in quantum mechanical simulations of spectroscopic signals (absorption, pump-probe, raman, etc). I was somewhat surprised not to see quantum mechanics mentioned at all in the article. The source that I'm currently following on this topic:
Shaul Mukamel's "Principles of Nonlinear Optical Spectroscopy" (ISBN: 0-19-513291-2).
The expansions are debuted in Ch2, "Magnus Expansion". Ch 8 is also devoted entirely to their practical use.
Side note: It amused me that there were "citation needed" marks on the phrase, "Note that expectation values are sometimes denoted by angle brackets". This notation is so ubiquitous in quantum mechanics that one could literally pick up any quantum textbook and insert it as a "source" to verify that this is common practice. Certainly the book I just mentioned could count as such a source. —Preceding unsigned comment added by 24.11.171.13 (talk) 23:26, 20 May 2011 (UTC)
I looked in the comments specifically to discuss the "citation needed" marks for angle bracket denotation of expectation values. It's like asking for a citation that addition is sometimes denoted with a plus sign. I'm going to remove it and someone can put it back in if they feel it's really necessary. Gregarobinson (talk) 16:14, 17 June 2011 (UTC)
## Should "Relation to statistical physics" be deleted?
Some formulas in section "Relation to statistical physics" are wrong. Instead of:
$Z(\beta) = \langle\exp(-\beta E)\rangle$
$Z(\beta) = \sum_i\exp(-\beta E_i)$
as found in any relevant textbook or the wiki page for the partition function itself. This breaks the following argument linking F(\beta) to the cumulant generating function for the energy \log Z, as Z is no longer an average. The same critic holds for the grand potential at the end of the section, which is also a sum, not an average.
The equations linking E and C to the corresponding cumulants of the energy are still valid, since the cumulants equal the moments (section "Some properties of cumulants"). However, the interpretation in terms of moments is quite widespread, and in fact the equation:
$E = \langle E_i \rangle,$
is considered a postulate, in which the energy is linked to an average. The addition of usage of cumulants in stat. mech. that can't be expressed more naturally in terms of moments should be made, if such an usage exists.
Futhermore, the section doesn't cite any source, and none of the article's sources seems relevant at first sight.
These three points make me feel the whole section is rather weak. I suggest it should be deleted. --Palatosa (talk) 19:53, 19 April 2013 (UTC)
## Minor error in the section "Some properties of the cumulant generating function" ?
There is something strange with the statement "The cumulant-generating function will have vertical asymptote(s) at the infimum of such c, if such an infimum exists etc". Note that x is a negative number here. Something being O(exp(2x)) is a tougher requirement than being O(exp(x)) when x tends to minus infinity. You get the toughest requirement possible by finding the supremeum over c. But changing infimum to supremum doesnt seem right either. Should there be some sign change also? — Preceding unsigned comment added by 89.236.1.222 (talk) 22:12, 17 October 2014 (UTC)
## Given definition is only a special case
The definition given requires the moment generating function to exist. Rather than change the definition to use the characteristic function, we just need a note that the relation between moments and cumulants given later, can be used as the definition. TerryM--re (talk) 04:18, 12 February 2015 (UTC)
If the definition given apply to the examples given, then generalizations may be postponed or omitted in order not to confuse the readers unnecessarily. Bo Jacoby (talk) 19:26, 25 May 2015 (UTC).
## Problems with statistics
This article is very good and informative but someone has a flag on it to improve citations. Standard approaches do not apply to mathematical subjects, where inline citations are not as frequent, and usually one cites a theorem or a result, with plenty of references in back. Limit-theorem (talk) | {"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": 24, "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": 0.8715348839759827, "perplexity": 1092.067362745936}, "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-2015-48/segments/1448398456289.53/warc/CC-MAIN-20151124205416-00037-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/decomposition-of-sl-2-c-weyl-spinors.469594/ | # Decomposition of SL(2,C) Weyl Spinors
• Start date
• #1
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## Homework Statement
Using
$$(\sigma^{\mu \nu})^{\beta}_{\alpha} (\sigma_{\mu \nu})^{\delta}_{\gamma} = \epsilon_{\alpha \gamma} \epsilon^{\beta \delta} + \delta^{\delta}_{\alpha} \delta^{\beta}_{\gamma}$$
show that
$$\Psi_{\alpha} X_{\beta} = \frac{1}{2} \epsilon_{\alpha \beta} (\Psi X) + \frac{1}{2} (\sigma^{\mu \nu} \epsilon^{T})_{\alpha \beta} (\Psi \sigma_{\mu \nu} X)$$
## The Attempt at a Solution
if I do $$(\sigma^{\mu \nu})^{\beta}_{\alpha} (\sigma_{\mu \nu})^{\delta}_{\gamma} \Psi _{\beta} X_{\delta}$$
I can get
$$\Psi_{\beta} X_{\alpha} = \epsilon_{\alpha \beta} (\Psi X) + (\sigma^{\mu \nu} \epsilon^{T})_{\alpha \beta} (\Psi \sigma_{\mu \nu} X)$$
so i don't know where the factors of a half come from and how to get the right index order
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• #2
fzero
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I find
$$(\sigma^{\mu \nu} \epsilon^{T})_{\alpha \beta} (\Psi \sigma_{\mu \nu} X) = \Psi_{\alpha} X_{\beta}+ \Psi_{\beta} X_{\alpha}.$$
• #3
dextercioby
Homework Helper
13,077
645
The 1/2 must come from the symmetrization
$$\Psi_{\beta}X_{\alpha} = \frac{1}{2} \left(\Psi_{\beta} X_{\alpha}+\Psi_{\alpha} X_{\beta}\right) + \frac{1}{2} \left(\Psi_{\beta} X_{\alpha}-\Psi_{\alpha} X_{\beta}\right)$$
The a-symmetric part must be proportional to the spinor metric, the symmetric one is what's left.
• #4
213
8
I find
$$(\sigma^{\mu \nu} \epsilon^{T})_{\alpha \beta} (\Psi \sigma_{\mu \nu} X) = \Psi_{\alpha} X_{\beta}+ \Psi_{\beta} X_{\alpha}.$$
how did you get that without a factor of two and if it is correct then symmetrising solves it
• #5
fzero
Homework Helper
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3,119
289
how did you get that without a factor of two and if it is correct then symmetrising solves it
That follows directly from the stated identity. Just put in the correct indices and keep track of the $$\epsilon$$ contractions.
• #6
213
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sorry for being dense but if I would have known how to manipulate this spinor algebra i wouldn't be asking, so please be more explicit
• #7
fzero
Homework Helper
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sorry for being dense but if I would have known how to manipulate this spinor algebra i wouldn't be asking, so please be more explicit
You should really give it try first or at least point out exactly which term you don't understand.
• #8
213
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I've been stuck on this for a day
what I don't understand is why it isn't
$$\bold 2 (\sigma^{\mu \nu} \epsilon^{T})_{\alpha \beta} (\Psi \sigma_{\mu \nu} X) = \Psi_{\alpha} X_{\beta}+ \Psi_{\beta} X_{\alpha}.$$
even looking at double epsilon identity in two dimensions you get
$$(\sigma^{\mu \nu})^{\beta}_{\alpha} (\sigma_{\mu \nu})^{\delta}_{\gamma} = \delta^{\beta}_{\alpha} \delta^{\delta}_{\gamma}$$
which gives you a factor of two in that calculation you did
at least give me the starting point of that calculation
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• #9
fzero
Homework Helper
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289
I've been stuck on this for a day
what I don't understand is why it isn't
$$\bold 2 (\sigma^{\mu \nu} \epsilon^{T})_{\alpha \beta} (\Psi \sigma_{\mu \nu} X) = \Psi_{\alpha} X_{\beta}+ \Psi_{\beta} X_{\alpha}.$$
even looking at double epsilon identity in two dimensions you get
$$(\sigma^{\mu \nu})^{\beta}_{\alpha} (\sigma_{\mu \nu})^{\delta}_{\gamma} = \delta^{\beta}_{\alpha} \delta^{\delta}_{\gamma}$$
You're missing part of this expression.
which gives you a factor of two in that calculation you did
at least give me the starting point of that calculation
$$(\sigma^{\mu \nu} \epsilon^{T})_{\alpha \beta} (\Psi \sigma_{\mu \nu} X) = {(\sigma^{\mu \nu})_\alpha}^\gamma \epsilon_{\beta \gamma} \Psi^\delta {( \sigma_{\mu \nu} )_\delta}^\epsilon X_\epsilon .$$
Use
$${(\sigma^{\mu \nu})_{\alpha}}^{\beta} {(\sigma_{\mu \nu})_{\gamma}}^{\delta} = \epsilon_{\alpha \gamma} \epsilon^{\beta \delta} + \delta^{\delta}_{\alpha} \delta^{\beta}_{\gamma}$$
and $$\Psi^\alpha = \epsilon^{\alpha\beta}\Psi_\beta$$.
• #10
213
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thank you it seems I made the mistake of having three repeated indicies
but this is correct
$$(\sigma^{\mu \nu})^{\beta}_{\alpha} (\sigma_{\mu \nu})^{\delta}_{\gamma} = \delta^{\beta}_{\alpha} \delta^{\delta}_{\gamma}$$
otherwise you can't derive the rest
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http://math.stackexchange.com/questions/170943/dedekinds-theorem-on-an-integrally-closed-algebra-over-a-commutative-ring-witho | # Dedekind's theorem on an integrally closed algebra over a commutative ring without Axiom of Choice
Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951.
Can we prove the following theorem without Axiom of Choice? If the answer is affirmative, by using this, we can get many examples of Dedekind domains without using Axiom of Choice. This is a related question.
Theorem Let $A$ be a commutative ring. Let $B$ be an integrally closed $A$-algebra. Suppose $B/fB$ has a composition series as an $A$-module for every non-zero element $f$ of $B$. Then the following assertions hold.
(1) Every ideal of $B$ is finitely generated.
(2) Every non-zero prime ideal of $B$ is maximal.
(3) Every non-zero ideal of $B$ is invertible.
(4) Every non-zero ideal of $B$ has a unique factorization as a product of prime ideals.
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Why are you asking a string of very closely related questions about whether theorems from commutative algebra hold without choice? – Alex Becker Jul 15 '12 at 1:19
I tried to solve the following problem presented by Weil. Most of my questions related AC came from my efforts to solve it. math.stackexchange.com/questions/155392/… – Makoto Kato Jul 15 '12 at 1:27
We use the definitions of my answer to this question.
Definition 1 Let $A$ be a commutative ring. Let $B$ be a commutative $A$-algebra. Suppose $leng_A B$ is finite. Then we say $B$ is an Artinian $A$-algebra.
Lemma 1 Let $A$ be a commutative ring. Let $B$ be an Artinian $A$-algebra. Let $\Lambda$ be nonempty set of ideals of $B$. Then there exist a maximal element and a minimal element in $\Lambda$.
Proof: We note that every ideal of $B$ can be regarded canonically as an A-module. Let $r = sup$ {$leng_A I; I \in \Lambda$}. Since $r$ is finite, there exists $I \in \Lambda$ such that $r = leng$ $I$. By Lemma 4 of my answer to this, $I$ is a maximal element of $\Lambda$.
The existence of a minimal element is proved similarly. QED
Lemma 2 Let $A$ be a commutative ring. Let $B$ be an Artinian $A$-algebra. Then $leng_B B$ is finite, namely $B$ is an Artinian ring as defined in Definition 1 in this.
Proof: Let $\Lambda$ be the set of ideals $I$ of $B$ such that $leng_B I$ is finite. Since $0 \in \Lambda$, $\Lambda$ is not empty. By Lemma 1, there exists a maximal element $I \in \Lambda$. Suppose $I \neq B$. Then, by Lemma 1, there exists an ideal $J$ of $B$ such that $I \subset J$ and $J/I$ is a simple $B$-module. Since $leng_B J = leng_B I + 1$, $J \in \Lambda$. This is a contradiction. Hence $B = I$. QED
Definition 2 Let $A$ be a commutative ring. Let $B$ be a commutative $A$-algebra. Suppose $leng_A B/fB$ is finite of for every non-zero element $f \in B$. Then we say $B$ is a weakly Artinian $A$-algebra. By Lemma 2, $B$ is a weakly Artinian ring.
Proof of the title theorem By Lemma 2, $B$ is a weakly Artinian ring. Hence the assertions of the title theorem follow immediately from Lemma 2 and this. QED
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http://stats.stackexchange.com/questions/38296/expected-value-of-a-natural-logarithm | # Expected value of a natural logarithm
I know $E(aX+b) = aE(X)+b$ with $a,b$ constants, so given $E(X)$, it's easy to solve. I also know that you can't apply that when its a nonlinear function, like in this case $E(1/X) \neq 1/E(X)$, and in order to solve that, I've got to do an approximation with Taylor's. So my question is how do I solve $E(\ln(1+X))$?? do I also approximate with Taylor?
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Yes you can apply the delta method in this case. – Michael Chernick Sep 29 '12 at 23:45
You should also look into the Jensen Inequality. – kjetil b halvorsen Sep 30 '12 at 19:58
In the paper
Y. W. Teh, D. Newman and M. Welling (2006), A Collapsed Variational Bayesian Inference Algorithm for Latent Dirichlet Allocation, NIPS 2006, 1353–1360.
a second order Taylor expansion around $x_0=\mathbb{E}[x]$ is used to approximate $\mathbb{E}[\log(x)]$:
$$\mathbb{E}[\log(x)]\approx\log(\mathbb{E}[x])-\frac{\mathbb{V}[x]}{2\mathbb{E}[x]^2} \>.$$
This approximation seems to work pretty well for their application.
Modifying this slightly to fit the question at hand yields, by linearity of expectation,
$$\mathbb{E}[\log(1+x)]\approx\log(1+\mathbb{E}[x])-\frac{\mathbb{V}[x]}{2(1+\mathbb{E}[x])^2} \>.$$
However, it can happen that either the left-hand side or the right-hand side does not exist while the other does, and so some care should be taken when employing this approximation.
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Interestingly, This can be used to get an approximation to the digamma function. – probabilityislogic Oct 3 '12 at 22:53
Suppose that $X$ has probability density $f_X$. Before you start approximating, remember that, for any measurable function $g$, you can prove that $$E[g(X)]=\int g(X)\,dP = \int_{-\infty}^\infty g(x)\,f_X(x)\,dx \, ,$$ in the sense that if the first integral exists, so does the second, and they have the same value.
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If the second integral exists. It needs not to. Take Cauchy distribution and $g(x)=x^2$. – mpiktas Sep 30 '12 at 13:34
I would add a second layer of pedantry by saying that you actually need $E[|g(X)|]<\infty$ for the expectation to be well defined. – probabilityislogic Oct 3 '12 at 22:45
@mpiktas - This expectation actually does exist but it is infinite. A better example is $g(x)=x$ for the Cauchy distribution. This expectation depends on how the lower and upper limits of integration tend to infinity. – probabilityislogic Oct 3 '12 at 22:49
@prob: No, you don't need that condition in your first comment, and even in a situation that may be very relevant to this question! (+1 to your second comment, though, which was something I had been meaning to comment on as well.) – cardinal Oct 3 '12 at 22:51
@prob: It is sufficient, but if you compare your first comment to your second one, you'll see why it's not necessary! :-) – cardinal Oct 3 '12 at 23:02
show 1 more comment
There are two usual approaches:
1. If you know the distribution of $X$, you may be able to find the distribution of $\ln(1+X)$ and hence its expectation; alternatively you may be able to use the law of the unconscious statistician directly (that is, integrate $\ln(1+x) f_{X}(x)$ over the domain of $x$).
2. As you suggest, if you know the first few moments you can compute a Taylor approximation.
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http://math.stackexchange.com/questions/164847/congruent-to-mod-p-1p-22p-2-cdots-left-fracp-12-rightp-2-equi/164900 | # congruent to mod p $1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.$
Let $p$ be an odd prime.How to prove that
$$1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.$$
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Let $f(x)=(x+1(\frac{p-1}{2}))(x+2(\frac{p-3}{2})...(x+(\frac{p-1}{2})(1))$,and $a=$ coefficient of $x$ of $f(x)$. Then the problem is equivalent to showing that $a\equiv 8(-1)^{\frac{p-1}{2}} (\frac{2^{p-1}-1}{p}) (mod p)$. I dunno if this would help us solve this problem. – Ben Jun 30 '12 at 9:53
Am I being stupid or does this not make sense since $p$ is not invertible mod $p$? – fretty Jun 30 '12 at 12:29
$2-2^p$ is divisible by $p$,so here $\frac{2-2^p}{p}$ is an integer and $\frac{1}{p}$ does not mean the inverse in the field $\mathbb{Z}/p\mathbb{Z}$. – Ben Jun 30 '12 at 12:30
Yes, I didn't look at the numerator in detail so I was being stupid. – fretty Jun 30 '12 at 12:40
I find that this identity is stated in this paper:
Lehmer, E. "On Congruences Involving Bernoulli Numbers and the Quotients of Fermat and Wilson." Ann. Math. 39, 350-360, 1938 : http://www.jstor.org/discover/10.2307/1968791?uid=2134&uid=2&uid=70&uid=4&sid=56284309973
EDIT:
In fact,we can use a simpler method to solve this problem.
Let $q(a)=\frac{a^{p-1}-1}{p}$ for all $a$ such that $\gcd(a,p)=1$. It is not difficult to show that $q(ab) \equiv q(a)+q(b) \pmod{p}...(1)$.
Let $a$ be an arbitrary integer which is relatively prime to $p$. For each $v \in \{1,2,...,p-1\}$,let $av=\lfloor{\frac{av}{p}} \rfloor p+r_v$. Then we can see that $r_v$,($r=1,2,...,p-1$) also runs over ${1,2,...,p-1}$. So $q(av)=((\lfloor{\frac{av}{p}} \rfloor p+r_v)^{p-1}-1)/p\equiv q(r_v)-\lfloor{\frac{av}{p}} \rfloor r_v^{p-2} \equiv q(r_v)-\frac{1}{av} \left \lfloor \frac{av}{p} \right \rfloor \pmod{p}$
So, $\sum_{v=1}^{p-1} q(av) \equiv \sum_{v=1}^{p-1} (q(r_v)-\frac{1}{av} \left \lfloor \frac{av}{p} \right \rfloor)\equiv \sum_{v=1}^{p-1} q(v)-\sum_{v=1}^{p-1}\frac{1}{av} \left \lfloor \frac{av}{p} \right \rfloor \pmod{p}...(2)$
Then by $(1),(2)$,we get $q(a) \equiv \sum_{v=1}^{p-1}\frac{1}{av} \left \lfloor \frac{av}{p} \right \rfloor \pmod{p}...(3)$
Take $a=2$, $\frac{2^p-2}{p} \equiv \sum_{v=1}^{p-1}\frac{1}{v} \left \lfloor \frac{2v}{p} \right \rfloor \equiv \sum_{p/2 <v} \frac{1}{v} \equiv -\sum_{v=1}^{(p-1)/2} \frac{1}{v} \equiv -\sum_{v=1}^{(p-1)/2} v^{p-2} \pmod{p}$
As a result,
$1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.$
-
– anon Jun 30 '12 at 12:45
BenLi,could you look at this solution : store2.up-00.com/June12/RoL51753.jpg – Frank Jul 2 '12 at 17:56
@MohammedAl-mubark, your solution looks alright. That's how I did it. Look below. – DonAntonio Jul 3 '12 at 1:50
@Mohammed Al-mubark:your solution looks much simpler and more direct. – Ben Jul 3 '12 at 2:00
$$2^p=(1+1)^p=\sum_{k=0}^p\binom {p}{k}=\left[\binom{p}{0}+\binom{p}{p}\right]+\ldots+\left[\binom{p}{\frac{p-1}{2}}+\binom{p}{\frac{p+1}{2}}\right]=$$ $$=2\left[\binom{p}{0}+\ldots+\binom{p}{\frac{p-1}{2}}\right]$$
and etc.
- | {"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": 0.929492175579071, "perplexity": 497.7416836302505}, "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-2015-22/segments/1432207927245.60/warc/CC-MAIN-20150521113207-00289-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1148484/ordinals-that-are-not-cardinals?noredirect=1 | # Ordinals that are not cardinals [duplicate]
I am reading Jech's set theory and he defines a cardinal number as an ordinal $\alpha$ (a cardinal) if $|\alpha| \neq |\beta|$ for all $\beta < \alpha$, and he says that all infinite cardinals are limit ordinals. My question is: are there any ordinals that are not cardinals? Although I know that an ordinal describe ordering and a cardinal the size of a set, I am a bit confused 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": 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": 0.9056689739227295, "perplexity": 184.9425586994823}, "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/1558232257156.50/warc/CC-MAIN-20190523063645-20190523085645-00023.warc.gz"} |
https://electronics.stackexchange.com/questions/148463/can-an-ammeter-damage-zeners | # can an ammeter damage zeners
So I got this small circuit that plugs to the mains as power source. as for its DC converter, it's one of those transformer-less ones that drops the voltage with a resistors and filters before rectifying and smoothing it. the ones that can only deliver small currents.
well, after that, it uses two IN4737 zeners in series to create a reference of 15V. I needed to find how much current is being drawn between the 1) DC converter (as described above) and 2) the zeners and the rest of the main larger circuit. so I disconnected the positive wire between these two and put in series an ammeter.
the measurements were odd. using my multimeter, it's here as follows (as I recall it properly):
20A range - 0.06 A
200mA range - 0.006 mA
20mA range - 0.060 mA
2mA range - I can't remember
200uA range - I can't remember
It was a weird set of readings (and yes the number I gave above for each range are exactly as they showed up in my display, even the units). I restored the original connection and the whole thing won't work anymore. I then found out that the IN4737's were blown and shorts at a few milliohms in both direction each.
so what went wrong?? the only difference was the ammeter in series for the positive wire between the converter and main circuit. aren't ammeters specifically designed to be passive and attempt to be a perfect conductor in all cases?? | {"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": 0.8171440958976746, "perplexity": 1683.5886401455505}, "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-2019-51/segments/1575540518337.65/warc/CC-MAIN-20191209065626-20191209093626-00263.warc.gz"} |
https://worldwidescience.org/topicpages/c/coupling+tensors+determined.html | #### Sample records for coupling tensors determined
1. Link prediction via generalized coupled tensor factorisation
DEFF Research Database (Denmark)
Ermiş, Beyza; Evrim, Acar Ataman; Taylan Cemgil, A.
2012-01-01
and higher-order tensors. We propose to use an approach based on probabilistic interpretation of tensor factorisation models, i.e., Generalised Coupled Tensor Factorisation, which can simultaneously fit a large class of tensor models to higher-order tensors/matrices with com- mon latent factors using...... different loss functions. Numerical experiments demonstrate that joint analysis of data from multiple sources via coupled factorisation improves the link prediction performance and the selection of right loss function and tensor model is crucial for accurately predicting missing links....
2. Couplings of self-dual tensor multiplet in six dimensions
NARCIS (Netherlands)
Bergshoeff, E.; Sezgin, E.; Sokatchev, E.
1996-01-01
The (1, 0) supersymmetry in six dimensions admits a tensor multiplet which contains a second-rank antisymmetric tensor field with a self-dual field strength and a dilaton. We describe the fully supersymmetric coupling of this multiplet to a Yang–Mills multiplet, in the absence of supergravity. The
3. Data fusion in metabolomics using coupled matrix and tensor factorizations
DEFF Research Database (Denmark)
Evrim, Acar Ataman; Bro, Rasmus; Smilde, Age Klaas
2015-01-01
of heterogeneous (i.e., in the form of higher order tensors and matrices) data sets with shared/unshared factors. In order to jointly analyze such heterogeneous data sets, we formulate data fusion as a coupled matrix and tensor factorization (CMTF) problem, which has already proved useful in many data mining...
4. Superconformal tensor calculus and matter couplings in six dimensions
International Nuclear Information System (INIS)
Bergshoeff, E.; Sezgin, E.; van Proeyen, A.
1989-01-01
Using superconformal tensor calculus the authors construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. They start from the superconformal algebra which they realize on a 40 + 40 Weyl multiplet and on several matter multiplets. A special role is played by the tensor multiplet, which cannot be treated as an ordinary matter multiplet, but leads to a second 40 + 40 version of the Weyl multiplet. The authors also obtain a 48 + 48 off-shell formulation of Poincare supergravity coupled to a tensor multiplet
5. Superconformal tensor calculus and matter couplings in six dimensions
International Nuclear Information System (INIS)
Bergshoeff, E.; Sezgin, E.; Proeyen, A. van
1986-01-01
Using superconformal tensor calculus we construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. We start from the superconformal algebra which we realize on a 40 + 40 Weyl multiplet and on several matter multiplets. A special role is played by the tensor multiplet, which cannot be treated as an ordinary matter multiplet, but leads to a second 40 + 40 version of the Weyl multiplet. We also obtain a 48 + 48 off-shell formulation of Poincare supergravity coupled to a tensor multiplet. (orig.)
6. Four dimensional sigma model coupled to the metric tensor field
International Nuclear Information System (INIS)
Ghika, G.; Visinescu, M.
1980-02-01
We discuss the four dimensional nonlinear sigma model with an internal O(n) invariance coupled to the metric tensor field satisfying Einstein equations. We derive a bound on the coupling constant between the sigma field and the metric tensor using the theory of harmonic maps. A special attention is paid to Einstein spaces and some new explicit solutions of the model are constructed. (author)
7. On the skew-symmetric character of the couple-stress tensor
OpenAIRE
2013-01-01
In this paper, the skew-symmetric character of the couple-stress tensor is established as the result of arguments from tensor analysis. Consequently, the couple-stress pseudo-tensor has a true vectorial character. The fundamental step in this development is that the isotropic couple-stress tensor cannot exist.
8. Relativistic New Yukawa-Like Potential and Tensor Coupling
International Nuclear Information System (INIS)
Ikhdair, S.M.; Hamzavi, M.
2012-01-01
We approximately solve the Dirac equation for a new suggested generalized inversely quadratic Yukawa potential including a Coulomb-like tensor interaction with arbitrary spin-orbit coupling quantum number κ. In the framework of the spin and pseudo spin (p-spin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions, in closed form, by using the parametric Nikiforov-Uvarov method. The numerical results show that the Coulomb-like tensor interaction, -T/r, removes degeneracies between spin and p-spin state doublets. The Dirac solutions in the presence of exact spin symmetry are reduced to Schroedinger solutions for Yukawa and inversely quadratic Yukawa potentials. (author)
9. QED approach to the nuclear spin-spin coupling tensor
International Nuclear Information System (INIS)
Romero, Rodolfo H.; Aucar, Gustavo A.
2002-01-01
A quantum electrodynamical approach for the calculation of the nuclear spin-spin coupling tensor of nuclear-magnetic-resonance spectroscopy is given. Quantization of radiation fields within the molecule is considered and expressions for the magnetic field in the neighborhood of a nucleus are calculated. Using a generalization of time-dependent response theory, an effective spin-spin interaction is obtained from the coupling of nuclear magnetic moments to a virtual quantized magnetic field. The energy-dependent operators obtained reduce to usual classical-field expressions at suitable limits
10. Complete stress tensor determination by microearthquake analysis
Science.gov (United States)
Slunga, R.
2010-12-01
the depth based on the assumptions of a fractured crust, widely vary ing stress field, and a general closeness to instability as found by stress measurements (Jamison and Cook 1976). Wheather this approach is working or not is best answered by applying it to real data. This was provided by the IMO network in Iceland. Along Southern Iceland Seismic Zone (SISZ) more than 200,000 microearthquakes and a few M 5 EQs and 2 M=6.6 EQs have been recorded. The results will be presented it is obvious that the use of the stresses determined from the microearthquake recordings may significa ntly improve earthquake warnings and will make it possible to use the absolute C FS method for more deterministic predictions. Note that the microearthquake meth od only shows the part of the stress field that has caused slip. Volumes with st able stress will not show up. However stress measurements (Brown and Hoek 1978, Slunga 1988) have shown that the crustal stresses in general are close to instabi lity and microearthquake source analysis has shown that a large number of differ ent fractures become unstable within longer time windows. This may explain the e xcellent results given by the Icelandic tests of the absolute stress tensor fiel d as given by the microearthquakes. However I prefer to call this stress apparen t.
11. Seismic moment tensor for anisotropic media: implication for Non-double-couple earthquakes
Science.gov (United States)
Cai, X.; Chen, X.; Chen, Y.; Cai, M.
2008-12-01
It is often found that the inversion results of seismic moment tensor from real seismic recorded data show the trace of seismic moment tensor M is not zero, a phenomenon called non-double-couple earthquake sources mechanism. Recently we have derived the analytical expressions of M in transversely isotropic media with the titled axis of symmetry and the results shows even only pure shear-motion of fault can lead to the implosive components determined by several combined anisotropic elastic constants. Many non-double-couple earthquakes from observations often appear in volcanic and geothermal areas (Julian, 1998), where there exist a mount of stress-aligned fluid-saturated parallel vertical micro-cracks identical to transversely isotropic media (Crampin, 2008), this stress-aligned crack will modify the seismic moment tensor. In another word, non-double-couple earthquakes don't mean to have a seismic failure movement perpendicular to the fault plane, while traditional research of seismic moment tensor focus on the case of isotropy, which cannot provide correct interpretation of seismic source mechanism. Reference: Julian, B.R., Miller, A.D. and Foulger, G.R., 1998. Non-double-couple earthquakes,1. Theory, Rev. Geophys., 36, 525¨C549. Crampin,S., Peacock,S., 2008, A review of the current understanding of seismic shear-wave splitting in the Earth's crust and common fallacies in interpretation, wave motion, 45,675-722
12. Coupling coefficients for tensor product representations of quantum SU(2)
International Nuclear Information System (INIS)
Groenevelt, Wolter
2014-01-01
We study tensor products of infinite dimensional irreducible * -representations (not corepresentations) of the SU(2) quantum group. We obtain (generalized) eigenvectors of certain self-adjoint elements using spectral analysis of Jacobi operators associated to well-known q-hypergeometric orthogonal polynomials. We also compute coupling coefficients between different eigenvectors corresponding to the same eigenvalue. Since the continuous spectrum has multiplicity two, the corresponding coupling coefficients can be considered as 2 × 2-matrix-valued orthogonal functions. We compute explicitly the matrix elements of these functions. The coupling coefficients can be considered as q-analogs of Bessel functions. As a results we obtain several q-integral identities involving q-hypergeometric orthogonal polynomials and q-Bessel-type functions
13. Coupling coefficients for tensor product representations of quantum SU(2)
Science.gov (United States)
Groenevelt, Wolter
2014-10-01
We study tensor products of infinite dimensional irreducible *-representations (not corepresentations) of the SU(2) quantum group. We obtain (generalized) eigenvectors of certain self-adjoint elements using spectral analysis of Jacobi operators associated to well-known q-hypergeometric orthogonal polynomials. We also compute coupling coefficients between different eigenvectors corresponding to the same eigenvalue. Since the continuous spectrum has multiplicity two, the corresponding coupling coefficients can be considered as 2 × 2-matrix-valued orthogonal functions. We compute explicitly the matrix elements of these functions. The coupling coefficients can be considered as q-analogs of Bessel functions. As a results we obtain several q-integral identities involving q-hypergeometric orthogonal polynomials and q-Bessel-type functions.
14. Limits on Tensor Coupling from Neutron $\\beta$-Decay
OpenAIRE
Pattie Jr, Robert W.; Hickerson, Kevin P.; Young, Albert R.
2013-01-01
Limits on the tensor couplings generating a Fierz interference term, b, in mixed Gamow-Teller Fermi decays can be derived by combining data from measurements of angular correlation parameters in neutron decay, the neutron lifetime, and $G_{\\text{V}}=G_{\\text{F}} V_{ud}$ as extracted from measurements of the $\\mathcal{F}t$ values from the $0^{+} \\to 0^{+}$ superallowed decays dataset. These limits are derived by comparing the neutron $\\beta$-decay rate as predicted in the standard model with t...
15. Electron paramagnetic resonance g-tensors from state interaction spin-orbit coupling density matrix renormalization group
Science.gov (United States)
Sayfutyarova, Elvira R.; Chan, Garnet Kin-Lic
2018-05-01
We present a state interaction spin-orbit coupling method to calculate electron paramagnetic resonance g-tensors from density matrix renormalization group wavefunctions. We apply the technique to compute g-tensors for the TiF3 and CuCl42 - complexes, a [2Fe-2S] model of the active center of ferredoxins, and a Mn4CaO5 model of the S2 state of the oxygen evolving complex. These calculations raise the prospects of determining g-tensors in multireference calculations with a large number of open shells.
16. Gauge theories of Yang-Mills vector fields coupled to antisymmetric tensor fields
International Nuclear Information System (INIS)
Anco, Stephen C.
2003-01-01
A non-Abelian class of massless/massive nonlinear gauge theories of Yang-Mills vector potentials coupled to Freedman-Townsend antisymmetric tensor potentials is constructed in four space-time dimensions. These theories involve an extended Freedman-Townsend-type coupling between the vector and tensor fields, and a Chern-Simons mass term with the addition of a Higgs-type coupling of the tensor fields to the vector fields in the massive case. Geometrical, field theoretic, and algebraic aspects of the theories are discussed in detail. In particular, the geometrical structure mixes and unifies features of Yang-Mills theory and Freedman-Townsend theory formulated in terms of Lie algebra valued curvatures and connections associated to the fields and nonlinear field strengths. The theories arise from a general determination of all possible geometrical nonlinear deformations of linear Abelian gauge theory for one-form fields and two-form fields with an Abelian Chern-Simons mass term in four dimensions. For this type of deformation (with typical assumptions on the allowed form considered for terms in the gauge symmetries and field equations), an explicit classification of deformation terms at first-order is obtained, and uniqueness of deformation terms at all higher orders is proven. This leads to a uniqueness result for the non-Abelian class of theories constructed here
17. Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination.
Science.gov (United States)
Zhao, Qibin; Zhang, Liqing; Cichocki, Andrzej
2015-09-01
CANDECOMP/PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance.
18. The effects of the tensor coupling term in the Zimanyi-Moszkowski model for unpolarized nuclear matter
International Nuclear Information System (INIS)
Ru-Keng Su; Li Li; Hong-Qiu Song
1998-01-01
The effects of the tensor coupling term on nuclear matter in the Zimanyi-Moszkowki (ZM) model are investigated. It is shown that the tensor coupling term in the ZM model leaves the thermodynamical properties of nuclear matter almost unchanged. The corrections of tensor coupling to the critical point of the liquid-gas phase transition are given. (author)
19. Superconformal tensor calculus and matter couplings in six dimensions
NARCIS (Netherlands)
Bergshoeff, E.; Sezgin, E.; Proeyen, A. Van
1986-01-01
Using superconformal tensor calculus we construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. We start from the superconformal algebra which we realize on a 40+40 Weyl multiplet and on several matter multiplets. A
20. All-at-once Optimization for Coupled Matrix and Tensor Factorizations
DEFF Research Database (Denmark)
Evrim, Acar Ataman; Kolda, Tamara G.; Dunlavy, Daniel M.
2011-01-01
.g., the person by person social network matrix or the restaurant by category matrix, and higher-order tensors, e.g., the "ratings" tensor of the form restaurant by meal by person. In this paper, we are particularly interested in fusing data sets with the goal of capturing their underlying latent structures. We...... formulate this problem as a coupled matrix and tensor factorization (CMTF) problem where heterogeneous data sets are modeled by fitting outer-product models to higher-order tensors and matrices in a coupled manner. Unlike traditional approaches solving this problem using alternating algorithms, we propose...... an all-at-once optimization approach called CMTF-OPT (CMTF-OPTimization), which is a gradient-based optimization approach for joint analysis of matrices and higher-order tensors. We also extend the algorithm to handle coupled incomplete data sets. Using numerical experiments, we demonstrate...
1. Core Polarization and Tensor Coupling Effects on Magnetic Moments of Hypernuclei
International Nuclear Information System (INIS)
Jiang-Ming, Yao; Jie, Meng; Hong-Feng, Lü; Greg, Hillhouse
2008-01-01
Effects of core polarization and tensor coupling on the magnetic moments in Λ 13 C, Λ 17 O, and Λ 41 Ca Λ-hypernuclei are studied by employing the Dirac equation with scalar, vector and tensor potentials. It is found that the effect of core polarization on the magnetic moments is suppressed by Λ tensor coupling. The Λ tensor potential reduces the spin-orbit splitting of p Λ states considerably. However, almost the same magnetic moments are obtained using the hyperon wavefunction obtained via the Dirac equation either with or without the A tensor potential in the electromagnetic current vertex. The deviations of magnetic moments for p Λ states from the Schmidt values are found to increase with nuclear mass number. (nuclear physics)
2. Symmetry rules for the indirect nuclear spin-spin coupling tensor revisited
Science.gov (United States)
Buckingham, A. D.; Pyykkö, P.; Robert, J. B.; Wiesenfeld, L.
The symmetry rules of Buckingham and Love (1970), relating the number of independent components of the indirect spin-spin coupling tensor J to the symmetry of the nuclear sites, are shown to require modification if the two nuclei are exchanged by a symmetry operation. In that case, the anti-symmetric part of J does not transform as a second-rank polar tensor under symmetry operations that interchange the coupled nuclei and may be called an anti-tensor. New rules are derived and illustrated by simple molecular models.
3. Non-Abelian formulation of a vector-tensor gauge theory with topological coupling
International Nuclear Information System (INIS)
Barcelos Neto, J.; Cabo, A.; Silva, M.B.D.
1995-08-01
We obtain a non-Abelian version of a theory involving vector and tensor and tensor gauge fields interacting via a massive topological coupling, besides the nonminimum one. The new fact is that the non-Abelian theory is not reducible and Stuckelberg fields are introduced in order to compatibilize gauge invariance, nontrivial physical degrees of freedom and the limit of the Abelian case. (author). 9 refs
4. The matter Lagrangian and the energy-momentum tensor in modified gravity with nonminimal coupling between matter and geometry
International Nuclear Information System (INIS)
Harko, T.
2010-01-01
We show that in modified f(R) type gravity models with nonminimal coupling between matter and geometry, both the matter Lagrangian and the energy-momentum tensor are completely and uniquely determined by the form of the coupling. This result is obtained by using the variational formulation for the derivation of the equations of motion in the modified gravity models with geometry-matter coupling, and the Newtonian limit for a fluid obeying a barotropic equation of state. The corresponding energy-momentum tensor of the matter in modified gravity models with nonminimal coupling is more general than the usual general-relativistic energy-momentum tensor for perfect fluids, and it contains a supplementary, equation of state dependent term, which could be related to the elastic stresses in the body, or to other forms of internal energy. Therefore, the extra force induced by the coupling between matter and geometry never vanishes as a consequence of the thermodynamic properties of the system, or for a specific choice of the matter Lagrangian, and it is nonzero in the case of a fluid of dust particles.
5. The modified indeterminate couple stress model: Why Yang et al.'s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless
OpenAIRE
Münch, Ingo; Neff, Patrizio; Madeo, Angela; Ghiba, Ionel-Dumitrel
2015-01-01
We show that the reasoning in favor of a symmetric couple stress tensor in Yang et al.'s introduction of the modified couple stress theory contains a gap, but we present a reasonable physical hypothesis, implying that the couple stress tensor is traceless and may be symmetric anyway. To this aim, the origin of couple stress is discussed on the basis of certain properties of the total stress itself. In contrast to classical continuum mechanics, the balance of linear momentum and the balance of...
6. SU(2)xSU(2) coupling rule and a tensor glueball candidate
International Nuclear Information System (INIS)
Lanik, J.
1984-01-01
The data on the decay of THETA(1640) particles are considered. It is shown that the SU(2)xSU(2) mechanism for coupling of theta(1640) tensor glueball candidate to pseudoscalar Gold-stone mesons is in a remarkable agreement with existing experimental data
7. The magnetic g-tensors for ion complexes with large spin-orbit coupling
International Nuclear Information System (INIS)
Chang, P.K.L.; Liu, Y.S.
1977-01-01
A nonperturbative method for calculating the magnetic g-tensors is presented and discussed for complexes of transition metal ions of large spin-orbit coupling, in the ground term 2 D. A numerical example for CuCl 2 .2H 2 O is given [pt
8. Coupled ADCPs can yield complete Reynolds stress tensor profiles in geophysical surface flows
NARCIS (Netherlands)
Vermeulen, B.; Hoitink, A.J.F.; Sassi, M.G.
2011-01-01
We introduce a new technique to measure profiles of each term in the Reynolds stress tensor using coupled acoustic Doppler current profilers (ADCPs). The technique is based on the variance method which is extended to the case with eight acoustic beams. Methods to analyze turbulence from a single
9. A high performance data parallel tensor contraction framework: Application to coupled electro-mechanics
Science.gov (United States)
Poya, Roman; Gil, Antonio J.; Ortigosa, Rogelio
2017-07-01
The paper presents aspects of implementation of a new high performance tensor contraction framework for the numerical analysis of coupled and multi-physics problems on streaming architectures. In addition to explicit SIMD instructions and smart expression templates, the framework introduces domain specific constructs for the tensor cross product and its associated algebra recently rediscovered by Bonet et al. (2015, 2016) in the context of solid mechanics. The two key ingredients of the presented expression template engine are as follows. First, the capability to mathematically transform complex chains of operations to simpler equivalent expressions, while potentially avoiding routes with higher levels of computational complexity and, second, to perform a compile time depth-first or breadth-first search to find the optimal contraction indices of a large tensor network in order to minimise the number of floating point operations. For optimisations of tensor contraction such as loop transformation, loop fusion and data locality optimisations, the framework relies heavily on compile time technologies rather than source-to-source translation or JIT techniques. Every aspect of the framework is examined through relevant performance benchmarks, including the impact of data parallelism on the performance of isomorphic and nonisomorphic tensor products, the FLOP and memory I/O optimality in the evaluation of tensor networks, the compilation cost and memory footprint of the framework and the performance of tensor cross product kernels. The framework is then applied to finite element analysis of coupled electro-mechanical problems to assess the speed-ups achieved in kernel-based numerical integration of complex electroelastic energy functionals. In this context, domain-aware expression templates combined with SIMD instructions are shown to provide a significant speed-up over the classical low-level style programming techniques.
10. Tensor calculus for the vector multiplet coupled to supergravity
International Nuclear Information System (INIS)
Stelle, K.S.
1978-01-01
An invariant coupling of a local vector multiplet to supergravity is constructed in analogy with the D term invariant of global supersymmetry. The rules for combining local vector and chiral scalar multiplets of opposite chirality are given. (Auth.)
11. Chiral primordial blue tensor spectra from the axion-gauge couplings
Energy Technology Data Exchange (ETDEWEB)
Obata, Ippei, E-mail: [email protected] [Department of Physics, Kyoto University, Kyoto, 606-8502 (Japan)
2017-06-01
We suggest the new feature of primordial gravitational waves sourced by the axion-gauge couplings, whose forms are motivated by the dimensional reduction of the form field in the string theory. In our inflationary model, as an inflaton we adopt two types of axion, dubbed the model-independent axion and the model-dependent axion, which couple with two gauge groups with different sign combination each other. Due to these forms both polarization modes of gauge fields are amplified and enhance both helicies of tensor modes during inflation. We point out the possibility that a primordial blue-tilted tensor power spectra with small chirality are provided by the combination of these axion-gauge couplings, intriguingly both amplitudes and chirality are potentially testable by future space-based gravitational wave interferometers such as DECIGO and BBO project.
12. On scalar and vector fields coupled to the energy-momentum tensor
Science.gov (United States)
Jiménez, Jose Beltrán; Cembranos, Jose A. R.; Sánchez Velázquez, Jose M.
2018-05-01
We consider theories for scalar and vector fields coupled to the energy-momentum tensor. Since these fields also carry a non-trivial energy-momentum tensor, the coupling prescription generates self-interactions. In analogy with gravity theories, we build the action by means of an iterative process that leads to an infinite series, which can be resumed as the solution of a set of differential equations. We show that, in some particular cases, the equations become algebraic and that is also possible to find solutions in the form of polynomials. We briefly review the case of the scalar field that has already been studied in the literature and extend the analysis to the case of derivative (disformal) couplings. We then explore theories with vector fields, distinguishing between gauge-and non-gauge-invariant couplings. Interactions with matter are also considered, taking a scalar field as a proxy for the matter sector. We also discuss the ambiguity introduced by superpotential (boundary) terms in the definition of the energy-momentum tensor and use them to show that it is also possible to generate Galileon-like interactions with this procedure. We finally use collider and astrophysical observations to set constraints on the dimensionful coupling which characterises the phenomenology of these models.
13. Double shadow of a regular phantom black hole as photons couple to the Weyl tensor
Energy Technology Data Exchange (ETDEWEB)
Huang, Yang; Chen, Songbai; Jing, Jiliang [Hunan Normal University, Institute of Physics and Department of Physics, Changsha, Hunan (China); Hunan Normal University, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha, Hunan (China); Hunan Normal University, Synergetic Innovation Center for Quantum Effects and Applications, Changsha, Hunan (China)
2016-11-15
We have studied the shadow of a regular phantom black hole as photons couple to the Weyl tensor. We find that due to the coupling photons with different polarization directions propagate along different paths in the spacetime so that there exists a double shadow for a black hole, which is quite different from that in the non-coupling case where only a single shadow emerges. The overlap region of the double shadow, the umbra, of the black hole increases with the phantom charge and decreases with the coupling strength. The dependence of the penumbra on the phantom charge and the coupling strength is converse to that of the umbra. Combining with the supermassive central object in our Galaxy, we estimated the shadow of the black hole as the photons couple to the Weyl tensor. Our results show that the coupling brings about richer behaviors of the propagation of coupled photon and the shadow of the black hole in the regular phantom black hole spacetime. (orig.)
14. Symmetry breaking nuclear quadrupole coupling tensor orientation for cesium-133 nuclei located in a mirror plane
Energy Technology Data Exchange (ETDEWEB)
Kim, Tae Ho; Kim, Jin Eun [Dept. of Chemistry (BK21 plus) and Research Institute of Natural Science, Gyeongsang National University, Jinju (Korea, Republic of); Lee, Kang Yeol [School of Mechanical Engineering, Korea University, Seoul (Korea, Republic of)
2016-11-15
Simultaneous multiple data set fits of all transition peaks of {sup 133}Cs nuclei enabled us to obtain accurate cesium-133 nuclear magnetic resonance (NMR) parameters and Euler angles between the principal axis systems of the chemical shift (CS) and quadrupole coupling (Q) tensors of {sup 133}Cs nuclei in Cs{sub 2}CrO{sub 4} . Although in a previous study of Cs{sub 2}CrO{sub 4} by Power et al. (W. P. Power, S. Mooibroek, R. E. Wasylishen, T. S. Cameron, J. Phys. Chem. 1994, 98, 1552), one central transition was observed for cesium sites 1 and 2 in the {sup 133}Cs NMR spectra and one Euler angle between the CS tensors and Q tensors was obtained as 52° and 7° for cesium sites 1 and 2, respectively, the present single-crystal {sup 133}Cs NMR measurements found two Euler angles (10(2)°, 51.9(1)°, 0°) for site 1 and two central transition peaks for site 2. Three principal components of the CS tensor for Cs1 are oriented along the crystallographic a, b, and c axes, whereas none of the principal components of the Q tensor for Cs1 are oriented along the crystal axes. The principal component V{sub 22} of the Q tensor for Cs1 is tilted 10° from the b axis in the bc plane, and the other two components are not located in the ac plane. Therefore, we have found that the requirement that “the quadrupole coupling tensor for a nucleus located in a mirror plane has one principal axis perpendicular to the mirror plane” cannot be applied to Cs1. On the other hand, δ{sub 11} and V{sub 22} for Cs2 are aligned along the b axis, and the other components of the CS and Q tensors deviate at an angle of 1.4(1)° and 10.1(1)°, respectively, from the a and c axes in the ac plane. A distortion-free powder {sup 133}Cs NMR spectrum of Cs{sub 2}CrO{sub 4} was measured using a solid-state spin echo technique.
15. 71Ga Chemical Shielding and Quadrupole Coupling Tensors of the Garnet Y(3)Ga(5)O(12) from Single-Crystal (71)Ga NMR
DEFF Research Database (Denmark)
Vosegaard, Thomas; Massiot, Dominique; Gautier, Nathalie
1997-01-01
A single-crystal (71)Ga NMR study of the garnet Y(3)Ga(5)O(12) (YGG) has resulted in the determination of the first chemical shielding tensors reported for the (71)Ga quadrupole. The single-crystal spectra are analyzed in terms of the combined effect of quadrupole coupling and chemical shielding ...
16. Structures and Nuclear Quadrupole Coupling Tensors of a Series of Chlorine-Containing Hydrocarbons
Science.gov (United States)
Dikkumbura, Asela S.; Webster, Erica R.; Dorris, Rachel E.; Peebles, Rebecca A.; Peebles, Sean A.; Seifert, Nathan A.; Pate, Brooks
2016-06-01
Rotational spectra for gauche-1,2-dichloroethane (12DCE), gauche-1-chloro-2-fluoroethane (1C2FE) and both anti- and gauche-2,3-dichloropropene (23DCP) have been observed using chirped-pulse Fourier-transform microwave (FTMW) spectroscopy in the 6-18 GHz region. Although the anti conformers for all three species are predicted to be more stable than the gauche forms, they are nonpolar (12DCE) or nearly nonpolar (predicted dipole components for anti-1C2FE: μ_a = 0.11 D, μ_b = 0.02 D and for anti-23DCP: μ_a = 0.25 D, μ_b = 0.02 D); nevertheless, it was also possible to observe and assign the spectrum of anti-23DCP. Assignments of parent spectra and 37Cl and 13C substituted isotopologues utilized predictions at the MP2/6-311++G(2d,2p) level and Pickett's SPCAT/SPFIT programs. For the weak anti-23DCP spectra, additional measurements also utilized a resonant-cavity FTMW spectrometer. Full chlorine nuclear quadrupole coupling tensors for gauche-12DCE and both anti- and gauche-23DCP have been diagonalized to allow comparison of coupling constants. Kraitchman's equations were used to determine r_s coordinates of isotopically substituted atoms and r_0 structures were also deduced for gauche conformers of 12DCE and 1C2FE. Structural details and chlorine nuclear quadrupole coupling constants of all three molecules will be compared, and effects of differing halogen substitution and carbon chain length on molecular properties will be evaluated.
17. Coupled Hartree-Fock calculation of {sup 13} C shielding tensors in acetylene clusters
Energy Technology Data Exchange (ETDEWEB)
Craw, John Simon; Nascimento, Marco Antonio Chaer [Universidade Federal, Rio de Janeiro, RJ (Brazil). Inst. de Quimica
1992-12-31
The coupled Hartree Fock method has been used to calculate ab-initio carbon magnetic shielding tensors for small clusters of acetylene molecules. The chemical shift increases from the monomer to the dimer and trimer. This is mainly due increased diamagnetism, which is imperfectly cancelled by increased paramagnetism due to loss of axial symmetry. Anisotropic effects are shown to be small in both the dimer the and trimer. (author) 21 refs., 2 tabs.
18. Improving the calculation of electron paramagnetic resonance hyperfine coupling tensors for d-block metals
DEFF Research Database (Denmark)
Hedegård, Erik Donovan; Kongsted, Jacob; Sauer, Stephan P. A.
2012-01-01
Calculation of hyperfine coupling constants (HFCs) of Electron Paramagnetic Resonance from first principles can be a beneficial compliment to experimental data in cases where the molecular structure is unknown. We have recently investigated basis set convergence of HFCs in d-block complexes...... and obtained a set of basis functions for the elements Sc–Zn, which were saturated with respect to both the Fermi contact and spin-dipolar components of the hyperfine coupling tensor [Hedeg°ard et al., J. Chem. Theory Comput., 2011, 7, pp. 4077-4087]. Furthermore, a contraction scheme was proposed leading...
19. Retrodictive determinism. [covariant and transformational behavior of tensor fields in hydrodynamics and thermodynamics
Science.gov (United States)
Kiehn, R. M.
1976-01-01
With respect to irreversible, non-homeomorphic maps, contravariant and covariant tensor fields have distinctly natural covariance and transformational behavior. For thermodynamic processes which are non-adiabatic, the fact that the process cannot be represented by a homeomorphic map emphasizes the logical arrow of time, an idea which encompasses a principle of retrodictive determinism for covariant tensor fields.
20. Weak deflection gravitational lensing for photons coupled to Weyl tensor in a Schwarzschild black hole
Science.gov (United States)
Cao, Wei-Guang; Xie, Yi
2018-03-01
Beyond the Einstein-Maxwell model, electromagnetic field might couple with gravitational field through the Weyl tensor. In order to provide one of the missing puzzles of the whole physical picture, we investigate weak deflection lensing for photons coupled to the Weyl tensor in a Schwarzschild black hole under a unified framework that is valid for its two possible polarizations. We obtain its coordinate-independent expressions for all observables of the geometric optics lensing up to the second order in the terms of ɛ which is the ratio of the angular gravitational radius to angular Einstein radius of the lens. These observables include bending angle, image position, magnification, centroid and time delay. The contributions of such a coupling on some astrophysical scenarios are also studied. We find that, in the cases of weak deflection lensing on a star orbiting the Galactic Center Sgr A*, Galactic microlensing on a star in the bulge and astrometric microlensing by a nearby object, these effects are beyond the current limits of technology. However, measuring the variation of the total flux of two weak deflection lensing images caused by the Sgr A* might be a promising way for testing such a coupling in the future.
1. Cross-scale Efficient Tensor Contractions for Coupled Cluster Computations Through Multiple Programming Model Backends
Energy Technology Data Exchange (ETDEWEB)
Ibrahim, Khaled Z. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division; Epifanovsky, Evgeny [Q-Chem, Inc., Pleasanton, CA (United States); Williams, Samuel W. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division; Krylov, Anna I. [Univ. of Southern California, Los Angeles, CA (United States). Dept. of Chemistry
2016-07-26
Coupled-cluster methods provide highly accurate models of molecular structure by explicit numerical calculation of tensors representing the correlation between electrons. These calculations are dominated by a sequence of tensor contractions, motivating the development of numerical libraries for such operations. While based on matrix-matrix multiplication, these libraries are specialized to exploit symmetries in the molecular structure and in electronic interactions, and thus reduce the size of the tensor representation and the complexity of contractions. The resulting algorithms are irregular and their parallelization has been previously achieved via the use of dynamic scheduling or specialized data decompositions. We introduce our efforts to extend the Libtensor framework to work in the distributed memory environment in a scalable and energy efficient manner. We achieve up to 240 speedup compared with the best optimized shared memory implementation. We attain scalability to hundreds of thousands of compute cores on three distributed-memory architectures, (Cray XC30&XC40, BlueGene/Q), and on a heterogeneous GPU-CPU system (Cray XK7). As the bottlenecks shift from being compute-bound DGEMM's to communication-bound collectives as the size of the molecular system scales, we adopt two radically different parallelization approaches for handling load-imbalance. Nevertheless, we preserve a uni ed interface to both programming models to maintain the productivity of computational quantum chemists.
2. Effects of Anomalous Tensor Couplings in B0s — B-bar0s Mixing
International Nuclear Information System (INIS)
Chang Qin; Han Lin; Yang Ya-Dong
2012-01-01
Motivated by the recently observed anomalous large dimuon charge asymmetry in neutral B decays, we study the effects of the anomalous tensor couplings to pursue a possible solution. With the constraints from the observables φ s J/ψ(φ,f 0 ) , a s sl and ΔM s , the new physics parameter spaces are severely restricted. We find that the contributions induced by the color-singlet or the color-octet tensor operators are helpful to moderate the anomaly in B 0 s — B-bar 0 s mixing. Numerically, the observable a s sl could be enhanced by about two orders of magnitude by the contributions of color-singlet or color-octet tensor operators with their respective nontrivial new weak phase φ T1 = 41deg ± 35deg or φ T8 = −47deg ± 33deg and relevant strength parameters |g T1 | = (2.89 ± 1.40) × 10 −2 or |g T8 | = (0.79 ± 0.34) × 10 −2 . However, due to the fact that the NP contributions are severely suppressed by the recent LHCb measurement for φ s J/ψ(φ,f 0 ) , our theoretical result of a s sl is still much smaller than the central value of the experimental data
3. The tensor calculus and matter coupling of the alternative minimal auxiliary field formulation of N = 1 supergravity
International Nuclear Information System (INIS)
Sohnius, M.; West, P.
1982-01-01
The tensor calculus for the new alternative minimal auxiliary field formulation of N = 1 supergravity is given. It is used to construct the couplings of this formulation of supergravity to matter. These couplings are found to be different, in several respects to those of the old minimal formulation of N = 1 supergravity. (orig.)
4. Turbo-SMT: Parallel Coupled Sparse Matrix-Tensor Factorizations and Applications
Science.gov (United States)
Papalexakis, Evangelos E.; Faloutsos, Christos; Mitchell, Tom M.; Talukdar, Partha Pratim; Sidiropoulos, Nicholas D.; Murphy, Brian
2016-01-01
How can we correlate the neural activity in the human brain as it responds to typed words, with properties of these terms (like ’edible’, ’fits in hand’)? In short, we want to find latent variables, that jointly explain both the brain activity, as well as the behavioral responses. This is one of many settings of the Coupled Matrix-Tensor Factorization (CMTF) problem. Can we enhance any CMTF solver, so that it can operate on potentially very large datasets that may not fit in main memory? We introduce Turbo-SMT, a meta-method capable of doing exactly that: it boosts the performance of any CMTF algorithm, produces sparse and interpretable solutions, and parallelizes any CMTF algorithm, producing sparse and interpretable solutions (up to 65 fold). Additionally, we improve upon ALS, the work-horse algorithm for CMTF, with respect to efficiency and robustness to missing values. We apply Turbo-SMT to BrainQ, a dataset consisting of a (nouns, brain voxels, human subjects) tensor and a (nouns, properties) matrix, with coupling along the nouns dimension. Turbo-SMT is able to find meaningful latent variables, as well as to predict brain activity with competitive accuracy. Finally, we demonstrate the generality of Turbo-SMT, by applying it on a Facebook dataset (users, ’friends’, wall-postings); there, Turbo-SMT spots spammer-like anomalies. PMID:27672406
5. Tensor contraction engine: Abstraction and automated parallel implementation of configuration-interaction, coupled-cluster, and many-body perturbation theories
International Nuclear Information System (INIS)
Hirata, So
2003-01-01
We develop a symbolic manipulation program and program generator (Tensor Contraction Engine or TCE) that automatically derives the working equations of a well-defined model of second-quantized many-electron theories and synthesizes efficient parallel computer programs on the basis of these equations. Provided an ansatz of a many-electron theory model, TCE performs valid contractions of creation and annihilation operators according to Wick's theorem, consolidates identical terms, and reduces the expressions into the form of multiple tensor contractions acted by permutation operators. Subsequently, it determines the binary contraction order for each multiple tensor contraction with the minimal operation and memory cost, factorizes common binary contractions (defines intermediate tensors), and identifies reusable intermediates. The resulting ordered list of binary tensor contractions, additions, and index permutations is translated into an optimized program that is combined with the NWChem and UTChem computational chemistry software packages. The programs synthesized by TCE take advantage of spin symmetry, Abelian point-group symmetry, and index permutation symmetry at every stage of calculations to minimize the number of arithmetic operations and storage requirement, adjust the peak local memory usage by index range tiling, and support parallel I/O interfaces and dynamic load balancing for parallel executions. We demonstrate the utility of TCE through automatic derivation and implementation of parallel programs for various models of configuration-interaction theory (CISD, CISDT, CISDTQ), many-body perturbation theory[MBPT(2), MBPT(3), MBPT(4)], and coupled-cluster theory (LCCD, CCD, LCCSD, CCSD, QCISD, CCSDT, and CCSDTQ)
6. Proton chemical shift tensors determined by 3D ultrafast MAS double-quantum NMR spectroscopy
International Nuclear Information System (INIS)
Zhang, Rongchun; Mroue, Kamal H.; Ramamoorthy, Ayyalusamy
2015-01-01
Proton NMR spectroscopy in the solid state has recently attracted much attention owing to the significant enhancement in spectral resolution afforded by the remarkable advances in ultrafast magic angle spinning (MAS) capabilities. In particular, proton chemical shift anisotropy (CSA) has become an important tool for obtaining specific insights into inter/intra-molecular hydrogen bonding. However, even at the highest currently feasible spinning frequencies (110–120 kHz), 1 H MAS NMR spectra of rigid solids still suffer from poor resolution and severe peak overlap caused by the strong 1 H– 1 H homonuclear dipolar couplings and narrow 1 H chemical shift (CS) ranges, which render it difficult to determine the CSA of specific proton sites in the standard CSA/single-quantum (SQ) chemical shift correlation experiment. Herein, we propose a three-dimensional (3D) 1 H double-quantum (DQ) chemical shift/CSA/SQ chemical shift correlation experiment to extract the CS tensors of proton sites whose signals are not well resolved along the single-quantum chemical shift dimension. As extracted from the 3D spectrum, the F1/F3 (DQ/SQ) projection provides valuable information about 1 H– 1 H proximities, which might also reveal the hydrogen-bonding connectivities. In addition, the F2/F3 (CSA/SQ) correlation spectrum, which is similar to the regular 2D CSA/SQ correlation experiment, yields chemical shift anisotropic line shapes at different isotropic chemical shifts. More importantly, since the F2/F1 (CSA/DQ) spectrum correlates the CSA with the DQ signal induced by two neighboring proton sites, the CSA spectrum sliced at a specific DQ chemical shift position contains the CSA information of two neighboring spins indicated by the DQ chemical shift. If these two spins have different CS tensors, both tensors can be extracted by numerical fitting. We believe that this robust and elegant single-channel proton-based 3D experiment provides useful atomistic-level structural and dynamical
7. Determination of mouse skeletal muscle architecture using three dimensional diffusion tensor imaging
NARCIS (Netherlands)
Heemskerk, A.M.; Strijkers, G.J.; Vilanova, A.; Drost, M.R.; Nicolaij, K.
2005-01-01
Muscle architecture is the main determinant of the mechanical behavior of skeletal muscles. This study explored the feasibility of diffusion tensor imaging (DTI) and fiber tracking to noninvasively determine the in vivo three-dimensional (3D) architecture of skeletal muscle in mouse hind leg. In six
8. Determination of mouse skeletal muscle architecture using three-dimensional diffusion tensor imaging
NARCIS (Netherlands)
Heemskerk, Anneriet M.; Strijkers, Gustav J.; Vilanova, Anna; Drost, Maarten R.; Nicolay, Klaas
2005-01-01
Muscle architecture is the main determinant of the mechanical behavior of skeletal muscles. This study explored the feasibility of diffusion tensor imaging (DTI) and fiber tracking to noninvasively determine the in vivo three-dimensional (3D) architecture of skeletal muscle in mouse hind leg. In six
9. Determination of 3D magnetic reluctivity tensor of soft magnetic composite material
International Nuclear Information System (INIS)
Guo Youguang; Zhu Jianguo; Lin Zhiwei; Zhong Jinjiang; Lu Haiyan; Wang Shuhong
2007-01-01
Soft magnetic composite (SMC) materials are especially suitable for construction of electrical machines with complex structures and three-dimensional (3D) magnetic fluxes. In the design and optimization of such 3D flux machines, the 3D vector magnetic properties of magnetic materials should be properly determined, modeled, and applied for accurate calculation of the magnetic field distribution, parameters, and performance. This paper presents the measurement of 3D vector magnetic properties and determination of 3D reluctivity tensor of SMC. The reluctivity tensor is a key factor for accurate numerical analysis of magnetic field in a 3D flux SMC motor
10. Solar System constraints on massless scalar-tensor gravity with positive coupling constant upon cosmological evolution of the scalar field
Science.gov (United States)
Anderson, David; Yunes, Nicolás
2017-09-01
Scalar-tensor theories of gravity modify general relativity by introducing a scalar field that couples nonminimally to the metric tensor, while satisfying the weak-equivalence principle. These theories are interesting because they have the potential to simultaneously suppress modifications to Einstein's theory on Solar System scales, while introducing large deviations in the strong field of neutron stars. Scalar-tensor theories can be classified through the choice of conformal factor, a scalar that regulates the coupling between matter and the metric in the Einstein frame. The class defined by a Gaussian conformal factor with a negative exponent has been studied the most because it leads to spontaneous scalarization (i.e. the sudden activation of the scalar field in neutron stars), which consequently leads to large deviations from general relativity in the strong field. This class, however, has recently been shown to be in conflict with Solar System observations when accounting for the cosmological evolution of the scalar field. We here study whether this remains the case when the exponent of the conformal factor is positive, as well as in another class of theories defined by a hyperbolic conformal factor. We find that in both of these scalar-tensor theories, Solar System tests are passed only in a very small subset of coupling parameter space, for a large set of initial conditions compatible with big bang nucleosynthesis. However, while we find that it is possible for neutron stars to scalarize, one must carefully select the coupling parameter to do so, and even then, the scalar charge is typically 2 orders of magnitude smaller than in the negative-exponent case. Our study suggests that future work on scalar-tensor gravity, for example in the context of tests of general relativity with gravitational waves from neutron star binaries, should be carried out within the positive coupling parameter class.
11. The continuous determination of spacetime geometry by the Riemann curvature tensor
International Nuclear Information System (INIS)
Rendall, A.D.
1988-01-01
It is shown that generically the Riemann tensor of a Lorentz metric on an n-dimensional manifold (n ≥ 4) determines the metric up to a constant factor and hence determines the associated torsion-free connection uniquely. The resulting map from Riemann tensors to connections is continuous in the Whitney Csup(∞) topology but, at least for some manifolds, constant factors cannot be chosen so as to make the map from Riemann tensors to metrics continuous in that topology. The latter map is, however, continuous in the compact open Csup(∞) topology so that estimates of the metric and its derivatives on a compact set can be obtained from similar estimates on the curvature and its derivatives. (author)
12. Dynamical compactification of D-dimensional gravity coupled to antisymmetric tensors in a 1/D expansion
International Nuclear Information System (INIS)
Foda, O.
1984-12-01
The effective potential of components of the curl of an antisymmetric tensor coupled to gravity in D dimensions is evaluated in a 1/D expansion. For large D, only highest-rank propagators contribute to leading order, while multiloop diagrams are suppressed by phase-space factors. Divergences are regulated by a cut-off LAMBDA, that we interpret as the mass-breaking scale of a larger theory that is finite. As an application we consider the bosonic sector of D=11, N=1 supergravity. If the full theory is finite, then LAMBDA is msub(SUSY): the scale below which the fermion sector decouples. For m 9 sub(SUSY)>1/akappa 2 , (kappa 2 : the D=11 Newton's coupling, a approx.= O(1)) the 11-dimensional symmetric vacuum is unstable under compactification. For m 9 sub(SUSY) 2 , it is metastable. To leading order in 1/D, all gauge dependence cancels identically, while ghosts as well as the graviton decouple. (author)
13. Current density tensors
Science.gov (United States)
Lazzeretti, Paolo
2018-04-01
It is shown that nonsymmetric second-rank current density tensors, related to the current densities induced by magnetic fields and nuclear magnetic dipole moments, are fundamental properties of a molecule. Together with magnetizability, nuclear magnetic shielding, and nuclear spin-spin coupling, they completely characterize its response to magnetic perturbations. Gauge invariance, resolution into isotropic, deviatoric, and antisymmetric parts, and contributions of current density tensors to magnetic properties are discussed. The components of the second-rank tensor properties are rationalized via relationships explicitly connecting them to the direction of the induced current density vectors and to the components of the current density tensors. The contribution of the deviatoric part to the average value of magnetizability, nuclear shielding, and nuclear spin-spin coupling, uniquely determined by the antisymmetric part of current density tensors, vanishes identically. The physical meaning of isotropic and anisotropic invariants of current density tensors has been investigated, and the connection between anisotropy magnitude and electron delocalization has been discussed.
14. TEV—A Program for the Determination of the Thermal Expansion Tensor from Diffraction Data
Directory of Open Access Journals (Sweden)
Thomas Langreiter
2015-02-01
Full Text Available TEV (Thermal Expansion Visualizing is a user-friendly program for the calculation of the thermal expansion tensor αij from diffraction data. Unit cell parameters determined from temperature dependent data collections can be provided as input. An intuitive graphical user interface enables fitting of the evolution of individual lattice parameters to polynomials up to fifth order. Alternatively, polynomial representations obtained from other fitting programs or from the literature can be entered. The polynomials and their derivatives are employed for the calculation of the tensor components of αij in the infinitesimal limit. The tensor components, eigenvalues, eigenvectors and their angles with the crystallographic axes can be evaluated for individual temperatures or for temperature ranges. Values of the tensor in directions parallel to either [uvw]’s of the crystal lattice or vectors (hkl of reciprocal space can be calculated. Finally, the 3-D representation surface for the second rank tensor and pre- or user-defined 2-D sections can be plotted and saved in a bitmap format. TEV is written in JAVA. The distribution contains an EXE-file for Windows users and a system independent JAR-file for running the software under Linux and Mac OS X. The program can be downloaded from the following link: http://www.uibk.ac.at/mineralogie/downloads/TEV.html (Institute of Mineralogy and Petrography, University of Innsbruck, Innsbruck, Austria
15. Determining chiral couplings at NLO
International Nuclear Information System (INIS)
Rosell, Ignasi
2007-01-01
We present a general method that allows to estimate the low-energy constants of Chiral Perturbation Theory up to next-to-leading corrections in the 1/N C expansion, that is, keeping full control of the renormalization scale dependence. As a first step we have determined L 8 and C 38 , the couplings related to the difference of the two-point correlation functions of two scalar and pseudoscalar currents, L 8 r (μ 0 ) = (0.6±0.4)·10 -3 and C 38 r (μ 0 ) = (2±6)·10 -6 , with μ 0 0.77 GeV. As in many effective approaches, one of the main ingredients of this method is the matching procedure: some comments related to this topic are presented here
16. Dynamical analysis for a scalar-tensor model with Gauss-Bonnet and non-minimal couplings
Energy Technology Data Exchange (ETDEWEB)
Granda, L.N.; Jimenez, D.F. [Universidad del Valle, Departamento de Fisica, Cali (Colombia)
2017-10-15
We study the autonomous system for a scalar-tensor model of dark energy with Gauss-Bonnet and non-minimal couplings. The critical points describe important stable asymptotic scenarios including quintessence, phantom and de Sitter attractor solutions. Two functional forms for the coupling functions and the scalar potential are considered: power-law and exponential functions of the scalar field. For the exponential functions the existence of stable quintessence, phantom or de Sitter solutions, allows for an asymptotic behavior where the effective Newtonian coupling becomes constant. The phantom solutions could be realized without appealing to ghost degrees of freedom. Transient inflationary and radiation-dominated phases can also be described. (orig.)
17. Interplay of tensor correlations and vibrational coupling for single-particle states in atomic nuclei
International Nuclear Information System (INIS)
Colo, G.; SAgawa, H.; Bortignon, P. F.
2009-01-01
To study the structure of atomic nuclei, the ab-initio methods can nowadays be applied only for mass number A smaller than ∼ 10-15. For heavier systems, the self-consistent mean-field (SCMF) approach is probably the most microscopic approach which can be systematically applied to stable and exotic nuclei. In practice, the SCMF is mostly based on parametrizations of an effective interaction. However, the are groups who are intensively working on the development of a general density functional (DF) which is not necessarily extracted from an Hamiltonian. The basic question is to what extent this allows improving on the existing functionals. In this contribution we analyze the performance of existing functionals as far as the reproduction of single-particle states is concerned. We start by analyzing the effect of the tensor terms, on which the attention of several groups have recently focused. Then we discuss the impact of the particle-vibration coupling (PVC). Although the basic idea of this approach dates back to long time ago, we present here for the first time calculations which are entirely based on microscopic interactions without dropping any term or introducing ad hoc parameters. We show results both for well-known, benchmark nuclei like 4 0C a and 2 08P b as well as unstable nuclei like 1 32S n. Both single-particle energies and spectroscopic factors are discussed.(author)
18. Determination and uncertainty of moment tensors for microearthquakes at Okmok Volcano, Alaska
Science.gov (United States)
Pesicek, J.D.; Sileny, J.; Prejean, S.G.; Thurber, C.H.
2012-01-01
Efforts to determine general moment tensors (MTs) for microearthquakes in volcanic areas are often hampered by small seismic networks, which can lead to poorly constrained hypocentres and inadequate modelling of seismic velocity heterogeneity. In addition, noisy seismic signals can make it difficult to identify phase arrivals correctly for small magnitude events. However, small volcanic earthquakes can have source mechanisms that deviate from brittle double-couple shear failure due to magmatic and/or hydrothermal processes. Thus, determining reliable MTs in such conditions is a challenging but potentially rewarding pursuit. We pursued such a goal at Okmok Volcano, Alaska, which erupted recently in 1997 and in 2008. The Alaska Volcano Observatory operates a seismic network of 12 stations at Okmok and routinely catalogues recorded seismicity. Using these data, we have determined general MTs for seven microearthquakes recorded between 2004 and 2007 by inverting peak amplitude measurements of P and S phases. We computed Green's functions using precisely relocated hypocentres and a 3-D velocity model. We thoroughly assessed the quality of the solutions by computing formal uncertainty estimates, conducting a variety of synthetic and sensitivity tests, and by comparing the MTs to solutions obtained using alternative methods. The results show that MTs are sensitive to station distribution and errors in the data, velocity model and hypocentral parameters. Although each of the seven MTs contains a significant non-shear component, we judge several of the solutions to be unreliable. However, several reliable MTs are obtained for a group of previously identified repeating events, and are interpreted as compensated linear-vector dipole events.
19. Highly Efficient and Scalable Compound Decomposition of Two-Electron Integral Tensor and Its Application in Coupled Cluster Calculations
Energy Technology Data Exchange (ETDEWEB)
Peng, Bo [William R. Wiley Environmental; Kowalski, Karol [William R. Wiley Environmental
2017-08-11
The representation and storage of two-electron integral tensors are vital in large- scale applications of accurate electronic structure methods. Low-rank representation and efficient storage strategy of integral tensors can significantly reduce the numerical overhead and consequently time-to-solution of these methods. In this paper, by combining pivoted incomplete Cholesky decomposition (CD) with a follow-up truncated singular vector decomposition (SVD), we develop a decomposition strategy to approximately represent the two-electron integral tensor in terms of low-rank vectors. A systematic benchmark test on a series of 1-D, 2-D, and 3-D carbon-hydrogen systems demonstrates high efficiency and scalability of the compound two-step decomposition of the two-electron integral tensor in our implementation. For the size of atomic basis set N_b ranging from ~ 100 up to ~ 2, 000, the observed numerical scaling of our implementation shows O(N_b^{2.5~3}) versus O(N_b^{3~4}) of single CD in most of other implementations. More importantly, this decomposition strategy can significantly reduce the storage requirement of the atomic-orbital (AO) two-electron integral tensor from O(N_b^4) to O(N_b^2 log_{10}(N_b)) with moderate decomposition thresholds. The accuracy tests have been performed using ground- and excited-state formulations of coupled- cluster formalism employing single and double excitations (CCSD) on several bench- mark systems including the C_{60} molecule described by nearly 1,400 basis functions. The results show that the decomposition thresholds can be generally set to 10^{-4} to 10^{-3} to give acceptable compromise between efficiency and accuracy.
20. TensorLy: Tensor Learning in Python
NARCIS (Netherlands)
Kossaifi, Jean; Panagakis, Yannis; Pantic, Maja
2016-01-01
Tensor methods are gaining increasing traction in machine learning. However, there are scant to no resources available to perform tensor learning and decomposition in Python. To answer this need we developed TensorLy. TensorLy is a state of the art general purpose library for tensor learning.
1. Full paleostress tensor reconstruction using quartz veins of Panasqueira Mine, central Portugal; part I: Paleopressure determination
Science.gov (United States)
Jaques, Luís; Pascal, Christophe
2017-09-01
Paleostress tensor restoration methods are traditionally limited to reconstructing geometrical parameters and are unable to resolve stress magnitudes. Based on previous studies we further developed a methodology to restore full paleostress tensors. We concentrated on inversion of Mode I fractures and acquired data in Panasqueira Mine, Portugal, where optimal exposures of mineralized quartz veins can be found. To carry out full paleostress restoration we needed to determine (1) pore (paleo)pressure and (2) vein attitudes. The present contribution focuses specifically on the determination of pore pressure. To these aims we conducted an extensive fluid inclusion study to derive fluid isochores from the quartz of the studied veins. To constrain P-T conditions, we combined these isochores with crystallisation temperatures derived from geochemical analyses of coeval arsenopyrite. We also applied the sphalerite geobarometer and considered two other independent pressure indicators. Our results point to pore pressures of ∼300 MPa and formation depths of ∼10 km. Such formation depths are in good agreement with the regional geological evolution. The obtained pore pressure will be merged with vein inversion results, in order to achieve full paleostress tensor restoration, in a forthcoming companion paper.
2. Spin and Pseudospin Symmetries with Trigonometric Pöschl-Teller Potential including Tensor Coupling
Directory of Open Access Journals (Sweden)
M. Hamzavi
2013-01-01
Full Text Available We study approximate analytical solutions of the Dirac equation with the trigonometric Pöschl-Teller (tPT potential and a Coulomb-like tensor potential for arbitrary spin-orbit quantum number κ under the presence of exact spin and pseudospin ( p -spin symmetries. The bound state energy eigenvalues and the corresponding two-component wave functions of the Dirac particle are obtained using the parametric generalization of the Nikiforov-Uvarov (NU method. We show that tensor interaction removes degeneracies between spin and pseudospin doublets. The case of nonrelativistic limit is studied too.
3. Generalized dielectric permittivity tensor
International Nuclear Information System (INIS)
Borzdov, G.N.; Barkovskii, L.M.; Fedorov, F.I.
1986-01-01
The authors deal with the question of what is to be done with the formalism of the electrodynamics of dispersive media based on the introduction of dielectric-permittivity tensors for purely harmonic fields when Voigt waves and waves of more general form exist. An attempt is made to broaden and generalize the formalism to take into account dispersion of waves of the given type. In dispersive media, the polarization, magnetization, and conduction current-density vectors of point and time are determined by the values of the electromagnetic field vectors in the vicinity of this point (spatial dispersion) in the preceding instants of time (time dispersion). The dielectric-permittivity tensor and other tensors of electrodynamic parameters of the medium are introduced in terms of a set of evolution operators and not the set of harmonic function. It is noted that a magnetic-permeability tensor and an elastic-modulus tensor may be introduced for an acoustic field in dispersive anisotropic media with coupling equations of general form
4. Benchmarking density-functional-theory calculations of rotational g tensors and magnetizabilities using accurate coupled-cluster calculations.
Science.gov (United States)
Lutnaes, Ola B; Teale, Andrew M; Helgaker, Trygve; Tozer, David J; Ruud, Kenneth; Gauss, Jürgen
2009-10-14
An accurate set of benchmark rotational g tensors and magnetizabilities are calculated using coupled-cluster singles-doubles (CCSD) theory and coupled-cluster single-doubles-perturbative-triples [CCSD(T)] theory, in a variety of basis sets consisting of (rotational) London atomic orbitals. The accuracy of the results obtained is established for the rotational g tensors by careful comparison with experimental data, taking into account zero-point vibrational corrections. After an analysis of the basis sets employed, extrapolation techniques are used to provide estimates of the basis-set-limit quantities, thereby establishing an accurate benchmark data set. The utility of the data set is demonstrated by examining a wide variety of density functionals for the calculation of these properties. None of the density-functional methods are competitive with the CCSD or CCSD(T) methods. The need for a careful consideration of vibrational effects is clearly illustrated. Finally, the pure coupled-cluster results are compared with the results of density-functional calculations constrained to give the same electronic density. The importance of current dependence in exchange-correlation functionals is discussed in light of this comparison.
5. TensorLy: Tensor Learning in Python
OpenAIRE
Kossaifi, Jean; Panagakis, Yannis; Pantic, Maja
2016-01-01
Tensors are higher-order extensions of matrices. While matrix methods form the cornerstone of machine learning and data analysis, tensor methods have been gaining increasing traction. However, software support for tensor operations is not on the same footing. In order to bridge this gap, we have developed \\emph{TensorLy}, a high-level API for tensor methods and deep tensorized neural networks in Python. TensorLy aims to follow the same standards adopted by the main projects of the Python scie...
6. Relativistic effect of pseudospin symmetry and tensor coupling on the Mie-type potential via Laplace transformation method
International Nuclear Information System (INIS)
Eshghi, M.; Ikhdair, S. M.
2014-01-01
A relativistic Mie-type potential for spin-1/2 particles is studied. The Dirac Hamiltonian contains a scalar S(r) and a vector V(r) Mie-type potential in the radial coordinates, as well as a tensor potential U(r) in the form of Coulomb potential. In the pseudospin (p-spin) symmetry setting Σ = C ps and Δ = V(r), an analytical solution for exact bound states of the corresponding Dirac equation is found. The eigenenergies and normalized wave functions are presented and particular cases are discussed with any arbitrary spin—orbit coupling number κ. Special attention is devoted to the case Σ = 0 for which p-spin symmetry is exact. The Laplace transform approach (LTA) is used in our calculations. Some numerical results are obtained and compared with those of other methods. (general)
7. Spin orbit coupling for molecular ab initio density matrix renormalization group calculations: Application to g-tensors
Energy Technology Data Exchange (ETDEWEB)
Roemelt, Michael, E-mail: [email protected] [Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, D-44780 Bochum, Germany and Max-Planck Institut für Kohlenforschung, Kaiser-Wilhelm-Platz 1, 45470 Mülheim an der Ruhr (Germany)
2015-07-28
Spin Orbit Coupling (SOC) is introduced to molecular ab initio density matrix renormalization group (DMRG) calculations. In the presented scheme, one first approximates the electronic ground state and a number of excited states of the Born-Oppenheimer (BO) Hamiltonian with the aid of the DMRG algorithm. Owing to the spin-adaptation of the algorithm, the total spin S is a good quantum number for these states. After the non-relativistic DMRG calculation is finished, all magnetic sublevels of the calculated states are constructed explicitly, and the SOC operator is expanded in the resulting basis. To this end, spin orbit coupled energies and wavefunctions are obtained as eigenvalues and eigenfunctions of the full Hamiltonian matrix which is composed of the SOC operator matrix and the BO Hamiltonian matrix. This treatment corresponds to a quasi-degenerate perturbation theory approach and can be regarded as the molecular equivalent to atomic Russell-Saunders coupling. For the evaluation of SOC matrix elements, the full Breit-Pauli SOC Hamiltonian is approximated by the widely used spin-orbit mean field operator. This operator allows for an efficient use of the second quantized triplet replacement operators that are readily generated during the non-relativistic DMRG algorithm, together with the Wigner-Eckart theorem. With a set of spin-orbit coupled wavefunctions at hand, the molecular g-tensors are calculated following the scheme proposed by Gerloch and McMeeking. It interprets the effective molecular g-values as the slope of the energy difference between the lowest Kramers pair with respect to the strength of the applied magnetic field. Test calculations on a chemically relevant Mo complex demonstrate the capabilities of the presented method.
8. Determination of mouse skeletal muscle architecture using three-dimensional diffusion tensor imaging.
Science.gov (United States)
Heemskerk, Anneriet M; Strijkers, Gustav J; Vilanova, Anna; Drost, Maarten R; Nicolay, Klaas
2005-06-01
Muscle architecture is the main determinant of the mechanical behavior of skeletal muscles. This study explored the feasibility of diffusion tensor imaging (DTI) and fiber tracking to noninvasively determine the in vivo three-dimensional (3D) architecture of skeletal muscle in mouse hind leg. In six mice, the hindlimb was imaged with a diffusion-weighted (DW) 3D fast spin-echo (FSE) sequence followed by the acquisition of an exercise-induced, T(2)-enhanced data set. The data showed the expected fiber organization, from which the physiological cross-sectional area (PCSA), fiber length, and pennation angle for the tibialis anterior (TA) were obtained. The values of these parameters ranged from 5.4-9.1 mm(2), 5.8-7.8 mm, and 21-24 degrees , respectively, which is in agreement with values obtained previously with the use of invasive methods. This study shows that 3D DT acquisition and fiber tracking is feasible for the skeletal muscle of mice, and thus enables the quantitative determination of muscle architecture.
9. Massively parallel implementations of coupled-cluster methods for electron spin resonance spectra. I. Isotropic hyperfine coupling tensors in large radicals
Energy Technology Data Exchange (ETDEWEB)
Verma, Prakash; Morales, Jorge A., E-mail: [email protected] [Department of Chemistry and Biochemistry, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061 (United States); Perera, Ajith [Department of Chemistry and Biochemistry, Texas Tech University, P.O. Box 41061, Lubbock, Texas 79409-1061 (United States); Department of Chemistry, Quantum Theory Project, University of Florida, Gainesville, Florida 32611 (United States)
2013-11-07
Coupled cluster (CC) methods provide highly accurate predictions of molecular properties, but their high computational cost has precluded their routine application to large systems. Fortunately, recent computational developments in the ACES III program by the Bartlett group [the OED/ERD atomic integral package, the super instruction processor, and the super instruction architecture language] permit overcoming that limitation by providing a framework for massively parallel CC implementations. In that scheme, we are further extending those parallel CC efforts to systematically predict the three main electron spin resonance (ESR) tensors (A-, g-, and D-tensors) to be reported in a series of papers. In this paper inaugurating that series, we report our new ACES III parallel capabilities that calculate isotropic hyperfine coupling constants in 38 neutral, cationic, and anionic radicals that include the {sup 11}B, {sup 17}O, {sup 9}Be, {sup 19}F, {sup 1}H, {sup 13}C, {sup 35}Cl, {sup 33}S,{sup 14}N, {sup 31}P, and {sup 67}Zn nuclei. Present parallel calculations are conducted at the Hartree-Fock (HF), second-order many-body perturbation theory [MBPT(2)], CC singles and doubles (CCSD), and CCSD with perturbative triples [CCSD(T)] levels using Roos augmented double- and triple-zeta atomic natural orbitals basis sets. HF results consistently overestimate isotropic hyperfine coupling constants. However, inclusion of electron correlation effects in the simplest way via MBPT(2) provides significant improvements in the predictions, but not without occasional failures. In contrast, CCSD results are consistently in very good agreement with experimental results. Inclusion of perturbative triples to CCSD via CCSD(T) leads to small improvements in the predictions, which might not compensate for the extra computational effort at a non-iterative N{sup 7}-scaling in CCSD(T). The importance of these accurate computations of isotropic hyperfine coupling constants to elucidate
10. Analytic determination at one loop of the energy-momentum tensor for lattice QCD
International Nuclear Information System (INIS)
Caracciolo, S.; Menotti, P.; Pelissetto, A.
1991-01-01
We give a completely analytical determinaton of the corrections to the naive energy-momentum tensor for lattice QCD at one loop. This tenor is conserved and gives rise to the correct trace anomaly. (orig.)
11. Chaos in the motion of a test scalar particle coupling to the Einstein tensor in Schwarzschild-Melvin black hole spacetime
Energy Technology Data Exchange (ETDEWEB)
Wang, Mingzhi [Hunan Normal University, Department of Physics, Institute of Physics, Changsha, Hunan (China); Hunan Normal University, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha, Hunan (China); Chen, Songbai; Jing, Jiliang [Hunan Normal University, Department of Physics, Institute of Physics, Changsha, Hunan (China); Hunan Normal University, Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha, Hunan (China); Hunan Normal University, Synergetic Innovation Center for Quantum Effects and Applications, Changsha, Hunan (China)
2017-04-15
We present firstly the equation of motion for a test scalar particle coupling to the Einstein tensor in the Schwarzschild-Melvin black hole spacetime through the short-wave approximation. Through analyzing Poincare sections, the power spectrum, the fast Lyapunov exponent indicator and the bifurcation diagram, we investigate the effects of the coupling parameter on the chaotic behavior of the particles. With the increase of the coupling strength, we find that the motion of the coupled particle for the chosen parameters becomes more regular and order for the negative couple constant. While, for the positive one, the motion of the coupled particles first undergoes a series of transitions betweens chaotic motion and regular motion and then falls into horizon or escapes to spatial infinity. Our results show that the coupling brings about richer effects for the motion of the particles. (orig.)
12. Mid-callosal plane determination using preferred directions from diffusion tensor images
Science.gov (United States)
Costa, André L.; Rittner, Letícia; Lotufo, Roberto A.; Appenzeller, Simone
2015-03-01
The corpus callosum is the major brain structure responsible for inter{hemispheric communication between neurons. Many studies seek to relate corpus callosum attributes to patient characteristics, cerebral diseases and psychological disorders. Most of those studies rely on 2D analysis of the corpus callosum in the mid-sagittal plane. However, it is common to find conflicting results among studies, once many ignore methodological issues and define the mid-sagittal plane based on precary or invalid criteria with respect to the corpus callosum. In this work we propose a novel method to determine the mid-callosal plane using the corpus callosum internal preferred diffusion directions obtained from diffusion tensor images. This plane is analogous to the mid-sagittal plane, but intended to serve exclusively as the corpus callosum reference. Our method elucidates the great potential the directional information of the corpus callosum fibers have to indicate its own referential. Results from experiments with five image pairs from distinct subjects, obtained under the same conditions, demonstrate the method effectiveness to find the corpus callosum symmetric axis relative to the axial plane.
13. Testing feasibility of scalar-tensor gravity by scale dependent mass and coupling to matter
International Nuclear Information System (INIS)
Mota, D. F.; Salzano, V.; Capozziello, S.
2011-01-01
We investigate whether there is any cosmological evidence for a scalar field with a mass and coupling to matter which change accordingly to the properties of the astrophysical system it ''lives in,'' without directly focusing on the underlying mechanism that drives the scalar field scale-dependent-properties. We assume a Yukawa type of coupling between the field and matter and also that the scalar-field mass grows with density, in order to overcome all gravity constraints within the Solar System. We analyze three different gravitational systems assumed as ''cosmological indicators'': supernovae type Ia, low surface brightness spiral galaxies and clusters of galaxies. Results show (i) a quite good fit to the rotation curves of low surface brightness galaxies only using visible stellar and gas-mass components is obtained; (ii) a scalar field can fairly well reproduce the matter profile in clusters of galaxies, estimated by x-ray observations and without the need of any additional dark matter; and (iii) there is an intrinsic difficulty in extracting information about the possibility of a scale-dependent massive scalar field (or more generally about a varying gravitational constant) from supernovae type Ia.
14. Determination of the plastic deformation and residual stress tensor distribution using surface and bulk intrinsic magnetic properties
International Nuclear Information System (INIS)
Hristoforou, E.; Svec, P. Sr.
2015-01-01
We have developed an unique method to provide the stress calibration curve in steels: performing flaw-less welding in the under examination steel, we obtained to determine the level of the local plastic deformation and the residual stress tensors. These properties where measured using both the X-ray and the neutron diffraction techniques, concerning their surface and bulk stresses type II (intra-grain stresses) respectively, as well as the stress tensor type III by using the electron diffraction technique. Measuring the distribution of these residual stresses along the length of a welded sample or structure, resulted in determining the local stresses from the compressive to tensile yield point. Local measurement of the intrinsic surface and bulk magnetic property tensors allowed for the un-hysteretic correlation. The dependence of these local magnetic tensors with the above mentioned local stress tensors, resulting in a unique and almost un-hysteretic stress calibration curve of each grade of steel. This calibration integrated the steel's mechanical and thermal history, as well as the phase transformations and the presence of precipitations occurring during the welding process.Additionally to that, preliminary results in different grade of steels reveal the existence of a universal law concerning the dependence of magnetic and magnetostrictive properties of steels on their plastic deformation and residual stress state, as they have been accumulated due to their mechanical and thermal fatigue and history. This universality is based on the unique dependence of the intrinsic magnetic properties of steels normalized with a certain magnetoelastic factor, upon the plastic deformation or residual stress state, which, in terms, is normalized with their yield point of stress. (authors)
15. Tensor-decomposed vibrational coupled-cluster theory: Enabling large-scale, highly accurate vibrational-structure calculations
Science.gov (United States)
Madsen, Niels Kristian; Godtliebsen, Ian H.; Losilla, Sergio A.; Christiansen, Ove
2018-01-01
A new implementation of vibrational coupled-cluster (VCC) theory is presented, where all amplitude tensors are represented in the canonical polyadic (CP) format. The CP-VCC algorithm solves the non-linear VCC equations without ever constructing the amplitudes or error vectors in full dimension but still formally includes the full parameter space of the VCC[n] model in question resulting in the same vibrational energies as the conventional method. In a previous publication, we have described the non-linear-equation solver for CP-VCC calculations. In this work, we discuss the general algorithm for evaluating VCC error vectors in CP format including the rank-reduction methods used during the summation of the many terms in the VCC amplitude equations. Benchmark calculations for studying the computational scaling and memory usage of the CP-VCC algorithm are performed on a set of molecules including thiadiazole and an array of polycyclic aromatic hydrocarbons. The results show that the reduced scaling and memory requirements of the CP-VCC algorithm allows for performing high-order VCC calculations on systems with up to 66 vibrational modes (anthracene), which indeed are not possible using the conventional VCC method. This paves the way for obtaining highly accurate vibrational spectra and properties of larger molecules.
16. The tensor hypercontracted parametric reduced density matrix algorithm: coupled-cluster accuracy with O(r(4)) scaling.
Science.gov (United States)
Shenvi, Neil; van Aggelen, Helen; Yang, Yang; Yang, Weitao; Schwerdtfeger, Christine; Mazziotti, David
2013-08-07
Tensor hypercontraction is a method that allows the representation of a high-rank tensor as a product of lower-rank tensors. In this paper, we show how tensor hypercontraction can be applied to both the electron repulsion integral tensor and the two-particle excitation amplitudes used in the parametric 2-electron reduced density matrix (p2RDM) algorithm. Because only O(r) auxiliary functions are needed in both of these approximations, our overall algorithm can be shown to scale as O(r(4)), where r is the number of single-particle basis functions. We apply our algorithm to several small molecules, hydrogen chains, and alkanes to demonstrate its low formal scaling and practical utility. Provided we use enough auxiliary functions, we obtain accuracy similar to that of the standard p2RDM algorithm, somewhere between that of CCSD and CCSD(T).
17. New perspectives in the PAW/GIPAW approach: J(P-O-Si) coupling constants, antisymmetric parts of shift tensors and NQR predictions.
Science.gov (United States)
Bonhomme, Christian; Gervais, Christel; Coelho, Cristina; Pourpoint, Frédérique; Azaïs, Thierry; Bonhomme-Coury, Laure; Babonneau, Florence; Jacob, Guy; Ferrari, Maude; Canet, Daniel; Yates, Jonathan R; Pickard, Chris J; Joyce, Siân A; Mauri, Francesco; Massiot, Dominique
2010-12-01
In 2001, Pickard and Mauri implemented the gauge including projected augmented wave (GIPAW) protocol for first-principles calculations of NMR parameters using periodic boundary conditions (chemical shift anisotropy and electric field gradient tensors). In this paper, three potentially interesting perspectives in connection with PAW/GIPAW in solid-state NMR and pure nuclear quadrupole resonance (NQR) are presented: (i) the calculation of J coupling tensors in inorganic solids; (ii) the calculation of the antisymmetric part of chemical shift tensors and (iii) the prediction of (14)N and (35)Cl pure NQR resonances including dynamics. We believe that these topics should open new insights in the combination of GIPAW, NMR/NQR crystallography, temperature effects and dynamics. Points (i), (ii) and (iii) will be illustrated by selected examples: (i) chemical shift tensors and heteronuclear (2)J(P-O-Si) coupling constants in the case of silicophosphates and calcium phosphates [Si(5)O(PO(4))(6), SiP(2)O(7) polymorphs and α-Ca(PO(3))(2)]; (ii) antisymmetric chemical shift tensors in cyclopropene derivatives, C(3)X(4) (X = H, Cl, F) and (iii) (14)N and (35)Cl NQR predictions in the case of RDX (C(3)H(6)N(6)O(6)), β-HMX (C(4)H(8)N(8)O(8)), α-NTO (C(2)H(2)N(4)O(3)) and AlOPCl(6). RDX, β-HMX and α-NTO are explosive compounds. Copyright © 2010 John Wiley & Sons, Ltd.
18. Simultaneous Mass Determination for Gravitationally Coupled Asteroids
Energy Technology Data Exchange (ETDEWEB)
Baer, James [Private address, 3210 Apache Road, Pittsburgh, PA 15241 (United States); Chesley, Steven R., E-mail: [email protected] [Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109 (United States)
2017-08-01
The conventional least-squares asteroid mass determination algorithm allows us to solve for the mass of a large subject asteroid that is perturbing the trajectory of a smaller test asteroid. However, this algorithm is necessarily a first approximation, ignoring the possibility that the subject asteroid may itself be perturbed by the test asteroid, or that the encounter’s precise geometry may be entangled with encounters involving other asteroids. After reviewing the conventional algorithm, we use it to calculate the masses of 30 main-belt asteroids. Compared to our previous results, we find new mass estimates for eight asteroids (11 Parthenope, 27 Euterpe, 51 Neimausa, 76 Freia, 121 Hermione, 324 Bamberga, 476 Hedwig, and 532 Herculina) and significantly more precise estimates for six others (2 Pallas, 3 Juno, 4 Vesta, 9 Metis, 16 Psyche, and 88 Thisbe). However, we also find that the conventional algorithm yields questionable results in several gravitationally coupled cases. To address such cases, we describe a new algorithm that allows the epoch state vectors of the subject asteroids to be included as solve-for parameters, allowing for the simultaneous solution of the masses and epoch state vectors of multiple subject and test asteroids. We then apply this algorithm to the same 30 main-belt asteroids and conclude that mass determinations resulting from current and future high-precision astrometric sources (such as Gaia ) should conduct a thorough search for possible gravitational couplings and account for their effects.
19. Simultaneous Mass Determination for Gravitationally Coupled Asteroids
Science.gov (United States)
Baer, James; Chesley, Steven R.
2017-08-01
The conventional least-squares asteroid mass determination algorithm allows us to solve for the mass of a large subject asteroid that is perturbing the trajectory of a smaller test asteroid. However, this algorithm is necessarily a first approximation, ignoring the possibility that the subject asteroid may itself be perturbed by the test asteroid, or that the encounter’s precise geometry may be entangled with encounters involving other asteroids. After reviewing the conventional algorithm, we use it to calculate the masses of 30 main-belt asteroids. Compared to our previous results, we find new mass estimates for eight asteroids (11 Parthenope, 27 Euterpe, 51 Neimausa, 76 Freia, 121 Hermione, 324 Bamberga, 476 Hedwig, and 532 Herculina) and significantly more precise estimates for six others (2 Pallas, 3 Juno, 4 Vesta, 9 Metis, 16 Psyche, and 88 Thisbe). However, we also find that the conventional algorithm yields questionable results in several gravitationally coupled cases. To address such cases, we describe a new algorithm that allows the epoch state vectors of the subject asteroids to be included as solve-for parameters, allowing for the simultaneous solution of the masses and epoch state vectors of multiple subject and test asteroids. We then apply this algorithm to the same 30 main-belt asteroids and conclude that mass determinations resulting from current and future high-precision astrometric sources (such as Gaia) should conduct a thorough search for possible gravitational couplings and account for their effects.
20. Simultaneous Mass Determination for Gravitationally Coupled Asteroids
International Nuclear Information System (INIS)
Baer, James; Chesley, Steven R.
2017-01-01
The conventional least-squares asteroid mass determination algorithm allows us to solve for the mass of a large subject asteroid that is perturbing the trajectory of a smaller test asteroid. However, this algorithm is necessarily a first approximation, ignoring the possibility that the subject asteroid may itself be perturbed by the test asteroid, or that the encounter’s precise geometry may be entangled with encounters involving other asteroids. After reviewing the conventional algorithm, we use it to calculate the masses of 30 main-belt asteroids. Compared to our previous results, we find new mass estimates for eight asteroids (11 Parthenope, 27 Euterpe, 51 Neimausa, 76 Freia, 121 Hermione, 324 Bamberga, 476 Hedwig, and 532 Herculina) and significantly more precise estimates for six others (2 Pallas, 3 Juno, 4 Vesta, 9 Metis, 16 Psyche, and 88 Thisbe). However, we also find that the conventional algorithm yields questionable results in several gravitationally coupled cases. To address such cases, we describe a new algorithm that allows the epoch state vectors of the subject asteroids to be included as solve-for parameters, allowing for the simultaneous solution of the masses and epoch state vectors of multiple subject and test asteroids. We then apply this algorithm to the same 30 main-belt asteroids and conclude that mass determinations resulting from current and future high-precision astrometric sources (such as Gaia ) should conduct a thorough search for possible gravitational couplings and account for their effects.
1. Energy-momentum tensor in theories with scalar fields and two coupling constants. I. Non-Abelian case
International Nuclear Information System (INIS)
Joglekar, S.D.; Misra, A.
1989-01-01
In this paper, we generalize our earlier discussion of renormalization of the energy-momentum tensor in scalar QED to that in non-Abelian gauge theories involving scalar fields. We show the need for adding an improvement term to the conventional energy-momentum tensor. We consider two possible forms for the improvement term: (i) one in which the improvement coefficient is a finite function of bare parameters of the theory (so that the energy-momentum tensor can be derived from an action that is a finite function of bare quantities); (ii) one in which the improvement coefficient is a finite quantity, i.e., a finite function of renormalized parameters. We establish a negative result; viz., neither form leads to a finite energy-momentum tensor to O(e 2 λ/sup n/)
2. De novo determination of internuclear vector orientations from residual dipolar couplings measured in three independent alignment media
International Nuclear Information System (INIS)
Ruan Ke; Briggman, Kathryn B.; Tolman, Joel R.
2008-01-01
The straightforward interpretation of solution state residual dipolar couplings (RDCs) in terms of internuclear vector orientations generally requires prior knowledge of the alignment tensor, which in turn is normally estimated using a structural model. We have developed a protocol which allows the requirement for prior structural knowledge to be dispensed with as long as RDC measurements can be made in three independent alignment media. This approach, called Rigid Structure from Dipolar Couplings (RSDC), allows vector orientations and alignment tensors to be determined de novo from just three independent sets of RDCs. It is shown that complications arising from the existence of multiple solutions can be overcome by careful consideration of alignment tensor magnitudes in addition to the agreement between measured and calculated RDCs. Extensive simulations as well applications to the proteins ubiquitin and Staphylococcal protein GB1 demonstrate that this method can provide robust determinations of alignment tensors and amide N-H bond orientations often with better than 10 o accuracy, even in the presence of modest levels of internal dynamics
3. Tensor spaces and exterior algebra
CERN Document Server
Yokonuma, Takeo
1992-01-01
This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra. Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. To facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. In particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them.
4. Determination of protein global folds using backbone residual dipolar coupling and long-range NOE restraints
International Nuclear Information System (INIS)
Giesen, Alexander W.; Homans, Steve W.; Brown, Jonathan Miles
2003-01-01
We report the determination of the global fold of human ubiquitin using protein backbone NMR residual dipolar coupling and long-range nuclear Overhauser effect (NOE) data as conformational restraints. Specifically, by use of a maximum of three backbone residual dipolar couplings per residue (N i -H N i , N i -C' i-1 , H N i - C' i-1 ) in two tensor frames and only backbone H N -H N NOEs, a global fold of ubiquitin can be derived with a backbone root-mean-square deviation of 1.4 A with respect to the crystal structure. This degree of accuracy is more than adequate for use in databases of structural motifs, and suggests a general approach for the determination of protein global folds using conformational restraints derived only from backbone atoms
5. Old tensor mesons in QCD sum rules
International Nuclear Information System (INIS)
Aliev, T.M.; Shifman, M.A.
1981-01-01
Tensor mesons f, A 2 and A 3 are analyzed within the framework of QCD sum rules. The effects of gluon and quark condensate is accounted for phenomenologically. Accurate estimates of meson masses and coupling constants of the lowest-lying states are obtained. It is shown that the masses are reproduced within theoretical uncertainty of about 80 MeV. The coupling of f meson to the corresponding quark current is determined. The results are in good aqreement with experimental data [ru
6. Tensor surgery and tensor rank
NARCIS (Netherlands)
M. Christandl (Matthias); J. Zuiddam (Jeroen)
2018-01-01
textabstractWe introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new vertices
7. Tensor surgery and tensor rank
NARCIS (Netherlands)
M. Christandl (Matthias); J. Zuiddam (Jeroen)
2016-01-01
textabstractWe introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new
8. Stress-energy tensor of a quark moving through a strongly-coupled N=4 supersymmetric Yang-Mills plasma: Comparing hydrodynamics and AdS/CFT duality
International Nuclear Information System (INIS)
Chesler, Paul M.; Yaffe, Laurence G.
2008-01-01
The stress-energy tensor of a quark moving through a strongly-coupled N=4 supersymmetric Yang-Mills plasma, at large N c , is evaluated using gauge/string duality. The accuracy with which the resulting wake, in position space, is reproduced by hydrodynamics is examined. Remarkable agreement is found between hydrodynamics and the complete result down to distances less than 2/T away from the quark. In performing the gravitational analysis, we use a relatively simple formulation of the bulk to boundary problem in which the linearized Einstein field equations are fully decoupled. Our analysis easily generalizes to other sources in the bulk.
9. Two-perfect fluid interpretation of an energy tensor
International Nuclear Information System (INIS)
Ferrando, J.J.; Morales, J.A.; Portilla, M.
1990-01-01
There are many topics in General Relativity where matter is represented by a mixture of two fluids. In fact, some astrophysical and cosmological situations need to be described by an energy tensor made up of the sum of two or more perfect fluids rather than that with only one. The paper contains the necessary and sufficient conditions for a given energy tensor to be interpreted as a sum of two perfect fluids. Given a tensor of this class, the decomposition in two perfect fluids (which is determined up to a couple of real functions) is obtained
10. Tensor Galileons and gravity
Energy Technology Data Exchange (ETDEWEB)
Chatzistavrakidis, Athanasios [Van Swinderen Institute for Particle Physics and Gravity, University of Groningen,Nijenborgh 4, 9747 AG Groningen (Netherlands); Khoo, Fech Scen [Department of Physics and Earth Sciences, Jacobs University Bremen,Campus Ring 1, 28759 Bremen (Germany); Roest, Diederik [Van Swinderen Institute for Particle Physics and Gravity, University of Groningen,Nijenborgh 4, 9747 AG Groningen (Netherlands); Schupp, Peter [Department of Physics and Earth Sciences, Jacobs University Bremen,Campus Ring 1, 28759 Bremen (Germany)
2017-03-13
The particular structure of Galileon interactions allows for higher-derivative terms while retaining second order field equations for scalar fields and Abelian p-forms. In this work we introduce an index-free formulation of these interactions in terms of two sets of Grassmannian variables. We employ this to construct Galileon interactions for mixed-symmetry tensor fields and coupled systems thereof. We argue that these tensors are the natural generalization of scalars with Galileon symmetry, similar to p-forms and scalars with a shift-symmetry. The simplest case corresponds to linearised gravity with Lovelock invariants, relating the Galileon symmetry to diffeomorphisms. Finally, we examine the coupling of a mixed-symmetry tensor to gravity, and demonstrate in an explicit example that the inclusion of appropriate counterterms retains second order field equations.
11. Measurement of the Higgs boson tensor coupling in $H \\rightarrow ZZ^{*} \\rightarrow 4\\ell$ decays with the ATLAS detector - How odd is the Higgs boson?
CERN Document Server
Ecker, Katharina Maria; Kortner, Sandra
The tensor structure of the Higgs boson couplings to gluons and heavy weak gauge bosons has been probed for small admixtures of non-Standard Model CP-odd and, only for heavy vector bosons, CP-even couplings to the CP-even Standard Model coupling. The Higgs boson candidates are reconstructed in the $\\HZZllll$ $(\\ell\\equiv e,\\mu)$ decay channel using proton-proton collision data recorded by the ATLAS detector at the Large Hadron Collider (LHC) in 2011 and 2012 at centre-of-mass energies of $\\sqrt{s}=7$ and $8\\,\\tev$ corresponding to an integrated luminosity of $\\intlumisetot\\,\\ifb$ and in 2015 and 2016 at $\\ecms$ corresponding to $\\intlumi\\,\\ifb$.\\\\ The non-Standard Model coupling parameters are defined within an effective field theory, the so-called Higgs characterisation framework. The relative contributions of the CP-even and CP-odd terms are described by the CP mixing angle $\\alpha$. The parameter $\\kaggnoma$ denotes the CP-odd non-Standard Model coupling at the Higgs to gluon interaction vertex and \\khvv... 12. Diffusion tensor image registration using hybrid connectivity and tensor features. Science.gov (United States) Wang, Qian; Yap, Pew-Thian; Wu, Guorong; Shen, Dinggang 2014-07-01 Most existing diffusion tensor imaging (DTI) registration methods estimate structural correspondences based on voxelwise matching of tensors. The rich connectivity information that is given by DTI, however, is often neglected. In this article, we propose to integrate complementary information given by connectivity features and tensor features for improved registration accuracy. To utilize connectivity information, we place multiple anchors representing different brain anatomies in the image space, and define the connectivity features for each voxel as the geodesic distances from all anchors to the voxel under consideration. The geodesic distance, which is computed in relation to the tensor field, encapsulates information of brain connectivity. We also extract tensor features for every voxel to reflect the local statistics of tensors in its neighborhood. We then combine both connectivity features and tensor features for registration of tensor images. From the images, landmarks are selected automatically and their correspondences are determined based on their connectivity and tensor feature vectors. The deformation field that deforms one tensor image to the other is iteratively estimated and optimized according to the landmarks and their associated correspondences. Experimental results show that, by using connectivity features and tensor features simultaneously, registration accuracy is increased substantially compared with the cases using either type of features alone. Copyright © 2013 Wiley Periodicals, Inc. 13. Single-shot full strain tensor determination with microbeam X-ray Laue diffraction and a two-dimensional energy-dispersive detector. Science.gov (United States) Abboud, A; Kirchlechner, C; Keckes, J; Conka Nurdan, T; Send, S; Micha, J S; Ulrich, O; Hartmann, R; Strüder, L; Pietsch, U 2017-06-01 The full strain and stress tensor determination in a triaxially stressed single crystal using X-ray diffraction requires a series of lattice spacing measurements at different crystal orientations. This can be achieved using a tunable X-ray source. This article reports on a novel experimental procedure for single-shot full strain tensor determination using polychromatic synchrotron radiation with an energy range from 5 to 23 keV. Microbeam X-ray Laue diffraction patterns were collected from a copper micro-bending beam along the central axis (centroid of the cross section). Taking advantage of a two-dimensional energy-dispersive X-ray detector (pnCCD), the position and energy of the collected Laue spots were measured for multiple positions on the sample, allowing the measurement of variations in the local microstructure. At the same time, both the deviatoric and hydrostatic components of the elastic strain and stress tensors were calculated. 14. The geomagnetic field gradient tensor DEFF Research Database (Denmark) Kotsiaros, Stavros; Olsen, Nils 2012-01-01 We develop the general mathematical basis for space magnetic gradiometry in spherical coordinates. The magnetic gradient tensor is a second rank tensor consisting of 3 × 3 = 9 spatial derivatives. Since the geomagnetic field vector B is always solenoidal (∇ · B = 0) there are only eight independent...... tensor elements. Furthermore, in current free regions the magnetic gradient tensor becomes symmetric, further reducing the number of independent elements to five. In that case B is a Laplacian potential field and the gradient tensor can be expressed in series of spherical harmonics. We present properties...... of the magnetic gradient tensor and provide explicit expressions of its elements in terms of spherical harmonics. Finally we discuss the benefit of using gradient measurements for exploring the Earth’s magnetic field from space, in particular the advantage of the various tensor elements for a better determination... 15. Multi-mode technique for the determination of the biaxial Y{sub 2}SiO{sub 5} permittivity tensor from 300 to 6 K Energy Technology Data Exchange (ETDEWEB) Carvalho, N. C., E-mail: [email protected]; Le Floch, J-M.; Tobar, M. E. [School of Physics, The University of Western Australia, Crawley 6009 (Australia); ARC Centre of Excellence for Engineered Quantum Systems (EQuS), 35 Stirling Hwy, Crawley 6009 (Australia); Krupka, J. [Instytut Mikroelektroniki i Optoelektroniki PW, Koszykowa 75, 00-662 Warsaw (Poland) 2015-05-11 The Y{sub 2}SiO{sub 5} (YSO) crystal is a dielectric material with biaxial anisotropy with known values of refractive index at optical frequencies. It is a well-known rare-earth (RE) host material for optical research and more recently has shown promising performance for quantum-engineered devices. In this paper, we report the first microwave characterization of the real permittivity tensor of a bulk YSO sample, as well as an investigation of the temperature dependence of the tensor components from 296 K down to 6 K. Estimated uncertainties were below 0.26%, limited by the precision of machining the cylindrical dielectric. Also, the electrical Q-factors of a few electromagnetic modes were recorded as a way to provide some information about the crystal losses over the temperature range. To solve the tensor components necessary for a biaxial crystal, we developed the multi-mode technique, which uses simultaneous measurement of low order whispering gallery modes. Knowledge of the permittivity tensor offers important data, essential for the design of technologies involving YSO, such as microwave coupling to electron and hyperfine transitions in RE doped samples at low temperatures. 16. Determinants In HIV Counselling And Testing In Couples In North ... African Journals Online (AJOL) Determinants In HIV Counselling And Testing In Couples In North Rift Kenya. PO Ayuo, E Were, K Wools-Kaloustian, J Baliddawa, J Sidle, K Fife. Abstract. Background: Voluntary HIV counselling and testing (VCT) has been shown to be an acceptable and effective tool in the fight against HIV/AIDS. Couple HIV Counselling ... 17. Determination of 13C CSA Tensors: Extension of the Model-independent Approach to an RNA Kissing Complex Undergoing Anisotropic Rotational Diffusion in Solution International Nuclear Information System (INIS) Ravindranathan, Sapna; Kim, Chul-Hyun; Bodenhausen, Geoffrey 2005-01-01 Chemical shift anisotropy (CSA) tensor parameters have been determined for the protonated carbons of the purine bases in an RNA kissing complex in solution by extending the model-independent approach [Fushman, D., Cowburn, D. (1998) J. Am. Chem. Soc. 120, 7109-7110]. A strategy for determining CSA tensor parameters of heteronuclei in isolated X-H two-spin systems (X = 13 C or 15 N) in molecules undergoing anisotropic rotational diffusion is presented. The original method relies on the fact that the ratio κ 2 =R 2 auto /R 2 cross of the transverse auto- and cross-correlated relaxation rates involving the X CSA and the X-H dipolar interaction is independent of parameters related to molecular motion, provided rotational diffusion is isotropic. However, if the overall motion is anisotropic κ 2 depends on the anisotropy D parallel /D -perpendicular of rotational diffusion. In this paper, the field dependence of both κ 2 and its longitudinal counterpart κ 1 =R 1 auto /R 1 cross are determined. For anisotropic rotational diffusion, our calculations show that the average κ av = 1/2 (κ 1 +κ 2 ), of the ratios is largely independent of the anisotropy parameter D parallel /D -perpendicular . The field dependence of the average ratio κ av may thus be utilized to determine CSA tensor parameters by a generalized model-independent approach in the case of molecules with an overall motion described by an axially symmetric rotational diffusion tensor 18. Determination of the Rotational Diffusion Tensor of Macromolecules in Solution from NMR Relaxation Data with a Combination of Exact and Approximate Methods—Application to the Determination of Interdomain Orientation in Multidomain Proteins Science.gov (United States) Ghose, Ranajeet; Fushman, David; Cowburn, David 2001-04-01 In this paper we present a method for determining the rotational diffusion tensor from NMR relaxation data using a combination of approximate and exact methods. The approximate method, which is computationally less intensive, computes values of the principal components of the diffusion tensor and estimates the Euler angles, which relate the principal axis frame of the diffusion tensor to the molecular frame. The approximate values of the principal components are then used as starting points for an exact calculation by a downhill simplex search for the principal components of the tensor over a grid of the space of Euler angles relating the diffusion tensor frame to the molecular frame. The search space of Euler angles is restricted using the tensor orientations calculated using the approximate method. The utility of this approach is demonstrated using both simulated and experimental relaxation data. A quality factor that determines the extent of the agreement between the measured and predicted relaxation data is provided. This approach is then used to estimate the relative orientation of SH3 and SH2 domains in the SH(32) dual-domain construct of Abelson kinase complexed with a consolidated ligand. 19. Determination of the rotational diffusion tensor of macromolecules in solution from nmr relaxation data with a combination of exact and approximate methods--application to the determination of interdomain orientation in multidomain proteins. Science.gov (United States) Ghose, R; Fushman, D; Cowburn, D 2001-04-01 In this paper we present a method for determining the rotational diffusion tensor from NMR relaxation data using a combination of approximate and exact methods. The approximate method, which is computationally less intensive, computes values of the principal components of the diffusion tensor and estimates the Euler angles, which relate the principal axis frame of the diffusion tensor to the molecular frame. The approximate values of the principal components are then used as starting points for an exact calculation by a downhill simplex search for the principal components of the tensor over a grid of the space of Euler angles relating the diffusion tensor frame to the molecular frame. The search space of Euler angles is restricted using the tensor orientations calculated using the approximate method. The utility of this approach is demonstrated using both simulated and experimental relaxation data. A quality factor that determines the extent of the agreement between the measured and predicted relaxation data is provided. This approach is then used to estimate the relative orientation of SH3 and SH2 domains in the SH(32) dual-domain construct of Abelson kinase complexed with a consolidated ligand. Copyright 2001 Academic Press. 20. Determination of the π3He3H coupling constant International Nuclear Information System (INIS) Nichitiu, F.; Sapozhnikov, M.G. 1977-01-01 Despersion relations for the real part of the antisymmetric amplitude of the π +-3 He scattering have been used in order to determine the π 3 He 3 H coupling constant. The coupling constant value determined by this method is larger than the elementary pion-nucleon coupling constant, but is in good agreement with the value obtained by another method. The obtained value is f 2 sub(π 3 He 3 H) = 0.12+-0.01. Shown is the importance of using the Coulomb corrections for dispersion relation calculations because the value of π 3 He 3 H coupling constant obtained with corrected total cross sections is larger by about 0.014 than the one obtained without these corrections. The best energy ranges for future π 3 He experiments are commented 1. Determination of BEACON Coupling Coefficients using data from Xenon transient International Nuclear Information System (INIS) Bozic, M.; Kurincic, B. 2007-01-01 NEK uses BEACO TM code (BEACO TM - Westinghouse Best Estimate Analyzer for Core Operating Nuclear) for core monitoring, analysis and core behaviour prediction. Coupling Coefficients determine relationship between core response and excore instrumentation. Measured power distribution using incore moveable detectors during Xenon transient with sufficient power axial offset change is the most important data for further analysis. Classic methodology and BEACO TM Conservative methodology using established Coupling Coefficients are compared on NPP Krsko case. BEACON TM Conservative methodology with predefined Coupling Coefficients is used as a surveillance tool for verification of relationship between core and excore instrumentation during power operation. (author) 2. Determining γ with B decays into a scalar/tensor meson International Nuclear Information System (INIS) Wang, Wei 2011-10-01 We propose a new way for determining the CP violation angle γ. The suggested method is to use the two triangles formed by the decay amplitudes of B ± →(D 0 , anti D 0 ,D 0 CP )K *± 0(2) (1430). The advantages are that large CP asymmetries are expected in these processes and only singly Cabibbo-suppressed D decay modes are involved. Measurements of the branching fractions of the neutral B d decays into DK * 0(2) (1430) and the time-dependent CP asymmetries in B s →(D 0 , anti D 0 )M (M=f 0 (980),f 0 (1370),f 2 ' (1525),f 1 (1285),f 1 (1420),h 1 (1180)) provide an alternative way to extract the angle γ, which will increase the statistical significance. No knowledge of the resonance structure in this method is required and therefore the angle γ can be extracted without any hadronic uncertainty. (orig.) 3. Determining {gamma} with B decays into a scalar/tensor meson Energy Technology Data Exchange (ETDEWEB) Wang, Wei 2011-10-15 We propose a new way for determining the CP violation angle {gamma}. The suggested method is to use the two triangles formed by the decay amplitudes of B{sup {+-}}{yields}(D{sup 0}, anti D{sup 0},D{sup 0}{sub CP})K{sup *{+-}}{sub 0(2)}(1430). The advantages are that large CP asymmetries are expected in these processes and only singly Cabibbo-suppressed D decay modes are involved. Measurements of the branching fractions of the neutral B{sub d} decays into DK{sup *}{sub 0(2)}(1430) and the time-dependent CP asymmetries in B{sub s}{yields}(D{sup 0}, anti D{sup 0})M (M=f{sub 0}(980),f{sub 0}(1370),f{sub 2}{sup '}(1525),f{sub 1}(1285),f{sub 1}(1420),h{sub 1}(1180)) provide an alternative way to extract the angle {gamma}, which will increase the statistical significance. No knowledge of the resonance structure in this method is required and therefore the angle {gamma} can be extracted without any hadronic uncertainty. (orig.) 4. Numerical estimates of the maximum sustainable pore pressure in anticline formations using the tensor based concept of pore pressure-stress coupling Directory of Open Access Journals (Sweden) Andreas Eckert 2015-02-01 Full Text Available The advanced tensor based concept of pore pressure-stress coupling is used to provide pre-injection analytical estimates of the maximum sustainable pore pressure change, ΔPc, for fluid injection scenarios into generic anticline geometries. The heterogeneous stress distribution for different prevailing stress regimes in combination with the Young's modulus (E contrast between the injection layer and the cap rock and the interbedding friction coefficient, μ, may result in large spatial and directional differences of ΔPc. A single value characterizing the cap rock as for horizontal layered injection scenarios is not obtained. It is observed that a higher Young's modulus in the cap rock and/or a weak mechanical coupling between layers amplifies the maximum and minimum ΔPc values in the valley and limb, respectively. These differences in ΔPc imposed by E and μ are further amplified by different stress regimes. The more compressional the stress regime is, the larger the differences between the maximum and minimum ΔPc values become. The results of this study show that, in general compressional stress regimes yield the largest magnitudes of ΔPc and extensional stress regimes provide the lowest values of ΔPc for anticline formations. Yet this conclusion has to be considered with care when folded anticline layers are characterized by flexural slip and the friction coefficient between layers is low, i.e. μ = 0.1. For such cases of weak mechanical coupling, ΔPc magnitudes may range from 0 MPa to 27 MPa, indicating imminent risk of fault reactivation in the cap rock. 5. Beyond Low Rank: A Data-Adaptive Tensor Completion Method OpenAIRE Zhang, Lei; Wei, Wei; Shi, Qinfeng; Shen, Chunhua; Hengel, Anton van den; Zhang, Yanning 2017-01-01 Low rank tensor representation underpins much of recent progress in tensor completion. In real applications, however, this approach is confronted with two challenging problems, namely (1) tensor rank determination; (2) handling real tensor data which only approximately fulfils the low-rank requirement. To address these two issues, we develop a data-adaptive tensor completion model which explicitly represents both the low-rank and non-low-rank structures in a latent tensor. Representing the no... 6. Energy-momentum tensor in quantum field theory International Nuclear Information System (INIS) Fujikawa, K. 1981-01-01 The definition of the energy-momentum tensor as a source current coupled to the background gravitational field receives an important modification in quantum theory. In the path-integral approach, the manifest covariance of the integral measure under general coordinate transformations dictates that field variables with weight 1/2 should be used as independent integration variables. An improved energy-momentum tensor is then generated by the variational derivative, and it gives rise to well-defined gravitational conformal (Weyl) anomalies. In the flat--space-time limit, all the Ward-Takahashi identities associated with space-time transformations including the global dilatation become free from anomalies in terms of this energy-momentum tensor, reflecting the general covariance of the integral measure; the trace of this tensor is thus finite at zero momentum transfer for renormalizable theories. The Jacobian for the local conformal transformation, however, becomes nontrivial, and it gives rise to an anomaly for the conformal identity. All the familiar anomalies are thus reduced to either chiral or conformal anomalies. The consistency of the dilatation and conformal identities at vanishing momentum transfer determines the trace anomaly of this energy-momentum tensor in terms of the renormalization-group b function and other parameters. In contrast, the trace of the conventional energy-momentum tensor generally diverges even at vanishing momentum transfer depending on the regularization scheme, and it is subtractively renormalized. We also explain how the apparently different renormalization properties of the chiral and trace anomalies arise 7. Weyl tensors for asymmetric complex curvatures International Nuclear Information System (INIS) Oliveira, C.G. Considering a second rank Hermitian field tensor and a general Hermitian connection the associated complex curvature tensor is constructed. The Weyl tensor that corresponds to this complex curvature is determined. The formalism is applied to the Weyl unitary field theory and to the Moffat gravitational theory. (Author) [pt 8. Model-independent determination of hadronic neutral-current couplings International Nuclear Information System (INIS) Claudson, M.; Paschos, E.A.; Strait, J.; Sulak, L.R. 1979-01-01 Completion of a second generation of experiments on neutrino-induced neutral-current reactions allows a more discriminating study of neutral-current couplings to hadrons. To minimize the sensitivity to model-dependent analyses of inclusive and exclusive pion data, we base our work on measurements of deep-inelastic and elastic reactions alone. Within the regions allowed by the deep-inelastic data for scattering on isoscalar targets, the coupling constants are fit to the q 2 dependence of the neutrino-proton elastic scattering data. This procedure initially yields two solutions for the couplings. One of these, at theta/sub L/ = 55 0 and theta/sub R/ = 205 0 , is predominantly isoscalar and therefore is ruled out by only qualitative consideration of exclusive pion data. The other solution at theta/sub D/ = 140 0 and and theta/sub R/ = 330 0 , is thus a unique determination of the hadronic neutral-current couplings. It coincides with solution A obtained in earlier work, and is insensitive to variations of M/sub A/ within 2 standard deviations of the world average. When constrained to the coupling constants required by the Weinberg-Salam model, the fit agrees with the data to within 1 standard deviation 9. Determination of the pion-nucleon coupling constant International Nuclear Information System (INIS) Samaranayake, V.K. 1977-06-01 Forward dispersion relations are used to determine the pion-nucleon coupling constant and S-wave scattering lengths using a least squares fit with additional parameters introduced to take account of the uncertainties in the calculation of dispersion integrals. The values obtained are: f 2 = (78.0+- 2.1).10 -3 , a 1 -a 3 = (272.4+- 12.3).10 -3 , a 1 +2a 3 = (15.1+-10.4).10 -3 10. The energy–momentum tensor(s in classical gauge theories Directory of Open Access Journals (Sweden) Daniel N. Blaschke 2016-11-01 Full Text Available We give an introduction to, and review of, the energy–momentum tensors in classical gauge field theories in Minkowski space, and to some extent also in curved space–time. For the canonical energy–momentum tensor of non-Abelian gauge fields and of matter fields coupled to such fields, we present a new and simple improvement procedure based on gauge invariance for constructing a gauge invariant, symmetric energy–momentum tensor. The relationship with the Einstein–Hilbert tensor following from the coupling to a gravitational field is also discussed. 11. Methods for the Determination of Currents and Fields in Steady Two-Dimensional MHD Flow With Tensor Conductivity International Nuclear Information System (INIS) Witalis, E.A. 1965-12-01 Rigorous derivations are given of the basic equations and methods available for the analysis of transverse MHD flow when Hall currents are not suppressed. The gas flow is taken to be incompressible and viscous with uniform tensor conductivity and arbitrary magnetic Reynold's number. The magnetic field is perpendicular to the flow and has variable strength. Analytical solutions can be obtained either in terms of the induced magnetic field or from two types of electric potential. The relevant set of suitable simplifications, restrictive conditions and boundary value considerations for each method is given 12. Methods for the Determination of Currents and Fields in Steady Two-Dimensional MHD Flow With Tensor Conductivity Energy Technology Data Exchange (ETDEWEB) Witalis, E A 1965-12-15 Rigorous derivations are given of the basic equations and methods available for the analysis of transverse MHD flow when Hall currents are not suppressed. The gas flow is taken to be incompressible and viscous with uniform tensor conductivity and arbitrary magnetic Reynold's number. The magnetic field is perpendicular to the flow and has variable strength. Analytical solutions can be obtained either in terms of the induced magnetic field or from two types of electric potential. The relevant set of suitable simplifications, restrictive conditions and boundary value considerations for each method is given. 13. Tensor Transpose and Its Properties OpenAIRE Pan, Ran 2014-01-01 Tensor transpose is a higher order generalization of matrix transpose. In this paper, we use permutations and symmetry group to define? the tensor transpose. Then we discuss the classification and composition of tensor transposes. Properties of tensor transpose are studied in relation to tensor multiplication, tensor eigenvalues, tensor decompositions and tensor rank. 14. Scalar-tensor linear inflation Energy Technology Data Exchange (ETDEWEB) Artymowski, Michał [Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków (Poland); Racioppi, Antonio, E-mail: [email protected], E-mail: [email protected] [National Institute of Chemical Physics and Biophysics, Rävala 10, 10143 Tallinn (Estonia) 2017-04-01 We investigate two approaches to non-minimally coupled gravity theories which present linear inflation as attractor solution: a) the scalar-tensor theory approach, where we look for a scalar-tensor theory that would restore results of linear inflation in the strong coupling limit for a non-minimal coupling to gravity of the form of f (φ) R /2; b) the particle physics approach, where we motivate the form of the Jordan frame potential by loop corrections to the inflaton field. In both cases the Jordan frame potentials are modifications of the induced gravity inflationary scenario, but instead of the Starobinsky attractor they lead to linear inflation in the strong coupling limit. 15. Energy-momentum tensor in quantum field theory International Nuclear Information System (INIS) Fujikawa, Kazuo. 1980-12-01 The definition of the energy-momentum tensor as a source current coupled to the background gravitational field receives an important modification in quantum theory. In the path integral approach, the manifest covariance of the integral measure under general coordinate transformations dictates that field variables with weight 1/2 should be used as independent integration variables. An improved energy-momentum tensor is then generated by the variational derivative, and it gives rise to well-defined gravitational conformal (Weyl) anomalies. In the flat space-time limit, all the Ward-Takahashi identities associate with space-time transformations including the global dilatation become free from anomalies, reflecting the general covariance of the integral measure; the trace of this energy-momentum tensor is thus finite at the zero momentum transfer. The Jacobian for the local conformal transformation however becomes non-trivial, and it gives rise to an anomaly for the conformal identity. All the familiar anomalies are thus reduced to either chiral or conformal anomalies. The consistency of the dilatation and conformal identities at the vanishing momentum transfer determines the trace anomaly of this energy-momentum tensor in terms of the renormalization group β-function and other parameters. In contrast, the trace of the conventional energy-momentum tensor generally diverges even at the vanishing momentum transfer depending on the regularization scheme, and it is subtractively renormalized. We also explain how the apparently different renormalization properties of the chiral and trace anomalies arise. (author) 16. (Ln-bar, g)-spaces. Special tensor fields International Nuclear Information System (INIS) Manoff, S.; Dimitrov, B. 1998-01-01 The Kronecker tensor field, the contraction tensor field, as well as the multi-Kronecker and multi-contraction tensor fields are determined and the action of the covariant differential operator, the Lie differential operator, the curvature operator, and the deviation operator on these tensor fields is established. The commutation relations between the operators Sym and Asym and the covariant and Lie differential operators are considered acting on symmetric and antisymmetric tensor fields over (L n bar, g)-spaces 17. Confinement through tensor gauge fields International Nuclear Information System (INIS) Salam, A.; Strathdee, J. 1977-12-01 Using the 0(3,2)-symmetric de Sitter solution of Einstein's equation describing a strongly interacting tensor field it is shown that hadronic bags confining quarks can be represented as de Sitter ''micro-universes'' with radii given 1/R 2 =lambdak 2 /6. Here k 2 and lambda are the strong coupling and the ''cosmological'' constant which apear in the Einstein equation used. Surprisingly the energy spectrum for the two-body hadronic states is the same as that for a harmonic oscillator potential, though the wave functions are completely different. The Einstein equation can be extended to include colour for the tensor fields 18. Tensor product of quantum logics Science.gov (United States) Pulmannová, Sylvia 1985-01-01 A quantum logic is the couple (L,M) where L is an orthomodular σ-lattice and M is a strong set of states on L. The Jauch-Piron property in the σ-form is also supposed for any state of M. A tensor product'' of quantum logics is defined. This definition is compared with the definition of a free orthodistributive product of orthomodular σ-lattices. The existence and uniqueness of the tensor product in special cases of Hilbert space quantum logics and one quantum and one classical logic are studied. 19. Tensors for physics CERN Document Server Hess, Siegfried 2015-01-01 This book presents the science of tensors in a didactic way. The various types and ranks of tensors and the physical basis is presented. Cartesian Tensors are needed for the description of directional phenomena in many branches of physics and for the characterization the anisotropy of material properties. The first sections of the book provide an introduction to the vector and tensor algebra and analysis, with applications to physics, at undergraduate level. Second rank tensors, in particular their symmetries, are discussed in detail. Differentiation and integration of fields, including generalizations of the Stokes law and the Gauss theorem, are treated. The physics relevant for the applications in mechanics, quantum mechanics, electrodynamics and hydrodynamics is presented. The second part of the book is devoted to tensors of any rank, at graduate level. Special topics are irreducible, i.e. symmetric traceless tensors, isotropic tensors, multipole potential tensors, spin tensors, integration and spin-... 20. Random tensors CERN Document Server Gurau, Razvan 2017-01-01 Written by the creator of the modern theory of random tensors, this book is the first self-contained introductory text to this rapidly developing theory. Starting from notions familiar to the average researcher or PhD student in mathematical or theoretical physics, the book presents in detail the theory and its applications to physics. The recent detections of the Higgs boson at the LHC and gravitational waves at LIGO mark new milestones in Physics confirming long standing predictions of Quantum Field Theory and General Relativity. These two experimental results only reinforce today the need to find an underlying common framework of the two: the elusive theory of Quantum Gravity. Over the past thirty years, several alternatives have been proposed as theories of Quantum Gravity, chief among them String Theory. While these theories are yet to be tested experimentally, key lessons have already been learned. Whatever the theory of Quantum Gravity may be, it must incorporate random geometry in one form or another.... 1. Tensor rank is not multiplicative under the tensor product DEFF Research Database (Denmark) Christandl, Matthias; Jensen, Asger Kjærulff; Zuiddam, Jeroen 2018-01-01 The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an ℓ-tensor. The tensor product of s and t is a (k+ℓ)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection b... 2. Tensor rank is not multiplicative under the tensor product NARCIS (Netherlands) M. Christandl (Matthias); A. K. Jensen (Asger Kjærulff); J. Zuiddam (Jeroen) 2018-01-01 textabstractThe tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an ℓ-tensor. The tensor product of s and t is a (k+ℓ)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the 3. Tensor rank is not multiplicative under the tensor product NARCIS (Netherlands) M. Christandl (Matthias); A. K. Jensen (Asger Kjærulff); J. Zuiddam (Jeroen) 2017-01-01 textabstractThe tensor rank of a tensor is the smallest number r such that the tensor can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor (not to be confused with the "tensor Kronecker product" used in 4. A new approach for applying residual dipolar couplings as restraints in structure elucidation International Nuclear Information System (INIS) Meiler, Jens; Blomberg, Niklas; Nilges, Michael; Griesinger, Christian 2000-01-01 Residual dipolar couplings are useful global structural restraints. The dipolar couplings define the orientation of a vector with respect to the alignment tensor. Although the size of the alignment tensor can be derived from the distribution of the experimental dipolar couplings, its orientation with respect to the coordinate system of the molecule is unknown at the beginning of structure determination. This causes convergence problems in the simulated annealing process. We therefore propose a protocol that translates dipolar couplings into intervector projection angles, which are independent of the orientation of the alignment tensor with respect to the molecule. These restraints can be used during the whole simulated annealing protocol 5. Applications of tensor functions in creep mechanics International Nuclear Information System (INIS) Betten, J. 1991-01-01 Within this contribution a short survey is given of some recent advances in the mathematical modelling of materials behaviour under creep conditions. The mechanical behaviour of anisotropic solids requires a suitable mathematical modelling. The properties of tensor functions with several argument tensors constitute a rational basis for a consistent mathematical modelling of complex material behaviour. This paper presents certain principles, methods, and recent successfull applications of tensor functions in solid mechanics. The rules for specifying irreducible sets of tensor invariants and tensor generators for material tensors of rank two and four are also discussed. Furthermore, it is very important that the scalar coefficients in constitutive and evolutional equations are determined as functions of the integrity basis and experimental data. It is explained in detail that these coefficients can be determined by using tensorial interpolation methods. Some examples for practical use are discussed. (orig./RHM) 6. The tensor distribution function. Science.gov (United States) Leow, A D; Zhu, S; Zhan, L; McMahon, K; de Zubicaray, G I; Meredith, M; Wright, M J; Toga, A W; Thompson, P M 2009-01-01 Diffusion weighted magnetic resonance imaging is a powerful tool that can be employed to study white matter microstructure by examining the 3D displacement profile of water molecules in brain tissue. By applying diffusion-sensitized gradients along a minimum of six directions, second-order tensors (represented by three-by-three positive definite matrices) can be computed to model dominant diffusion processes. However, conventional DTI is not sufficient to resolve more complicated white matter configurations, e.g., crossing fiber tracts. Recently, a number of high-angular resolution schemes with more than six gradient directions have been employed to address this issue. In this article, we introduce the tensor distribution function (TDF), a probability function defined on the space of symmetric positive definite matrices. Using the calculus of variations, we solve the TDF that optimally describes the observed data. Here, fiber crossing is modeled as an ensemble of Gaussian diffusion processes with weights specified by the TDF. Once this optimal TDF is determined, the orientation distribution function (ODF) can easily be computed by analytic integration of the resulting displacement probability function. Moreover, a tensor orientation distribution function (TOD) may also be derived from the TDF, allowing for the estimation of principal fiber directions and their corresponding eigenvalues. 7. The Riemann-Lovelock Curvature Tensor OpenAIRE Kastor, David 2012-01-01 In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann-Lovelock tensor as a certain quantity having a total 4k-indices, which is kth-order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth-order Riemann-Lovelock tensor is determined by its traces in dimensions 2k \\le D 8. (Ln-bar, g)-spaces. Ordinary and tensor differentials International Nuclear Information System (INIS) Manoff, S.; Dimitrov, B. 1998-01-01 Different types of differentials as special cases of differential operators acting on tensor fields over (L n bar, g)-spaces are considered. The ordinary differential, the covariant differential as a special case of the covariant differential operator, and the Lie differential as a special case of the Lie differential operator are investigated. The tensor differential and its special types (Covariant tensor differential, and Lie tensor differential) are determined and their properties are discussed. Covariant symmetric and antisymmetric (external) tensor differentials, Lie symmetric, and Lie antisymmetric (external) tensor differentials are determined and considered over (L n bar, g)-spaces 9. Measurement of thepp\\to H\\to ZZ^* \\to 4 \\ell$Production and$HZZ$Tensor Coupling with the ATLAS Detector at 13 TeV Centre-of-Mass Energy CERN Document Server Walbrecht, Verena Maria; Kortner, Sandra In this master thesis the measurement of the Higgs boson production in the$H\\to~ZZ^*~\\to 4~\\ell$decay channel ($\\ell=e,\\mu$) is performed together with the measurement of the tensor structure of the Higgs boson couplings to$Z$bosons. The results are based on the Run~II dataset of LHC's proton-proton collisions at a centre-of-mass energy of 13~TeV, with the ATLAS detector and corresponding to a total integrated luminosity of$14.78$~fb$^{-1}$. Special emphasis is given to the estimation of the reducible background contribution. Based on the signal and background estimations, there are$32.0\\pm3.2$Higgs boson candidates expected after the final event selection, while$44$candidates are observed. The difference is compatible at the level of about$2$standard derivations with the Standard Model predictions. All selected candidates are used in the study of the tensor structure of the$HZZ$coupling between the Higgs boson and the two$Zbosons. For this study a dedicated signal model is introduced to desc... 10. Tensor rank is not multiplicative under the tensor product OpenAIRE Christandl, Matthias; Jensen, Asger Kjærulff; Zuiddam, Jeroen 2017-01-01 The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. Specif... 11. Tensor gauge condition and tensor field decomposition Science.gov (United States) Zhu, Ben-Chao; Chen, Xiang-Song 2015-10-01 We discuss various proposals of separating a tensor field into pure-gauge and gauge-invariant components. Such tensor field decomposition is intimately related to the effort of identifying the real gravitational degrees of freedom out of the metric tensor in Einstein’s general relativity. We show that as for a vector field, the tensor field decomposition has exact correspondence to and can be derived from the gauge-fixing approach. The complication for the tensor field, however, is that there are infinitely many complete gauge conditions in contrast to the uniqueness of Coulomb gauge for a vector field. The cause of such complication, as we reveal, is the emergence of a peculiar gauge-invariant pure-gauge construction for any gauge field of spin ≥ 2. We make an extensive exploration of the complete tensor gauge conditions and their corresponding tensor field decompositions, regarding mathematical structures, equations of motion for the fields and nonlinear properties. Apparently, no single choice is superior in all aspects, due to an awkward fact that no gauge-fixing can reduce a tensor field to be purely dynamical (i.e. transverse and traceless), as can the Coulomb gauge in a vector case. 12. Tensor structure for Nori motives OpenAIRE Barbieri-Viale, Luca; Huber, Annette; Prest, Mike 2018-01-01 We construct a tensor product on Freyd's universal abelian category attached to an additive tensor category or a tensor quiver and establish a universal property. This is used to give an alternative construction for the tensor product on Nori motives. 13. Radiative corrections in a vector-tensor model International Nuclear Information System (INIS) Chishtie, F.; Gagne-Portelance, M.; Hanif, T.; Homayouni, S.; McKeon, D.G.C. 2006-01-01 In a recently proposed model in which a vector non-Abelian gauge field interacts with an antisymmetric tensor field, it has been shown that the tensor field possesses no physical degrees of freedom. This formal demonstration is tested by computing the one-loop contributions of the tensor field to the self-energy of the vector field. It is shown that despite the large number of Feynman diagrams in which the tensor field contributes, the sum of these diagrams vanishes, confirming that it is not physical. Furthermore, if the tensor field were to couple with a spinor field, it is shown at one-loop order that the spinor self-energy is not renormalizable, and hence this coupling must be excluded. In principle though, this tensor field does couple to the gravitational field 14. Tensor eigenvalues and their applications CERN Document Server Qi, Liqun; Chen, Yannan 2018-01-01 This book offers an introduction to applications prompted by tensor analysis, especially by the spectral tensor theory developed in recent years. It covers applications of tensor eigenvalues in multilinear systems, exponential data fitting, tensor complementarity problems, and tensor eigenvalue complementarity problems. It also addresses higher-order diffusion tensor imaging, third-order symmetric and traceless tensors in liquid crystals, piezoelectric tensors, strong ellipticity for elasticity tensors, and higher-order tensors in quantum physics. This book is a valuable reference resource for researchers and graduate students who are interested in applications of tensor eigenvalues. 15. Harmonic d-tensors Energy Technology Data Exchange (ETDEWEB) Hohmann, Manuel [Physikalisches Institut, Universitaet Tartu (Estonia) 2016-07-01 Tensor harmonics are a useful mathematical tool for finding solutions to differential equations which transform under a particular representation of the rotation group SO(3). In order to make use of this tool also in the setting of Finsler geometry, where the objects of relevance are d-tensors instead of tensors, we construct a set of d-tensor harmonics for both SO(3) and SO(4) symmetries and show how these can be used for calculations in Finsler geometry and gravity. 16. Abelian gauge theories with tensor gauge fields International Nuclear Information System (INIS) Kapuscik, E. 1984-01-01 Gauge fields of arbitrary tensor type are introduced. In curved space-time the gravitational field serves as a bridge joining different gauge fields. The theory of second order tensor gauge field is developed on the basis of close analogy to Maxwell electrodynamics. The notion of tensor current is introduced and an experimental test of its detection is proposed. The main result consists in a coupled set of field equations representing a generalization of Maxwell theory in which the Einstein equivalence principle is not satisfied. (author) 17. Monograph On Tensor Notations Science.gov (United States) Sirlin, Samuel W. 1993-01-01 Eight-page report describes systems of notation used most commonly to represent tensors of various ranks, with emphasis on tensors in Cartesian coordinate systems. Serves as introductory or refresher text for scientists, engineers, and others familiar with basic concepts of coordinate systems, vectors, and partial derivatives. Indicial tensor, vector, dyadic, and matrix notations, and relationships among them described. 18. Nonlocal elasticity tensors in dislocation and disclination cores International Nuclear Information System (INIS) Taupin, V.; Gbemou, K.; Fressengeas, C.; Capolungo, L. 2017-01-01 We introduced nonlocal elastic constitutive laws for crystals containing defects such as dislocations and disclinations. Additionally, the pointwise elastic moduli tensors adequately reflect the elastic response of defect-free regions by relating stresses to strains and couple-stresses to curvatures, elastic cross-moduli tensors relating strains to couple-stresses and curvatures to stresses within convolution integrals are derived from a nonlocal analysis of strains and curvatures in the defects cores. Sufficient conditions are derived for positive-definiteness of the resulting free energy, and stability of elastic solutions is ensured. The elastic stress/couple stress fields associated with prescribed dislocation/disclination density distributions and solving the momentum and moment of momentum balance equations in periodic media are determined by using a Fast Fourier Transform spectral method. Here, the convoluted cross-moduli bring the following results: (i) Nonlocal stresses and couple stresses oppose their local counterparts in the defects core regions, playing the role of restoring forces and possibly ensuring spatio-temporal stability of the simulated defects, (ii) The couple stress fields are strongly affected by nonlocality. Such effects favor the stability of the simulated grain boundaries and allow investigating their elastic interactions with extrinsic defects, (iii) Driving forces inducing grain growth or refinement derive from the self-stress and couple stress fields of grain boundaries in nanocrystalline configurations. 19. The Topology of Symmetric Tensor Fields Science.gov (United States) Levin, Yingmei; Batra, Rajesh; Hesselink, Lambertus; Levy, Yuval 1997-01-01 Combinatorial topology, also known as "rubber sheet geometry", has extensive applications in geometry and analysis, many of which result from connections with the theory of differential equations. A link between topology and differential equations is vector fields. Recent developments in scientific visualization have shown that vector fields also play an important role in the analysis of second-order tensor fields. A second-order tensor field can be transformed into its eigensystem, namely, eigenvalues and their associated eigenvectors without loss of information content. Eigenvectors behave in a similar fashion to ordinary vectors with even simpler topological structures due to their sign indeterminacy. Incorporating information about eigenvectors and eigenvalues in a display technique known as hyperstreamlines reveals the structure of a tensor field. The simplify and often complex tensor field and to capture its important features, the tensor is decomposed into an isotopic tensor and a deviator. A tensor field and its deviator share the same set of eigenvectors, and therefore they have a similar topological structure. A a deviator determines the properties of a tensor field, while the isotopic part provides a uniform bias. Degenerate points are basic constituents of tensor fields. In 2-D tensor fields, there are only two types of degenerate points; while in 3-D, the degenerate points can be characterized in a Q'-R' plane. Compressible and incompressible flows share similar topological feature due to the similarity of their deviators. In the case of the deformation tensor, the singularities of its deviator represent the area of vortex core in the field. In turbulent flows, the similarities and differences of the topology of the deformation and the Reynolds stress tensors reveal that the basic addie-viscosity assuptions have their validity in turbulence modeling under certain conditions. 20. The determination of young couples educational needs in Yazd Directory of Open Access Journals (Sweden) 2014-07-01 Full Text Available Abstract Introduction:one of the essential needs of young couples is to have knowledge in the field of health reproduction that will have significant impact on health improvement and family bonds.recognization of young couples are lead more attention of policy maker towards issues which is contained low level of knowledge for the young couples. Presenting the required training to young couples will help to have better undersanding of their thoughts and feelings and make decisions more corrective and appropriate for themselves about reproductive health issues. Methods: this is a descriptive study. The statistical population is volentier couples who have reffering to the premartial counseling centers that they had been married since last year to express their training needs based on experince of the life. The samples are included 240 couples and data instrument was the questionnaires that directly were interviewed and data were analyzed by use of SPSS software. Results:less educated, rurals, home worker and laborershas more training need than the others and there is a significant difference. (p = 0.000. but training need is not related to the marriageable age, gender and there was no significant difference(p > 0.50. The young couples are explained the lessamount of training needs in case of form and function of the reproductive system, method of contraception from prognancy, preparation needs before sextual, simultanios and how they can have sexual relation and how they have sexual and intercourse intercourse and statidfy. While the need of training based on healthy issue during menstruation and time of intercourse, how to become pregnant, sexually transmitted, high risk prognancies, congenital diseases, common cancerns, sexual dysfunction are more and more mentioned. Conclusion :Training needs of couples should be more considered. For the rurals and less educated groupit should be perform supplementory classes after the marriage. Based on the 1. iDC: A comprehensive toolkit for the analysis of residual dipolar couplings for macromolecular structure determination International Nuclear Information System (INIS) Wei Yufeng; Werner, Milton H. 2006-01-01 Measurement of residual dipolar couplings (RDCs) has become an important method for the determination and validation of protein or nucleic acid structures by NMRf spectroscopy. A number of toolkits have been devised for the handling of RDC data which run in the Linux/Unix operating environment and require specifically formatted input files. The outputs from these programs, while informative, require format modification prior to the incorporation of this data into commonly used personal computer programs for manuscript preparation. To bridge the gap between analysis and publication, an easy-to-use, comprehensive toolkit for RDC analysis has been created, iDC. iDC is written for the WaveMetrics Igor Pro mathematics program, a widely used graphing and data analysis software program that runs on both Windows PC and Mac OS X computers. Experimental RDC values can be loaded into iDC using simple data formats accessible to Igor's tabular data function. The program can perform most useful RDC analyses, including alignment tensor estimation from a histogram of RDC occurrence versus values and order tensor analysis by singular value decomposition (SVD). SVD analysis can be performed on an entire structure family at once, a feature missing in other applications of this kind. iDC can also import from and export to several different commonly used programs for the analysis of RDC data (DC, PALES, REDCAT) and can prepare formatted files for RDC-based refinement of macromolecular structures using XPLOR-NIH, CNS and ARIA. The graphical user interface provides an easy-to-use I/O for data, structures and formatted outputs 2. Bowen-York tensors International Nuclear Information System (INIS) Beig, Robert; Krammer, Werner 2004-01-01 For a conformally flat 3-space, we derive a family of linear second-order partial differential operators which sends vectors into trace-free, symmetric 2-tensors. These maps, which are parametrized by conformal Killing vectors on the 3-space, are such that the divergence of the resulting tensor field depends only on the divergence of the original vector field. In particular, these maps send source-free electric fields into TT tensors. Moreover, if the original vector field is the Coulomb field on R 3 {0}, the resulting tensor fields on R 3 {0} are nothing but the family of TT tensors originally written by Bowen and York 3. A framework for fast probabilistic centroid-moment-tensor determination-inversion of regional static displacement measurements NARCIS (Netherlands) Käufl, P.J.; Valentine, A.P.; O'Toole, T.B.; Trampert, J. 2014-01-01 The determination of earthquake source parameters is an important task in seismology. For many applications, it is also valuable to understand the uncertainties associated with these determinations, and this is particularly true in the context of earthquake early warning (EEW) and hazard mitigation. 4. 3D reconstruction of tensors and vectors International Nuclear Information System (INIS) Defrise, Michel; Gullberg, Grant T. 2005-01-01 Here we have developed formulations for the reconstruction of 3D tensor fields from planar (Radon) and line-integral (X-ray) projections of 3D vector and tensor fields. Much of the motivation for this work is the potential application of MRI to perform diffusion tensor tomography. The goal is to develop a theory for the reconstruction of both Radon planar and X-ray or line-integral projections because of the flexibility of MRI to obtain both of these type of projections in 3D. The development presented here for the linear tensor tomography problem provides insight into the structure of the nonlinear MRI diffusion tensor inverse problem. A particular application of tensor imaging in MRI is the potential application of cardiac diffusion tensor tomography for determining in vivo cardiac fiber structure. One difficulty in the cardiac application is the motion of the heart. This presents a need for developing future theory for tensor tomography in a motion field. This means developing a better understanding of the MRI signal for diffusion processes in a deforming media. The techniques developed may allow the application of MRI tensor tomography for the study of structure of fiber tracts in the brain, atherosclerotic plaque, and spine in addition to fiber structure in the heart. However, the relations presented are also applicable to other fields in medical imaging such as diffraction tomography using ultrasound. The mathematics presented can also be extended to exponential Radon transform of tensor fields and to other geometric acquisitions such as cone beam tomography of tensor fields 5. Determination of beam coupling impedance in the frequency domain Energy Technology Data Exchange (ETDEWEB) Niedermayer, Uwe 2016-07-01 The concept of beam coupling impedance describes the electromagnetic interaction of uniformly moving charged particles with their surrounding structures in the Frequency Domain (FD). In synchrotron accelerators, beam coupling impedances can lead to beam induced component heating and coherent beam instabilities. Thus, in order to ensure the stable operation of a synchrotron, its impedances have to be quantified and their effects have to be controlled. Nowadays, beam coupling impedances are mostly obtained by Fourier transform of wake potentials, which are the results of Time Domain (TD) simulations. However, at low frequencies, low beam velocity, or for dispersive materials, TD simulations become unhandy. In this area, analytical calculations of beam coupling impedance in the FD, combined with geometry approximations, are still widely used. This thesis describes the development of two electromagnetic field solvers to obtain the beam coupling impedance directly in the FD, where the beam velocity is only a parameter and dispersive materials can be included easily. One solver is based on the Finite Integration Technique (FIT) on a staircase mesh. It is implemented both in 2D and 3D. However, the staircase mesh is inefficient on curved structures, which is particularly problematic for the modeling of a dipole source, that is required for the computation of the transverse beam coupling impedance. This issue is overcome by the second solver developed in this thesis, which is based on the Finite Element Method (FEM) on an unstructured triangular mesh. It is implemented in 2D and includes an optional Surface Impedance Boundary Condition (SIBC). Thus, it is well suited for the computation of longitudinal and transverse impedances of long beam pipe structures of arbitrary cross-section. Besides arbitrary frequency and beam velocity, also dispersive materials can be chosen, which is crucial for the computation of the impedance of ferrite kicker magnets. Numerical impedance 6. Convenient method for resolving degeneracies due to symmetry of the magnetic susceptibility tensor and its application to pseudo contact shift-based protein–protein complex structure determination International Nuclear Information System (INIS) Kobashigawa, Yoshihiro; Saio, Tomohide; Ushio, Masahiro; Sekiguchi, Mitsuhiro; Yokochi, Masashi; Ogura, Kenji; Inagaki, Fuyuhiko 2012-01-01 Pseudo contact shifts (PCSs) induced by paramagnetic lanthanide ions fixed in a protein frame provide long-range distance and angular information, and are valuable for the structure determination of protein–protein and protein–ligand complexes. We have been developing a lanthanide-binding peptide tag (hereafter LBT) anchored at two points via a peptide bond and a disulfide bond to the target proteins. However, the magnetic susceptibility tensor displays symmetry, which can cause multiple degenerated solutions in a structure calculation based solely on PCSs. Here we show a convenient method for resolving this degeneracy by changing the spacer length between the LBT and target protein. We applied this approach to PCS-based rigid body docking between the FKBP12-rapamycin complex and the mTOR FRB domain, and demonstrated that degeneracy could be resolved using the PCS restraints obtained from two-point anchored LBT with two different spacer lengths. The present strategy will markedly increase the usefulness of two-point anchored LBT for protein complex structure determination. 7. Convenient method for resolving degeneracies due to symmetry of the magnetic susceptibility tensor and its application to pseudo contact shift-based protein-protein complex structure determination Energy Technology Data Exchange (ETDEWEB) Kobashigawa, Yoshihiro; Saio, Tomohide [Hokkaido University, Department of Structural Biology, Faculty of Advanced Life Science (Japan); Ushio, Masahiro [Hokkaido University, Graduate School of Life Science (Japan); Sekiguchi, Mitsuhiro [Astellas Pharma Inc., Analysis and Pharmacokinetics Research Labs, Department of Drug Discovery (Japan); Yokochi, Masashi; Ogura, Kenji; Inagaki, Fuyuhiko, E-mail: [email protected] [Hokkaido University, Department of Structural Biology, Faculty of Advanced Life Science (Japan) 2012-05-15 Pseudo contact shifts (PCSs) induced by paramagnetic lanthanide ions fixed in a protein frame provide long-range distance and angular information, and are valuable for the structure determination of protein-protein and protein-ligand complexes. We have been developing a lanthanide-binding peptide tag (hereafter LBT) anchored at two points via a peptide bond and a disulfide bond to the target proteins. However, the magnetic susceptibility tensor displays symmetry, which can cause multiple degenerated solutions in a structure calculation based solely on PCSs. Here we show a convenient method for resolving this degeneracy by changing the spacer length between the LBT and target protein. We applied this approach to PCS-based rigid body docking between the FKBP12-rapamycin complex and the mTOR FRB domain, and demonstrated that degeneracy could be resolved using the PCS restraints obtained from two-point anchored LBT with two different spacer lengths. The present strategy will markedly increase the usefulness of two-point anchored LBT for protein complex structure determination. 8. Supergravity tensor calculus in 5D from 6D International Nuclear Information System (INIS) Kugo, Taichiro; Ohashi, Keisuke 2000-01-01 Supergravity tensor calculus in five spacetime dimensions is derived by dimensional reduction from the d=6 superconformal tensor calculus. In particular, we obtain an off-shell hypermultiplet in 5D from the on-shell hypermultiplet in 6D. Our tensor calculus retains the dilatation gauge symmetry, so that it is a trivial gauge fixing to make the Einstein term canonical in a general matter-Yang-Mills-supergravity coupled system. (author) 9. Categorical Tensor Network States Directory of Open Access Journals (Sweden) Jacob D. Biamonte 2011-12-01 Full Text Available We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unification of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition problem which has several appealing features. Specifically, given an n-body quantum state |ψ〉, we present a new and general method to factor |ψ〉 into a tensor network of clearly defined building blocks. We use the solution to expose a previously unknown and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states. 10. Classification of the Ricci and Plebanski tensors in general relativity using Newman--Penrose formalism International Nuclear Information System (INIS) McIntosh, C.B.G.; Foyster, J.M.; Lun, A.W.h. 1981-01-01 A list is given of a canonical set of the Newman--Penrose quantities Phi/sub A/B, the tetrad components of the trace-free Ricci tensor, for each Plebanski class according to Plebanski's classification of this tensor. This comparative list can easily be extended to cover the classification in tetrad language of any second-order, trace-free, symmetric tensor in a space-time. A fourth-order tensor which is the product of two such tensors was defined by Plebanski and used in his classification. This has the same symmetries as the Weyl tensor. The Petrov classification of this tensor, here called the Plebanski tensor, is discussed along with the classification of the Ricci tensor. The use of the Plebanski tensor in a couple of areas of general relativity is also briefly discussed 11. Heterogeneity of time delays determines synchronization of coupled oscillators. Science.gov (United States) Petkoski, Spase; Spiegler, Andreas; Proix, Timothée; Aram, Parham; Temprado, Jean-Jacques; Jirsa, Viktor K 2016-07-01 Network couplings of oscillatory large-scale systems, such as the brain, have a space-time structure composed of connection strengths and signal transmission delays. We provide a theoretical framework, which allows treating the spatial distribution of time delays with regard to synchronization, by decomposing it into patterns and therefore reducing the stability analysis into the tractable problem of a finite set of delay-coupled differential equations. We analyze delay-structured networks of phase oscillators and we find that, depending on the heterogeneity of the delays, the oscillators group in phase-shifted, anti-phase, steady, and non-stationary clusters, and analytically compute their stability boundaries. These results find direct application in the study of brain oscillations. 12. Direct determinations of the πNN coupling constants International Nuclear Information System (INIS) Ericson, T.E.O.; ); Loiseau, B. 1998-01-01 A novel extrapolation method has been used to deduce directly the charged πNN coupling constant from backward np differential scattering cross sections. The extracted value, g c 2 = 14.52(026)is higher than the indirectly deduced values obtained in nucleon-nucleon energy-dependent partial-wave analyses. Our preliminary direct value from a reanalysis of the GMO sum-rule points to an intermediate value of g c 2 about 13.97(30). (author) 13. Cartesian tensors an introduction CERN Document Server Temple, G 2004-01-01 This undergraduate text provides an introduction to the theory of Cartesian tensors, defining tensors as multilinear functions of direction, and simplifying many theorems in a manner that lends unity to the subject. The author notes the importance of the analysis of the structure of tensors in terms of spectral sets of projection operators as part of the very substance of quantum theory. He therefore provides an elementary discussion of the subject, in addition to a view of isotropic tensors and spinor analysis within the confines of Euclidean space. The text concludes with an examination of t 14. [Determination of a Friction Coefficient for THA Bearing Couples]. Science.gov (United States) Vrbka, M; Nečas, D; Bartošík, J; Hartl, M; Křupka, I; Galandáková, A; Gallo, J 2015-01-01 The wear of articular surfaces is considered one of the most important factors limiting the life of total hip arthroplasty (THA). It is assumed that the particles released from the surface of a softer material induce a complex inflammatory response, which will eventually result in osteolysis and aseptic loosening. Implant wear is related to a friction coefficient which depends on combination of the materials used, roughness of the articulating surfaces, internal clearance, and dimensions of the prosthesis. The selected parameters of the bearing couples tested were studied using an experimental device based on the principle of a pendulum. Bovine serum was used as a lubricant and the load corresponded to a human body mass of 75 kg. The friction coefficient was derived from a curve of slowdown of pendulum oscillations. Roughness was measured with a device working on the principle of interferometry. Clearance was assessed by measuring diameters of the acetabular and femoral heads with a 3D optical scanner. The specimens tested included unused metal-on-highly cross-linked polyethylene, ceramic-on-highly cross-linked polyethylene and ceramic-on-ceramic bearing couples with the diameters of 28 mm and 36 mm. For each measured parameter, an arithmetic mean was calculated from 10 measurements. 1) The roughness of polyethylene surfaces was higher by about one order of magnitude than the roughness of metal and ceramic components. The Protasul metal head had the least rough surface (0.003 μm). 2) The ceramic-on-ceramic couples had the lowest clearance. Bearing couples with polyethylene acetabular liners had markedly higher clearances ranging from 150 μm to 545 μm. A clearance increased with large femoral heads (up to 4-fold in one of the couple tested). 3) The friction coefficient was related to the combination of materials; it was lowest in ceramic-on-ceramic surfaces (0.11 to 0.12) and then in ceramic-on-polyethylene implants (0.13 to 0.14). The friction coefficient is 15. Determination of the seismic moment tensor for local events in the South Shetland Islands and Bransfield Strait International Nuclear Information System (INIS) Guidarelli, M.; Panza, G.F. 2005-06-01 We present the results of the analysis for a set of earthquakes recorded in the Bransfield Strait and the South Shetland Islands in the period 1997-1998, to determine focal mechanisms and source time functions. Events with magnitudes between 3 and 5.6 have been analysed, and the source parameters have been retrieved using a robust methodology (INPAR) that allows the reliable inversion of a limited number of noisy records. This methodology is particularly important in oceanic environments, where the presence of seismic noise and the small number of stations makes it difficult to analyse small magnitude events. (author) 16. Two-photon couplings of 1 = 0 scalars and tensors from analysis of new γγ → ππ data International Nuclear Information System (INIS) Morgan, D.; Pennington, M.R. 1989-11-01 New data on γγ→π + π - and π 0 π 0 admit an amplitude analysis whereby two-photon couplings of the I = 0 scalars and of the f 2 (1270) can be extracted in a much more model independent way than hitherto. Alternative trial forms respecting known properties at low energies and of final state interactions are fitted to the data. The ensuing resonance couplings span a much wider range than is commonly supposed. The best fits correspond to solutions with a relatively large S-wave coupling (∼ 8keV) through the f 2 -region. All fits have an S* coupling of about 1/2 keV. (author) 17. Restricted magnetically balanced basis applied for relativistic calculations of indirect nuclear spin-spin coupling tensors in the matrix Dirac-Kohn-Sham framework International Nuclear Information System (INIS) Repisky, Michal; Komorovsky, Stanislav; Malkina, Olga L.; Malkin, Vladimir G. 2009-01-01 The relativistic four-component density functional approach based on the use of restricted magnetically balanced basis (mDKS-RMB), applied recently for calculations of NMR shielding, was extended for calculations of NMR indirect nuclear spin-spin coupling constants. The unperturbed equations are solved with the use of a restricted kinetically balanced basis set for the small component while to solve the second-order coupled perturbed DKS equations a restricted magnetically balanced basis set for the small component was applied. Benchmark relativistic calculations have been carried out for the X-H and H-H spin-spin coupling constants in the XH 4 series (X = C, Si, Ge, Sn and Pb). The method provides an attractive alternative to existing approximate two-component methods with transformed Hamiltonians for relativistic calculations of spin-spin coupling constants of heavy-atom systems. In particular, no picture-change effects arise in our method for property calculations 18. The effects of noise over the complete space of diffusion tensor shape. Science.gov (United States) Gahm, Jin Kyu; Kindlmann, Gordon; Ennis, Daniel B 2014-01-01 Diffusion tensor magnetic resonance imaging (DT-MRI) is a technique used to quantify the microstructural organization of biological tissues. Multiple images are necessary to reconstruct the tensor data and each acquisition is subject to complex thermal noise. As such, measures of tensor invariants, which characterize components of tensor shape, derived from the tensor data will be biased from their true values. Previous work has examined this bias, but over a narrow range of tensor shape. Herein, we define the mathematics for constructing a tensor from tensor invariants, which permits an intuitive and principled means for building tensors with a complete range of tensor shape and salient microstructural properties. Thereafter, we use this development to evaluate by simulation the effects of noise on characterizing tensor shape over the complete space of tensor shape for three encoding schemes with different SNR and gradient directions. We also define a new framework for determining the distribution of the true values of tensor invariants given their measures, which provides guidance about the confidence the observer should have in the measures. Finally, we present the statistics of tensor invariant estimates over the complete space of tensor shape to demonstrate how the noise sensitivity of tensor invariants varies across the space of tensor shape as well as how the imaging protocol impacts measures of tensor invariants. Copyright © 2013 Elsevier B.V. All rights reserved. 19. Off-shell N = 2 tensor supermultiplets International Nuclear Information System (INIS) Wit, Bernard de; Saueressig, Frank 2006-01-01 A multiplet calculus is presented for an arbitrary number n of N = 2 tensor supermultiplets. For rigid supersymmetry the known couplings are reproduced. In the superconformal case the target spaces parametrized by the scalar fields are cones over (3n-1)-dimensional spaces encoded in homogeneous SU(2) invariant potentials, subject to certain constraints. The coupling to conformal supergravity enables the derivation of a large class of supergravity Lagrangians with vector and tensor multiplets and hypermultiplets. Dualizing the tensor fields into scalars leads to hypermultiplets with hyperkaehler or quaternion-Kaehler target spaces with at least n abelian isometries. It is demonstrated how to use the calculus for the construction of Lagrangians containing higher-derivative couplings of tensor multiplets. For the application of the c-map between vector and tensor supermultiplets to Lagrangians with higher-order derivatives, an off-shell version of this map is proposed. Various other implications of the results are discussed. As an example an elegant derivation of the classification of 4-dimensional quaternion-Kaehler manifolds with two commuting isometries is given 20. Linear Invariant Tensor Interpolation Applied to Cardiac Diffusion Tensor MRI Science.gov (United States) Gahm, Jin Kyu; Wisniewski, Nicholas; Kindlmann, Gordon; Kung, Geoffrey L.; Klug, William S.; Garfinkel, Alan; Ennis, Daniel B. 2015-01-01 Purpose Various methods exist for interpolating diffusion tensor fields, but none of them linearly interpolate tensor shape attributes. Linear interpolation is expected not to introduce spurious changes in tensor shape. Methods Herein we define a new linear invariant (LI) tensor interpolation method that linearly interpolates components of tensor shape (tensor invariants) and recapitulates the interpolated tensor from the linearly interpolated tensor invariants and the eigenvectors of a linearly interpolated tensor. The LI tensor interpolation method is compared to the Euclidean (EU), affine-invariant Riemannian (AI), log-Euclidean (LE) and geodesic-loxodrome (GL) interpolation methods using both a synthetic tensor field and three experimentally measured cardiac DT-MRI datasets. Results EU, AI, and LE introduce significant microstructural bias, which can be avoided through the use of GL or LI. Conclusion GL introduces the least microstructural bias, but LI tensor interpolation performs very similarly and at substantially reduced computational cost. PMID:23286085 1. New approach to the determination phosphorothioate oligonucleotides by ultra high performance liquid chromatography coupled with inductively coupled plasma mass spectrometry. Science.gov (United States) Studzińska, Sylwia; Mounicou, Sandra; Szpunar, Joanna; Łobiński, Ryszard; Buszewski, Bogusław 2015-01-15 This text presents a novel method for the separation and detection of phosphorothioate oligonucleotides with the use of ion pair ultra high performance liquid chromatography coupled with inductively coupled plasma mass spectrometry The research showed that hexafluoroisopropanol/triethylamine based mobile phases may be successfully used when liquid chromatography is coupled with such elemental detection. However, the concentration of both HFIP and TEA influences the final result. The lower concentration of HFIP, the lower the background in ICP-MS and the greater the sensitivity. The method applied for the analysis of serum samples was based on high resolution inductively coupled plasma mass spectrometry. Utilization of this method allows determination of fifty times lower quantity of phosphorothioate oligonucleotides than in the case of quadrupole mass analyzer. Monitoring of (31)P may be used to quantify these compounds at the level of 80 μg L(-1), while simultaneous determination of sulfur is very useful for qualitative analysis. Moreover, the results presented in this paper demonstrate the practical applicability of coupling LC with ICP-MS in determining phosphorothioate oligonucleotides and their metabolites in serum within 7 min with a very good sensitivity. The method was linear in the concentration range between 0.2 and 3 mg L(-1). The limit of detection was in the range of 0.07 and 0.13 mg L(-1). Accuracy varied with concentration, but was in the range of 3%. Copyright © 2014 Elsevier B.V. All rights reserved. 2. Superspace actions and duality transformations for N=2 tensor multiplets International Nuclear Information System (INIS) Galperin, A.; Ivanov, E.; Ogievetsky, V. 1985-01-01 General actions for self-interacting N=2 tensor multiplets are considered in the harmonic superspace approach. All of them are shown to be equivalent, by superfield duality transformations, to some restricted class of the hypermultiplets actions. In particular, the improved tensor multiplet theory is dual to a free hypermultiplet one. Superspace couplings of these improved matter multiplets against conformal supergravity are also constructed 3. (2, 0) tensor multiplets and conformal supergravity in D = 6 NARCIS (Netherlands) Bergshoeff, Eric; Sezgin, Ergin; Proeyen, Antoine Van 1999-01-01 We construct the supercurrent multiplet that contains the energy–momentum tensor of the (2, 0) tensor multiplet. By coupling this multiplet of currents to the fields of conformal supergravity, we first construct the linearized superconformal transformations rules of the (2, 0) Weyl multiplet. 4. [An Improved Spectral Quaternion Interpolation Method of Diffusion Tensor Imaging]. Science.gov (United States) Xu, Yonghong; Gao, Shangce; Hao, Xiaofei 2016-04-01 Diffusion tensor imaging(DTI)is a rapid development technology in recent years of magnetic resonance imaging.The diffusion tensor interpolation is a very important procedure in DTI image processing.The traditional spectral quaternion interpolation method revises the direction of the interpolation tensor and can preserve tensors anisotropy,but the method does not revise the size of tensors.The present study puts forward an improved spectral quaternion interpolation method on the basis of traditional spectral quaternion interpolation.Firstly,we decomposed diffusion tensors with the direction of tensors being represented by quaternion.Then we revised the size and direction of the tensor respectively according to different situations.Finally,we acquired the tensor of interpolation point by calculating the weighted average.We compared the improved method with the spectral quaternion method and the Log-Euclidean method by the simulation data and the real data.The results showed that the improved method could not only keep the monotonicity of the fractional anisotropy(FA)and the determinant of tensors,but also preserve the tensor anisotropy at the same time.In conclusion,the improved method provides a kind of important interpolation method for diffusion tensor image processing. 5. Mean template for tensor-based morphometry using deformation tensors. Science.gov (United States) Leporé, Natasha; Brun, Caroline; Pennec, Xavier; Chou, Yi-Yu; Lopez, Oscar L; Aizenstein, Howard J; Becker, James T; Toga, Arthur W; Thompson, Paul M 2007-01-01 Tensor-based morphometry (TBM) studies anatomical differences between brain images statistically, to identify regions that differ between groups, over time, or correlate with cognitive or clinical measures. Using a nonlinear registration algorithm, all images are mapped to a common space, and statistics are most commonly performed on the Jacobian determinant (local expansion factor) of the deformation fields. In, it was shown that the detection sensitivity of the standard TBM approach could be increased by using the full deformation tensors in a multivariate statistical analysis. Here we set out to improve the common space itself, by choosing the shape that minimizes a natural metric on the deformation tensors from that space to the population of control subjects. This method avoids statistical bias and should ease nonlinear registration of new subjects data to a template that is 'closest' to all subjects' anatomies. As deformation tensors are symmetric positive-definite matrices and do not form a vector space, all computations are performed in the log-Euclidean framework. The control brain B that is already the closest to 'average' is found. A gradient descent algorithm is then used to perform the minimization that iteratively deforms this template and obtains the mean shape. We apply our method to map the profile of anatomical differences in a dataset of 26 HIV/AIDS patients and 14 controls, via a log-Euclidean Hotelling's T2 test on the deformation tensors. These results are compared to the ones found using the 'best' control, B. Statistics on both shapes are evaluated using cumulative distribution functions of the p-values in maps of inter-group differences. 6. Algebraic and computational aspects of real tensor ranks CERN Document Server Sakata, Toshio; Miyazaki, Mitsuhiro 2016-01-01 This book provides comprehensive summaries of theoretical (algebraic) and computational aspects of tensor ranks, maximal ranks, and typical ranks, over the real number field. Although tensor ranks have been often argued in the complex number field, it should be emphasized that this book treats real tensor ranks, which have direct applications in statistics. The book provides several interesting ideas, including determinant polynomials, determinantal ideals, absolutely nonsingular tensors, absolutely full column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. In addition to reviews of methods to determine real tensor ranks in details, global theories such as the Jacobian method are also reviewed in details. The book includes as well an accessible and comprehensive introduction of mathematical backgrounds, with basics of positive polynomials and calculations by using the Groebner basis. Furthermore, this book provides insights into numerical methods of finding tensor ranks through... 7. Improved tensor multiplets International Nuclear Information System (INIS) Wit, B. de; Rocek, M. 1982-01-01 We construct a conformally invariant theory of the N = 1 supersymmetric tensor gauge multiplet and discuss the situation in N = 2. We show that our results give rise to the recently proposed variant of Poincare supergravity, and provide the complete tensor calculus for the theory. Finally, we argue that this theory cannot be quantized sensibly. (orig.) 8. Time integration of tensor trains OpenAIRE Lubich, Christian; Oseledets, Ivan; Vandereycken, Bart 2014-01-01 A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train / matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formul... 9. Generalized tensor-based morphometry of HIV/AIDS using multivariate statistics on deformation tensors. Science.gov (United States) Lepore, N; Brun, C; Chou, Y Y; Chiang, M C; Dutton, R A; Hayashi, K M; Luders, E; Lopez, O L; Aizenstein, H J; Toga, A W; Becker, J T; Thompson, P M 2008-01-01 This paper investigates the performance of a new multivariate method for tensor-based morphometry (TBM). Statistics on Riemannian manifolds are developed that exploit the full information in deformation tensor fields. In TBM, multiple brain images are warped to a common neuroanatomical template via 3-D nonlinear registration; the resulting deformation fields are analyzed statistically to identify group differences in anatomy. Rather than study the Jacobian determinant (volume expansion factor) of these deformations, as is common, we retain the full deformation tensors and apply a manifold version of Hotelling'sT(2) test to them, in a Log-Euclidean domain. In 2-D and 3-D magnetic resonance imaging (MRI) data from 26 HIV/AIDS patients and 14 matched healthy subjects, we compared multivariate tensor analysis versus univariate tests of simpler tensor-derived indices: the Jacobian determinant, the trace, geodesic anisotropy, and eigenvalues of the deformation tensor, and the angle of rotation of its eigenvectors. We detected consistent, but more extensive patterns of structural abnormalities, with multivariate tests on the full tensor manifold. Their improved power was established by analyzing cumulative p-value plots using false discovery rate (FDR) methods, appropriately controlling for false positives. This increased detection sensitivity may empower drug trials and large-scale studies of disease that use tensor-based morphometry.
10. Abelian tensor models on the lattice
Science.gov (United States)
Chaudhuri, Soumyadeep; Giraldo-Rivera, Victor I.; Joseph, Anosh; Loganayagam, R.; Yoon, Junggi
2018-04-01
We consider a chain of Abelian Klebanov-Tarnopolsky fermionic tensor models coupled through quartic nearest-neighbor interactions. We characterize the gauge-singlet spectrum for small chains (L =2 ,3 ,4 ,5 ) and observe that the spectral statistics exhibits strong evidence in favor of quasi-many-body localization.
11. Dark energy in scalar-tensor theories
Energy Technology Data Exchange (ETDEWEB)
Moeller, J.
2007-12-15
We investigate several aspects of dynamical dark energy in the framework of scalar-tensor theories of gravity. We provide a classification of scalar-tensor coupling functions admitting cosmological scaling solutions. In particular, we recover that Brans-Dicke theory with inverse power-law potential allows for a sequence of background dominated scaling regime and scalar field dominated, accelerated expansion. Furthermore, we compare minimally and non-minimally coupled models, with respect to the small redshift evolution of the dark energy equation of state. We discuss the possibility to discriminate between different models by a reconstruction of the equation-of-state parameter from available observational data. The non-minimal coupling characterizing scalar-tensor models can - in specific cases - alleviate fine tuning problems, which appear if (minimally coupled) quintessence is required to mimic a cosmological constant. Finally, we perform a phase-space analysis of a family of biscalar-tensor models characterized by a specific type of {sigma}-model metric, including two examples from recent literature. In particular, we generalize an axion-dilaton model of Sonner and Townsend, incorporating a perfect fluid background consisting of (dark) matter and radiation. (orig.)
12. Primordial tensor modes from quantum corrected inflation
DEFF Research Database (Denmark)
Joergensen, Jakob; Sannino, Francesco; Svendsen, Ole
2014-01-01
. Finally we confront these theories with the Planck and BICEP2 data. We demonstrate that the discovery of primordial tensor modes by BICEP2 require the presence of sizable quantum departures from the $\\phi^4$-Inflaton model for the non-minimally coupled scenario which we parametrize and quantify. We...
13. Dark energy in scalar-tensor theories
International Nuclear Information System (INIS)
Moeller, J.
2007-12-01
We investigate several aspects of dynamical dark energy in the framework of scalar-tensor theories of gravity. We provide a classification of scalar-tensor coupling functions admitting cosmological scaling solutions. In particular, we recover that Brans-Dicke theory with inverse power-law potential allows for a sequence of background dominated scaling regime and scalar field dominated, accelerated expansion. Furthermore, we compare minimally and non-minimally coupled models, with respect to the small redshift evolution of the dark energy equation of state. We discuss the possibility to discriminate between different models by a reconstruction of the equation-of-state parameter from available observational data. The non-minimal coupling characterizing scalar-tensor models can - in specific cases - alleviate fine tuning problems, which appear if (minimally coupled) quintessence is required to mimic a cosmological constant. Finally, we perform a phase-space analysis of a family of biscalar-tensor models characterized by a specific type of σ-model metric, including two examples from recent literature. In particular, we generalize an axion-dilaton model of Sonner and Townsend, incorporating a perfect fluid background consisting of (dark) matter and radiation. (orig.)
14. [Determinants of sterilization among married couples in Korea].
Science.gov (United States)
Kim, Ju Hee; Chung, Woojin; Lee, Sunmi; Suh, Moonhee; Kang, Dae Ryong
2007-11-01
The purpose of this study was to examine the determinants of sterilization in South Korea. This study was based on the data from the Korea National Fertility Survey carried out in the year 2000 by the Korea Institute of Health and Social Affairs. The subjects of the analysis were 4,604 women and their husbands who were in their first marriage, in the age group of 15-49 years. The data were analyzed by multiple logistic regression analysis. Consistent with the findings of previous studies, the woman's age and the number of total children increased the likelihood of sterilization. In addition, the year of marriage had a strong positive association with sterilization. Interestingly, the number of surviving sons tended to increase the likelihood of sterilization, whereas the woman's education level and age at the time of marriage showed a negative association with sterilization. Religion, place of residence, son preference, and the husband's education level, age and type of occupation were not significant determinants of sterilization. The sex of previous children and lower level of education are distinct determinants of sterilization among women in South Korea. More studies are needed in order to determine the associations between sterilization rate and decreased fertility.
15. Renormalization of nonabelian gauge theories with tensor matter fields
International Nuclear Information System (INIS)
Lemes, Vitor; Renan, Ricardo; Sorella, Silvio Paolo
1996-03-01
The renormalizability of a nonabelian model describing the coupling between antisymmetric second rank tensor matter fields and Yang-Mills gauge fields is discussed within the BRS algebraic framework. (author). 12 refs
16. On the energy-momentum tensor in Moyal space
International Nuclear Information System (INIS)
Balasin, Herbert; Schweda, Manfred; Blaschke, Daniel N.; Gieres, Francois
2015-01-01
We study the properties of the energy-momentum tensor of gauge fields coupled to matter in non-commutative (Moyal) space. In general, the non-commutativity affects the usual conservation law of the tensor as well as its transformation properties (gauge covariance instead of gauge invariance). It is well known that the conservation of the energy-momentum tensor can be achieved by a redefinition involving another star-product. Furthermore, for a pure gauge theory it is always possible to define a gauge invariant energy-momentum tensor by means of a Wilson line. We show that the last two procedures are incompatible with each other if couplings of gauge fields to matter fields (scalars or fermions) are considered: The gauge invariant tensor (constructed via Wilson line) does not allow for a redefinition assuring its conservation, and vice versa the introduction of another star-product does not allow for gauge invariance by means of a Wilson line. (orig.)
17. Tensor form factor for the D → π(K) transitions with Twisted Mass fermions.
Science.gov (United States)
Lubicz, Vittorio; Riggio, Lorenzo; Salerno, Giorgio; Simula, Silvano; Tarantino, Cecilia
2018-03-01
We present a preliminary lattice calculation of the D → π and D → K tensor form factors fT (q2) as a function of the squared 4-momentum transfer q2. ETMC recently computed the vector and scalar form factors f+(q2) and f0(q2) describing D → π(K)lv semileptonic decays analyzing the vector current and the scalar density. The study of the weak tensor current, which is directly related to the tensor form factor, completes the set of hadronic matrix element regulating the transition between these two pseudoscalar mesons within and beyond the Standard Model where a non-zero tensor coupling is possible. Our analysis is based on the gauge configurations produced by the European Twisted Mass Collaboration with Nf = 2 + 1 + 1 flavors of dynamical quarks. We simulated at three different values of the lattice spacing and with pion masses as small as 210 MeV and with the valence heavy quark in the mass range from ≃ 0.7 mc to ≃ 1.2mc. The matrix element of the tensor current are determined for a plethora of kinematical conditions in which parent and child mesons are either moving or at rest. As for the vector and scalar form factors, Lorentz symmetry breaking due to hypercubic effects is clearly observed in the data. We will present preliminary results on the removal of such hypercubic lattice effects.
18. Tensor spherical harmonics and tensor multipoles. II. Minkowski space
International Nuclear Information System (INIS)
Daumens, M.; Minnaert, P.
1976-01-01
The bases of tensor spherical harmonics and of tensor multipoles discussed in the preceding paper are generalized in the Hilbert space of Minkowski tensor fields. The transformation properties of the tensor multipoles under Lorentz transformation lead to the notion of irreducible tensor multipoles. We show that the usual 4-vector multipoles are themselves irreducible, and we build the irreducible tensor multipoles of the second order. We also give their relations with the symmetric tensor multipoles defined by Zerilli for application to the gravitational radiation
19. Determination of g-tensors of low-symmetry Nd{sup 3+} centers in LiNbO{sub 3} by rectification of angular dependence of electron paramagnetic resonance spectra
Energy Technology Data Exchange (ETDEWEB)
Grachev, V., E-mail: [email protected]; Malovichko, G. [Physics Department, Montana State University, Bozeman, Montana 59717 (United States); Munro, M. [Quantel Laser, Bozeman, Montana 59715 (United States); Kokanyan, E. [Institute of Physical Researches, Ashtarak (Armenia)
2015-07-28
Two procedures for facilitation of line tracing and deciphering of complicated spectra of electron paramagnetic resonance (EPR) were developed: a correction of microwave frequencies for every orientation of external magnetic field on the base of known values of g-tensor components for a reference paramagnetic center and followed rectification of measured angular dependences using plots of effective deviation of g{sup 2}-factors of observed lines from effective g{sup 2}-factors of the reference center versus angles or squared cosines of angles describing magnetic field orientations. Their application to EPR spectra of nearly stoichiometric lithium niobate crystals doped with neodymium allowed identifying two axial and six different low-symmetry Nd{sup 3+} centers, to determine all components of their g-tensors, and to propose common divacancy models for a whole family of Nd{sup 3+} centers.
20. Tensor modes in pure natural inflation
Science.gov (United States)
Nomura, Yasunori; Yamazaki, Masahito
2018-05-01
We study tensor modes in pure natural inflation [1], a recently-proposed inflationary model in which an axionic inflaton couples to pure Yang-Mills gauge fields. We find that the tensor-to-scalar ratio r is naturally bounded from below. This bound originates from the finiteness of the number of metastable branches of vacua in pure Yang-Mills theories. Details of the model can be probed by future cosmic microwave background experiments and improved lattice gauge theory calculations of the θ-angle dependence of the vacuum energy.
1. Determination of neutral current couplings from neutrino-induced semi-inclusive pion and inclusive reactions
International Nuclear Information System (INIS)
Hung, P.Q.
1977-01-01
It is shown that by looking at data from neutrino-induced semi-inclusive pion and inclusive reactions on isoscalar targets along, one can determine completely the neutral current couplings. Predictions for various models are also presented. (Auth.)
2. Determination of trimethyllead reference material using high performance liquid chromatography-inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Lu Hai; Wei Chao; Wang Jun; Chao Jingbo; Zhou Tao; Chen Dazhou
2005-01-01
A high-performance liquid chromatography-inductively coupled plasma mass spectrometry (HPLC-ICPMS) was combined, and the chromatography conditions were optimized. The stability and homogeneity of a trimethyllead reference material were determined using this method. (authors)
3. Complementary and alternative medicine usage and its determinant factors among Iranian infertile couples.
Science.gov (United States)
Dehghan, Mahlagha; Mokhtarabadi, Sima; Heidari, Fatemeh Ghaedi
2018-04-04
Background The aim of this study was to determine the status of utilizing some complementary and alternative medicine techniques in infertile couples. Methods This was a cross-sectional study conducted on 250 infertile couples referred to a hospital in Kerman using convenience sampling. A researcher-made questionnaire was used to study the prevalence and user satisfaction of complementary and alternative medicines. Results Results indicated that 49.6% of the infertile couples used at least one of the complementary and alternative medicines during the past year. Most individuals used spiritual techniques (71.8% used praying and 70.2% used Nazr) and medicinal plants (54.8%). Safety is the most important factor affecting the satisfaction of infertile couples with complementary treatments (couples think that such treatments are safe (54.8%)). Discussion Concerning high prevalence of complementary and alternative treatments in infertile couples, incorporating such treatments into the healthcare education and promoting the awareness of infertile individuals seem crucial.
4. Tensors and their applications
CERN Document Server
Islam, Nazrul
2006-01-01
About the Book: The book is written is in easy-to-read style with corresponding examples. The main aim of this book is to precisely explain the fundamentals of Tensors and their applications to Mechanics, Elasticity, Theory of Relativity, Electromagnetic, Riemannian Geometry and many other disciplines of science and engineering, in a lucid manner. The text has been explained section wise, every concept has been narrated in the form of definition, examples and questions related to the concept taught. The overall package of the book is highly useful and interesting for the people associated with the field. Contents: Preliminaries Tensor Algebra Metric Tensor and Riemannian Metric Christoffels Symbols and Covariant Differentiation Riemann-Christoffel Tensor The e-Systems and the Generalized Krönecker Deltas Geometry Analytical Mechanics Curvature of a Curve, Geodesic Parallelism of Vectors Riccis Coefficients of Rotation and Congruence Hyper Surfaces
5. Symmetric Tensor Decomposition
DEFF Research Database (Denmark)
Brachat, Jerome; Comon, Pierre; Mourrain, Bernard
2010-01-01
We present an algorithm for decomposing a symmetric tensor, of dimension n and order d, as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables...... of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation...... of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions and for detecting the rank....
International Nuclear Information System (INIS)
Scheunert, M.
1982-10-01
We develop a graded tensor calculus corresponding to arbitrary Abelian groups of degrees and arbitrary commutation factors. The standard basic constructions and definitions like tensor products, spaces of multilinear mappings, contractions, symmetrization, symmetric algebra, as well as the transpose, adjoint, and trace of a linear mapping, are generalized to the graded case and a multitude of canonical isomorphisms is presented. Moreover, the graded versions of the classical Lie algebras are introduced and some of their basic properties are described. (orig.)
7. On an uninterpretated tensor in Dirac's theory
International Nuclear Information System (INIS)
Costa de Beauregard, O.
1989-01-01
Franz, in 1935, deduced systematically from the Dirac equation 10 tensorial equations, 5 with a mechanical interpretation, 5 with an electromagnetic interpretation, which are also consequences of Kemmer's formalism for spins 1 and 0; Durand, in 1944, operating similarly with the second order Dirac equation, obtained, 10 equations, 5 of which expressing the divergences of the Gordon type tensors. Of these equations, together with the tensors they imply, some are easily interpreted by reference to the classical theories, some other remain uniterpreted. Recently (1988) we proposed a theory of the coupling between Einstein's gravity field and the 5 Franz mechanical equations, yielding as a bonus the complete interpretation of the 5 Franz mechanical equations. This is an incitation to reexamine the 5 electromagnetic equations. We show here that two of these, together with one of the Durand equations, implying the same tensor, remain uninterpreted. This is proposed as a challenge to the reader's sagacity [fr
8. A Review of Tensors and Tensor Signal Processing
Science.gov (United States)
Cammoun, L.; Castaño-Moraga, C. A.; Muñoz-Moreno, E.; Sosa-Cabrera, D.; Acar, B.; Rodriguez-Florido, M. A.; Brun, A.; Knutsson, H.; Thiran, J. P.
Tensors have been broadly used in mathematics and physics, since they are a generalization of scalars or vectors and allow to represent more complex properties. In this chapter we present an overview of some tensor applications, especially those focused on the image processing field. From a mathematical point of view, a lot of work has been developed about tensor calculus, which obviously is more complex than scalar or vectorial calculus. Moreover, tensors can represent the metric of a vector space, which is very useful in the field of differential geometry. In physics, tensors have been used to describe several magnitudes, such as the strain or stress of materials. In solid mechanics, tensors are used to define the generalized Hooke’s law, where a fourth order tensor relates the strain and stress tensors. In fluid dynamics, the velocity gradient tensor provides information about the vorticity and the strain of the fluids. Also an electromagnetic tensor is defined, that simplifies the notation of the Maxwell equations. But tensors are not constrained to physics and mathematics. They have been used, for instance, in medical imaging, where we can highlight two applications: the diffusion tensor image, which represents how molecules diffuse inside the tissues and is broadly used for brain imaging; and the tensorial elastography, which computes the strain and vorticity tensor to analyze the tissues properties. Tensors have also been used in computer vision to provide information about the local structure or to define anisotropic image filters.
9. Diffusion tensor optical coherence tomography
Science.gov (United States)
Marks, Daniel L.; Blackmon, Richard L.; Oldenburg, Amy L.
2018-01-01
In situ measurements of diffusive particle transport provide insight into tissue architecture, drug delivery, and cellular function. Analogous to diffusion-tensor magnetic resonance imaging (DT-MRI), where the anisotropic diffusion of water molecules is mapped on the millimeter scale to elucidate the fibrous structure of tissue, here we propose diffusion-tensor optical coherence tomography (DT-OCT) for measuring directional diffusivity and flow of optically scattering particles within tissue. Because DT-OCT is sensitive to the sub-resolution motion of Brownian particles as they are constrained by tissue macromolecules, it has the potential to quantify nanoporous anisotropic tissue structure at micrometer resolution as relevant to extracellular matrices, neurons, and capillaries. Here we derive the principles of DT-OCT, relating the detected optical signal from a minimum of six probe beams with the six unique diffusion tensor and three flow vector components. The optimal geometry of the probe beams is determined given a finite numerical aperture, and a high-speed hardware implementation is proposed. Finally, Monte Carlo simulations are employed to assess the ability of the proposed DT-OCT system to quantify anisotropic diffusion of nanoparticles in a collagen matrix, an extracellular constituent that is known to become highly aligned during tumor development.
10. Determination of structural fluctuations of proteins from structure-based calculations of residual dipolar couplings
International Nuclear Information System (INIS)
Montalvao, Rinaldo W.; De Simone, Alfonso; Vendruscolo, Michele
2012-01-01
Residual dipolar couplings (RDCs) have the potential of providing detailed information about the conformational fluctuations of proteins. It is very challenging, however, to extract such information because of the complex relationship between RDCs and protein structures. A promising approach to decode this relationship involves structure-based calculations of the alignment tensors of protein conformations. By implementing this strategy to generate structural restraints in molecular dynamics simulations we show that it is possible to extract effectively the information provided by RDCs about the conformational fluctuations in the native states of proteins. The approach that we present can be used in a wide range of alignment media, including Pf1, charged bicelles and gels. The accuracy of the method is demonstrated by the analysis of the Q factors for RDCs not used as restraints in the calculations, which are significantly lower than those corresponding to existing high-resolution structures and structural ensembles, hence showing that we capture effectively the contributions to RDCs from conformational fluctuations.
11. Determinants in HIV counselling and testing in couples in North Rift Kenya.
Science.gov (United States)
Ayuo, P O; Were, E; Wools-Kaloustian, K; Baliddawa, J; Sidle, J; Fife, K
2009-02-01
Voluntary HIV counselling and testing (VCT) has been shown to be an acceptable and effective tool in the fight against HIV/AIDS. Couple HIV Counselling and Testing (CHCT) however, is a relatively new concept whose acceptance and efficacy is yet to be determined. To describe factors that motivate couples to attend VCT as a couple. A cross sectional qualitative study. Moi Teaching and Referral Hospital and Moi University, School of Medicine, Eldoret, Kenya Seventy one individuals were interviewed during KII (9) and dyad interviews (31 couples). Ten FGDs involving a total of 109 individuals were held. Cultural practices, lack of CHCT awareness, stigma and fear of results deter CHCT utilisation. Location of centre where it is unlikely to be associated with HIV testing, qualified professional staff and minimal waiting times would enhance CHCT utilisation. CHCT as a tool in the fight against HIV/AIDS in this region of Kenya is feasible as the factors that would deter couples are not insurmountable.
12. Tensor analysis for physicists
CERN Document Server
Schouten, J A
1989-01-01
This brilliant study by a famed mathematical scholar and former professor of mathematics at the University of Amsterdam integrates a concise exposition of the mathematical basis of tensor analysis with admirably chosen physical examples of the theory. The first five chapters incisively set out the mathematical theory underlying the use of tensors. The tensor algebra in EN and RN is developed in Chapters I and II. Chapter II introduces a sub-group of the affine group, then deals with the identification of quantities in EN. The tensor analysis in XN is developed in Chapter IV. In chapters VI through IX, Professor Schouten presents applications of the theory that are both intrinsically interesting and good examples of the use and advantages of the calculus. Chapter VI, intimately connected with Chapter III, shows that the dimensions of physical quantities depend upon the choice of the underlying group, and that tensor calculus is the best instrument for dealing with the properties of anisotropic media. In Chapte...
13. Precision determination of the strong coupling constant within a global PDF analysis
NARCIS (Netherlands)
Ball, Richard D.; Carrazza, Stefano; Debbio, Luigi Del; Forte, Stefano; Kassabov, Zahari; Rojo, Juan; Slade, Emma; Ubiali, Maria
2018-01-01
We present a determination of the strong coupling constant $\\alpha_s(m_Z)$ based on the NNPDF3.1 determination of parton distributions, which for the first time includes constraints from jet production, top-quark pair differential distributions, and the $Z$ $p_T$ distributions using exact NNLO
14. Determination of technetium-99 in soil samples by high performance liquid chromatography coupled to inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Muto, Toshio; Shimokawa, Toshinari
1997-01-01
A new powerful analytical technique viz. high performance liquid chromatography(HPLC) coupled to inductively coupled plasma mass spectrometry(HPLC/ICP-MS) has been applied to the determination of technetium-99( 99 Tc) in soils as a typical environmental sample. Technetium was enriched in a solution from incinerated soil samples by leaching in HNO 3 and passed through 'TEVA resin' column. The solution was injected into HPLC/ICP-MS system to eliminate the interfering elements (i.e. Ru and Mo) and to determine the 99 Tc concentration at the same time. The concentrations of 99 Tc in the incinerated soils were found to be 0.49Bq/kg(0.77ng/kg)-1.4Bq/kg(2.2ng/kg) with the determination limit of 0.02Bq/kg(0.03ng/kg(0.03ppt)). The results indicate the following findings; 1) the determination of 99 Tc by ICP-MS after strict elimination of the interfering elements by HPLC brings about the improvement in their reliability; 2) the detection limits identified are much lower compared with those by conventional ICP-MS methods because of the concentration of 99 Tc to smaller volume, which is due to only 100μl of samples could be measured by HPLC/ICP-MS system; 3) sample preparation could be simplified because of strict elimination of the interfering elements by HPLC. This research showed that HPLC/ICP-MS system is very effective to determine 99 Tc in environmental samples. (author)
15. Susceptibility tensor imaging (STI) of the brain.
Science.gov (United States)
Li, Wei; Liu, Chunlei; Duong, Timothy Q; van Zijl, Peter C M; Li, Xu
2017-04-01
Susceptibility tensor imaging (STI) is a recently developed MRI technique that allows quantitative determination of orientation-independent magnetic susceptibility parameters from the dependence of gradient echo signal phase on the orientation of biological tissues with respect to the main magnetic field. By modeling the magnetic susceptibility of each voxel as a symmetric rank-2 tensor, individual magnetic susceptibility tensor elements as well as the mean magnetic susceptibility and magnetic susceptibility anisotropy can be determined for brain tissues that would still show orientation dependence after conventional scalar-based quantitative susceptibility mapping to remove such dependence. Similar to diffusion tensor imaging, STI allows mapping of brain white matter fiber orientations and reconstruction of 3D white matter pathways using the principal eigenvectors of the susceptibility tensor. In contrast to diffusion anisotropy, the main determinant factor of the susceptibility anisotropy in brain white matter is myelin. Another unique feature of the susceptibility anisotropy of white matter is its sensitivity to gadolinium-based contrast agents. Mechanistically, MRI-observed susceptibility anisotropy is mainly attributed to the highly ordered lipid molecules in the myelin sheath. STI provides a consistent interpretation of the dependence of phase and susceptibility on orientation at multiple scales. This article reviews the key experimental findings and physical theories that led to the development of STI, its practical implementations, and its applications for brain research. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.
16. Susceptibility Tensor Imaging (STI) of the Brain
Science.gov (United States)
Li, Wei; Liu, Chunlei; Duong, Timothy Q.; van Zijl, Peter C.M.; Li, Xu
2016-01-01
Susceptibility tensor imaging (STI) is a recently developed MRI technique that allows quantitative determination of orientation-independent magnetic susceptibility parameters from the dependence of gradient echo signal phase on the orientation of biological tissues with respect to the main magnetic field. By modeling the magnetic susceptibility of each voxel as a symmetric rank-2 tensor, individual magnetic susceptibility tensor elements as well as the mean magnetic susceptibility (MMS) and magnetic susceptibility anisotropy (MSA) can be determined for brain tissues that would still show orientation dependence after conventional scalar-based quantitative susceptibility mapping (QSM) to remove such dependence. Similar to diffusion tensor imaging (DTI), STI allows mapping of brain white matter fiber orientations and reconstruction of 3D white matter pathways using the principal eigenvectors of the susceptibility tensor. In contrast to diffusion anisotropy, the main determinant factor of susceptibility anisotropy in brain white matter is myelin. Another unique feature of susceptibility anisotropy of white matter is its sensitivity to gadolinium-based contrast agents. Mechanistically, MRI-observed susceptibility anisotropy is mainly attributed to the highly ordered lipid molecules in myelin sheath. STI provides a consistent interpretation of the dependence of phase and susceptibility on orientation at multiple scales. This article reviews the key experimental findings and physical theories that led to the development of STI, its practical implementations, and its applications for brain research. PMID:27120169
17. Killing tensors and conformal Killing tensors from conformal Killing vectors
International Nuclear Information System (INIS)
Rani, Raffaele; Edgar, S Brian; Barnes, Alan
2003-01-01
Koutras has proposed some methods to construct reducible proper conformal Killing tensors and Killing tensors (which are, in general, irreducible) when a pair of orthogonal conformal Killing vectors exist in a given space. We give the completely general result demonstrating that this severe restriction of orthogonality is unnecessary. In addition, we correct and extend some results concerning Killing tensors constructed from a single conformal Killing vector. A number of examples demonstrate that it is possible to construct a much larger class of reducible proper conformal Killing tensors and Killing tensors than permitted by the Koutras algorithms. In particular, by showing that all conformal Killing tensors are reducible in conformally flat spaces, we have a method of constructing all conformal Killing tensors, and hence all the Killing tensors (which will in general be irreducible) of conformally flat spaces using their conformal Killing vectors
18. Tensors, relativity, and cosmology
CERN Document Server
2015-01-01
Tensors, Relativity, and Cosmology, Second Edition, combines relativity, astrophysics, and cosmology in a single volume, providing a simplified introduction to each subject that is followed by detailed mathematical derivations. The book includes a section on general relativity that gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes and Penrose processes), and considers the energy-momentum tensor for various solutions. In addition, a section on relativistic astrophysics discusses stellar contraction and collapse, neutron stars and their equations of state, black holes, and accretion onto collapsed objects, with a final section on cosmology discussing cosmological models, observational tests, and scenarios for the early universe. This fully revised and updated second edition includes new material on relativistic effects, such as the behavior of clocks and measuring rods in m...
19. Generalized Tensor-Based Morphometry of HIV/AIDS Using Multivariate Statistics on Deformation Tensors
OpenAIRE
Lepore, Natasha; Brun, Caroline; Chou, Yi-Yu; Chiang, Ming-Chang; Dutton, Rebecca A.; Hayashi, Kiralee M.; Luders, Eileen; Lopez, Oscar L.; Aizenstein, Howard J.; Toga, Arthur W.; Becker, James T.; Thompson, Paul M.
2008-01-01
This paper investigates the performance of a new multivariate method for tensor-based morphometry (TBM). Statistics on Riemannian manifolds are developed that exploit the full information in deformation tensor fields. In TBM, multiple brain images are warped to a common neuroanatomical template via 3-D nonlinear registration; the resulting deformation fields are analyzed statistically to identify group differences in anatomy. Rather than study the Jacobian determinant (volume expansion factor...
20. QCD vacuum tensor susceptibility and properties of transversely polarized mesons
International Nuclear Information System (INIS)
Bakulev, A.P.; Mikhajlov, S.V.
1999-01-01
We re-estimate the tensor susceptibility of QCD vacuum, χ, and to this end, we re-estimate the leptonic decay constants for transversely polarized ρ-, ρ'- and b 1 -mesons. The origin of the susceptibility is analyzed using duality between ρ- and b 1 -channels in a 2-point correlator of tensor currents and disagree with [2] on both OPE expansion and the value of QCD vacuum tensor susceptibility. Using our value for the latter we determine new estimations of nucleon tensor charges related to the first moment of the transverse structure functions h 1 of a nucleon
1. Prescribed curvature tensor in locally conformally flat manifolds
Science.gov (United States)
Pina, Romildo; Pieterzack, Mauricio
2018-01-01
A global existence theorem for the prescribed curvature tensor problem in locally conformally flat manifolds is proved for a special class of tensors R. Necessary and sufficient conditions for the existence of a metric g ¯ , conformal to Euclidean g, are determined such that R ¯ = R, where R ¯ is the Riemannian curvature tensor of the metric g ¯ . The solution to this problem is given explicitly for special cases of the tensor R, including the case where the metric g ¯ is complete on Rn. Similar problems are considered for locally conformally flat manifolds.
2. Thermodynamical inequivalence of quantum stress-energy and spin tensors
International Nuclear Information System (INIS)
Becattini, F.; Tinti, L.
2011-01-01
It is shown that different couples of stress-energy and spin tensors of quantum-relativistic fields, which would be otherwise equivalent, are in fact inequivalent if the second law of thermodynamics is taken into account. The proof of the inequivalence is based on the analysis of a macroscopic system at full thermodynamical equilibrium with a macroscopic total angular momentum and a specific instance is given for the free Dirac field, for which we show that the canonical and Belinfante stress-energy tensors are not equivalent. For this particular case, we show that the difference between the predicted angular momentum densities for a rotating system at full thermodynamical equilibrium is a quantum effect, persisting in the nonrelativistic limit, corresponding to a polarization of particles of the order of (ℎ/2π)ω/KT (ω being the angular velocity) and could in principle be measured experimentally. This result implies that specific stress-energy and spin tensors are physically meaningful even in the absence of gravitational coupling and raises the issue of finding the thermodynamically right (or the right class of) tensors. We argue that the maximization of the thermodynamic potential theoretically allows us to discriminate between two different couples, yet for the present we are unable to provide a theoretical method to single out the best couple of tensors in a given quantum field theory. The existence of a nonvanishing spin tensor would have major consequences in hydrodynamics, gravity and cosmology.
3. Determination of the components of three dimensional vector and tensor anisotropy of cosmic radiation with application to the results of the Musala experiment
International Nuclear Information System (INIS)
Somogyi, A.J.
1976-09-01
The paper proves that it is possible to interpret the experimental results of the Musala experiment as being consequences of a vector anisotropy with maximum in the direction of the galactic centre and a tensor anisotropy with principal axes in the physically plausible directions of the galactic arm, the normal direction of the galactic plane and the direction perpendicular them, respectively. It is underlined that the interpretation is not the only possible one and, in addition to this, statistical errors are rather large. The results favour the galactic origin of the particles concerned (E=6x10 13 eV). (Sz.N.Z.)
4. Couplings
Science.gov (United States)
Stošić, Dušan; Auroux, Aline
Basic principles of calorimetry coupled with other techniques are introduced. These methods are used in heterogeneous catalysis for characterization of acidic, basic and red-ox properties of solid catalysts. Estimation of these features is achieved by monitoring the interaction of various probe molecules with the surface of such materials. Overview of gas phase, as well as liquid phase techniques is given. Special attention is devoted to coupled calorimetry-volumetry method. Furthermore, the influence of different experimental parameters on the results of these techniques is discussed, since it is known that they can significantly influence the evaluation of catalytic properties of investigated materials.
5. Aspects of the Antisymmetric Tensor Field
Science.gov (United States)
Lahiri, Amitabha
1991-02-01
With the possible exception of gravitation, fundamental interactions are generally described by theories of point particles interacting via massless gauge fields. Since the advent of string theories the picture of physical interaction has changed to accommodate one in which extended objects interact with each other. The generalization of the gauge theories to extended objects leads to theories of antisymmetric tensor fields. At scales corresponding to present-day laboratory experiments one expects to see only point particles, their interactions modified by the presence of antisymmetric tensor fields in the theory. Therefore, in order to establish the validity of any theory with antisymmetric tensor fields one needs to look for manifestations of these fields at low energies. The principal problem of gauge theories is the failure to provide a suitable explanation for the generation of masses for the fields in the theory. While there is a known mechanism (spontaneous symmetry breaking) for generating masses for both the matter fields and the gauge fields, the lack of experimental evidence in support of an elementary scalar field suggests that one look for alternative ways of generating masses for the fields. The interaction of gauge fields with an antisymmetric tensor field seems to be an attractive way of doing so, especially since all indications point to the possibility that there will be no remnant degrees of freedom. On the other hand the interaction of such a field with black holes suggest an independent way of verifying the existence of such fields. In this dissertation the origins of the antisymmetric tensor field are discussed in terms of string theory. The interaction of black holes with such a field is discussed next. The last chapter discusses the effects of an antisymmetric tensor field on quantum electrodynamics when the fields are minimally coupled.
6. Tensor hypercontraction. II. Least-squares renormalization
Science.gov (United States)
Parrish, Robert M.; Hohenstein, Edward G.; Martínez, Todd J.; Sherrill, C. David
2012-12-01
The least-squares tensor hypercontraction (LS-THC) representation for the electron repulsion integral (ERI) tensor is presented. Recently, we developed the generic tensor hypercontraction (THC) ansatz, which represents the fourth-order ERI tensor as a product of five second-order tensors [E. G. Hohenstein, R. M. Parrish, and T. J. Martínez, J. Chem. Phys. 137, 044103 (2012)], 10.1063/1.4732310. Our initial algorithm for the generation of the THC factors involved a two-sided invocation of overlap-metric density fitting, followed by a PARAFAC decomposition, and is denoted PARAFAC tensor hypercontraction (PF-THC). LS-THC supersedes PF-THC by producing the THC factors through a least-squares renormalization of a spatial quadrature over the otherwise singular 1/r12 operator. Remarkably, an analytical and simple formula for the LS-THC factors exists. Using this formula, the factors may be generated with O(N^5) effort if exact integrals are decomposed, or O(N^4) effort if the decomposition is applied to density-fitted integrals, using any choice of density fitting metric. The accuracy of LS-THC is explored for a range of systems using both conventional and density-fitted integrals in the context of MP2. The grid fitting error is found to be negligible even for extremely sparse spatial quadrature grids. For the case of density-fitted integrals, the additional error incurred by the grid fitting step is generally markedly smaller than the underlying Coulomb-metric density fitting error. The present results, coupled with our previously published factorizations of MP2 and MP3, provide an efficient, robust O(N^4) approach to both methods. Moreover, LS-THC is generally applicable to many other methods in quantum chemistry.
7. On the SU2 unit tensor
International Nuclear Information System (INIS)
Kibler, M.; Grenet, G.
1979-07-01
The SU 2 unit tensor operators tsub(k,α) are studied. In the case where the spinor point group G* coincides with U 1 , then tsub(k α) reduces up to a constant to the Wigner-Racah-Schwinger tensor operator tsub(kqα), an operator which produces an angular momentum state. One first investigates those general properties of tsub(kα) which are independent of their realization. The tsub(kα) in terms of two pairs of boson creation and annihilation operators are realized. This leads to look at the Schwinger calculus relative to one angular momentum of two coupled angular momenta. As a by-product, a procedure is given for producing recursion relationships between SU 2 Wigner coefficients. Finally, some of the properties of the Wigner and Racah operators for an arbitrary compact group and the SU 2 coupling coefficients are studied
8. Second rank direction cosine spherical tensor operators and the nuclear electric quadrupole hyperfine structure Hamiltonian of rotating molecules
Science.gov (United States)
di Lauro, C.
2018-03-01
Transformations of vector or tensor properties from a space-fixed to a molecule-fixed axis system are often required in the study of rotating molecules. Spherical components λμ,ν of a first rank irreducible tensor can be obtained from the direction cosines between the two axis systems, and a second rank tensor with spherical components λμ,ν(2) can be built from the direct product λ × λ. It is shown that the treatment of the interaction between molecular rotation and the electric quadrupole of a nucleus is greatly simplified, if the coefficients in the axis-system transformation of the gradient of the electric field of the outer charges at the coupled nucleus are arranged as spherical components λμ,ν(2). Then the reduced matrix elements of the field gradient operators in a symmetric top eigenfunction basis, including their dependence on the molecule-fixed z-angular momentum component k, can be determined from the knowledge of those of λ(2) . The hyperfine structure Hamiltonian Hq is expressed as the sum of terms characterized each by a value of the molecule-fixed index ν, whose matrix elements obey the rule Δk = ν. Some of these terms may vanish because of molecular symmetry, and the specific cases of linear and symmetric top molecules, orthorhombic molecules, and molecules with symmetry lower than orthorhombic are considered. Each ν-term consists of a contraction of the rotational tensor λ(2) and the nuclear quadrupole tensor in the space-fixed frame, and its matrix elements in the rotation-nuclear spin coupled representation can be determined by the standard spherical tensor methods.
9. The simplicial Ricci tensor
International Nuclear Information System (INIS)
Alsing, Paul M; McDonald, Jonathan R; Miller, Warner A
2011-01-01
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The three-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The four-dimensional Ric is the Einstein tensor for such spacetimes. More recently, the Ric was used by Hamilton to define a nonlinear, diffusive Ricci flow (RF) that was fundamental to Perelman's proof of the Poincare conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area-an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimensions.
10. The simplicial Ricci tensor
Science.gov (United States)
Alsing, Paul M.; McDonald, Jonathan R.; Miller, Warner A.
2011-08-01
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The three-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The four-dimensional Ric is the Einstein tensor for such spacetimes. More recently, the Ric was used by Hamilton to define a nonlinear, diffusive Ricci flow (RF) that was fundamental to Perelman's proof of the Poincarè conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area—an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimensions.
11. Applied tensor stereology
DEFF Research Database (Denmark)
Ziegel, Johanna; Nyengaard, Jens Randel; Jensen, Eva B. Vedel
In the present paper, statistical procedures for estimating shape and orientation of arbitrary three-dimensional particles are developed. The focus of this work is on the case where the particles cannot be observed directly, but only via sections. Volume tensors are used for describing particle s...
12. J-Spectroscopy in the presence of residual dipolar couplings: determination of one-bond coupling constants and scalable resolution
International Nuclear Information System (INIS)
Furrer, Julien; John, Michael; Kessler, Horst; Luy, Burkhard
2007-01-01
The access to weak alignment media has fuelled the development of methods for efficiently and accurately measuring residual dipolar couplings (RDCs) in NMR-spectroscopy. Among the wealth of approaches for determining one-bond scalar and RDC constants only J-modulated and J-evolved techniques retain maximum resolution in the presence of differential relaxation. In this article, a number of J-evolved experiments are examined with respect to the achievable minimum linewidth in the J-dimension, using the peptide PA 4 and the 80-amino-acid-protein Saposin C as model systems. With the JE-N-BIRD d,X -HSQC experiment, the average full-width at half height could be reduced to approximately 5 Hz for the protein, which allows the additional resolution of otherwise unresolved peaks by the active (J+D)-coupling. Since RDCs generally can be scaled by the choice of alignment medium and alignment strength, the technique introduced here provides an effective resort in cases when chemical shift differences alone are insufficient for discriminating signals. In favorable cases even secondary structure elements can be distinguished
13. The evolution of tensor polarization
International Nuclear Information System (INIS)
Huang, H.; Lee, S.Y.; Ratner, L.
1993-01-01
By using the equation of motion for the vector polarization, the spin transfer matrix for spin tensor polarization, the spin transfer matrix for spin tensor polarization is derived. The evolution equation for the tensor polarization is studied in the presence of an isolate spin resonance and in the presence of a spin rotor, or snake
14. Tensor Calculus: Unlearning Vector Calculus
Science.gov (United States)
Lee, Wha-Suck; Engelbrecht, Johann; Moller, Rita
2018-01-01
Tensor calculus is critical in the study of the vector calculus of the surface of a body. Indeed, tensor calculus is a natural step-up for vector calculus. This paper presents some pitfalls of a traditional course in vector calculus in transitioning to tensor calculus. We show how a deeper emphasis on traditional topics such as the Jacobian can…
15. On deformed tensor potential for inelastic deuteron scattering
International Nuclear Information System (INIS)
Raynal, Jacques.
1980-08-01
Tensor analysing powers for inelastic deuteron scattering have been measured around 12 to 15 MeV. There is no problem to use such a tensor potential for the excited states in coupled channel calculations. However, for transition potentials, form factors are very different. A fit has been done with the first order vibrational model for 64 Ni(d,d') 64 Ni*, 2 + at 1,344 MeV
16. Determination of rare earth elements in aluminum by inductively coupled plasma-atomic emission spectroscopy
International Nuclear Information System (INIS)
Mahanti, H.S.; Barnes, R.M.
1983-01-01
Inductively coupled plasma-atomic emission spectroscopy is evaluated for the determination of 14 rare earth elements in aluminum. Spectral line interference, limit of detection, and background equivalent concentration values are evaluated, and quantitative recovery is obtained from aluminum samples spiked with rare earth elements. The procedure is simple and suitable for routine process control analysis. 20 references, 5 tables
17. Determination of platinum in human subcellular microsamples by inductively coupled plasma mass spectrometry
DEFF Research Database (Denmark)
Björn, Erik; Nygren, Yvonne; Nguyen, Tam T. T. N.
2007-01-01
A fast and robust method for the determination of platinum in human subcellular microsamples by inductively coupled plasma mass spectrometry was developed, characterized, and validated. Samples of isolated DNA and exosome fractions from human ovarian (2008) and melanoma (T289) cancer cell lines w...
18. The Riemann-Lovelock curvature tensor
International Nuclear Information System (INIS)
Kastor, David
2012-01-01
In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann-Lovelock tensor as a certain quantity having a total 4k-indices, which is kth order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth-order Riemann-Lovelock tensor is determined by its traces in dimensions 2k ≤ D < 4k. In D = 2k + 1 this identity implies that all solutions of pure kth-order Lovelock gravity are 'Riemann-Lovelock' flat. It is verified that the static, spherically symmetric solutions of these theories, which are missing solid angle spacetimes, indeed satisfy this flatness property. This generalizes results from Einstein gravity in D = 3, which corresponds to the k = 1 case. We speculate about some possible further consequences of Riemann-Lovelock curvature. (paper)
19. Gogny interactions with tensor terms
Energy Technology Data Exchange (ETDEWEB)
Anguiano, M.; Lallena, A.M.; Bernard, R.N. [Universidad de Granada, Departamento de Fisica Atomica, Molecular y Nuclear, Granada (Spain); Co' , G. [INFN, Lecce (Italy); De Donno, V. [Universita del Salento, Dipartimento di Matematica e Fisica ' ' E. De Giorgi' ' , Lecce (Italy); Grasso, M. [Universite Paris-Sud, Institut de Physique Nucleaire, IN2P3-CNRS, Orsay (France)
2016-07-15
We present a perturbative approach to include tensor terms in the Gogny interaction. We do not change the values of the usual parameterisations, with the only exception of the spin-orbit term, and we add tensor terms whose only free parameters are the strengths of the interactions. We identify observables sensitive to the presence of the tensor force in Hartree-Fock, Hartree-Fock-Bogoliubov and random phase approximation calculations. We show the need of including two tensor contributions, at least: a pure tensor term and a tensor-isospin term. We show results relevant for the inclusion of the tensor term for single-particle energies, charge-conserving magnetic excitations and Gamow-Teller excitations. (orig.)
20. Singular Poisson tensors
International Nuclear Information System (INIS)
Littlejohn, R.G.
1982-01-01
The Hamiltonian structures discovered by Morrison and Greene for various fluid equations were obtained by guessing a Hamiltonian and a suitable Poisson bracket formula, expressed in terms of noncanonical (but physical) coordinates. In general, such a procedure for obtaining a Hamiltonian system does not produce a Hamiltonian phase space in the usual sense (a symplectic manifold), but rather a family of symplectic manifolds. To state the matter in terms of a system with a finite number of degrees of freedom, the family of symplectic manifolds is parametrized by a set of Casimir functions, which are characterized by having vanishing Poisson brackets with all other functions. The number of independent Casimir functions is the corank of the Poisson tensor J/sup ij/, the components of which are the Poisson brackets of the coordinates among themselves. Thus, these Casimir functions exist only when the Poisson tensor is singular
1. TensorFlow Distributions
OpenAIRE
Dillon, Joshua V.; Langmore, Ian; Tran, Dustin; Brevdo, Eugene; Vasudevan, Srinivas; Moore, Dave; Patton, Brian; Alemi, Alex; Hoffman, Matt; Saurous, Rif A.
2017-01-01
The TensorFlow Distributions library implements a vision of probability theory adapted to the modern deep-learning paradigm of end-to-end differentiable computation. Building on two basic abstractions, it offers flexible building blocks for probabilistic computation. Distributions provide fast, numerically stable methods for generating samples and computing statistics, e.g., log density. Bijectors provide composable volume-tracking transformations with automatic caching. Together these enable...
2. Raman scattering tensors of tyrosine.
Science.gov (United States)
Tsuboi, M; Ezaki, Y; Aida, M; Suzuki, M; Yimit, A; Ushizawa, K; Ueda, T
1998-01-01
Polarized Raman scattering measurements have been made of a single crystal of L-tyrosine by the use of a Raman microscope with the 488.0-nm exciting beam from an argon ion laser. The L-tyrosine crystal belongs to the space group P2(1)2(1)2(1) (orthorhombic), and Raman scattering intensities corresponding to the aa, bb, cc, ab and ac components of the crystal Raman tensor have been determined for each prominent Raman band. A similar set of measurements has been made of L-tyrosine-d4, in which four hydrogen atoms on the benzene ring are replaced by deuterium atoms. The effects of NH3-->ND3 and OH-->OD on the Raman spectrum have also been examined. In addition, depolarization ratios of some bands of L-tyrosine in aqueous solutions of pH 13 and pH 1 were examined. For comparison with these experimental results, on the other hand, ab initio molecular orbital calculations have been made of the normal modes of vibration and their associated polarizability oscillations of the L-tyrosine molecule. On the basis of these experimental data and by referring to the results of the calculations, discussions have been presented on the Raman tensors associated to some Raman bands, including those at 829 cm-1 (benzene ring breathing), 642 cm-1 (benzene ring deformation), and 432 cm-1 (C alpha-C beta-C gamma bending).
3. The determination of transition probabilities with an inductively-coupled plasma discharge
International Nuclear Information System (INIS)
Nieuwoudt, G.
1984-03-01
The 27 MHz inductively-coupled plasma discharge (ICP) is used for the determination of relative transition probabilities of the 451, 459 and 470 nm argon spectral lines. The temperature of the argon plasma is determined with hydrogen as thermometric specie, because of the accurate transition probabilities ( approximately 1% uncertainty) there of. The relative transition probabilities of the specific argon spectral lines were determined by substitution of the measured spectral radiances thereof, together with the hydrogen temperature, in the two-line equation of temperature measurement
4. Determination of trace amounts of cerium in paint by inductively coupled plasma atomic emission spectrometry
International Nuclear Information System (INIS)
Wong, K.L.
1981-01-01
The determination of Ce in paint by inductively coupled plasma atomic emission spectrometry (ICP-OES) is described, and the detection limit of ICP-OES of 0.0004 ppM is compared with that of other methods. The effects of the major elemental components of paint, Si, Pb, Cr, and Na on the ICP-OES determination of Ce were studied. The interference of 400 ppM of the other ions on the determination of 10 ppM Ce was small (0 to 3% error). The method is applicable to the range of 0.2 to 700 ppM Ce
5. A fiber-coupled displacement measuring interferometer for determination of the posture of a reflective surface
International Nuclear Information System (INIS)
Mao, Shuai; Hu, Peng-Cheng; Ding, Xue-Mei; Tan, Jiu-Bin
2016-01-01
A fiber-coupled displacement measuring interferometer capable of determining of the posture of a reflective surface of a measuring mirror is proposed. The newly constructed instrument combines fiber-coupled displacement and angular measurement technologies. The proposed interferometer has advantages of both the fiber-coupled and the spatially beam-separated interferometer. A portable dual-position sensitive detector (PSD)-based unit within this proposed interferometer measures the parallelism of the two source beams to guide the fiber-coupling adjustment. The portable dual PSD-based unit measures not only the pitch and yaw of the retro-reflector but also measures the posture of the reflective surface. The experimental results of displacement calibration show that the deviations between the proposed interferometer and a reference one, Agilent 5530, at two different common beam directions are both less than ±35 nm, thus verifying the effectiveness of the beam parallelism measurement. The experimental results of angular calibration show that deviations of pitch and yaw with the auto-collimator (as a reference) are less than ±2 arc sec, thus proving the proposed interferometer’s effectiveness for determination of the posture of a reflective surface.
6. Tensor Permutation Matrices in Finite Dimensions
OpenAIRE
Christian, Rakotonirina
2005-01-01
We have generalised the properties with the tensor product, of one 4x4 matrix which is a permutation matrix, and we call a tensor commutation matrix. Tensor commutation matrices can be constructed with or without calculus. A formula allows us to construct a tensor permutation matrix, which is a generalisation of tensor commutation matrix, has been established. The expression of an element of a tensor commutation matrix has been generalised in the case of any element of a tensor permutation ma...
7. Tensor Factorization for Low-Rank Tensor Completion.
Science.gov (United States)
Zhou, Pan; Lu, Canyi; Lin, Zhouchen; Zhang, Chao
2018-03-01
Recently, a tensor nuclear norm (TNN) based method was proposed to solve the tensor completion problem, which has achieved state-of-the-art performance on image and video inpainting tasks. However, it requires computing tensor singular value decomposition (t-SVD), which costs much computation and thus cannot efficiently handle tensor data, due to its natural large scale. Motivated by TNN, we propose a novel low-rank tensor factorization method for efficiently solving the 3-way tensor completion problem. Our method preserves the low-rank structure of a tensor by factorizing it into the product of two tensors of smaller sizes. In the optimization process, our method only needs to update two smaller tensors, which can be more efficiently conducted than computing t-SVD. Furthermore, we prove that the proposed alternating minimization algorithm can converge to a Karush-Kuhn-Tucker point. Experimental results on the synthetic data recovery, image and video inpainting tasks clearly demonstrate the superior performance and efficiency of our developed method over state-of-the-arts including the TNN and matricization methods.
8. Determination of long-lived actinides in soil leachates by inductively coupled plasma: Mass spectrometry
International Nuclear Information System (INIS)
Crain, J.S.; Smith, L.L.; Yaeger, J.S.; Alvarado, J.A.
1994-01-01
Inductively coupled plasma -- mass spectrometry (ICP-MS) was used to concurrently determine multiple long-lived (t 1/2 > 10 4 y) actinide isotopes in soil samples. Ultrasonic nebulization was found to maximize instrument sensitivity. Instrument detection limits for actinides in solution ranged from 50 mBq L -1 ( 239 Pu) to 2 μBq L -1 ( 235 U) Hydride adducts of 232 Th and 238 U interfered with the determinations of 233 U and 239 Pu; thus, extraction chromatography was, used to eliminate the sample matrix, concentrate the analytes, and separate uranium from the other actinides. Alpha spectrometric determinations of 230 Th, 239 Pu, and the 234 U/ 238 U activity ratio in soil leachates compared well with ICP-MS determinations; however, there were some small systematic differences (ca. 10%) between ICP-MS and a-spectrometric determinations of 234 U and 238 U activities
9. Extracting the diffusion tensor from molecular dynamics simulation with Milestoning
International Nuclear Information System (INIS)
Mugnai, Mauro L.; Elber, Ron
2015-01-01
We propose an algorithm to extract the diffusion tensor from Molecular Dynamics simulations with Milestoning. A Kramers-Moyal expansion of a discrete master equation, which is the Markovian limit of the Milestoning theory, determines the diffusion tensor. To test the algorithm, we analyze overdamped Langevin trajectories and recover a multidimensional Fokker-Planck equation. The recovery process determines the flux through a mesh and estimates local kinetic parameters. Rate coefficients are converted to the derivatives of the potential of mean force and to coordinate dependent diffusion tensor. We illustrate the computation on simple models and on an atomically detailed system—the diffusion along the backbone torsions of a solvated alanine dipeptide
10. Microseismic Full Waveform Modeling in Anisotropic Media with Moment Tensor Implementation
Science.gov (United States)
Shi, Peidong; Angus, Doug; Nowacki, Andy; Yuan, Sanyi; Wang, Yanyan
2018-03-01
Seismic anisotropy which is common in shale and fractured rocks will cause travel-time and amplitude discrepancy in different propagation directions. For microseismic monitoring which is often implemented in shale or fractured rocks, seismic anisotropy needs to be carefully accounted for in source location and mechanism determination. We have developed an efficient finite-difference full waveform modeling tool with an arbitrary moment tensor source. The modeling tool is suitable for simulating wave propagation in anisotropic media for microseismic monitoring. As both dislocation and non-double-couple source are often observed in microseismic monitoring, an arbitrary moment tensor source is implemented in our forward modeling tool. The increments of shear stress are equally distributed on the staggered grid to implement an accurate and symmetric moment tensor source. Our modeling tool provides an efficient way to obtain the Green's function in anisotropic media, which is the key of anisotropic moment tensor inversion and source mechanism characterization in microseismic monitoring. In our research, wavefields in anisotropic media have been carefully simulated and analyzed in both surface array and downhole array. The variation characteristics of travel-time and amplitude of direct P- and S-wave in vertical transverse isotropic media and horizontal transverse isotropic media are distinct, thus providing a feasible way to distinguish and identify the anisotropic type of the subsurface. Analyzing the travel-times and amplitudes of the microseismic data is a feasible way to estimate the orientation and density of the induced cracks in hydraulic fracturing. Our anisotropic modeling tool can be used to generate and analyze microseismic full wavefield with full moment tensor source in anisotropic media, which can help promote the anisotropic interpretation and inversion of field data.
11. Black holes in vector-tensor theories
Energy Technology Data Exchange (ETDEWEB)
Heisenberg, Lavinia [Institute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich (Switzerland); Kase, Ryotaro; Tsujikawa, Shinji [Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601 (Japan); Minamitsuji, Masato, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Centro Multidisciplinar de Astrofisica—CENTRA, Departamento de Fisica, Instituto Superior Tecnico—IST, Universidade de Lisboa—UL, Avenida Rovisco Pais 1, 1049-001 Lisboa (Portugal)
2017-08-01
We study static and spherically symmetric black hole (BH) solutions in second-order generalized Proca theories with nonminimal vector field derivative couplings to the Ricci scalar, the Einstein tensor, and the double dual Riemann tensor. We find concrete Lagrangians which give rise to exact BH solutions by imposing two conditions of the two identical metric components and the constant norm of the vector field. These exact solutions are described by either Reissner-Nordström (RN), stealth Schwarzschild, or extremal RN solutions with a non-trivial longitudinal mode of the vector field. We then numerically construct BH solutions without imposing these conditions. For cubic and quartic Lagrangians with power-law couplings which encompass vector Galileons as the specific cases, we show the existence of BH solutions with the difference between two non-trivial metric components. The quintic-order power-law couplings do not give rise to non-trivial BH solutions regular throughout the horizon exterior. The sixth-order and intrinsic vector-mode couplings can lead to BH solutions with a secondary hair. For all the solutions, the vector field is regular at least at the future or past horizon. The deviation from General Relativity induced by the Proca hair can be potentially tested by future measurements of gravitational waves in the nonlinear regime of gravity.
12. Trilinear Higgs coupling determination via single-Higgs differential measurements at the LHC
Energy Technology Data Exchange (ETDEWEB)
Maltoni, Fabio; Shivaji, Ambresh; Zhao, Xiaoran [Universite Catholique de Louvain, Centre for Cosmology, Particle Physics and Phenomenology (CP3), Louvain-la-Neuve (Belgium); Pagani, Davide [Technische Universitaet Muenchen, Garching (Germany)
2017-12-15
We study one-loop effects induced by an anomalous Higgs trilinear coupling on total and differential rates for the H → 4l decay and some of the main single-Higgs production channels at the LHC, namely, VBF, VH, t anti tH and tHj. Our results are based on a public code that calculates these effects by simply reweighting samples of Standard-Model-like events for a given production channel. For VH and t anti tH production, where differential effects are particularly relevant, we include Standard Model electroweak corrections, which have similar sizes but different kinematic dependences. Finally, we study the sensitivity of future LHC runs to determine the trilinear coupling via inclusive and differential measurements, considering also the case where the Higgs couplings to vector bosons and the top quark is affected by new physics. We find that the constraints on the couplings and the relevance of differential distributions critically depend on the expected experimental and theoretical uncertainties. (orig.)
13. Optical determination of the electronic coupling and intercalation geometry of thiazole orange homodimer in DNA
Science.gov (United States)
Cunningham, Paul D.; Bricker, William P.; Díaz, Sebastián A.; Medintz, Igor L.; Bathe, Mark; Melinger, Joseph S.
2017-08-01
Sequence-selective bis-intercalating dyes exhibit large increases in fluorescence in the presence of specific DNA sequences. This property makes this class of fluorophore of particular importance to biosensing and super-resolution imaging. Here we report ultrafast transient anisotropy measurements of resonance energy transfer (RET) between thiazole orange (TO) molecules in a complex formed between the homodimer TOTO and double-stranded (ds) DNA. Biexponential homo-RET dynamics suggest two subpopulations within the ensemble: 80% intercalated and 20% non-intercalated. Based on the application of the transition density cube method to describe the electronic coupling and Monte Carlo simulations of the TOTO/dsDNA geometry, the dihedral angle between intercalated TO molecules is estimated to be 81° ± 5°, corresponding to a coupling strength of 45 ± 22 cm-1. Dye intercalation with this geometry is found to occur independently of the underlying DNA sequence, despite the known preference of TOTO for the nucleobase sequence CTAG. The non-intercalated subpopulation is inferred to have a mean inter-dye separation distance of 19 Å, corresponding to coupling strengths between 0 and 25 cm-1. This information is important to enable the rational design of energy transfer systems that utilize TOTO as a relay dye. The approach used here is generally applicable to determining the electronic coupling strength and intercalation configuration of other dimeric bis-intercalators.
14. Model-independent determination of the triple Higgs coupling at e+e- colliders
Science.gov (United States)
Barklow, Tim; Fujii, Keisuke; Jung, Sunghoon; Peskin, Michael E.; Tian, Junping
2018-03-01
The observation of Higgs pair production at high-energy colliders can give evidence for the presence of a triple Higgs coupling. However, the actual determination of the value of this coupling is more difficult. In the context of general models for new physics, double Higgs production processes can receive contributions from many possible beyond-Standard-Model effects. This dependence must be understood if one is to make a definite statement about the deviation of the Higgs field potential from the Standard Model. In this paper, we study the extraction of the triple Higgs coupling from the process e+e-→Z h h . We show that, by combining the measurement of this process with other measurements available at a 500 GeV e+e- collider, it is possible to quote model-independent limits on the effective field theory parameter c6 that parametrizes modifications of the Higgs potential. We present precise error estimates based on the anticipated International Linear Collider physics program, studied with full simulation. Our analysis also gives new insight into the model-independent extraction of the Higgs boson coupling constants and total width from e+e- data.
15. Trilinear Higgs coupling determination via single-Higgs differential measurements at the LHC
Science.gov (United States)
Maltoni, Fabio; Pagani, Davide; Shivaji, Ambresh; Zhao, Xiaoran
2017-12-01
We study one-loop effects induced by an anomalous Higgs trilinear coupling on total and differential rates for the H→ 4ℓ decay and some of the main single-Higgs production channels at the LHC, namely, VBF, VH, t{\\bar{t}}H and tHj. Our results are based on a public code that calculates these effects by simply reweighting samples of Standard-Model-like events for a given production channel. For VH and t{\\bar{t}}H production, where differential effects are particularly relevant, we include Standard Model electroweak corrections, which have similar sizes but different kinematic dependences. Finally, we study the sensitivity of future LHC runs to determine the trilinear coupling via inclusive and differential measurements, considering also the case where the Higgs couplings to vector bosons and the top quark is affected by new physics. We find that the constraints on the couplings and the relevance of differential distributions critically depend on the expected experimental and theoretical uncertainties.
16. Reduction schemes for one-loop tensor integrals
International Nuclear Information System (INIS)
Denner, A.; Dittmaier, S.
2006-01-01
We present new methods for the evaluation of one-loop tensor integrals which have been used in the calculation of the complete electroweak one-loop corrections to e + e - ->4 fermions. The described methods for 3-point and 4-point integrals are, in particular, applicable in the case where the conventional Passarino-Veltman reduction breaks down owing to the appearance of Gram determinants in the denominator. One method consists of different variants for expanding tensor coefficients about limits of vanishing Gram determinants or other kinematical determinants, thereby reducing all tensor coefficients to the usual scalar integrals. In a second method a specific tensor coefficient with a logarithmic integrand is evaluated numerically, and the remaining coefficients as well as the standard scalar integral are algebraically derived from this coefficient. For 5-point tensor integrals, we give explicit formulas that reduce the corresponding tensor coefficients to coefficients of 4-point integrals with tensor rank reduced by one. Similar formulas are provided for 6-point functions, and the generalization to functions with more internal propagators is straightforward. All the presented methods are also applicable if infrared (soft or collinear) divergences are treated in dimensional regularization or if mass parameters (for unstable particles) become complex
17. Tensor Train Neighborhood Preserving Embedding
Science.gov (United States)
Wang, Wenqi; Aggarwal, Vaneet; Aeron, Shuchin
2018-05-01
In this paper, we propose a Tensor Train Neighborhood Preserving Embedding (TTNPE) to embed multi-dimensional tensor data into low dimensional tensor subspace. Novel approaches to solve the optimization problem in TTNPE are proposed. For this embedding, we evaluate novel trade-off gain among classification, computation, and dimensionality reduction (storage) for supervised learning. It is shown that compared to the state-of-the-arts tensor embedding methods, TTNPE achieves superior trade-off in classification, computation, and dimensionality reduction in MNIST handwritten digits and Weizmann face datasets.
18. Notes on super Killing tensors
Energy Technology Data Exchange (ETDEWEB)
Howe, P.S. [Department of Mathematics, King’s College London,The Strand, London WC2R 2LS (United Kingdom); Lindström, University [Department of Physics and Astronomy, Theoretical Physics, Uppsala University,SE-751 20 Uppsala (Sweden); Theoretical Physics, Imperial College London,Prince Consort Road, London SW7 2AZ (United Kingdom)
2016-03-14
The notion of a Killing tensor is generalised to a superspace setting. Conserved quantities associated with these are defined for superparticles and Poisson brackets are used to define a supersymmetric version of the even Schouten-Nijenhuis bracket. Superconformal Killing tensors in flat superspaces are studied for spacetime dimensions 3,4,5,6 and 10. These tensors are also presented in analytic superspaces and super-twistor spaces for 3,4 and 6 dimensions. Algebraic structures associated with superconformal Killing tensors are also briefly discussed.
19. Tensor norms and operator ideals
CERN Document Server
Defant, A; Floret, K
1992-01-01
The three chapters of this book are entitled Basic Concepts, Tensor Norms, and Special Topics. The first may serve as part of an introductory course in Functional Analysis since it shows the powerful use of the projective and injective tensor norms, as well as the basics of the theory of operator ideals. The second chapter is the main part of the book: it presents the theory of tensor norms as designed by Grothendieck in the Resumé and deals with the relation between tensor norms and operator ideals. The last chapter deals with special questions. Each section is accompanied by a series of exer
20. Determination of rare earth elements in tomato plants by inductively coupled plasma mass spectrometry techniques.
Science.gov (United States)
Spalla, S; Baffi, C; Barbante, C; Turetta, C; Turretta, C; Cozzi, G; Beone, G M; Bettinelli, M
2009-10-30
In recent years identification of the geographical origin of food has grown more important as consumers have become interested in knowing the provenance of the food that they purchase and eat. Certification schemes and labels have thus been developed to protect consumers and genuine producers from the improper use of popular brand names or renowned geographical origins. As the tomato is one of the major components of what is considered to be the healthy Mediterranean diet, it is important to be able to determine the geographical origin of tomatoes and tomato-based products such as tomato sauce. The aim of this work is to develop an analytical method to determine rare earth elements (RRE) for the control of the geographic origin of tomatoes. The content of REE in tomato plant samples collected from an agricultural area in Piacenza, Italy, was determined, using four different digestion procedures with and without HF. Microwave dissolution with HNO3 + H2O2 proved to be the most suitable digestion procedure. Inductively coupled plasma quadrupole mass spectrometry (ICPQMS) and inductively coupled plasma sector field plasma mass spectrometry (ICPSFMS) instruments, both coupled with a desolvation system, were used to determine the REE in tomato plants in two different laboratories. A matched calibration curve method was used for the quantification of the analytes. The detection limits (MDLs) of the method ranged from 0.03 ng g(-1) for Ho, Tm, and Lu to 2 ng g(-1) for La and Ce. The precision, in terms of relative standard deviation on six replicates, was good, with values ranging, on average, from 6.0% for LREE (light rare earth elements) to 16.5% for HREE (heavy rare earth elements). These detection limits allowed the determination of the very low concentrations of REE present in tomato berries. For the concentrations of REE in tomato plants, the following trend was observed: roots > leaves > stems > berries. Copyright 2009 John Wiley & Sons, Ltd.
1. Temperature dependence of the dielectric tensor of monoclinic Ga2O3 single crystals in the spectral range 1.0-8.5 eV
Science.gov (United States)
Sturm, C.; Schmidt-Grund, R.; Zviagin, V.; Grundmann, M.
2017-08-01
The full dielectric tensor of monoclinic Ga2O3 (β-phase) was determined by generalized spectroscopic ellipsometry in the spectral range from 1.0 eV up to 8.5 eV and temperatures in the range from 10 K up to 300 K. By using the oriented dipole approach, the energies and broadenings of the excitonic transitions are determined as a function of the temperature, and the exciton-phonon coupling properties are deduced.
2. Data on final calcium concentration in native gel reagents determined accurately through inductively coupled plasma measurements
Directory of Open Access Journals (Sweden)
Jeffrey Viviano
2016-03-01
Full Text Available In this article we present data on the concentration of calcium as determined by Inductively Coupled Plasma (ICP measurements. Calcium was estimated in the reagents used for native gel electrophoresis of Neuronal Calcium Sensor (NCS proteins. NCS proteins exhibit calcium-dependent mobility shift in native gels. The sensitivity of this shift to calcium necessitated a precise determination of calcium concentrations in all reagents used. We determined the calcium concentrations in different components used along with the samples in the native gel experiments. These were: 20 mM Tris pH 7.5, loading dye and running buffer, with distilled water as reference. Calcium determinations were through ICP measurements. It was found that the running buffer contained calcium (244 nM over the blank. Keywords: Neuronal calcium sensor proteins, Electrophoresis, Mobility shift, Calcium, Magnesium
3. A new approach using artificial neural networks for determination of the thermodynamic properties of fluid couples
International Nuclear Information System (INIS)
Sencan, Arzu; Kalogirou, Soteris A.
2005-01-01
This paper presents a new approach using artificial neural networks (ANN) to determine the thermodynamic properties of two alternative refrigerant/absorbent couples (LiCl-H 2 O and LiBr + LiNO 3 + LiI + LiCl-H 2 O). These pairs can be used in absorption heat pump systems, and their main advantage is that they do not cause ozone depletion. In order to train the network, limited experimental measurements were used as training and test data. Two feedforward ANNs were trained, one for each pair, using the Levenberg-Marquardt algorithm. The training and validation were performed with good accuracy. The correlation coefficient obtained when unknown data were applied to the networks was 0.9997 and 0.9987 for the two pairs, respectively, which is very satisfactory. The present methodology proved to be much better than linear multiple regression analysis. Using the weights obtained from the trained network, a new formulation is presented for determination of the vapor pressures of the two refrigerant/absorbent couples. The use of this new formulation, which can be employed with any programming language or spreadsheet program for estimation of the vapor pressures of fluid couples, as described in this paper, may make the use of dedicated ANN software unnecessary
4. Endoscopic Anatomy of the Tensor Fold and Anterior Attic.
Science.gov (United States)
Li, Bin; Doan, Phi; Gruhl, Robert R; Rubini, Alessia; Marchioni, Daniele; Fina, Manuela
2018-02-01
Objectives The objectives of the study were to (1) study the anatomical variations of the tensor fold and its anatomic relation with transverse crest, supratubal recess, and anterior epitympanic space and (2) explore the most appropriate endoscopic surgical approach to each type of the tensor fold variants. Study Design Cadaver dissection study. Setting Temporal bone dissection laboratory. Subjects and Methods Twenty-eight human temporal bones (26 preserved and 2 fresh) were dissected through an endoscopic transcanal approach between September 2016 and June 2017. The anatomical variations of the tensor fold, transverse crest, supratubal recess, and anterior epitympanic space were studied before and after removing ossicles. Results Three different tensor fold orientations were observed: vertical (type A, 11/28, 39.3%) with attachment to the transverse crest, oblique (type B, 13/28, 46.4%) with attachment to the anterior tegmen tympani, and horizontal (type C, 4/28, 14.3%) with attachment to the tensor tympani canal. The tensor fold was a complete membrane in 20 of 28 (71.4%) specimens, preventing direct ventilation between the supratubal recess and anterior epitympanic space. We identified 3 surgical endoscopic approaches, which allowed visualization of the tensor fold without removing the ossicles. Conclusions The orientation of the tensor fold is the determining structure that dictates the conformation and limits of the epitympanic space. We propose a classification of the tensor fold based on 3 anatomical variants. We also describe 3 different minimally invasive endoscopic approaches to identify the orientation of the tensor fold while maintaining ossicular chain continuity.
5. The strong coupling from a nonperturbative determination of the Λ parameter in three-flavor QCD
Energy Technology Data Exchange (ETDEWEB)
Bruno, Mattia [Brookhaven National Laboratory, Upton, NY (United States). Physics Dept.; Dalla Brida, Mattia [Univ. di Milano-Bicocca (Italy). Dipt. di Fisica; INFN, Sezione di Milano-Bicocca (Italy); Fritzsch, Patrick; Ramos, Alberto [CERN, Geneva (Switzerland). Theoretical Physics Dept.; Korzec, Tomasz [Wuppertal Univ. (Germany). Dept. of Physics; Schaefer, Stefan; Simma, Hubert [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Sint, Stefan [Trinity College Dublin (Ireland). School of Mathematics and Hamilton Mathematics Inst.; Sommer, Rainer [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik; Collaboration: ALPHA Collaboration
2017-07-15
We present a lattice determination of the Λ parameter in three-flavor QCD and the strong coupling at the Z pole mass. Computing the nonperturbative running of the coupling in the range from 0.2 GeV to 70 GeV, and using experimental input values for the masses and decay constants of the pion and the kaon, we obtain Λ{sup (3)}{sub MS}=341(12) MeV. The nonperturbative running up to very high energies guarantees that systematic effects associated with perturbation theory are well under control. Using the four-loop prediction for Λ{sup (5)}{sub MS}/Λ{sup (3)}{sub MS} yields α{sup (5)}{sub MS}(m{sub Z})=0.11852(84).
6. Determination of diffusion profiles in thin film couples by means of X-ray-diffraction
International Nuclear Information System (INIS)
Wagendristel, A.
1975-01-01
An X-ray method for the determination of concentration profiles in thin film diffusion couples is presented. This method is based on the theory of Fourier analysis of X-ray diffraction profiles which is generalized to polycrystalline samples showing non-uniform lattice parameter. A Fourier synthesis of the concentration spectrum is possible when the influences of the particle size and the strain in the sample as well as the instrumental function are eliminated from the measured diffraction profile. This can be done by means of reference profiles obtained from layers of the diffusion components. Absorption of the radiation in the sample is negligible when diffusion couples of symmetrical sandwich structure are used. The method is tested experimentally in the system Au-Cu. (orig.) [de
7. Effective field theory approaches for tensor potentials
Energy Technology Data Exchange (ETDEWEB)
Jansen, Maximilian
2016-11-14
Effective field theories are a widely used tool to study physical systems at low energies. We apply them to systematically analyze two and three particles interacting via tensor potentials. Two examples are addressed: pion interactions for anti D{sup 0}D{sup *0} scattering to dynamically generate the X(3872) and dipole interactions for two and three bosons at low energies. For the former, the one-pion exchange and for the latter, the long-range dipole force induce a tensor-like structure of the potential. We apply perturbative as well as non-perturbative methods to determine low-energy observables. The X(3872) is of major interest in modern high-energy physics. Its exotic characteristics require approaches outside the range of the quark model for baryons and mesons. Effective field theories represent such methods and provide access to its peculiar nature. We interpret the X(3872) as a hadronic molecule consisting of neutral D and D{sup *} mesons. It is possible to apply an effective field theory with perturbative pions. Within this framework, we address chiral as well as finite volume extrapolations for low-energy observables, such as the binding energy and the scattering length. We show that the two-point correlation function for the D{sup *0} meson has to be resummed to cure infrared divergences. Moreover, next-to-leading order coupling constants, which were introduced by power counting arguments, appear to be essential to renormalize the scattering amplitude. The binding energy as well as the scattering length display a moderate dependence on the light quark masses. The X(3872) is most likely deeper bound for large light quark masses. In a finite volume on the other hand, the binding energy significantly increases. The dependence on the light quark masses and the volume size can be simultaneously obtained. For bosonic dipoles we apply a non-perturbative, numerical approach. We solve the Lippmann-Schwinger equation for the two-dipole system and the Faddeev
8. Progress in determination of long-lived radionuclides by inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Becker, J.S.; Dietze, H.J.
2000-01-01
Mass spectrometric methods (such as inductively coupled plasma mass spectrometry - ICP-MS and laser ablation (LA)-ICP-MS) with their ability to provide a very sensitive multielemental and precise isotopic analysis have become established for the determination of radionuclides in quite different sample materials. The determination of long-lived radionuclides is of increasing interest for the characterization of radioactive waste materials and for the detection of radionuclide contamination in environmental materials in which several radioactive nuclides are present from fallout due to nuclear weapons testing, nuclear power plants or nuclear accidents. Due to its multielement capability, excellent sensitivity, low detection limits (up to sub pg I 1 range), very good precision, easy sample preparation and measurement procedures ICP-MS of aqueous solutions has been increasingly applied for the ultrasensitive determination of long-lived radionuclides such as 99 Tc, 129 I, 230 Th, 232 Th, 234 U, 235 U, 236 U, 239 Pu, 240 Pu and 241 Am and precise isotope ratio measurements of U,Th and Pu. The application especially of microanalytical methods (analysis of some MU by flow injection and on-line coupling techniques as capillary electrophoresis (CE-ICP-MS) or HPLC-ICP-MS) for the precise determination nuclide abundances and concentration of long-lived radionuclides at ultra trace concentration levels in radioactive waste and also for controlling contamination from radioactive waste in the environment is a challenging task
9. Diffusion tensor studies and voxel-based morphometry of the temporal lobe to determine the cognitive prognosis in cases of Alzheimer's disease and mild cognitive impairment: Do white matter changes precede gray matter changes?
Science.gov (United States)
Taoka, Toshiaki; Yasuno, Fumihiko; Morikawa, Masayuki; Inoue, Makoto; Kiuchi, Kuniaki; Kitamura, Soichiro; Matsuoka, Kiwamu; Kishimoto, Toshifumi; Kichikawa, Kimihiko; Naganawa, Shinji
2016-01-01
The purpose of the current study was to assess the feasibility of diffusion tensor imaging (DTI) parameters for determining the prognosis of Alzheimer's disease (AD). We also analyzed the correlation among DTI, voxel-based morphometry (VBM), and results of the mini-mental state examination (MMSE). The subjects of this prospective study were patients with AD and mild cognitive impairment. We performed annual follow-ups with DTI, VBM, and MMSE for 2 or 3 years. On DTI, the apparent diffusion coefficient (ADC) and fractional anisotropy (FA) of the uncinate fascicles were measured. VBM was performed to provide a z-score for the parahippocampal gyrus. The correlations among these factors were evaluated in the same period and the next period of the follow-up study. For evaluation of the same period, both DTI parameters and z-scores showed statistically significant correlations with the MMSE score. Also for evaluation of the next period, both DTI parameters and z-scores showed statistically significant correlations with the MMSE score of the next period. We observed a statistically significant correlation between the ADC value of the uncinate fascicles and the z-score of the next period. Diffusion tensor parameters (ADC and FA) of the uncinate fascicles correlated well with cognitive function in the next year and seemed to be feasible for use as biomarkers for predicting the progression of AD. In addition, the white matter changes observed in the ADC seemed to precede changes in the gray matter volume of the parahippocampal gyrus that were represented by z-scores of VBM.
10. Typesafe Abstractions for Tensor Operations
OpenAIRE
Chen, Tongfei
2017-01-01
We propose a typesafe abstraction to tensors (i.e. multidimensional arrays) exploiting the type-level programming capabilities of Scala through heterogeneous lists (HList), and showcase typesafe abstractions of common tensor operations and various neural layers such as convolution or recurrent neural networks. This abstraction could lay the foundation of future typesafe deep learning frameworks that runs on Scala/JVM.
11. Indicial tensor manipulation on MACSYMA
International Nuclear Information System (INIS)
Bogen, R.A.; Pavelle, R.
1977-01-01
A new computational tool for physical calculations is described. It is the first computer system capable of performing indicial tensor calculus (as opposed to component tensor calculus). It is now operational on the symbolic manipulation system MACSYMA. The authors outline the capabilities of the system and describe some of the physical problems considered as well as others being examined at this time. (Auth.)
12. Tensor integrand reduction via Laurent expansion
Energy Technology Data Exchange (ETDEWEB)
Hirschi, Valentin [SLAC, National Accelerator Laboratory,2575 Sand Hill Road, Menlo Park, CA 94025-7090 (United States); Peraro, Tiziano [Higgs Centre for Theoretical Physics, School of Physics and Astronomy,The University of Edinburgh,Edinburgh EH9 3JZ, Scotland (United Kingdom)
2016-06-09
We introduce a new method for the application of one-loop integrand reduction via the Laurent expansion algorithm, as implemented in the public C++ library Ninja. We show how the coefficients of the Laurent expansion can be computed by suitable contractions of the loop numerator tensor with cut-dependent projectors, making it possible to interface Ninja to any one-loop matrix element generator that can provide the components of this tensor. We implemented this technique in the Ninja library and interfaced it to MADLOOP, which is part of the public MADGRAPH5{sub A}MC@NLO framework. We performed a detailed performance study, comparing against other public reduction tools, namely CUTTOOLS, SAMURAI, IREGI, PJFRY++ and GOLEM95. We find that Ninja outperforms traditional integrand reduction in both speed and numerical stability, the latter being on par with that of the tensor integral reduction tool GOLEM95 which is however more limited and slower than Ninja. We considered many benchmark multi-scale processes of increasing complexity, involving QCD and electro-weak corrections as well as effective non-renormalizable couplings, showing that Ninja’s performance scales well with both the rank and multiplicity of the considered process.
13. Multielement determination of rare earth elements by liquid chromatography/inductively coupled plasma atomic emission spectrometry
International Nuclear Information System (INIS)
Sawatari, Hideyuki; Asano, Takaaki; Hu, Xincheng; Saizuka, Tomoo; Itoh, Akihide; Hirose, Akio; Haraguchi, Hiroki
1995-01-01
The rapid determination of rare earth elements (REEs) has been investigated by an on-line system of high performance liquid chromatography/multichannel inductively coupled plasma atomic emission spectrometry. In the present system, all REEs could be detected simultaneously in a single chromatographic measurement without spectral interferences. Utilizing a cation exchange column and 2-hydroxy-2-methylpropanoic acid aqueous solution as the mobile phase, the detection limits of 0.4-30 ng ml -1 for all REEs were obtained. The system was applied to the determination of REEs in geological standard rock samples and rare earth impurities in high purity rare earth oxides. The REEs in standard rocks could be determined by the present HPLC/ICP-AES system without pretreatment after acid digestion, although the detection limits were not sufficient for the analysis of rare earth oxides. (author)
14. [Study on the determination of 14 inorganic elements in coffee by inductively coupled plasma mass spectrometry].
Science.gov (United States)
Nie, Xi-Du; Fu, Liang
2013-07-01
Samples of coffee were digested by microwave digestion, and inorganic elements amounts of Na, Mg, P, Ca, Cr, Mn, Fe, Co, Cu, Zn, As, Se, Mo and Pb in sample solutions were determined by inductively coupled plasma mass spectrometry (ICP-MS). HNO3 + H2O2 was used to achieve the complete decomposition of the organic matrix in a closed-vessel microwave oven. The working parameters of the instrument were optimized. The results showed that the relative standard deviation (RSD) was less than 3.84% for all the elements, and the recovery was found to be 92.00% -106.52% by adding standard recovery experiment. This method was simple, sensitive and precise and can perform simultaneous multi-elements determination of coffee, which could satisfy the sample examination request and provide scientific rationale for determining inorganic elements of coffee.
15. Tensor polarized deuteron targets for intermediate energy physics experiments
International Nuclear Information System (INIS)
Meyer, W.; Schilling, E.
1985-03-01
At intermediate energies measurements from a tensor polarized deuteron target are being prepared for the following reactions: the photodisintegration of the deuteron, the elastic pion-deuteron scattering and the elastic electron-deuteron scattering. The experimental situation of the polarization experiments for these reactions is briefly discussed in section 2. In section 3 the definitions of the deuteron polarization and the possibilities to determine the vector and tensor polarization are given. Present tensor polarization values and further improvements in this field are reported in section 4. (orig.)
16. arXiv Hybrid Fluid Models from Mutual Effective Metric Couplings
CERN Document Server
Kurkela, Aleksi; Preis, Florian; Rebhan, Anton; Soloviev, Alexander
Motivated by a semi-holographic approach to the dynamics of quark-gluon plasma which combines holographic and perturbative descriptions of a strongly coupled infrared and a more weakly coupled ultraviolet sector, we construct a hybrid two-fluid model where interactions between its two sectors are encoded by their effective metric backgrounds, which are determined mutually by their energy-momentum tensors. We derive the most general consistent ultralocal interactions such that the full system has a total conserved energy-momentum tensor in flat Minkowski space and study its consequences in and near thermal equilibrium by working out its phase structure and its hydrodynamic modes.
17. Atomic-batched tensor decomposed two-electron repulsion integrals
Science.gov (United States)
Schmitz, Gunnar; Madsen, Niels Kristian; Christiansen, Ove
2017-04-01
We present a new integral format for 4-index electron repulsion integrals, in which several strategies like the Resolution-of-the-Identity (RI) approximation and other more general tensor-decomposition techniques are combined with an atomic batching scheme. The 3-index RI integral tensor is divided into sub-tensors defined by atom pairs on which we perform an accelerated decomposition to the canonical product (CP) format. In a first step, the RI integrals are decomposed to a high-rank CP-like format by repeated singular value decompositions followed by a rank reduction, which uses a Tucker decomposition as an intermediate step to lower the prefactor of the algorithm. After decomposing the RI sub-tensors (within the Coulomb metric), they can be reassembled to the full decomposed tensor (RC approach) or the atomic batched format can be maintained (ABC approach). In the first case, the integrals are very similar to the well-known tensor hypercontraction integral format, which gained some attraction in recent years since it allows for quartic scaling implementations of MP2 and some coupled cluster methods. On the MP2 level, the RC and ABC approaches are compared concerning efficiency and storage requirements. Furthermore, the overall accuracy of this approach is assessed. Initial test calculations show a good accuracy and that it is not limited to small systems.
18. Quantum size effects in Pb layers with absorbed Kondo adatoms: Determination of the exchange coupling constant
KAUST Repository
Schwingenschlö gl, Udo; Shelykh, I. A.
2009-01-01
We consider the magnetic interaction of manganese phtalocyanine (MnPc) absorbed on Pb layers that were grown on a Si substrate. We perform an ab initio calculation of the density of states and Kondo temperature as a function of the number of Pb monolayers. Comparison to experimental data [Y.-S. Fu et al., Phys. Rev. Lett. 99, 256601 (2007)] then allows us to determine the exchange coupling constant J between the spins of the adsorbed molecules and those of the Pb host. This approach gives rise to a general and reliable method for obtaining J by combining experimental and numerical results.
19. Determination of the Axial-Vector Weak Coupling Constant with Ultracold Neutrons
International Nuclear Information System (INIS)
Liu, J.; Mendenhall, M. P.; Carr, R.; Filippone, B. W.; Hickerson, K. P.; Perez Galvan, A.; Russell, R.; Holley, A. T.; Hoagland, J.; VornDick, B.; Back, H. O.; Pattie, R. W. Jr.; Young, A. R.; Bowles, T. J.; Clayton, S.; Currie, S.; Hogan, G. E.; Ito, T. M.; Makela, M.; Morris, C. L.
2010-01-01
A precise measurement of the neutron decay β asymmetry A 0 has been carried out using polarized ultracold neutrons from the pulsed spallation ultracold neutron source at the Los Alamos Neutron Science Center. Combining data obtained in 2008 and 2009, we report A 0 =-0.119 66±0.000 89 -0.00140 +0.00123 , from which we determine the ratio of the axial-vector to vector weak coupling of the nucleon g A /g V =-1.275 90 -0.00445 +0.00409 .
20. Quantum size effects in Pb layers with absorbed Kondo adatoms: Determination of the exchange coupling constant
KAUST Repository
Schwingenschlögl, Udo
2009-07-01
We consider the magnetic interaction of manganese phtalocyanine (MnPc) absorbed on Pb layers that were grown on a Si substrate. We perform an ab initio calculation of the density of states and Kondo temperature as a function of the number of Pb monolayers. Comparison to experimental data [Y.-S. Fu et al., Phys. Rev. Lett. 99, 256601 (2007)] then allows us to determine the exchange coupling constant J between the spins of the adsorbed molecules and those of the Pb host. This approach gives rise to a general and reliable method for obtaining J by combining experimental and numerical results.
1. A supersymmetric SYK-like tensor model
Energy Technology Data Exchange (ETDEWEB)
Peng, Cheng; Spradlin, Marcus; Volovich, Anastasia [Department of Physics, Brown University,Providence, RI, 02912 (United States)
2017-05-11
We consider a supersymmetric SYK-like model without quenched disorder that is built by coupling two kinds of fermionic N=1 tensor-valued superfields, “quarks” and “mesons”. We prove that the model has a well-defined large-N limit in which the (s)quark 2-point functions are dominated by mesonic “melon” diagrams. We sum these diagrams to obtain the Schwinger-Dyson equations and show that in the IR, the solution agrees with that of the supersymmetric SYK model.
2. Tensor glueball-meson mixing phenomenology
International Nuclear Information System (INIS)
Burakovsky, L.; Page, P.R.
2000-01-01
The overpopulated isoscalar tensor states are sifted using Schwinger-type mass relations. Two solutions are found: one where the glueball is the f J (2220), and one where the glueball is more distributed, with f 2 (1820) having the largest component. The f 2 (1565) and f J (1710) cannot be accommodated as glueball-(hybrid) meson mixtures in the absence of significant coupling to decay channels. f 2 '(1525)→ππ is in agreement with experiment. The f J (2220) decays neither flavour democratically nor is narrow. (orig.)
3. A supersymmetric SYK-like tensor model
International Nuclear Information System (INIS)
Peng, Cheng; Spradlin, Marcus; Volovich, Anastasia
2017-01-01
We consider a supersymmetric SYK-like model without quenched disorder that is built by coupling two kinds of fermionic N=1 tensor-valued superfields, “quarks” and “mesons”. We prove that the model has a well-defined large-N limit in which the (s)quark 2-point functions are dominated by mesonic “melon” diagrams. We sum these diagrams to obtain the Schwinger-Dyson equations and show that in the IR, the solution agrees with that of the supersymmetric SYK model.
4. Killing-Yano tensors and Nambu mechanics
International Nuclear Information System (INIS)
Baleanu, D.
1998-01-01
Killing-Yano tensors were introduced in 1952 by Kentaro-Yano from mathematical point of view. The physical interpretation of Killing-Yano tensors of rank higher than two was unclear. We found that all Killing-Yano tensors η i 1 i 2 . .. i n with covariant derivative zero are Nambu tensors. We found that in the case of flat space case all Killing-Yano tensors are Nambu tensors. In the case of Taub-NUT and Kerr-Newmann metric Killing-Yano tensors of order two generate Nambu tensors of rank 3
5. Comparison of alignment tensors generated for native tRNAVal using magnetic fields and liquid crystalline media
International Nuclear Information System (INIS)
Latham, Michael P.; Hanson, Paul; Brown, Darin J.; Pardi, Arthur
2008-01-01
Residual dipolar couplings (RDCs) complement standard NOE distance and J-coupling torsion angle data to improve the local and global structure of biomolecules in solution. One powerful application of RDCs is for domain orientation studies, which are especially valuable for structural studies of nucleic acids, where the local structure of a double helix is readily modeled and the orientations of the helical domains can then be determined from RDC data. However, RDCs obtained from only one alignment media generally result in degenerate solutions for the orientation of multiple domains. In protein systems, different alignment media are typically used to eliminate this orientational degeneracy, where the combination of RDCs from two (or more) independent alignment tensors can be used to overcome this degeneracy. It is demonstrated here for native E. coli tRNA Val that many of the commonly used liquid crystalline alignment media result in very similar alignment tensors, which do not eliminate the 4-fold degeneracy for orienting the two helical domains in tRNA. The intrinsic magnetic susceptibility anisotropy (MSA) of the nucleobases in tRNA Val was also used to obtain RDCs for magnetic alignment at 800 and 900 MHz. While these RDCs yield a different alignment tensor, the specific orientation of this tensor combined with the high rhombicity for the tensors in the liquid crystalline media only eliminates two of the four degenerate orientations for tRNA Val . Simulations are used to show that, in optimal cases, the combination of RDCs obtained from liquid crystalline medium and MSA-induced alignment can be used to obtain a unique orientation for the two helical domains in tRNA Val
6. Effect on Tensor Correlations on Gamow- Teller States in 90Zr and 208Pb
International Nuclear Information System (INIS)
Bai, C. L.; Sagawa, H.; Zhang, H. Q.
2009-01-01
The tensor terms of the Skyrme effective interaction are included in the self-consistent Hartree-Fock plus Random Phase Approximation (HF-RPA) model. The Gamow-Teller (GT) strength function of 9 0Z r and 2 08P b are calculated with and without the tensor terms. The main peaks are moved downwards by about 2 MeV when including the tensor contribution. About 10% of the non-energy weighted sum rule is shifted to the excitation energy region above 30 MeV by the RPA tensor correlations. The contribution of the tensor terms to the energy weighted sum rule is given analytically, and compared to the outcome of RPA. A microscopic origin of the quenching of GT sum rule is discussed in relation with the coupling to giant spin-quadrupole excitations by the tensor interactions.(author)
7. Tensor-GMRES method for large sparse systems of nonlinear equations
Science.gov (United States)
Feng, Dan; Pulliam, Thomas H.
1994-01-01
This paper introduces a tensor-Krylov method, the tensor-GMRES method, for large sparse systems of nonlinear equations. This method is a coupling of tensor model formation and solution techniques for nonlinear equations with Krylov subspace projection techniques for unsymmetric systems of linear equations. Traditional tensor methods for nonlinear equations are based on a quadratic model of the nonlinear function, a standard linear model augmented by a simple second order term. These methods are shown to be significantly more efficient than standard methods both on nonsingular problems and on problems where the Jacobian matrix at the solution is singular. A major disadvantage of the traditional tensor methods is that the solution of the tensor model requires the factorization of the Jacobian matrix, which may not be suitable for problems where the Jacobian matrix is large and has a 'bad' sparsity structure for an efficient factorization. We overcome this difficulty by forming and solving the tensor model using an extension of a Newton-GMRES scheme. Like traditional tensor methods, we show that the new tensor method has significant computational advantages over the analogous Newton counterpart. Consistent with Krylov subspace based methods, the new tensor method does not depend on the factorization of the Jacobian matrix. As a matter of fact, the Jacobian matrix is never needed explicitly.
8. Tensor Rank Preserving Discriminant Analysis for Facial Recognition.
Science.gov (United States)
Tao, Dapeng; Guo, Yanan; Li, Yaotang; Gao, Xinbo
2017-10-12
Facial recognition, one of the basic topics in computer vision and pattern recognition, has received substantial attention in recent years. However, for those traditional facial recognition algorithms, the facial images are reshaped to a long vector, thereby losing part of the original spatial constraints of each pixel. In this paper, a new tensor-based feature extraction algorithm termed tensor rank preserving discriminant analysis (TRPDA) for facial image recognition is proposed; the proposed method involves two stages: in the first stage, the low-dimensional tensor subspace of the original input tensor samples was obtained; in the second stage, discriminative locality alignment was utilized to obtain the ultimate vector feature representation for subsequent facial recognition. On the one hand, the proposed TRPDA algorithm fully utilizes the natural structure of the input samples, and it applies an optimization criterion that can directly handle the tensor spectral analysis problem, thereby decreasing the computation cost compared those traditional tensor-based feature selection algorithms. On the other hand, the proposed TRPDA algorithm extracts feature by finding a tensor subspace that preserves most of the rank order information of the intra-class input samples. Experiments on the three facial databases are performed here to determine the effectiveness of the proposed TRPDA algorithm.
9. Nonperturbative loop quantization of scalar-tensor theories of gravity
International Nuclear Information System (INIS)
Zhang Xiangdong; Ma Yongge
2011-01-01
The Hamiltonian formulation of scalar-tensor theories of gravity is derived from their Lagrangian formulation by Hamiltonian analysis. The Hamiltonian formalism marks off two sectors of the theories by the coupling parameter ω(φ). In the sector of ω(φ)=-(3/2), the feasible theories are restricted and a new primary constraint generating conformal transformations of spacetime is obtained, while in the other sector of ω(φ)≠-(3/2), the canonical structure and constraint algebra of the theories are similar to those of general relativity coupled with a scalar field. By canonical transformations, we further obtain the connection-dynamical formalism of the scalar-tensor theories with real su(2) connections as configuration variables in both sectors. This formalism enables us to extend the scheme of nonperturbative loop quantum gravity to the scalar-tensor theories. The quantum kinematical framework for the scalar-tensor theories is rigorously constructed. Both the Hamiltonian constraint operator and master constraint operator are well defined and proposed to represent quantum dynamics. Thus the loop quantum gravity method is also valid for general scalar-tensor theories.
10. Radionuclide determination in environmental samples by inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Lariviere, Dominic; Taylor, Vivien F.; Evans, R. Douglas; Cornett, R. Jack
2006-01-01
The determination of naturally occurring and anthropogenic radionuclides in the environment by inductively coupled plasma mass spectrometry has gained recognition over the last fifteen years, relative to radiometric techniques, as the result of improvement in instrumental performance, sample introduction equipment, and sample preparation. With the increase in instrumental sensitivity, it is now possible to measure ultratrace levels (fg range) of many radioisotopes, including those with half-lives between 1 and 1000 years, without requiring very complex sample pre-concentration schemes. However, the identification and quantification of radioisotopes in environmental matrices is still hampered by a variety of analytical issues such as spectral (both atomic and molecular ions) and non-spectral (matrix effect) interferences and instrumental limitations (e.g., abundance sensitivity). The scope of this review is to highlight recent analytical progress and issues associated with the determination of radionuclides by inductively coupled plasma mass spectrometry. The impact of interferences, instrumental limitations (e.g., degree of ionization, abundance sensitivity, detection limits) and low sample-to-plasma transfer efficiency on the measurement of radionuclides by inductively coupled plasma mass spectrometry will be described. Solutions that overcome these issues will be discussed, highlighting their pros and cons and assessing their impact on the measurement of environmental radioactivity. Among the solutions proposed, mass and chemical resolution through the use of sector-field instruments and chemical reactions/collisions in a pressurized cell, respectively, will be described. Other methods, such as unique sample introduction equipment (e.g., laser ablation, electrothermal vaporisation, high efficiency nebulization) and instrumental modifications/optimizations (e.g., instrumental vacuum, radiofrequency power, guard electrode) that improve sensitivity and performance
11. Tensor-optimized shell model for the Li isotopes with a bare nucleon-nucleon interaction
Science.gov (United States)
Myo, Takayuki; Umeya, Atsushi; Toki, Hiroshi; Ikeda, Kiyomi
2012-08-01
We study the Li isotopes systematically in terms of the tensor-optimized shell model (TOSM) by using a bare nucleon-nucleon interaction as the AV8' interaction. The short-range correlation is treated in the unitary correlation operator method (UCOM). Using the TOSM + UCOM approach, we investigate the role of the tensor force on each spectrum of the Li isotopes. It is found that the tensor force produces quite a characteristic effect on various states in each spectrum and those spectra are affected considerably by the tensor force. The energy difference between the spin-orbit partner, the p1/2 and p3/2 orbits of the last neutron, in 5Li is caused by opposite roles of the tensor correlation. In 6Li, the spin-triplet state in the LS coupling configuration is favored energetically by the tensor force in comparison with jj coupling shell-model states. In 7,8,9Li, the low-lying states containing extra neutrons in the p3/2 orbit are favored energetically due to the large tensor contribution to allow the excitation from the 0s, orbit to the p1/2 orbit by the tensor force. Those three nuclei show the jj coupling character in their ground states which is different from 6Li.
12. Determination of Dibutyltin in Sediments Using Isotope Dilution Liquid Chromatography-Inductively Coupled Plasma Mass Spectrometry
International Nuclear Information System (INIS)
Yim, Yong Hyeon; Park, Ji Youn; Han, Myung Sub; Park, Mi Kyung; Kim, Byung Joo; Lim, Young Ran; Hwang, Eui Jin; So, Hun Young
2005-01-01
A method is described for the determination of dibutyltin (DBT) in sediment by isotope dilution using liquid chromatography inductively-coupled plasma/mass spectrometry (LC-ICP/MS). To achieve the highest accuracy and precision, special attentions are paid in optimization and evaluation of overall processes of the analysis including extraction of analytes, characterization of the standards used for calibration and LC-ICP/MS conditions. An approach for characterization of natural abundance DBT standard has been developed by combining inductively-coupled plasma/optical emission spectrometry (ICP/OES) and LC-ICP/MS for the total Sn assay and the analysis of Sn species present as impurities, respectively. An excellent LC condition for separation of organotin species was found, which is suitable for simultaneous DBT and tributyltin (TBT) analysis as well as impurity analysis of DBT standards. Microwave extraction condition was also optimized for high efficiency while preventing species transformation. The present method determines the amount contents of DBT in sediments with expanded uncertainty of less than 5% and its result shows high degree of equivalence with reference values of an international inter-comparison and a certified reference material (CRM) within stated uncertainties
13. Determination of Dibutyltin in Sediments Using Isotope Dilution Liquid Chromatography-Inductively Coupled Plasma Mass Spectrometry
Energy Technology Data Exchange (ETDEWEB)
Yim, Yong Hyeon; Park, Ji Youn; Han, Myung Sub; Park, Mi Kyung; Kim, Byung Joo; Lim, Young Ran; Hwang, Eui Jin; So, Hun Young [Korea Research Institute of Standards and Science, Daejeon (Korea, Republic of)
2005-03-15
A method is described for the determination of dibutyltin (DBT) in sediment by isotope dilution using liquid chromatography inductively-coupled plasma/mass spectrometry (LC-ICP/MS). To achieve the highest accuracy and precision, special attentions are paid in optimization and evaluation of overall processes of the analysis including extraction of analytes, characterization of the standards used for calibration and LC-ICP/MS conditions. An approach for characterization of natural abundance DBT standard has been developed by combining inductively-coupled plasma/optical emission spectrometry (ICP/OES) and LC-ICP/MS for the total Sn assay and the analysis of Sn species present as impurities, respectively. An excellent LC condition for separation of organotin species was found, which is suitable for simultaneous DBT and tributyltin (TBT) analysis as well as impurity analysis of DBT standards. Microwave extraction condition was also optimized for high efficiency while preventing species transformation. The present method determines the amount contents of DBT in sediments with expanded uncertainty of less than 5% and its result shows high degree of equivalence with reference values of an international inter-comparison and a certified reference material (CRM) within stated uncertainties.
14. Trace determination of Pu by LIF in an inductively coupled plasma
International Nuclear Information System (INIS)
Mauchien, P.; Briand, A.; Moulin, C.
1989-01-01
Inductively Coupled Plasma/Emission Spectrometry (ICP/ES) technique is largely used in the nuclear industry as an elementary analytical technique. Nevertheless, when the sample to analyse presents elements with a lot of emission spectral lines, spectral interferences lead to limited sensitivity. This is the case for Pu determination in presence of large U concentration. In pure aqueous solution, the limit of detection (LOD) for Pu is 10 μg/1. In presence of U, the LOD is determined by a ratio U/Pu = 1000. Pulsed Laser Induced Fluorescence (LIF) spectrometry is known to be a very selective technique when associated with an Inductively Coupled Plasma source. The absolute sensitivity is better by 2 or 3 orders of magnitude; its principle is based on selective excitation of the ionic species in the plasma followed by fluorescence radiation detection of these species; this radiation being practically free from spectral interferences, it is possible to improve the relative LOD. In this presentation, experimental results performed at Cogema/Marcoule laboratory are presented. After the experimental set-up description, first results of LIF are shown: - very good selectivity is effectively obtained, - a series of analytical results obtained with excitation scanning from the visible to the U.V. show that sensitivity of LIF technique is strictly related to the spectroscopic scheme
15. Determination of the pion-nucleon coupling constant and scattering lengths
CERN Document Server
Ericson, Torleif Eric Oskar; Thomas, A W
2002-01-01
We critically evaluate the isovector GMO sum rule for forward pion-nucleon scattering using the recent precision measurements of negatively charged pion-proton and pion-deuteron scattering lengths from pionic atoms. We deduce the charged-pion-nucleon coupling constant, with careful attention to systematic and statistical uncertainties. This determination gives, directly from data a pseudoscalar coupling constant of 14.17+-0.05(statistical)+-0.19(systematic) or a pseudovector one of 0.0786(11). This value is intermediate between that of indirect methods and the direct determination from backward neutron-proton differential scattering cross sections. We also use the pionic atom data to deduce the coherent symmetric and antisymmetric sums of the negatively charged pion-proton and pion-neutron scattering lengths with high precision. The symmetric sum gives 0.0017+-0.0002(statistical)+-0.0008 (systematic) and the antisymmetric one 0.0900+-0.0003(statistical)+-0.0013(systematic), both in units of inverse charged pi...
16. MATLAB tensor classes for fast algorithm prototyping.
Energy Technology Data Exchange (ETDEWEB)
Bader, Brett William; Kolda, Tamara Gibson (Sandia National Laboratories, Livermore, CA)
2004-10-01
Tensors (also known as mutidimensional arrays or N-way arrays) are used in a variety of applications ranging from chemometrics to psychometrics. We describe four MATLAB classes for tensor manipulations that can be used for fast algorithm prototyping. The tensor class extends the functionality of MATLAB's multidimensional arrays by supporting additional operations such as tensor multiplication. The tensor as matrix class supports the 'matricization' of a tensor, i.e., the conversion of a tensor to a matrix (and vice versa), a commonly used operation in many algorithms. Two additional classes represent tensors stored in decomposed formats: cp tensor and tucker tensor. We descibe all of these classes and then demonstrate their use by showing how to implement several tensor algorithms that have appeared in the literature.
17. Determination of rare earth elements by liquid chromatography/inductively coupled plasma atomic emission spectrometry
International Nuclear Information System (INIS)
Yoshida, K.; Haraguchi, H.
1984-01-01
Inductively coupled plasma atomic emission spectrometry (ICP-AES) interfaced with high-performance liquid chromatography (HPLC) has been applied to the determination of rare earth elements. ICP-AES was used as an element-selective detector for HPLC. The separation of rare earth elements with HPLC helped to avoid erroneous analytical results due to spectral interferences. Fifteen rare earth elements (Y and 14 lanthanides) were determined selectively with the HPLC/ICP-AES system using a concentration gradient method. The detection limits with the present HPLC/ICP-AES system were about 0.001-0.3 μg/mL with a 100-μL sample injection. The calibration curves obtained by the peak height measurements showed linear relationships in the concentration range below 500 μg/mL for all rare earth elements. A USGS rock standard sample, rare earth ores, and high-purity lanthanide reagents (>99.9%) were successfully analyzed without spectral interferences
18. Two-dimensional NMR measurement and point dipole model prediction of paramagnetic shift tensors in solids
Energy Technology Data Exchange (ETDEWEB)
Walder, Brennan J.; Davis, Michael C.; Grandinetti, Philip J. [Department of Chemistry, Ohio State University, 100 West 18th Avenue, Columbus, Ohio 43210 (United States); Dey, Krishna K. [Department of Physics, Dr. H. S. Gour University, Sagar, Madhya Pradesh 470003 (India); Baltisberger, Jay H. [Division of Natural Science, Mathematics, and Nursing, Berea College, Berea, Kentucky 40403 (United States)
2015-01-07
A new two-dimensional Nuclear Magnetic Resonance (NMR) experiment to separate and correlate the first-order quadrupolar and chemical/paramagnetic shift interactions is described. This experiment, which we call the shifting-d echo experiment, allows a more precise determination of tensor principal components values and their relative orientation. It is designed using the recently introduced symmetry pathway concept. A comparison of the shifting-d experiment with earlier proposed methods is presented and experimentally illustrated in the case of {sup 2}H (I = 1) paramagnetic shift and quadrupolar tensors of CuCl{sub 2}⋅2D{sub 2}O. The benefits of the shifting-d echo experiment over other methods are a factor of two improvement in sensitivity and the suppression of major artifacts. From the 2D lineshape analysis of the shifting-d spectrum, the {sup 2}H quadrupolar coupling parameters are 〈C{sub q}〉 = 118.1 kHz and 〈η{sub q}〉 = 0.88, and the {sup 2}H paramagnetic shift tensor anisotropy parameters are 〈ζ{sub P}〉 = − 152.5 ppm and 〈η{sub P}〉 = 0.91. The orientation of the quadrupolar coupling principal axis system (PAS) relative to the paramagnetic shift anisotropy principal axis system is given by (α,β,γ)=((π)/2 ,(π)/2 ,0). Using a simple ligand hopping model, the tensor parameters in the absence of exchange are estimated. On the basis of this analysis, the instantaneous principal components and orientation of the quadrupolar coupling are found to be in excellent agreement with previous measurements. A new point dipole model for predicting the paramagnetic shift tensor is proposed yielding significantly better agreement than previously used models. In the new model, the dipoles are displaced from nuclei at positions associated with high electron density in the singly occupied molecular orbital predicted from ligand field theory.
19. Efficient tensor completion for color image and video recovery: Low-rank tensor train
OpenAIRE
Bengua, Johann A.; Phien, Ho N.; Tuan, Hoang D.; Do, Minh N.
2016-01-01
This paper proposes a novel approach to tensor completion, which recovers missing entries of data represented by tensors. The approach is based on the tensor train (TT) rank, which is able to capture hidden information from tensors thanks to its definition from a well-balanced matricization scheme. Accordingly, new optimization formulations for tensor completion are proposed as well as two new algorithms for their solution. The first one called simple low-rank tensor completion via tensor tra...
20. Random SU(2) invariant tensors
Science.gov (United States)
Li, Youning; Han, Muxin; Ruan, Dong; Zeng, Bei
2018-04-01
SU(2) invariant tensors are states in the (local) SU(2) tensor product representation but invariant under the global group action. They are of importance in the study of loop quantum gravity. A random tensor is an ensemble of tensor states. An average over the ensemble is carried out when computing any physical quantities. The random tensor exhibits a phenomenon known as ‘concentration of measure’, which states that for any bipartition the average value of entanglement entropy of its reduced density matrix is asymptotically the maximal possible as the local dimensions go to infinity. We show that this phenomenon is also true when the average is over the SU(2) invariant subspace instead of the entire space for rank-n tensors in general. It is shown in our earlier work Li et al (2017 New J. Phys. 19 063029) that the subleading correction of the entanglement entropy has a mild logarithmic divergence when n = 4. In this paper, we show that for n > 4 the subleading correction is not divergent but a finite number. In some special situation, the number could be even smaller than 1/2, which is the subleading correction of random state over the entire Hilbert space of tensors.
1. A new procedure for coupling antibody to paper discs for radioimmunoassay: application to the determination of alpha-fetoprotein
International Nuclear Information System (INIS)
Sasaki, T.; Tsukada, Y.; Hirai, H.
1983-01-01
Horse anti-alpha-fetoprotein was coupled to CM-cellulose discs by a modified carbodiimide reaction. The resulting coupled CM-discs were used in solid-phase radioimmunoassay of human alpha-fetoprotein. The sensitivity of these discs and conventional BrCN activated filter paper discs coupled anti-alpha-fetoprotein was approximately the same. A fair correlation between the alpha-fetoprotein levels determined by both methods was observed. The coupling procedure with carbodiimide is simple and the use of hazardous BrCN is eliminated. (Auth.)
2. Extended pure Yang-Mills gauge theories with scalar and tensor gauge fields
International Nuclear Information System (INIS)
Gabrielli, E.
1991-01-01
The usual abelian gauge theory is extended to an interacting Yang-Mills-like theory containing vector, scalar and tensor gauge fields. These gauge fields are seen as components along the Clifford algebra basis of a gauge vector-spinorial field. Scalar fields φ naturally coupled to vector and tensor fields have been found, leading to a natural φ 4 coupling in the lagrangian. The full expression of the lagrangian for the euclidean version of the theory is given. (orig.)
3. Multielement determination of rare earth elements in rock sample by liquid chromatography / inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Hamanaka, Tadashi; Itoh, Akihide; Itoh, Shinya; Sawatari, Hideyuki; Haraguchi, Hiroki.
1995-01-01
Rare earth elements in geological standard rock sample JG-1 (granodiolite)issued from the Geological Survey of Japan have been determined by a combined system of liquid chromatography and inductively coupled plasma mass spectrometry. (author)
4. Complete algebraic reduction of one-loop tensor Feynman integrals
International Nuclear Information System (INIS)
Fleischer, J.; Riemann, T.
2011-01-01
We set up a new, flexible approach for the tensor reduction of one-loop Feynman integrals. The 5-point tensor integrals up to rank R=5 are expressed by 4-point tensor integrals of rank R-1, such that the appearance of the inverse 5-point Gram determinant is avoided. The 4-point tensor coefficients are represented in terms of 4-point integrals, defined in d dimensions, 4-2ε≤d≤4-2ε+2(R-1), with higher powers of the propagators. They can be further reduced to expressions which stay free of the inverse 4-point Gram determinants but contain higher-dimensional 4-point integrals with only the first power of scalar propagators, plus 3-point tensor coefficients. A direct evaluation of the higher-dimensional 4-point functions would avoid the appearance of inverse powers of the Gram determinants completely. The simplest approach, however, is to apply here dimensional recurrence relations in order to reduce them to the familiar 2- to 4-point functions in generic dimension d=4-2ε, introducing thereby coefficients with inverse 4-point Gram determinants up to power R for tensors of rank R. For small or vanishing Gram determinants--where this reduction is not applicable--we use analytic expansions in positive powers of the Gram determinants. Improving the convergence of the expansions substantially with Pade approximants we close up to the evaluation of the 4-point tensor coefficients for larger Gram determinants. Finally, some relations are discussed which may be useful for analytic simplifications of Feynman diagrams.
5. The application of the inductively coupled plasma system to the simultaneous determination of precious metals
International Nuclear Information System (INIS)
Watson, A.E.; Russell, G.M.; Middleton, H.R.; Davenport, F.F.
1983-01-01
This report describes the development of a spectrochemical technique using excitation by an inducticely coupled plasma (ICP) source for the simultaneous determination of the precious metals (defined here as gold, silver, and all the platinum-group metals except osmium) in a wide variety of samples from a plant for the extraction and refining of platinum metal. The limits of detection for the analytes were determined in various acid and salt media and, under the conditions used, ranged from 20 to 100ng/l. The analytes were determined in the presence of a thousandfold excess of each of the other precious metals used as a matrix element. Some severe interferences were noted but were ascribed to spectral-line overlap or to contamination of the matrix material. Various dissolution techniques, based upon standard procedures applied in the precious-metals industry, were used, depending on the particular type of material treated. The spectrometer was calibrated by the use of solutions containing the analytes, sodium chloride, and acid, with scandium as the internal standard. The accuracy and precision of the technique, established by the analysis of many samples of each type, were found to be satisfactory when close attention was paid to detail in the preparation of the analytical solution. The relative standard deviation of the method ranges from 0,005 to 0,05, depending on the element being determined
6. General scalar-tensor theories for induced gravity inflation
International Nuclear Information System (INIS)
Boutaleb J, H.; Marrakchi, A.L.
1992-07-01
Some cosmological implications of a general scalar-tensor theory for induced gravity are discussed. The model exhibits a slow-rolling phase provided that the coupling function ε(φ) varies slowly enough such that φ dlnε(φ)/dφ much less than 2 during almost the inflationary epoch. It is then shown that, as in the ordinary induced gravity inflation, the chaotic scenario is more natural than the new scenario which proves to be even not self-consistent. The results are applied, for illustration, to a scalar-tensor theory of the Barker type. (author). 25 refs
7. Quark-gluon mixing in pseudoscalar and tensor mesons
International Nuclear Information System (INIS)
Eremyan, Sh.S.; Nazaryan, A.E.
1986-01-01
A mixing model of quark-antiquark ang gluonium states in η, η', i(1440) pseudoscalar and f, f', Θ(1690) tensor mesons is considered. Description of and predictions for 68 two-particle decays with these particles taking part in them are obtained. It is shown that i(1440) by 85% consists of gluonium and Θ(1690) is a pure gluonic state. The quark-gluon and gluon-gluon couplings in the pseudoscalar sector are obtained to be stronger as compared to the corresponding ones in the tensor case
8. Complete Cubic and Quartic Couplings of 16 and $\\bar{16}$ in SO(10) Unification
CERN Document Server
Syed, R M; Nath, Pran; Syed, Raza M.
2001-01-01
A recently derived basic theorem on the decomposition of SO(2N) vertices is used to obtain a complete analytic determination of all SO(10) invariant cubic superpotential couplings involving $16_{\\pm}$ semispinors of SO(10) chirality $\\pm$ and tensor representations. In addition to the superpotential couplings computed previously using the basic theorem involving the 10, 120 and $\\bar{126}$ tensor representations we compute here couplings involving the 1, 45 and 210 dimensional tensor representations, i.e., we compute the $\\bar{16}_{\\mp}16_{\\pm}1$,$\\bar{16}_{\\mp}16_{\\pm}45$ and $\\bar{16}_{\\mp}16_{\\pm}210$ Higgs couplings in the superpotential. A complete determination of dimension five operators in the superpotential arising from the mediation of the 1, 45 and 210 dimensional representations is also given. The vector couplings $\\bar{16}_{\\pm}16_{\\pm}1$, $\\bar{16}_{\\pm}16_{\\pm}45$ and $\\bar{16}_{\\pm}16_{\\pm}210$ are also analyzed. The role of large tensor representations and the possible application of results ...
9. Complete cubic and quartic couplings of 16 and 16-bar in SO(10) unification
International Nuclear Information System (INIS)
Nath, Pran; Syed, Raza M.
2001-01-01
A recently derived basic theorem on the decomposition of SO(2N) vertices is used to obtain a complete analytic determination of all SO(10)-invariant cubic superpotential couplings involving 16 ± semispinors of SO(10) chirality ± and tensor representations. In addition to the superpotential couplings computed previously using the basic theorem involving the 10, 120 and 126-bar tensor representations we compute here couplings involving the 1-, 45- and 210-dimensional tensor representations, i.e., we compute the 16-bar -+ 16 ± 1, 16-bar -+ 16 ± 45 and 16-bar -+ 16 ± 210 Higgs couplings in the superpotential. A complete determination of dimension five operators in the superpotential arising from the mediation of the 1-, 45- and 210-dimensional representations is also given. The vector couplings 16-bar ± 16 ± 1, 16-bar ± 16 ± 45 and 16-bar ± 16 ± 210 are also analyzed. The role of large tensor representations and the possible application of results derived here in model building are discussed
10. Holographic stress tensor for non-relativistic theories
International Nuclear Information System (INIS)
Ross, Simon F.; Saremi, Omid
2009-01-01
We discuss the calculation of the field theory stress tensor from the dual geometry for two recent proposals for gravity duals of non-relativistic conformal field theories. The first of these has a Schroedinger symmetry including Galilean boosts, while the second has just an anisotropic scale invariance (the Lifshitz case). For the Lifshitz case, we construct an appropriate action principle. We propose a definition of the non-relativistic stress tensor complex for the field theory as an appropriate variation of the action in both cases. In the Schroedinger case, we show that this gives physically reasonable results for a simple black hole solution and agrees with an earlier proposal to determine the stress tensor from the familiar AdS prescription. In the Lifshitz case, we solve the linearised equations of motion for a general perturbation around the background, showing that our stress tensor is finite on-shell.
11. Tensor Product of Polygonal Cell Complexes
OpenAIRE
Chien, Yu-Yen
2017-01-01
We introduce the tensor product of polygonal cell complexes, which interacts nicely with the tensor product of link graphs of complexes. We also develop the unique factorization property of polygonal cell complexes with respect to the tensor product, and study the symmetries of tensor products of polygonal cell complexes.
12. The Einstein tensor characterizing some Riemann spaces
International Nuclear Information System (INIS)
Rahman, M.S.
1993-07-01
A formal definition of the Einstein tensor is given. Mention is made of how this tensor plays a role of expressing certain conditions in a precise form. The cases of reducing the Einstein tensor to a zero tensor are studied on its merit. A lucid account of results, formulated as theorems, on Einstein symmetric and Einstein recurrent spaces is then presented. (author). 5 refs
13. Colored Tensor Models - a Review
Directory of Open Access Journals (Sweden)
Razvan Gurau
2012-04-01
Full Text Available Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating two-dimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1/N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger-Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions, non-trivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.
14. Inflation in non-minimal matter-curvature coupling theories
Energy Technology Data Exchange (ETDEWEB)
Gomes, C.; Bertolami, O. [Departamento de Física e Astronomia and Centro de Física do Porto, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto (Portugal); Rosa, J.G., E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Departamento de Física da Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro (Portugal)
2017-06-01
We study inflationary scenarios driven by a scalar field in the presence of a non-minimal coupling between matter and curvature. We show that the Friedmann equation can be significantly modified when the energy density during inflation exceeds a critical value determined by the non-minimal coupling, which in turn may considerably modify the spectrum of primordial perturbations and the inflationary dynamics. In particular, we show that these models are characterised by a consistency relation between the tensor-to-scalar ratio and the tensor spectral index that can differ significantly from the predictions of general relativity. We also give examples of observational predictions for some of the most commonly considered potentials and use the results of the Planck collaboration to set limits on the scale of the non-minimal coupling.
15. Determination of rare earth elements by liquid chromatographic separation using inductively coupled plasma mass spectrometric detection
International Nuclear Information System (INIS)
Braverman, D.S.
1992-01-01
High-performance liquid chromatography (HPLC) is used to separate the rare earth elements (REEs) prior to detection by inductively coupled plasma mass spectrometry (ICP-MS). The use of HPLC-ICP-MS in series combines the separation power and speed of HPLC with the sensitivity, isotopic selectivity and speed of ICP-MS. The detection limits for the REEs are in the sub-ng ml -1 range and the response is linear over four orders of magnitude. A preliminary comparison of isotope dilution and external standard results for the determination of REEs in National Institute of Standards and Technology (NIST) Standard Reference Material (SRM 1633a) Fly Ash is presented. (author)
16. New azo coupling reactions for visible spectrophotometric determination of salbutamol in bulk and pharmaceutical preparations
International Nuclear Information System (INIS)
Dhahir, S. A.
2011-01-01
The purpose of the present study was to develop a new, simple, cheap, fast, accurate, and sensitive colorimetric methods that can be used for the determination of salbutamol sulphate drug in pure from as well as in pharmaceutical formulations. The method is based on the reaction 2-chloro-4-nitroaniline with nitrite in acid medium to form diazonium ion, which is coupled with of salbutamol in basic medium to form azo dyes, showing yellow color and absorption maxima at 463 nm. Beer's law is obeyed in the concentration of 4-48μg/ml. The molar absorptivity and san dell's sensitivity are 5.27x103 L mole-1 cm-1, 0.015 μgcm-2, respectively. The optimum reaction conditions and other analytical parameters were evaluated. (author).
17. Determination of long-chain fatty acids in serum by gas chromatography coupled to mass spectrometry
International Nuclear Information System (INIS)
Nuevas Paz, Lauro; Camayd Viera, Ivette
2014-01-01
The quantification of long-chain fatty acids is fundamental for the diagnosis of several peroxisome disorders, particularly those in which the β-oxidation peroxisome of fatty acids is affected. In this work the implementation of an analytical method for the determination of these markers in serum by gas chromatography coupled to mass spectrometry is described. Besides, samples from patients with a diagnostic impression of adrenoleukodystrophy linked to the X chromosome were analyzed. The necessary experimental conditions were achieved for the separation and quantification of C22:0, C24:0 and C26:0 fatty acids in serum, which are biochemical markers of various peroxisome diseases. The application of this method allowed confirming the diagnosis of three patients with a diagnostic impression of adrenoleukodystrophy linked to the X chromosome. The application of the method in daily practice will allow the Cuban medical system to count on a new laboratory parameter for the diagnosis of peroxisome disorders
18. Determination of Rare Earth Elements in Thai Monazite by Inductively Coupled Plasma and Nuclear Analytical techniques
International Nuclear Information System (INIS)
Busamongkol, Arporn; Ratanapra, Dusadee; Sukharn, Sumalee; Laoharojanaphand, Sirinart
2003-10-01
The inductively coupled plasma atomic emission spectroscopy (ICP-AES) for the determination of individual rare-earth elements (REE) was evaluated by comparison with instrumental neutron activation analysis (INAA) and x-ray fluorescence spectrometry (XRF). The accuracy and precision of INAA and ICP-AES were evaluated by using standard reference material IGS-36, a monazite concentrate. For INAA, the results were close to the certified value while ICP-AES were in good agreement except for some low concentration rare earth. The techniques were applied for the analysis of some rare earth elements in two Thai monazite samples preparing as the in-house reference material for the Rare Earth Research and Development Center, Chemistry Division, Office of Atoms for Peace. The analytical results obtained by these techniques were in good agreement with each other
19. Serum/plasma methylmercury determination by isotope dilution gas chromatography-inductively coupled plasma mass spectrometry
Energy Technology Data Exchange (ETDEWEB)
Baxter, Douglas C., E-mail: [email protected] [ALS Scandinavia AB, Aurorum 10, 977 75 Lulea (Sweden); Faarinen, Mikko [ALS Scandinavia AB, Aurorum 10, 977 75 Lulea (Sweden); Osterlund, Helene; Rodushkin, Ilia [ALS Scandinavia AB, Aurorum 10, 977 75 Lulea (Sweden); Division of Geosciences, Lulea University of Technology, 977 87 Lulea (Sweden); Christensen, Morten [ALS Scandinavia AB, Maskinvaegen 2, 183 53 Taeby (Sweden)
2011-09-09
Highlights: {center_dot} We determine methylmercury in serum and plasma using isotope dilution calibration. {center_dot} Separation by gas chromatography and detection by inductively coupled plasma mass spectrometry. {center_dot} Data for 50 specimens provides first reference range for methylmercury in serum. {center_dot} Serum samples shown to be stable for 11 months in refrigerator. - Abstract: A method for the determination of methylmercury in plasma and serum samples was developed. The method uses isotope dilution with {sup 198}Hg-labeled methylmercury, extraction into dichloromethane, back-extraction into water, aqueous-phase ethylation, purge and trap collection, thermal desorption, separation by gas chromatography, and mercury isotope specific detection by inductively coupled plasma mass spectrometry. By spiking 2 mL sample with 1.2 ng tracer, measurements in a concentration interval of (0.007-2.9) {mu}g L{sup -1} could be performed with uncertainty amplification factors <2. A limit of quantification of 0.03 {mu}g L{sup -1} was estimated at 10 times the standard deviation of concentrations measured in preparation blanks. Within- and between-run relative standard deviations were <10% at added concentration levels of 0.14 {mu}g L{sup -1}, 0.35 {mu}g L{sup -1} and 2.8 {mu}g L{sup -1}, with recoveries in the range 82-110%. Application of the method to 50 plasma/serum samples yielded a median (mean; range) concentration of methylmercury of 0.081 (0.091; <0.03-0.19) {mu}g L{sup -1}. This is the first time methylmercury has been directly measured in this kind of specimen, and is therefore the first estimate of a reference range.
20. Plutonium determination in seawater by inductively coupled plasma mass spectrometry: A review.
Science.gov (United States)
Cao, Liguo; Bu, Wenting; Zheng, Jian; Pan, Shaoming; Wang, Zhongtang; Uchida, Shigeo
2016-05-01
1. Laser ablation inductively coupled plasma mass spectrometry for the determination of trace elements in soil
International Nuclear Information System (INIS)
Lee Yiling; Chang Chaochiang; Jiang Shiuhjen
2003-01-01
Laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) has been applied to the determination of Cr, Cu, Zn, Cd and Pb in soil samples. The dried soil powder was pressed into a pellet for LA-ICP-MS analysis. Triton X-100 was added to work as the modifier to enhance the ion signals. The influences of instrument operating conditions (LA and ICP-MS) and pellet preparation on the ion signals were reported. For Cr determination, the ICP-MS was operated under the dynamic reaction cell mode which alleviated the mass overlap interference. Standard addition method and isotope dilution method were used for the quantitation work. The powder sample was spiked with suitable amounts of element standards and/or enriched isotopes, well-mixed, dried and then pressed into a pellet for LA-ICP-MS analysis. This method has been applied to determine Cr, Cu, Zn, Cd and Pb in NIST SRM 2711 Montana soil and NIST SRM 2709 San Joaquin soil reference materials. The analysis results were in agreement with the certified values. The precision between sample replicates was better than 5% with LA-ICP-MS method. Detection limits estimated from standard addition curves were approximately 0.9, 2, 9, 0.7 and 0.3 ng g -1 for Cr, Cu, Zn, Cd and Pb, respectively
2. Tensor Completion Algorithms in Big Data Analytics
OpenAIRE
Song, Qingquan; Ge, Hancheng; Caverlee, James; Hu, Xia
2017-01-01
Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications have received wide attention and achievement in areas like data mining, computer vision, signal processing, and neuroscience. In this survey, we provide a modern overview of recent advances in tensor completion algorithms from the perspective of big data an...
3. A model for soft high-energy scattering: Tensor pomeron and vector odderon
Energy Technology Data Exchange (ETDEWEB)
Ewerz, Carlo, E-mail: [email protected] [Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg (Germany); ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, D-64291 Darmstadt (Germany); Maniatis, Markos, E-mail: [email protected] [Departamento de Ciencias Básicas, Universidad del Bío-Bío, Avda. Andrés Bello s/n, Casilla 447, Chillán 3780000 (Chile); Nachtmann, Otto, E-mail: [email protected] [Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg (Germany)
2014-03-15
A model for soft high-energy scattering is developed. The model is formulated in terms of effective propagators and vertices for the exchange objects: the pomeron, the odderon, and the reggeons. The vertices are required to respect standard rules of QFT. The propagators are constructed taking into account the crossing properties of amplitudes in QFT and the power-law ansätze from the Regge model. We propose to describe the pomeron as an effective spin 2 exchange. This tensor pomeron gives, at high energies, the same results for the pp and pp{sup -bar} elastic amplitudes as the standard Donnachie–Landshoff pomeron. But with our tensor pomeron it is much more natural to write down effective vertices of all kinds which respect the rules of QFT. This is particularly clear for the coupling of the pomeron to particles carrying spin, for instance vector mesons. We describe the odderon as an effective vector exchange. We emphasise that with a tensor pomeron and a vector odderon the corresponding charge-conjugation relations are automatically fulfilled. We compare the model to some experimental data, in particular to data for the total cross sections, in order to determine the model parameters. The model should provide a starting point for a general framework for describing soft high-energy reactions. It should give to experimentalists an easily manageable tool for calculating amplitudes for such reactions and for obtaining predictions which can be compared in detail with data. -- Highlights: •A general model for soft high-energy hadron scattering is developed. •The pomeron is described as effective tensor exchange. •Explicit expressions for effective reggeon–particle vertices are given. •Reggeon–particle and particle–particle vertices are related. •All vertices respect the standard C parity and crossing rules of QFT.
4. Development of the Tensoral Computer Language
Science.gov (United States)
Ferziger, Joel; Dresselhaus, Eliot
1996-01-01
The research scientist or engineer wishing to perform large scale simulations or to extract useful information from existing databases is required to have expertise in the details of the particular database, the numerical methods and the computer architecture to be used. This poses a significant practical barrier to the use of simulation data. The goal of this research was to develop a high-level computer language called Tensoral, designed to remove this barrier. The Tensoral language provides a framework in which efficient generic data manipulations can be easily coded and implemented. First of all, Tensoral is general. The fundamental objects in Tensoral represent tensor fields and the operators that act on them. The numerical implementation of these tensors and operators is completely and flexibly programmable. New mathematical constructs and operators can be easily added to the Tensoral system. Tensoral is compatible with existing languages. Tensoral tensor operations co-exist in a natural way with a host language, which may be any sufficiently powerful computer language such as Fortran, C, or Vectoral. Tensoral is very-high-level. Tensor operations in Tensoral typically act on entire databases (i.e., arrays) at one time and may, therefore, correspond to many lines of code in a conventional language. Tensoral is efficient. Tensoral is a compiled language. Database manipulations are simplified optimized and scheduled by the compiler eventually resulting in efficient machine code to implement them.
5. The tensor part of the Skyrme energy density functional. I. Spherical nuclei
Energy Technology Data Exchange (ETDEWEB)
Lesinski, T.; Meyer, J. [Universite de Lyon, F-69003 Lyon (France)]|[Institut de Physique Nucleaire de Lyon, CNRS/IN2P3, Universite Lyon 1, F-69622 Villeurbanne (France); Bender, M. [DSM/DAPNIA/SPhN, CEA Saclay, F-91191 Gif-sur-Yvette Cedex (France)]|[Universite Bordeaux, CNRS/IN2P3, Centre d' Etudes Nucleaires de Bordeaux Gradignan, UMR5797, Chemin du Solarium, BP120, F-33175 Gradignan (France); Bennaceur, K. [Universite de Lyon, F-69003 Lyon (France)]|[Institut de Physique Nucleaire de Lyon, CNRS/IN2P3, Universite Lyon 1, F-69622 Villeurbanne (France)]|[DSM/DAPNIA/SPhN, CEA Saclay, F-91191 Gif-sur-Yvette Cedex (France); Duguet, T. [National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824 (United States)
2007-04-15
We perform a systematic study of the impact of the J-vector{sup 2} tensor term in the Skyrme energy functional on properties of spherical nuclei. In the Skyrme energy functional, the tensor terms originate both from zero-range central and tensor forces. We build a set of 36 parameterizations which cover a wide range of the parameter space of the isoscalar and isovector tensor term coupling constants with a fit protocol very similar to that of the successful SLy parameterizations. We analyze the impact of the tensor terms on a large variety of observables in spherical mean-field calculations, such as the spin-orbit splittings and single-particle spectra of doubly-magic nuclei, the evolution of spin-orbit splittings along chains of semi-magic nuclei, mass residuals of spherical nuclei, and known anomalies of radii. The major findings of our study are (i) tensor terms should not be added perturbatively to existing parameterizations, a complete refit of the entire parameter set is imperative. (ii) The free variation of the tensor terms does not lower the {chi}{sup 2} within a standard Skyrme energy functional. (iii) For certain regions of the parameter space of their coupling constants, the tensor terms lead to instabilities of the spherical shell structure, or even the coexistence of two configurations with different spherical shell structure. (iv) The standard spin-orbit interaction does not scale properly with the principal quantum number, such that single-particle states with one or several nodes have too large spin-orbit splittings, while those of node-less intruder levels are tentatively too small. Tensor terms with realistic coupling constants cannot cure this problem. (v) Positive values of the coupling constants of proton-neutron and like-particle tensor terms allow for a qualitative description of the evolution of spin-orbit splittings in chains of Ca, Ni and Sn isotopes. (vi) For the same values of the tensor term coupling constants, however, the overall
6. Physical and Geometric Interpretations of the Riemann Tensor, Ricci Tensor, and Scalar Curvature
OpenAIRE
Loveridge, Lee C.
2004-01-01
Various interpretations of the Riemann Curvature Tensor, Ricci Tensor, and Scalar Curvature are described. Also, the physical meanings of the Einstein Tensor and Einstein's Equations are discussed. Finally a derivation of Newtonian Gravity from Einstein's Equations is given.
7. Direct rare earth determination by inductively coupled plasma optical emission spectrometry
International Nuclear Information System (INIS)
Marin, Sergio; Cornejo, Silvia; Rojas, Jacqueline
2003-01-01
In the present work, the use of the inductively coupled plasma optical emission spectrometry (ICP-OES), for the sequential determination of Rare Earth elements in the metallurgical process samples is described. In the first place, the optimum parameters for the determination of the elements in study are established, like instrumental calibration, wavelengths spectral selection and interference of matrix. Next, the methodology for the digestion of solid samples (system of digestion with pressure) and the recovery of the interest elements are presented. Two material rocks as of reference Syenite SY3 are used. In order to assure the validity of the obtained data, the reference materials SY2 and SY3 were analyzed by means of two different techniques, ICP-OES and ICP-Mass, this last one was made by an international laboratory and a fusion with lithium metaborate was used with digestion method. Finally, the obtained results demonstrate that the reproducibility in the recovery of rare earth analyzed by both techniques is comparable, and that the methodology of digestion used for these elements is statistically valid (author)
8. Process monitored spectrophotometric titration coupled with chemometrics for simultaneous determination of mixtures of weak acids.
Science.gov (United States)
Liao, Lifu; Yang, Jing; Yuan, Jintao
2007-05-15
A new spectrophotometric titration method coupled with chemometrics for the simultaneous determination of mixtures of weak acids has been developed. In this method, the titrant is a mixture of sodium hydroxide and an acid-base indicator, and the indicator is used to monitor the titration process. In a process of titration, both the added volume of titrant and the solution acidity at each titration point can be obtained simultaneously from an absorption spectrum by least square algorithm, and then the concentration of each component in the mixture can be obtained from the titration curves by principal component regression. The method only needs the information of absorbance spectra to obtain the analytical results, and is free of volumetric measurements. The analyses are independent of titration end point and do not need the accurate values of dissociation constants of the indicator and the acids. The method has been applied to the simultaneous determination of the mixtures of benzoic acid and salicylic acid, and the mixtures of phenol, o-chlorophenol and p-chlorophenol with satisfactory results.
9. Inductively coupled plasma atomic emission spectrometric determination of tin in canned food.
Science.gov (United States)
Sumitani, H; Suekane, S; Nakatani, A; Tatsuka, K
1993-01-01
Various canned foods were digested sequentially with HNO3 and HCl, diluted to 100 mL, and filtered, and then tin was determined by inductively coupled plasma atomic emission spectrometry (ICP/AES). Samples of canned Satsuma mandarin, peach, apricot, pineapple, apple juice, mushroom, asparagus, evaporated milk, short-necked clam, spinach, whole tomato, meat, and salmon were evaluated. Sample preparations did not require time-consuming dilutions, because ICP/AES has wide dynamic range. The standard addition method was used to determine tin concentration. Accuracy of the method was tested by analyzing analytical standards containing tin at 2 levels (50 and 250 micrograms/g). The amounts of tin found for the 50 and 250 micrograms/g levels were 50.5 and 256 micrograms/g, respectively, and the repeatability coefficients of variation were 4.0 and 3.8%, respectively. Recovery of tin from 13 canned foods spiked at 2 levels (50 and 250 micrograms/g) ranged from 93.9 to 109.4%, with a mean of 99.2%. The quantitation limit for tin standard solution was about 0.5 microgram/g.
10. Accurate determination of silver nanoparticles in animal tissues by inductively coupled plasma mass spectrometry
Energy Technology Data Exchange (ETDEWEB)
2014-12-01
This study examined recoveries of silver determination in animal tissues after wet digestion by inductively coupled plasma mass spectrometry. The composition of the mineralization mixture for microwave assisted digestion was optimized and the best recoveries were obtained for mineralization with HNO{sub 3} and addition of HCl promptly after digestion. The optimization was performed on model samples of chicken meat spiked with silver nanoparticles and a solution of ionic silver. Basic calculations of theoretical distribution of Ag among various silver-containing species were implemented and the results showed that most of the silver is in the form of soluble complexes AgCl{sub 2}{sup −} and AgCl{sub 3}{sup 2−} for the optimized composition of the mineralization mixture. Three animal tissue certified reference materials were then analyzed to verify the trueness and precision of the results. - Highlights: • We performed detailed optimization of microwave assisted digestion procedure of animal tissue used prior to Ag determination by ICP-MS. • We provide basic equilibrium calculations to give theoretical explanation of results from optimization of tested mineralization mixtures. • Results from method validation that was done by analysis of several matrix CRMs are presented.
11. Determination of the pion-nucleon coupling constant and scattering lengths
International Nuclear Information System (INIS)
Ericson, T.E.O.; Loiseau, B.; Thomas, A.W.
2002-01-01
We critically evaluate the isovector Goldberger-Miyazawa-Oehme (GMO) sum rule for forward πN scattering using the recent precision measurements of π - p and π - d scattering lengths from pionic atoms. We deduce the charged-pion-nucleon coupling constant, with careful attention to systematic and statistical uncertainties. This determination gives, directly from data, g c 2 (GMO)/4π=14.11±0.05(statistical)±0.19(systematic) or f c 2 /4π=0.0783(11). This value is intermediate between that of indirect methods and the direct determination from backward np differential scattering cross sections. We also use the pionic atom data to deduce the coherent symmetric and antisymmetric sums of the pion-proton and pion-neutron scattering lengths with high precision, namely, (a π - p +a π - n )/2=[-12±2(statistical)±8(systematic)]x10 -4 m π -1 and (a π - p -a π - n )/2=[895±3(statistical)±13 (systematic)]x10 -4 m π -1 . For the need of the present analysis, we improve the theoretical description of the pion-deuteron scattering length
12. Determining Li+-Coupled Redox Targeting Reaction Kinetics of Battery Materials with Scanning Electrochemical Microscopy.
Science.gov (United States)
Yan, Ruiting; Ghilane, Jalal; Phuah, Kia Chai; Pham Truong, Thuan Nguyen; Adams, Stefan; Randriamahazaka, Hyacinthe; Wang, Qing
2018-02-01
The redox targeting reaction of Li + -storage materials with redox mediators is the key process in redox flow lithium batteries, a promising technology for next-generation large-scale energy storage. The kinetics of the Li + -coupled heterogeneous charge transfer between the energy storage material and redox mediator dictates the performance of the device, while as a new type of charge transfer process it has been rarely studied. Here, scanning electrochemical microscopy (SECM) was employed for the first time to determine the interfacial charge transfer kinetics of LiFePO 4 /FePO 4 upon delithiation and lithiation by a pair of redox shuttle molecules FcBr 2 + and Fc. The effective rate constant k eff was determined to be around 3.70-6.57 × 10 -3 cm/s for the two-way pseudo-first-order reactions, which feature a linear dependence on the composition of LiFePO 4 , validating the kinetic process of interfacial charge transfer rather than bulk solid diffusion. In addition, in conjunction with chronoamperometry measurement, the SECM study disproves the conventional "shrinking-core" model for the delithiation of LiFePO 4 and presents an intriguing way of probing the phase boundary propagations induced by interfacial redox reactions. This study demonstrates a reliable method for the kinetics of redox targeting reactions, and the results provide useful guidance for the optimization of redox targeting systems for large-scale energy storage.
13. [Determination of Heavy Metal Elements in Diatomite Filter Aid by Inductively Coupled Plasma Mass Spectrometry].
Science.gov (United States)
Nie, Xi-du; Fu, Liang
2015-11-01
This study established a method for determining Be, Cr, Ni, As, Cd, Sb, Sn, Tl, Hg and Pb, total 10 heavy metals in diatomite filter aid. The diatomite filter aid was digested by using the mixture acid of HNO₃ + HF+ H₃PO₄ in microwave system, 10 heavy metals elements were determined by inductively coupled plasma mass spectrometry (ICP-MS). The interferences of mass spectrometry caused by the high silicon substrate were optimized, first the equipment parameters and isotopes of test metals were selected to eliminate these interferences, the methane was selected as reactant gas, and the mass spectral interferences were eliminated by dynamic reaction cell (DRC). Li, Sc, Y, In and Bi were selected as the internal standard elements to correct the interferences caused by matrix and the drift of sensitivity. The results show that the detection limits for analyte is in the range of 3.29-15.68 ng · L⁻¹, relative standard deviations (RSD) is less than 4.62%, and the recovery is in the range of 90.71%-107.22%. The current method has some advantages such as, high sensitivity, accurate, and precision, which can be used in diatomite filter aid quality control and safety estimations.
14. Determination of metals content from wines by inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Iordache, Andreea-Maria; Geana, Elisabeta-Irina
2009-01-01
Full text: Wine is a widely consumed beverage with thousands of years of tradition. Wine composition strongly determines its quality besides having a great relevance on wine characterization, tipyfication and frauds detection. Wine composition is influenced by many and diverse factors corresponding to the specific production area, such as grape variety, soil and climate, culture, yeast, winemaking practices, transport and storage. Daily consumption of wine in moderate quantities contributes significantly to the requirements of the human organism for essential elements such as Cr, Cu, Zn, Fe, Mn, Co, Ni and Sr. On the other hand, several metals, such as Pb and Cd , are known to be potentially toxic. The objective of this work was to develop a method to determine the metals content in wine samples from Romania. Three samples of difference white wines available in the supermarket was analyzed for identify the presence of: Cr, Cu, Zn, Fe, Mn, Pb, Cd, Co, Ni and Sr by inductively coupled plasma mass spectrometry (ICP-MS). (authors)
15. The Possible Role of Dentin as a Piezoelectric Signal Generator by Determining the Elec-tromechanical Coupling Factor of Dentin
Directory of Open Access Journals (Sweden)
Atabak Shahidi
2011-08-01
Full Text Available Introduction: This article aimed at calculation of the electromechanical coupling factor of dentin which is an indicator of the effectiveness with which a piezoelectric material converts electrical en-ergy into mechanical energy, or vice versa. The hypothesis: The electro-mechanical coupling factor of dentin was determined in mode 11 and 33 by calculating the ratio of the produced electrical energy to the stored elastic energy in dentin under applied pressure. This study showed that the electromechanical coupling factor of dentin was affected by the direction of the applied force and the moisture content of dentin. Also dentin was a weak electromechanical energy converter which might be categorized as a piezoelectric pressure sensor.Evaluation of the hypothesis: Determination of the electrome-chanical coupling factor of dentin and its other piezoelectric constants is essential to investigate the biologic role of piezoelectricity in tooth.
16. Determination of coupling coefficients at various zenith angles of the basis of the cosmic ray azimuth effect
Science.gov (United States)
Belskiy, S. A.; Dmitriev, B. A.; Romanov, A. M.
1975-01-01
The value of EW asymmetry and coupling coefficients at different zenith angles were measured by means of a double coincidence crossed telescope which gives an opportunity to measure simultaneously the intensity of the cosmic ray hard component at zenith angles from 0 to 84 deg in opposite azimuths. The advantages of determining the coupling coefficients by the cosmic ray azimuth effect as compared to their measurement by the latitudinal effect are discussed.
17. How Precisely can we Determine the $\\piNN$ Coupling Constant from the Isovector GMO Sum Rule?
CERN Document Server
Loiseau, B; Thomas, A W
1999-01-01
The isovector GMO sum rule for zero energy forward pion-nucleon scattering iscritically studied to obtain the charged pion-nucleon coupling constant usingthe precise negatively charged pion-proton and pion-deuteron scattering lengthsdeduced recently from pionic atom experiments. This direct determination leadsto a pseudoscalar charged pion-nucleon coupling constant of 14.23 +- 0.09(statistic) +- 0.17 (systematic). We obtain also accurate values for thepion-nucleon scattering lengths.
18. The tensor rank of tensor product of two three-qubit W states is eight
OpenAIRE
Chen, Lin; Friedland, Shmuel
2017-01-01
We show that the tensor rank of tensor product of two three-qubit W states is not less than eight. Combining this result with the recent result of M. Christandl, A. K. Jensen, and J. Zuiddam that the tensor rank of tensor product of two three-qubit W states is at most eight, we deduce that the tensor rank of tensor product of two three-qubit W states is eight. We also construct the upper bound of the tensor rank of tensor product of many three-qubit W states.
19. [Determination of 24 minerals in human milk by inductively coupled plasma mass spectrometry with microwave digestion].
Science.gov (United States)
Sun, Zhongqing; Yue, Bing; Yang, Zhenyu; Li, Xiaowei; Wu, Yongning; Yin, Shian
2013-05-01
To determine the levels of 24 minerals in human milk by inductively coupled plasma mass spectrometry with microwave digestion. The samples were digested by microwave. The contents of minerals were determined by inductively coupled plasma mass spectrometry. The standard reference minerals of 1849a and 1568a from National Institute of Science and Technology were used for quality control. The accuracy and reproduability for this method were evaluated with mix standards and 1849a and 1568a standard reference materials. The ranges of the levels of sodium, magnesium, phosphorus, potassium, calcium, aluminum, chromium, arsenic, selenium, iron, zinc, manganese, copper, molybdenum, vanadium, cobalt, nickel, gallium, cadmium, silver, strontium, cesium, barium, lead in human milk was 34.97-415.83 mg/kg, 19.00-39.52 mg/kg, 102.13-274.53 mg/kg, 351.19-713.99 mg/kg, 180.08-349.64 mg/kg, 0.06-0.44 mg/kg, 0.9-7.37 microg/kg, 0.92-2.72 microg/kg, 0.20-21.15 microg/kg, 0.10-0.70 mg/kg, 0.56-3.25 mg/kg, 3.00-16.12 micro.g/kg, 62.16-591.69 microg/kg, 0.02-6.91 microg/kg, 5.99-13.70 microg/kg, 0.07-2.11 microg/kg, 0.77-209.26 microg/kg, 0.005-0.28 microg/kg, 0.02-0.23 microg/kg, 0.02-0.71 microg/kg, 36.89-132.26 microg/kg, 0.01-4.72 microg/kg, 0.83-28.16 microg/kg, 2.5-5.3 microg/kg, respectively. The levels of minerals in human milk in present study were consisted with other similar studies. The experiment examined the levels of minerals in human milk satisfactorily. The method has high accuracy and good reproducibility, which could be used for understanding the levels of minerals in human milk.
20. Top-down approach in protein RDC data analysis: de novo estimation of the alignment tensor
International Nuclear Information System (INIS)
Chen Kang; Tjandra, Nico
2007-01-01
In solution NMR spectroscopy the residual dipolar coupling (RDC) is invaluable in improving both the precision and accuracy of NMR structures during their structural refinement. The RDC also provides a potential to determine protein structure de novo. These procedures are only effective when an accurate estimate of the alignment tensor has already been made. Here we present a top-down approach, starting from the secondary structure elements and finishing at the residue level, for RDC data analysis in order to obtain a better estimate of the alignment tensor. Using only the RDCs from N-H bonds of residues in α-helices and CA-CO bonds in β-strands, we are able to determine the offset and the approximate amplitude of the RDC modulation-curve for each secondary structure element, which are subsequently used as targets for global minimization. The alignment order parameters and the orientation of the major principal axis of individual helix or strand, with respect to the alignment frame, can be determined in each of the eight quadrants of a sphere. The following minimization against RDC of all residues within the helix or strand segment can be carried out with fixed alignment order parameters to improve the accuracy of the orientation. For a helical protein Bax, the three components A xx , A yy and A zz , of the alignment order can be determined with this method in average to within 2.3% deviation from the values calculated with the available atomic coordinates. Similarly for β-sheet protein Ubiquitin they agree in average to within 8.5%. The larger discrepancy in β-strand parameters comes from both the diversity of the β-sheet structure and the lower precision of CA-CO RDCs. This top-down approach is a robust method for alignment tensor estimation and also holds a promise for providing a protein topological fold using limited sets of RDCs
1. Piezo-optic tensor of crystals from quantum-mechanical calculations.
Science.gov (United States)
Erba, A; Ruggiero, M T; Korter, T M; Dovesi, R
2015-10-14
An automated computational strategy is devised for the ab initio determination of the full fourth-rank piezo-optic tensor of crystals belonging to any space group of symmetry. Elastic stiffness and compliance constants are obtained as numerical first derivatives of analytical energy gradients with respect to the strain and photo-elastic constants as numerical derivatives of analytical dielectric tensor components, which are in turn computed through a Coupled-Perturbed-Hartree-Fock/Kohn-Sham approach, with respect to the strain. Both point and translation symmetries are exploited at all steps of the calculation, within the framework of periodic boundary conditions. The scheme is applied to the determination of the full set of ten symmetry-independent piezo-optic constants of calcium tungstate CaWO4, which have recently been experimentally reconstructed. Present calculations unambiguously determine the absolute sign (positive) of the π61 constant, confirm the reliability of 6 out of 10 experimentally determined constants and provide new, more accurate values for the remaining 4 constants.
2. Stress tensor fluctuations in de Sitter spacetime
Energy Technology Data Exchange (ETDEWEB)
Pérez-Nadal, Guillem; Verdaguer, Enric [Departament de Física Fonamental and Institut de Ciències del Cosmos, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona (Spain); Roura, Albert, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Golm (Germany)
2010-05-01
The two-point function of the stress tensor operator of a quantum field in de Sitter spacetime is calculated for an arbitrary number of dimensions. We assume the field to be in the Bunch-Davies vacuum, and formulate our calculation in terms of de Sitter-invariant bitensors. Explicit results for free minimally coupled scalar fields with arbitrary mass are provided. We find long-range stress tensor correlations for sufficiently light fields (with mass m much smaller than the Hubble scale H), namely, the two-point function decays at large separations like an inverse power of the physical distance with an exponent proportional to m{sup 2}/H{sup 2}. In contrast, we show that for the massless case it decays at large separations like the fourth power of the physical distance. There is thus a discontinuity in the massless limit. As a byproduct of our work, we present a novel and simple geometric interpretation of de Sitter-invariant bitensors for pairs of points which cannot be connected by geodesics.
3. New Diazo Coupling Reactions for Visible Spectrophotometric Determination of Alfuzosin in Pharmaceutical Preparations
Directory of Open Access Journals (Sweden)
M. Vamsi Krishna
2007-01-01
Full Text Available Simple, rapid and sensitive spectrophotometric procedures were developed for the analysis of Alfuzosin hydrochloride (AFZ in pure form as well as in pharmaceutical formulations. The methods are based on the reaction of AFZ with nitrite in acid medium to form diazonium ion, which is coupled with ethoxyethylenemaleic ester (Method A or ethylcyanoacetate (Method B or acetyl acetone (method C in basic medium to form azo dyes, showing absorption maxima at 440, 465 and 490 nm respectively. Beer’s law is obeyed in the concentration of 4-20 μg/mL of AFZ for methods A, B and 3-15 μg/mL of AFZ for method C. The molar absorptivity and sandell’s sensitivity of AFZ- ethoxyethylenemaleic ester, AFZ- ethylcyanoacetate and AFZ-acetyl acetone are1.90 × 104, 0.022; 1.93 × 104, 0.021 and 2.67 × 104 L mole-1 cm-1, 0.015 μg cm-2 respectively. The optimum reaction conditions and other analytical parameters were evaluated. The methods were successfully applied to the determination of AFZ in pharmaceutical formulations.
4. [Determination of arsenic speciation in Scomberomorus niphonius by capillary electrophoresis-inductively coupled plasma mass spectrometry].
Science.gov (United States)
Chen, Fa-rong; Zheng, Li; Wang, Zhi-Guang; Sun, Jie; Han, Li-Hui; Wang, Xiao-ru
2014-06-01
A method for the detection of arsenocholine (AsC), arsenobetaine (AsB), As(III), dimethylarsinic (DMA), monomethylarsonic (MMA) and As (V) by capillary electrophoresis-inductively coupled plasma mass spectrometry (CE-ICP-MS) was established. The results showed that the six species of arsenic were separated within 20 min under the optimized conditions. Good linearities of 6 arsenic species were observed in the range from 2 to 50 μg x L(-1) with the linear correlation greater than 0.996, the detection limits were 0.10-1.08 μg x L(-1) and the RSDs (n = 5) of the peak areas were smaller than 7%. The method was successfully adopted to the determination of the species in Scomberomorus niphonius. The recoveries were between 93% and 98%, and we found the arsenobetaine (AsB) was the main species in the sample. The method was suitable for the analysis of other biological samples with the advantages of good stability, less sample consumption, short analysis time and convenience.
5. Determination of 238U in marine organisms by inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Ishii, Toshiaki; Nakahara, Motokazu; Matsuba, Mitsue; Ishikawa, Masafumi
1991-01-01
Determination of 238 U in fifty-five species of marine organisms was carried out by inductively coupled plasma mass spectrometry which showed some advantages such as high sensitivity, wide dynamic range and small interferences from matrices for the analysis of high mass elements. The concentrations of 238 U in soft tissues of marine animals ranged from 0.076 to 5000 ng/g wet wt. Especially, the branchial heart of cephalopod molluscs showed the specific accumulation of 238 U. The concentration factor of the branchial heart of Octopus vulgaris, which indicated the highest value, was calculated to be about 10 3 by comparing it with the concentration of 238 U (3.2±0.2 ng/ml) in coastal seawaters of Japan. The concentrations of 238 U in hard tissues of marine invertebrates were similar to those in soft tissues. In contrast, hard tissues like bone, scale, fin, etc. of fishes showed much higher concentrations of 238 U than soft tissues like muscle and liver. The concentrations of 238 U of twenty species of algae ranged from 10 to 3700 ng/g dry wt. (author)
6. Radiochemical determination of zirconium by inductively coupled plasma mass spectrometry (ICPMS)
International Nuclear Information System (INIS)
Oliveira, Thiago C.; Oliveira, Arno Heeren de
2013-01-01
The zirconium isotope 93 Zr is a long-lived pure β-particle-emitting radionuclide thus occurring as one of the radionuclides found in nuclear reactors. It's produced from 235 U fission and from 92 Zr neutron activation. Due to its long half-life, 93 Zr is one of the interest radionuclides for assessment studies performance of waste storage or disposal. Measurement of 93 Zr is difficult owing to its trace level concentration and its low activity in nuclear wastes and further because its certified standards are not frequently available. The aim of this work was to apply a selective radiochemical separation methodology for 93 Zr determination in nuclear waste and analyze it by Inductively Coupled Plasma Mass Spectrometry (ICPMS). To set up the zirconium radiochemical separation procedure, a zirconium tracer solution was used in order to follow the zirconium behavior during the radiochemical separation. A tracer solution containing the main interferences, Ba, Co, Eu, Fe, Mn, Nb, Ni, Sr, and Y was used in order to verify the decontamination factor during separation process. The limit of detection of 0,039 ppb was obtained for zirconium standard solutions by ICPMS. Then, the protocol will be applied to low level waste (LLW) and intermediate level waste (ILW) from nuclear power plants. (author)
7. Levels of Essential Elements in Different Medicinal Plants Determined by Using Inductively Coupled Plasma Mass Spectrometry
Directory of Open Access Journals (Sweden)
Eid I. Brima
2018-01-01
Full Text Available The objective of this study was to investigate the content of essential elements in medicinal plants in the Kingdom of Saudi Arabia (KSA. Five different medical plants (mahareeb (Cymbopogon schoenanthus, sheeh (Artemisia vulgaris, harjal (Cynanchum argel delile, nabipoot (Equisetum arvense, and cafmariam (Vitex agnus-castus were collected from Madina city in the KSA. Five elements Fe, Mn, Zn, Cu, and Se were determined by using inductively coupled plasma mass spectrometry (ICP-MS. Fe levels were the highest and Se levels were the lowest in all plants. The range levels of all elements in all plants were as follows: Fe 193.4–1757.9, Mn 23.6–143.7, Zn 15.4–32.7, Se 0.13–0.92, and Cu 11.3–21.8 µg/g. Intakes of essential elements from the medical plants in infusion were calculated: Fe 4.6–13.4, Mn 6.7–123.2, Zn 7.0–42.7, Se 0.14–1.5, and Cu 1.5–5.0 µg/dose. The calculated intakes of essential elements for all plants did not exceed the daily intake set by the World Health Organization (WHO and European Food Safety Authority (EFSA. These medicinal plants may be useful sources of essential elements, which are vital for health.
8. Fluorescent Biosensor for Phosphate Determination Based on Immobilized Polyfluorene-Liposomal Nanoparticles Coupled with Alkaline Phosphatase.
Science.gov (United States)
Kahveci, Zehra; Martínez-Tomé, Maria José; Mallavia, Ricardo; Mateo, C Reyes
2017-01-11
This work describes the development of a novel fluorescent biosensor based on the inhibition of alkaline phosphatase (ALP). The biosensor is composed of the enzyme ALP and the conjugated cationic polyfluorene HTMA-PFP. The working principle of the biosensor is based on the fluorescence quenching of this polyelectrolyte by p-nitrophenol (PNP), a product of the hydrolysis reaction of p-nitrophenyl phosphate (PNPP) catalyzed by ALP. Because HTMA-PFP forms unstable aggregates in buffer, with low fluorescence efficiency, previous stabilization of the polyelectrolyte was required before the development of the biosensor. HTMA-PFP was stabilized through its interaction with lipid vesicles to obtain stable blue-emitting nanoparticles (NPs). Fluorescent NPs were characterized, and the ability to be quenched by PNP was evaluated. These nanoparticles were coupled to ALP and entrapped in a sol-gel matrix to produce a biosensor that can serve as a screening platform to identify ALP inhibitors. The components of the biosensor were examined before and after sol-gel entrapment, and the biosensor was optimized to allow the determination of phosphate ion in aqueous medium.
9. Determination of stable cesium and strontium in rice samples by inductively coupled plasma mass spectrometry
Science.gov (United States)
Srinuttrakul, W.; Yoshida, S.
2017-06-01
For long-term radiation dose assessment models, food ingestion is one of the major exposure pathways to human. In general, the stable isotopes can serve as analogues of radioisotopes. In this study, rice samples were collected from 30 paddy fields in Si Sa Ket, Yasothon and Roi Et in the northeast of Thailand in November 2014. The concentrations of stable cesium (Cs-133) and strontium (Sr-88) in polished rice were determined by inductively coupled plasma mass spectrometry (ICP-MS). The standard reference material of rice flour (NIST 1568a) with spiked Cs and Sr was used to validate the analytical method. The concentration of Cs in polished rice from Si Sa Ket, Yasothon and Roi Et was 0.158 ± 0.167 mg kg-1, 0.090 ± 0.117 mg kg-1 and 0.054 ± 0.031 mg kg-1, respectively. The concentration of Sr in polished rice from Si Sa Ket, Yasothon and Roi Et was 0.351 ± 0.108 mg kg-1, 0.364 ± 0.215 mg kg-1 and 0.287 ± 0.102 mg kg-1, respectively. Comparison of the results with Japanese data before the Fukushima Di-ichi nuclear power plant accident showed that the concentrations of both Cs and Sr for Thai rice were higher than those for Japanese rice.
10. Determination of stable cesium and strontium in rice samples by inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Srinuttrakul, W; Yoshida, S
2017-01-01
For long-term radiation dose assessment models, food ingestion is one of the major exposure pathways to human. In general, the stable isotopes can serve as analogues of radioisotopes. In this study, rice samples were collected from 30 paddy fields in Si Sa Ket, Yasothon and Roi Et in the northeast of Thailand in November 2014. The concentrations of stable cesium (Cs-133) and strontium (Sr-88) in polished rice were determined by inductively coupled plasma mass spectrometry (ICP-MS). The standard reference material of rice flour (NIST 1568a) with spiked Cs and Sr was used to validate the analytical method. The concentration of Cs in polished rice from Si Sa Ket, Yasothon and Roi Et was 0.158 ± 0.167 mg kg -1 , 0.090 ± 0.117 mg kg -1 and 0.054 ± 0.031 mg kg -1 , respectively. The concentration of Sr in polished rice from Si Sa Ket, Yasothon and Roi Et was 0.351 ± 0.108 mg kg -1 , 0.364 ± 0.215 mg kg -1 and 0.287 ± 0.102 mg kg -1 , respectively. Comparison of the results with Japanese data before the Fukushima Di-ichi nuclear power plant accident showed that the concentrations of both Cs and Sr for Thai rice were higher than those for Japanese rice. (paper)
11. Determination of zinc stable isotopes in biological materials using isotope dilution inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Patterson, K.Y.; Veillon, Claude
1992-01-01
A method is described for using isotope dilution to determine both the amount of natural zinc and enriched isotopes of zinc in biological samples. Isotope dilution inductively coupled plasma mass spectrometry offers a way to quantify not only the natural zinc found in a sample but also the enriched isotope tracers of zinc. Accurate values for the enriched isotopes and natural zinc are obtained by adjusting the mass count rate data for measurable instrumental biases. Analytical interferences from the matrix are avoided by extracting the zinc from the sample matrix using diethylammonium diethyldithiocarbamate. The extraction technique separates the zinc from elements which form interfering molecular ions at the same nominal masses as the zinc isotopes. Accuracy of the method is verified using standard reference materials. The detection limit is 0.06 μg Zn per sample. Precision of the abundance ratios range from 0.3-0.8%. R.S.D. for natural zinc concentrations is about 200-600 μg g -1 . The accuracy and precision of the measurements make it possible to follow enriched isotopic tracers of zinc in biological samples in metabolic tracer studies. (author). 19 refs.; 1 fig., 4 tabs
12. Copper Determination in Gunshot Residue by Cyclic Voltammetric and Inductive Coupled Plasma-Optical Emission Spectroscopy
Directory of Open Access Journals (Sweden)
Mohd Hashim Nurul’Afiqah Hashimah
2016-01-01
Full Text Available Analysis of gunshot residue (GSR is a crucial evidences for a forensic analyst in the fastest way. GSR analysis insists a suitable method provides a relatively simple, rapid and precise information on the spot at the crime scene. Therefore, the analysis of Cu(II in GSR using cyclic voltammetry (CV on screen printed carbon electrode (SPCE is a better choice compared to previous alternative methods such as Inductive Coupled Plasma-Optical Emission Spectroscopy (ICP-OES those required a long time for analysis. SPCE is specially designed to handle with microvolumes of sample such as GSR sample. It gives advantages for identification of copper in GSR on-site preliminary test to prevent the sample loss on the process to be analyzed in the laboratory. SPCE was swabbed directly on the shooter’s arm immediately after firing and acetate buffer was dropped on SPCE before CV analysis. For ICP-OES analysis, cotton that had been soaked in 0.5 M nitric acid was swabbed on the shooter’s arm immediately after firing and kept in a tightly closed sampling tube. Gold coated SPCE that had been through nanoparticles modification exhibits excellent performance on voltammograms. The calibration was linear from 1 to 50 ppm of copper, the limit of detection for copper was 0.3 ppm and a relative standard deviation was 6.1 %. The method was successfully applied to the determination of copper in GSR. The Cu determination on SPCE was compared and validated by ICP-OES method with 94 % accuracy.
13. Determination of the quark coupling strength vertical bar V-ub vertical bar using baryonic decays
NARCIS (Netherlands)
Aaij, R.; Adeva, B.; Adinolfi, M.; Older, A. A.; Ajaltouni, Z.; Akar, S.; Albrecht, J.; Alessio, F.; Alexander, M.; Ali, S.; Alkhazov, G.; Cartelle, P. Alvarez; Alves, A. A.; Amato, S.; Amerio, S.; Amhis, Y.; An, L.; Anderlini, L.; Andreotti, M.; Andrews, J. E.; Appleby, R. B.; Gutierrez, O. Aquines; Archilli, F.; Artamonov, A.; Artuso, M.; Aslanides, E.; Auriemma, G.; Baalouch, M.; Bachmann, S.; Back, J. J.; Badalov, A.; Baesso, C.; Baldini, W.; Barlow, R. J.; Barschel, C.; Barsuk, S.; Barter, W.; Batozskaya, V.; Battista, V.; Beaucourt, L.; Beddow, J.; Bedeschi, F.; Bediaga, I.; Bel, L. J.; Belyaev, I.; Ben-Haim, E.; Bencivenni, G.; Onderwater, C. J. G.; Pellegrino, A.; Tolk, S.
In the Standard Model of particle physics, the strength of the couplings of the b quark to the u and c quarks, vertical bar V-ub vertical bar and vertical bar V-ub vertical bar, are governed by the coupling of the quarks to the Higgs boson. Using data from the LHCb experiment at the Large Hadron
14. Couples' Career Orientation, Gender Role Orientation, and Perceived Equity as Determinants of Marital Power.
Science.gov (United States)
Sexton, Christine S.; Perlman, Daniel S.
1989-01-01
Investigated influence of resource exchanges and gender role on marital power. Compared dual-career (N=50) and single-career (N=50) couples. Found two couple types did not differ in perceived power nor in self-reported strategies for influencing spouses. Found gender role orientation did not affect marital power. (Author/CM)
15. Spectral Tensor-Train Decomposition
DEFF Research Database (Denmark)
Bigoni, Daniele; Engsig-Karup, Allan Peter; Marzouk, Youssef M.
2016-01-01
The accurate approximation of high-dimensional functions is an essential task in uncertainty quantification and many other fields. We propose a new function approximation scheme based on a spectral extension of the tensor-train (TT) decomposition. We first define a functional version of the TT...... adaptive Smolyak approach. The method is also used to approximate the solution of an elliptic PDE with random input data. The open source software and examples presented in this work are available online (http://pypi.python.org/pypi/TensorToolbox/)....
16. Analytical effective tensor for flow-through composites
Science.gov (United States)
Sviercoski, Rosangela De Fatima [Los Alamos, NM
2012-06-19
A machine, method and computer-usable medium for modeling an average flow of a substance through a composite material. Such a modeling includes an analytical calculation of an effective tensor K.sup.a suitable for use with a variety of media. The analytical calculation corresponds to an approximation to the tensor K, and follows by first computing the diagonal values, and then identifying symmetries of the heterogeneity distribution. Additional calculations include determining the center of mass of the heterogeneous cell and its angle according to a defined Cartesian system, and utilizing this angle into a rotation formula to compute the off-diagonal values and determining its sign.
17. Electrical conductivity tensor of an irradiated metal
International Nuclear Information System (INIS)
Corciovei, A.; Dumitru, R.D.
1979-01-01
A method to calculate the electrical conductivity tensor of an irradiated metal is presented. The proposed method relies on the use of the Kubo formula, evaluated by a perturbation method. The one electron Hamiltonian is written as a sum of two terms: the Hamiltonian of the conduction electrons moving in a periodic lattice and the perturbation, namely, the scattering potential due to the irradiation defects of the ideal crystal. Then, the lowest order of the conductivity is determined by the lowest order of the Laplace transform of the current. An integral equation is written for this last quantity. (author)
18. Tensor ghosts in the inflationary cosmology
International Nuclear Information System (INIS)
Clunan, Tim; Sasaki, Misao
2010-01-01
Theories with curvature-squared terms in the action are known to contain ghost modes in general. However, if we regard curvature-squared terms as quantum corrections to the original theory, the emergence of ghosts may be simply due to the perturbative truncation of a full non-perturbative theory. If this is the case, there should be a way to live with ghosts. In this paper, we take the Euclidean path integral approach, in which ghost degrees of freedom can be, and are integrated out in the Euclideanized spacetime. We apply this procedure to Einstein gravity with a Weyl curvature-squared correction in the inflationary background. We find that the amplitude of tensor perturbations is modified by a term of O(α 2 H 2 ) where α 2 is a coupling constant in front of the Weyl-squared term and H is the Hubble parameter during inflation.
19. Diffusion tensor MR microscopy of tissues with low diffusional anisotropy.
Science.gov (United States)
Bajd, Franci; Mattea, Carlos; Stapf, Siegfried; Sersa, Igor
2016-06-01
Diffusion tensor imaging exploits preferential diffusional motion of water molecules residing within tissue compartments for assessment of tissue structural anisotropy. However, instrumentation and post-processing errors play an important role in determination of diffusion tensor elements. In the study, several experimental factors affecting accuracy of diffusion tensor determination were analyzed. Effects of signal-to-noise ratio and configuration of the applied diffusion-sensitizing gradients on fractional anisotropy bias were analyzed by means of numerical simulations. In addition, diffusion tensor magnetic resonance microscopy experiments were performed on a tap water phantom and bovine articular cartilage-on-bone samples to verify the simulation results. In both, the simulations and the experiments, the multivariate linear regression of the diffusion-tensor analysis yielded overestimated fractional anisotropy with low SNRs and with low numbers of applied diffusion-sensitizing gradients. An increase of the apparent fractional anisotropy due to unfavorable experimental conditions can be overcome by applying a larger number of diffusion sensitizing gradients with small values of the condition number of the transformation matrix. This is in particular relevant in magnetic resonance microscopy, where imaging gradients are high and the signal-to-noise ratio is low.
20. The 'gravitating' tensor in the dualistic theory
International Nuclear Information System (INIS)
Mahanta, M.N.
1989-01-01
The exact microscopic system of Einstein-type field equations of the dualistic gravitation theory is investigated as well as an analysis of the modified energy-momentum tensor or so called 'gravitating' tensor is presented
1. Tensor calculus for physics a concise guide
CERN Document Server
Neuenschwander, Dwight E
2015-01-01
Understanding tensors is essential for any physics student dealing with phenomena where causes and effects have different directions. A horizontal electric field producing vertical polarization in dielectrics; an unbalanced car wheel wobbling in the vertical plane while spinning about a horizontal axis; an electrostatic field on Earth observed to be a magnetic field by orbiting astronauts—these are some situations where physicists employ tensors. But the true beauty of tensors lies in this fact: When coordinates are transformed from one system to another, tensors change according to the same rules as the coordinates. Tensors, therefore, allow for the convenience of coordinates while also transcending them. This makes tensors the gold standard for expressing physical relationships in physics and geometry. Undergraduate physics majors are typically introduced to tensors in special-case applications. For example, in a classical mechanics course, they meet the "inertia tensor," and in electricity and magnetism...
2. Endomorphism Algebras of Tensor Powers of Modules for Quantum Groups
DEFF Research Database (Denmark)
Andersen, Therese Søby
We determine the ring structure of the endomorphism algebra of certain tensor powers of modules for the quantum group of sl2 in the case where the quantum parameter is allowed to be a root of unity. In this case there exists -- under a suitable localization of our ground ring -- a surjection from...... the group algebra of the braid group to the endomorphism algebra of any tensor power of the Weyl module with highest weight 2. We take a first step towards determining the kernel of this map by reformulating well-known results on the semisimplicity of the Birman-Murakami-Wenzl algebra in terms of the order...... of the quantum parameter. Before we arrive at these main results, we investigate the structure of the endomorphism algebra of the tensor square of any Weyl module....
3. Performance Optimization of Tensor Contraction Expressions for Many Body Methods in Quantum Chemistry
International Nuclear Information System (INIS)
Hartono, Albert; Lu, Qingda; Henretty, Thomas; Krishnamoorthy, Sriram; Zhang, Huaijian; Baumgartner, Gerald; Bernholdt, David E.; Nooijen, Marcel; Pitzer, Russell M.; Ramanujam, J.; Sadayappan, Ponnuswamy
2009-01-01
Complex tensor contraction expressions arise in accurate electronic structure models in quantum chemistry, such as the coupled cluster method. This paper addresses two complementary aspects of performance optimization of such tensor contraction expressions. Transformations using algebraic properties of commutativity and associativity can be used to significantly decrease the number of arithmetic operations required for evaluation of these expressions. The identification of common subexpressions among a set of tensor contraction expressions can result in a reduction of the total number of operations required to evaluate the tensor contractions. The first part of the paper describes an effective algorithm for operation minimization with common subexpression identification and demonstrates its effectiveness on tensor contraction expressions for coupled cluster equations. The second part of the paper highlights the importance of data layout transformation in the optimization of tensor contraction computations on modern processors. A number of considerations such as minimization of cache misses and utilization of multimedia vector instructions are discussed. A library for efficient index permutation of multi-dimensional tensors is described and experimental performance data is provided that demonstrates its effectiveness.
4. Performance Optimization of Tensor Contraction Expressions for Many Body Methods in Quantum Chemistry
International Nuclear Information System (INIS)
Krishnamoorthy, Sriram; Bernholdt, David E.; Pitzer, R.M.; Sadayappan, Ponnuswamy
2009-01-01
Complex tensor contraction expressions arise in accurate electronic structure models in quantum chemistry, such as the coupled cluster method. This paper addresses two complementary aspects of performance optimization of such tensor contraction expressions. Transformations using algebraic properties of commutativity and associativity can be used to significantly decrease the number of arithmetic operations required for evaluation of these expressions. The identification of common subexpressions among a set of tensor contraction expressions can result in a reduction of the total number of operations required to evaluate the tensor contractions. The first part of the paper describes an effective algorithm for operation minimization with common subexpression identification and demonstrates its effectiveness on tensor contraction expressions for coupled cluster equations. The second part of the paper highlights the importance of data layout transformation in the optimization of tensor contraction computations on modern processors. A number of considerations, such as minimization of cache misses and utilization of multimedia vector instructions, are discussed. A library for efficient index permutation of multidimensional tensors is described, and experimental performance data is provided that demonstrates its effectiveness.
5. Reciprocal mass tensor : a general form
International Nuclear Information System (INIS)
Roy, C.L.
1978-01-01
Using the results of earlier treatment of wave packets, a general form of reciprocal mass tensor has been obtained. The elements of this tensor are seen to be dependent on momentum as well as space coordinates of the particle under consideration. The conditions under which the tensor would reduce to the usual space-independent form, are discussed and the impact of the space-dependence of this tensor on the motion of Bloch electrons, is examined. (author)
6. A new deteriorated energy-momentum tensor
International Nuclear Information System (INIS)
Duff, M.J.
1982-01-01
The stress-tensor of a scalar field theory is not unique because of the possibility of adding an 'improvement term'. In supersymmetric field theories the stress-tensor will appear in a super-current multiplet along with the sypersymmetry current. The general question of the supercurrent multiplet for arbitrary deteriorated stress tensors and their relationship to supercurrent multiplets for models with gauge antisymmetric tensors is answered for various models of N = 1, 2 and 4 supersymmetry. (U.K.)
7. Tensor-based spatiotemporal saliency detection
Science.gov (United States)
Dou, Hao; Li, Bin; Deng, Qianqian; Zhang, LiRui; Pan, Zhihong; Tian, Jinwen
2018-03-01
This paper proposes an effective tensor-based spatiotemporal saliency computation model for saliency detection in videos. First, we construct the tensor representation of video frames. Then, the spatiotemporal saliency can be directly computed by the tensor distance between different tensors, which can preserve the complete temporal and spatial structure information of object in the spatiotemporal domain. Experimental results demonstrate that our method can achieve encouraging performance in comparison with the state-of-the-art methods.
8. Anti-symmetric rank-two tensor matter field on superspace for NT=2
International Nuclear Information System (INIS)
Spalenza, Wesley; Ney, Wander G.; Helayel-Neto, J.A.
2004-01-01
In this work, we discuss the interaction between anti-symmetric rank-two tensor matter and topological Yang-Mills fields. The matter field considered here is the rank-2 Avdeev-Chizhov tensor matter field in a suitably extended N T =2 SUSY. We start off from the N T =2, D=4 superspace formulation and we go over to Riemannian manifolds. The matter field is coupled to the topological Yang-Mills field. We show that both actions are obtained as Q-exact forms, which allows us to express the energy-momentum tensor as Q-exact observables
9. Relativistic symmetries in the Hulthén scalar—vector—tensor interactions
International Nuclear Information System (INIS)
Hamzavi Majid; Rajabi Ali Akbar
2013-01-01
In the presence of spin and pseudospin (p-spin) symmetries, the approximate analytical bound states of the Dirac equation for scalar—vector—tensor Hulthén potentials are obtained with any arbitrary spin—orbit coupling number κ using the Pekeris approximation. The Hulthén tensor interaction is studied instead of the commonly used Coulomb or linear terms. The generalized parametric Nikiforov—Uvarov (NU) method is used to obtain energy eigenvalues and corresponding wave functions in their closed forms. It is shown that tensor interaction removes degeneracy between spin and p-spin doublets. Some numerical results are also given. (general)
10. Induced vacuum energy-momentum tensor in the background of a cosmic string
OpenAIRE
Sitenko, Yu. A.; Vlasii, N. D.
2011-01-01
A massive scalar field is quantized in the background of a cosmic string which is generalized to a static flux-carrying codimension-2 brane in the locally flat multidimensional space-time. We find that the finite energy-momentum tensor is induced in the vacuum. The dependence of the tensor components on the brane flux and tension, as well as on the coupling to the space-time curvature scalar, is comprehensively analyzed. The tensor components are holomorphic functions of space dimension, decr...
11. The direct tensor solution and higher-order acquisition schemes for generalized diffusion tensor imaging
NARCIS (Netherlands)
Akkerman, Erik M.
2010-01-01
Both in diffusion tensor imaging (DTI) and in generalized diffusion tensor imaging (GDTI) the relation between the diffusion tensor and the measured apparent diffusion coefficients is given by a tensorial equation, which needs to be inverted in order to solve the diffusion tensor. The traditional
12. Determination of uranium from nuclear fuel in environmental samples using inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Boulyga, S.F.; Becker, J.S.
2000-01-01
As a result of the accident at the Chernobyl nuclear power plant (NPP) the environment was contaminated with spent nuclear fuel. The 236 U isotope was used in this study to monitor the spent uranium from nuclear fallout in soil samples collected in the vicinity of the Chernobyl NPP. A rapid and sensitive analytical procedure was developed for uranium isotopic ratio measurement in environmental samples based on inductively coupled plasma quadrupole mass spectrometry with a hexapole collision cell (HEX-ICP-QMS). The figures of merit of the HEX-ICP-QMS were studied with a plasma-shielded torch using different nebulizers (such as an ultrasonic nebulizer (USN) and Meinhard nebulizer) for solution introduction. A 238 U + ion intensity of up to 27000 MHz/ppm in HEX-ICP-QMS with USN was observed by introducing helium into the hexapole collision cell as the collision gas at a flow rate of 10 ml min -1 . The formation rate of uranium hydride ions UH + /U + of 2 x 10 -6 was obtained by using USN with a membrane desolvator. The limit of 236 U/ 238 U ratio determination in 10 μg 1 -1 uranium solution was 3 x 10 -7 corresponding to the detection limit for 236 U of 3 pg 1 -1 . The precision of uranium isotopic ratio measurements in 10 μg 1 -1 laboratory uranium isotopic standard solution was 0.13% ( 235 U/ 238 U) and 0.33% ( 236 U/ 238 U) using a Meinhard nebulizer and 0.45% ( 235 U/ 238 U) and 0.88% ( 236 U/ 238 U) using a USN. The isotopic composition of all investigated Chernobyl soil samples differed from those of natural uranium; i.e. in these samples the 236 U/ 238 U ratio ranged from 10 -5 to 10 -3 . (orig.)
13. Tucker Tensor analysis of Matern functions in spatial statistics
KAUST Repository
Litvinenko, Alexander
2018-03-09
In this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes. Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and store, especially in 3D. Therefore, we approximate covariance functions by cheap surrogates in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of Matern- and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence. We prove the exponential convergence of the Tucker and canonical approximations in tensor rank parameters. Several statistical operations are performed in this low-rank tensor format, including evaluating the conditional covariance matrix, spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations reduce the computing and storage costs essentially. For example, the storage cost is reduced from an exponential O(n^d) to a linear scaling O(drn), where d is the spatial dimension, n is the number of mesh points in one direction, and r is the tensor rank. Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations, and measurements lie on a tensor (axes-parallel) grid and that the covariance function depends on a distance, ||x-y||.
14. Cosmological simulations using a static scalar-tensor theory
Energy Technology Data Exchange (ETDEWEB)
RodrIguez-Meza, M A [Depto. de Fisica, Instituto Nacional de Investigaciones Nucleares, Col. Escandon, Apdo. Postal 18-1027, 11801 Mexico D.F (Mexico); Gonzalez-Morales, A X [Departamento Ingenierias, Universidad Iberoamericana, Prol. Paseo de la Reforma 880 Lomas de Santa Fe, Mexico D.F. Mexico (Mexico); Gabbasov, R F [Depto. de Fisica, Instituto Nacional de Investigaciones Nucleares, Col. Escandon, Apdo. Postal 18-1027, 11801 Mexico D.F (Mexico); Cervantes-Cota, Jorge L [Depto. de Fisica, Instituto Nacional de Investigaciones Nucleares, Col. Escandon, Apdo. Postal 18-1027, 11801 Mexico D.F (Mexico)
2007-11-15
We present {lambda}CDM N-body cosmological simulations in the framework of of a static general scalar-tensor theory of gravity. Due to the influence of the non-minimally coupled scalar field, the gravitational potential is modified by a Yukawa type term, yielding a new structure formation dynamics. We present some preliminary results and, in particular, we compute the density and velocity profiles of the most massive group.
15. Compact stars in vector-tensor-Horndeski theory of gravity
Energy Technology Data Exchange (ETDEWEB)
Momeni, Davood; Myrzakulov, Kairat; Myrzakulov, Ratbay [Eurasian National University, Department of General and Theoretical Physics, Eurasian International Center for Theoretical Physics, Astana (Kazakhstan); Faizal, Mir [University of British Columbia-Okanagan, Irving K. Barber School of Arts and Sciences, Kelowna, BC (Canada); University of Lethbridge, Department of Physics and Astronomy, Lethbridge, AB (Canada)
2017-01-15
In this paper, we will analyze a theory of modified gravity, in which the field content of general relativity will be increased to include a vector field. We will use the Horndeski formalism to non-minimally couple this vector field to the metric. As we will be using the Horndeski formalism, this theory will not contain Ostrogradsky ghost degree of freedom. We will analyze compact stars using this vector-tensor-Horndeski theory. (orig.)
16. Werner-Wheeler mass tensor for fusionlike configuration
International Nuclear Information System (INIS)
Gherghescu, R.A.; Poenaru, D.N.
2005-01-01
The Werner-Wheeler approach is used to calculate the components of the mass tensor for a binary configuration of two intersected spheroids. Four free coordinates form the deformation space: the small semiaxis of the projectile, the two semiaxis ratios of the spheroids, and the distance between centers. A correction term is also calculated, due to the center of mass motion. Final results are presented for the fusion channel 54 Cr+ 240 Pu, and all possible couplings are analyzed
17. Scalar-tensor theory of fourth-order gravity
International Nuclear Information System (INIS)
Accioly, A.J.; Goncalves, A.T.
1986-04-01
A scalar-tensor theory of fourth-order gravity is considered. Some cosmological consequences, due to the presence of the scalar field, as well as of metric derivatives higher than second order, are analysed. In particular, upperbpunds are obtained for the coupling constant α and for the scale factor of the universe, respectively. The discussion is restricted to Robertson-Walker universes. (Author) [pt
18. Efficient Tensor Strategy for Recommendation
Directory of Open Access Journals (Sweden)
Aboagye Emelia Opoku
2017-07-01
Full Text Available The era of big data has witnessed the explosion of tensor datasets, and large scale Probabilistic Tensor Factorization (PTF analysis is important to accommodate such increasing trend of data. Sparsity, and Cold-Start are some of the inherent problems of recommender systems in the era of big data. This paper proposes a novel Sentiment-Based Probabilistic Tensor Analysis technique senti-PTF to address the problems. The propose framework first applies a Natural Language Processing technique to perform sentiment analysis taking advantage of the huge sums of textual data generated available from the social media which are predominantly left untouched. Although some current studies do employ review texts, many of them do not consider how sentiments in reviews influence recommendation algorithm for prediction. There is therefore this big data text analytics gap whose modeling is computationally expensive. From our experiments, our novel machine learning sentiment-based tensor analysis is computationally less expensive, and addresses the cold-start problem, for optimal recommendation prediction.
19. Spherical Tensor Calculus for Local Adaptive Filtering
Science.gov (United States)
Reisert, Marco; Burkhardt, Hans
In 3D image processing tensors play an important role. While rank-1 and rank-2 tensors are well understood and commonly used, higher rank tensors are rare. This is probably due to their cumbersome rotation behavior which prevents a computationally efficient use. In this chapter we want to introduce the notion of a spherical tensor which is based on the irreducible representations of the 3D rotation group. In fact, any ordinary cartesian tensor can be decomposed into a sum of spherical tensors, while each spherical tensor has a quite simple rotation behavior. We introduce so called tensorial harmonics that provide an orthogonal basis for spherical tensor fields of any rank. It is just a generalization of the well known spherical harmonics. Additionally we propose a spherical derivative which connects spherical tensor fields of different degree by differentiation. Based on the proposed theory we present two applications. We propose an efficient algorithm for dense tensor voting in 3D, which makes use of tensorial harmonics decomposition of the tensor-valued voting field. In this way it is possible to perform tensor voting by linear-combinations of convolutions in an efficient way. Secondly, we propose an anisotropic smoothing filter that uses a local shape and orientation adaptive filter kernel which can be computed efficiently by the use spherical derivatives.
20. Determination of nonlinear nanomechanical resonator-qubit coupling coefficient in a hybrid quantum system.
Science.gov (United States)
Geng, Qi; Zhu, Ka-Di
2016-07-10
We have theoretically investigated a hybrid system that is composed of a traditional optomechanical component and an additional charge qubit (Cooper pair box) that induces a new nonlinear interaction. It is shown that the peak in optomechanically induced transparency has been split by the new nonlinear interaction, and the width of the splitting is proportional to the coupling coefficient of this nonlinear interaction. This may give a way to measure the nanomechanical oscillator-qubit coupling coefficient in hybrid quantum systems.
1. Determination of the quark coupling strength $|V_{ub}|$ using baryonic decays
CERN Document Server
2015-01-01
In the Standard Model of particle physics, the strength of the couplings of the $b$ quark to the $u$ and $c$ quarks, $|V_{ub}|$ and $|V_{cb}|$, are governed by the coupling of the quarks to the Higgs boson. Using data from the LHCb experiment at the Large Hadron Collider, the probability for the $\\Lambda^0_b$ baryon to decay into the p \\mu^- \\overline{\ 2. Plexcitons: The Role of Oscillator Strengths and Spectral Widths in Determining Strong Coupling Energy Technology Data Exchange (ETDEWEB) Thomas, Reshmi [School; Thomas, Anoop [School; Pullanchery, Saranya [School; Joseph, Linta [School; Somasundaran, Sanoop Mambully [School; Swathi, Rotti Srinivasamurthy [School; Gray, Stephen K. [Center; Thomas, K. George [School 2018-01-05 Strong coupling interactions between plasmon and exciton-based excitations have been proposed to be useful in the design of optoelectronic systems. However, the role of various optical parameters dictating the plasmon-exciton (plexciton) interactions is less understood. Herein, we propose an inequality for achieving strong coupling between plasmons and excitons through appropriate variation of their oscillator strengths and spectral widths. These aspects are found to be consistent with experiments on two sets of free-standing plexcitonic systems obtained by (i) linking fluorescein isothiocyanate on Ag nanoparticles of varying sizes through silane coupling and (ii) electrostatic binding of cyanine dyes on polystyrenesulfonate-coated Au nanorods of varying aspect ratios. Being covalently linked on Ag nanoparticles, fluorescein isothiocyanate remains in monomeric state, and its high oscillator strength and narrow spectral width enable us to approach the strong coupling limit. In contrast, in the presence of polystyrenesulfonate, monomeric forms of cyanine dyes exist in equilibrium with their aggregates: Coupling is not observed for monomers and H-aggregates whose optical parameters are unfavorable. The large aggregation number, narrow spectral width, and extremely high oscillator strength of J-aggregates of cyanines permit effective delocalization of excitons along the linear assembly of chromophores, which in turn leads to efficient coupling with the plasmons. Further, the results obtained from experiments and theoretical models are jointly employed to describe the plexcitonic states, estimate the coupling strengths, and rationalize the dispersion curves. The experimental results and the theoretical analysis presented here portray a way forward to the rational design of plexcitonic systems attaining the strong coupling limits. 3. FI/SI on-line solvent extraction/back extraction preconcentration coupled to direct injection nebulization inductively coupled plasma mass spectrometry for determination of copper and lead DEFF Research Database (Denmark) Wang, Jianhua; Hansen, Elo Harald 2002-01-01 An automated sequential injection on-line preconcentration procedure for determination of trace levels of copper and lead via solvent extraction/back extraction coupled to ICP-MS is described. In citrate buffer of pH 3, neutral complexes between the analytes and the chelating reagent, ammonium...... loop, the content of which is subsequently introduced into the ICP-MS, via a direct injection high efficiency nebulizer (DIHEN), for quantification. Enrichment factors of 29.6 (Cu) and 23.3 (Pb), detection limits of 17 ng/l (Cu) and 11 ng/l (Pb), along with a sampling frequency of 13 s/h were obtained... 4. Determinations of the QCD strong coupling αsub(s) and the scale Λsub(QCD) International Nuclear Information System (INIS) Duke, D.W.; Roberts, R.G. 1984-08-01 The authors review determinations, via experiment of the strong coupling of QCD, αsub(s). In almost every case, the results are used of perturbative QCD to make the necessary extraction from data. These include scaling violations of deep inelastic scattering, e + e - annihilation experiments (including quarkonium decays) and lepton pair production. Finally estimates for Λ from lattice calculations are listed. (author) 5. Determination of uranium in urine - Measurement of isotope ratios and quantification by use of inductively coupled plasma mass spectrometry NARCIS (Netherlands) Krystek, Petra; Ritsema, R. 2002-01-01 For analysis of uranium in urine determination of the isotope ratio and quantification were investigated by high-resolution inductively coupled plasma mass spectrometry (HR ICP-MS). The instrument used (ThermoFinniganMAT ELEMENT2) is a single-collector MS and, therefore, a stable sample-introduction 6. Implementation of Fully Coupled Heat and Mass Transport Model to Determine Temperature and Moisture State at Elevated Temperatures DEFF Research Database (Denmark) Pecenko, R.; Hozjan, Tomaz; Svensson, Staffan 2014-01-01 The aim of this study is to present precise numerical formulation to determine temperature and moisture state of timber in the situation prior pyrolysis. The strong formulations needed for an accurate description of the physics are presented and discussed as well as their coupling terms. From... 7. Determination of the ratio of axial-vector-to-vector weak coupling constants for beta decay of triton CERN Document Server Akulov, Y A 2002-01-01 Data on the chemical shifts of half-lives for atomic and molecular tritium were used to determine the ratio of axial-vector-to-vector weak coupling constants for beta decay of triton (G sub A /G sub V) sub t = -1.2646 +- 0.0035 8. A new Weyl-like tensor of geometric origin Science.gov (United States) Vishwakarma, Ram Gopal 2018-04-01 A set of new tensors of purely geometric origin have been investigated, which form a hierarchy. A tensor of a lower rank plays the role of the potential for the tensor of one rank higher. The tensors have interesting mathematical and physical properties. The highest rank tensor of the hierarchy possesses all the geometrical properties of the Weyl tensor. 9. Conflicts Within the Family and Within the Couple as Contextual Factors in the Determinism of Male Sexual Dysfunction. Science.gov (United States) Boddi, Valentina; Fanni, Egidia; Castellini, Giovanni; Fisher, Alessandra Daphne; Corona, Giovanni; Maggi, Mario 2015-12-01 The deterioration of a couple's relationship has been previously associated with impairment in male sexual function. Besides a couple's dystonic relationship, other stressors can unfavorably influence dyadic intimacy. A largely neglected etiopathogenetic factor affecting couple sexuality is the frustration caused by conflicts within the family. To evaluate the possible associations between male sexual dysfunction (SD) and conflictual relationships within the couple or the family. A consecutive series of 3,975 men, attending the Outpatient Clinic for SD for the first time, was retrospectively studied. Conflicts within the family and within the couple were assessed using two standard questions: "Are there any conflicts at home," and "Do you have a difficult relationship with your partner?" respectively, rating 0 = normal relationships, 1 = occasional quarrels, and 2 = frequent quarrels or always. Several clinical, biochemical, and psychological (Middlesex Hospital Questionnaire) parameters were studied. Among the 3,975 patients studied, we observed a high prevalence of conflicts within the family and within the couple (32% vs. 21.2%). When compared with the rest of the sample, subjects reporting both type of conflicts showed a higher prevalence of psychiatric comorbidities. Hence, all data were adjusted for this parameter and for age. Family and couple conflicts were significantly associated with free floating anxiety, depression symptoms, and with a higher risk of subjective (self-reported) and objective (peak systolic velocity at the penile color Doppler ultrasound conflicts. This study indicates that the presence of often unexplored issues, like conflicts within the family or within the couple, can represent an important contextual factor in the determinism of male SD. © 2015 International Society for Sexual Medicine. 10. Renormalized energy-momentum tensor of λΦ4 theory in curved ... Indian Academy of Sciences (India) Divergenceless expression for the energy-momentum tensor of scalar field is obtained using the momentum cut-off regularization technique. We consider a scalar field with quartic self-coupling in a spatially flat (3+1)-dimensional Robertson–Walker space-time, having arbitrary mass and coupled to gravity. As special cases ... 11. Comparison of Magnetic Susceptibility Tensor and Diffusion Tensor of the Brain. Science.gov (United States) Li, Wei; Liu, Chunlei 2013-10-01 Susceptibility tensor imaging (STI) provides a novel approach for noninvasive assessment of the white matter pathways of the brain. Using mouse brain ex vivo , we compared STI with diffusion tensor imaging (DTI), in terms of tensor values, principal tensor values, anisotropy values, and tensor orientations. Despite the completely different biophysical underpinnings, magnetic susceptibility tensors and diffusion tensors show many similarities in the tensor and principal tensor images, for example, the tensors perpendicular to the fiber direction have the highest gray-white matter contrast, and the largest principal tensor is along the fiber direction. Comparison to DTI fractional anisotropy, the susceptibility anisotropy provides much higher sensitivity to the chemical composition of the white matter, especially myelin. The high sensitivity can be further enhanced with the perfusion of ProHance, a gadolinium-based contrast agent. Regarding the tensor orientations, the direction of the largest principal susceptibility tensor agrees with that of diffusion tensors in major white matter fiber bundles. The STI fiber tractography can reconstruct the fiber pathways for the whole corpus callosum and for white matter fiber bundles that are in close contact but in different orientations. There are some differences between susceptibility and diffusion tensor orientations, which are likely due to the limitations in the current STI reconstruction. With the development of more accurate reconstruction methods, STI holds the promise for probing the white matter micro-architectures with more anatomical details and higher chemical sensitivity. 12. Dilaton and second-rank tensor fields as supersymmetric compensators International Nuclear Information System (INIS) Nishino, Hitoshi; Rajpoot, Subhash 2007-01-01 We formulate a supersymmetric theory in which both a dilaton and a second-rank tensor play roles of compensators. The basic off-shell multiplets are a linear multiplet (B μν ,χ,φ) and a vector multiplet (A μ ,λ;C μνρ ), where φ and B μν are, respectively, a dilaton and a second-rank tensor. The third-rank tensor C μνρ in the vector multiplet is ''dual'' to the conventional D field with 0 on-shell or 1 off-shell degree of freedom. The dilaton φ is absorbed into one longitudinal component of A μ , making it massive. Initially, B μν has 1 on-shell or 3 off-shell degrees of freedom, but it is absorbed into the longitudinal components of C μνρ . Eventually, C μνρ with 0 on-shell or 1 off-shell degree of freedom acquires in total 1 on-shell or 4 off-shell degrees of freedom, turning into a propagating massive field. These basic multiplets are also coupled to chiral multiplets and a supersymmetric Dirac-Born-Infeld action. Some of these results are also reformulated in superspace. The proposed mechanism may well provide a solution to the long-standing puzzle of massless dilatons and second-rank tensors in supersymmetric models inspired by string theory 13. Tensor voting for robust color edge detection OpenAIRE Moreno, Rodrigo; García, Miguel Ángel; Puig, Domenec 2014-01-01 The final publication is available at Springer via http://dx.doi.org/10.1007/978-94-007-7584-8_9 This chapter proposes two robust color edge detection methods based on tensor voting. The first method is a direct adaptation of the classical tensor voting to color images where tensors are initialized with either the gradient or the local color structure tensor. The second method is based on an extension of tensor voting in which the encoding and voting processes are specifically tailored to ... 14. The Physical Interpretation of the Lanczos Tensor OpenAIRE Roberts, Mark D. 1999-01-01 The field equations of general relativity can be written as first order differential equations in the Weyl tensor, the Weyl tensor in turn can be written as a first order differential equation in a three index tensor called the Lanczos tensor. The Lanczos tensor plays a similar role in general relativity to that of the vector potential in electro-magnetic theory. The Aharonov-Bohm effect shows that when quantum mechanics is applied to electro-magnetic theory the vector potential is dynamicall... 15. Estimation of full moment tensors, including uncertainties, for earthquakes, volcanic events, and nuclear explosions Science.gov (United States) Alvizuri, Celso R. We present a catalog of full seismic moment tensors for 63 events from Uturuncu volcano in Bolivia. The events were recorded during 2011-2012 in the PLUTONS seismic array of 24 broadband stations. Most events had magnitudes between 0.5 and 2.0 and did not generate discernible surface waves; the largest event was Mw 2.8. For each event we computed the misfit between observed and synthetic waveforms, and we used first-motion polarity measurements to reduce the number of possible solutions. Each moment tensor solution was obtained using a grid search over the six-dimensional space of moment tensors. For each event we show the misfit function in eigenvalue space, represented by a lune. We identify three subsets of the catalog: (1) 6 isotropic events, (2) 5 tensional crack events, and (3) a swarm of 14 events southeast of the volcanic center that appear to be double couples. The occurrence of positively isotropic events is consistent with other published results from volcanic and geothermal regions. Several of these previous results, as well as our results, cannot be interpreted within the context of either an oblique opening crack or a crack-plus-double-couple model. Proper characterization of uncertainties for full moment tensors is critical for distinguishing among physical models of source processes. A seismic moment tensor is a 3x3 symmetric matrix that provides a compact representation of a seismic source. We develop an algorithm to estimate moment tensors and their uncertainties from observed seismic data. For a given event, the algorithm performs a grid search over the six-dimensional space of moment tensors by generating synthetic waveforms for each moment tensor and then evaluating a misfit function between the observed and synthetic waveforms. 'The' moment tensor M0 for the event is then the moment tensor with minimum misfit. To describe the uncertainty associated with M0, we first convert the misfit function to a probability function. The uncertainty, or 16. Determination of uranium from nuclear fuel in environmental samples using inductively coupled plasma mass spectrometry Energy Technology Data Exchange (ETDEWEB) Boulyga, S.F. [Forschungszentrum Juelich GmbH (Germany). Zentralabteilung fuer Chemische Analysen]|[Radiation Physics and Chemistry Problems Inst., Minsk (Belarus); Becker, J.S. [Forschungszentrum Juelich GmbH (Germany). Zentralabteilung fuer Chemische Analysen 2000-11-01 As a result of the accident at the Chernobyl nuclear power plant (NPP) the environment was contaminated with spent nuclear fuel. The {sup 236}U isotope was used in this study to monitor the spent uranium from nuclear fallout in soil samples collected in the vicinity of the Chernobyl NPP. A rapid and sensitive analytical procedure was developed for uranium isotopic ratio measurement in environmental samples based on inductively coupled plasma quadrupole mass spectrometry with a hexapole collision cell (HEX-ICP-QMS). The figures of merit of the HEX-ICP-QMS were studied with a plasma-shielded torch using different nebulizers (such as an ultrasonic nebulizer (USN) and Meinhard nebulizer) for solution introduction. A {sup 238}U{sup +} ion intensity of up to 27000 MHz/ppm in HEX-ICP-QMS with USN was observed by introducing helium into the hexapole collision cell as the collision gas at a flow rate of 10 ml min{sup -1}. The formation rate of uranium hydride ions UH{sup +}/U{sup +} of 2 x 10{sup -6} was obtained by using USN with a membrane desolvator. The limit of {sup 236}U/{sup 238}U ratio determination in 10 {mu}g 1{sup -1} uranium solution was 3 x 10{sup -7} corresponding to the detection limit for {sup 236}U of 3 pg 1{sup -1}. The precision of uranium isotopic ratio measurements in 10 {mu}g 1{sup -1} laboratory uranium isotopic standard solution was 0.13% ({sup 235}U/{sup 238}U) and 0.33% ({sup 236}U/{sup 238}U) using a Meinhard nebulizer and 0.45% ({sup 235}U/{sup 238}U) and 0.88% ({sup 236}U/{sup 238}U) using a USN. The isotopic composition of all investigated Chernobyl soil samples differed from those of natural uranium; i.e. in these samples the {sup 236}U/{sup 238}U ratio ranged from 10{sup -5} to 10{sup -3}. (orig.) 17. Torsion tensor and covector in a unified field theory International Nuclear Information System (INIS) Chernikov, N.A. 1976-01-01 The Einstein unified field theory is used to solve a tensor equation to provide the unambiguous definition of affine connectedness. In the process of solving the Einstein equation limitations imposed by symmetry on the tensor and the torsion covector as well as on affine connectedness are elucidated. It is demonstrated that in a symmetric case the connectedness is unambiguously determined by the Einstein equation. By means of the Riemann geometry a formula for the torsion covector is derived. The equivalence of Einstein equations to those of the nonlinear Born-Infeld electrodynamics is proved 18. The 1/ N Expansion of Tensor Models Beyond Perturbation Theory Science.gov (United States) Gurau, Razvan 2014-09-01 We analyze in full mathematical rigor the most general quartically perturbed invariant probability measure for a random tensor. Using a version of the Loop Vertex Expansion (which we call the mixed expansion) we show that the cumulants write as explicit series in 1/ N plus bounded rest terms. The mixed expansion recasts the problem of determining the subleading corrections in 1/ N into a simple combinatorial problem of counting trees decorated by a finite number of loop edges. As an aside, we use the mixed expansion to show that the (divergent) perturbative expansion of the tensor models is Borel summable and to prove that the cumulants respect an uniform scaling bound. In particular the quartically perturbed measures fall, in the N→ ∞ limit, in the universality class of Gaussian tensor models. 19. Robust estimation of adaptive tensors of curvature by tensor voting. Science.gov (United States) Tong, Wai-Shun; Tang, Chi-Keung 2005-03-01 Although curvature estimation from a given mesh or regularly sampled point set is a well-studied problem, it is still challenging when the input consists of a cloud of unstructured points corrupted by misalignment error and outlier noise. Such input is ubiquitous in computer vision. In this paper, we propose a three-pass tensor voting algorithm to robustly estimate curvature tensors, from which accurate principal curvatures and directions can be calculated. Our quantitative estimation is an improvement over the previous two-pass algorithm, where only qualitative curvature estimation (sign of Gaussian curvature) is performed. To overcome misalignment errors, our improved method automatically corrects input point locations at subvoxel precision, which also rejects outliers that are uncorrectable. To adapt to different scales locally, we define the RadiusHit of a curvature tensor to quantify estimation accuracy and applicability. Our curvature estimation algorithm has been proven with detailed quantitative experiments, performing better in a variety of standard error metrics (percentage error in curvature magnitudes, absolute angle difference in curvature direction) in the presence of a large amount of misalignment noise. 20. Antisymmetric tensor generalizations of affine vector fields. Science.gov (United States) Houri, Tsuyoshi; Morisawa, Yoshiyuki; Tomoda, Kentaro 2016-02-01 Tensor generalizations of affine vector fields called symmetric and antisymmetric affine tensor fields are discussed as symmetry of spacetimes. We review the properties of the symmetric ones, which have been studied in earlier works, and investigate the properties of the antisymmetric ones, which are the main theme in this paper. It is shown that antisymmetric affine tensor fields are closely related to one-lower-rank antisymmetric tensor fields which are parallelly transported along geodesics. It is also shown that the number of linear independent rank- p antisymmetric affine tensor fields in n -dimensions is bounded by ( n + 1)!/ p !( n - p )!. We also derive the integrability conditions for antisymmetric affine tensor fields. Using the integrability conditions, we discuss the existence of antisymmetric affine tensor fields on various spacetimes. 1. Scattering tensors and optical transitions in Si and Ge CSIR Research Space (South Africa) Kunert, HW 2012-08-01 Full Text Available and L high symmetry points and the highest maximum of the valence band (VB) in the Brillouin zone of Oh7 space group symmetry are determined. The elements of El-Ph scattering tensors are linear combinations of the Clebsch-Gordon coefficients (CGC... 2. ULTRASONIC NEBULIZATION AND ARSENIC VALENCE STATE CONSIDERATIONS PRIOR TO DETERMINATION VIA INDUCTIVELY COUPLED PLASMA MASS SPECTROMETRY Science.gov (United States) An ultrasonic nebulizer (USN) was utilized as a sample introduction device for an inductively coupled plasma mass spectrometer in an attempt to increase the sensitivity for As. The USN produced a valence state response difference for As. The As response was suppressed approximate... 3. Determination of Arsenic in Sinus Wash and Tap Water by Inductively Coupled Plasma-Mass Spectrometry Science.gov (United States) Donnell, Anna M.; Nahan, Keaton; Holloway, Dawone; Vonderheide, Anne P. 2016-01-01 Arsenic is a toxic element to which humans are primarily exposed through food and water; it occurs as a result of human activities and naturally from the earth's crust. An experiment was developed for a senior level analytical laboratory utilizing an Inductively Coupled Plasma-Mass Spectrometer (ICP-MS) for the analysis of arsenic in household… 4. Residual dipolar couplings : a new technique for structure determination of proteins in solution NARCIS (Netherlands) van Lune, Frouktje Sapke 2004-01-01 The aim of the work described in this thesis was to investigate how residual dipolar couplings can be used to resolve or refine the three-dimensional structure of one of the proteins of the phosphoenol-pyruvate phosphotransferase system (PTS), the main transport system for carbohydrates in 5. Determination of tetrabromobisphenol-A/S and their main derivatives in water samples by high performance liquid chromatography coupled with inductively coupled plasma tandem mass spectrometry. Science.gov (United States) Liu, Lihong; Liu, Aifeng; Zhang, Qinghua; Shi, Jianbo; He, Bin; Yun, Zhaojun; Jiang, Guibin 2017-05-12 As the most widely used brominated flame retardants (BFRs), Tetrabromobisphenol-A (TBBPA) as well as its alternative Tetrabromobisphenol-S (TBBPS) and their derivatives have raised wide concerns due to their adverse effects on human health and hence the sensitive detection of those BFRs was urgently needed. Herein, a novel analytical method based on high-performance liquid chromatography (HPLC) coupled with inductively coupled plasma tandem mass spectrometry (ICP-MS/MS) has been developed for the determination of TBBPA/S and their derivatives, including TBBPA-bis(2-hydroxyethyl ether) (TBBPA-BHEE), TBBPA-bis(allylether) (TBBPA-BAE), TBBPA-bis(glycidyl ether) (TBBPA-BGE), TBBPA-bis(2,3-dibromopropyl ether) (TBBPA-BDBPE) and TBBPS-bis(2,3-dibromopropyl ether) (TBBPS-BDBPE) in water samples. After optimization, the TBBPA/S and their derivatives, especially the TBBPA-BAE and TBBPA-BDBPE were simultaneously and sensitively quantified by determination of bromine (m/z=79) by using the ICP-MS. The instrument limits of detection (LODs) for the TBBPA, TBBPA-BHEE, TBBPA-BGE, TBBPA-BAE, TBBPA-BDBPE, TBBPS and TBBPS-BDBPE were determined to be 0.12, 0.14, 0.19, 0.14, 0.12, 0.17 and 0.13μgL -1 , respectively, which was close to or much better than the reported methods. The relative standard deviations (RSDs, n=5) of peak area and retention time were better than 2.2% and 0.2% for intra-day analysis, indicating good repeatability and high precision. The proposed method had been successfully applied for the analysis of TBBPA/S and their derivatives in water samples with satisfactory recoveries (67.7%-113%). Copyright © 2017 Elsevier B.V. All rights reserved. 6. Extended vector-tensor theories Energy Technology Data Exchange (ETDEWEB) Kimura, Rampei; Naruko, Atsushi; Yoshida, Daisuke, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551 (Japan) 2017-01-01 Recently, several extensions of massive vector theory in curved space-time have been proposed in many literatures. In this paper, we consider the most general vector-tensor theories that contain up to two derivatives with respect to metric and vector field. By imposing a degeneracy condition of the Lagrangian in the context of ADM decomposition of space-time to eliminate an unwanted mode, we construct a new class of massive vector theories where five degrees of freedom can propagate, corresponding to three for massive vector modes and two for massless tensor modes. We find that the generalized Proca and the beyond generalized Proca theories up to the quartic Lagrangian, which should be included in this formulation, are degenerate theories even in curved space-time. Finally, introducing new metric and vector field transformations, we investigate the properties of thus obtained theories under such transformations. 7. Correlating the P-31 NMR Chemical Shielding Tensor and the (2)J(P,C) Spin-Spin Coupling Constants with Torsion Angles zeta and alpha in the Backbone of Nucleic Acids Czech Academy of Sciences Publication Activity Database Benda, Ladislav; Sochorová Vokáčová, Zuzana; Straka, Michal; Sychrovský, Vladimír 2012-01-01 Roč. 116, č. 12 (2012), s. 3823-3833 ISSN 1520-6106 R&D Projects: GA ČR GAP205/10/0228; GA ČR GPP208/10/P398; GA ČR GA203/09/2037 Institutional research plan: CEZ:AV0Z40550506 Keywords : nucleic acids * phosphorus NMR * NMR calculations * cross-correlated relaxation * spin–spin coupling constants Subject RIV: CF - Physical ; Theoretical Chemistry Impact factor: 3.607, year: 2012 8. On-line preconcentration and determination of chromium in parenteral solutions by inductively coupled plasma optical emission spectrometry International Nuclear Information System (INIS) Gil, R.A.; Cerutti, S.; Gasquez, J.A.; Olsina, R.A.; Martinez, L.D. 2005-01-01 A method for the preconcentration and speciation of chromium was developed. On-line preconcentration and determination were obtained using inductively coupled plasma optical emission spectrometry (ICP-OES) coupled with flow injection. To determinate the chromium (III) present in parenteral solutions, chromium was retained on activated carbon at pH 5.0. On the other hand, a step of reduction was necessary in order to determine total chromium content. The Cr(VI) concentration was then determined by difference between the total chromium concentration and that of Cr(III). A sensitivity enrichment factor of 70-fold was obtained with respect to the chromium determination by ICP-OES without preconcentration. The detection limit for the preconcentration of 25 ml of sample was 29 ng l -1 . The precision for the 10 replicate determinations at the 5 μg l -1 Cr level was 2.3% relative standard deviation, calculated with the peak heights. The calibration graph using the preconcentration method for chromium species was linear with a correlation coefficient of 0.9995 at levels near the detection limits up to at least 60 μg l -1 . The method can be applied to the determination and speciation of chromium in parenteral solutions 9. Extraction inductively coupled plasma-optical emission spectrometry (ICP-OES). Determination of traces of phosphorus in tungsten International Nuclear Information System (INIS) Bauer, G.; Wegscheider, W.; Mueller, K. 1989-01-01 A method for the separation and preconcentration of traces of phosphorus from tungsten was developed. Solid phase extraction of the phosphovanadomolybdate complex performed on a micro-column was applied. Phosphorus was determined by optical emission spectroscopy (OES) with inductively coupled plasma (ICP) excitation. A limit of detection of 0,4 μg/g P with respect to the solid phase is obtained. By directly coupling the extraction/elution step to the ICP instrument a detection limit of 0,06 μg/g P in W was achieved. Besides, the complexity of spectral evaluation in ICP-OES determinations of traces in spectralline-rich matrices is discussed. (Authors) 10. Ab initio determination of effective electron-phonon coupling factor in copper Science.gov (United States) Ji, Pengfei; Zhang, Yuwen 2016-04-01 The electron temperature Te dependent electron density of states g (ε), Fermi-Dirac distribution f (ε), and electron-phonon spectral function α2 F (Ω) are computed as prerequisites before achieving effective electron-phonon coupling factor Ge-ph. The obtained Ge-ph is implemented into a molecular dynamics (MD) and two-temperature model (TTM) coupled simulation of femtosecond laser heating. By monitoring temperature evolutions of electron and lattice subsystems, the result utilizing Ge-ph from ab initio calculation shows a faster decrease of Te and increase of Tl than those using Ge-ph from phenomenological treatment. The approach of calculating Ge-ph and its implementation into MD-TTM simulation is applicable to other metals. 11. Cα chemical shift tensors in helical peptides by dipolar-modulated chemical shift recoupling NMR International Nuclear Information System (INIS) Yao Xiaolan; Yamaguchi, Satoru; Hong Mei 2002-01-01 The Cα chemical shift tensors of proteins contain information on the backbone conformation. We have determined the magnitude and orientation of the Cα chemical shift tensors of two peptides with α-helical torsion angles: the Ala residue in G*AL (φ=-65.7 deg., ψ=-40 deg.), and the Val residue in GG*V (φ=-81.5 deg., ψ=-50.7 deg.). The magnitude of the tensors was determined from quasi-static powder patterns recoupled under magic-angle spinning, while the orientation of the tensors was extracted from Cα-Hα and Cα-N dipolar modulated powder patterns. The helical Ala Cα chemical shift tensor has a span of 36 ppm and an asymmetry parameter of 0.89. Its σ 11 axis is 116 deg. ± 5 deg. from the Cα-Hα bond while the σ 22 axis is 40 deg. ± 5 deg. from the Cα-N bond. The Val tensor has an anisotropic span of 25 ppm and an asymmetry parameter of 0.33, both much smaller than the values for β-sheet Val found recently (Yao and Hong, 2002). The Val σ 33 axis is tilted by 115 deg. ± 5 deg. from the Cα-Hα bond and 98 deg. ± 5 deg. from the Cα-N bond. These represent the first completely experimentally determined Cα chemical shift tensors of helical peptides. Using an icosahedral representation, we compared the experimental chemical shift tensors with quantum chemical calculations and found overall good agreement. These solid-state chemical shift tensors confirm the observation from cross-correlated relaxation experiments that the projection of the Cα chemical shift tensor onto the Cα-Hα bond is much smaller in α-helices than in β-sheets 12. Sparse alignment for robust tensor learning. Science.gov (United States) Lai, Zhihui; Wong, Wai Keung; Xu, Yong; Zhao, Cairong; Sun, Mingming 2014-10-01 Multilinear/tensor extensions of manifold learning based algorithms have been widely used in computer vision and pattern recognition. This paper first provides a systematic analysis of the multilinear extensions for the most popular methods by using alignment techniques, thereby obtaining a general tensor alignment framework. From this framework, it is easy to show that the manifold learning based tensor learning methods are intrinsically different from the alignment techniques. Based on the alignment framework, a robust tensor learning method called sparse tensor alignment (STA) is then proposed for unsupervised tensor feature extraction. Different from the existing tensor learning methods, L1- and L2-norms are introduced to enhance the robustness in the alignment step of the STA. The advantage of the proposed technique is that the difficulty in selecting the size of the local neighborhood can be avoided in the manifold learning based tensor feature extraction algorithms. Although STA is an unsupervised learning method, the sparsity encodes the discriminative information in the alignment step and provides the robustness of STA. Extensive experiments on the well-known image databases as well as action and hand gesture databases by encoding object images as tensors demonstrate that the proposed STA algorithm gives the most competitive performance when compared with the tensor-based unsupervised learning methods. 13. Shape anisotropy: tensor distance to anisotropy measure Science.gov (United States) Weldeselassie, Yonas T.; El-Hilo, Saba; Atkins, M. S. 2011-03-01 Fractional anisotropy, defined as the distance of a diffusion tensor from its closest isotropic tensor, has been extensively studied as quantitative anisotropy measure for diffusion tensor magnetic resonance images (DT-MRI). It has been used to reveal the white matter profile of brain images, as guiding feature for seeding and stopping in fiber tractography and for the diagnosis and assessment of degenerative brain diseases. Despite its extensive use in DT-MRI community, however, not much attention has been given to the mathematical correctness of its derivation from diffusion tensors which is achieved using Euclidean dot product in 9D space. But, recent progress in DT-MRI has shown that the space of diffusion tensors does not form a Euclidean vector space and thus Euclidean dot product is not appropriate for tensors. In this paper, we propose a novel and robust rotationally invariant diffusion anisotropy measure derived using the recently proposed Log-Euclidean and J-divergence tensor distance measures. An interesting finding of our work is that given a diffusion tensor, its closest isotropic tensor is different for different tensor distance metrics used. We demonstrate qualitatively that our new anisotropy measure reveals superior white matter profile of DT-MR brain images and analytically show that it has a higher signal to noise ratio than fractional anisotropy. 14. Transposes, L-Eigenvalues and Invariants of Third Order Tensors OpenAIRE Qi, Liqun 2017-01-01 Third order tensors have wide applications in mechanics, physics and engineering. The most famous and useful third order tensor is the piezoelectric tensor, which plays a key role in the piezoelectric effect, first discovered by Curie brothers. On the other hand, the Levi-Civita tensor is famous in tensor calculus. In this paper, we study third order tensors and (third order) hypermatrices systematically, by regarding a third order tensor as a linear operator which transforms a second order t... 15. Determination of trace elements in petroleum products by inductively coupled plasma techniques: A critical review International Nuclear Information System (INIS) Sánchez, Raquel; Todolí, José Luis; Lienemann, Charles-Philippe; Mermet, Jean-Michel 2013-01-01 The fundamentals, applications and latter developments of petroleum products analysis through inductively coupled plasma optical emission spectrometry (ICP-OES) and mass spectrometry (ICP-MS) are revisited in the present bibliographic survey. Sample preparation procedures for the direct analysis of fuels by using liquid sample introduction systems are critically reviewed and compared. The most employed methods are sample dilution, emulsion or micro-emulsion preparation and sample decomposition. The first one is the most widely employed due to its simplicity. Once the sample has been prepared, an organic matrix is usually present. The performance of the sample introduction system (i.e., nebulizer and spray chamber) depends strongly upon the nature and properties of the solution finally obtained. Many different devices have been assayed and the obtained results are shown. Additionally, samples can be introduced into the plasma by using an electrothermal vaporization (ETV) device or a laser ablation system (LA). The recent results published in the literature showing the feasibility, advantages and drawbacks of latter alternatives are also described. Therefore, the main goal of the review is the discussion of the different approaches developed for the analysis of crude oil and its derivates by inductively coupled plasma (ICP) techniques. - Highlights: • Analysis of petroleum products by inductively coupled plasma techniques is revisited. • Fundamental studies are included together with reports dealing with applications. • Conventional and non-conventional sample introduction methods are considered. • Sample preparation methods are critically compared and described 16. Gradients estimation from random points with volumetric tensor in turbulence Science.gov (United States) Watanabe, Tomoaki; Nagata, Koji 2017-12-01 We present an estimation method of fully-resolved/coarse-grained gradients from randomly distributed points in turbulence. The method is based on a linear approximation of spatial gradients expressed with the volumetric tensor, which is a 3 × 3 matrix determined by a geometric distribution of the points. The coarse grained gradient can be considered as a low pass filtered gradient, whose cutoff is estimated with the eigenvalues of the volumetric tensor. The present method, the volumetric tensor approximation, is tested for velocity and passive scalar gradients in incompressible planar jet and mixing layer. Comparison with a finite difference approximation on a Cartesian grid shows that the volumetric tensor approximation computes the coarse grained gradients fairly well at a moderate computational cost under various conditions of spatial distributions of points. We also show that imposing the solenoidal condition improves the accuracy of the present method for solenoidal vectors, such as a velocity vector in incompressible flows, especially when the number of the points is not large. The volumetric tensor approximation with 4 points poorly estimates the gradient because of anisotropic distribution of the points. Increasing the number of points from 4 significantly improves the accuracy. Although the coarse grained gradient changes with the cutoff length, the volumetric tensor approximation yields the coarse grained gradient whose magnitude is close to the one obtained by the finite difference. We also show that the velocity gradient estimated with the present method well captures the turbulence characteristics such as local flow topology, amplification of enstrophy and strain, and energy transfer across scales. 17. Validation of diffusion tensor MRI measurements of cardiac microstructure with structure tensor synchrotron radiation imaging. Science.gov (United States) Teh, Irvin; McClymont, Darryl; Zdora, Marie-Christine; Whittington, Hannah J; Davidoiu, Valentina; Lee, Jack; Lygate, Craig A; Rau, Christoph; Zanette, Irene; Schneider, Jürgen E 2017-03-10 Diffusion tensor imaging (DTI) is widely used to assess tissue microstructure non-invasively. Cardiac DTI enables inference of cell and sheetlet orientations, which are altered under pathological conditions. However, DTI is affected by many factors, therefore robust validation is critical. Existing histological validation is intrinsically flawed, since it requires further tissue processing leading to sample distortion, is routinely limited in field-of-view and requires reconstruction of three-dimensional volumes from two-dimensional images. In contrast, synchrotron radiation imaging (SRI) data enables imaging of the heart in 3D without further preparation following DTI. The objective of the study was to validate DTI measurements based on structure tensor analysis of SRI data. One isolated, fixed rat heart was imaged ex vivo with DTI and X-ray phase contrast SRI, and reconstructed at 100 μm and 3.6 μm isotropic resolution respectively. Structure tensors were determined from the SRI data and registered to the DTI data. Excellent agreement in helix angles (HA) and transverse angles (TA) was observed between the DTI and structure tensor synchrotron radiation imaging (STSRI) data, where HA DTI-STSRI = -1.4° ± 23.2° and TA DTI-STSRI = -1.4° ± 35.0° (mean ± 1.96 standard deviation across all voxels in the left ventricle). STSRI confirmed that the primary eigenvector of the diffusion tensor corresponds with the cardiomyocyte long-axis across the whole myocardium. We have used STSRI as a novel and high-resolution gold standard for the validation of DTI, allowing like-with-like comparison of three-dimensional tissue structures in the same intact heart free of distortion. This represents a critical step forward in independently verifying the structural basis and informing the interpretation of cardiac DTI data, thereby supporting the further development and adoption of DTI in structure-based electro-mechanical modelling and routine clinical 18. Polarized Neutron Diffraction as a Tool for Mapping Molecular Magnetic Anisotropy: Local Susceptibility Tensors in Co(II) Complexes. Science.gov (United States) Ridier, Karl; Gillon, Béatrice; Gukasov, Arsen; Chaboussant, Grégory; Cousson, Alain; Luneau, Dominique; Borta, Ana; Jacquot, Jean-François; Checa, Ruben; Chiba, Yukako; Sakiyama, Hiroshi; Mikuriya, Masahiro 2016-01-11 Polarized neutron diffraction (PND) experiments were carried out at low temperature to characterize with high precision the local magnetic anisotropy in two paramagnetic high-spin cobalt(II) complexes, namely [Co(II) (dmf)6 ](BPh4 )2 (1) and [Co(II) 2 (sym-hmp)2 ](BPh4 )2 (2), in which dmf=N,N-dimethylformamide; sym-hmp=2,6-bis[(2-hydroxyethyl)methylaminomethyl]-4-methylphenolate, and BPh4 (-) =tetraphenylborate. This allowed a unique and direct determination of the local magnetic susceptibility tensor on each individual Co(II) site. In compound 1, this approach reveals the correlation between the single-ion easy magnetization direction and a trigonal elongation axis of the Co(II) coordination octahedron. In exchange-coupled dimer 2, the determination of the individual Co(II) magnetic susceptibility tensors provides a clear outlook of how the local magnetic properties on both Co(II) sites deviate from the single-ion behavior because of antiferromagnetic exchange coupling. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. 19. Determination of low cadmium concentrations in wine by on-line preconcentration in a knotted reactor coupled to an inductively coupled plasma optical emission spectrometer with ultrasonic nebulization Energy Technology Data Exchange (ETDEWEB) Lara, R.F. [Inst. de Investigaciones Mineras, Universidad Nacional de San Juan (Argentina); Wuilloud, R.G.; Salonia, J.A. [Dept. of Analytical Chemistry, National University of San Luis (Argentina); Olsina, R.A.; Martinez, L.D. [Dept. of Analytical Chemistry, National University of San Luis (Argentina); Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET) (Argentina) 2001-12-01 An on-line cadmium preconcentration and determination system implemented with inductively coupled plasma optical emission spectrometry (ICP-OES) associated to flow injection (FI) with ultrasonic nebulization system (USN) was studied. The cadmium was retained as the cadmium-2-(5-bromo-2-pyridylazo)-5-diethylaminophenol, Cd-(5-Br-PADAP), complex, at pH 9.5. The cadmium complex was removed from the knotted reactor (KR) with 3.0 mol/L nitric acid. A total enhancement factor of 216 was obtained with respect to ICP-OES using pneumatic nebulization (12 for USN and 18 for KR) with a preconcentration time of 60 s. The value of the detection limit for the preconcentration of 5 mL of sample solution was 5 ng/L. The precision for 10 replicate determinations at the 5 {mu}g/L Cd level was 2.9% relative standard deviation (RSD), calculated from the peak heights obtained. The calibration graph using the preconcentration system for cadmium was linear with a correlation coefficient of 0.9998 at levels near the detection limits up to at least 1000 {mu}g/L. The method was successfully applied to the determination of cadmium in wine samples. (orig.) 20. Unified cosmology with scalar-tensor theory of gravity Energy Technology Data Exchange (ETDEWEB) Tajahmad, Behzad [Faculty of Physics, University of Tabriz, Tabriz (Iran, Islamic Republic of); Sanyal, Abhik Kumar [Jangipur College, Department of Physics, Murshidabad (India) 2017-04-15 Unlike the Noether symmetry, a metric independent general conserved current exists for non-minimally coupled scalar-tensor theory of gravity if the trace of the energy-momentum tensor vanishes. Thus, in the context of cosmology, a symmetry exists both in the early vacuum and radiation dominated era. For slow roll, symmetry is sacrificed, but at the end of early inflation, such a symmetry leads to a Friedmann-like radiation era. Late-time cosmic acceleration in the matter dominated era is realized in the absence of symmetry, in view of the same decayed and redshifted scalar field. Thus, unification of early inflation with late-time cosmic acceleration with a single scalar field may be realized. (orig.) 1. Unified cosmology with scalar-tensor theory of gravity International Nuclear Information System (INIS) Tajahmad, Behzad; Sanyal, Abhik Kumar 2017-01-01 Unlike the Noether symmetry, a metric independent general conserved current exists for non-minimally coupled scalar-tensor theory of gravity if the trace of the energy-momentum tensor vanishes. Thus, in the context of cosmology, a symmetry exists both in the early vacuum and radiation dominated era. For slow roll, symmetry is sacrificed, but at the end of early inflation, such a symmetry leads to a Friedmann-like radiation era. Late-time cosmic acceleration in the matter dominated era is realized in the absence of symmetry, in view of the same decayed and redshifted scalar field. Thus, unification of early inflation with late-time cosmic acceleration with a single scalar field may be realized. (orig.) 2. Tensor SOM and tensor GTM: Nonlinear tensor analysis by topographic mappings. Science.gov (United States) Iwasaki, Tohru; Furukawa, Tetsuo 2016-05-01 In this paper, we propose nonlinear tensor analysis methods: the tensor self-organizing map (TSOM) and the tensor generative topographic mapping (TGTM). TSOM is a straightforward extension of the self-organizing map from high-dimensional data to tensorial data, and TGTM is an extension of the generative topographic map, which provides a theoretical background for TSOM using a probabilistic generative model. These methods are useful tools for analyzing and visualizing tensorial data, especially multimodal relational data. For given n-mode relational data, TSOM and TGTM can simultaneously organize a set of n-topographic maps. Furthermore, they can be used to explore the tensorial data space by interactively visualizing the relationships between modes. We present the TSOM algorithm and a theoretical description from the viewpoint of TGTM. Various TSOM variations and visualization techniques are also described, along with some applications to real relational datasets. Additionally, we attempt to build a comprehensive description of the TSOM family by adapting various data structures. Copyright © 2016 Elsevier Ltd. All rights reserved. 3. A brief summary on formalizing parallel tensor distributions redistributions and algorithm derivations. Energy Technology Data Exchange (ETDEWEB) Schatz, Martin D. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Kolda, Tamara G. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); van de Geijn, Robert [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States) 2015-09-01 Large-scale datasets in computational chemistry typically require distributed-memory parallel methods to perform a special operation known as tensor contraction. Tensors are multidimensional arrays, and a tensor contraction is akin to matrix multiplication with special types of permutations. Creating an efficient algorithm and optimized im- plementation in this domain is complex, tedious, and error-prone. To address this, we develop a notation to express data distributions so that we can apply use automated methods to find optimized implementations for tensor contractions. We consider the spin-adapted coupled cluster singles and doubles method from computational chemistry and use our methodology to produce an efficient implementation. Experiments per- formed on the IBM Blue Gene/Q and Cray XC30 demonstrate impact both improved performance and reduced memory consumption. 4. Dielectric tensor elements for the description of waves in rotating inhomogeneous magnetized plasma spheroids Science.gov (United States) Abdoli-Arani, A.; Ramezani-Arani, R. 2012-11-01 The dielectric permittivity tensor elements of a rotating cold collisionless plasma spheroid in an external magnetic field with toroidal and axial components are obtained. The effects of inhomogeneity in the densities of charged particles and the initial toroidal velocity on the dielectric permittivity tensor and field equations are investigated. The field components in terms of their toroidal components are calculated and it is shown that the toroidal components of the electric and magnetic fields are coupled by two differential equations. The influence of thermal and collisional effects on the dielectric tensor and field equations in the rotating plasma spheroid are also investigated. In the limiting spherical case, the dielectric tensor of a stationary magnetized collisionless cold plasma sphere is presented. 5. Parameterized Post-Newtonian Expansion of Scalar-Vector-Tensor Theory of Gravity International Nuclear Information System (INIS) Arianto; Zen, Freddy P.; Gunara, Bobby E.; Hartanto, Andreas 2010-01-01 We investigate the weak-field, post-Newtonian expansion to the solution of the field equations in scalar-vector-tensor theory of gravity. In the calculation we restrict ourselves to the first post Newtonian. The parameterized post Newtonian (PPN) parameters are determined by expanding the modified field equations in the metric perturbation. Then, we compare the solution to the PPN formalism in first PN approximation proposed by Will and Nordtvedt and read of the coefficients (the PPN parameters) of post Newtonian potentials of the theory. We find that the values of γ PPN and β PPN are the same as in General Relativity but the coupling functions β 1 , β 2 , and β 3 are the effect of the preferred frame. 6. New results for algebraic tensor reduction of Feynman integrals Energy Technology Data Exchange (ETDEWEB) Fleischer, Jochem [Bielefeld Univ. (Germany). Fakultaet fuer Physik; Riemann, Tord [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Yundin, Valery [Copenhagen Univ. (Denmark). Niels Bohr International Academy and Discovery Center 2012-02-15 We report on some recent developments in algebraic tensor reduction of one-loop Feynman integrals. For 5-point functions, an efficient tensor reduction was worked out recently and is now available as numerical C++ package, PJFry, covering tensor ranks until five. It is free of inverse 5- point Gram determinants and inverse small 4-point Gram determinants are treated by expansions in higher-dimensional 3-point functions. By exploiting sums over signed minors, weighted with scalar products of chords (or, equivalently, external momenta), extremely efficient expressions for tensor integrals contracted with external momenta were derived. The evaluation of 7-point functions is discussed. In the present approach one needs for the reductions a (d +2)-dimensional scalar 5-point function in addition to the usual scalar basis of 1- to 4-point functions in the generic dimension d=4-2{epsilon}. When exploiting the four-dimensionality of the kinematics, this basis is sufficient. We indicate how the (d+2)-dimensional 5-point function can be evaluated. (orig.) 7. New results for algebraic tensor reduction of Feynman integrals International Nuclear Information System (INIS) Fleischer, Jochem; Yundin, Valery 2012-02-01 We report on some recent developments in algebraic tensor reduction of one-loop Feynman integrals. For 5-point functions, an efficient tensor reduction was worked out recently and is now available as numerical C++ package, PJFry, covering tensor ranks until five. It is free of inverse 5- point Gram determinants and inverse small 4-point Gram determinants are treated by expansions in higher-dimensional 3-point functions. By exploiting sums over signed minors, weighted with scalar products of chords (or, equivalently, external momenta), extremely efficient expressions for tensor integrals contracted with external momenta were derived. The evaluation of 7-point functions is discussed. In the present approach one needs for the reductions a (d +2)-dimensional scalar 5-point function in addition to the usual scalar basis of 1- to 4-point functions in the generic dimension d=4-2ε. When exploiting the four-dimensionality of the kinematics, this basis is sufficient. We indicate how the (d+2)-dimensional 5-point function can be evaluated. (orig.) 8. Seamless warping of diffusion tensor fields DEFF Research Database (Denmark) Xu, Dongrong; Hao, Xuejun; Bansal, Ravi 2008-01-01 To warp diffusion tensor fields accurately, tensors must be reoriented in the space to which the tensors are warped based on both the local deformation field and the orientation of the underlying fibers in the original image. Existing algorithms for warping tensors typically use forward mapping...... of seams, including voxels in which the deformation is extensive. Backward mapping, however, cannot reorient tensors in the template space because information about the directional orientation of fiber tracts is contained in the original, unwarped imaging space only, and backward mapping alone cannot...... transfer that information to the template space. To combine the advantages of forward and backward mapping, we propose a novel method for the spatial normalization of diffusion tensor (DT) fields that uses a bijection (a bidirectional mapping with one-to-one correspondences between image spaces) to warp DT... 9. Trilinear self couplings of vector bosons and their determination in e+e-→W+W- International Nuclear Information System (INIS) Gounaris, G.; Kneur, J.L.; Schildknecht, D.; Layssac, J.; Moultaka, G.; Renard, F.M. 1992-01-01 The constraints on the γW + W - and Z 0 W + W - couplings are summarized. They essentially follow from various symmetry requirements which are weaker than the ones embodied in the SU(2) L *U(1) Y theory. The theoretical considerations lead to a well-defined systematic procedure for the analysis of future data. The helicity amplitudes for e + e - →W + W - are discussed and, in particular, the high-energy limit of s>>4M W 2 , relevant at an energy of 500 GeV, is analyzed. (K.A.) 18 refs., 4 figs., 4 tabs 10. SAID analysis of meson photoproduction: Determination of neutron and proton EM couplings Directory of Open Access Journals (Sweden) Strakovsky Igor 2014-06-01 Full Text Available We present an overview of the GW SAID group effort to analyze on new pion photoproduction on both proton- and neutron-targets. The main database contribution came from the recent CLAS and MAMI unpolarized and polarized measurements. The differential cross section for the processes γn → π−p was extracted from new measurements accounting for Fermi motion effects in the impulse approximation (IA as well as NN- and πN effects beyond the IA. The electromagnetic coupling results are compared to other recent studies. 11. The presentation of the nonabelian tensor square of a Bieberbach group of dimension five with dihedral point group Science.gov (United States) Fauzi, Wan Nor Farhana Wan Mohd; Idrus, Nor'ashiqin Mohd; Masri, Rohaidah; Ting, Tan Yee; Sarmin, Nor Haniza; Hassim, Hazzirah Izzati Mat 2014-12-01 One of the homological functors of a group, is the nonabelian tensor square. It is important in the determination of the other homological functors of a group. In order to compute the nonabelian tensor square, we need to get its independent generators and its presentation. In this paper, we present the calculation of getting the presentation of the nonabelian tensor square of the group. The presentation is computed based on its independent generators by using the polycyclic method. 12. On improving the efficiency of tensor voting OpenAIRE Moreno, Rodrigo; Garcia, Miguel Angel; Puig, Domenec; Pizarro, Luis; Burgeth, Bernhard; Weickert, Joachim 2011-01-01 This paper proposes two alternative formulations to reduce the high computational complexity of tensor voting, a robust perceptual grouping technique used to extract salient information from noisy data. The first scheme consists of numerical approximations of the votes, which have been derived from an in-depth analysis of the plate and ball voting processes. The second scheme simplifies the formulation while keeping the same perceptual meaning of the original tensor voting: The stick tensor v... 13. Should I use TensorFlow OpenAIRE Schrimpf, Martin 2016-01-01 Google's Machine Learning framework TensorFlow was open-sourced in November 2015 [1] and has since built a growing community around it. TensorFlow is supposed to be flexible for research purposes while also allowing its models to be deployed productively. This work is aimed towards people with experience in Machine Learning considering whether they should use TensorFlow in their environment. Several aspects of the framework important for such a decision are examined, such as the heterogenity,... 14. Efficient Low Rank Tensor Ring Completion OpenAIRE Wang, Wenqi; Aggarwal, Vaneet; Aeron, Shuchin 2017-01-01 Using the matrix product state (MPS) representation of the recently proposed tensor ring decompositions, in this paper we propose a tensor completion algorithm, which is an alternating minimization algorithm that alternates over the factors in the MPS representation. This development is motivated in part by the success of matrix completion algorithms that alternate over the (low-rank) factors. In this paper, we propose a spectral initialization for the tensor ring completion algorithm and ana... 15. Composite antisymmetric tensor bosons in a four-fermion interaction model International Nuclear Information System (INIS) Dmitrasinovic, V. 2000-01-01 We discuss the phenomenological consequences of the U A (1) symmetry-breaking two-flavour four-fermion antisymmetric (AS) Lorentz tensor interaction Lagrangians. We use the recently developed methods that respect the 'duality' symmetry of this interaction. Starting from the Fierz transform of the two-flavour 't Hooft interaction (a four-fermion Lagrangian with AS tensor interaction terms augmented by Nambu and Jona-Lasinio (NJL)-type Lorentz scalar interaction responsible for dynamical symmetry breaking and quark mass generation), we find the following. (a) Four antisymmetric tensor and four AS pseudotensor bosons exist which satisfy a mass relation previously derived for scalar and pseudoscalar mesons from the 't Hooft interaction. (b) Antisymmetric tensor bosons mix with vector bosons via one-fermion-loop effective couplings so that both kinds of bosons have their masses shifted and the fermions (quarks) acquire anomalous magnetic moment form factors that explicitly violate chiral symmetry. (c) The mixing of massive AS tensor fields with vector fields leads to two sets of spin-1 states. The second set of spin-1 mesons is heavy and has not been observed. Moreover, at least one member of this second set is tachyonic, under standard assumptions about the source and strength of the AS tensor interaction. The tachyonic state also shows up as a pole in the space-like region of the electromagnetic form factors. (d) The mixing of axial-vector fields with antisymmetric tensor bosons is proportional to the (small) isospin-breaking up-down quark mass difference, so the mixing-induced mass shift is negligible. (e) The AS tensor version of the Veneziano-Witten U A (1) symmetry-breaking interaction does not lead to tachyons, or any AS tensor field propagation to leading order in N C . (author) 16. The 1/ N Expansion of Tensor Models with Two Symmetric Tensors Science.gov (United States) Gurau, Razvan 2018-06-01 It is well known that tensor models for a tensor with no symmetry admit a 1/ N expansion dominated by melonic graphs. This result relies crucially on identifying jackets, which are globally defined ribbon graphs embedded in the tensor graph. In contrast, no result of this kind has so far been established for symmetric tensors because global jackets do not exist. In this paper we introduce a new approach to the 1/ N expansion in tensor models adapted to symmetric tensors. In particular we do not use any global structure like the jackets. We prove that, for any rank D, a tensor model with two symmetric tensors and interactions the complete graph K D+1 admits a 1/ N expansion dominated by melonic graphs. 17. AGREEMENT BETWEEN THE WHITE MATTER CONNECTIVITY BASED ON THE TENSOR-BASED MORPHOMETRY AND THE VOLUMETRIC WHITE MATTER PARCELLATIONS BASED ON DIFFUSION TENSOR IMAGING OpenAIRE Kim, Seung-Goo; Lee, Hyekyoung; Chung, Moo K.; Hanson, Jamie L.; Avants, Brian B.; Gee, James C.; Davidson, Richard J.; Pollak, Seth D. 2012-01-01 We are interested in investigating white matter connectivity using a novel computational framework that does not use diffusion tensor imaging (DTI) but only uses T1-weighted magnetic resonance imaging. The proposed method relies on correlating Jacobian determinants across different voxels based on the tensor-based morphometry (TBM) framework. In this paper, we show agreement between the TBM-based white matter connectivity and the DTI-based white matter atlas. As an application, altered white ... 18. Determination of coupled-lattice properties using turn-by-turn data International Nuclear Information System (INIS) Bourianoff, G.; Hunt, S.; Mathieson, D.; Pilat, F.; Talman, R.; Morpurgo, G. 1992-01-01 A formalism for extracting coupled betatron parameters from multiturn, shock excited, beam position monitor data is described. The most important results are nonperturbative in that they do not rely on the underlying ideal lattice model. Except for damping, which is assumed to be exponential and small enough to be removed empirically, the description is symplectic. As well as simplifying the description, this leads to self-consistency checks that are applied to the data. The most important of these is a ''magic ratio'' of Fourier coefficients that is required to be a lattice invariant, the same at every beam position monitor. All formulas are applied to both real and simulated data. The real data were acquired June 1992 at LEP as part of decoupling studies, using the LEP beam orbit measurement system. Simulated data, obtained by numerical tracking (TEAPOT) in the same (except for unknown errors) lattice, agrees well with real data when subjected to identical analysis. For both datasets, deviations between extracted and design parameters and deviations from self-consistency can be accounted for by noise and signal-processing limitations. This investigation demonstrates that the LEP beam position system yields reliable local coupling measurements. It can be conservatively assumed that systems of similar design at the SSC and LHC will provide the measurements needed for local decoupling 19. Loss of incoherence and determination of coupling constants in quantum gravity International Nuclear Information System (INIS) Giddings, S.B.; Strominger, A. 1988-01-01 The wave function of an interacting 'family' of one large 'parent' and many Planck-sized 'baby' universes is computed in a semiclassical approximation using an adaptation of Hartle-Hawking initial conditions. A recently discovered gravitational instanton which exists for general relativity coupled to axions is employed. The outcome of a single experiment in the parent universe is in general described by a mixed state, even if the initial state is pure. However, a sequence of measurements rapidly collapses the wave function of the family of universes into one of an infinite number of 'coherent' states for which quantum incoherence is not observed in the parent universe. This provides a concrete illustration of an unexpected phenomena whose existence has been argued for on quite general grounds by Coleman: Quantum incoherence due to information loss to baby universes is not experimentally observable. We further argue that all coupling constants governing dynamics in the parent universe depend on the parameters describing the particular coherent state into which the family wave function collapses. In particular, generically terms that violate any global symmetries will be induced in the effective action for the parent universe. These last results have much broader applicability than our specific model. (orig.) 20. Determination of coupled-lattice properties using turn-by-turn data International Nuclear Information System (INIS) Bourianoff, G.; Hunt, S.; Mathieson, D. 1992-12-01 A formalism for extracting coupled betatron parameters from multiturn, shock excited, beam position monitor data is described. The most important results are nonperturbative in that they do not rely on the underlying ideal lattice model. Except for damping, which is assumed to be exponential and small enough to be removed empirically, the description is symplectic. As well as simplifying the description, this leads to self-consistency checks that are applied to the data. The most important of these is a open-quotes magic ratioclose quotes of Fourier coefficients that is required to be a lattice invariant, the same at every beam position monitor. All formulas are applied to both real and simulated data. The real data was acquired June, 1992 at LEP as part of decoupling studies, using the LEP beam orbit measurement system. Simulated data, obtained by numerical tracking (TEAPOT) in the same (except for unknown errors) lattice, agrees well with real data when subjected to identical analysis. For both datasets, deviations between extracted and design parameters and deviations from self-consistency can be accounted for by noise and signal processing limitations. This investigation demonstrates that the LEP beam position system yields reliable local coupling measurements. It can be conservatively assumed that systems of similar design at the SSC and LHC will provide the measurements needed for local decoupling 1. Study of the 23Na EFG (Electrostatic Field Gradient) tensor on single crystals of Na2S.9H2O by wideline NMR International Nuclear Information System (INIS) Miksche, G. 1982-01-01 The quadrupole coupling constant |e 2 qQ/n| if 23 Na has been determined by measuring single crystals of Na 2 S.9H 2 O at room temperature. A value of 687.5 +- 1.2 kHz was found. The asymmetry parameter eta = (qsub(x'x') - qsub(y'y')) / qsub(z'z') of the efg-tensor is zero, there is axial symmetry. The principle axis of the efg-tensor runs parallel to the main crystallographic axis c, the value of the main component of the efg-tensor in c-direction is 171.875 +- 0.6 kHz. The longitudinal relaxation time T 1 has been evaluated as 1.8 s. On this account, the mean distance between two Na-atoms has been determined by measuring the splitting of the central line due to dipole-dipole interaction. The Na-Na distance was found with 0.36 +- 0.007 nm. This value is in good agreement with results from neutron diffraction studies. It was not possible to determine direction and length of hydrogen bonds by NMR-results. A method of growing single crystals of Na 2 S.9H 2 O of demanded size and purity has been described. Constructional details and technical data of a self-made wideline-NMR-spectrometer are added in an appendix. (Author) 2. Dictionary-Based Tensor Canonical Polyadic Decomposition Science.gov (United States) Cohen, Jeremy Emile; Gillis, Nicolas 2018-04-01 To ensure interpretability of extracted sources in tensor decomposition, we introduce in this paper a dictionary-based tensor canonical polyadic decomposition which enforces one factor to belong exactly to a known dictionary. A new formulation of sparse coding is proposed which enables high dimensional tensors dictionary-based canonical polyadic decomposition. The benefits of using a dictionary in tensor decomposition models are explored both in terms of parameter identifiability and estimation accuracy. Performances of the proposed algorithms are evaluated on the decomposition of simulated data and the unmixing of hyperspectral images. 3. Bayesian regularization of diffusion tensor images DEFF Research Database (Denmark) Frandsen, Jesper; Hobolth, Asger; Østergaard, Leif 2007-01-01 Diffusion tensor imaging (DTI) is a powerful tool in the study of the course of nerve fibre bundles in the human brain. Using DTI, the local fibre orientation in each image voxel can be described by a diffusion tensor which is constructed from local measurements of diffusion coefficients along...... several directions. The measured diffusion coefficients and thereby the diffusion tensors are subject to noise, leading to possibly flawed representations of the three dimensional fibre bundles. In this paper we develop a Bayesian procedure for regularizing the diffusion tensor field, fully utilizing... 4. A RENORMALIZATION PROCEDURE FOR TENSOR MODELS AND SCALAR-TENSOR THEORIES OF GRAVITY OpenAIRE SASAKURA, NAOKI 2010-01-01 Tensor models are more-index generalizations of the so-called matrix models, and provide models of quantum gravity with the idea that spaces and general relativity are emergent phenomena. In this paper, a renormalization procedure for the tensor models whose dynamical variable is a totally symmetric real three-tensor is discussed. It is proven that configurations with certain Gaussian forms are the attractors of the three-tensor under the renormalization procedure. Since these Gaussian config... 5. Full moment tensor retrieval and fluid dynamics in volcanic areas: The case of phlegraean field (south Italy) International Nuclear Information System (INIS) Campus, P.; Cespuglio, G. 1994-04-01 When studying seismicity in volcanic areas it is appropriate to treat the seismic source in a form a priori not restricted to a double couple, since its mechanism may reflect not only small scale tectonics but also fluid dynamics. The monitoring of fluid dynamics can be therefore attempted from the retrieval of the rupture processes. It is not possible to use standard methods, based on the distribution of polarities of first arrivals to determine the non double-couple components of the seismic source. The new method presented here is based on the wave form inversion of the dominant part of the seismograms, where the signal to noise ratio is very large and allows the inversion of the full seismic moment tensor. The results of a pilot study in the Phlegraean Fields (South Italy) are presented. 13 refs, 10 figs, 4 tabs 6. The tensor network theory library Science.gov (United States) Al-Assam, S.; Clark, S. R.; Jaksch, D. 2017-09-01 In this technical paper we introduce the tensor network theory (TNT) library—an open-source software project aimed at providing a platform for rapidly developing robust, easy to use and highly optimised code for TNT calculations. The objectives of this paper are (i) to give an overview of the structure of TNT library, and (ii) to help scientists decide whether to use the TNT library in their research. We show how to employ the TNT routines by giving examples of ground-state and dynamical calculations of one-dimensional bosonic lattice system. We also discuss different options for gaining access to the software available at www.tensornetworktheory.org. 7. Dirac tensor with heavy photon Energy Technology Data Exchange (ETDEWEB) Bytev, V.V.; Kuraev, E.A. [Joint Institute of Nuclear Research, Moscow (Russian Federation). Bogoliubov Lab. of Theoretical Physics; Scherbakova, E.S. [Hamburg Univ. (Germany). 1. Inst. fuer Theoretische Physik 2012-01-15 For the large-angles hard photon emission by initial leptons in process of high energy annihilation of e{sup +}e{sup -} {yields} to hadrons the Dirac tensor is obtained, taking into account the lowest order radiative corrections. The case of large-angles emission of two hard photons by initial leptons is considered. This result is being completed by the kinematics case of collinear hard photons emission as well as soft virtual and real photons and can be used for construction of Monte-Carlo generators. (orig.) 8. Principles and determinants of G-protein coupling by the rhodopsin-like thyrotropin receptor. Directory of Open Access Journals (Sweden) Gunnar Kleinau Full Text Available In this study we wanted to gain insights into selectivity mechanisms between G-protein-coupled receptors (GPCR and different subtypes of G-proteins. The thyrotropin receptor (TSHR binds G-proteins promiscuously and activates both Gs (cAMP and Gq (IP. Our goal was to dissect selectivity patterns for both pathways in the intracellular region of this receptor. We were particularly interested in the participation of poorly investigated receptor parts.We systematically investigated the amino acids of intracellular loop (ICL 1 and helix 8 using site-directed mutagenesis alongside characterization of cAMP and IP accumulation. This approach was guided by a homology model of activated TSHR in complex with heterotrimeric Gq, using the X-ray structure of opsin with a bound G-protein peptide as a structural template.We provide evidence that ICL1 is significantly involved in G-protein activation and our model suggests potential interactions with subunits G alpha as well as G betagamma. Several amino acid substitutions impaired both IP and cAMP accumulation. Moreover, we found a few residues in ICL1 (L440, T441, H443 and helix 8 (R687 that are sensitive for Gq but not for Gs activation. Conversely, not even one residue was found that selectively affects cAMP accumulation only. Together with our previous mutagenesis data on ICL2 and ICL3 we provide here the first systematically completed map of potential interfaces between TSHR and heterotrimeric G-protein. The TSHR/Gq-heterotrimer complex is characterized by more selective interactions than the TSHR/Gs complex. In fact the receptor interface for binding Gs is a subset of that for Gq and we postulate that this may be true for other GPCRs coupling these G-proteins. Our findings support that G-protein coupling and preference is dominated by specific structural features at the intracellular region of the activated GPCR but is completed by additional complementary recognition patterns between receptor and G 9. Principles and determinants of G-protein coupling by the rhodopsin-like thyrotropin receptor. Science.gov (United States) Kleinau, Gunnar; Jaeschke, Holger; Worth, Catherine L; Mueller, Sandra; Gonzalez, Jorge; Paschke, Ralf; Krause, Gerd 2010-03-18 In this study we wanted to gain insights into selectivity mechanisms between G-protein-coupled receptors (GPCR) and different subtypes of G-proteins. The thyrotropin receptor (TSHR) binds G-proteins promiscuously and activates both Gs (cAMP) and Gq (IP). Our goal was to dissect selectivity patterns for both pathways in the intracellular region of this receptor. We were particularly interested in the participation of poorly investigated receptor parts.We systematically investigated the amino acids of intracellular loop (ICL) 1 and helix 8 using site-directed mutagenesis alongside characterization of cAMP and IP accumulation. This approach was guided by a homology model of activated TSHR in complex with heterotrimeric Gq, using the X-ray structure of opsin with a bound G-protein peptide as a structural template.We provide evidence that ICL1 is significantly involved in G-protein activation and our model suggests potential interactions with subunits G alpha as well as G betagamma. Several amino acid substitutions impaired both IP and cAMP accumulation. Moreover, we found a few residues in ICL1 (L440, T441, H443) and helix 8 (R687) that are sensitive for Gq but not for Gs activation. Conversely, not even one residue was found that selectively affects cAMP accumulation only. Together with our previous mutagenesis data on ICL2 and ICL3 we provide here the first systematically completed map of potential interfaces between TSHR and heterotrimeric G-protein. The TSHR/Gq-heterotrimer complex is characterized by more selective interactions than the TSHR/Gs complex. In fact the receptor interface for binding Gs is a subset of that for Gq and we postulate that this may be true for other GPCRs coupling these G-proteins. Our findings support that G-protein coupling and preference is dominated by specific structural features at the intracellular region of the activated GPCR but is completed by additional complementary recognition patterns between receptor and G-protein subtypes. 10. Determination of rare earth elements in uranium bearing samples using Inductively Coupled Plasma Mass Spectrometry (ICPMS) International Nuclear Information System (INIS) Mishra, S.; Chaudhury, P.; Pradeepkumar, K.S.; Sahoo, S.K. 2017-01-01 In the present study a methodology has been described for determination of REEs without involving separation and the method is successfully applied for determination of REE concentration in uranium ore as well as in soil samples from a uranium mining site 11. Determination of the quark coupling strength|V_{ub}|$using baryonic decays CERN Document Server AUTHOR|(INSPIRE)INSPIRE-00392760 This thesis presents the first determination of$|V_{ub}|$at a hadron collider and in a baryonic decay. The determination is made by measuring the ratio of branching fractions of the baryonic decays$\\Lambda^0_b \\to p \\mu^-\\overline{\
12. The determination of the weak neutral current coupling constants and limits on the electromagnetic properties of the muon neutrino
International Nuclear Information System (INIS)
Callas, J.L.
1987-05-01
The goal of this thesis is to determine experimentally the cross section for nu/sub μ/e → nu/sub μ/e scattering from a sample of over 100 expected nu/sub μ/e → nu/sub μ/e events collected by the E734 neutrino detector in BNL wide band neutrino beam. By combining these results with results from an anti-neutrino determination of the cross section for anti nu/sub μ/e → anti nu/sub μ/e scattering in the form of a ratio of cross sections, the weak coupling constants for the electron, g/sub V/ and g/sub A/ can be determined in a model independent way to within a four fold ambiguity where three of the ambiguities can be eliminated by results from e + e - experiments. The predictions of the Standard Model for the weak coupling constants can then be tested and a precise determination of the electroweak mixing parameter, sin 2 θ/sub W/ can be made
13. NK sensitivity of neuroblastoma cells determined by a highly sensitive coupled luminescent method
International Nuclear Information System (INIS)
Ogbomo, Henry; Hahn, Anke; Geiler, Janina; Michaelis, Martin; Doerr, Hans Wilhelm; Cinatl, Jindrich
2006-01-01
The measurement of natural killer (NK) cells toxicity against tumor or virus-infected cells especially in cases with small blood samples requires highly sensitive methods. Here, a coupled luminescent method (CLM) based on glyceraldehyde-3-phosphate dehydrogenase release from injured target cells was used to evaluate the cytotoxicity of interleukin-2 activated NK cells against neuroblastoma cell lines. In contrast to most other methods, CLM does not require the pretreatment of target cells with labeling substances which could be toxic or radioactive. The effective killing of tumor cells was achieved by low effector/target ratios ranging from 0.5:1 to 4:1. CLM provides highly sensitive, safe, and fast procedure for measurement of NK cell activity with small blood samples such as those obtained from pediatric patients
14. Precision determination of the $\\pi N$ scattering lengths and the charged $\\pi NN$ coupling constant
CERN Document Server
Ericson, Torleif Eric Oskar; Thomas, A W
2000-01-01
We critically evaluate the isovector GMO sumrule for the charged $\\pi N N$ coupling constant using recent precision data from $\\pi ^-$p and $\\pi^-$d atoms and with careful attention to systematic errors. From the $\\pi ^-$d scattering length we deduce the pion-proton scattering lengths ${1/2}(a_{\\pi ^-p}+a_{\\pi ^-n})=(-20\\pm 6$(statistic)$\\pm 10$ (systematic))~$\\cdot 10^{-4}m_{\\pi_c}^{-1}$ and ${1/2}(a_{\\pi ^-p}-a_{\\pi ^-n})=(903 \\pm 14)\\cdot 10^{-4}m_{\\pi_c}^{-1}$. From this a direct evaluation gives $g^2_c(GMO) =14.20\\pm 0.07$(statistic)$\\pm 0.13$(systematic) or $f^2_c= 0.0786\\pm 0.0008$.
15. Simultaneous determination of 2 aconitum alkaloids and 12 ginsenosides in Shenfu injection by ultraperformance liquid chromatography coupled with a photodiode array detector with few markers to determine multicomponents
Directory of Open Access Journals (Sweden)
Ai-Hua Ge
2015-06-01
Full Text Available A method with few markers to determine multicomponents was established and validated to evaluate the quality of Shenfu injection by ultraperformance liquid chromatography coupled with a photodiode array detector. The separations were performed on an ACQUITY UPLC BEH C18 (2.1 × 50 mm2, 1.7 μm column. Methanol and 0.1% formic acid aqueous solution were used as the mobile phase. The flow rate was 0.3 mL/min. 2 aconitum alkaloids and 12 ginsenosides could be perfectly separated within 15 minutes. Ginsenoside Rg1 and benzoylmesaconine, the easily available active components, were employed as the maker components to calculate the relative correction factors of other components in Shenfu injection, Panax ginseng and Aconitum carmichaeli. The external standard method was also established to validate the feasibility of the method with few markers to determine multicomponents. Parameter p and the principal component analysis method were employed to investigate the disparities among batches for the effective quality control of Shenfu injection. The results demonstrated that the ultraperformance liquid chromatography coupled with a photodiode array detector method with few markers to determine multicomponents could be used as a powerful tool for the quality evaluation of traditional Chinese medicines and their preparations.
16. Anomalous coupling of scalars to gauge fields
Energy Technology Data Exchange (ETDEWEB)
Brax, Philippe [CEA, IPhT, CNRS, URA 2306, Gif-sur-Yvette (France). Inst. de Physique Theorique; Burrage, Clare [Geneve Univ. (Switzerland). Dept. de Physique Theorique; Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Davis, Anne-Christine [Centre for Mathematical Sciences, Cambridge (United Kingdom). Dept. of Applied Mathematics and Theoretical Physics; Seery, David [Sussex Univ., Brighton (United Kingdom). Dept. of Physics and Astronomy; Weltman, Amanda [Cape Town Univ., Rondebosch (South Africa). Astronomy, Cosmology and Gravity Centre
2010-10-15
We study the transformation properties of a scalar-tensor theory, coupled to fermions, under the Weyl rescaling associated with a transition from the Jordan to the Einstein frame. We give a simple derivation of the corresponding modification to the gauge couplings. After changing frames, this gives rise to a direct coupling between the scalar and the gauge fields. (orig.)
17. Anomalous coupling of scalars to gauge fields
International Nuclear Information System (INIS)
Brax, Philippe; Davis, Anne-Christine; Seery, David; Weltman, Amanda
2010-10-01
We study the transformation properties of a scalar-tensor theory, coupled to fermions, under the Weyl rescaling associated with a transition from the Jordan to the Einstein frame. We give a simple derivation of the corresponding modification to the gauge couplings. After changing frames, this gives rise to a direct coupling between the scalar and the gauge fields. (orig.)
18. Development of an analytical method for the determination of polybrominated diphenyl ethers in sewage sludge by the use of gas chromatography coupled to inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Novak, Petra; Zuliani, Tea; Milačič, Radmila; Ščančar, Janez
2016-01-01
Polybrominated diphenyl ethers (PBDEs) are flame retardants. As a consequence of their widespread use, they have been released into the environment. PBDEs are lipophilic organic contaminants that enter wastewater treatment plants (WWTPs) from urban, agricultural and industrial discharges. Because of their low aqueous solubility and resistance to biodegradation, up to 90% of the PBDEs are accumulated in the sewage sludge during the wastewater treatment. To assess the possibilities for sludge re-use, a reliable determination of the concentrations of these PBDEs is of crucial importance. Six PBDE congeners (BDE 28, BDE 47, BDE 99, BDE 100, BDE 153 and BDE 154) are listed as priority substances under the EU Water Framework Directive. In the present work a simple analytical method with minimal sample-preparation steps was developed for a sensitive and reliable determination of the six PBDEs in sewage sludge by the use of gas chromatography coupled to inductively coupled plasma mass spectrometry (GC-ICP-MS). For this purpose an extraction procedure was optimised. Different extracting agents (methanol (MeOH), acetic acid (AcOH)/MeOH mixture (3:1) and 0.1 mol L"−"1 hydrochloric acid (HCl) in MeOH) followed by the addition of a Tris-citrate buffer (co-extracting agent) and iso-octane were applied under different modes of extraction (mechanical shaking, microwave- and ultrasound-assisted extraction). Mechanical shaking or the microwave-assisted extraction of sewage sludge with 0.1 mol L"−"1 HCl in MeOH and the subsequent addition of the Tris-citrate buffer and the iso-octane extracted the PBDEs from the complex sludge matrix most effectively. However, due to easier sample manipulation during the extraction step, mechanical shaking was used. The PBDEs in the organic phase were quantified with GC-ICP-MS by applying a standard addition calibration method. The spike recovery test (recoveries between 95 and 104%) and comparative analyses with the species-specific isotope
19. Development of an analytical method for the determination of polybrominated diphenyl ethers in sewage sludge by the use of gas chromatography coupled to inductively coupled plasma mass spectrometry
Energy Technology Data Exchange (ETDEWEB)
Novak, Petra [Department of Environmental Sciences, Jožef Stefan Institute, Jamova 39, 1000, Ljubljana (Slovenia); Jožef Stefan International Postgraduate School, Jamova 39, 1000, Ljubljana (Slovenia); Zuliani, Tea [Department of Environmental Sciences, Jožef Stefan Institute, Jamova 39, 1000, Ljubljana (Slovenia); Milačič, Radmila [Department of Environmental Sciences, Jožef Stefan Institute, Jamova 39, 1000, Ljubljana (Slovenia); Jožef Stefan International Postgraduate School, Jamova 39, 1000, Ljubljana (Slovenia); Ščančar, Janez, E-mail: [email protected] [Department of Environmental Sciences, Jožef Stefan Institute, Jamova 39, 1000, Ljubljana (Slovenia); Jožef Stefan International Postgraduate School, Jamova 39, 1000, Ljubljana (Slovenia)
2016-04-07
Polybrominated diphenyl ethers (PBDEs) are flame retardants. As a consequence of their widespread use, they have been released into the environment. PBDEs are lipophilic organic contaminants that enter wastewater treatment plants (WWTPs) from urban, agricultural and industrial discharges. Because of their low aqueous solubility and resistance to biodegradation, up to 90% of the PBDEs are accumulated in the sewage sludge during the wastewater treatment. To assess the possibilities for sludge re-use, a reliable determination of the concentrations of these PBDEs is of crucial importance. Six PBDE congeners (BDE 28, BDE 47, BDE 99, BDE 100, BDE 153 and BDE 154) are listed as priority substances under the EU Water Framework Directive. In the present work a simple analytical method with minimal sample-preparation steps was developed for a sensitive and reliable determination of the six PBDEs in sewage sludge by the use of gas chromatography coupled to inductively coupled plasma mass spectrometry (GC-ICP-MS). For this purpose an extraction procedure was optimised. Different extracting agents (methanol (MeOH), acetic acid (AcOH)/MeOH mixture (3:1) and 0.1 mol L{sup −1} hydrochloric acid (HCl) in MeOH) followed by the addition of a Tris-citrate buffer (co-extracting agent) and iso-octane were applied under different modes of extraction (mechanical shaking, microwave- and ultrasound-assisted extraction). Mechanical shaking or the microwave-assisted extraction of sewage sludge with 0.1 mol L{sup −1} HCl in MeOH and the subsequent addition of the Tris-citrate buffer and the iso-octane extracted the PBDEs from the complex sludge matrix most effectively. However, due to easier sample manipulation during the extraction step, mechanical shaking was used. The PBDEs in the organic phase were quantified with GC-ICP-MS by applying a standard addition calibration method. The spike recovery test (recoveries between 95 and 104%) and comparative analyses with the species
20. The influence of fragmentation models on the determination of the strong coupling constant in e+e- annihilation into hadrons
International Nuclear Information System (INIS)
Behrend, H.J.; Chen, C.; Fenner, H.; Schachter, M.J.; Schroeder, V.; Sindt, H.; D'Agostini, G.; Apel, W.D.; Banerjee, S.; Bodenkamp, J.; Chrobaczek, D.; Engler, J.; Fluegge, G.; Fries, D.C.; Fues, W.; Gamerdinger, K.; Hopp, G.; Kuester, H.; Mueller, H.; Randoll, H.; Schmidt, G.; Schneider, H.; Boer, W. de; Buschhorn, G.; Grindhammer, G.; Grosse-Wiesmann, P.; Gunderson, B.; Kiesling, C.; Kotthaus, R.; Kruse, U.; Lierl, H.; Lueers, D.; Oberlack, H.; Schacht, P.; Colas, P.; Cordier, A.; Davier, M.; Fournier, D.; Grivaz, J.F.; Haissinski, J.; Journe, V.; Klarsfeld, A.; Laplanche, F.; Le Diberder, F.; Mallik, U.; Veillet, J.J.; Field, J.H.; George, R.; Goldberg, M.; Grossetete, B.; Hamon, O.; Kapusta, F.; Kovacs, F.; London, G.; Poggioli, L.; Rivoal, M.; Aleksan, R.; Bouchez, J.; Carnesecchi, G.; Cozzika, G.; Ducros, Y.; Gaidot, A.; Jadach, S.; Lavagne, Y.; Pamela, J.; Pansart, J.P.; Pierre, F.
1983-01-01
Hadronic events obtained with the CELLO detector at PETRA were compared with first-order QCD predictions using two different models for the fragmentation of quarks and gluons, the Hoyer model and the Lund model. Both models are in reasonable agreement with the data, although they do not completely reproduce the details of many distributions. Several methods have been applied to determine the strong coupling constant αsub(s). Although within one model the value of αsub(s) varies by 20% among the different methods, the values determined using the Lund model are 30% or more larger (depending on the method used) than the values determined with the Hoyer model. Our results using the Hoyer model are in agreement with previous results based on this approach. (orig.)
1. Determination of strontium and lead isotope ratios of grains using high resolution inductively coupled plasma mass spectrometer with single collector
International Nuclear Information System (INIS)
Shinozaki, Miyuki; Ariyama, Kaoru; Kawasaki, Akira; Hirata, Takafumi
2010-01-01
A method for determining strontium and lead isotope ratios of grains was developed. The samples investigated in this study were rice, barley and wheat. The samples were digested with nitric acid and hydrogen peroxide, and heated in a heating block. Strontium and lead were separated from the matrix by adding an acid digested solution into a column packed with Sr resin, which has selectivity for the absorption of strontium and lead. Strontium and lead isotope ratios were determined using a high-resolution inductively coupled plasma mass spectrometer (HR-ICP-MS) with a single collector. The intraday relative standard deviations of 87 Sr/ 86 Sr and lead isotope ratios ( 204 Pb/ 206 Pb, 207 Pb/ 206 Pb, 208 Pb/ 206 Pb) by HR-ICP-MS measurements were < 0.06% and around 0.1%, respectively. This method enabled us to determine strontium and lead isotope ratios in two days. (author)
2. Standard test method for determining elements in waste streams by inductively coupled plasma-atomic emission spectroscopy
International Nuclear Information System (INIS)
Anon.
1989-01-01
This test method covers the determination of trace, minor, and major elements in waste streams by inductively coupled plasma-atomic emission spectroscopy (ICP-AES) following an acid digestion of the specimen. Waste streams from manufacturing processes of nuclear and nonnuclear materials can be analyzed. This test method is applicable to the determination of total metals. Results from this test method can be used to characterize waste received by treatment facilities and to formulate appropriate treatment recipes. The results are also usable to process control within waste treatment facilities. This test method is applicable only to waste streams that contain radioactivity levels which do not require special personnel or environmental protection. A list of the elements determined in waste streams and the corresponding lower reporting limit is included
3. The nonabelian tensor square of Bieberbach group of dimension five with dihedral point group of order eight
Science.gov (United States)
Fauzi, Wan Nor Farhana Wan Mohd; Idrus, Nor'ashiqin Mohd; Masri, Rohaidah; Sarmin, Nor Haniza
2014-07-01
The nonabelian tensor product was originated in homotopy theory as well as in algebraic K-theory. The nonabelian tensor square is a special case of the nonabelian tensor product where the product is defined if the two groups act on each other in a compatible way and their action are taken to be conjugation. In this paper, the computation of nonabelian tensor square of a Bieberbach group, which is a torsion free crystallographic group, of dimension five with dihedral point group of order eight is determined. Groups, Algorithms and Programming (GAP) software has been used to assist and verify the results.
4. Monitoring of the tensor polarization of high energy deuteron beams; Monitoring tenzornoj polyarizatsii dejtronnykh puchkov vysokoj ehnergii
Energy Technology Data Exchange (ETDEWEB)
Zolin, L S; Litvinenko, A G; Pilipenko, Yu K; Reznikov, S G; Rukoyatkin, P A; Fimushkin, V V
1998-12-01
The method of determining the tensor component of high energy polarized deuteron beams, based on measuring of the tensor analyzing power in the deuteron stripping reaction, is discussed. This method is convenient for monitoring during long time runs on the tensor polarized deuteron beams. The method was tested in the 5-days run at the LHE JINR accelerator with the 3 and 9 GeV/c tensor polarized deuterons. The results made it possible to estimate the beam polarization stability in time 5 refs., 4 figs., 1 tab.
5. A forgotten argument by Gordon uniquely selects Abraham's tensor as the energy-momentum tensor for the electromagnetic field in homogeneous, isotropic matter
International Nuclear Information System (INIS)
Antoci, S.; Mihich, L.
1997-01-01
Given the present status of the problem of the electromagnetic energy tensor in matter, there is perhaps use in recalling a forgotten argument given in 1923 by W. Gordon. Let us consider a material medium which is homogeneous and isotropic when observed in its rest frame. For such a medium, Gordon's argument allows to reduce the above-mentioned problem to an analogous one, defined in a general relativistic vacuum. For the latter problem the form of the Lagrangian is known already, hence the determination of the energy tensor is a straightforward matter. One just performs the Hamiltonian derivative of the Lagrangian chosen in this way with respect to the true metric g ik . Abraham's tensor is thus selected as the electromagnetic energy tensor for a medium which is homogeneous and isotropic in its rest frame
6. Simultaneous determination of macronutrients, micronutrients and trace elements in mineral fertilizers by inductively coupled plasma optical emission spectrometry
International Nuclear Information System (INIS)
Oliveira Souza, Sidnei de; Silvério Lopes da Costa, Silvânio; Santos, Dayane Melo; Santos Pinto, Jéssica dos; Garcia, Carlos Alexandre Borges
2014-01-01
An analytical method for simultaneous determination of macronutrients (Ca, Mg, Na and P), micronutrients (Cu, Fe, Mn and Zn) and trace elements (Al, As, Cd, Pb and V) in mineral fertilizers was optimized. Two-level full factorial design was applied to evaluate the optimal proportions of reagents used in the sample digestion on hot plate. A Doehlert design for two variables was used to evaluate the operating conditions of the inductively coupled plasma optical emission spectrometer in order to accomplish the simultaneous determination of the analyte concentrations. The limits of quantification (LOQs) ranged from 2.0 mg kg −1 for Mn to 77.3 mg kg −1 for P. The accuracy and precision of the proposed method were evaluated by analysis of standard reference materials (SRMs) of Western phosphate rock (NIST 694), Florida phosphate rock (NIST 120C) and Trace elements in multi-nutrient fertilizer (NIST 695), considered to be adequate for simultaneous determination. Twenty-one samples of mineral fertilizers collected in Sergipe State, Brazil, were analyzed. For all samples, the As, Ca, Cd and Pb concentrations were below the LOQ values of the analytical method. For As, Cd and Pb the obtained LOQ values were below the maximum limit allowed by the Brazilian Ministry of Agriculture, Livestock and Food Supply (Ministério da Agricultura, Pecuária e Abastecimento — MAPA). The optimized method presented good accuracy and was effectively applied to quantitative simultaneous determination of the analytes in mineral fertilizers by inductively coupled plasma optical emission spectrometry (ICP OES). - Highlights: • Determination of inorganic constituents in mineral fertilizers was proposed. • Experimental design methodology was used to optimize analytical method. • The sample preparation procedure using diluted reagents (HNO 3 and H 2 O 2 ) was employed. • The analytical method was satisfactorily to the determination of thirteen elements. • The ICP OES technique can be
7. Determination of Tributyltin in Seafood Based on Magnetic Molecularly Imprinted Polymers Coupled with High-Performance Liquid Chromatography-Inductively Coupled Plasma Mass Spectrometry
Directory of Open Access Journals (Sweden)
Hua Yang
2017-01-01
Full Text Available In this study, Fe3O4 was adopted as a carrier for surface molecular imprinting with two-stage polymerization. First, the functional monomer (methacrylic acid, MAA was modified on the surface of Fe3O4, which was then polymerized with the template molecule (tributyltin, TBT, cross linking agent (ethylene glycol dimethacrylate, EGDMA, and porogen (acetonitrile, hereby successfully preparing Fe3O4@MIPs prone to specifically identify TBT. The physical properties of Fe3O4@MIPs were then characterized, and adsorption and selection capacities were also assessed. Compared with conventional imprinting polymers, this magnetic molecular imprinting polymer (MIP displayed significantly increased and more specific adsorption. Meanwhile, its pretreatment was simpler and faster due to magnetic separation characteristics. Using magnetic MIPs as adsorbents for enrichment and separation, detection limit, recovery rate, and linear range were 1.0 ng g−1, 79.74–95.72%, and 5 ng g−1~1000 ng g−1, respectively, for a number of seafood samples. High-performance liquid chromatography-inductively coupled plasma mass spectrometry (HPLC-ICP-MS was used to analyze Tegillarca granosa, mussels, large yellow croaker, and other specimens, with recovery rates of 79.74–95.72% and RSD of 1.3%–4.7%. Overall, this method has a shorter total analysis time, lower detection limit, and wider linear range and can be more effectively applied to determine MAA in seawater and seafood.
8. In-coupled syringe assisted octanol-water partition microextraction coupled with high-performance liquid chromatography for simultaneous determination of neonicotinoid insecticide residues in honey.
Science.gov (United States)
Vichapong, Jitlada; Burakham, Rodjana; Srijaranai, Supalax
2015-07-01
A simple and fast method namely in-coupled syringe assisted octanol-water partition microextraction combined with high performance liquid chromatography (HPLC) has been developed for the extraction, preconcentration and determination of neonicotinoid insecticide residues (e.g. imidacloprid, acetamiprid, clothianidin, thiacloprid, thiamethoxam, dinotefuran, and nitenpyram) in honey. The experimental parameters affected the extraction efficiency, including kind and concentration of salt, kind of disperser solvent and its volume, kind of extraction solvent and its volume, shooting times and extraction time were investigated. The extraction process was carried out by rapid shooting of two syringes. Therefore, rapid dispersion and mass transfer processes was created between phases, and thus affects the extraction efficiency of the proposed method. The optimum extraction conditions were 10.00 mL of aqueous sample, 10% (w/v) Na2SO4, 1-octanol (100µL) as an extraction solvent, shooting 4 times and extraction time 2min. No disperser solvent and centrifugation step was necessary. Linearity was obtained within the range of 0.1-3000 ngmL(-1), with the correlation coefficients greater than 0.99. The high enrichment factor of the target analytes was 100 fold and low limit of detection (0.25-0.50 ngmL(-1)) could be obtained. This proposed method has been successfully applied in the analysis of neonicotinoid residues in honey, and good recoveries in the range of 96.93-107.70% were obtained. Copyright © 2015 Elsevier B.V. All rights reserved.
9. Anharmonic vibrational properties in periodic systems: energy, electron-phonon coupling, and stress
OpenAIRE
Monserrat, Bartomeu; Drummond, N. D.; Needs, R. J.
2013-01-01
A unified approach is used to study vibrational properties of periodic systems with first-principles methods and including anharmonic effects. Our approach provides a theoretical basis for the determination of phonon-dependent quantities at finite temperatures. The low-energy portion of the Born-Oppenheimer energy surface is mapped and used to calculate the total vibrational energy including anharmonic effects, electron-phonon coupling, and the vibrational contribution to the stress tensor. W...
10. Algebraic classification of the Weyl tensor in higher dimensions based on its 'superenergy' tensor
International Nuclear Information System (INIS)
Senovilla, Jose M M
2010-01-01
The algebraic classification of the Weyl tensor in the arbitrary dimension n is recovered by means of the principal directions of its 'superenergy' tensor. This point of view can be helpful in order to compute the Weyl aligned null directions explicitly, and permits one to obtain the algebraic type of the Weyl tensor by computing the principal eigenvalue of rank-2 symmetric future tensors. The algebraic types compatible with states of intrinsic gravitational radiation can then be explored. The underlying ideas are general, so that a classification of arbitrary tensors in the general dimension can be achieved. (fast track communication)
11. Urinary elimination of molybdenum by healthy subjects as determined by inductively coupled plasma mass spectrometry.
Science.gov (United States)
Allain, P; Berre, S; Prémel-Cabic, A; Mauras, Y; Cledes, A; Cournot, A
The concentration of molybdenum was measured by inductively coupled plasma mass spectrometry (ICPMS) in the urines of two groups of healthy people living in two areas of France, Brest and Paris, about 500 km away. The concentration of Mo in the 24-hour urines of 10 healthy subjects from the Brest region was 25 +/- 10 micrograms/l, 38 +/- 20 micrograms/24 h and 21 +/- 9 micrograms/g creatinine. The concentration of Mo in the morning urines of 23 healthy men of the Paris region was 41 +/- 34 micrograms/l and 21 +/- 15 micrograms/g creatinine. Thus the mean elimination of Mo per gram of creatinine was the same in the two groups (21 +/- 9 and 21 +/- 15). Since the three main isotopes of Mo m/z = 95, 96 and 98, corresponding to an abundance percentage of 16, 17 and 24.5, respectively, were simultaneously analyzed in each sample and led to similar results, the ICPMS method seems reliable.
12. Determination of Oxidized Phosphatidylcholines by Hydrophilic Interaction Liquid Chromatography Coupled to Fourier Transform Mass Spectrometry
Directory of Open Access Journals (Sweden)
Pia Sala
2015-04-01
Full Text Available A novel liquid chromatography-mass spectrometry (LC-MS approach for analysis of oxidized phosphatidylcholines by an Orbitrap Fourier Transform mass spectrometer in positive electrospray ionization (ESI coupled to hydrophilic interaction liquid chromatography (HILIC was developed. This method depends on three selectivity criteria for separation and identification: retention time, exact mass at a resolution of 100,000 and collision induced dissociation (CID fragment spectra in a linear ion trap. The process of chromatography development showed the best separation properties with a silica-based Kinetex column. This type of chromatography was able to separate all major lipid classes expected in mammalian samples, yielding increased sensitivity of oxidized phosphatidylcholines over reversed phase chromatography. Identification of molecular species was achieved by exact mass on intact molecular ions and CID tandem mass spectra containing characteristic fragments. Due to a lack of commercially available standards, method development was performed with copper induced oxidation products of palmitoyl-arachidonoyl-phosphatidylcholine, which resulted in a plethora of lipid species oxidized at the arachidonoyl moiety. Validation of the method was done with copper oxidized human low-density lipoprotein (LDL prepared by ultracentrifugation. In these LDL samples we could identify 46 oxidized molecular phosphatidylcholine species out of 99 possible candidates.
13. Applications of inductively coupled plasma-mass spectrometry to radionuclide determinations: Second volume
International Nuclear Information System (INIS)
Morrow, R.W.; Crain, J.S.
1998-01-01
Even from its early conception, inductively coupled plasma-mass spectrometry (ICP-MS) was thought to be well-suited to the unique measurement problems facing the nuclear industry. These thoughts were well-founded; indeed, one might consider it unusual if a modern nuclear research center did not have access to one or more ICP mass spectrometers (quadrupole or otherwise). However, as ICP-MS has matured, improvements in sensitivity and precision have made possible measurements that were inconceivable to the founding fathers of the technology. Therefore, there is a periodic need to gather information and obtain a snapshot in time of the technology and its applications in nuclear energy. This second symposium was an international event in which speakers from the US, Europe, and the Middle East described new developments in ICP-MS relevant to the nuclear energy community. The papers presented at the 1998 symposium are published herein. Several papers have been processed separately for inclusion on the data base
14. How precisely can the difference method determine the $\\pi$NN coupling constant?
CERN Document Server
Loiseau, B
2000-01-01
The Coulomb-like backward peak of the neutron-proton scattering differentialcross section is due to one-pion exchange. Extrapolation to the pion pole ofprecise data should allow to obtain the value of the charged pion-nucleoncoupling constant. This was classically attempted by the use of a smoothphysical function, the Chew function, built from the cross section. To improveaccuracy of such an extrapolation one has introduced a difference method. Itconsists of extrapolating the difference between the Chew function based onexperimental data and that built from a model where the pion-nucleon couplingis exactly known. Here we cross-check to which precision can work this novelextrapolation method by applying it to differences between models and betweendata and models. With good reference models and for the 162 MeV neutron-protonUppsala single energy precise data with a normalisation error of 2.3 , thevalue of the charged pion-nucleon coupling constant is obtained with anaccuracy close to 1.8
15. Determination of diphenylether herbicides in water samples by solid-phase microextraction coupled to liquid chromatography.
Science.gov (United States)
Sheu, Hong-Li; Sung, Yu-Hsiang; Melwanki, Mahaveer B; Huang, Shang-Da
2006-11-01
Solid-phase microextraction (SPME) coupled to LC for the analysis of five diphenylether herbicides (aclonifen, bifenox, fluoroglycofen-ethyl, oxyfluorfen, and lactofen) is described. Various parameters of extraction of analytes onto the fiber (such as type of fiber, extraction time and temperature, pH, impact of salt and organic solute) and desorption from the fiber in the desorption chamber prior to separation (such as type and composition of desorption solvent, desorption mode, soaking time, and flush-out time) were studied and optimized. Four commercially available SPME fibers were studied. PDMS/divinylbenzene (PDMS/DVB, 60 microm) and carbowax/ templated resin (CW/TPR, 50 microm) fibers were selected due to better extraction efficiencies. Repeatability (RSD, 0.994), and detection limit (0.33-1.74 and 0.22-1.94 ng/mL, respectively, for PDMS/DVB and CW/TPR) were investigated. Relative recovery (81-104% for PDMS/DVB and 83-100% for CW/TPR fiber) values have also been calculated. The developed method was successfully applied to the analysis of river water and water collected from a vegetable garden.
16. Supersymmetry breaking and determination of the unification gauge coupling constant in string theories
International Nuclear Information System (INIS)
Carlos, B. de; Casas, J.A.; Munoz, C.
1993-01-01
We study in a systematic and modular invariant way gaugino condensation in the hidden sector as a potential source of hierarchical supersymmetry breaking and a non-trivial potential for the dilaton S whose real part corresponds to the tree-level gauge coupling constant (Re S∝g gut -2 ). For the case of pure Yang-Mills condensation, we show that no realistic results (in particular no reasonable values for Re S) can emerge, even if the hidden gauge group is not simple. However, in the presence of hidden matter (i.e. the most frequent case) there arises a very interesting class of scenarios with two or more hidden condensing groups for which the dilaton dynamically acquires a reasonable value (Re S∝2) and supersymmetry is broken at the correct scale (m 3/2 ∝10 3 GeV) with no need of fine-tuning. Actually, good values for Re S and m 3/2 are correlated. We make an exhaustive classification of the working possibilities. Remarkably, the results are basically independent from the value of δ GS (the contributions from the Green-Schwarz mechanism). The radius of the compactified space also acquires an expectation value, breaking duality spontaneously. (orig.)
17. 'Age' determination of irradiated materials utilizing inductively coupled plasma mass spectrometric (ICP-MS) detection
International Nuclear Information System (INIS)
Sommers, J.; Cummings, D.; Giglio, J.; Carney, K.
2009-01-01
A gas pressurized extraction chromatography (GPEC) system has been developed to perform elemental separations on radioactive samples to determine total and isotopic compositions of Cs and Ba from an irradiated salt sample, fuel sample and two sealed radiation sources. The GPEC system employs compressed nitrogen to move liquid through the system, compared to gravity or pumped liquids that are typically used for separations. A commercially available Sr-Resin TM was used to perform the separation for the above mentioned analytes. A 1% acetic acid solution was determined to be the best extractant for Ba. A flow rate of 0.1 mL/min was determined to be optimal for the separation of Ba. Complete recovery of the Cs and Ba was achieved, within the systematic uncertainties of the experiments. (author)
18. A separation method to overcome the interference of aluminium on zinc determination by inductively coupled plasma atomic emission spectroscopy
OpenAIRE
Jesus, Djane S. de; Korn, Maria das Graças Andrade; Ferreira, Sergio Luis Costa; Carvalho, Marcelo Souza de
2000-01-01
Texto completo: acesso restrito. p.389–394 The use of polyurethane foam (PUF) to separate zinc from large amounts of aluminium and its determination by inductively coupled plasma atomic emission spectroscopy technique (ICP-AES) in aluminium matrices is described. The proposed method is based on the solid-phase extraction of the zinc(II) cation as a thiocyanate complex. Parameters such as effect of pH on zinc sorption, zinc desorption from the foam and analytical features of the procedure w...
19. Determination of Cr(VI) and Cr(III) in urine and dextrose by inductively coupled plasma emission spectroscopy
Science.gov (United States)
Mianzhi, Zhuang; Barnes, Ramon M.
The determination of Cr(VI) and Cr(III) in human urine and in commercial dextrose solution is performed by induclively coupled plasma-atomic emission spectroscopy after selective preconcentration of the chromium species at different pH values by poly(dithiocarbamate) and poly(acrylamidoxime) chelating resins. The chelating properties of these resins with chromium, including the kinetics of uptake and removal of Cr(III), and the influence of matrix concentrations were evaluated. Chromium in human urine was found to exist exclusively as Cr(III).
20. Distinct Phosphorylation Clusters Determine the Signaling Outcome of Free Fatty Acid Receptor 4/G Protein-Coupled Receptor 120
DEFF Research Database (Denmark)
Prihandoko, Rudi; Alvarez-Curto, Elisa; Hudson, Brian D
2016-01-01
of these phosphoacceptor sites to alanine completely prevented phosphorylation of mFFA4 but did not limit receptor coupling to extracellular signal regulated protein kinase 1 and 2 (ERK1/2) activation. Rather, an inhibitor of Gq/11proteins completely prevented receptor signaling to ERK1/2. By contrast, the recruitment...... activation. These unique observations define differential effects on signaling mediated by phosphorylation at distinct locations. This hallmark feature supports the possibility that the signaling outcome of mFFA4 activation can be determined by the pattern of phosphorylation (phosphorylation barcode...
1. Determination of sulfonamides in meat by liquid chromatography coupled with atmospheric pressure chemical ionization mass spectrometry
International Nuclear Information System (INIS)
Kim, Dal Ho; Choi, Jong Oh; Kim, Jin Seog; Lee, Dai Woon
2002-01-01
Liquid chromatography/atmospheric pressure chemical ionization-mass spectrometry (LC-APCI-MS) has been used for the determination of sulfonamides in meat. Five typical sulfonamides were selected as target compounds, and beef meat was selected as a matrix sample. As internal standards, sulfapyridine and isotope labeled sulfamethazine ( 13 C 6 -SMZ) were used. Compared to the results of recent reports, our results have shown improved precision to a RSD of 1.8% for the determination of sulfamethazine spiked with 75 ng/g level in meat
2. Studies in the determination of lead isotope ratios by inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Date, A.R.; Yuk Ying Cheung
1987-01-01
The application of ICP-MS to the determination of lead isotope ratios in geological materials is described. Data presented for a series of lead mineral concentrates are compared with reference values obtained by conventional solid source thermal ionisation mass spectrometry. The simultaneous determination of lead isotope ratios and trace elements is carried out in a rapid analysis mode. The application of an electrothermal vaporisation technique for small solution aliquots is described. Lead isotope ratio data for the United States Geological Survey standard reference silicate rock BCR-1, obtained without separation of lead from the matrix, are compared with previously published values obtained after separation. (author)
3. Mixed symmetry tensors in the worldline formalism
Energy Technology Data Exchange (ETDEWEB)
Corradini, Olindo [Dipartimento di Scienze Fisiche, Informatiche e Matematiche,Università degli Studi di Modena e Reggio Emilia, via Campi 213/A, I-41125 Modena (Italy); INFN - Sezione di Bologna,via Irnerio 46, I-40126 Bologna (Italy); Edwards, James P. [Department of Mathematical Sciences, University of Bath,Claverton Down, Bath BA2 7AY (United Kingdom)
2016-05-10
We consider the first quantised approach to quantum field theory coupled to a non-Abelian gauge field. Representing the colour degrees of freedom with a single family of auxiliary variables the matter field transforms in a reducible representation of the gauge group which — by adding a suitable Chern-Simons term to the particle action — can be projected onto a chosen fully (anti-)symmetric representation. By considering F families of auxiliary variables, we describe how to extend the model to arbitrary tensor products of F reducible representations, which realises a U(F) “flavour” symmetry on the worldline particle model. Gauging this symmetry allows the introduction of constraints on the Hilbert space of the colour fields which can be used to project onto an arbitrary irreducible representation, specified by a certain Young tableau. In particular the occupation numbers of the wavefunction — i.e. the lengths of the columns (rows) of the Young tableau — are fixed through the introduction of Chern-Simons terms. We verify this projection by calculating the number of colour degrees of freedom associated to the matter field. We suggest that, using the worldline approach to quantum field theory, this mechanism will allow the calculation of one-loop scattering amplitudes with the virtual particle in an arbitrary representation of the gauge group.
4. Gravitational Metric Tensor Exterior to Rotating Homogeneous ...
African Journals Online (AJOL)
The covariant and contravariant metric tensors exterior to a homogeneous spherical body rotating uniformly about a common φ axis with constant angular velocity ω is constructed. The constructed metric tensors in this gravitational field have seven non-zero distinct components.The Lagrangian for this gravitational field is ...
5. Tensor Network Quantum Virtual Machine (TNQVM)
Energy Technology Data Exchange (ETDEWEB)
2016-11-18
There is a lack of state-of-the-art quantum computing simulation software that scales on heterogeneous systems like Titan. Tensor Network Quantum Virtual Machine (TNQVM) provides a quantum simulator that leverages a distributed network of GPUs to simulate quantum circuits in a manner that leverages recent results from tensor network theory.
6. Tensor product varieties and crystals. GL case
OpenAIRE
Malkin, Anton
2001-01-01
The role of Spaltenstein varieties in the tensor product for GL is explained. In particular a direct (non-combinatorial) proof of the fact that the number of irreducible components of a Spaltenstein variety is equal to a Littlewood-Richardson coefficient (i.e. certain tensor product multiplicity) is obtained.
7. Energy momentum tensor in theories with scalar field
International Nuclear Information System (INIS)
Joglekar, S.D.
1992-01-01
The renormalization of energy momentum tensor in theories with scalar fields and two coupling constants is considered. The need for addition of an improvement term is shown. Two possible forms for the improvement term are: (i) One in which the improvement coefficient is a finite function of bare parameters of the theory (so that the energy-momentum tensor can be derived from an action that is a finite function of bare quantities), (ii) One in which the improvement coefficient is a finite quantity, i.e. finite function of the renormalized quantities are considered. Four possible model of such theories are (i) Scalar Q.E.D. (ii) Non-Abelian theory with scalars, (iii) Yukawa theory, (iv) A model with two scalars. In all these theories a negative conclusion is established: neither forms for the improvement terms lead to a finite energy momentum tensor. Physically this means that when interaction with external gravity is incorporated in such a model, additional experimental input in the form of root mean square mass radius must be given to specify the theory completely, and the flat space parameters are insufficient. (author). 12 refs
8. Differential invariants for higher-rank tensors. A progress report
International Nuclear Information System (INIS)
Tapial, V.
2004-07-01
We outline the construction of differential invariants for higher-rank tensors. In section 2 we outline the general method for the construction of differential invariants. A first result is that the simplest tensor differential invariant contains derivatives of the same order as the rank of the tensor. In section 3 we review the construction for the first-rank tensors (vectors) and second-rank tensors (metrics). In section 4 we outline the same construction for higher-rank tensors. (author)
9. Unique characterization of the Bel-Robinson tensor
International Nuclear Information System (INIS)
Bergqvist, G; Lankinen, P
2004-01-01
We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson-type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors
10. Gas-diffusion microextraction coupled with spectrophotometry for the determination of formaldehyde in cork agglomerates.
Science.gov (United States)
Brandão, Pedro F; Ramos, Rui M; Valente, Inês M; Almeida, Paulo J; Carro, Antonia M; Lorenzo, Rosa A; Rodrigues, José A
2017-04-01
In this work, a simple methodology was developed for the extraction and determination of free formaldehyde content in cork agglomerate samples. For the first time, gas-diffusion microextraction was used for the extraction of volatile formaldehyde directly from samples, with simultaneous derivatization with acetylacetone (Hantzsch reaction). The absorbance of the coloured solution was read in a spectrophotometer at 412 nm. Different extraction parameters were studied and optimized (extraction temperature, sample mass, volume of acceptor solution, extraction time and concentration of derivatization reagent) by means of an asymmetric screening. The developed methodology proved to be a reliable tool for the determination of formaldehyde in cork agglomerates with the following suitable method features: low LOD (0.14 mg kg -1 ) and LOQ (0.47 mg kg -1 ), r 2 = 0.9994, and intraday and interday precision of 3.5 and 4.9%, respectively. The developed methodology was applied to the determination of formaldehyde in different cork agglomerate samples, and contents between 1.9 and 9.4 mg kg -1 were found. Furthermore, formaldehyde was also determined by the standard method EN 717-3 for comparison purposes; no significant differences between the results of both methods were observed. Graphical abstract Representation of the GDME system and its main components.
11. 'Age' Determination of Irradiated Materials Utilizing Inductively Coupled Plasma Mass Spectrometric (ICP-MS) Detection
International Nuclear Information System (INIS)
Sommers, James; Giglio, Jeffrey J.; Cummings, Daniel; Carney, Kevin P.
2009-01-01
A gas pressurized extraction chromatography (GPEC) system has been developed to perform elemental separations on radioactive samples to determine total and isotopic compositions of Cs and Ba from an irradiated salt sample, fuel sample and two sealed radiation sources. The separation is necessary to remove isobaric interferences in the determination of 137Cs, 135Cs, 137Ba, 135Ba, which are used to determine the age of a sample from radioactive decay or purification. The micro-column extraction chromatography system employs compressed nitrogen to move liquid through the system, compared to gravity or pumped liquids that are typically used for separations. The use of compressed gas allows for accurate and precise recovery of all liquids put into the chromatography system, enabling very accurate dilutions. The use of a small analytical column permits the use of very small amounts of liquids to be used. As a benefit, the amount of radiological waste that is generated in the separation process is minimized. For this work, a commercially available Sr-Resin(trademark) was used to perform the separation for the above mentioned analytes. The column consists of a 7 inch piece of 1/16 in. O.D. x 0.030 in I.D. Teflon(trademark) tubing having an internal volume of 81 (micro)L. To this column, 49 mg of resin was added. The columns are re-usable after regeneration with 3 M HNO3. All samples were separated using batch collection, although real time analysis is possible with the current experimental design. A 1 % acetic acid solution was determined to be the best extractant for Ba. A flow rate of 0.1 mL/min was determined to be optimal for the separation of Ba. Complete recovery of the Cs and Ba was achieved, within the systematic error of the experiments.
12. Tensor completion and low-n-rank tensor recovery via convex optimization
International Nuclear Information System (INIS)
Gandy, Silvia; Yamada, Isao; Recht, Benjamin
2011-01-01
In this paper we consider sparsity on a tensor level, as given by the n-rank of a tensor. In an important sparse-vector approximation problem (compressed sensing) and the low-rank matrix recovery problem, using a convex relaxation technique proved to be a valuable solution strategy. Here, we will adapt these techniques to the tensor setting. We use the n-rank of a tensor as a sparsity measure and consider the low-n-rank tensor recovery problem, i.e. the problem of finding the tensor of the lowest n-rank that fulfills some linear constraints. We introduce a tractable convex relaxation of the n-rank and propose efficient algorithms to solve the low-n-rank tensor recovery problem numerically. The algorithms are based on the Douglas–Rachford splitting technique and its dual variant, the alternating direction method of multipliers
13. Determination of total tin in canned food using inductively coupled plasma atomic emission spectroscopy
Energy Technology Data Exchange (ETDEWEB)
Perring, Loic; Basic-Dvorzak, Marija [Department of Quality and Safety Assurance, Nestle Research Centre, P.O. Box 44, Vers chez-les-Blanc, 1000, Lausanne (Switzerland)
2002-09-01
Tin is considered to be a priority contaminant by the Codex Alimentarius Commission. Tin can enter foods either from natural sources, environmental pollution, packaging material or pesticides. Higher concentrations are found in processed food and canned foods. Dissolution of the tinplate depends on the of food matrix, acidity, presence of oxidising reagents (anthocyanin, nitrate, iron and copper) presence of air (oxygen) in the headspace, time and storage temperature. To reduce corrosion and dissolution of tin, nowadays cans are usually lacquered, which gives a marked reduction of tin migration into the food product. Due to the lack of modern validated published methods for food products, an ICP-AES (Inductively coupled plasma-atomic emission spectroscopy) method has been developed and evaluated. This technique is available in many laboratories in the food industry and is more sensitive than atomic absorption. Conditions of sample preparation and spectroscopic parameters for tin measurement by axial ICP-AES were investigated for their ruggedness. Two methods of preparation involving high-pressure ashing or microwave digestion in volumetric flasks were evaluated. They gave complete recovery of tin with similar accuracy and precision. Recoveries of tin from spiked products with two levels of tin were in the range 99{+-}5%. Robust relative repeatabilities and intermediate reproducibilities were <5% for different food matrices containing >30 mg/kg of tin. Internal standard correction (indium or strontium) did not improve the method performance. Three emission lines for tin were tested (189.927, 283.998 and 235.485 nm) but only 189.927 nm was found to be robust enough with respect to interferences, especially at low tin concentrations. The LOQ (limit of quantification) was around 0.8 mg/kg at 189.927 nm. A survey of tin content in a range of canned foods is given. (orig.)
14. Accurate determination and certification of bromine in plastic by isotope dilution inductively coupled plasma mass spectrometry
Energy Technology Data Exchange (ETDEWEB)
Ohata, Masaki, E-mail: [email protected]; Miura, Tsutomu
2014-07-21
Highlights: • Accurate analytical method of Br in plastic was studied by isotope dilution ICPMS. • A microwave acid digestion using quartz vessel was suitable for Br analysis. • Sample dilution by NH{sub 3} solution could remove memory effect for ICPMS measurement. • The analytical result of the ID-ICPMS showed consistency with that of INAA. • The ID-ICPMS developed could apply to certification of Br in candidate plastic CRM. - Abstract: The accurate analytical method of bromine (Br) in plastic was developed by an isotope dilution inductively coupled plasma mass spectrometry (ID-ICPMS). The figures of merit of microwave acid digestion procedures using polytetrafluoroethylene (PTFE) or quartz vessels were studied and the latter one was suitable for Br analysis since its material was free from Br contamination. The sample dilution procedures using Milli-Q water or ammonium (NH{sub 3}) solution were also studied to remove memory effect for ICPMS measurement. Although severe memory effect was observed on Milli-Q water dilution, NH{sub 3} solution could remove it successfully. The accuracy of the ID-ICPMS was validated by a certified reference material (CRM) as well as the comparison with the analytical result obtained by an instrumental neutron activation analysis (INAA) as different analytical method. From these results, the ID-ICPMS developed in the present study could be evaluated as accurate analytical method of Br in plastic materials and it could apply to certification of Br in candidate plastic CRM with respect to such regulations related to RoHS (restriction of the use of hazardous substances in electrical and electronics equipment) directive.
15. Sulfonated polystyrene magnetic nanobeads coupled with immunochromatographic strip for clenbuterol determination in pork muscle.
Science.gov (United States)
Wu, Kesheng; Guo, Liang; Xu, Wei; Xu, Hengyi; Aguilar, Zoraida P; Xu, Guomao; Lai, Weihua; Xiong, Yonghua; Wan, Yiqun
2014-11-01
16. Accurate determination and certification of bromine in plastic by isotope dilution inductively coupled plasma mass spectrometry
International Nuclear Information System (INIS)
Ohata, Masaki; Miura, Tsutomu
2014-01-01
Highlights: • Accurate analytical method of Br in plastic was studied by isotope dilution ICPMS. • A microwave acid digestion using quartz vessel was suitable for Br analysis. • Sample dilution by NH 3 solution could remove memory effect for ICPMS measurement. • The analytical result of the ID-ICPMS showed consistency with that of INAA. • The ID-ICPMS developed could apply to certification of Br in candidate plastic CRM. - Abstract: The accurate analytical method of bromine (Br) in plastic was developed by an isotope dilution inductively coupled plasma mass spectrometry (ID-ICPMS). The figures of merit of microwave acid digestion procedures using polytetrafluoroethylene (PTFE) or quartz vessels were studied and the latter one was suitable for Br analysis since its material was free from Br contamination. The sample dilution procedures using Milli-Q water or ammonium (NH 3 ) solution were also studied to remove memory effect for ICPMS measurement. Although severe memory effect was observed on Milli-Q water dilution, NH 3 solution could remove it successfully. The accuracy of the ID-ICPMS was validated by a certified reference material (CRM) as well as the comparison with the analytical result obtained by an instrumental neutron activation analysis (INAA) as different analytical method. From these results, the ID-ICPMS developed in the present study could be evaluated as accurate analytical method of Br in plastic materials and it could apply to certification of Br in candidate plastic CRM with respect to such regulations related to RoHS (restriction of the use of hazardous substances in electrical and electronics equipment) directive
17. Determination of serum calcium levels by 42Ca isotope dilution inductively coupled plasma mass spectrometry.
Science.gov (United States)
Han, Bingqing; Ge, Menglei; Zhao, Haijian; Yan, Ying; Zeng, Jie; Zhang, Tianjiao; Zhou, Weiyan; Zhang, Jiangtao; Wang, Jing; Zhang, Chuanbao
2017-11-27
Serum calcium level is an important clinical index that reflects pathophysiological states. However, detection accuracy in laboratory tests is not ideal; as such, a high accuracy method is needed. We developed a reference method for measuring serum calcium levels by isotope dilution inductively coupled plasma mass spectrometry (ID ICP-MS), using 42Ca as the enriched isotope. Serum was digested with 69% ultrapure nitric acid and diluted to a suitable concentration. The 44Ca/42Ca ratio was detected in H2 mode; spike concentration was calibrated by reverse IDMS using standard reference material (SRM) 3109a, and sample concentration was measured by a bracketing procedure. We compared the performance of ID ICP-MS with those of three other reference methods in China using the same serum and aqueous samples. The relative expanded uncertainty of the sample concentration was 0.414% (k=2). The range of repeatability (within-run imprecision), intermediate imprecision (between-run imprecision), and intra-laboratory imprecision were 0.12%-0.19%, 0.07%-0.09%, and 0.16%-0.17%, respectively, for two of the serum samples. SRM909bI, SRM909bII, SRM909c, and GBW09152 were found to be within the certified value interval, with mean relative bias values of 0.29%, -0.02%, 0.10%, and -0.19%, respectively. The range of recovery was 99.87%-100.37%. Results obtained by ID ICP-MS showed a better accuracy than and were highly correlated with those of other reference methods. ID ICP-MS is a simple and accurate candidate reference method for serum calcium measurement and can be used to establish and improve serum calcium reference system in China.
18. Weyl curvature tensor in static spherical sources
International Nuclear Information System (INIS)
Ponce de Leon, J.
1988-01-01
The role of the Weyl curvature tensor in static sources of the Schwarzschild field is studied. It is shown that in general the contribution from the Weyl curvature tensor (the ''purely gravitational field energy'') to the mass-energy inside the body may be positive, negative, or zero. It is proved that a positive (negative) contribution from the Weyl tensor tends to increase (decrease) the effective gravitational mass, the red-shift (from a point in the sphere to infinity), as well as the gravitational force which acts on a constituent matter element of a body. It is also proved that the contribution from the Weyl tensor always is negative in sources with surface gravitational potential larger than (4/9. It is pointed out that large negative contributions from the Weyl tensor could give rise to the phenomenon of gravitational repulsion. A simple example which illustrates the results is discussed
19. A recursive reduction of tensor Feynman integrals
International Nuclear Information System (INIS)
Diakonidis, T.; Riemann, T.; Tausk, J.B.; Fleischer, J.
2009-07-01
We perform a recursive reduction of one-loop n-point rank R tensor Feynman integrals [in short: (n,R)-integrals] for n≤6 with R≤n by representing (n,R)-integrals in terms of (n,R-1)- and (n-1,R-1)-integrals. We use the known representation of tensor integrals in terms of scalar integrals in higher dimension, which are then reduced by recurrence relations to integrals in generic dimension. With a systematic application of metric tensor representations in terms of chords, and by decomposing and recombining these representations, we find the recursive reduction for the tensors. The procedure represents a compact, sequential algorithm for numerical evaluations of tensor Feynman integrals appearing in next-to-leading order contributions to massless and massive three- and four-particle production at LHC and ILC, as well as at meson factories. (orig.)
20. On Lovelock analogs of the Riemann tensor
Science.gov (United States)
2016-03-01
It is possible to define an analog of the Riemann tensor for Nth order Lovelock gravity, its characterizing property being that the trace of its Bianchi derivative yields the corresponding analog of the Einstein tensor. Interestingly there exist two parallel but distinct such analogs and the main purpose of this note is to reconcile both formulations. In addition we will introduce a simple tensor identity and use it to show that any pure Lovelock vacuum in odd d=2N+1 dimensions is Lovelock flat, i.e. any vacuum solution of the theory has vanishing Lovelock-Riemann tensor. Further, in the presence of cosmological constant it is the Lovelock-Weyl tensor that vanishes.
1. Effective gravitational wave stress-energy tensor in alternative theories of gravity
International Nuclear Information System (INIS)
Stein, Leo C.; Yunes, Nicolas
2011-01-01
The inspiral of binary systems in vacuum is controlled by the stress-energy of gravitational radiation and any other propagating degrees of freedom. For gravitational waves, the dominant contribution is characterized by an effective stress-energy tensor at future null infinity. We employ perturbation theory and the short-wavelength approximation to compute this stress-energy tensor in a wide class of alternative theories. We find that this tensor is generally a modification of that first computed by Isaacson, where the corrections can dominate over the general relativistic term. In a wide class of theories, however, these corrections identically vanish at asymptotically flat, future, null infinity, reducing the stress-energy tensor to Isaacson's. We exemplify this phenomenon by first considering dynamical Chern-Simons modified gravity, which corrects the action via a scalar field and the contraction of the Riemann tensor and its dual. We then consider a wide class of theories with dynamical scalar fields coupled to higher-order curvature invariants and show that the gravitational wave stress-energy tensor still reduces to Isaacson's. The calculations presented in this paper are crucial to perform systematic tests of such modified gravity theories through the orbital decay of binary pulsars or through gravitational wave observations.
2. Filtering overpopulated isoscalar tensor states with mass relations
International Nuclear Information System (INIS)
Burakovsky, Leonid; Page, Philip R.
2000-01-01
Schwinger-type mass formulas are used to analyze glueball-meson mixing for isoscalar tensor mesons. In one solution, the f J (2220) is the physical glueball, and in the other the glueball is distributed over various states, with f 2 (1810) having the largest glueball component. Neither the f 2 (1565) nor the f J (1710) are among the physical states without assuming significant coupling to decay channels. The decay f 2 (1525)→ππ is consistent with experiment, and f J (2220) is neither narrow nor decays flavor democratically. (c) 2000 The American Physical Society
3. Cosmology and a general scalar-tensor theory of gravity
International Nuclear Information System (INIS)
Bishop, N.T.
1976-01-01
The cosmological models resulting from a general scalar-tensor theory of gravity are discussed. Those models for which the scalar field varies as a power of the cosmological expansion factor (i.e. phi varies as Rsup(n)) are considered in detail, leading to a set of such models compatible with observation. This set includes models in which the scalar coupling parameter ω is negative. The models described here are similar to those of Newtonian cosmology obtained from an impotence principle. (author)
4. Nonuniversal scalar-tensor theories and big bang nucleosynthesis
International Nuclear Information System (INIS)
Coc, Alain; Olive, Keith A.; Uzan, Jean-Philippe; Vangioni, Elisabeth
2009-01-01
We investigate the constraints that can be set from big bang nucleosynthesis on two classes of models: extended quintessence and scalar-tensor theories of gravity in which the equivalence principle between standard matter and dark matter is violated. In the latter case, and for a massless dilaton with quadratic couplings, the phase space of theories is investigated. We delineate those theories where attraction toward general relativity occurs. It is shown that big bang nucleosynthesis sets more stringent constraints than those obtained from Solar System tests.
5. Nonuniversal scalar-tensor theories and big bang nucleosynthesis
Science.gov (United States)
Coc, Alain; Olive, Keith A.; Uzan, Jean-Philippe; Vangioni, Elisabeth
2009-05-01
We investigate the constraints that can be set from big bang nucleosynthesis on two classes of models: extended quintessence and scalar-tensor theories of gravity in which the equivalence principle between standard matter and dark matter is violated. In the latter case, and for a massless dilaton with quadratic couplings, the phase space of theories is investigated. We delineate those theories where attraction toward general relativity occurs. It is shown that big bang nucleosynthesis sets more stringent constraints than those obtained from Solar System tests.
6. Headspace single-drop microextraction coupled to microvolume UV-vis spectrophotometry for iodine determination
International Nuclear Information System (INIS)
Pena-Pereira, Francisco; Lavilla, Isela; Bendicho, Carlos
2009-01-01
Headspace single-drop microextraction has been combined with microvolume UV-vis spectrophotometry for iodine determination. Matrix separation and preconcentration of iodide following in situ volatile iodine generation and extraction into a microdrop of N,N'-dimethylformamide is performed. An exhaustive characterization of the microextraction system and the experimental variables affecting iodine generation from iodide was carried out. The procedure employed consisted of exposing 2.5 μL of N,N'-dimethylformamide to the headspace of a 10 mL acidic (H 2 SO 4 2 mol L -1 ) aqueous solution containing 1.7 mol L -1 Na 2 SO 4 for 7 min. Addition of 1 mL of H 2 O 2 1 mol L -1 for in situ iodine generation was performed. The limit of detection was determined as 0.69 μg L -1 . The repeatability, expressed as relative standard deviation, was 4.7% (n = 6). The calibration working range was from 5 to 200 μg L -1 (r 2 = 0.9991). The large preconcentration factor obtained, ca. 623 in only 7 min, compensate for the 10-fold loss in sensitivity caused by the decreased optical path, which results in improved detection limits as compared to spectrophotometric measurements carried out with conventional sample cells. The method was successfully applied to the determination of iodine in water, pharmaceutical and food samples
7. Anti-symmetric rank-two tensor matter field on superspace for N{sub T}=2
Energy Technology Data Exchange (ETDEWEB)
Spalenza, Wesley; Ney, Wander G; Helayel-Neto, J A
2004-05-06
In this work, we discuss the interaction between anti-symmetric rank-two tensor matter and topological Yang-Mills fields. The matter field considered here is the rank-2 Avdeev-Chizhov tensor matter field in a suitably extended N{sub T}=2 SUSY. We start off from the N{sub T}=2, D=4 superspace formulation and we go over to Riemannian manifolds. The matter field is coupled to the topological Yang-Mills field. We show that both actions are obtained as Q-exact forms, which allows us to express the energy-momentum tensor as Q-exact observables.
8. Coupled nutrient cycling determines tropical forest trajectory under elevated CO2.
Science.gov (United States)
Bouskill, N.; Zhu, Q.; Riley, W. J.
2017-12-01
Tropical forests have a disproportionate capacity to affect Earth's climate relative to their areal extent. Despite covering just 12 % of land surface, tropical forests account for 35 % of global net primary productivity and are among the most significant of terrestrial carbon stores. As atmospheric CO2 concentrations increase over the next century, the capacity of tropical forests to assimilate and sequester anthropogenic CO2 depends on limitation by multiple factors, including the availability of soil nutrients. Phosphorus availability has been considered to be the primary factor limiting metabolic processes within tropical forests. However, recent evidence points towards strong spatial and temporal co-limitation of tropical forests by both nitrogen and phosphorus. Here, we use the Accelerated Climate Modeling for Energy (ACME) Land Model (ALMv1-ECA-CNP) to examine how nutrient cycles interact and affect the trajectory of the tropical forest carbon sink under, (i) external nutrient input, (ii) climate (iii) elevated CO2, and (iv) a combination of 1-3. ALMv1 includes recent theoretical advances in representing belowground competition between roots, microbes and minerals for N and P uptake, explicit interactions between the nitrogen and phosphorus cycles (e.g., phosphatase production and nitrogen fixation), the dynamic internal allocation of plant N and P resources, and the integration of global datasets of plant physiological traits. We report nutrient fertilization (N, P, N+P) predictions for four sites in the tropics (El Verde, Puerto Rico, Barro Colorado Island, Panama, Manaus, Brazil and the Osa Peninsula, Coast Rica) to short-term nutrient fertilization (N, P, N+P), and benchmarking of the model against a meta-analysis of forest fertilization experiments. Subsequent simulations focus on the interaction of the carbon, nitrogen, and phosphorus cycles across the tropics with a focus on the implications of coupled nutrient cycling and the fate of the tropical
9. Avoiding bias effects in NMR experiments for heteronuclear dipole-dipole coupling determinations: principles and application to organic semiconductor materials.
Science.gov (United States)
Kurz, Ricardo; Cobo, Marcio Fernando; de Azevedo, Eduardo Ribeiro; Sommer, Michael; Wicklein, André; Thelakkat, Mukundan; Hempel, Günter; Saalwächter, Kay
2013-09-16
Carbon-proton dipole-dipole couplings between bonded atoms represent a popular probe of molecular dynamics in soft materials or biomolecules. Their site-resolved determination, for example, by using the popular DIPSHIFT experiment, can be challenged by spectral overlap with nonbonded carbon atoms. The problem can be solved by using very short cross-polarization (CP) contact times, however, the measured modulation curves then deviate strongly from the theoretically predicted shape, which is caused by the dependence of the CP efficiency on the orientation of the CH vector, leading to an anisotropic magnetization distribution even for isotropic samples. Herein, we present a detailed demonstration and explanation of this problem, as well as providing a solution. We combine DIPSHIFT experiments with the rotor-directed exchange of orientations (RODEO) method, and modifications of it, to redistribute the magnetization and obtain undistorted modulation curves. Our strategy is general in that it can also be applied to other types of experiments for heteronuclear dipole-dipole coupling determinations that rely on dipolar polarization transfer. It is demonstrated with perylene-bisimide-based organic semiconductor materials, as an example, in which measurements of dynamic order parameters reveal correlations of the molecular dynamics with the phase structure and functional properties. Copyright © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
10. Determination of ammonium in wastewaters by capillary electrophoresis on a column-coupling chip with conductivity detection.
Science.gov (United States)
Luc, Milan; Kruk, Pavol; Masár, Marián
2011-07-01
Analytical potentialities of a chip-based CE in determination of ammonium in wastewaters were investigated. CZE with the electric field and/or ITP sample stacking was performed on a column-coupling (CC) chip with integrated conductivity detectors. Acetate background electrolytes (pH ∼3) including 18-crown-6-ether (18-crown-6) and tartaric acid were developed to reach rapid (in 7-8 min) CZE and ITP-CZE resolutions of ammonium from other cations (sodium, potassium, calcium and magnesium) present in wastewater samples. Under preferred working conditions (suppressed hydrodynamic flow (HDF) and EOF on the column-coupling chip), both the employed methods did provide very good repeatabilities of the migration (RSD of 0.2-0.8% for the migration time) and quantitative (RSD of 0.3-4.9% for the peak area) parameters in the model and wastewater samples. Using a 900-nL sample injection volume, LOD for ammonium were obtained at 20 and 40 μg/L concentrations in CZE and ITP-CZE separations, respectively. Very good agreements of the CZE and ITP-CZE determinations of ammonium in six untreated wastewater samples (only filtration and dilution) with the results obtained by a reference spectrometric method indicate a very good accuracy of both the CE methods presented. Copyright © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
11. Classification of Antarctic algae by applying Kohonen neural network with 14 elements determined by inductively coupled plasma optical emission spectrometry
Energy Technology Data Exchange (ETDEWEB)
Balbinot, L. [Departamento de Quimica Analitica-Instituto de Quimica-Unicamp, PO Box 6154, CEP: 13083-971, Campinas, SP (Brazil); Smichowski, P. [Comision Nacional de Energia Atomica, Unidad de Actividad Quimica, Centro Atomico Constituyentes, Av. Gral Paz 1499, B1650KNA, San Martin, Provincia de Buenos Aires (Argentina); Farias, S. [Comision Nacional de Energia Atomica, Unidad de Actividad Quimica, Centro Atomico Constituyentes, Av. Gral Paz 1499, B1650KNA, San Martin, Provincia de Buenos Aires (Argentina); Arruda, M.A.Z. [Departamento de Quimica Analitica-Instituto de Quimica-Unicamp, PO Box 6154, CEP: 13083-971, Campinas, SP (Brazil); Vodopivez, C. [Instituto Antartico Argentino, Cerrito 1010, C1248AAZ, Buenos Aires (Argentina); Poppi, R.J. [Departamento de Quimica Analitica-Instituto de Quimica-Unicamp, PO Box 6154, CEP: 13083-971, Campinas, SP (Brazil)]. E-mail: [email protected]
2005-06-30
Optical emission spectrometers can generate results, which sometimes are not easy to interpret, mainly when the analyses involve classifications. To make simultaneous data interpretation possible, the Kohonen neural network is used to classify different Antarctic algae according to their taxonomic groups from the determination of 14 analytes. The Kohonen neural network architecture used was 5x5 neurons, thus reducing 14-dimension input data to two-dimensional space. The input data were 14 analytes (As, Co, Cu, Fe, Mn, Sr, Zn, Cd, Cr, Mo, Ni, Pb, Se, V) with their concentrations, determined by inductively coupled plasma optical emission spectrometry in 11 different species of algae. Three taxonomic groups (Rhodophyta, Phaeophyta and Cholorophyta) can be differentiated and classified through only their Cu content.
12. Major constituent quantitative determination in uranium alloys by coupled plasma atomic emission spectrometry and X ray fluorescence wavelength dispersive spectrometry
International Nuclear Information System (INIS)
Oliveira, Luis Claudio de; Silva, Adriana Mascarenhas Martins da; Gomide, Ricardo Goncalves; Silva, Ieda de Souza
2013-01-01
A wavelength-dispersive X-ray fluorescence (WD-XRF) spectrometric method for determination of major constituents elements (Zr, Nb, Mo) in Uranium/Zirconium/Niobium and Uranium/Molybdenum alloy samples were developed. The methods use samples taken in the form of chips that were dissolved in hot nitric acid and precipitate particles melted with lithium tetraborate and dissolved in hot nitric acid and finally analyzed as a solution. Studies on the determination by inductively coupled plasma optic emission spectrometry (ICP OES) using matched matrix in calibration curve were developed. The same samples solution were analyzed in both methods. The limits of detection (LOD), linearity of the calibrations curves, recovery study, accuracy and precision of the both techniques were carried out. The results were compared. (author)
13. Classification of Antarctic algae by applying Kohonen neural network with 14 elements determined by inductively coupled plasma optical emission spectrometry
International Nuclear Information System (INIS)
Balbinot, L.; Smichowski, P.; Farias, S.; Arruda, M.A.Z.; Vodopivez, C.; Poppi, R.J.
2005-01-01
Optical emission spectrometers can generate results, which sometimes are not easy to interpret, mainly when the analyses involve classifications. To make simultaneous data interpretation possible, the Kohonen neural network is used to classify different Antarctic algae according to their taxonomic groups from the determination of 14 analytes. The Kohonen neural network architecture used was 5x5 neurons, thus reducing 14-dimension input data to two-dimensional space. The input data were 14 analytes (As, Co, Cu, Fe, Mn, Sr, Zn, Cd, Cr, Mo, Ni, Pb, Se, V) with their concentrations, determined by inductively coupled plasma optical emission spectrometry in 11 different species of algae. Three taxonomic groups (Rhodophyta, Phaeophyta and Cholorophyta) can be differentiated and classified through only their Cu content
14. The determination of low level trace elements in coals by laser ablation-inductively coupled plasma-mass spectrometry
Energy Technology Data Exchange (ETDEWEB)
Booth, C.A.; Spears, D.A.; Krause, P.; Cox, A.G. [University of Sheffield, Sheffield (United Kingdom). Dept. of Earth Sciences
1999-11-01
The rapid determination of elements present in low level concentrations in bituminous coals is possible using laser abalation-inductively coupled plasma-mass spectrometry (l.a.-i.c.p.-m.s.). A wide range of trace elements can routinely be determined using this technique but it is for environmentally sensitive elements, such as As, Cd, Mo, Sb, Se and Hg, that it is of most use due to the low levels of detection. Calibration of the i.c.p.-m.s. was achieved using a series of uncertified coals and the method evaluated using the South African certified coals, Sarm 18, 19 and 20. A critical evaluation of the data obtained shows that for many of the elements studied the results obtained are both accurate and precise, even at very low concentrations, with the limits of detection for all of the elements being in the {mu}g/kg (parts per billion) range. 6 refs., 3 figs., 9 tabs.
15. One-loop tensor integrals in dimensional regularisation
International Nuclear Information System (INIS)
Campbell, J.M.; Glover, E.W.N.; Miller, D.J.
1997-01-01
We show how to evaluate tensor one-loop integrals in momentum space avoiding the usual plague of Gram determinants. We do this by constructing combinations of n- and (n-1)-point scalar integrals that are finite in the limit of vanishing Gram determinant. These non-trivial combinations of dilogarithms, logarithms and constants are systematically obtained by either differentiating with respect to the external parameters - essentially yielding scalar integrals with Feynman parameters in the numerator - or by developing the scalar integral in D=6-2ε or higher dimensions. An additional advantage is that other spurious kinematic singularities are also controlled. As an explicit example, we develop the tensor integrals and associated scalar integral combinations for processes where the internal particles are massless and where up to five (four massless and one massive) external particles are involved. For more general processes, we present the equations needed for deriving the relevant combinations of scalar integrals. (orig.)
16. Spatial Mapping of Translational Diffusion Coefficients Using Diffusion Tensor Imaging: A Mathematical Description.
Science.gov (United States)
Shetty, Anil N; Chiang, Sharon; Maletic-Savatic, Mirjana; Kasprian, Gregor; Vannucci, Marina; Lee, Wesley
2014-01-01
In this article, we discuss the theoretical background for diffusion weighted imaging and diffusion tensor imaging. Molecular diffusion is a random process involving thermal Brownian motion. In biological tissues, the underlying microstructures restrict the diffusion of water molecules, making diffusion directionally dependent. Water diffusion in tissue is mathematically characterized by the diffusion tensor, the elements of which contain information about the magnitude and direction of diffusion and is a function of the coordinate system. Thus, it is possible to generate contrast in tissue based primarily on diffusion effects. Expressing diffusion in terms of the measured diffusion coefficient (eigenvalue) in any one direction can lead to errors. Nowhere is this more evident than in white matter, due to the preferential orientation of myelin fibers. The directional dependency is removed by diagonalization of the diffusion tensor, which then yields a set of three eigenvalues and eigenvectors, representing the magnitude and direction of the three orthogonal axes of the diffusion ellipsoid, respectively. For example, the eigenvalue corresponding to the eigenvector along the long axis of the fiber corresponds qualitatively to diffusion with least restriction. Determination of the principal values of the diffusion tensor and various anisotropic indices provides structural information. We review the use of diffusion measurements using the modified Stejskal-Tanner diffusion equation. The anisotropy is analyzed by decomposing the diffusion tensor based on symmetrical properties describing the geometry of diffusion tensor. We further describe diffusion tensor properties in visualizing fiber tract organization of the human brain.
17. Solitons in a six-dimensional super Yang-Mills-tensor system and non-critical strings
International Nuclear Information System (INIS)
Nair, V.P.; Randjbar-Daemi, S.
1997-11-01
In this letter we study a coupled system of six-dimensional N = 1 tensor and super Yang-Mills multiplets. We identify some of the solitonic states of this system which exhibit stringy behaviour in six dimensions. A discussion of the supercharges and energy for the tensor multiples as well as zero modes is also given. We speculate about the possible relationship between our solution and what is known as tensionless strings. (author)
18. Determination of zearalenone content in cereals and feedstuffs by immunoaffinity column coupled with liquid chromatography.
Science.gov (United States)
Fazekas, B; Tar, A
2001-01-01
The zearalenone content of maize, wheat, barley, swine feed, and poultry feed samples was determined by immunoaffinity column cleanup followed by liquid chromatography (IAC-LC). Samples were extracted in methanol-water (8 + 2, v/v) solution. The filtered extract was diluted with distilled water and applied to immunoaffinity columns. Zearalenone was eluted with methanol, dried by evaporation, and dissolved in acetonitrile-water (3 + 7, v/v). Zearalenone was separated by isocratic elution of acetonitrile-water (50 + 50, v/v) on reversed-phase C18 column. The quantitative analysis was performed by fluorescence detector and confirmation was based on the UV spectrum obtained by a diode array detector. The mean recovery rate of zearalenone was 82-97% (RSD, 1.4-4.1%) on the original (single-use) immunoaffinity columns. The limit of detection of zearalenone by fluorescence was 10 ng/g at a signal-to-noise ratio of 10:1 and 30 ng/g by spectral confirmation in UV. A good correlation was found (R2 = 0.89) between the results obtained by IAC-LC and by the official AOAC-LC method. The specificity of the method was increased by using fluorescence detection in parallel with UV detection. This method was applicable to the determination of zearalenone content in cereals and other kinds of feedstuffs. Reusability of immunoaffinity columns was examined by washing with water after sample elution and allowing columns to stand for 24 h at room temperature. The zearalenone recovery rate of the regenerated columns varied between 79 and 95% (RSD, 3.2-6.3%). Columns can be regenerated at least 3 times without altering their performance and without affecting the results of repeated determinations.
19. Simultaneous determination of macronutrients, micronutrients and trace elements in mineral fertilizers by inductively coupled plasma optical emission spectrometry
Science.gov (United States)
de Oliveira Souza, Sidnei; da Costa, Silvânio Silvério Lopes; Santos, Dayane Melo; dos Santos Pinto, Jéssica; Garcia, Carlos Alexandre Borges; Alves, José do Patrocínio Hora; Araujo, Rennan Geovanny Oliveira
2014-06-01
An analytical method for simultaneous determination of macronutrients (Ca, Mg, Na and P), micronutrients (Cu, Fe, Mn and Zn) and trace elements (Al, As, Cd, Pb and V) in mineral fertilizers was optimized. Two-level full factorial design was applied to evaluate the optimal proportions of reagents used in the sample digestion on hot plate. A Doehlert design for two variables was used to evaluate the operating conditions of the inductively coupled plasma optical emission spectrometer in order to accomplish the simultaneous determination of the analyte concentrations. The limits of quantification (LOQs) ranged from 2.0 mg kg- 1 for Mn to 77.3 mg kg- 1 for P. The accuracy and precision of the proposed method were evaluated by analysis of standard reference materials (SRMs) of Western phosphate rock (NIST 694), Florida phosphate rock (NIST 120C) and Trace elements in multi-nutrient fertilizer (NIST 695), considered to be adequate for simultaneous determination. Twenty-one samples of mineral fertilizers collected in Sergipe State, Brazil, were analyzed. For all samples, the As, Ca, Cd and Pb concentrations were below the LOQ values of the analytical method. For As, Cd and Pb the obtained LOQ values were below the maximum limit allowed by the Brazilian Ministry of Agriculture, Livestock and Food Supply (Ministério da Agricultura, Pecuária e Abastecimento - MAPA). The optimized method presented good accuracy and was effectively applied to quantitative simultaneous determination of the analytes in mineral fertilizers by inductively coupled plasma optical emission spectrometry (ICP OES).
20. A Coupled CFD/FEM Structural Analysis to Determine Deformed Shapes of the RSRM Inhibitors
Science.gov (United States)
Dill, Richard A.; Whitesides, R. Harold
1996-01-01
Recent trends towards an increase in the stiffness of the acrylonitrile butadiene rubber (NBR) insulation material used in the construction of the redesigned solid rocket motor (RSRM) propellant inhibitors prompted questions about possible effects on RSRM performance. The specific objectives of the computational fluid dynamics (CFD) task included: (1) the definition of pressure loads to calculate the deformed shape of stiffer inhibitors, (2) the calculation of higher port velocities over the inhibitors to determine shifts in the vortex shedding or edge tone frequencies, and (3) the quantification of higher slag impingement and collection rates on the inhibitors and in the submerged nose nozzle cavity. | {"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": 0.8821082711219788, "perplexity": 2899.809245159416}, "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/1566027320156.86/warc/CC-MAIN-20190824084149-20190824110149-00469.warc.gz"} |
http://www.phy.duke.edu/~rgb/General/latex/ltx-223.html | ## eqnarray
\begin{eqnarray[*]}
var_1 & rel_1 & eq1 \\
var_2 & rel_2 & eq2 \\
....
\end{eqnarray[*]}
The eqnarray environment is typically used to display a sequence of equations or inequalities; it may also be used to manage spacing for long equations. The \lefteqn command is useful in this environment for splitting long equations over several lines.
It is very much like a three-column array environment, with position argument rcl, i.e., the columns are justified right, center, and left, respectively. (However, \multicolumn may not be used.)
Consecutive rows are separated by \\ commands and consecutive items within a row separated by an &. Any item may be empty, i.e., no text.
A separate equation number is placed on every line unless that line has a \nonumber command. The optional eqnarray* form does not generate any equation numbers.
A \label command anywhere within a row generates a reference to that row's number.
Related topics | {"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": 0.9888563752174377, "perplexity": 3511.1382479038275}, "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-2014-23/segments/1406510276353.59/warc/CC-MAIN-20140728011756-00103-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://de.maplesoft.com/support/help/errors/view.aspx?path=PDEtools/ChangeSymmetry | ChangeSymmetry - Maple Help
PDEtools
ChangeSymmetry
perform a change of variables on the infinitesimals of a symmetry generator
Calling Sequence ChangeSymmetry(TR, S, ITR, DepVars, NewVars, 'options'='value')
Parameters
TR - a transformation equation or a set of them S - a list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator differential operator ITR - optional - the inverse transformation equation or a set of them DepVars - optional - may be required, a function or a list of them indicating the (old) dependent variables of the problem NewDepVars - optional - a function or a list of them representing the new dependent variables jetnotation = ... - (optional) can be true (default, the notation found in S), false, jetvariables, jetvariableswithbrackets, jetnumbers or jetODE; to respectively return or not using the different jet notations available output = ... - optional - can be list or operator, indicating the output to be a list of infinitesimal components or the corresponding infinitesimal generator differential operator simplifier = ... - optional - indicates the simplifier to be used instead of the default simplify/size
Description
• The ChangeSymmetry command performs changes of variables in a list of infinitesimals of a symmetry generator or its corresponding infinitesimal generator differential operator. This transformation takes into account that the infinitesimals are coefficients of differentiation operators which are also changed by the transformation, thus contributing to the resulting infinitesimals in the new variables.
• To avoid having to remember the optional keywords if you misspell the keyword, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.
Examples
> $\mathrm{with}\left(\mathrm{PDEtools},\mathrm{ChangeSymmetry},\mathrm{CanonicalCoordinates},\mathrm{InfinitesimalGenerator}\right)$
$\left[{\mathrm{ChangeSymmetry}}{,}{\mathrm{CanonicalCoordinates}}{,}{\mathrm{InfinitesimalGenerator}}\right]$ (1)
Consider a PDE problem with two independent variables and one dependent variable, u(x, t), and consider the list of infinitesimals of a symmetry group
> $S≔\left[\mathrm{_ξ}\left[x\right]=x,\mathrm{_ξ}\left[t\right]=1,\mathrm{_η}\left[u\right]=u\right]$
${S}{≔}\left[{{\mathrm{_ξ}}}_{{x}}{=}{x}{,}{{\mathrm{_ξ}}}_{{t}}{=}{1}{,}{{\mathrm{_η}}}_{{u}}{=}{u}\right]$ (2)
In the input above you can also pass the symmetry as without infinitesimals' labels, as in $\left[x,1,u\right]$. The corresponding infinitesimal generator is
> $G≔\mathrm{InfinitesimalGenerator}\left(S,u\left(x,t\right)\right)$
${G}{≔}{f}{→}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{f}\right){+}\frac{{\partial }}{{\partial }{t}}{}{f}{+}{u}{}\left(\frac{{\partial }}{{\partial }{u}}{}{f}\right)$ (3)
Consider now the following transformation to be applied to the infinitesimals S
> $\mathrm{TR}≔\left\{t=r+v\left(r,s\right),x=\mathrm{exp}\left(v\left(r,s\right)\right),u\left(x,t\right)=s\mathrm{exp}\left(v\left(r,s\right)\right)\right\}$
${\mathrm{TR}}{≔}\left\{{t}{=}{r}{+}{v}{}\left({r}{,}{s}\right){,}{x}{=}{{ⅇ}}^{{v}{}\left({r}{,}{s}\right)}{,}{u}{}\left({x}{,}{t}\right){=}{s}{}{{ⅇ}}^{{v}{}\left({r}{,}{s}\right)}\right\}$ (4)
A direct application of this transformation to each component of $S$ is incorrect because these infinitesimals are coefficients of differentiation operators in the infinitesimal generator $G$ above. That fact is taken into account by ChangeSymmetry; the syntax it uses is the same as that of PDEtools[dchange] and DEtools[Xchange]
> $\mathrm{ChangeSymmetry}\left(\mathrm{TR},S\right)$
$\left[{{\mathrm{_ξ}}}_{{r}}{=}{0}{,}{{\mathrm{_ξ}}}_{{s}}{=}{0}{,}{{\mathrm{_η}}}_{{v}}{=}{1}\right]$ (5)
You can change variables directly in the infinitesimal generator differential operator, in which case the output has the same format, is also a differential operator
> $\mathrm{ChangeSymmetry}\left(\mathrm{TR},G\right)$
${f}{→}\frac{{\partial }}{{\partial }{v}}{}{f}$ (6)
You can also optionally request the output to be in list or operator format to override returning in the same format of the symmetry.
The transformation used in this example introduces the canonical coordinates of the symmetry group with infinitesimals S. That is why the result above is the normal form of the generator, all infinitesimals equal to 0 but for one equal to 1.
Consider now changing variables in a different symmetry, using the same transformation $\mathrm{TR}$
> $\mathrm{ChangeSymmetry}\left(\mathrm{TR},\left[\mathrm{_ξ}\left[x\right]=u,\mathrm{_ξ}\left[t\right]=x,\mathrm{_η}\left[u\right]=t\right]\right)$
$\left[{{\mathrm{_ξ}}}_{{r}}{=}{{ⅇ}}^{{v}}{-}{s}{,}{{\mathrm{_ξ}}}_{{s}}{=}\frac{{-}{{ⅇ}}^{{v}}{}{{s}}^{{2}}{+}{r}{+}{v}}{{{ⅇ}}^{{v}}}{,}{{\mathrm{_η}}}_{{v}}{=}{s}\right]$ (7)
Compare with the output in different jetnotation or in function notation (jetnotation = false); we also pass the symmetry without the infinitesimals' labels to save some keystrokes; correspondingly the output also comes without infinitesimals' labels
> $\mathrm{ChangeSymmetry}\left(\mathrm{TR},\left[u,x,t\right],\mathrm{jetnotation}=\mathrm{jetnumbers}\right)$
$\left[{{ⅇ}}^{{v}\left[\right]}{-}{s}{,}\frac{{-}{{ⅇ}}^{{v}\left[\right]}{}{{s}}^{{2}}{+}{r}{+}{v}\left[\right]}{{{ⅇ}}^{{v}\left[\right]}}{,}{s}\right]$ (8)
> $\mathrm{ChangeSymmetry}\left(\mathrm{TR},\left[u,x,t\right],\mathrm{jetnotation}=\mathrm{false}\right)$
$\left[{{ⅇ}}^{{v}{}\left({r}{,}{s}\right)}{-}{s}{,}\frac{{-}{{ⅇ}}^{{v}{}\left({r}{,}{s}\right)}{}{{s}}^{{2}}{+}{r}{+}{v}{}\left({r}{,}{s}\right)}{{{ⅇ}}^{{v}{}\left({r}{,}{s}\right)}}{,}{s}\right]$ (9) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 22, "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": 0.8093209862709045, "perplexity": 882.9785986055333}, "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/1679296948765.13/warc/CC-MAIN-20230328042424-20230328072424-00261.warc.gz"} |
http://math.stackexchange.com/questions/229184/what-is-the-effect-of-axis-rotation-on-functions-defined-on-mathbbr2 | # What is the effect of axis rotation on functions defined on $\mathbb{R}^{2}$
I haven't studied multivariable calculus yet but I have a question that bothers me. Let $F$ be a function $\mathbb{R}^2 \to \mathbb{R}$. Imagine that we rotate the co-ordinate axis by an angle $\theta$. I think the shape of the function should change. How should this function change if we make a rotation of the co-ordinate axis by some angle?
-
Do you mean to rotate in the plane of inputs $R^2$? If so this will only rotate the entire graph of $F$. If you want to rotate the entire $R^3$ in which the graph lies, the rotation may not even give the graph of a function. – coffeemath Nov 4 '12 at 23:04
Yes, I mean to rotate the domain . How can we express the new function in terms of the old function and the angle $\theta$? – Nabil Nov 5 '12 at 8:45
Let $F : \mathbb{R}^2 \to \mathbb{R}$ be the given function, and let $G_{\theta} : \mathbb{R}^2\to\mathbb{R}^2$ be rotation by $\theta$, then you want to consider the function $F\circ G_{\theta}$. Note that $G_{\theta}$ is a linear transformation and its standard matrix is a rotation matrix so we obtain
$$G_{\theta}\left(\left[\begin{array}\ x\\ y\end{array}\right]\right) = \left[\begin{array}\ \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{array}\right]\left[\begin{array}\ x\\ y\end{array}\right] = \left[\begin{array}\ x\cos\theta - y\sin\theta\\ x\sin\theta + y\cos\theta\end{array}\right].$$
$$(F\circ G_{\theta})(x, y) = F(G_{\theta}(x, y)) = F(x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta).$$
Note that the graph of $F\circ G_{\theta}$ is just the graph of $F$ rotated around the $z$-axis by $\theta$. | {"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": 0.8725774884223938, "perplexity": 128.34993006483535}, "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-2015-22/segments/1432207929956.54/warc/CC-MAIN-20150521113209-00144-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Supermanifold | # Supermanifold
Jump to: navigation, search
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.
## Physics
In physics, a supermanifold is a manifold with both bosonic and fermionic coordinates. These coordinates are usually denoted by
$(x,\theta,\bar{\theta})$
where x is the usual spacetime vector, and $\theta\,$ and $\bar{\theta}$ are Grassmann-valued spinors.
Whether these extra coordinates have any physical meaning is debatable. However this formalism is very useful for writing down supersymmetric Lagrangians.
## Supermanifold: a definition
Although supermanifolds are special cases of noncommutative manifolds, the local structure of supermanifolds make them better suited to study with the tools of standard differential geometry and locally ringed spaces.
A supermanifold M of dimension (p,q) is a topological space M with a sheaf of superalgebras, usually denoted OM or C(M), that is locally isomorphic to $C^\infty(\mathbb{R}^p)\otimes\Lambda^\bullet(\xi_1,\dots\xi_q).$
Note that the definition of a supermanifold is similar to that of a differentiable manifold, except that the model space Rp has been replaced by the model superspace Rp|q.
### Side comment
This is different from the alternative definition where, using a fixed Grassmann algebra generated by a countable number of generators Λ, one defines a supermanifold as a point set space using charts with the "even coordinates" taking values in the linear combination of elements of Λ with even grading and the "odd coordinates" taking values which are linear combinations of elements of Λ with odd grading. This raises the question of the physical meaning of all these Grassmann-valued variables. Many physicists claim that they have none and that they are purely formal; if this is the case, this may make the definition in the main part of the article preferable.
## Properties
Unlike a regular manifold, a supermanifold is not entirely composed of a set of points. Instead, one takes the dual point of view that the structure of a supermanifold M is contained in its sheaf OM of "smooth functions". In the dual point of view, an injective map corresponds to a surjection of sheaves, and a surjective map corresponds to an injection of sheaves.
An alternative approach to the dual point of view is to use the functor of points.
If M is a supermanifold of dimension (p,q), then the underlying space M inherits the structure of a differentiable manifold whose sheaf of smooth functions is OM/I, where I is the ideal generated by all odd functions. Thus M is called the underlying space, or the body, of M. The quotient map OMOM/I corresponds to an injective map MM; thus M is a submanifold of M.
## Examples
• Let M be a manifold. The odd tangent bundle ΠTM is a supermanifold given by the sheaf Ω(M) of differential forms on M.
• More generally, let EM be a vector bundle. Then ΠE is a supermanifold given by the sheaf Γ(ΛE*). In fact, Π is a functor from the category of vector bundles to the category of supermanifolds.
## Batchelor's theorem
Batchelor's theorem states that every supermanifold is noncanonically isomorphic to a supermanifold of the form ΠE. The word "noncanonically" prevents one from concluding that supermanifolds are simply glorified vector bundles; although the functor Π maps surjectively onto the isomorphism classes of supermanifolds, it is not an equivalence of categories.
The proof of Batchelor's theorem relies in an essential way on the existence of a partition of unity, so it does not hold for complex or real-analytic supermanifolds.
## Odd symplectic structures
### Odd symplectic form
In many physical and geometric applications, a supermanifold comes equipped with an Grassmann-odd symplectic structure. All natural geometric objects on a supermanifold are graded. In particular, the bundle of two-forms is equipped with a grading. An odd symplectic form ω on a supermanifold is a closed, odd form, inducing a non-degenerate pairing on TM. Such a supermanifold is called a P-manifold. Its graded dimension is necessarily (n,n), because the odd symplectic form induces a pairing of odd and even variables. There is a version of the Darboux theorem for P-manifolds, which allows one to equip a P-manifold locally with a set of coordinates where the odd symplectic form ω is written as
$\omega = \sum_{i} d\xi_i \wedge dx_i ,$
where $x_i$ are even coordinates, and $\xi_i$ odd coordinates. (An odd symplectic form should not be confused with a Grassmann-even symplectic form on a supermanifold. In contrast, the Darboux version of an even symplectic form is
$\sum_i dp_i \wedge dq_i+\sum_j \frac{\varepsilon_j}{2}(d\xi_j)^2,$
where $p_i,q_i$ are even coordinates, $\xi_i$ odd coordinates and $\varepsilon_j$ are either +1 or -1.)
### Antibracket
Given an odd symplectic 2-form ω one may define a Poisson bracket known as the antibracket of any two functions F and G on a supermanifold by
$\{F,G\}=\frac{\partial_rF}{\partial z^i}\omega^{ij}(z)\frac{\partial_lG}{\partial z^j}.$
Here $\partial_r$ and $\partial_l$ are the right and left derivatives respectively and z are the coordinates of the supermanifold. Equipped with this bracket, the algebra of functions on a supermanifold becomes an antibracket algebra.
A coordinate transformation that preserves the antibracket is called a P-transformation. If the Berezinian of a P-transformation is equal to one then it is called an SP-transformation.
### P and SP-manifolds
Using the Darboux theorem for odd symplectic forms one can show that P-manifolds are constructed from open sets of superspaces ${\mathcal{R}}^{n|n}$ glued together by P-transformations. A manifold is said to be an SP-manifold if these transition functions can be chosen to be SP-transformations. Equivalently one may define an SP-manifold as a supermanifold with a nondegenerate odd 2-form ω and a density function ρ such that on each coordinate patch there exist Darboux coordinates in which ρ is identically equal to one.
### Laplacian
One may define a Laplacian operator Δ on an SP-manifold as the operator which takes a function H to one half of the divergence of the corresponding Hamiltonian vector field. Explicitly one defines
$\Delta H=\frac{1}{2\rho}\frac{\partial_r}{\partial z^a}\left(\rho\omega^{ij}(z)\frac{\partial_l H}{\partial z^j}\right)$.
In Darboux coordinates this definition reduces to
$\Delta=\frac{\partial_r}{\partial x^a}\frac{\partial_l}{\partial \theta_a}$
where xa and θa are even and odd coordinates such that
$\omega=dx^a\wedge d\theta_a$.
The Laplacian is odd and nilpotent
$\Delta^2=0$.
One may define the cohomology of functions H with respect to the Laplacian. In Geometry of Batalin-Vilkovisky quantization, Albert Schwarz has proven that the integral of a function H over a Lagrangian submanifold L depends only on the cohomology class of H and on the homology class of the body of L in the body of the ambient supermanifold.
## SUSY
A pre-SUSY-structure on a supermanifold of dimension (n,m) is an odd m-dimensional distribution $P\subset TM$. With such a distribution one associates its Frobenius tensor $S^2 P \mapsto TM/P$ (since P is odd, the skew-symmetric Frobenius tensor is a symmetric operation). If this tensor is non-degenerate, e.g. lies in an open orbit of $GL(P)\times GL(TM/P)$, M is called a SUSY-manifold. SUSY-structure in dimension (1, k) is the same as odd contact structure.
## References
[1] Joseph Bernstein, Lectures on Supersymmetry (notes by Dennis Gaitsgory) [1], "Quantum Field Theory program at IAS: Fall Term"
[2] A. Schwarz, Geometry of Batalin-Vilkovisky quantization, hep-th/9205088
[3] C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez, The Geometry of Supermanifolds (Kluwer, 1991) ISBN 0-7923-1440-9
[4] A. Rogers, Supermanifolds: Theory and Applications (World Scientific, 2007) ISBN 981-02-1228-3
[5] L. Mangiarotti, G. Sardanashvily, Connections in Classical and Quantum Field Theory (World Scientific, 2000) ISBN 981-02-2013-8 (arXiv: 0910.0092) | {"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": 22, "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": 0.9806106686592102, "perplexity": 672.8363656055512}, "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-2015-35/segments/1440645167592.45/warc/CC-MAIN-20150827031247-00261-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://blog.plover.com/math/sqrt-2-addendum.html | # The Universe of Discourse
Thu, 26 Jan 2006
More irrational numbers
Gaal Yahas has written in with a delightfully simple proof that a particular number is irrational. Let x = log2 3; that is, such that 2x = 3. If x is rational, then we have 2a/b = 3 and 2a = 3b, where a and b are integers. But the left side is even and the right side is odd, so there are no such integers, and x must be irrational.
As long as I am on the subject, undergraduates are sometimes asked whether there are irrational numbers a and b such that ab is rational. It's easy to prove that there are. First, consider a = b = √2. If √2√2 is rational, then we are done. Otherwise, take a = √2√2 and b = √2. Both are irrational, but ab = 2.
This is also a standard example of a non-constructive proof: it demonstrates conclusively that the numbers in question exist, but it does not tell you which of the two constructed pairs is actually the one that is wanted. Pinning down the real answer is tricky. The Gelfond-Schneider theorem establishes that it is in fact the second pair, as one would expect. | {"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": 0.941335916519165, "perplexity": 263.3169071009548}, "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-2016-30/segments/1469257831771.10/warc/CC-MAIN-20160723071031-00250-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/i-have-a-conservation-of-momentum-equation-q.25491/ | # I have a conservation of momentum equation Q
1. May 13, 2004
### Divergent13
I have a conservation of momentum equation Q!!
A 0.25kg skeet (clay target) is fired at an angle of 30 degrees to the horizon with a speed of 30 m/s. When it reaches its maximum height, it is hit from below by a 15g pellet traveling vertically upward at a speed 200m/s, the pellet is imbedded in the skeet.
So they want to know how much higher the skeet would go, and how much EXTRA horizontal distance the skeet would travel because of that collision...
I believe you can get the height from using conservation of energy. Then you can use what you know about projectiile motion to find all the different time intervals involved. From there, Distance in the x direction is simply Vx*t
I am just not sure if im getting the correct numbers? What would you guys do?
2. May 13, 2004
### arildno
Step 1.
Energy conservation up to maximal height h for the skeet:
$$\frac{1}{2}V_{0}^{2}=\frac{1}{2}V_{0,x}^{2}+gh$$
($$V_{0}$$ is initial velocity, with components $$V_{0,x},V_{0,y})$$
Step 2. Conservation of momentum in inelastic collision.
This will yield a non-zero upwards velocity component.
Step 3. Book-keeping:
Record horizontal distance already traveled, $$X_{1}$$, and vertical and horizontal velocities after collision.
Step 4.
Solve for landing position as a function of time, using parameters obtained in Step 3.
3. May 13, 2004
### Theelectricchild
But since Vo x is constant why would you use that equation to solve it?
4. May 13, 2004
### arildno
Because I forgot to eliminate it on both sides of the equation..
5. May 13, 2004
### Divergent13
I get an impact height of 11.43 m. Is this correct? (Using your equation.)
6. May 13, 2004
### arildno
If $$11.43=\frac{(30\sin30)^{2}}{2g}$$, then it is correct, as long as you with "impact height" means the height of the sleet when the proctile hits it.
7. May 13, 2004
### Divergent13
Cool then I just use conservation of momentum to find the new Vx and Vy and treat it like an ordinary 2D kinematics problem.
8. May 13, 2004
### arildno
Yes; that would be it.
Similar Discussions: I have a conservation of momentum equation Q | {"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": 0.8576478362083435, "perplexity": 1297.0360986762787}, "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-2017-47/segments/1510934807650.44/warc/CC-MAIN-20171124104142-20171124124142-00146.warc.gz"} |
http://math.stackexchange.com/questions/149361/is-every-semi-simple-ring-a-product-of-simple-rings | # Is every semi-simple ring a product of simple rings
I was wondering if the following statements were true;
1) Every semi-simple ring is a product of simple rings.
2) Every module over a division ring $R$ is free.
I think both of these statements are false but cannot come up with any counterexamples. Does anyone have any ideas?
-
Both statements are true. The first follows from Artin-Wedderburn and the proof of the second is the same as the corresponding proof for fields (Zorn's lemma). – Qiaochu Yuan May 24 '12 at 20:30
If you use "semisimple" as I do (to mean "$R$ is the direct sum of its simple right ideals" or "$R$ is Artinian with $rad(R)=0$") then the Artin-Wedderburn theorem proves the first statement true.
For the second question, I assume you are familiar with the proof that every vector space over a field $F$ has a basis. Once you know that is true, and $V$ has a basis $\{v_i\mid i\in I\}$, then you can map elements of $V$ to their coefficients in $\bigoplus_{i\in I} F$ to produce an isomorphism, showing that $V$ is free. If you review try this with division rings, you will find that commutativity was not necessary, and everything goes through here as well.
You also can try to work out the converse: if all right $R$ modules are free, then $R$ is a division ring. | {"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": 0.8427186012268066, "perplexity": 104.0037981855936}, "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-2016-26/segments/1466783392159.3/warc/CC-MAIN-20160624154952-00195-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/106719/nearly-constant-curvature-implies-nearly-isometric-to-a-space-form/106728 | # Nearly constant curvature implies “nearly isometric” to a space form?
It is well known a Riemannian manifold with constant sectional curvature is a quotient of the Euclidean space, hyperbolic space or sphere. In particular we know how their metric looks like locally.
My question is, is there a quantitative version of the above result? By this, I mean for example, given $\varepsilon>0$, there exists $\delta$ such that if $(M,g)$ satisfies $\left|Rm\right|_g<\delta$ `, does there exist a local diffeomorphism $\phi$ to $\mathbb R^n$ such that $|\phi_*g-g_0|_{g_0}< \varepsilon$, where $g_0$ is the standard metric on $\mathbb R^n$?
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This question is two-sided, and I'm not sure what you mean by "local diffeomorphism" so I'll treat both aspects. There is a local and a global version :
## Local version :
Q1 : Given a point $x$ in a Riemannian manifold $(M,g)$, can we find a constant curvature metric on a neighborhood of $x$ which is close to $g$ ?
First remark : since we want a local statement, zero curvature is as good as constant curvature here.
The answer to Q1 is "yes", without any restriction on the curvature. This can be seen using normal coordinates centered at $x$ : in these coordinates, the metric at $x$ is euclidean and its distortion from being euclidean as one moves away from $x$ can be controlled (using that the curvature is bounded in a neighborhood of $x$, and that the injectivity radius at $x$ is positive).
Edit : The above statement is too complicated. As Anton's say in his comment to Agol's answer, use the exponential map ! The pull back of $g$ by the exponential map defines a riemannian metric on some neighborhood of the origin in $T_xM$ which is equal to $g_x$ at $x$, by continuity, this pullback stay close to the euclidean metric $g_x$ on $T_xM$ and this does the job.
We can refine the question then :
Q1' : What can be said about the size of the neighborhood we obtained ?
In this case we need to impose geometric restrictions on $(M,g)$. For instance, Cheeger and Anderson proved the following :
For $n\in\mathbb{N}$, $k\in\mathbb{R}$, $i>0$ and $\varepsilon>0$, one can find $\delta>0$ such that in a $n$-manifold of Ricci curvature greater than $k$ and injectivity radius greater than $i$, any ball of radius $\delta$ admits a flat metric which is $\varepsilon$-close to $g$ in $C^0$-norm.
See "$C^\alpha$-compactness for manifolds with Ricci curvature and injectivity radius bounded below."
The proof uses more elaborate machinery than just normal coordinates : harmonic coordinates are used. If you stick to normal coordinates you can obtain a similar result but with stronger geometric assumptions.
## Global version :
Q2 : Under which condition does a manifold with almost $k$ curvature admit a $C^0$-close metric of constant curvature $k$ ?
If you consider large spheres, they have almost zero cuvrvature but don't admit any flat metric, so you need to put some restrictions on the side of the manifolds.
An example of theorem you can get is the following :
For any $n\in\mathbb{N}$, $k\in\mathbb{R}$, $V>0$, $D>0$ and $\varepsilon>0$, there is a $\delta>0$ such that any $n$-manifold $(M,g)$ of diameter less than $D$, volume more than $V$, and sectional curvature between $k-\delta$ and $k+\delta$ admits a metric af constant sectional curvature $k$ which is $\varepsilon$-close to $g$ in $C^0$-norm.
The proof relies on Cheeger-Gromov compactness theorem for sequences of Riemannian manifolds. A (really) sketchy goes like that : we argue by contradiction, you take a sequence $\delta_i$ going to $0$, and you assume you can find a sequence of manifolds $(M_i,g_i)$ satisfying the hypothesis of the theorem with $\delta=\delta_i$ and not satisfying the conclusion of the theorem. Then up to a subsequence, the sequence has a limit which is a manifold of constant curvature $k$, by the very definition of Cheeger-Gromov convergence, this imply the for some $i$ large enough, $M_i$ admit a constant curvature $k$ metric $\varepsilon$-close to $g_i$, a contradiction.
The lower bound on the volume is necessary (at least in the $k=0$ case) the so called "infranilmanifolds" admit metrics of curvature as close as wanted to $0$ with diameter bounded above but no flat metric.
For the $k=1$ case, the bounds on the diameter is unnecessary because of Myers theorem.
For the $k=-1$ case, I don't know if the hypothesis can be weakened.
-
For $k=-1$ you do not need the lower volume bound, but you need the upper diameter bound. See Pinching constants for hyperbolic manifolds. by Gromov and Thurston math.psu.edu/petrunin/teach-old/minicourse-china-2008/… – Anton Petrunin Sep 9 '12 at 16:15
Thnks ! It's funny that the situation is opposite to that of positive curvaure. Another question came to my mind about the positive case : if we assume that $M$ is simply connected, then the lower bound on the volume isn't necessary (Klingenberg's Lemma). With Synge, it also shows that the volume is not needed in even dimensions. What about odd dimensions ? – Thomas Richard Sep 9 '12 at 16:27
Thanks for the very detailed reply! The question in the "global version" comes close to what's in my mind. Can you suggest a reference for me to look up your stated (or similar) result? (I am indeed not very familiar with Cheeger-Gromov compactness, but I can take it for granted and take a look at the proof of these types of results. ) – Kwong Sep 10 '12 at 13:40
In fact, almost everything is in the proof of the Cheeger-Gromov compactness theorem. The only additional observation is to show that the limit space has constant curvature, which is not that obvious because the limit space is only a $C^{1,\alpha}$ riemannian manifold. The trick is to see that the comparison results with spaceforms of constant curvature $k$ hold in both directions, and to use this to build a local isometry with a constant curvature space. For the proof of Cheeger-Gromov compactness, see the papers by Greene and Wu, and by S. Peters. I don't remember the titles, I'll check. – Thomas Richard Sep 11 '12 at 17:52
There's an unpublished preprint of Tian which gives a criterion on when a manifold is close to being Einstein may actually be deformed to being Einstein (see Theorem 6.1 of his paper - sorry, this copy has only odd pages!). In 3 dimensions, this means that if one is close to being hyperbolic in his sense, then there is a deformation of the metric to a hyperbolic metric. '
There's also the $1/4$ pinching theorem of Brendle-Schoen.
-
Ian, the answer to Kwong's question is "exponential map" and you are answering different (and more advanced) question. – Anton Petrunin Sep 9 '12 at 13:19
@Anton : I didn't see your comment when I was writing my answer. – Thomas Richard Sep 9 '12 at 13:27
For the global question, and hyperbolic metrics, in dimension > 3 this is a result of Gromov, stated in his 1978 JDG paper, and in dimension 3 it is an unpublished result of Daryl Cooper, from the late nineties, and Gromov, independently, so while Tian might have a more general result, he is far from the first.
- | {"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": 0.9713099002838135, "perplexity": 206.90902354161832}, "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-2016-26/segments/1466783392069.78/warc/CC-MAIN-20160624154952-00092-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/145239/a-dynamic-programming-problem-with-continuous-states-and-observations | # A dynamic programming problem with continuous states and observations
I have a dynamic programming expressed in the following Bellman backup equation form,
$$V(\boldsymbol{\theta},T)=\max_{i \in N} \mathbb{E} \left[ x_i + V(\boldsymbol{\theta}_{x_i}, T-1) \right]$$
where $\theta_i$ is the expectation of $x_i$, and the $\boldsymbol{\theta}$ vector is updated with observation $x_i$ using Bayes rules.
So the backup could be expanded recursively as,
\begin{align} V(\boldsymbol{\theta},T)&=\max_{i \in N} \theta_i + \int p(x_i) V(\boldsymbol{\theta}_{x_i}, T-1) dx\\ &=\max_{i \in N} \theta_i + \int p(x_i) \max_{j \in N} \left( \theta_{j,x_i} + V(\boldsymbol{\theta}_{x_i, x_j}, T-2) \right) dx\\ &=\dots \end{align}
The integral part, for example,
$$\int p(x) \max_{j\in N} \theta_{j, x} dx$$ is complicated because the integrand is changing with different $x$, although $\theta$ is simply a linear function of $x$. I can surely find a section where certain $\theta$ should be choosed via max operator for the above equation, by solving a group of inequalities. But for the whole $T$ iteration it becomes impossible.
I don't know how I can solve this equation, or approximate the optimal solution. Any help is appreciated! Thanks!
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An inequity is an injustice; you mean inequalities. – joriki May 15 '12 at 0:35
@joriki Typo edited. Thanks. – shuaiyuancn May 16 '12 at 8:07 | {"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": 0.9988376498222351, "perplexity": 1424.1569844381352}, "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/1438042987034.19/warc/CC-MAIN-20150728002307-00290-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.math.nyu.edu/dynamic/calendars/seminars/analysis-seminar/1279/ | # Analysis Seminar
#### Instability, Index Theorems, and Exponential Dichotomy of Hamiltonian PDEs
Speaker: Chongchun Zeng, Georgia Tech
Location: Warren Weaver Hall 1302
Date: Thursday, April 6, 2017, noon
Synopsis:
Motivated by the stability/instability analysis of coherent states (standing waves, traveling waves, etc.) in nonlinear Hamiltonian PDEs such as BBM, GP, and 2-D Euler equations, we consider a general linear Hamiltonian system $$u_t = JL u$$ in a real Hilbert space $$X$$ -- the energy space. The main assumption is that the energy functional $$\frac 12 \langle Lu, u\rangle$$ has only finitely many negative dimensions -- $$n^-(L) < \infty$$. Our first result is an $$L$$-orthogonal decomposition of $$X$$ into closed subspaces so that $$JL$$ has a nice structure. Consequently, we obtain an index theorem which relates $$n^-(L)$$ and the dimensions of subspaces of generalized eigenvectors of some eigenvalues of $$JL$$, along with some information on such subspaces. Our third result is the linear exponential trichotomy of the group $$e^{tJL}$$. This includes the nonexistence of exponential growth in the finite co-dimensional invariant center subspace and the optimal bounds on the algebraic growth rate there. Next we consider the robustness of the stability/instability under small Hamiltonian perturbations. In particular, we give a necessary and sufficient condition on whether a purely imaginary eigenvalues may become hyperbolic under small perturbations. Finally we revisit some nonlinear Hamiltonian PDEs. This is a joint work with Zhiwu Lin. | {"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": 0.9534855484962463, "perplexity": 470.4902471047502}, "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-2018-13/segments/1521257650685.77/warc/CC-MAIN-20180324132337-20180324152337-00100.warc.gz"} |
http://www.zora.uzh.ch/16558/ | # Formation and accretion history of terrestrial planets from runaway growth through to late time: implications for orbital eccentricity - Zurich Open Repository and Archive
Morishima, R; Schmidt, M W; Stadel, J; Moore, B (2008). Formation and accretion history of terrestrial planets from runaway growth through to late time: implications for orbital eccentricity. Astrophysical Journal, 685(2):1247-1261.
## Abstract
Remnant planetesimals might have played an important role in reducing the orbital eccentricities of the terrestrial planets after their formation via giant impacts. However, the population and the size distribution of remnant planetesimals during and after the giant impact stage are unknown, because simulations of planetary accretion in the runaway growth and giant impact stages have been conducted independently. Here we report results of direct N-body simulations of the formation of terrestrial planets beginning with a compact planetesimal disk. The initial planetesimal disk has a total mass and angular momentum as observed for the terrestrial planets, and we vary the width (0.3 and 0.5 AU) and the number of planetesimals (1000-5000). This initial configuration generally gives rise to three final planets of similar size, and sometimes a fourth small planet forms near the location of Mars. Since a sufficient number of planetesimals remains, even after the giant impact phase, the final orbital eccentricities are as small as those of the Earth and Venus.
## Abstract
Remnant planetesimals might have played an important role in reducing the orbital eccentricities of the terrestrial planets after their formation via giant impacts. However, the population and the size distribution of remnant planetesimals during and after the giant impact stage are unknown, because simulations of planetary accretion in the runaway growth and giant impact stages have been conducted independently. Here we report results of direct N-body simulations of the formation of terrestrial planets beginning with a compact planetesimal disk. The initial planetesimal disk has a total mass and angular momentum as observed for the terrestrial planets, and we vary the width (0.3 and 0.5 AU) and the number of planetesimals (1000-5000). This initial configuration generally gives rise to three final planets of similar size, and sometimes a fourth small planet forms near the location of Mars. Since a sufficient number of planetesimals remains, even after the giant impact phase, the final orbital eccentricities are as small as those of the Earth and Venus.
## Citations
35 citations in Web of Science®
33 citations in Scopus®
## Altmetrics
Detailed statistics
Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute for Computational Science 530 Physics English October 2008 06 Mar 2009 10:36 05 Apr 2016 13:06 Institute of Physics Publishing 0004-637X Publisher DOI. An embargo period may apply. https://doi.org/10.1086/590948 http://arxiv.org/abs/0806.1689 | {"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": 0.9642753601074219, "perplexity": 2020.7978258671155}, "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-2017-17/segments/1492917124478.77/warc/CC-MAIN-20170423031204-00597-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/does-the-probability-collapse-theory-of-qm-imply-backward-in-time.861651/ | I Does the probability collapse theory of QM imply backward in time
1. Mar 11, 2016
fredt17
In the thought experiment known as Schrodinger's cat a cat is placed in a sealed box, and its life or death is tied to an uncertain quantum event such as radioactive decay. If the radioactive particle decays, the cat dies. If not, nothing happens.
According to probability collapse QM, as I understand it, the cat is in an uncertain state until we open the box and collapse the probability wave created by the quantum event. But what if we wait ten days to open the box? Will we discover that, if the cat died, its body has decomposed for up to ten days?
But when did the ten days of decomposition occur (or become certain)? Does probability collapse theory claim that the death of the cat does not occur (or become certain) until we open the box and the decay too is uncertain even though the biological process will appear to have taken up to ten days?
Or does the probability collapse theory of QM imply a backward in time causation?
2. Mar 11, 2016
Staff: Mentor
Sure.
They occured during the ten days - in parallel to the cat living there for 10 days. They became certain when you opened the box.
Anyway, living systems are too complex for such a superposition to happen. With some atoms, that scenario is possible.
No.
3. Mar 11, 2016
StevieTNZ
In 'Quantum Enigma' by Bruce Rosenblum and Fred Kuttner, they describe a measurement occurring. Using your example, ten days after setting up the experiment, upon measurement the appropriate history of the quantum systems is created. In principle, macroscopic objects (despite experiencing decoherence, which does -not- cause a definite reality to arise [only -apparent- collapse) are quantum systems also.
4. Mar 12, 2016
the_pulp
Depending on the interpretation. In time symmetric interpretation the answer is yes. The beta decay interact with a decomposed cat and that decomposed cat transforms continuously backward in time into a present alive cat. Of course in not being too precise but this is the general idea within this interpretation. The"no"answer is also possible and it is related to the more traditional collapse interpretation. I tend to like more the time symmetric interpretation because it preserves locality and determinism (but loses causality -as the cat"first decomposes and then goes back to life backward in time", you know what I mean-). Anyway is just a matter of taste.
5. Mar 12, 2016
the_pulp
Sorry I wrote something wrong. What I tried to s say, generally speaking, is that we open the box and we interact with a decomposed dead cat which transforms continuously backward in time in a dead not decomposed cat which interacts with beta decay and transforms backward in time in a present alive carry. Sorry!!
6. Mar 12, 2016
Demystifier
No, we are in an uncertain state. The cat is in a certain state.
7. Mar 12, 2016
fredt17
:
Well, my understanding (like the pulp and StevieTNZ) is that the point of the Schrodinger's Cat experiment was to tie a micro event (such as the death of a cat) to a quantum event, and so the complexity of the biological event is irrelevant. Thus, the probability theory of QM does imply backward in time causation, or perhaps more accurately, that time is suspended in the quantum system until we measure the system. Have I got that right?
8. Mar 12, 2016
David Lewis
The backward-in-time interpretation for Schroedinger's Cat Paradox was detailed in a book by John Gribbin (Schroedinger's Kittens).
9. Mar 12, 2016
Staff: Mentor
It is still relevant. You get decoherence.
The time-symmetric interpretation is one of many. You do not need backwards causation and most interpretations do not have that.
10. Mar 13, 2016
AlexCaledin
Perhaps it's better to think of the quantum system as of an essentially spatiotemporal (existing in its space-time) object ?
It can be considered as a superposition of histories - until it (or our uncertainty?) is reduced by observation to one "actual" decoherent history.
Our problem seems to be the habit of imposing temporal evolution on Nature too much...
Last edited: Mar 13, 2016
11. Mar 13, 2016
Demystifier
That is right only in the interpretation of QM that stipulates that everything (not only time) is suspended until we measure the system.
12. Mar 13, 2016
eloheim
Which interpretation is this? Doesn't evolution within the isolated system continue until you open the box?
13. Mar 14, 2016
Demystifier
Draft saved Draft deleted
Similar Discussions: Does the probability collapse theory of QM imply backward in time | {"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": 0.8066986799240112, "perplexity": 1778.2973847943813}, "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/1508187825147.83/warc/CC-MAIN-20171022060353-20171022080353-00300.warc.gz"} |
https://mathzsolution.com/compact-sets-are-closed/ | # Compact sets are closed?
I feel really ignorant in asking this question but I am really just don’t understand how a compact set can be considered closed.
By definition of a compact set it means that given an open cover we can find a finite subcover the covers the topological space.
I think the word “open cover” is bothering me because if it is an open cover doesn’t that mean it consists of open sets in the topology? If that is the case how can we have a “closed compact set”?
I know a topology can be defined with the notion of closed sets rather than open sets but I guess I am just really confused by this terminology. Please any explanation would be helpful to help clear up this confusion. Thank you!
I think that what you’re missing is that an open cover of a compact set can cover more than just that set. Let $X$ be a topological space, and let $K$ be a compact subset of $X$. A family $\mathscr{U}$ of open subsets of $X$ is an open cover of $K$ if $K\subseteq\bigcup\mathscr{U}$; it’s not required that $K=\bigcup\mathscr{U}$. You’re right that $\bigcup\mathscr{U}$, being a union of open sets, must be open in $X$, but it needn’t be equal to $K$.
For example, suppose that $X=\Bbb R$ and $K=[0,3]$; the family $\{(-1,2),(1,4)\}$ is an open cover of $[0,3]$: it’s a family of open sets, and $[0,3]\subseteq(-1,2)\cup(1,4)=(-1,4)$. And yes, $(-1,4)$ is certainly open in $\Bbb R$, but $[0,3]$ is not.
Note, by the way, that it’s not actually true that a compact subset of an arbitrary topological space is closed. For example, let $\tau$ be the cofinite topology on $\Bbb Z$: the open sets are $\varnothing$ and the sets whose complements in $\Bbb Z$ are finite. It’s a straightforward exercise to show that every subset of $\Bbb Z$ is compact in this topology, but the only closed sets are the finite ones and $\Bbb Z$ itself. Thus, for example, $\Bbb Z^+$ is a compact subset that isn’t closed. | {"extraction_info": {"found_math": true, "script_math_tex": 26, "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": 0.9215176105499268, "perplexity": 55.263594546376915}, "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/1664030337490.6/warc/CC-MAIN-20221004085909-20221004115909-00185.warc.gz"} |
http://talkstats.com/search/583729/ | # Search results
1. ### Logistic Regression: Evaluate model & Check for Multicollinearity
Thank you! Where can I find the calibration and c-statistics?
2. ### Mean center binary categorical variables for logistic regression?
I am wondering if mean-centering is possible, makes sense and needs to be or should not be done for a logistic regression with interaction term and only binary categorical variables. They all have values of 0 or 1. Is it technically possible to mean center (because there is no mean for...
3. ### Logistic Regression: Evaluate model & Check for Multicollinearity
Hello, I have conducted a logistic regression with a binary categorical outcome and a binary categorical moderator + binary categorical independent variable (or more specific: two independent variables and in the logistic regression entered the interaction factor). How do I now check how good...
4. ### Using PROCESS macro vs. logistic regression
Ok, thank you for your answer. I was not sure if it yields the same results as in the PROCESS tool that officially measures moderation effects. Do you have any info about that?
5. ### Using PROCESS macro vs. logistic regression
Hi all, I am trying to investigate a moderation effect of a binary categorical moderator variable on the effect of another independent variable (categorical, binary) on a categorical binary outcome variable. With binary categorical I mean that the variables can have basically two values (0 and... | {"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": 0.8445343375205994, "perplexity": 1602.3953087126254}, "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/1560628000266.39/warc/CC-MAIN-20190626094111-20190626120111-00212.warc.gz"} |
https://study.com/academy/answer/a-2-50-kg-fireworks-shell-is-fired-straight-up-from-a-mortar-and-reaches-a-height-of-110-m-a-neglecting-air-resistance-a-poor-assumption-but-we-will-make-it-for-this-example-calculate-the-shell-s-velocity-when-it-leaves-the-mortar-b-the-mortar.html | # A 2.50 kg fireworks shell is fired straight up from a mortar and reaches a height of 110. m. (a)...
## Question:
A 2.50 kg fireworks shell is fired straight up from a mortar and reaches a height of 110. m.
(a) Neglecting air resistance (a poor assumption, but we will make it for this example), calculate the shell's velocity when it leaves the mortar.
(b) The mortar itself is a tube 0.450 m long. Calculate the average acceleration of the shell in the tube as it goes from zero to the velocity found in (a).
(c) What is the average net force on the shell in the mortar? How does this force compare to the weight of the shell?
## Acceleration:
Assume a particle on which a force is imposed and the velocity of the particle changes constantly. In this scenario, the rate of change of velocity for the particle would be termed as its acceleration.
Given data:
• Mass of the shell, {eq}m = 2.50 \ kg {/eq}
• Height, {eq}h = 110 \ m {/eq}
• Length of the barrel, {eq}d = 0.450 \ m {/eq}
• Initial speed of the shell in the barrel, {eq}u = 0 {/eq}
Part (a):
Let the lauched speed of the sheel be v.
From the conservation law o mechanical energy,
{eq}\begin{align*} \frac{1}{2}mv^{2} &= mgh\\ \Rightarrow \ v &= \sqrt{2gh}\\ v &= \sqrt{2 \times 9.80 \times 110}\\ v &= 46.43 \ m/s.\\ \end{align*} {/eq}
Part (b):
The average acceleration of the sheel can be given as,
{eq}\begin{align*} a &= \frac{v^{2}-u^{2}}{2d}\\ a &= \frac{46.43^{2}-0^{2}}{2 \times 0.450}\\ a &= 2395.27 \ m/s^{2}.\\ \end{align*} {/eq}
Part (c):
The average net force on sheel can be given as,
{eq}\begin{align*} F &= ma\\ F &=2.50 \times 2395.27 \\ F &= 5988.18 \ \rm N.\\ \end{align*} {/eq}
On comparing this force with the weight of the sheel,
{eq}\begin{align*} \frac{F}{W} &= \frac{5988.18 }{2.50 \times 9.80}\\ F &= 244.41 W.\\ \end{align*} {/eq} | {"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": 1.000004768371582, "perplexity": 3028.9997328279783}, "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-2020-24/segments/1590347388758.12/warc/CC-MAIN-20200525130036-20200525160036-00111.warc.gz"} |
http://stats.stackexchange.com/questions/41891/find-the-partial-correlation-coefficient-r-1p-2468 | # Find the partial correlation coefficient $r_{1p.2468}.$
Suppose all the simple correlations between $x_i$ and $x_j$ are $r$ for all $i,j=1,2,\dots,p, i\neq j.p>8$. Find the partial correlation coefficient $r_{1p.2468}.$
By definition, $$r_{1p.2468}=\frac{cov(e_{1.2468},e_{p.2468})}{\sqrt{var(e_{1.2468})}\sqrt{var(e_{p.2468})}}$$ how can I find $cov(e_{1.2468},e_{p.2468}),var(e_{1.2468}),var(e_{p.2468})$ in terms of $r$?
- | {"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": 0.8662495017051697, "perplexity": 155.1374746629457}, "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-2013-48/segments/1386164936474/warc/CC-MAIN-20131204134856-00059-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://stacks.math.columbia.edu/tag/01CX | Definition 17.24.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. The Picard group $\mathop{\mathrm{Pic}}\nolimits (X)$ of $X$ is the abelian group whose elements are isomorphism classes of invertible $\mathcal{O}_ X$-modules, with addition corresponding to tensor product.
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": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9648633003234863, "perplexity": 370.1134911831945}, "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-2021-49/segments/1637964362891.54/warc/CC-MAIN-20211203151849-20211203181849-00637.warc.gz"} |
http://mathhelpforum.com/geometry/25167-circle-theorems-print.html | # Circle theorems
• Dec 21st 2007, 09:43 AM
Geometor
Circle theorems
Hi, could anybody help me with this?
the question:
what is the distance from the centre (O) to C
http://img45.imageshack.us/img45/678/21870685kw9.png
if you need any clarification just ask
Any help appreciated!
• Dec 21st 2007, 10:09 AM
Plato
1 Attachment(s)
Looking at the modified drawing, recall the secant theorem.
(AC)(BC)=(EC)(DC). Then solve for x.
• Dec 21st 2007, 10:28 AM
Soroban
Hello, Geometor!
I have a solution . . . hope it's acceptable.
Quote:
What is the distance from the centre (O) to C?
http://img45.imageshack.us/img45/678/21870685kw9.png
Let $x \:=\:OC.$
Draw an altitude from $O$ to $AB$; call it $OD.$
In right triangle $ODA,\:OA = 5,\:AD = 4\quad\Rightarrow\quad \cos A \:=\:\frac{4}{5}$
Law of Cosines: . $OC^2 \;=\;AC^2 + OA^2 - 2(AC)(OA)\cos A$
So we have: . $x^2 \;=\;18^2 + 5^2 - 2(18)(5)\left(\frac{4}{5}\right) \;=\;205$
Therefore: . $x\;=\;\sqrt{205}$
• Dec 21st 2007, 10:33 AM
Geometor
thank you plato!
and soroban!
however i find using plato's secant theorem being the easier:
since:
(AC)(BC)=(EC)(DC).
(18)(10)=(10+x)(x)
=x^2+10x
x^2+10x-180=0
(-10 +/- sq.root 82) / 2
and since we need a positive value we do:
(-10 + sq.root 82) / 2
= 9.317821063....
and add the 5cm radius to get OC
= 14.3cm(1dp)
=sq. root 205 as soroban said :D
Thanks for the help!
• Dec 21st 2007, 12:05 PM
JaneBennet
Quote:
Originally Posted by Soroban
Hello, Geometor!
I have a solution . . . hope it's acceptable.
Let $x \:=\:OC.$
Draw an altitude from $O$ to $AB$; call it $OD.$
In right triangle $ODA,\:OA = 5,\:AD = 4\quad\Rightarrow\quad \cos A \:=\:\frac{4}{5}$
Law of Cosines: . $OC^2 \;=\;AC^2 + OA^2 - 2(AC)(OA)\cos A$
So we have: . $x^2 \;=\;18^2 + 5^2 - 2(18)(5)\left(\frac{4}{5}\right) \;=\;205$
Therefore: . $x\;=\;\sqrt{205}$
That is also my method except that I didn’t use trigonometry. By Pythagoras’ theorem on triangle ODA, OD = $\sqrt{5^2-4^2}$ = 3 cm. By Pythagoras’ theorem on triangle ODC, OC = $\sqrt{(4+10)^2+3^2}=\sqrt{205}$ cm.
Quote:
Originally Posted by Geometor
thank you plato!
and soroban!
however i find using plato's secant theorem being the easier:
since:
(AC)(BC)=(EC)(DC).
(18)(10)=(10+x)(x)
=x^2+10x
x^2+10x-180=0
(-10 +/- sq.root 82) / 2
and since we need a positive value we do:
(-10 + sq.root 82) / 2
= 9.317821063....
and add the 5cm radius to get OC
= 14.3cm(1dp)
=sq. root 205 as soroban said
Thanks for the help!
You made a mistake there; it should be 820, not 82. It’s also easier to make mistakes with your calculations using the secant method. My recommendation: use the method that Soroban and I used. :rolleyes:
• Dec 21st 2007, 12:58 PM
Geometor
haha thanks for pointing it out :D
i wrote 820 in my calculations though phew
• Dec 21st 2007, 04:04 PM
loui1410
Quote:
Originally Posted by Soroban
Hello, Geometor!
I have a solution . . . hope it's acceptable.
Let $x \:=\:OC.$
Draw an altitude from $O$ to $AB$; call it $OD.$
In right triangle $ODA,\:OA = 5,\:AD = 4\quad\Rightarrow\quad \cos A \:=\:\frac{4}{5}$
Law of Cosines: . $OC^2 \;=\;AC^2 + OA^2 - 2(AC)(OA)\cos A$
So we have: . $x^2 \;=\;18^2 + 5^2 - 2(18)(5)\left(\frac{4}{5}\right) \;=\;205$
Therefore: . $x\;=\;\sqrt{205}$
How do you know AD=4?
• Dec 21st 2007, 04:08 PM
JaneBennet
Because OAB is an isosceles triangle (so the perpendicular from O to AB bisects AB).
• Dec 21st 2007, 04:09 PM
loui1410
Oh right, sorry :o it's 2:07 AM here lol
• Dec 21st 2007, 05:25 PM
Plato
Quote:
Originally Posted by JaneBennet
My recommendation: use the method that Soroban and I used.
Reading the title of the posting “Circle Theorems” why would you recommend against using a very basic theorem about circles? The approach that you advocate is not unique to theorems about circles but rather belongs to the general discussion about triangles.
• Dec 21st 2007, 06:04 PM
JaneBennet
I didn’t realize you had to use the theorem to solve this problem. My apologies. | {"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": 26, "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": 0.9321427345275879, "perplexity": 4371.221648990578}, "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-2016-50/segments/1480698541864.24/warc/CC-MAIN-20161202170901-00139-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/147490/sequential-probability-ratio-test-or-other-sequential-sampling-techniques-for | # Sequential Probability Ratio Test (or other Sequential Sampling techniques) for testing difference
I have the results from running two algorithms and I want to be able to say that there is, say, a 95% probability that one of the sets of results is different to the other where different means A > B or B > A ("better" or "worse" in practical terms).
Basically I want to try and reject the null hypothesis that both sets of results are drawn from the same distribution in the same manner as a 2 tailed T Test or Wilcoxon test (yes I know there is a slight difference in the null hypothesis between parametric and non parametric but that's not important right now).
I want to do this with sequential sampling in which you run an initial, for example, 20 runs, carry out the test, and if there's no significant difference yet you run another run of each and repeat. The sequential sampling technique I can find the most information on is the Sequential Probability Ratio Test:
Although if people know of an alternative way of achieving the same basic goal of minimizing number of runs to prove significance that would also be helpful.
For SPRT Log L(...)/L(....) in those slides is the log likelihood ratio and my problem is I have no idea how to calculate it. It seems to be the probability of your data given the alternative hypothesis divided by the prob given the null hypothesis - but when your alternative hypothesis is just that the two data sets are different there's not enough info for you to actually calculate this probability. I'm getting the feeling that I was misled to believe SPRT was designed for this sort of hypothesis testing. So if anyone would be so kind as to either confirm that this isn't what SPRT is for or give me a concrete example for how to do this with SPRT or suggest an alternative then any info appreciated!
I should point out that in this case I am assuming that you have a rough idea of the distribution of the data (e.g. normally distributed).
Many thanks!
• For Binomial data, formulas and a simple worked example are provided by William Q. Meeker Jr., A Conditional Sequential Test for the Equality of Two Binomial Proportions. Appl. Stat. (1981) 30, No. 2, pp 109-115 (available at JSTOR). I recently applied this to a Web "AB test" involving very long sequences; it performed as claimed and was reasonably efficient to compute.
– whuber
Apr 21, 2015 at 15:54 | {"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": 0.885973334312439, "perplexity": 256.2847708259822}, "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-2022-21/segments/1652662521041.0/warc/CC-MAIN-20220518021247-20220518051247-00261.warc.gz"} |
http://mathhelpforum.com/differential-geometry/125783-complex-integrals.html | # Math Help - complex integrals
1. ## complex integrals
hello,
I wonder if someone can help me solve:
1.what are the singularic points of f(z)
2.how to solve the integral
thank's
Attached Thumbnails
2. Originally Posted by avazim
hello,
I wonder if someone can help me solve:
1.what are the singularic points of f(z)
2.how to solve the integral
thank's
z = 0 is a pole of order 3.
3. i still not so understand why it is a pol in order 3?
and after that to sole the intgeral i just need to find 2*pi*i*RES(f,0); 0 as order 3 ?
4. Originally Posted by avazim
i still not so understand why it is a pol in order 3?
and after that to sole the intgeral i just need to find 2*pi*i*RES(f,0); 0 as order 3 ?
To see why z = 0 is a pole of order 3, substitute the Maclaurin series for sin z and simplify .... | {"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": 0.8048449754714966, "perplexity": 1807.812230862885}, "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/1438042989331.34/warc/CC-MAIN-20150728002309-00136-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/31749/explanation-of-notation-a-probability-space-equipped-with-measure-p/31754 | # Explanation of notation: a probability space equipped with measure P( . )
In a lecture I attended today, the professor made an off-hand comment of:
"Suppose we have the set $S_n$ of permutations of $\{1, 2, ..., n\}$, which we can think of as a probability space equipped with measure $P( . )$."
I'm not sure what this means - does it mean we have a probability of picking a random permutation with some probability, or something different...?
-
I assume it means that the probability of picking a given permutation is $\frac{1}{n!}$. – Qiaochu Yuan Apr 8 '11 at 15:54
Any finite set $S$ can be equipped with a natural probability measure $P\$ by setting, for any subset $A\subseteq S$,
$$P(A)={\mbox{number of elements in }A\over \mbox{number of elements in }S}.$$
This corresponds to selecting an item from $S$ uniformly or at random. I suspect that your professor was thinking of applying this idea to the set of permutations $S_n$.
@Undercover Mathematician Yes, some people use a dot as a place holder for a variable. Occasionally you see $f(\cdot)$ instead of $f(x)$, for instance. – Byron Schmuland Apr 9 '11 at 13:37 | {"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": 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": 0.9862378239631653, "perplexity": 258.3140440315493}, "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/1461860110356.23/warc/CC-MAIN-20160428161510-00188-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/116881/evaluate-sum-n-1-infty-ln-left-frac7n17n-right | # Evaluate $\sum_{n=1}^{\infty }\ln \left (\frac{7^n+1}{7^n} \right )$
Evaluate $\sum_{n=1}^{\infty }\ln \left (\frac{7^n+1}{7^n} \right )$ .
Found this question on Art of Problem Solving. It was stuck in the "solved" section, but I couldn't find a solution, and I myself am stumped.
Apparently it could also be simplified to $\sum_{k=0}^{\infty }\frac{\left ( -1 \right )^{k+1}}{k\left ( 7^{k}-1 \right )}$ , but I don't follow this either.
-
AOPS say $$\sum_{n=1}^{\infty} \ln \left( \frac{7^n+1}{7^n} \right)$$ – user17762 Mar 6 '12 at 0:01
@SivaramAmbikasaran: I see. Very strange of OP to miss that! – Aryabhata Mar 6 '12 at 0:10
You'll want $k$ to go from $1$ to $\infty$, not from $0$. – Robert Israel Mar 6 '12 at 2:51
@SivaramAmbikasaran I'm new to LaTeX. Give me a bit of leniency. :P – badreferences Mar 6 '12 at 17:54
## 2 Answers
This is $$\log \prod_{n=1}^\infty (1 + 7^{-n}) = \log \phi(1/49) - \log \phi(1/7)$$
where $$\phi(q) = \prod_{n=1}^\infty (1 - q^n)$$ is the Euler function. I doubt that you can get a much simpler "closed form" than that.
-
Using the inequality
$$\log (1+x) \ge x - \frac{x^2}{2}$$
we see that the series diverges: $$\log\left(\frac{7n + 1}{7n}\right) \ge \frac{1}{7n} - \frac{1}{98n^2}$$
EDIT:
If the series is $\sum_{n=1}^{\infty}\log\left(\frac{7^n + 1}{7^n}\right)$ (as pointed out in the comments), then, using the Taylor series expansion of $\log(1+x)$:
$$\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$$
and the geometric series sum
$$\sum_{n=1}^{\infty} r^n = \frac{r}{1-r}$$
we get the sum which you state.
(Of course, that would need some justification, but I believe it is doable).
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https://eprints.lancs.ac.uk/id/eprint/63441/ | Q-ball formation in the wake of Hubble-induced radiative corrections
Allahverdi, Rouzbeh and Mazumdar, Anupam and Ozpineci, Altug (2002) Q-ball formation in the wake of Hubble-induced radiative corrections. Physical Review D, 65 (12). ISSN 1550-7998
Preview
PDF
PhysRevD.65.125003.pdf - Published Version
Abstract
We discuss some interesting aspects of the $\rm Q$-ball formation during the early oscillations of the flat directions. These oscillations are triggered by the running of soft $({\rm mass})^2$ stemming from the nonzero energy density of the Universe. However, this is quite different from the standard $\rm Q$-ball formation. The running in presence of gauge and Yukawa couplings becomes strong if $m_{1/2}/m_0$ is sufficiently large. Moreover, the $\rm Q$-balls which are formed during the early oscillations constantly evolve, due to the redshift of the Hubble-induced soft mass, until the low-energy supersymmtery breaking becomes dominant. For smaller $m_{1/2}/m_0$, $\rm Q$-balls are not formed during early oscillations because of the shrinking of the instability band due to the Hubble expansion. In this case the $\rm Q$-balls are formed only at the weak scale, but typically carry smaller charges, as a result of their amplitude redshift. Therefore, the Hubble-induced corrections to the flat directions give rise to a successful $\rm Q$-ball cosmology.
Item Type:
Journal Article
Journal or Publication Title:
Physical Review D | {"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": 0.8407973647117615, "perplexity": 1858.213251685402}, "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-2020-45/segments/1603107866404.1/warc/CC-MAIN-20201019203523-20201019233523-00499.warc.gz"} |
https://physicscatalyst.com/magnetism/diamagnetism.php | # Diamagnetism
## Diamagnetism
• Diamagnetic effects occurs in materials where magnetic field due to electronic motions i.e orbiting and spinning completely cancels each other
• Thus for diamagnetic materials intrinsic magnetic moments of all the atoms is zero and such materials are weakly affected by the magnetic field
• The diamagnetic effects in material is a result of inductive action of the externally applied field on the molecular currents
• To explain the occurrence of this effect ,we first consider the Lenz law accordingly to which, whenever there is a change in a flux in a circuit, an induced current is setup to oppose the change in flux linked by the circuit
• Here the circuit under consideration is orbiting electrons in an atom, ions or molecules constituting the material under consideration
• we know that moving electron are equivalent to current and when there is a current ,there is a flux
• On application of external field ,the current changes to oppose the change in flux and this appear as a change in the frequency of the revolution
• The change in frequency gives rise to magnetization as a result of which each atom will get additional magnetic moment ,aligned opposite to the external field causing it
• it is this additional magnetic moment which gives diamagnetic susceptibility a negative sign which is order of 10-5 for most diamagnetic material (e g. bismith,lead,copper,silicon,diamond etc)
• All substances are diamagnetic ,although diamagnetism may vary frequently be masked by a stronger positive paramagnetic effect on the part of external magnetic field and as a result of internal interactions
• Diamagnetic susceptibility is independent of temperature as effect of thermal motion is very less on electron orbits as long as it deform them
Note to our visitors :-
Thanks for visiting our website. | {"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": 0.9352778196334839, "perplexity": 837.6938338492005}, "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-2020-40/segments/1600400238038.76/warc/CC-MAIN-20200926071311-20200926101311-00105.warc.gz"} |
https://www.physicsforums.com/threads/the-universes-size-always-infinite.507327/ | # The universe's size : always infinite?
1. Jun 15, 2011
### pedersean
I came across a startling position on more than one occasion while reading "The Fabric of the Cosmos: Space, Time, and the Texture of Reality" by Brian Greene. The position is that our immeasurable universe is infinite. He continues by writing that any mathematical modification to the size of the universe will always result infinite. Perhaps my understanding of infinity is misleading, but I've always held the position of infinity being obtainably intangible and impossible. Instead I prefer to hold that infinity is instead a special numerical placeholder of the extraordinarily large, and with subtracting from infinity a given tolerance becomes greater. Adding to infinity decreases that tolerance by bringing the actual number closer to its equivalent of the all powerful forever number. Am I nuts to argue that an expanding universe can not persist its size?
2. Jun 16, 2011
It basically which side of the fence you are on
If you think that the universe was created by the big bang, then it is physically impossible to fit infinite mass inside a ball the size of a pea (yes I know, laws of physics has been broken before). Then I would say that the universe is expading an it does have it's limits if you consider the theory that energy cannot be created or destroyed.
If you are one of those people who say that the universe has been here since the dawn of time and will always be here, then no... the universe isn't expanding (somehow :P)
as for the question is it infinite, I believe that rather than the universe being never ending, I think its in a big loop (like how people thought that earth was flat, but it's in a sphere, if you go in one direction for long enough you will eventually arrive in the exact same place that you have started from. ;)
Live Long and Prosper \ m /
3. Jun 16, 2011
### sicarius
Infinity is a funny thing. For example, lets say you have an infinite amount of water. That means for every 1 oxygen atom you have 2 hydrogen atoms. While you obviosly have 2x as many hydrogen atoms, you at the same time have equal amounts, since there are infinite amounts of both hydrogen and oxygen. (1*∞)/3 = (2*∞)/3
If you take 1/2 of an infinite volume you still end up with an infinite volume. ∞/2 = ∞. Only if you divide an infinite volume infinitely do you get a finite number: (2*∞)/∞ = 2
So, if the universe is infinite it would have to be expanding infinitely fast, as there is an infinite amount of space to contribute to the expansion. But when you take an infinitely small portion of the universe (like what we can measure) that expansion rate can be finite. So if the universe were to gain 10% size over a given time, it could do so and still be infinite as 1.1*∞ = ∞.
All this seems counter intuitive and hard to accept, but the math is the math. Without accepting infinity as a real possibility than things like a singularity become impossible. An infinite universe also nicely explains expansion. While the universe was a finite size each point had an infinite amount of energy contributing to its expansion. Only once it reached an infinite size would there be a finite amount of energy at each point and be able to start to cool down.
Hope this helps.
4. Jun 16, 2011
### BruceW
The current theory of the universe says that the universe is of finite size.
But there are models of the universe that say it could be infinite.
At the moment, they think its finite.
5. Jun 16, 2011
### DragonPetter
Wouldn't an infinite universe imply infinite energy? For example, two objects infinitely away from each other would have infinite potential energy. What if 2 objects fall at each other from opposite "sides" of an infinite universe, would they accelerate more and more as they got closer to each other? Would they be infinitesimally approaching the speed of light as they accelerate closer to each other, or would it just take infinite time for this to ever happen anyway?
6. Jun 16, 2011
### Haroldingo
Well for all intents and purposes the universe is infinite, in the fact that if we travel from one end to the other we find ourselves back at the place from which we started.
7. Jun 16, 2011
### WannabeNewton
WMAP has found, to some degree of experimental error, that the universe is flat. This means that when one looks at the Friedmann model for a flat universe it is infinite in extent not finite. However, we can only view a finite portion (observable universe) of it because there are regions of the universe that are expanding faster than the speed of signals from those areas.
8. Jun 16, 2011
### sicarius
Yes an infinite universe implies infinite energy. If the universe was empty execpt for those two obejects then yes they theoretically would fall towards at speeds approaching the speed of light, and yes they would never actually reach eachother. It is also possible that gravitons from one object would never reach the other and the falling would never even begin. We don't know enough about gravity to say for sure.
9. Jun 16, 2011
### Nano-Passion
Wow, that was interesting.
But can there be an infinite velocity? I mean, Einstein proved that the speed of light can not be breached. Unless you are telling me that space needs not follow that rule.
If so it would be pose very interesting questions to space and its affects.
10. Jun 16, 2011
### WannabeNewton
Precisely, space does not follow that rule.
11. Jun 16, 2011
### Nano-Passion
Space is indeed interesting and mysterious. It really challenges your imagination and reasoning to the extreme. 0__o
Too many people take the word "space" for granted. We all grew up in it. Space to most people is just the room in their kitchen. Most people think of space as nothing. But in physics space actually has a life and physics of its own. Wow, I only got to appreciate it when I started to deeply ponder..
I wonder how much we truly understand about it?
12. Jun 17, 2011
### sicarius
Thank you.
The speed of light is a measure of movement through space, and that speed cannot be breached. Expanding space is not "moving through space" and does not hit the same limitations.
13. Jun 17, 2011
### Nano-Passion
Yes I've heard that before -- though I wonder about the mathematics behind these sort of things and I'm dumbfounded how we can use math to describe phenomena such as that. Unless there is no rigorous math to it and it is philosophical reasoning to the idea of expanding space.
14. Jun 17, 2011
### sicarius
I may be wrong here, but I think that it is more like they have not found any math that disallows this, not so much as they have mathematically proved it.
15. Jun 17, 2011
### BruceW
There is a rigorous mathematical explanation for the universe being able to expand faster than the speed of light. It is called general relativity.
The universe is highly curved at large scale, which is why two faraway objects can be moving away from each other faster than the speed of light.
16. Jun 17, 2011
### WannabeNewton
The universe is flat, according to observations, at the large scale and two objects don't really move faster than each at the speed of light but rather the space between them expands faster than the speed of light.
17. Jun 17, 2011
### ZapperZ
Staff Emeritus
Closed, pending moderation.
Zz.
18. Jun 18, 2011
### bcrowell
Staff Emeritus
Hi, pedersean,
Welcome to PF!
It would have been better to post this in Cosmology rather than in General Physics. Typically people who are most knowledgeable about a particular field will only pay attention to posts in the relevant forum. Because the discussion had drifted off track, the thread was temporarily locked. I've moved it to Cosmology and opened it back up again.
This is not really right, and since Brian Greene is a competent physicist, I think probably what's happened is that you misinterpreted or oversimplified something he wrote. We have an entry on this topic in the cosmology FAQ: https://www.physicsforums.com/showthread.php?t=506986 We actually don't know whether the universe is spatially finite or spatially infinite.
This sounds like another case where the message got garbled somewhere along the line. This would depend on what was meant by "mathematical modification."
The Math FAQ has a good entry on infinity:
https://www.physicsforums.com/showthread.php?t=507003 [Broken] The truth or falsehood of your statement would depend on what you meant by "obtainably intangible and impossible."
This kind of statement really can't be decided, because it uses undefined terms like "all powerful forever number." The real number system doesn't include infinite numbers. The math FAQ entry gives some examples of number systems that do include infinite numbers.
Not nuts, just incorrect :-) I'm not clear here on why you use the word "persist." Are you discussing the possibility that it would start out infinite and then become finite at some later time? (This would seem to go along with what you said above about "mathematical modification.") According to general relativity, if the universe is finite at one time, then it's finite at all earlier and later times; if it's infinite at one time, then it's infinite at all earlier and later times. This can be proved mathematically based on the Einstein field equations plus some other very reasonable physical assumptions that we have good reason to believe hold in our universe: http://arxiv.org/abs/gr-qc/9406053 The term for this is "topology change."
This actually doesn't quite work in cosmology. There is no principle of conservation of energy in cosmology. We have a FAQ entry about this: https://www.physicsforums.com/showthread.php?t=506985
Mass and energy are equivalent in relativity, so we actually can't define the total mass of the universe (regardless of whether it's spatially finite or spatially infinite). However, we can discuss things like how many hydrogen atoms there are. "The size of a pea" would only apply to cosmologies that are spatially finite (and therefore spatially finite at all times). In these cosmologies, there is only a finite number of hydrogen atoms (or any other particle) in the universe.
The thing to be careful about here is that unless you specify a particular number system (with certain axioms), these statements about arithmetic operations involving infinity are neither true not false. You also have to be careful about your implicit assumption that there is only one infinite number, which is not true in all number systems that include infinite numbers. The math FAQ entry does a good job of explaining this.
This is sort of right, except that you haven't really defined what you meant by "infinitely fast." Maybe you mean the velocity of one galaxy relative to another galaxy that is at a cosmological distance from it? In this case, there is actually no uniquely defined way to talk about the velocity in GR. However, one reasonable way to talk about it is to let $v=\Delta L/\Delta t$, where L and t are the quantities defined in this cosmology FAQ entry: https://www.physicsforums.com/showthread.php?t=506990 In that case, v is finite for any two galaxies, but in an infinite universe there is no upper bound on v (and v can be greater than c).
This is incorrect, because, as discussed above, GR says changes of topology aren't possible.
Nope. The cosmology FAQ entry discusses this.
No, the wrap-around thing would apply to a spatially finite universe (one with finite volume), but as explained in the FAQ, we don't know if it's spatially finite or spatially infinite.
This is not quite right. As explained in the FAQ entry, the universe is within error bars of being flat. Therefore it could have either positive curvature (with finite spatial volume) or negative curvature (with infinite spatial volume).
Sorry, but this is basically all wrong.
It's not an question of space versus physical objects, it's a question of local versus global. Relativity only prohibits objects from zooming right past each other at >c. For cosmologically distant objects, velocity isn't even uniquely well defined (see above).
It is rigorous math. It's how general relativity works.
This is a common way of explaining it nonmathematically. Mathematically, "speed" is just not defined in this context, and expansion of space, although a possible verbal description, is not the only way of verbally describing the mathematics of an expanding universe.
-Ben
Last edited by a moderator: May 5, 2017
19. Jun 18, 2011
### Bob3141592
No, not if the universe is a mixture of positive and negative energy, unevenly distributed on small scales. Then you could have an infinite universe with zero or really any finite amount of energy.
20. Jun 18, 2011
### bcrowell
Staff Emeritus | {"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": 0.802178680896759, "perplexity": 633.2789494645596}, "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-2018-43/segments/1539583512592.60/warc/CC-MAIN-20181020055317-20181020080817-00502.warc.gz"} |
https://www.physicsforums.com/threads/velocity-from-position-vector-in-rotating-object.921143/ | # Velocity from position vector in rotating object
1. Jul 26, 2017
### Nikstykal
1. The problem statement, all variables and given/known data
I am trying to solve for change in velocity for the center of a rim with respect to the contact patch of a tire that has some degree of camber. The equation finalized is shown in the image below, equation 2.6.
http://imgur.com/a/oHucp
2. Relevant equations
3. The attempt at a solution
I understand how to get the position vector shown in 2.5. The first part of 2.6 is just deriving 2.5 with respect to h. The 3rd and 4th terms are what confuse me. In regards to dj/dt = -wz i, I understand that the change in j with respect to time is directly related to the yaw moment (wz) but what is the mathematical reasoning for using the i unit vector? Further, the 4th term demonstrates that dk/dt = -γ' / cos2 γ j, showing that k = -tanγ.
Sorry for the improper notation, was hoping to get further insight into how these terms are being derived.
Last edited: Jul 26, 2017
2. Jul 26, 2017
### Dr.D
I can't see your figure, and without that, there is no good way to reply to you. Please get the figure into the post.
3. Jul 26, 2017
### Nikstykal
Sorry about that, should be fixed.
Draft saved Draft deleted
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https://www.physicsforums.com/threads/subsequence-converging.231439/ | # Subsequence converging
1. Apr 26, 2008
### Doom of Doom
1. The problem statement, all variables and given/known data
Consider the sequence $$\left\{ x_{n} \right\}$$.
Then $$x_{n}$$ is convergent and $$\lim x_{n}=a$$ if and only if, for every non-trivial convergent subsequence, $$x_{n_{i}}$$, of $$x_{n}$$, $$\lim x_{n_{i}}=a$$.
2. Relevant equations
The definition of the limit of a series:
$$\lim {x_{n}} = a \Leftrightarrow$$ for every $$\epsilon > 0$$, there exists $$N \in \mathbb{N}$$ such that for every $$n>N$$, $$\left| x_{n} - a \right| < \epsilon$$.
3. The attempt at a solution
Ok, so I easily see how to show that it $$\lim {x_{n}} = a$$, then every convergent subsequence must also converge to $$a$$.
But I'm stuck on how to show the other way.
2. Apr 26, 2008
### Dick
I would say, well isn't a_n a subsequence of itself? But you also said 'non-trivial'. I'm not sure exactly what that means, but can't you split a_n into two 'non-trivial' subsequences, which then converge, but when put together make all of a_n?
3. Apr 26, 2008
### Doom of Doom
Yeah, I asked my prof about this one. To him, apparently "non-trivial" just means that the subsequence is not equal to the original sequence. I don't think it actually has any bearing on the problem.
The trick, he said, is that you have to consider every non-trivial (convergent) subsequence.
I'm not sure I know what that means.
4. Apr 26, 2008
### Dick
Ok, then suppose a_n has two convergent subsequences with different limits. Then does a_n have a limit?
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https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Introductory_Electrical_Engineering/Electrical_Engineering_(Johnson)/05%3A_Digital_Signal_Processing/5.12%3A_Discrete-Time_Systems_in_the_Time-Domain | # 5.12: Discrete-Time Systems in the Time-Domain
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##### Learning Objectives
• Discrete-time systems allow for mathematically specified processes like the difference equation.
A discrete-time signal $$s(n)$$ is delayed by $$n_0$$ samples when we write $$s(n-n)0)$$ with $$n_0>0$$. Choosing $$n_0$$ to be negative advances the signal along the integers. As opposed to analog delays, discrete-time delays can only be integer valued. In the frequency domain, delaying a signal corresponds to a linear phase shift of the signal's discrete-time Fourier transform:
$s(n=-n_{0}\leftrightarrow e^{-(i2\pi fn_{0})}S(e^{i2\pi f}) \nonumber$
Linear discrete-time systems have the superposition property.
$S\left ( a_{1}x_{1}(n)+a_{2}x_{2}(n) \right )=a_{1}S\left ( x_{1}(n) \right )+a_{2}S\left ( x_{2}(n) \right ) \nonumber$
A discrete-time system is called shift-invariant (analogous to time-invariant analog systems) if delaying the input delays the corresponding output. If
$S\left ( x(n) \right )=y(n) \nonumber$
Then a shift-invariant system has the property
$S\left ( x(n-n_{0}) \right )=y(n-n_{0}) \nonumber$
We use the term shift-invariant to emphasize that delays can only have integer values in discrete-time, while in analog signals, delays can be arbitrarily valued.
We want to concentrate on systems that are both linear and shift-invariant. It will be these that allow us the full power of frequency-domain analysis and implementations. Because we have no physical constraints in "constructing" such systems, we need only a mathematical specification. In analog systems, the differential equation specifies the input-output relationship in the time-domain. The corresponding discrete-time specification is the difference equation.
$y(n)=a_{1}y(n-1)+...+a_{p}y(n-p)+b_{0}x(n)+b_{1}x(n-1)+...+b_{q}x(n-q) \nonumber$
Here, the output signal $$y(n)$$ is related to its past values
$y(n-1),l=\left \{ 1,...,p \right \} \nonumber$
and to the current and past values of the input signal $$x(n)$$. The system's characteristics are determined by the choices for the number of coefficients $$p$$ and $$q$$ and the coefficients' values
$\left \{ a_{1},...,a_{p} \right \}\; and\; \left \{ b_{0},b_{1},...,b_{q} \right \} \nonumber$
##### Note
There is an asymmetry in the coefficients: where is $$a_0$$? This coefficient would multiply the $$y(n)$$ term in the above equation. We have essentially divided the equation by it, which does not change the input-output relationship. We have thus created the convention that $$a_0$$ is always one.
As opposed to differential equations, which only provide an implicit description of a system (we must somehow solve the differential equation), difference equations provide an explicit way of computing the output for any input. We simply express the difference equation by a program that calculates each output from the previous output values, and the current and previous inputs.
Difference equations are usually expressed in software with for loops. A MATLAB program that would compute the first 1000 values of the output has the form for n=1:1000 y(n) = sum(a.*y(n-1:-1:n-p)) + sum(b.*x(n:-1:n-q)); end An important detail emerges when we consider making this program work; in fact, as written it has (at least) two bugs. What input and output values enter into the computation of y(1)? We need values for y(0), y(-1),..., values we have not yet computed. To compute them, we would need more previous values of the output, which we have not yet computed. To compute these values, we would need even earlier values, ad infinitum. The way out of this predicament is to specify the system's initial conditions: we must provide the p output values that occurred before the input started. These values can be arbitrary, but the choice does impact how the system responds to a given input. One choice gives rise to a linear system: Make the initial conditions zero. The reason lies in the definition of a linear system: The only way that the output to a sum of signals can be the sum of the individual outputs occurs when the initial conditions in each case are zero.
##### Exercise $$\PageIndex{1}$$
The initial condition issue resolves making sense of the difference equation for inputs that start at some index. However, the program will not work because of a programming, not conceptual, error. What is it? How can it be "fixed?"
Solution
The indices can be negative, and this condition is not allowed in MATLAB. To fix it, we must start the signals later in the array.
##### Example $$\PageIndex{1}$$
Let's consider the simple system having $$p = 1$$ and $$q = 0$$.
$y(n)=ay(n-1)+bx(n) \nonumber$
To compute the output at some index, this difference equation says we need to know what the previous output y(n-1) and what the input signal is at that moment of time. In more detail, let's compute this system's output to a unit-sample input:
$x(n)=\delta (n) \nonumber$
Because the input is zero for negative indices, we start by trying to compute the output at n = 0.
$y(0)=ay(-1)+b \nonumber$
What is the value of y(-1)? Because we have used an input that is zero for all negative indices, it is reasonable to assume that the output is also zero. Certainly, the difference equation would not describe a linear system if the input that is zero for all time did not produce a zero output. With this assumption, y(-1) = 0, leaving y(0) = b. For n > 0, the input unit-sample is zero, which leaves us with the difference equation
$\forall n,n> 0:\left ( y(n)=ay(n-1) \right ) \nonumber$
We can envision how the filter responds to this input by making a table.
$y(n)=ay(n-1)+b\delta (n) \nonumber$
n x(n) y(n)
-1 0 0
0 1 b
1 0 ba
2 0 ba2
: 0 :
n 0 ban
Coefficient values determine how the output behaves. The parameter b can be any value, and serves as a gain. The effect of the parameter a is more complicated (see Table above). If it equals zero, the output simply equals the input times the gain b. For all non-zero values of a, the output lasts forever; such systems are said to be IIR (Infinite Impulse Response). The reason for this terminology is that the unit sample also known as the impulse (especially in analog situations), and the system's response to the "impulse" lasts forever. If a is positive and less than one, the output is a decaying exponential. When a = 1, the output is a unit step. If a is negative and greater than -1, the output oscillates while decaying exponentially. When a = -1, the output changes sign forever, alternating between b and -b. More dramatic effects when |a| > 1; whether positive or negative, the output signal becomes larger and larger, growing exponentially.
Positive values of
##### Exercise $$\PageIndex{1}$$
Note that the difference equation
$y(n)=a_{1}y(n-1)+...+a_{p}y(n-p)+b_{0}x(n)+b_{1}x(n-1)+...+b_{q}x(n-q) \nonumber$
does not involve terms like $$y(n+1)$$ or $$x(n+1)$$ on the equation's right side. Can such terms also be included? Why or why not?
Solution
Such terms would require the system to know what future input or output values would be before the current value was computed. Thus, such terms can cause difficulties.
##### Example $$\PageIndex{1}$$:
A somewhat different system has no "a" coefficients. Consider the difference equation
$y(n)=\frac{1}{q}\left ( x(n)+...+x(n-q+1) \right ) \nonumber$
Because this system's output depends only on current and previous input values, we need not be concerned with initial conditions. When the input is a unit-sample, the output equals
$\frac{1}{q}\; for\; n=\left \{ 0,...,q-1 \right \} \nonumber$
then equals zero thereafter. Such systems are said to be FIR (Finite Impulse Response) because their unit sample responses have finite duration. Plotting this response (Figure 5.12.2) shows that the unit-sample response is a pulse of width q and height 1/q. This waveform is also known as a boxcar, hence the name boxcar filter given to this system. We'll derive its frequency response and develop its filtering interpretation in the next section. For now, note that the difference equation says that each output value equals the average of the input's current and previous values. Thus, the output equals the running average of input's previous q values. Such a system could be used to produce the average weekly temperature (q = 7) that could be updated daily.
## Contributor
• ContribEEOpenStax
This page titled 5.12: Discrete-Time Systems in the Time-Domain is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by Don H. Johnson via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. | {"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": 0.8880831003189087, "perplexity": 552.5273732847138}, "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/1674764500719.31/warc/CC-MAIN-20230208060523-20230208090523-00348.warc.gz"} |
https://www.askmehelpdesk.com/mathematics/how-can-find-area-inner-circle-area-large-outer-circle-583845.html?s=626e1606a2595a557abdbd30af5e3013 | Joe's centrepiece is a simple but very effective use of two circles and
Two regular hexagons rotated to give the effect of a medieval dial. The
Radius of the inner circle is 10 cm, half the length of the sides of the
Regular hexagon. AC is a side of one of the hexagons and BD is a side
Of the second, which is obtained from the first by rotation.
(I) Find the area of the inner circle, giving your answer in terms of π.
(ii) Find the area of the large outer circle also in terms of π and
Hence express the area of the inner circle as a percentage of the
Area of the large outer circle.
Hint: Each hexagon can be divided into six congruent equilateral
Triangles, for example the triangle AMC is one of the six
Equilateral triangles that make up one hexagon and the triangle
BMD is one of six equilateral triangles that make up the second
Hexagon. | {"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": 0.8103160262107849, "perplexity": 454.4559555860714}, "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-2018-13/segments/1521257648178.42/warc/CC-MAIN-20180323044127-20180323064127-00023.warc.gz"} |
https://www.arxiv-vanity.com/papers/1008.4579/ | Nonlinear as Asymptotic Symmetry of Three-Dimensional Higher Spin AdS Gravity
Marc Henneaux & Soo-Jong Rey
Université Libre de Bruxelles and International Solvay Institutes
ULB-Campus Plaine CP231, 1050 Brussels, BELGIUM
Centro de Estudios Científicos (CECS), Casilla 1469, Valdivia, CHILE
School of Natural Sciences, Institute for Advanced Study, Princeton NJ 08540 USA
School of Physics and Astronomy & Center for Theoretical Physics
Seoul National University, Seoul 151-747 KOREA
abstract
We investigate the asymptotic symmetry algebra of (2+1)-dimensional higher spin, anti-de Sitter gravity. We use the formulation of the theory as a Chern-Simons gauge theory based on the higher spin algebra . Expanding the gauge connection around asymptotically anti-de Sitter spacetime, we specify consistent boundary conditions on the higher spin gauge fields. We then study residual gauge transformation, the corresponding surface terms and their Poisson bracket algebra. We find that the asymptotic symmetry algebra is a nonlinearly realized algebra with classical central charges. We discuss implications of our results to quantum gravity and to various situations in string theory.
## 1 Introduction
Higher spin (HS) anti-de Sitter (AdS) gravity [1, 2, 3] is an interesting extension of AdS Einstein-Hilbert gravity, whose various properties turn out to be highly nontrivial compared to the latter.
This HS theory is also expected to be relevant to a variety of situations in string theory. For example, in Maldacena’s anti-de Sitter / conformal field theory (AdS/CFT) correspondence [4], one would like to understand the holographic dual of CFT at weak ’t Hooft coupling regime. CFTs in this regime are known to possess infinitely many towers of HS currents [5]. By holography, this would mean that the putative closed string dual is at small string tension or large spacetime curvature, and must contain infinitely many towers of HS gauge fields in addition to gravity. One expects that HS AdS gravity is the simplest framework for studying the AdS/CFT correspondence in this regime. In this context, asymptotic symmetry was studied extensively for AdS (black hole) spacetime as the holographic dual of symmetries of CFT at strong ’t Hooft coupling regime. An interesting question is whether the symmetry persists as the correspondence is interpolated to small ’t Hooft coupling regime and, if so, how we may identify it as an asymptotic symmetry of the holographic dual, HS AdS gravity.
(2+1)-dimensional AdS gravity is particularly interesting since the theory is simple yet possesses a rich asymptotic symmetry [6] and provides a concrete framework for studying the AdS/CFT correspondence. It was shown in [6] that the asymptotic symmetry algebra is the infinite-dimensional conformal algebra in two dimensions, viz. two copies of the Virasoro algebra , with central charge
c=3ℓ2G , (1.1)
where is the anti-de Sitter radius and is the Newton’s constant. Extension to (2+1)-dimensional AdS supergravities [7] was considered in [8, 9]. In this case, the asymptotic symmetry algebra is enlarged to appropriate extended superconformal algebras with quadratic nonlinearities in the currents [10, 11, 12, 13, 14, 15].
The purpose of this work is to report results on the asymptotic symmetry algebra of HS AdS gravity in (2+1) dimensional spacetime. The reason we focus on (2+1) dimensions is because the HS AdS gravity again takes a particularly simple form — it can be formulated as a Chern-Simons theory based on so-called infinite-dimensional HS algebra [16, 17]. This algebra contains as a subalgebra, and hence its Chern-Simons formulation automatically contains three-dimensional AdS gravity [7, 18]. After briefly reviewing the theory, we provide boundary conditions on the fields that are asymptotically invariant under an infinite-dimensional set of transformations that contains the conformal group at infinity, and whose generators are shown to close according to a classical nonlinear algebra. This algebra is an extension of the classical version of the algebras of [19]. Classical [20, 21] and quantum [22] nonlinear algebras have appeared previously, but unlike the classical algebra of [20, 21], the asymptotic algebra uncovered here has a nontrivial central charge set by the AdS radius scale measured in unit of the Newton’s constant. In particular, the central charge in the Virasoro subalgebra remains equal to that of the pure gravity (1.1).
A more detailed presentation of our results as well as supersymmetric extensions will be presented in separate works [23].
## 2 Higher Spin Anti-de Sitter Gravity
We first recapitulate (2+1)-dimensional AdS Einstein-Hilbert gravity coupled to an infinite tower of HS gauge fields. It is well-known that the (2+1)-dimensional AdS Einstein-Hilbert gravity can be reformulated as a Chern-Simons gauge theory with gauge group . Following the pioneering work of Blencowe [16], an approach incorporating the HS gauge fields simply replaces the gauge gauge group by a suitable infinite-dimensional extension of it. In this work, we shall follow this approach. We should, however, emphasize that our analysis is strictly at classical level and there will make no difference between the Chern-Simons and the Einstein-Hilbert formulations.
The action describing the HS extension is a difference of two Chern-Simons actions [16]:
S[Γ,~Γ]=SCS[Γ]−SCS[~Γ] (2.1)
where , are connections taking values in the algebra . This algebra is a ’higher spin algebra’ of the class introduced in [24]. Its properties needed for the present discussion are reviewed in Appendix A, to which we refer for notations and conventions. The connections contain all HS gauge field components as well as the metric and spin connection. In (2.1), is the Chern-Simons action, defined by
SCS[Γ]=k4π∫MTr(Γ∧dΓ+23Γ∧Γ∧Γ) . (2.2)
The 3-manifold is assumed to have topology where is a 2-manifold with at least one boundary on which we shall focus our analysis and which we refer as ’infinity’. The parameter is related to the (2+1)-dimensional Newton constant as , where is the AdS radius of curvature.
It is well known that the HS theory (2.1) embeds the AdS gravity by truncation. Truncating the connections to the components and identifying them with the triad and the spin connection:
Aai=ωai+1ℓeaiand~Aai=ωai−1ℓeai, (2.3)
one finds that the action takes the form
S[Γ,~Γ]=18πG∫Md3x(12eR+eℓ2+2LHS) , (2.4)
the Einstein-Hilbert gravity with negative cosmological constant. The equations of motion read
dea+ϵabcωb∧ec=0 dωa+12ϵabcωb∧ωc+12ℓ2ϵabceb∧ec=0. (2.5)
The last term in (2.4) denotes contribution of higher spin fields. For instance, retaining the components as well and identifying them with
Aabi=ωabi+1ℓeabi% and~Aabi=ωabi−1ℓeabi, (2.6)
one easily find that the last term in (2.4) takes the form
LHS=ϵabcea∧(ωbd∧ωce+ebd∧ece)ηde+eab∧(dωab+ϵdeaωd∧ωbe)] . (2.7)
These are precisely the spin-3 field equations in the background of negative cosmological constant, expressed in the first-order formalism. We should, however, note that the HS theory (2.1) is not a smooth extrapolation of the AdS gravity - for example, integrating out the massless HS gauge fields does not lead to the AdS gravity in any direct and obvious way.
## 3 Asymptotic symmetries
### 3.1 boundary conditions and surface terms
With Chern-Simons formulation of the (2+1)-dimensional HS AdS gravity at hand, we are ready to study global gauge symmetries at asymptotic infinity. We shall from now on focus on either chiral sector in (2.1). The analysis for the other chiral sector proceeds in exactly the same way. We shall also work in units of , unless otherwise stated.
In the case of the (2+1)-dimensional AdS Einstein-Hilbert gravity, it was shown in [25, 8, 9] that the boundary conditions of [6] describing asymptotically AdS metrics is given in terms of the connections of the Chern-Simons formulation by
A∼[−1r2πkL(ϕ,t)X11+rX22]dx+−[1rX122]dr (3.1)
with the other chirality sector fulfilling a similar condition. Here, are chiral coordinates, and is an arbitrary function of and . We denoted the generators as . See appendix A.2 for our conventions and notations.
It is convenient to eliminate the leading -dependence by performing the gauge transformation [25, 8, 9]
Γi→Δi=Ω∂iΩ−1+ΩΓiΩ−1,~Γi→~Δi=Ω∂iΩ−1+Ω~ΓiΩ−1 , (3.2)
where depends only on and is given by
Ω=⎛⎝r1200r−12⎞⎠. (3.3)
In the new connection , the only component that does not vanish asymptotically is , given by
Δ∼X22−2πkL(ϕ,t)X11. (3.4)
We see that the asymptotic boundary conditions are encoded entirely to the highest-weight component, spanned in the present case by the generator .
We shall generalize these boundary conditions to HS gauge fields. In the conventions and notations of appendix A.3, we proceed by allowing non-zero components of the . Intuitively, it suffices to vary only the highest-weight components spanned by the generators whose indices are all , viz. . Thus, we require that, after gauge transformation (3.2), the connection behaves asymptotically as
Δ∼X22−2πkLX11+122πkMX1111+% higher" . (3.5)
Here, “higher” denotes terms involving the generators of higher spin () and , , … are arbitrary functions of and . The numerical factors are chosen to get correct normalization in the gauge functional (3.9) below.
The boundary conditions are preserved by the residual gauge transformations
δΔ=Λ′+[Δ,Λ] (3.6)
that maintain the behavior at asymptotic infinity. Here, the prime denotes derivative with respect to . Recall that does not depend asymptotically on in order to preserve at asymptotic infinity, so the derivative with respect to is also the derivative with respect to . As shown in the next subsection, these asymptotic symmetries are spanned by the gauge parameter
Λ=εX22+∑s≥2ηs+1X(0,2s)+λ . (3.7)
Here, and are mutually independent arbitrary functions of . Also, involves only the generators with at least one index equal to (i.e., ) and is completely determined through the asymptotic conditions in terms of and . The lower order terms in take the form
λ = (12ε′′−2πkεL+a(2,0))X11+(12ε′+a(1,1))X12 (3.8) +∑p≥1,q≥0,p+q=2k≥4A(p,q)X(p,q) ,
where and are determined by ’s (independent of ) and where the coefficients are also completely determined by and .
Therefore, we see that the asymptotic symmetries are completely encoded to the independent functions and in the gauge parameter (3.7). We stress again that they are arbitrary functions of and thus arbitrary functions of at a given time .
According to the general principle of gauge theory, these asymptotic symmetries are generated in the equal-time Poisson bracket by the spatial integral , where (i) are the Chern-Simons-Gauss constraints, equal to minus the factor of the temporal components of the connection in the action, and (ii) is a boundary term at asymptotic infinity chosen such that the variation of the generator contains only un-differentiated field variations under the given boundary conditions [26]. This is the requirement that has well-defined functional derivatives. Applying this procedure and using the fact that the generators (which are the only ones that appear in except for ) are paired in the scalar product with , one gets
G[Λ]=∮dϕ (εL+ηM+⋯) , (3.9)
up to bulk terms that vanish on-shell. Here, is abbreviation of and the ellipses denote contribution of HS terms involving for . The normalization factors in (3.5) were chosen chosen so as not to have factors in (3.9). In the next section, we shall show that the asymptotic symmetry generated by is a nonlinearly realized algebra with classical central charges.
### 3.2 general structure of symmetry transformations
In the previous subsection, we argued that the gauge parameter generating the asymptotic symmetry algebra takes the form of (3.7) where is given by (3.8). Here, we prove this and further identify the general structure generating the sought-for HS symmetry algebra.
The condition that , with given by (3.5) and given by (3.6), should take the same form as leads to conditions on the gauge parameter . This gauge parameter has a priori the general form (3.7) but with not yet known. We want to prove that the conditions on yield no restriction on and , while completely determine in terms of and . To that end, we first observe that reads asymptotically
δΔ=−2πkδLX11+122πkδMX1111+More" , (3.10)
where “More” denotes terms involving the generators of higher orders (i.e., ). Thus, involves only the generators (no index ). We must therefore require that all terms proportional to the generators with at least one index equal to should cancel in (3.6).
To analyze this requirement, it is useful to have a notation that counts the number of indices and in the generators. Therefore, we rewrite as
Δ=X22+∑k≥1N(2k,0)X(2k,0) (3.11)
where the coefficients and are evidently proportional to and , respectively. We also rewrite as
Λ = ∑k≥1ρ(0,2k)X(0,2k)+∑k≥1ρ(1,2k−1)X(1,2k−1)+∑k≥1ρ(2,2k−2)X(2,2k−2) + ∑k≥2ρ(3,2k−3)X(3,2k−3)+∑k≥2ρ(4,2k−4)X(4,2k−4) + ∑k≥3ρ(5,2k−5)X(5,2k−5)+⋯
The first term in this expansion is a rewriting of part, while is the sum of all the other terms.
The idea now is to investigate consequences of the requirement that all terms proportional to the generators with at least one index equal to ought to cancel in (3.6) by examining (i) first the terms containing the generators with no index equal to 1 in (viz. all indices equal to 2), (ii) next those with only one index equal to 1, (iii) next those with only two indices equal to 1, etc.
A simple calculation shows that the coefficient of in (no indices equal to 1) is given by
c(0,2k)∼ρ′(0,2k)+ρ(1,2k−1)+f0(ρ(0,2i),N(2j,0)) , (3.12)
As our goal is to explicitly indicate how the structure emerges, we presented the terms only schematically by dropping numerical factors. The term is an infinite sum of bilinears in the ’s and the ’s. The first contribution to comes from the bracket of with (one s replaced by one ), while the second contribution to arises from the bracket which is the only bracket in yielding generators with no index equal to . This bracket yields other generators as well, but they only contribute to the equations at the subsequent levels. Thus, we can regard the condition as determining the coefficients of in in terms of the ’s and the ’s of the connection .
Note that even though is an infinite sum, there is only a finite number of terms involving a given because one must have for the bracket to yield a non vanishing term involving (). It is also easy to check that is explicitly given by
c(0,2)=ρ′(0,2)−2ρ(1,1)+more" , (3.13)
where numerical factors are reinstated and “more” denotes terms independent of . The condition then implies the expression (3.8) for the coefficient of in .
The next step is to examine the coefficient of in (only one index equal to ). By a similar reasoning, one finds
c(1,2k−1)∼ρ′(1,2k−1)+ρ(2,2k−2)+f1(ρ(0,2i),ρ(1,2l−1),N(2j,0)). (3.14)
Therefore, the requirement determines the ’s in terms of the ’s and the ’s. Since the ’s are functions of the ’s that have been determined at the previous step, the ’s are determined in terms of the ’s.
Note again that even though there is an infinite number of terms in because of , there is only a finite number of terms containing a given . One finds in particular that takes the schematic form
c(1,1)=ρ′(1,1)−ρ(2,0)+N(2,0)ρ(0,2)+more" (3.15)
so that the equation implies the expression (3.8) for the coefficient of in .
The triangular pattern of the procedure is now evident and proceeds similarly at the next levels. One determines in this fashion recursively not only the coefficients , but also , , viz. the complete functional form of , in terms of the coefficients ’s, which remain unconstrained. The procedure terminates once one has imposed the conditions . Consequently, there is no condition imposed on . Rather, the coefficient determines the variation of the connection through . Notice that the procedure introduces nonlinearities through the ’s.
We have thus established that the gauge parameter generating asymptotic symmetry takes precisely the form given in (3.7) and (3.8).
## 4 Nonlinear W∞ Symmetry Algebra
As we explained above, the variations of the coefficients of the connection under the asymptotic symmetries are given by the equation
δN(2k,0)=c(2k,0) , (4.1)
where the are the unconstrained coefficients of the generator in . The recursive method explained in the previous section enables to determine these coefficients in terms of the independent parameters ’s parametrizing the asymptotic symmetry.
We have recalled in the previous section that the coefficients of the connection are themselves the generators of the gauge transformations and hence of the asymptotic symmetries. In fact, in more compact notations, (3.9) has the form
G[Λ]=∮dϕ(∑k≥1ρ(0,2k)N(2k,0)) (4.2)
up to bulk terms that vanish on-shell. Again, for clarity, we kept the expression schematic regarding normalization of the generators. They will not affect foregoing argument and result. (We shall work out the normalization explicitly in the next section for the truncation to ).
In general, the variation of any phase-space function under the gauge transformation with parameter is equal to where is classical Poisson bracket. Thus, in the present case, we have
δN(2k,0)={N(2k,0)(ϕ),∫dϕ′(∑m≥1ρ(0,2m)(ϕ′)N(2m,0)(ϕ′))}PB . (4.3)
This observation enables us to read the Poisson bracket commutators of the ’s in (4.2) among themselves from their variations (4.1) 111If one drops the bulk terms as can be done by fixing the gauge in the bulk, the Poisson bracket in question is the corresponding Dirac bracket. The form of the symmetry algebra does not depend on how one fixes the gauge because the generators are first class.:
{N(2k,0)(ϕ),∫dϕ′(∑m≥1ρ(0,2m)(ϕ′)N(2m,0)(ϕ′))}PB=c(2k,0)(ϕ) , (4.4)
where we have made it explicit for the angular dependence at a fixed time. By identifying the coefficients of the arbitrary parameter on both sides of this equation, one can read off the Poisson brackets
{N(2k,0)(ϕ),N(2m,0)(ϕ′)}PB (4.5)
and resulting algebra . In the rest of this section, we sketch the general procedure of extracting . To illustrate the procedure concretely, in the next section, we will work out the case corresponding to the truncation of .
It is evident from the above analysis that the expression obtained for is closed, in the sense that it is expressed entirely in terms of the ’s. Terms that are generated from the Poisson bracket are in fact nonlinear polynomials in the ’s. Therefore, the resulting gauge algebra is not a Lie-type but a nonlinear realization thereof. Furthermore, by construction, the Jacobi identity holds for because it always holds for the Poisson brackets or the corresponding Dirac brackets after the bulk terms are gauge-fixed.
We claim that the resulting algebra is a classical, nonlinearly realized with classical central charges. It is a classical algebra because we are using the Poisson-Dirac bracket of classical quantities and not the commutator of corresponding operators. It also has nontrivial classical central charges.
To support this claim, it suffices to prove that
1. The algebra contains the Virasoro algebra at lowest degree , viz. the generators form a Virasoro algebra with central charge :
{L(ϕ),L(ϕ′)}PB=−k4π∂3ϕδ(ϕ−ϕ′)+(L(ϕ)+L(ϕ′))∂ϕδ(ϕ−ϕ′) (4.6)
2. The generators have the conformal weight :
{L(ϕ),Mj+1(ϕ′)}PB=(Mj+1(ϕ)+jMj+1(ϕ′))∂ϕδ(ϕ−ϕ′) . (4.7)
To establish these statements, we pick up the terms proportional to in and . This is done by first determining the form of in the particular case when the only non-vanishing free parameter is . In that case, the solution is easily determined to be
Λ=εX22+12ε′X12+(12ε′′−2πkεL)X11+ε∑j≥2N(2j,0)X(2j,0) , (4.8)
since with this , the expression contains only generators .
The coefficients of the generators in give furthermore the variations of and (). These are easily derived from (4.8) using
[ X(2j,0),X12 ]=(2j)X(2j,0) . (4.9)
δL = −k4πε′′′+(εL)′+ε′L (4.10) δN(2j,0) = (εN(2j,0))′+jε′N(2j,0)(j>1). (4.11)
The relations (4.6) and (4.7) follow immediately from these.
Explicit form of the Poisson-Dirac brackets of the resulting algebra and classical central charges therein are obtainable by straightforward though tedious computations.
## 5 Truncation to W3 Algebra
To illustrate the above procedure explicitly, we truncate the theory by assuming that all the generators with are zero. i.e., we keep only and . This amounts to truncating the HS algebra by keeping only , and setting all the other generators to zero. As shown in appendix B, this is a unique, consistent truncation as the Jacobi identity remains to hold. The resulting algebra is , albeit not in a Chevalley-Serre basis 222The relation between the Chern-Simons formulation and the symmetric tensor formulation is implicit in [16] (using the vielbein/spin connection-like Vasiliev formulation of higher spins), and underlies the fact that [16] is a theory of higher spins coupled to gravity. Truncating this general relation to , one gets the ’metric + 3-index symmetric tensor’ formulation of the coupled ‘spin-2 + spin-3’ system. This was briefly recalled at the end of section 2.. We now show that and fulfill the classical nonlinear algebra with classical central charges.
The condition that , with given by
Δ∼X22−2πkLX11+122πkMX1111 (5.1)
and given by (3.6), should take the same form as leads to conditions on the coefficients of the gauge parameter in the expansion
Λ=aX11+bX12+εX22+mX1111+nX1112+pX1122+qX1222+ηX2222 (5.2)
These conditions are explicitly that are determined as
b=12ε′,a=12ε′′−2πkεL−22πkηM . (5.3)
and that are determined as
m = 124η′′′′−16⋅2πk(ηL)′′−14⋅2πk(η′L)′ −2πk(14η′′−2πkηL)L+122πkεM n = 124η′′′−16⋅2πk(ηL)′−14⋅2πkη′L p = 112η′′−13⋅2πkηL q = 14η′ . (5.4)
One also obtains the gauge variations of and as
δL=−k4πε′′′+(Lε)′+ε′L+2(ηM)′+η′M (5.5)
and
δM = 1288⋅k2πη′′′′′−172(ηL)′′′−148(η′L)′′ (5.6) −112((14η′′−2πkηL)L)′ −112(16η′′′−23⋅2πk(ηL)′−2πkη′L)L +(εM)′+2ε′M .
Now, as already recalled above in the general case, the variation of any phase space function under the gauge transformation with parameter is equal to where is the classical Poisson bracket. One can use this to find the Poisson brackets of and from their variations, taking (3.9) into account.
One finds explicitly that
{L(ϕ),L(ϕ′)}PB=−k4π∂3ϕδ(ϕ−ϕ′)+(L(ϕ)+L(ϕ′))∂ϕδ(ϕ−ϕ′) {L(ϕ),M(ϕ′)}PB=(M(ϕ)+2M(ϕ′))∂ϕδ(ϕ−ϕ′) {M(ϕ),M(ϕ′)}PB=1288⋅k2π∂5ϕδ(ϕ−ϕ′)−5144(L(ϕ)+L(ϕ′))∂3ϕδ(ϕ−ϕ′) +148(L′′(ϕ)+L′′(ϕ′))∂ϕδ(ϕ−ϕ′) +19⋅2πk(L2(ϕ)+L2(ϕ′))∂ϕδ(ϕ−ϕ′) .
This is the classical algebra studied previously in various different contexts [19], [27], [28].
Upon Fourier mode decomposition, the nonlinear algebra is given by [27]
i[ Lm,Ln ] = (n−m)Lm+n+c12m(m2−1)δm+n,0 (5.7) i[ Lm,Vn ] = (2m−n)Vm+n i[ Vm,Vn ] = c360m(m2−1)(m2−4)δm+n,0+165c(m−n)Λm+n + (m−n)(115(m+n+2)(m+n+3)−16(m+2)(n+2))Lm+n
where
Λm=+∞∑n=−∞Lm−n Ln . (5.8)
sums quadratic nonlinear terms. The classical central charges are given by . The quantum counterpart of this algebra was studied by Zamolodchikov [19] in a different context. Comparing it with the above classical algebra, one sees that the quantum effects enter to regularization of the quadratic nonlinear terms ’s and to the shift of the overall coefficient of the quadratic terms. This fits with the fact that classical limit takes .
## 6 Discussions
In this paper, we have established that the asymptotic symmetries of the HS AdS gravity form a nonlinear algebra. A salient feature of the emerging classical algebra is that it is determined in a unique manner from the gauge algebra and the Chern-Simons parameter , without an extra free parameter. Moreover, this classical algebra has from the outset definite nontrivial central charges expressed solely in terms of the AdS radius and the Newton’s constant (and nothing else). In particular, the central charge appearing in the Virasoro subalgebra is just the AdS central charge (1.1).
Truncation of the higher spin gauge algebra up to a finite spin is inconsistent when , since the corresponding generators do not form a subalgebra (except when ). The Poisson-Dirac commutators of with involve indeed generators of degree , which is strictly higher than and when , . One may try to ignore these higher degree terms but this brutal truncation yields commutators that do not fulfill the Jacobi identity (except for as we pointed out, see appendix A) and so this cannot be done (except for )333That the case works is somewhat unanticipated and should be considered exceptional from the point of view.. On the other hand, from the purely algebraic viewpoint, one might opt to start from the algebra with finite obtained from algebra of gauge invariance and take the limit to obtain a universal -algebra. However, these two approaches are completely different in spirit since in general the truncation of up to a finite spin does not yield the algebra for any finite .
Nonlinearity of the Poisson-Dirac brackets or commutation relations (compared to the Lie-type algebra) is an important and distinguishing characteristic of the operator algebra for spin . However, in the usual large- limit, this nonlinearity is typically lost [29], [30], [31], [32]: the resulting algebra is usually linear (and also in some cases the classical central charge is absent). Our approach obtains the gauge algebra in a completely different way, and in particular does not rely on such a limiting procedure. We note that the nonlinearity of the algebra puts strong constraints through the Jacobi identities. Thus, the nonlinear algebra derived in this work, which is inherent to the -based HS extension of the (2+1)-dimensional AdS Einstein-Hilbert gravity – in the sense that it uniquely determined by it – has a rich and interesting structure. More detailed analysis of this algebraic structure, extensions to supersymmetry and inclusion of spin-1 currents will be reported elsewhere [23].
Related to this, it has been known previously that the linear version of the algebra is related to the first Hamiltonian structure of the KP hierarchy [33]. In our nonlinearly realized version of the algebra, we speculate the relation goes to the second Hamiltonian structure of the KP hierarchy, the structure proposed by Dickey [34] from generalizing the Gelfand-Dickey brackets [35] to pseudo-differential operators. It is an interesting question what these relations tell us about the spectrum of classical solutions of the HS AdS gravity.
It is tempting to interpret that the presence of the algebra at infinity implies that the classical solutions of the (2+1)-dimensional HS AdS gravity are labeled by infinitely many conserved charges, among which mass and angular momentum are just the first two. If the interpretation is correct, we expect that these charges play a central role in understanding microstates responsible for the black hole entropy in the regime where the spacetime curvature is large or, in string theory context, the string scale is very low444One should also mention here the intriguing appearance of the linear algebra found in [37, 38] in the context of black holes and Hawking radiation.. A possible holographic dual in this regime was explored recently by Witten [36] for pure AdS gravity. There, an indication was found that two-dimensional CFT duals are the monster theory of Frenkel, Lepowsky and Meurman or discrete series extensions thereof. Once embedded to string theory, one expects this regime must includes (nearly) massless HS gauge fields in addition to the gravity. This brings in a host of intriguing questions: Are there HS extensions of the monster theory and, if so, what are they? Can the extension be related or interpreted physically to condensation of long strings?
In the context of string theory, the massless HS gauge fields were interpreted to arise via a sort of inverse Higgs mechanism in the limit of vanishing string mass scale (viz. string tension) [39]. If so, the HS gauge fields would become massive at large but finite string mass scale [40]. In CFT dual, this would be reflected to anomalous violation of conservation laws of the HS currents. Nevertheless, the symmetry algebra discovered in this work would be an approximate symmetry of the CFT duals and should still be useful for understanding these theories.
In addition to the weak ’ t Hooft coupling regime alluded in the Introduction, there is another situation in string theory where the result of this paper may be applicable. The near-horizon geometry of the small black strings carrying one or two charges is singular in Einstein-Hilbert gravity. One expects that, by the stretched horizon mechanism [41], string corrections resolve it to the (2+1)-dimensional AdS spacetime times a compact 7-dimensional manifold characterizing the black string horizon with residual chiral supersymmetries. [42]. A concrete suggestion like this was put forward for the ’stretched horizon’, near-horizon geometry of the macroscopic Type II and heterotic strings [43], [44], [45]. In both cases, the near-horizon geometry has curvature radius of order the string scale. So, not just the massless but also all HS string states are equally important for finite energy excitations. This suggests that (2+1)-dimensional HS AdS supergravity theories are appropriate frameworks. It would then be very interesting to identify the origin of the symmetry algebra as well as the classical central charges associated with HS currents from the macroscopic superstring viewpoint.
On a more speculative side, our result may also find a potentially novel connection of the (2+1)-dimensional HS AdS gravity to higher-dimensional gravity. It has been known [46], [47], [48] that 4-dimensional self-dual gravity is equivalent to a large limit of 2-dimensional nonlinear sigma model with Wess-Zumino terms only. The self-dual sector has an infinite-dimensional symmetry algebra which includes the algebra. This hints that (2+1)-dimensional HS AdS gravity might be ’holographically dual’ to 4-dimensional self-dual gravity, providing a concrete example of heretofore unexplored gravity-gravity correspondence.
Centered to all these issues, the most outstanding question posed by our work is:
What are the black holes carrying hairs in HS AdS gravity?
We are currently exploring these issues and intend to report progress elsewhere.
## Acknowledgement
We thank Nima Arkani-Hamed, Juan Maldacena and Edward Witten for useful discussions. MH is grateful to the Institute for Advanced Study (Princeton) for hospitality during this work and to the Max-Planck-Institut für Gravitationphysik (Potsdam) where it was completed. SJR is grateful to the Max-Planck-Institut für Gravitationphysik (Potsdam) during this work and to the Institute for Advanced Study (Princeton) where it was completed. We both acknowledge support from the Alexander von Humboldt Foundation through a Humboldt Research Award (MH) and a Bessel Research Award (SJR). The work of MH is partially supported by IISN - Belgium (conventions 4.4511.06 and 4.4514.08), by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P6/11 and by an ARC research grant 2010-2015. The work of SJR is partially supported in part by the National Research Foundation of Korea grants KRF-2005-084-C00003, KRF-2010-220-C00003, KOSEF-2009-008-0372, EU-FP Marie Curie Training Program (KICOS-2009-06318), and the U.S. Department of Energy grant DE-FG02-90ER40542.
## Appendix A Higher Spin Algebra
### a.1 Definition
The higher spin algebra in (2+1)-dimensional spacetime is the direct sum of two chiral copies of :
A=hs(1,1)L⊕hs(1,1)R (A.1)
The infinite-dimensional algebra itself is defined as follows. Consider an auxiliary space of polynomials of even degree in two commuting spinors , . One defines the trace of a polynomial as
Trf=2f(0)≡2f(ξ)∣∣ξ=0. (A.2)
Here, the factor 2 is included to match the traces of matrices considered below. Then, the elements of the algebra are the elements of with no constant term, viz. traceless polynomials.
To define the Lie bracket, one first considers the star-product defined by
(f⋆g)(ξ)=exp[i(∂∂η1∂∂ζ2−∂∂η2∂∂ζ1)] f(η)g(ζ)∣∣η=ζ=ξ (A.3)
The star-product is associative. Although non-commutative, the star-product is trace-commutative:
(f⋆g)(0)=(g⋆f)(0), (A.4)
viz.
Tr(f⋆g)=Tr(g⋆f) (A.5)
because and are polynomials of even degree. The Lie bracket in the algebra is just the -commutator (modulo a numerical factor chosen for convenience to be ):
[f,g]≡12i(f⋆g−g⋆f)=sin(∂∂η1∂∂ζ2−∂∂η2∂∂ζ1)f(η)g(ζ)∣∣η=ζ=ξ. (A.6)
It fulfills the Jacobi identity because the star-product is associative.
The Lie algebra possesses a symmetric and invariant bilinear form denoted , defined by
(f,g)≡Tr(f⋆g). (A.7)
This bilinear form is symmetric because of the trace-commutativity (A.5) and invariant
(f,[g,h])=([f,g],h) (A.8)
because of the associativity of the star-product and (A.5) again. The invariant symmetric bilinear form is non-degenerate.
### a.2 sl(2,R) subalgebra
The polynomials of degree 2 form a subalgebra isomorphic to . Taking as a basis of this subspace as
X11=12(ξ1)2,X12=ξ1ξ2,X22=12(ξ2)2 , (A.9)
one finds
[X11,X12]=2X11,[X11,X22]=X12,[X12,X22]=2X22 . (A.10)
One can thus identify the with the standard Chevalley-Serre generators as follows: , and .
Moreover, traces of the products of ’s match with traces of the products of the corresponding matrices. The non-zero scalar products are
(X12,X12)=2,(X11,X22)=−1,(X22,X11)=−1. (A.11)
The subalgebra splits into a direct sum of representations of :
hs(1,1)=⊕k≥1Dk , (A.12)
where the spin representation corresponds to the homogeneous polynomials of degree . The trivial representation does not appear because we consider traceless polynomials. It is straightforward to verify that the subspaces and are orthogonal for and that the scalar product is non-degenerate on each .
We emphasize that, as showed in the text, the representation yields asymptotically the generators of conformal spin . Notice the shift of the spin label by one unit.
### a.3 more commutation relations
We list here the commutation relations involving and . A basis of the representation of , polynomials of order 4, may be taken to be
X1111=14!(ξ1)4,X1112=13!(ξ1)3ξ2,X1122=14(ξ1)2(ξ2)2,
X1222=13!ξ1(ξ2)3,X2222=14!(ξ2)4 .
More generally, we define
X(p,q)≡X1⋯1p2⋯2q=1p!1q!(ξ1)p(ξ2)q with p+q even (A.13)
The vectors with form a basis of . We use the collective notation for the ’s with .
The brackets of the ’s with the ’s are given by
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https://portlandpress.com/biochemj/article/245/3/723/23552/Effects-of-Mg2-anions-and-cations-on-the-Ca2-Mg2 | In a previous paper [Gould, East, Froud, McWhirter, Stefanova & Lee (1986) Biochem. J. 237, 217-227] we presented a kinetic model for the activity of the Ca2+ + Mg2+-activated ATPase of sarcoplasmic reticulum. Here we extend the model to account for the effects on ATPase activity of Mg2+, cations and anions. We find that Mg2+ concentrations in the millimolar range inhibit ATPase activity, which we attribute to competition between Mg2+ and MgATP for binding to the nucleotide-binding site on the E1 and E2 conformations of the ATPase and on the phosphorylated forms of the ATPase. Competition is also suggested between Mg2+ and MgADP for binding to the phosphorylated form of the ATPase. ATPase activity is increased by low concentrations of K+, Na+ and NH4+, but inhibited by higher concentrations. It is proposed that these effects follow from an increase in the rate of dephosphorylation but a decrease in the rate of the conformational transition E1′PCa2-E2′PCa2 with increasing cation concentration. Li+ and choline+ decrease ATPase activity. Anions also decrease ATPase activity, the effects of I- and SCN- being more marked than that of Cl-. These effects are attributed to binding at the nucleotide-binding site, with a decrease in binding affinity and an increase in ‘off’ rate constant for the nucleotide.
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https://www.physicsforums.com/threads/find-all-roots-of-x-3-3x-2-10x-6.240450/ | # Homework Help: Find all roots of x^3 + 3x^2 - 10x + 6
1. Jun 15, 2008
### stat643
find all roots of x^3 + 3x^2 - 10x + 6
the solution:
identify the easy root of x=1,
find the remaining roots from (x-1)(x^2+4x) using quadratic formula.
The only thing i dont understand here is how to factorize to (x-1)(x^2+4x)... namely the (x^2+4x) part.
2. Jun 15, 2008
### rocomath
factor out a common term in x^2+4x
and you're pretty much done
3. Jun 15, 2008
### stat643
but how did i get to x^2+4x in the first place?... the original equation was x^3 + 3x^2 - 10x + 6.. i merely copied the solution... so i find the easy root of 1.. then what?
Last edited: Jun 15, 2008
4. Jun 15, 2008
### rocomath
use synthetic division
5. Jun 15, 2008
### matt grime
No, don't use synthetic division (just yet). Pause for a moment and think: is it plausible that x^2+4x is a factor? It isn't. Copying out the answer is never a good idea.
6. Jun 15, 2008
### stat643
i just looked up synthetic devision on wikipedia and tried it but it didnt work
7. Jun 15, 2008
### stat643
should i take out the common term x first?
8. Jun 15, 2008
### arildno
Then you should practice synthetic division once more!
Further, ponder over matt grime's words:
WHY should you be suspicious of that particular factorization?
Hint:
How could you ascertain whether the factorization is correct or false?
9. Jun 15, 2008
### stat643
oh sorry i copied it wrong, it should be (x-1)(x^2+4x-6).. now expanding that get: x^3 + 4x^2 -6x -x^2 -4x + 6 = x^3 + 3x^2 - 10x + 6.. so yeh its right now.. though i still cant get the synthetic devision right (its new to me)
i tried to learn it now from http://en.wikipedia.org/wiki/Synthetic_division
though i keep getting 1,2,-12,18
can someone help show how i would use synthetic devision for the original polynomial ?
Last edited: Jun 15, 2008
10. Jun 15, 2008
### arildno
Okay, we wish to find a second-order polynomial so that:
$$(x-1)(ax^{2}+bx+c)=x^{3}+3x^{2}-10x+6$$ holds for all x.
I.e, we must determine a,b and c!!
Multiplying out the left-hand side, and organizing in powers of x, the lefthandside can be rewritten as:
$$ax^{3}+(b-a)x^{2}+(c-b)x-c= x^{3}+3x^{2}-10x+6$$
NOw, the coefficients of each power must be equal on the right and left sides, yielding the system of equations:
a=1
b-a=3
c-b=-10
-c=6
This yields:
a=1
b=4
c=-6
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http://www.cfd-online.com/Forums/main/115443-viscosity-term-discetrization-momentum-equation.html | # viscosity term in discetrization momentum equation
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March 31, 2013, 06:38 viscosity term in discetrization momentum equation #1 New Member hans Join Date: Sep 2012 Posts: 7 Rep Power: 4 Hi all, I have a question regarding the discretization of the momentum equation for use in a simple solver. I'm having trouble in understanding where the viscosity term goes. I'm currently reading Versteeg 2007 , An introduction to CFD, and can't figure it out. At some point the momentum equation, including the body surface forces caused by viscosity is discretisized to: As I understand it, the viscosity term is not present here, coefficients only have a velocity and density dependence. I can't seem to locate a viscosity in the source term,S ,either. If find that on the web it is often pointed out that the discretization is similar to that of the general transport equation where the viscosity term can be handled equal to the diffusion term and the velocity term to the property term. In this case the diffusion term is preserved in the discretization, why than not (or at least lost to me) in the momentum equation? Can anyone explain what i'm missing or point me in a direction where i can find some answers?I would be very happy to find the complete derivation of the discretization of the momentum equation so i can see what's happening step by step. Kind regards,
March 31, 2013, 06:51
#2
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,588
Rep Power: 20
Quote:
Originally Posted by hans-186 Hi all, I have a question regarding the discretization of the momentum equation for use in a simple solver. I'm having trouble in understanding where the viscosity term goes. I'm currently reading Versteeg 2007 , An introduction to CFD, and can't figure it out. At some point the momentum equation, including the body surface forces caused by viscosity is discretisized to: As I understand it, the viscosity term is not present here, coefficients only have a velocity and density dependence. I can't seem to locate a viscosity in the source term,S ,either. If find that on the web it is often pointed out that the discretization is similar to that of the general transport equation where the viscosity term can be handled equal to the diffusion term and the velocity term to the property term. In this case the diffusion term is preserved in the discretization, why than not (or at least lost to me) in the momentum equation? Can anyone explain what i'm missing or point me in a direction where i can find some answers?I would be very happy to find the complete derivation of the discretization of the momentum equation so i can see what's happening step by step. Kind regards,
in § 6.3 it is explicitly stated that the coefficients contain combination of convective and diffusive terms..
March 31, 2013, 06:56 #3 New Member hans Join Date: Sep 2012 Posts: 7 Rep Power: 4 FMDenaro, Thanks for your reply! I've seen this in chapter 6.3 .But I'm a bit confused in how to interpret this, the viscosity and diffusive terms are the same? I presume this is the diffusion of momentum from one cell to the other? Hoe to couple this momentum diffusion term to viscosity then? Last edited by hans-186; March 31, 2013 at 07:25.
April 1, 2013, 12:17 #4 New Member Aniket Sachdeva Join Date: Mar 2012 Posts: 22 Rep Power: 5 Viscosity by definition is the momentum diffusivity.. The rate at which momentum of one layer of the fluid is diffused to other layers is decided by the viscosity..
April 1, 2013, 13:34 #5 New Member hans Join Date: Sep 2012 Posts: 7 Rep Power: 4 Yeah, I'm back on track . Got somewhat lost in the maze I guess. Thanks for your replies.
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All times are GMT -4. The time now is 18:15. | {"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": 0.8177857995033264, "perplexity": 1032.2256672127464}, "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/1435375102712.76/warc/CC-MAIN-20150627031822-00073-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/39002/optimizing-over-matrices-with-spectral-radius-1/39013 | Optimizing over matrices with spectral radius <1?
Suppose $F(x)$ is a convex objective function on $n\times n$ matrices, and I need to numerically optimize $F$ with the condition that $x$ has spectral radius less than $1$. This might be too hard, so an approximation would be needed. Has this problem been studied before?
Motivation: Boltzmann machines are hard to evaluate when spectral radius of the weight matrix is large, especially if it's above $1$ so best fit to data subject to this constraint would give a useful model.
Example: Let $X=\{1,-1\}^d$ and $\hat{X}$ some list of $\{1,-1\}$ $d$-tuples. Find $$\max_A \sum_{x\in \hat{X}} \mathbf{x}'A\mathbf{x} - |\hat{X}|\log \sum_{x\in X} \exp(\mathbf{x}'A\mathbf{x})$$ Where $A$ is symmetric real-valued $d\times d$ matrix with spectral radius < 1. This needs to be done in time polynomial in $d$ and linear in $|\hat{X}|$. When spectral radius is <1, belief propagation gives a reasonably accurate way to approximate gradient of this objective in $O(|\hat{X}|d^2)$ time
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If the function is convex, can't you restrict trivially to spectral radius=1? – Federico Poloni Sep 16 '10 at 19:21
Well, I want <1, but restricting to, say, 1/2 would be useful, how would I do that? – Yaroslav Bulatov Sep 16 '10 at 20:45
By convexity, $f(\frac{a+b}2) \leq \frac{f(a)+f(b)}2$, therefore one among $f(a)$ and $f(b)$ is larger (or equal) than $f(\frac{a+b}2)$. So the supremum cannot occur on an inner point (i.e., one that you can write as the midpoint of two other points in the set). So you are can restrict wlog to the boundary of your (convex) set. – Federico Poloni Sep 17 '10 at 2:04
If your matrices are symmetric, the set of matrices with spectral radius $\le 1$ is convex, and can be modelled using a linear matrix inequality (LMI), see e.g. page 147 in Lectures on Modern Convex Optimization by Ben-Tal and Nemirovski. If you wanted to minimize a convex objective that is also semidefinite-representable, you could in principle formulate and solve your problem as a semidefinite programming problem. However, maximizing a convex objective over a convex set is a much more difficult problem.
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Good observation, I re-examined the underlying problem, and looks like it comes down to minimizing a convex objective (or maximizing concave negation, as it's more frequently presented) – Yaroslav Bulatov Sep 17 '10 at 18:03
Thanks, that formula seems pretty useful. But I still don't know how to turn it into Semidefinite Programming since my objective is not linear...do you have a good reference for some examples of how that's done? – Yaroslav Bulatov Sep 21 '10 at 20:59
Ben-Tan and Nemirovski is the bible for semidefinite modelling, so you should start looking there. – F_G Sep 22 '10 at 10:03
This started as a comment, but it's too long.
Is the objective function invariant under conjugation?
Spectral radius is far from a convex function of matrices. If you take two non-negative matrices with 1's on the diagonal, one zero below the diagonal and positive above, the other vice versa, any convex combination has spectral radius bigger than 1. It's easy to make the spectral radius as large as you like.
The set of characteristic polynomials for matrices of spectral radius 1 isn't convex either. For example, the average of $(x - .99)^2$ and $(x - .99i)^2$ has roots outside the unit circle.
However, every conjugacy class in $GL(n,\mathbb C)$ has a representative that is upper triangular, and the upper triangular matrices of spectral radius 1 form a convex set. This may make it easier to find the optimum (depending what it is, which you didn't say). There are convex sets containing just conjugacy classes of spectral radius < 1 for various other kinds of matrices.
Even if the function is not invariant under conjugation, it may help to break it up by conjugacy class: for each conjugacy class, find the optimum, then find the optimum among all conjugacy classes.
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Does invariance under conjugation matter if my objective and constraints only involve real numbers? Also, I added the complete problem description – Yaroslav Bulatov Sep 17 '10 at 4:42
No, with the added information, you're using $A$ as a quadratic form, and as FG noted, spectral radius is a convex function among those, and the potential function is not invariant by a significant group. But: as you've phrased it, $\hat(X)$ might contain exponentially many $d$-tuples, so to evaluate the function just once already could take exponential time. Are you asking about probabilistic algorithms? – Bill Thurston Sep 17 '10 at 5:23
Actually $O(|\hat{X}|)$ time is acceptable because $\hat{X}$ represents the input to the optimizer which is small compared to $2^d$. A typical value would be $|\hat{X}|<10,000$, $d<1000$ – Yaroslav Bulatov Sep 17 '10 at 8:03 | {"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": 0.9196537137031555, "perplexity": 265.00962032291847}, "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-2015-06/segments/1422122238694.20/warc/CC-MAIN-20150124175718-00222-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://mathhelpforum.com/differential-equations/222159-first-order-autonomous-equations.html | # Math Help - First order autonomous equations
1. ## First order autonomous equations
Hello,
I've been working more out of Christian Constanda's "Differential Equations: A Primer for Scientists and Engineers" ISBN 978-1-4614-7296-4. This is from section 3.3, which is on autonomous equations and their models. In my particular section of Diff-EQ, all homework is 'optional'.
I'm having what is likely a stupid problem. I've been working later sections all day - second order homogenous differential equations - and have been fine. It's only now that I'm back to first order autonomous equations that I'm hitting a brick wall. This chapter is killing me every time I look at any of its sections!
The exercise numbers are 1 and 15.
The instructions are: "Find the critical points and the equilibrium solutions of the given equation and solve the equation with each of the prescribed initial conditions."
Additionally we're supposed to sketch the graphs of the solutions and comment on the stability/instability of the equilibrium solutions as well as identify what model is governed by the problem (I.E. population with logistic growth, population with ac ritical threshold, chemical reactions, etc). The part in quotes is what I'm looking for help with.
#1: y'=300y - 2y^2; y(0) = 50; y(0) = 100; and y(0) = 200
Correct answer: Critical points are at 0, 150. y(t)=0 (unstable, y(t) = 150 (asymptotically stable);
y=(150y_0)/(y_0-(y_0-150)e^(-300t)).
This models a population with logistic growth, tau=300 and Beta = 150.
The given equation in the section for population with logistic growth is y'=((tau)-(alpha)*y)*y OR y'=(tau)*(1-y/(Beta))*y. Alpha is a constant > 0.
My work is:
dy/dt = 300y-2y^2
dy/(y^2-150y) = -2*dt
1/150 * LN((y-150)/y) = -2T + C
(y-150)/y = e^(-300T+150C)
y=(-150*e^(300t))/(e^300t(e^c-1)+1)
At this point I realized I was swinging my shovel in the air trying to dig myself up out of a hole. Where'd I take a wrong turn/what rule did I break? Is anyone willing to work this problem through to solution?
The second question I have is for good measure to make certain I get the rest of the section. It's exercise 15, I think I need to see it solved for y.
y'=y^2+y-6; y(0) = -4; y(0) = -2; y(0) = 1; and y(0) = 3.
-3, 2; y(t) = -3 (asymptotically stable), y(t) = 2 (unstable), and y=[-3(y_0 - 2) - 2(y_0+3)e^(-5t)] / (y_0 - 2 - (y_0 + 3) e^(-5t)]. It does not, by the book's answer, model any specific system.
Thank you very much for your time!
2. ## Re: First order autonomous equations
Hey AnotherGeek.
This line:
dy/(y^2-150y) = -2*dt
looks ok but the next one doesn't. You should try completing the square and then solving the integral. It will be in terms of
Integral 1/[(y-75)^2 - b] for some non-zero b (Complete the square to get b). (Also look up arctanh function). | {"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": 0.8755000829696655, "perplexity": 1103.2625272690025}, "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-2014-42/segments/1413507445886.27/warc/CC-MAIN-20141017005725-00288-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/117441/the-compactness-of-the-unit-sphere-in-finite-dimensional-normed-vector-space | The compactness of the unit sphere in finite dimensional normed vector space
We define $(\mathbb{R}^m, \|.\|)$ to be a finite dimensional normed vector space with $\|.\|$ is defined to be any norm in $\mathbb{R^m}$. Let $S = \lbrace x \in \mathbb{R}^m: \| x\| = 1 \rbrace.$ Prove that $S$ is compact in $(\mathbb{R}^m, \|.\|).$
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have you heard of Heine-Borel? – Holdsworth88 Mar 7 '12 at 8:16
@Sulaiman: Since you are happy with the answers for this questions, I suggest you accept one of them by ticking it. It makes this site much more organised. While I'm at it, I also suggest to you to accept the (good) answers to your other questions. – Michalis Mar 7 '12 at 19:09
@Michalis Thank you but I already did. In fact I ticked up both answers if this is not a problem. – Zizo Mar 7 '12 at 19:32
@Sulaiman: What you did is upvote. For the OP there is also another option, that marks a question as "answered". It is the tick right below the up/downvote buttons. It is important that you mark answered questions, as the users of this site will be able to concentrate on the unanswered ones. – Michalis Mar 7 '12 at 19:38
Thank you @Michalis! sure! done!! – Zizo Mar 7 '12 at 19:51
You can use induction on $m$ and properties of $\mathbb R$ to show compacity using sequential compactness, which means the same thing for metric spaces. Now consider the norm induced on the space $\mathbb R \cong \mathbb R \times \{ 0 \} \times \dots \times \{ 0 \}$ viewed as a sub-metric-space of $\mathbb R^m$, and also consider the subsequence $^1 x_n$ induced by putting all the other components but the first equal to $0$. Therefore the first component is a sequence of real numbers. Since in the reals, every metric is equivalent to the absolute value metric in the following sense $$\forall (\mathbb R,d), \quad \exists c_1, c_2 > 0 \quad s.t. \quad \forall x,y \in \mathbb R, \quad c_1 d(x,y) \le |x-y| \le c_2 d(x,y).$$ One can deduce that the Bolzano-Weierstrass theorem also holds if we replace $| \cdot |$ by the induced metric from the norm in $\mathbb R^m$. Since the sequence $x_n$ is bounded, the sequence $^1x_n$ is also bounded in $\mathbb R$. Therefore there exists a subsequence of the sequence $x_n$ such that the first component converges. Repeat this procedure with the other components $^kx_n$ with $1 \le k \le n$, and you will get a subsequence that converges component by component, hence converges. This gives you for every sequence an element $x$ and a subsequence for which $x_n \to x$. Since $\| x_n \| = 1$ for every $n$, clearly $\|x \| = 1$, so that your subsequence converges in $S$ and we are done.
That is one way to do it ; if you have seen theorems in class that might help, perhaps they might make this less complicated.
Hope that helps,
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Thanks, this answered my question. Nice work. – Zizo Mar 7 '12 at 12:50
Is there any reference for this proof? – Zizo Mar 12 '12 at 3:30
I just wrote it ; so I guess you could call me the reference. – Patrick Da Silva Mar 12 '12 at 15:10
Use the Bolzano-Weierstrass theorem:
Since all the norms on $\mathbb{R}^m$ are equivalent, your subset will be closed and bounded in the euclidian norm $||\cdot||_2$, and hence compact.
Here is an exercise I found, that shows that all norms on $\mathbb{R}^m$ are equivalent: http://math.bu.edu/people/paul/771/equivalent_norms.pdf
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The compacity depends on the metric space, thus on the metric ; your way is one way to go, but it's not complete yet ; you need to show that compacity is a property invariant by equivalent metrics induced by norms. – Patrick Da Silva Mar 7 '12 at 8:56
The equivalence essentially relies on the fact that equivalent metrics induce open balls that can be included in one another. – Patrick Da Silva Mar 7 '12 at 9:17
@RagibZaman: This is not generally true, it depends on your base field. It is true if you have a complete valued field like $\mathbb{R},\mathbb{C}$ or $\mathbb{Q}_p$, but fails for example if you look at vector spaces over $\mathbb{Q}$ (e.g. number fields). – Michalis Mar 9 '12 at 11:21
@Patrick Da Silva: You are right, proving that the balls are included in each other you can show that two equivalent norms (with the "inequality"-definition) induce the same topology, now I understand your remark. – Michalis Mar 9 '12 at 11:25
@Michalis Sorry, I should have remembered that! Every normed vector space I've been studying lately has been over $\mathbb{R}$ or $\mathbb{C}$. – Ragib Zaman Mar 9 '12 at 13:31 | {"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": 0.9623887538909912, "perplexity": 270.017729502257}, "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-2013-48/segments/1387345759442/warc/CC-MAIN-20131218054919-00083-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://mathhelpforum.com/trigonometry/469-finding-length-arc-print.html | # Finding the length of an arc
• June 19th 2005, 06:18 AM
Justardms
Finding the length of an arc
:) Hi, just wanted to make sure that I am doing this correctly? Can someone please help me with Finding the length of an arc if the radius is 10" and the central angle is 45 deg. I need to find the answer in terms of pie. Thanks!
Also, I need to find a positive co-terminal angle for 120 deg. and one for pie/5
and a negative co-terminal angle for 420 deg. and one for 3 pie/5
can someone verify what the complement of angle of 20 deg is. and the supplement of an angle of 3pie / 5
How do I change a the following into decimal form? 23 deg. 30' 15"
THANK YOU VERY MUCH
• June 19th 2005, 10:10 AM
MathGuru
arc length give radius and angle
Welcom Justardms,
Just a note of advice, try to keep questions to a minimum of 1 or 2 per post as you will get better results. In keeping with my advice I will answer your first question only;)
To find the arc length you can use the equation $s=r\theta$
where s = arc length, r = radius, and $\theta$ = angle in radians
so
$s = 10"(45*\frac{\pi}{180})$
• June 19th 2005, 11:49 AM
Justardms
Is this correct?
So is the answer 10pie/4 ? First I have to convert deg. in to rads? correct? which is pie / 4 and then mutiple that by 10? correct? So the Arc lenght is 10 pie / 4? Or would you write 10 times pie / 4? Does it matter!?!?!?! THANKS and thanks for the tip about how to submit questions!
• June 19th 2005, 11:51 AM
Justardms
Since it is 10 inches * pie / 4, I am just not sure how I would write that? Thanks again
• June 19th 2005, 03:28 PM
Math Help
pi or 3.14
you can either answer with $\frac{10\pi}{4}inches = \frac{5\pi}{2}inches$ or substitue 3.14 for pi and multiply it out to get the answer in inches. | {"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": 0.8272402882575989, "perplexity": 941.5829893750247}, "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/1424936464809.62/warc/CC-MAIN-20150226074104-00269-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://www.ams.org/mathscinet-getitem?mr=35:2283 | MathSciNet bibliographic data MR211402 55.40 Wall, C. T. C. Finiteness conditions for ${\rm CW}$${\rm CW}$ complexes. II. Proc. Roy. Soc. Ser. A 295 1966 129–139. Links to the journal or article are not yet available
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews. | {"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": 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": 0.9535884857177734, "perplexity": 3543.6663480215525}, "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-2016-50/segments/1480698543170.25/warc/CC-MAIN-20161202170903-00472-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://publications.mfo.de/handle/mfo/1350/browse?type=author&value=Ingalls%2C+Colin | Now showing items 1-2 of 2
• #### The Magic Square of Reflections and Rotations
[OWP-2018-13] (Mathematisches Forschungsinstitut Oberwolfach, 2018-07-01)
We show how Coxeter’s work implies a bijection between complex reflection groups of rank two and real reflection groups in 0(3). We also consider this magic square of reflections and rotations in the framework of Clifford ...
• #### A McKay Correspondence for Reflection Groups
[OWP-2018-14] (Mathematisches Forschungsinstitut Oberwolfach, 2018-07-02)
We construct a noncommutative desingularization of the discriminant of a finite reflection group $G$ as a quotient of the skew group ring $A=S*G$. If $G$ is generated by order two reflections, then this quotient identifies ... | {"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": 0.8565515279769897, "perplexity": 905.1415472635522}, "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-2020-29/segments/1593655886706.29/warc/CC-MAIN-20200704201650-20200704231650-00068.warc.gz"} |
https://hidden-facts.info/relationship-between-and/what-is-the-relationship-between-wind-velocity-and-wave-height.php | # What is the relationship between wind velocity and wave height
In fluid dynamics, wind waves, or wind-generated waves, are surface waves that occur on the Given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a . The relationship between the wavelength, period, and velocity of any wave is. Predicting the height of the waves depending on the wind strength. It's time to tell you about forecasting wave height depending on the wind speed. the relationship between the dimensionless parameters of the waves obey universal laws. characterize wind, wave and currents, are taken from a m height meteorological mast, Relation between wave height, wave period and wind speed.
They are often found where there is a sudden rise in the sea floor, such as a reef or sandbar.
Deceleration of the wave base is sufficient to cause upward acceleration and a significant forward velocity excess of the upper part of the crest. The peak rises and overtakes the forward face, forming a "barrel" or "tube" as it collapses.
They tend to form on steep shorelines. These waves can knock swimmers over and drag them back into deeper water. When the shoreline is near vertical, waves do not break, but are reflected. Most of the energy is retained in the wave as it returns to seaward.
### Wind wave - Wikipedia
Interference patterns are caused by superposition of the incident and reflected waves, and the superposition may cause localised instability when peaks cross, and these peaks may break due to instability.
Airy wave theory Stokes drift in shallow water waves Animation Wind waves are mechanical waves that propagate along the interface between water and air ; the restoring force is provided by gravity, and so they are often referred to as surface gravity waves. As the wind blows, pressure and friction perturb the equilibrium of the water surface and transfer energy from the air to the water, forming waves. The initial formation of waves by the wind is described in the theory of Phillips fromand the subsequent growth of the small waves has been modeled by Milesalso in The wave conditions are: As a result, the surface of the water forms not an exact sine wavebut more a trochoid with the sharper curves upwards—as modeled in trochoidal wave theory.
Wind waves are thus a combination of transversal and longitudinal waves. When waves propagate in shallow waterwhere the depth is less than half the wavelength the particle trajectories are compressed into ellipses.
The empirical relation for the fully formed waves height, which can serve as the upper limit of assessment of wave height for any wind speed has been derived. Everything got more complicated. At the place of the wave prediction models of the first generation came second-generation model using the energy spectrum.
### Online calculator: The waves and the wind. Wave height statistical forecasting
In the early s, there were wave models of the third-generation 3G. Actually, we hadn't reached the fourth-generation models yet, but the most commonly used model is the third generation WAM model Hasselmann, S. Of course, there are still shortcomings, for example, these models can not predict the waves in a rapidly changing wind situations, but still 3G models provide a good result.
In the pre-computer era, you could use a model built on the nomogram for wave heights forecasting in relatively simple situations, such as pre-assessment or for small projects which have been given, for example, in Shore Protection Manual. There are 3 situations possible when the simplified prediction will give quite an exact estimation. The wind is blowing in a constant direction over some distance and not limited by time enough time - then the growth of the wave is determined and limited by the length of acceleration fetch-limited.
The wind rapidly increases within a short period of time and not limited by distance enough distance - then the growth of the wave is determined and limited by elapsed time duration-limited.
This occurs very rarely in nature. The wind is blowing in a constant direction at a sufficient distance and for a sufficient time so the wave will be fully formed fully developed wave under these conditions. Note that even in the open ocean waves rarely reach the limit values at wind speeds greater than 50 knots. Empirically, we obtained the following dependence for the case when wave growth is limited by the length of the acceleration.
The time waves require under the wind influence at the velocity on the distance to achieve the maximum possible for a given distance heights. The relationship between the significant wave height and the distance The relationship between the period of the wave and the distance The drag coefficient For a fully developed waves Also the transition from the duration of the wind to the length of the acceleration i.
Thus, if the duration of action and length of the acceleration of the wind is known, it is necessary to select the most restrictive value. If the wave generation height is limited by the time it is necessary to replace it by an equivalent distance and calculate the wave height based on it.
In case of shallow water equations remain valid except for the additional limitations under which the wave period can not exceed the following ratiosThen the order of the wave height prediction for the shallow water is as follows: Assess the wave period for a given distance and wind speed using conventional formula.
In the case of shallow water verify the conditions of the period and depth.
## The waves and the wind. Wave height statistical forecasting
If they are exceeded take the boundary value. In the case of the wave boundary value, find the distance corresponding to the generation of waves with such period.
Calculate the height in accordance with the value of the distance. If the wave height exceeds 0. Some more important notes These empirical formulas derived for relatively normal weather conditions, and are not applicable for the assessment of the wave height in the event of, for example, a hurricane.
Nomograms contained in the directory is built for the wind speed no higher than These empirical formulas are used for statistical forecasting of wave heights, so the height of these formulas is nothing more than a significant wave height determined by the dispersion of the wave spectrum as follows: This is a more modern definition of significant height of the waves, and the very first definition, which was given to Walter Munk during World War II, was: | {"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": 0.975754976272583, "perplexity": 558.5117314997462}, "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-2019-30/segments/1563195527204.71/warc/CC-MAIN-20190721205413-20190721231413-00459.warc.gz"} |
http://motls.blogspot.com/2015/10/picard-number-for-pedestrian-physicists.html?m=1 | ## Thursday, October 22, 2015
### Picard number for pedestrian physicists
Guest blog by Monster from a U.K. research university founded in 1907
Let me try to explain in a maybe more physicists-friendly way essentially the same things that dalpezzo already mentioned beneath the "blog post about 1729". It is an exercise for experts to notice the hidden assumptions and oversimplified explanations.
Let $$X$$ be a compact manifold. Let us consider $$U(1)$$ gauge theory on $$X$$, i.e usual Maxwell's electromagnetism. Locally on $$X$$, the gauge field is a 1-form $$A$$ and the field strength $$F=dA$$ is a 2-form. If $$X$$ has non-trivial 2-cycles, one can have non-trivial fluxes of $$F$$ through these 2-cycles. The number of topologically inequivalent 2-cycles in $$X$$ is $$B_2$$, the second Betti number of $$X$$ and, as the fluxes are quantized, a flux configuration is given by a collection $$(n_1,\dots, n_{B_2})$$ of $$B_2$$ integers. It is easy to show that for any flux configuration, there exists a gauge field with the prescribed fluxes.
Assume that $$X$$ has the extra structure of a complex manifold. It means that locally on $$X$$, we have a notion of holomorphic coordinates $$z_i$$ and antiholomorphic coordinates $$\bar{z}_i$$. It is then possible to decompose the field strength as$F=F^{2,0}+F^{1,1}+F^{0,2}$ where the $$(2,0)$$ part $$F^{2,0}$$ only contains terms proportional to $$dz_i \wedge dz_j$$, the $$(1,1)$$ part $$F^{1,1}$$ only contains term proportional to $$dz_i \wedge d\bar{z}_j$$, and the $$(0,2)$$ part $$F^{0,2}$$ only contains terms proportional to $$d\bar{z}_i \wedge d\bar{z}_j$$. In terms of indices, the $$(p,q)$$ part has $$p$$ holomorphic indices and $$q$$ antiholomorphic indices.
When the complex structures changes continuously, the holomorphic/antiholomorphic coordinates change continuously and so the above decomposition of the field strength changes continuously too. Now, given a complex structure on $$X$$ and a flux configuration, one can ask the following question; is there a gauge field with the prescribed fluxes such that the associated field strength satisfies$F^{2,0}=F^{0,2}=0 \quad ?$ This condition is a first order linear partial differential equation on the gauge field. It is always possible to solve it locally but an obstruction to glue these local solutions and to obtain a global solution can exist.
As the equation is linear, the set of flux configurations such that there exists a solution is a sublattice of the lattice of flux configurations. The Picard number of $$X$$ is the rank of this sublattice, i.e. the number of independent flux configurations generating the space of flux configurations such that there exists a solution to the equation $$F^{2,0}=F^{0,2}=0$$. The Picard number is an integer between $$0$$ and $$B_2$$ and in general depends on the complex structure of $$X$$.
The field strength $$F$$ of a gauge field configuration satisfies flux quantization but it is not in general the case of $$F^{2,0}$$, $$F^{1,1}$$ or $$F^{0,2}$$. In general their fluxes are complex numbers. So it is useful to introduce the space of complexified flux configurations made of $$B_2$$-uples $$(a_1,...,a_{B_2})$$ of complex numbers. The space of integral fluxes is a discrete subset of this complex vector space. One can show that the decomposition in $$(2,0)$$, $$(1,1)$$, and $$(0,2)$$ parts extend to the space of complexified flux configurations. The complex dimension of the space
of complexified flux configurations of type $$(p,q)$$ is called the Hodge number $$h^{p,q}$$ of $$X$$. One has$B_2=h^{2,0}+h^{1,1}+h^{0,2}$ and $h^{2,0}=h^{0,2}.$ One can show that the Hodge numbers do not change when the complex structure of $$X$$ moves continuously but the corresponding subspaces of the space of complexified flux configurations in general moves continuously. The (integral) flux configurations such that there exists a gauge
field with these prescribed fluxes such that $$F^{2,0}=F^{0,2}=0$$ are exactly the (integral) flux configurations leaving inside the $$(1,1)$$ subspace of the space of complexified flux configurations. In particular, the Picard number is always between $$0$$ and $$h^{1,1}$$.
So the picture to have is mind is the following: a big complex vector space, a discrete lattice of integral points and a specific subspace inside it. The Picard number measures the amount of integral points in the specific subspace. When the complex subspace moves, due to a change in the complex structure of $$X$$, the Picard number in general changes. More precisely, when the complex subspace only moves a bit, an integral point which was not inside cannot become suddenly inside but an integral point inside can suddenly moves out. It means that the Picard number can get enhanced at special points of the moduli space of complex structures on $$X$$. The whole subtlety of the Picard number comes from this interplay between the discrete set of (integral) flux configurations and the continuous subspace of complexified flux configurations of type $$(1,1)$$.
If $$h^{2,0}=0$$, the above subtlety is not here: any complexified configuration is of type $$(1,1)$$, the equation $$F^{2,0}=F^{0,2}=0$$ has always a solution and the Picard number is simply the second Betti number $$B_2$$ and in particular does not depend on the complex structure. It is what happens for projective spaces, del Pezzo surfaces or Calabi-Yau manifolds (in the strict sense: holonomy equal to $$SU(n)$$ and not just contained in $$SU(n)$$). In all these examples, the Picard number is not something interesting: it is something we already knew, i.e. $$B_2$$.
To have the full subtle story of variations of Picard numbers, one needs to have $$h^{2,0}$$ non zero. It is for example the case for complex tori or $$K3$$ surfaces. For $$K3$$ surfaces, we have$B_2=22, \quad h^{2,0}=h^{0,2}=1, \quad h^{1,1}=20.$ The moduli space of complex structures on a $$K3$$ surface is of complex dimension 20. A generic $$K3$$ surface has Picard number $$0$$. There is a special locus of complex codimension $$1$$ at which the Picard number is enhanced to $$1$$. There is a special locus of complex codimension $$2$$ at which the Picard number is enhanced to $$2$$ and so on until a special locus of complex codimension $$20$$, i.e. dimension $$0$$, at which the Picard number is enhanced to $$20$$, its maximal possible value. Each of these special locus of complex dimension $$k$$ is fairly complicated: it is a countable union of varieties of dimension $$k$$.
For example, the space of $$K3$$ surfaces of Picard number $$20$$ is of dimension $$0$$ but it is made of a countably infinite number of points and is in fact dense in the full moduli space of $$K3$$ surfaces: it is as the rational numbers in the real numbers and so the full picture of the moduli space of $$K3$$ surfaces with the various loci of given Picard numbers is extremely intricate.
Under nice hypothesis ($$X$$ algebraic), there is a more geometric interpretation of the Picard number. A flux configuration is the data for each 2-cycle of an integer. If $$X$$ is of (real) dimension $$n$$, one can interpret these integers as prescribed intersection numbers with the various 2-cycles for a $$(n-2)$$-cycle. If $$X$$ is a (algebraic) complex manifold, one can show that the existence of a gauge field with $$F^{2,0}=F^{0,2}=0$$ and prescribed flux configuration is equivalent to the existence of an holomorphic representative for the $$(n-2)$$-cycle determined by the flux configuration. In other words, the Picard number measures the amount of holomorphic hypersurfaces (complex codimension $$1$$, i.e. real codimension $$2$$) in $$X$$. For a $$K3$$ surface, of real dimension $$4$$, i.e. complex dimension $$2$$, an holomorphic hypersurface is the same thing that an holomorphic curve (complex dimension $$1$$, i.e. real dimension $$2$$). For example, a generic $$K3$$ surface has Picard number $$0$$ and so has no holomorphic curves in it. In contrary, a $$K3$$ surface with high Picard number has many holomorphic curves in it and so a rich complex geometry. Any non-trivial holomorphic geometry in a $$K3$$ surface in general requires a high enough Picard number.
For instance, to compactify F-theory on a $$K3$$ surface, one needs an elliptic fibration with a section. The fiber of the elliptic fibration is a non-trivial holomorphic curve in the $$K3$$ surface and similarly for the image of the section, and so such $$K3$$ surface has at least Picard number $$2$$. This kind on restriction on the allowed $$K3$$ surfaces for a F-theory compactification is not very surprising: IIB superstring compactified on a $$K3$$ surface is dual to heterotic string on $$T^4$$ and F-theory compactified on a $$K3$$ surface is dual to heterotic string on $$T^2$$. As there are clearly less parameters in $$T^2$$ that in $$T^4$$, the range of allowed $$K3$$ on the F-theory side has to be somehow limited.
Similarly, I think that when one writes a $$K3$$ surface in a relatively simple explicit form, something one wants to do for explicit computations and explicit checks of various dualities, one generally obtains a $$K3$$ surface with relatively high Picard number precisely because the ability to write a simple description of an object is a sign of its deeper and richer structure.
But in general, it seems quite difficult to find a direct physical meaning to the Picard number or a jump in the Picard number. For example, moving in the moduli space of $$K3$$ surfaces, when the Picard number jumps, nothing happens to the topology, nothing becomes singular, it is a really a subtle modification of the complex geometry and so does not correspond to something as brutal as a topology change transition or gauge symmetry enhancement. It is has a physical meaning, this one has to be relatively subtle. I can think of two examples of such physical meaning and both are about $$K3$$ surfaces of maximal Picard number, i.e. Picard number $$20$$.
The first one is due to Moore and is about the attractor mechanism. Let us have a look at type IIB superstring compactified on the Calabi-Yau 3-fold obtained as a product of a $$K3$$ surface by an elliptic curve. The vector multiplet moduli space is the complex moduli space of this Calabi-Yau
and so in particular contains the moduli space of $$K3$$ surfaces. One can construct BPS black holes in four dimensions by wrapping D3-branes over 3-cycles of the Calabi-Yau. The possible charges for this BPS black holes form a infinite discrete set. The choice of a vacua of the theory is a choice of asymptotic value for the vectomultiplet moduli at infinity. But when we have a BPS black hole and when we move from infinity toward the black hole, the value of the vector multiplet get modified, as prescribed by the supergravity equation of motion. The attractor mechanism describes this evolution of the moduli and asserts that the value at the horizon of the black hole only depends on the charge of the black hole and not of the asymptotic value (as expected by general black hole entropy considerations: the entropy only depends on the charge and so the local geometry near the horizon should also only depends on the charge). So for each choice of charge, there should be a particular $$K3$$ surface describing the compactified geometry at the horizon. The result is that these $$K3$$ surfaces are exactly the $$K3$$ surfaces of maximal Picard number.
Second, Gukov and Vafa have speculated about when a non-linear sigma model, defining a two dimensional conformal field theory, is in fact a rational conformal field theory. A rational CFT is a CFT with the local fields organized in finitely many irreducible representations of a chiral algebra. The rationality of a CFT is a quite subtle property, not preserved in general under deformations. Applied to the case of $$K3$$ surfaces, the proposal of Gukov and Vafa asserts that the supersymmetric sigma model of target a $$K3$$ surface is a rational SCFT if and only the $$K3$$ surface has maximal Picard number. I think that the validity of this proposal is an open 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": 0.9370518922805786, "perplexity": 210.05170399440243}, "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/1498128320841.35/warc/CC-MAIN-20170626170406-20170626190406-00669.warc.gz"} |
http://math.stackexchange.com/questions/244341/calculus-of-variation-didos-problem?answertab=votes | # Calculus of Variation (Dido's Problem?)
Given the length L of a curve going two given point $(a,\alpha)$, $(b,\beta)$ find the equation of the curve so that the curve together with the interval $[a,b]$ encloses the largest area. Am I correct in thinking this is Dido's problem? Is it possible to use Green's theorem to find the equation?
-
Yes I think this is dido's problem or some variation (sorry for the pun). But think about what you're trying to extremize and with what constraint. You are trying to extremize $\int^b_a y(x) \text{d}x$ together with the constraint that $\int^b_a \sqrt{1+y'(x)^2}dx = L$ and you are also given that the endpoints are fixed i.e, $y(a) = \alpha$ and $y(b) = \beta$. What method does one usually employ if they want to extremize a functional subject to a constraint and given that the endpoints are fixed? You use Lagrange multipliers together with the Euler-Lagrange equation. | {"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": 0.8907087445259094, "perplexity": 193.8433524464498}, "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-2014-10/segments/1394010352519/warc/CC-MAIN-20140305090552-00043-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1569900/determine-whether-a-matrix-is-othrogonal | # Determine whether a matrix is othrogonal
I need to determine if the following matrix is orthogonal
$A = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 0 & 0 \\ 1 & -1 \end{pmatrix}$
Here is what I did:
$u \cdot v = (\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}) + (\frac{1}{\sqrt{2}} \cdot -\frac{1}{\sqrt{2}}) = 0$
$||u|| = {\sqrt{(\frac{1}{\sqrt{2}})^2 + 0 + (\frac{1}{\sqrt{2}}}})^2 = 1$
$||v|| = {\sqrt{(\frac{1}{\sqrt{2}})^2 + 0 + (-\frac{1}{\sqrt{2}}}})^2 = 1$
This should indicate that the matrix is orthogonal, however, the answer in the book says it is not orthogonal and I can't see where I went wrong
• Shouldn't orthogonal matices be $n\times n$? – user228113 Dec 10 '15 at 21:14
• I suppose that is where my mistake is. I guess I got a bit too caught up and overlooked something as simple as that – user273323 Dec 10 '15 at 21:16
• Orthogonal matrices are square matrices that have the property: $A^T = A^{-1}$. – Nathan Marianovsky Dec 10 '15 at 21:16
Your matrix satisfies $A^TA=I_2$, that is the columns are orthonormal. Such matrices are useful when you want to define the polar decomposition of a $m\times n$ matrix $M$ with $m\geq n$ and $rank(M)=n$. The decomposition is $M=US$ where $U$ is a $m\times n$ matrix with orthonormal columns and $S$ is a $n\times n$ SDP matrix. More precisely $S=\sqrt{M^TM}$ and $U=MS^{-1}$. | {"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": 0.9702929258346558, "perplexity": 161.53241151967183}, "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-2021-25/segments/1623487608856.6/warc/CC-MAIN-20210613131257-20210613161257-00428.warc.gz"} |
https://www.physicsforums.com/threads/elastic-collision-symbolic-question.222342/ | # Elastic Collision - Symbolic Question
1. Mar 16, 2008
### kanavulator
1. The problem statement, all variables and given/known data
Suppose you hold a small ball of mass m1 in contact with, and directly over, the center of mass of a large ball of mass m2. If you then drop the small ball a short time after dropping the large ball, the small ball rebounds with surprising speed. If we ignore air resistance and assume the large ball makes an elastic collision with the floor and then makes an elastic collision with the still descending small ball and that large ball has much larger mass than the small ball then:
a) If the velocity of the small ball immediately before the collision is v, what is the velocity of the large ball? (in terms of v)
b) What is the velocity of the small ball immediately after its collision with the large ball? (in terms of v)
c) What is the ratio of the small ball's rebound distance to the distance it fell before the collision? (a number)
2. Relevant equations
1/2mv^2 + mgh = 1/2mv^2 + mgh
Elastic collision: V1 = -V2
3. The attempt at a solution
a. -V
b. ?
c. ?
2. Mar 16, 2008
### Staff: Mentor
Hint: Analyze the problem in the center of mass frame, then transform back to the lab frame. (Assume m2 >> m1.)
3. Mar 16, 2008
### kanavulator
That...doesn't make a bit of sense to me. Pardon my lack of knowledge, but I'm not really familiar with the terms you were using, Doc Al.
4. Mar 16, 2008
### Staff: Mentor
No problem. Sometimes problems are easier to solve in certain frames of reference--but let's forget that for the moment.
What do you know about elastic collision?
You'll need this for part c.
What does this mean? If you mean the relative velocity reverses: Great! Use it.
Assuming you meant -v (the same v as the small ball): Good!
Here's a hint for part b: If a ping pong ball hits a bowling ball, what happens to the velocity of the bowling ball? (To a good approximation.) | {"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": 0.900945246219635, "perplexity": 906.814050199842}, "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-2016-50/segments/1480698541322.19/warc/CC-MAIN-20161202170901-00224-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/141115-parametric-equations.html | 1. ## Parametric equations.
The following problem problem I need help with:-
The parametric curve de ned by the equations
x
(t) = cos(t); y(t) = sin(3t); 0 <=t <= 2:
a) Find a formula which represents the slope of the tangent line to the curve at the point (
x(t); y(t)):
b) Find the points (values of
t and corresponding (x; y) coordinates) where the curve has a horizontal tangent line, and nd the points where the curve has a vertical tangent line.
c) Use the information you found in part b) to draw a sketch of the curve.
d) Set-up,
but do not compute, an integral that represents the length of this parametric curve.
2. [QUOTE=Sally_Math;500410]
The following problem problem I need help with:-
The parametric curve de ned by the equations
x
(t) = cos(t); y(t) = sin(3t); 0 <=t <= 2:
a) Find a formula which represents the slope of the tangent line to the curve at the point (
x(t); y(t)):
The "slope of the tangent line" is $\frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
b) Find the points (values of
t and corresponding (x; y) coordinates) where the curve has a horizontal tangent line, and nd the points where the curve has a vertical tangent line.
The tangent line will be horizontal when $\frac{dy}{dt}= 0$ and vertical when $\frac{dx}{dt}= 0$.
c) Use the information you found in part b) to draw a sketch of the curve.
d) Set-up,
but do not compute, an integral that represents the length of this parametric curve.
$\int \sqrt{(\frac{dx}{dt})^2+ \frac{dy}{dt})^2} dt$ | {"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": 0.9107535481452942, "perplexity": 848.8243251333419}, "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/1498128319688.9/warc/CC-MAIN-20170622181155-20170622201155-00233.warc.gz"} |
https://math.stackexchange.com/questions/570954/simple-undergraduate-series-quesiton | consider $\displaystyle \sum_{n=1}^\infty (-1)^{n-1}a_n$ where $(a_n)$ is a monotone decreasing sequence of nonnegative numbers with $a_n \rightarrow 0$ by the alternating series test, series of this form always converge.
Show that $0 \leq \displaystyle\sum_{n=1}^\infty (-1)^{n-1}a_n \leq a_1$
There is a hint in the question:
if $(S_N)$ denotes the sequence of partial sums, consider the subsequences $(S_{2N})$ and $(S_{2N-1})$ can you show that one is decreasing while the other is increasing?
I have proven the hint - however I am unable to proceed from here - Could someone please direct me to the correct direction?
We will show that $0\leq S_{n}\leq a_{1}$ for all $n\in\mathbb{N}$ (and hence $0\leq\lim S_{n}\leq a_{1}$).
We know that $0\leq a_{1}-a_{2}=S_{2}\leq S_{1}= a_{1}$ and you already showed that $S_{2n}$ is increasing while $S_{2n-1}$ is decreasing.
Now let $k\in\mathbb{N}$ and notice that $S_{2k}\leq S_{2k-1}$. Because $S_{2n}$ is increasing and $S_{2n-1}$ is decreasing we now know that $0\leq S_{2}\leq S_{2k}\leq S_{2k-1}\leq S_{1}=a_{1}$. To finish note that, for any $n\in\mathbb{N}$, $S_{n}=S_{2k}$ or $S_{n}=S_{2k-1}$ for some $k\in\mathbb{N}$.
$$S_{n}=a_1\underbrace{-a_2+a_3}_{\leq 0}\underbrace{-a_4+a_5}_{\leq 0}\cdot\cdot \cdot\underbrace{-a_{n-1}+a_{n}}_{\leq 0}\leq a_1$$.
On the other hand if n is even than $$S_{n}=a_1\underbrace{-a_2+a_3}_{\leq 0}\underbrace{-a_4+a_5}_{\leq 0}\cdot\cdot \cdot\underbrace{-a_{n-2}+a_{n-1}}_{\leq 0}\underbrace{-a_{n}}_{\leq 0}\leq a_1$$ So in any case $S_n\leq a_1$ | {"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": 0.9956572651863098, "perplexity": 56.5382320681935}, "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/1614178358203.43/warc/CC-MAIN-20210227054852-20210227084852-00425.warc.gz"} |
http://csd.newcastle.edu.au/chapters/chapter17.html | You are here : Control System Design - Index | Book Contents | Chapter 17
## 17. Linear State Space Models
### Preview
We have seen that there are many alternative model formats that can be used for linear dynamic systems. In simple SISO problems, any representation is probably as good as any other. However, as we move to more complex problems (especially multivariable problems) it is desirable to use special model formats. One of the most flexible and useful structures is the state space model. As we saw in Chapter 3, this model takes the form of a coupled set of first order differential (or difference) equations. This model format is particularly useful with regard to numerical computations.
State space models were briefly introduced in Chapter 3. Here we will examine linear state space models in a little more depth for the SISO case. Note, however, that many of the ideas will directly carry over to the multivariable case presented later. In particular, we will study
• similarity transformations and equivalent state representations
• state space model properties
• controllability, reachability and stabililizability
• observability, reconstructability and detectability
• special (canonical) model formats
The key tools used in studying linear state space methods are linear algebra and vector space methods. The reader is thus encouraged to briefly review these concepts as a prelude to reading this chapter.
### Summary
• State variables are system internal variables, upon which a full model for the system behavior can be built. The state variables can be ordered in a state vector.
• Given a linear system, the choice of state variables is not unique. However,
• the minimal dimension of the state vector is a system invariant,
• there exists a nonsingular matrix which defines a similarity transformation between any two state vectors, and
• any designed system output can be expressed as a linear combination of the states variables and the inputs.
• For linear, time invariant systems the state space model is expressed in the following equations:
• Stability and natural response characteristics of the system can be studied from the eigenvalues of the matrix ( , ).
• State space models facilitate the study of certain system properties which are paramount in the solution control design problem. These properties relate to the following questions
• By proper choice of the input u, can we steer the system state to a desired state (point value)? controllability
• If some states are or uncontrollable, will these states generate a time decaying component? (stabilizability)
• If one knows the input, u(t) for , can we infer the state at time t=t0 by measuring the system output, y(t) for ? (observability)
• If some of the states are unobservable, do these states generate a time decaying signal? (detectability)
• Controllability tells us about the feasibility to control a plant.
• Observability tells us about whether it is possible to know what is happening in a given system by observing its outputs.
• The above system properties are system invariants. However, changes in the number of inputs, in their injection points, in the number of measurements and in the choice of variables to be measured may yield different properties.
• A transfer function can always be derived from a state space model.
• A state space model can be built from a transfer function model. However, only the completely controllable and observable part of the system is described in that state space model. Thus the transfer function model might be only a partial description of the system.
• The properties of individual systems do not necessarily translate unmodified to composed systems. In particular, given two systems completely reachable, observable, controllable and reconstructible, their cascade connection:
• is not completely observable if a pole of the first system coincides with a zero of the second system (pole-zero cancellation),
• is not detectable if the pole-zero cancellation affects an unstable pole,
• is not completely controllable if a zero of the first system coincides with a pole of the second system (zero-pole cancellation), and
• is not stabilizable if the zero-pole cancellation affects a NMP zero
• this chapter provides a foundation for the design criteria which states that one should never attempt to cancel unstable poles and zeros.
Previous - Chapter 16 Up - Book Contents Next - Chapter 18 | {"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": 0.8766963481903076, "perplexity": 596.6161507918613}, "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-2018-13/segments/1521257644271.19/warc/CC-MAIN-20180317035630-20180317055630-00781.warc.gz"} |
http://www.formuladirectory.com/user/formula/285 | HOSTING A TOTAL OF 318 FORMULAS WITH CALCULATORS
Length Of The Hypotenuse Of Right Triangle
In mathematics, the Pythagorean theorem — or Pythagoras' theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle).
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
$\sqrt{{a}^{2}+{b}^{2}}$
Here,a=base,b=perpendicular and c=Hypotenuse.
ENTER THE VARIABLES TO BE USED IN THE FORMULA
SOLVE FORMULA | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.9647494554519653, "perplexity": 290.0779505337279}, "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-2018-51/segments/1544376824119.26/warc/CC-MAIN-20181212203335-20181212224835-00495.warc.gz"} |
http://www.researchgate.net/publication/252304359_Baxter%27sT-Q_equation_SU%28_N%29SU%282%29_N_-_3_correspondence_and_Omega-deformed_Seiberg-Witten_prepotential | Article
# Baxter'sT-Q equation, SU( N)/SU(2) N - 3 correspondence and Omega-deformed Seiberg-Witten prepotential
Journal of High Energy Physics (Impact Factor: 5.62). 01/2011; 9. DOI:10.1007/JHEP09(2011)125
ABSTRACT We study Baxter's T-Q equation of XXX spin-chain models under the semiclassical limit where an intriguing SU( N)/SU(2) N-3 correspondence is found. That is, two kinds of 4D {N} = 2 superconformal field theories having the above different gauge groups are encoded simultaneously in one Baxter's T-Q equation which captures their spectral curves. For example, while one is SU( N c ) with N f = 2 N c flavors the other turns out to be {{SU}}{(2)^{{N_c} - 3}} with N c hyper-multiplets ( N c > 3). It is seen that the corresponding Seiberg-Witten differential supports our proposal.
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##### Article: On non-stationary Lam\'e equation from WZW model and spin-1/2 XYZ chain
[hide abstract]
ABSTRACT: We study the link between WZW model and the spin-1/2 XYZ chain. This is achieved by comparing the second-order differential equations from them. In the former case, the equation is the Ward-Takahashi identity satisfied by one-point toric conformal blocks. In the latter case, it arises from Baxter's TQ relation. We find that the dimension of the representation space w.r.t. the V-valued primary field in these conformal blocks gets mapped to the total number of chain sites. By doing so, Stroganov's "The Importance of being Odd" (cond-mat/0012035) can be consistently understood in terms of WZW model language. We first confirm this correspondence by taking a trigonometric limit of the XYZ chain. That eigenstates of the resultant two-body Sutherland model from Baxter's TQ relation can be obtained by deforming toric conformal blocks supports our proposal.
Journal of High Energy Physics 02/2012; 2012(6). · 5.62 Impact Factor
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##### Article: Liouville theory, \mathcal{N} = 2 gauge theories and accessory parameters
[hide abstract]
ABSTRACT: The correspondence between the semiclassical limit of the DOZZ quantum Liouville theory and the Nekrasov-Shatashvili limit of the $\mathcal{N} = 2$ (Ω-deformed) U(2) super-Yang-Mills theories is used to calculate the unknown accessory parameter of the Fuchsian uniformization of the 4-punctured sphere. The computation is based on the saddle point method. This allows to find an analytic expression for the N f = 4, U(2) instanton twisted superpotential and, in turn, to sum up the 4-point classical block. It is well known that the critical value of the Liouville action functional is the generating function of the accessory parameters. This statement and the factorization property of the 4-point action allow to express the unknown accessory parameter as the derivative of the 4-point classical block with respect to the modular parameter of the 4-punctured sphere. It has been found that this accessory parameter is related to the sum of all rescaled column lengths of the so-called ’critical’ Young diagram extremizing the instanton ’free energy’. It is shown that the sum over the ’critical’ column lengths can be rewritten in terms of a contour integral in which the integrand is built out of certain special functions closely related to the ordinary Gamma function.
Journal of High Energy Physics 01/2012; 2012(5). · 5.62 Impact Factor | {"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": 0.909515380859375, "perplexity": 1003.455439536521}, "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-2014-15/segments/1397609537186.46/warc/CC-MAIN-20140416005217-00018-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/help-for-the-stochastic-differential-equations.540552/ | # Help for the stochastic differential equations
1. Oct 15, 2011
### ptc_scr
Hi,
Could some one help me to solve the equations ?
dX =sqrt(X) dB
where X is a process; B is a Brownian motion with B(0,w) =0;sqrt(X) is squart root of X.
2. Oct 15, 2011
### chiro
Hey ptc_scr and welcome to the forums.
In these forums, we require the poster to show any work that they have done before we can help them. We do this so that you can actually learn for yourself what is going on so that you do the work and end up understanding it yourself.
So first I ask you to show any working, and secondly what do you know about solving SDE's with Brownian motion? Do you know about Ito's lemma and its assumptions?
3. Oct 16, 2011
### ptc_scr
Hi,
I just try to assign Y=sqrt(X) and use Ito lemma to solve the problem. so
dY= 1/2 dB+ 1/(4Y) dt.
Obviously, we cannot put Y one left side. So the substitution is failed.
ANy one can show me how to find a good substitution or show me it is impossible to solve the problem ?
But for existence and uniqueness theorm for Ito-diffusion, it seems that the problem can be solve ?
because sqrt(X) <= C(1+|X|) for some certain C.
Thanks | {"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": 0.9128122925758362, "perplexity": 1180.4061280547812}, "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/1505818689900.91/warc/CC-MAIN-20170924081752-20170924101752-00093.warc.gz"} |
https://brilliant.org/problems/volume-of-the-tetrahedron/ | # Volume of the tetrahedron
Geometry Level pending
In a regular hexagonal prism $$ABCDEF-GHIJKL$$, whose volume is $$90 m^3$$, calculates the volume of the tetrahedron $$ADHK$$.
× | {"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": 0.8461540937423706, "perplexity": 1838.345706205665}, "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-2017-13/segments/1490218189252.15/warc/CC-MAIN-20170322212949-00581-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/54078/ratio-of-distance-between-mirror-and-person/54079 | # Ratio of distance between mirror and person
In perspective of a given example, if a man was to stand $2\ m$ away from a mirror which was $0.9\ m$ in height and was able to see his full reflection, what would the height of the mirror have to be if the man was now $6\ m$ away from the mirror and was to maintain a full reflection? Would the mirror be equal to or less than the original height and why? This scenario seems to have caught me out multiple times. So, some reasoning would be much appreciated.
-
You haven't said what the condition for the height the mirror would have to be is. Perhaps the requirement is that the man can see his full reflection? This is not an uncommon example problem in geometric optics and the answer is slightly surprising but very easy to get using the rules of reflection. – dmckee Feb 15 '13 at 20:47
Ah thank you for pointing out that critical fact! Post was edited to show that requirement – Jared Ping Feb 15 '13 at 20:49 | {"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": 0.8817617893218994, "perplexity": 330.6988372620073}, "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/1454701166570.91/warc/CC-MAIN-20160205193926-00204-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/linear-algebra-inverse-of-the-sum-of-two-matrices.421434/ | # Homework Help: Linear algebra: inverse of the sum of two matrices
1. Aug 10, 2010
### degs2k4
1. The problem statement, all variables and given/known data
Show that $$(I-A)^{-1} = I + A + A^2 + A^3$$ if $$A^4=0$$
3. The attempt at a solution
I found at Google Books some kind of formula for it:
However, I think I should develop some kind of series for it using I = A(A^-1), I tried but I haven't been successful...
2. Aug 10, 2010
### Dick
Just multiply (I-A) by I+A+A^2+A^3 and see if you get I.
3. Aug 10, 2010
### degs2k4
Thanks for your reply, got it solved! | {"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": 0.9424253702163696, "perplexity": 1806.5628791691822}, "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-2018-26/segments/1529267864019.29/warc/CC-MAIN-20180621020632-20180621040632-00054.warc.gz"} |
http://repo.scoap3.org/record/32971 | # Testing production scenarios for (anti-)(hyper-)nuclei and exotica at energies available at the CERN Large Hadron Collider
Bellini, Francesca (European Organisation for Nuclear Research (CERN), 1211 Geneva, Switzerland) ; Kalweit, Alexander P. (European Organisation for Nuclear Research (CERN), 1211 Geneva, Switzerland)
28 May 2019
Abstract: We present a detailed comparison of coalescence and thermal-statistical models for the production of (anti-) (hyper-)nuclei in high-energy collisions. For the first time, such a study is carried out as a function of the size of the object relative to the size of the particle emitting source. Our study reveals large differences between the two scenarios for the production of objects with extended wave functions. While both models give similar predictions and show similar agreement with experimental data for (anti-)deuterons and (anti-)${}^{3}\mathrm{He}$ nuclei, they largely differ in their description of (anti-)hypertriton production. We propose to address experimentally the comparison of the production models by measuring the coalescence parameter systematically for different (anti-)(hyper-)nuclei in different collision systems and differentially in multiplicity. Such measurements are feasible with the current and upgraded Large Hadron Collider experiments. Our findings highlight the unique potential of ultrarelativistic heavy-ion collisions as a laboratory to clarify the internal structure of exotic QCD objects and can serve as a basis for more refined calculations in the future.
Published in: Physical Review C 99 (2019) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.8573830127716064, "perplexity": 1794.9020492557142}, "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/1560627999163.73/warc/CC-MAIN-20190620065141-20190620091141-00079.warc.gz"} |
http://math.stackexchange.com/questions/255454/about-sylow-systems | "$G$ is a solvable group if and only if $G$ has a Sylow system"
(Sylow system: a set $S$ of Sylow subgroups of $G$, one for each prime dividing $|G|$, so that if $P$, $Q$ $\in{S}$, then $PQ=QP$). | {"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": 0.9918639659881592, "perplexity": 85.36948538091323}, "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-2014-23/segments/1406510264270.11/warc/CC-MAIN-20140728011744-00058-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/simple-classic-action-integral.658356/ | # Simple classic action integral
1. Dec 11, 2012
### dydxforsn
I'm trying to solve this simple problem (it's the first problem of Quantum Mechanics and Path Integrals by Feynman, I feel like an idiot not being able to do it....) It's just solving for the action, S, of a free particle (no potential, only kinetic energy..)
So it should just be $$S = \int_{t_a}^{t_b}{\frac{m}{2} (\frac{dx}{dt})^2 dt}$$
which according to the book is simply $$S = \frac{m}{2} \frac{(x_b - x_a)^2}{t_b - t_a}$$
I've tried a couple of different ways to reason myself into this solution but I can't seem to figure it out.
2. Dec 11, 2012
### Mute
What have you tried so far? What did you plug in for $dx/dt$?
3. Dec 11, 2012
### dydxforsn
Incredibly wrong stuff, heh..
Yeah I'm an idiot. I was supposed to just plug in $v = \left ( \frac{x_{b} - x_{a}}{t_{b} - t_{a}} \right )$ because 'v' is constant from the Euler-Lagrange equation..
Thanks for helping me see what should have been obvious >_< I was hell bent on doing things symbolically and didn't seem to care about the appearance of the end point 'x' values.. These should have been very suggestive.
Last edited: Dec 11, 2012
4. Dec 11, 2012
### Mute
Great! You figured it out! Yeah, with a problem like this it helps to remember that the action is a functional of $x(t)$ and $\dot{x}(t)$, so you get different answers depending on which function x(t) you use. Of course, varying the action with respect to x(t) (giving the Euler-Lagrange equations) yields the equation of motion for the classical path. The problem wanted the action of a classical path with boundary values $x(t_a) = x_a$ and $x(t_b) = x_b$.
It can take some practice seeing these sorts of problems a few times before it clicks. =)
Similar Discussions: Simple classic action integral | {"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": 0.8906840085983276, "perplexity": 541.9356910328266}, "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/1508187824325.29/warc/CC-MAIN-20171020192317-20171020212317-00755.warc.gz"} |
https://chemrxiv.org/articles/Machine-Learnt_Fragment-Based_Energies_for_Crystal_Structure_Prediction/7583294/1 | ## Machine-Learnt Fragment-Based Energies for Crystal Structure Prediction
2019-01-14T19:10:39Z (GMT) by
Crystal structure prediction involves a search of a complex configurational space for local minima corresponding to stable crystal structures, which can be performed efficiently using atom-atom force fields for the assessment of intermolecular interactions. However, for challenging systems, the limitations in the accuracy of force fields prevents a reliable assessment of the relative thermodynamic stability of potential structures. Here we present a method to rapidly improve force field lattice energies by correcting two-body interactions with a higher level of theory in a fragment-based approach, and predicting these corrections with machine learning. We find corrected lattice energies with commonly used density functionals and second order perturbation theory (MP2) all significantly improve the ranking of experimentally known polymorphs where the rigid molecule model is applicable. The relative lattice energies of known polymorphs are also found to systematically improve towards experimentally determined values and more comprehensive energy models when using MP2 corrections, despite remaining at the force field geometry. Predicting two-body interactions with atom-centered symmetry functions in a Gaussian process is found to give highly accurate results with as little as 10-20% of the training data, reducing the cost of the energy correction by up to an order of magnitude. The machine learning approach opens up the possibility of using fragment-based methods to a greater degree in crystal structure prediction, providing alternative energy models where standard approaches are insufficient. | {"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": 0.8085839152336121, "perplexity": 1276.7927197952847}, "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-2019-09/segments/1550249578748.86/warc/CC-MAIN-20190224023850-20190224045850-00401.warc.gz"} |
https://www.spruceid.dev/treeldr/treeldr-basics/properties | SpruceID
Search…
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# Properties
There exists two ways of defining a property. One is to embed the property definition inside a type definition:
base <https://example.com/>;
type MyType {
myProperty: Type // embedded property definition.
}
This will define the `https://example.com/MyType/myProperty` property. Note how the base IRI changes inside the braces to match the IRI of the type. The `myProperty` relative IRI is resolved into `https://example.com/MyType/myProperty` and not `https://example.com/myProperty`. For this reason, one may prefer to define properties independently. This can be done using the `property` keyword:
base <https://example.com/>;
property myProperty: Type; // independent property definition.
It can then be referred to using an absolute, relative, or compact IRI:
use <https://example.com/> as ex;
type MyType {
<https://example.com/myProperty>,
<../myProperty>, // same as above
ex:myProperty // same as above
}
As showed in this example, when a property is defined outside the type definition, it is not required to specify its type again. | {"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": 0.9361906051635742, "perplexity": 2948.384688312572}, "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/1674764500028.12/warc/CC-MAIN-20230202133541-20230202163541-00069.warc.gz"} |
http://copilot.caltech.edu/events/50517 | Search
# Boson condensation and instability in the tensor network representation of topological states
Friday, June 3, 2016
4:00 PM - 5:00 PM
Location: East Bridge 114
Sujeet Shukla, Graduate Student, Preskill Group
Abstract:The tensor network representation of many-body quantum states, given by local tensors provides a promising numerical tool for the study of strongly correlated topological phases in two dimension. However, the topological order in tensor network representations of the Toric code ground state has been shown to be unstable under certain small variations of the local tensor, if these small variations does not obey the local Z2 symmetry of the local tensor. In this work we ask the questions of whether other types of topological orders suffer from similar kinds of instability and if so, whether we can protect the order by enforcing certain symmetry on the tensor. We answer these questions by showing that the tensor network representation of all string-net models are indeed unstable, but the matrix product operator (MPO) symmetry identified by Burak et al. can help to protect the order. We find that, `stand-alone' variations that break MPO symmetry lead to instability because they induce the condensation of bosonic quasi-particles and destroy the topological order in the system. Therefore, such variations must be forbidden for the encoded topological order to be reliably extracted from the local tensor. On the other hand, if a tensor network algorithm is used to simulate the phase transition due to boson condensation, then such variation directions must be allowed in order to access the continuous phase transition process correctly.
Series: IQIM Postdoctoral and Graduate Student Seminar Series | {"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": 0.9232136011123657, "perplexity": 644.4123858929327}, "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-2018-05/segments/1516084889681.68/warc/CC-MAIN-20180120182041-20180120202041-00384.warc.gz"} |
https://aas.aanda.org/articles/aas/full/2000/14/h2123/node5.html | Up: Pulsars in the Westerbork
5 Non-detected pulsars
Table 4 lists the pulsars that have no counterpart in the WENSS. In 14 cases the expected pulsar flux density is higher than three times the local noise level. Still, the pulsar was not detected. In two cases no reliable pulsar flux density estimate at 325 MHz is available. In three other cases the estimate is based on 400 MHz observations. In case of PSRs B0841+80 and B1839+36A this was done, because there was no spectral information available. The spectrum of PSR J1012+5307 might also have a low frequency turnover (see its spectrum as plotted by Kramer et al. 1999). Its flux density is known to vary by up to a factor four from its mean value of 30 mJy (Nicastro et al. 1995).
Five (and possibly eight) pulsars are detected, which were expected to be not detectable. The number of non-detected sources that were expected to have a flux density greater than the detection limit, should be roughly the same as the number of unexpected detections. The difference may be due to Poisson fluctuations in the (small) number of pulsars in this study.
Up: Pulsars in the Westerbork
Copyright The European Southern Observatory (ESO) | {"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": 0.9785544276237488, "perplexity": 1161.328193790026}, "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-2021-25/segments/1623487630518.38/warc/CC-MAIN-20210617162149-20210617192149-00051.warc.gz"} |
http://www1.maths.leeds.ac.uk/~pmtemf/web/gallery-ADE.html | $A_1$ (aka The Cone) $x^2+y^2-z^2=0$
$A_1$ in different coordinates $z^2+x^2-y^2=0$
$A_2$ (aka The Cusp) $z^2+y^2+x^3=0$
$A_4$ $z^2+y^2+x^5=0$
$A_6$ $z^2+y^2+x^7=0$
$A_3$ $z^2+y^2-x^4=0$
$A_5$ $z^2+y^2-x^6=0$
$A_7$ $z^2+y^2-x^8=0$
$D_4$ $z^2+x(y^2-x^2)=0$
$D_6$ $z^2+x(y^2-x^4)=0$
$D_8$ $z^2+x(y^2-x^6)=0$
$D_5$ $z^2+x(y^2-x^3)=0$
$D_7$ $z^2+x(y^2-x^5)=0$
$D_9$ $z^2+x(y^2-x^7)=0$
$E_6$ $z^2+x^3+y^4=0$
$E_7$ $z^2+x(x^2+y^3)=0$
$E_8$ $z^2+x^3+y^5=0$
*: Real pictures of some complex quotient singularities $\mathbb{C}^2 / \Gamma$, where $\Gamma$ is a finite subgroup of $SL_2(\mathbb{C})$. These surfaces are also known as ADE surfaces or Kleinian surfaces or $2$-dimensional rational double points or ...
In each surface a curve is highlighted. This curve is the intersection of the surface with the plane $\{ z =0 \}$, or, if we stay in the context of groups, it is the discriminant curve of the complex reflection group $G$ such that $[G:\Gamma]=2$, that is, $\Gamma=G \cap SL_2(\mathbb{C})$. | {"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": 0.9421966671943665, "perplexity": 77.98870501253671}, "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-2020-05/segments/1579251690095.81/warc/CC-MAIN-20200126165718-20200126195718-00029.warc.gz"} |
http://mathhelpforum.com/differential-geometry/194557-lusin-theorem-2.html | ## Re: Lusin Theorem.
Maybe we can avoid the case $C_0=\emptyset$ using the regualirity of Lebesgue measure: we can find a compact $K_{\varepsilon}$ contained in $A$ such that $m(A\setminus K_{\varepsilon})<\varepsilon /2$, then work with $K_{\varepsilon}$. The fact that an intersection of non empty compacts is non empty. | {"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": 5, "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": 0.9925181269645691, "perplexity": 319.44691085797336}, "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/1422115926769.79/warc/CC-MAIN-20150124161206-00107-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://formulasearchengine.com/wiki/Bernoulli_differential_equation | # Bernoulli differential equation
Template:No footnotes In mathematics, an ordinary differential equation of the form
${\displaystyle y'+P(x)y=Q(x)y^{n}\,}$
is called a Bernoulli equation when n≠1, 0, which is named after Jacob Bernoulli, who discussed it in 1695 Template:Harv. Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.
## Solution
${\displaystyle \left\{{\begin{array}{ll}z:(a,b)\rightarrow (0,\infty )\ ,&{\textrm {if}}\ \alpha \in {\mathbb {R} }\setminus \{1,2\},\\z:(a,b)\rightarrow {\mathbb {R} }\setminus \{0\}\ ,&{\textrm {if}}\ \alpha =2,\\\end{array}}\right.}$
be a solution of the linear differential equation
${\displaystyle z'(x)=(1-\alpha )P(x)z(x)+(1-\alpha )Q(x).}$
Then we have that ${\displaystyle y(x):=[z(x)]^{\frac {1}{1-\alpha }}}$ is a solution of
${\displaystyle y'(x)=P(x)y(x)+Q(x)y^{\alpha }(x)\ ,\ y(x_{0})=y_{0}:=[z(x_{0})]^{\frac {1}{1-\alpha }}.}$
And for every such differential equation, for all ${\displaystyle \alpha >0}$ we have ${\displaystyle y\equiv 0}$ as solution for ${\displaystyle y_{0}=0}$.
## Example
Consider the Bernoulli equation (more specifically Riccati's equation).[1]
${\displaystyle y'-{\frac {2y}{x}}=-x^{2}y^{2}}$
We first notice that ${\displaystyle y=0}$ is a solution. Division by ${\displaystyle y^{2}}$ yields
${\displaystyle y'y^{-2}-{\frac {2}{x}}y^{-1}=-x^{2}}$
Changing variables gives the equations
${\displaystyle w={\frac {1}{y}}}$
${\displaystyle w'={\frac {-y'}{y^{2}}}.}$
${\displaystyle w'+{\frac {2}{x}}w=x^{2}}$
which can be solved using the integrating factor
${\displaystyle M(x)=e^{2\int {\frac {1}{x}}dx}=e^{2\ln x}=x^{2}.}$
Multiplying by ${\displaystyle M(x)}$,
${\displaystyle w'x^{2}+2xw=x^{4},\,}$
Note that left side is the derivative of ${\displaystyle wx^{2}}$. Integrating both sides results in the equations
${\displaystyle \int d[wx^{2}]=\int x^{4}dx}$
${\displaystyle wx^{2}={\frac {1}{5}}x^{5}+C}$
${\displaystyle {\frac {1}{y}}x^{2}={\frac {1}{5}}x^{5}+C}$
The solution for ${\displaystyle y}$ is
${\displaystyle y={\frac {x^{2}}{{\frac {1}{5}}x^{5}+C}}}$
## References
• {{#invoke:citation/CS1|citation
|CitationClass=citation }}. Cited in Template:Harvtxt.
• {{#invoke:citation/CS1|citation
|CitationClass=citation }}.
1. y'-2*y/x=-x^2*y^2, Wolfram Alpha, 01-06-2013 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 26, "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": 0.9842096567153931, "perplexity": 929.9906630832862}, "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/1603107894203.73/warc/CC-MAIN-20201027140911-20201027170911-00690.warc.gz"} |
https://www.sarthaks.com/8206/particle-starts-from-rest-with-constant-ratio-space-average-velocity-time-average-velocity | # Particle starts from rest with constant a. Ratio of space average velocity to time average velocity is
1.9k views
in Physics
edited
Particle starts from rest with constant a. Ratio of space average velocity to time average velocity is
by (128k points)
by (2.8k points)
Can you pls show the whole method?
by (2.8k points)
I don't understand what you're trying to say.
by (2.8k points)
Can u tell me how u got the answer? | {"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": 0.9981523752212524, "perplexity": 4584.083663754558}, "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-2021-25/segments/1623487617599.15/warc/CC-MAIN-20210615053457-20210615083457-00526.warc.gz"} |
http://www.ma.utexas.edu/mediawiki/index.php?title=List_of_results_that_are_fundamentally_different_to_the_local_case&oldid=1293&diff=prev | # List of results that are fundamentally different to the local case
(Difference between revisions)
Revision as of 16:28, 6 October 2015 (view source)Luis (Talk | contribs)← Older edit Revision as of 16:33, 6 October 2015 (view source)Luis (Talk | contribs) (→Improved differentiability of solutions to integro-differential equations in divergence form)Newer edit → Line 71: Line 71: $\mathrm{div} \, ( A(x) \nabla u) = 0,$ $\mathrm{div} \, ( A(x) \nabla u) = 0,$ where $\lambda I \leq A(x) \leq \Lambda I$ belong to the space $W^{1,2+\varepsilon}$ for some $\varepsilon > 0$. This is a nontrivial result, since the variational formulation of the problem only gives us a solution in $W^{1,2}$. The result provides an improvement in the integrability of $|\nabla u|$ from $L^2$ to $L^{2+\varepsilon}$. where $\lambda I \leq A(x) \leq \Lambda I$ belong to the space $W^{1,2+\varepsilon}$ for some $\varepsilon > 0$. This is a nontrivial result, since the variational formulation of the problem only gives us a solution in $W^{1,2}$. The result provides an improvement in the integrability of $|\nabla u|$ from $L^2$ to $L^{2+\varepsilon}$. + + The fractional version of the equation consists in a function $u$ so that + $\int (u(x)-u(y)) (\eta(x)-\eta(y)) K(x,y) \, dx dy = 0,$ + for all compactly supported, smooth enough, functions $\eta$. It turns out that under the uniform ellipticity assumption $\lambda |x-y|^{-d-2s} \leq K(x,y) \leq \Lambda |x-y|^{-d-2s}$, the solution $u$ turns out to belong to the space $W^{s+\varepsilon,2+\varepsilon}$ . The surprising part of the result is that there is an improvement of differentiability. Not only is the power of integrability improved from $2$ to $2+\varepsilon$, but also the order of differentiability is improved from $s$ to $s+\varepsilon$. == References == == References ==
## Revision as of 16:33, 6 October 2015
In this page we collect some results in nonlocal equations that contradict the intuition built in analogy with the local case.
## Contents
### Traveling fronts in Fisher-KPP equations with fractional diffusion have exponential speed
Let us consider the reaction diffusion equation $u_t + (-\Delta)^s u = f(u),$ with a Fisher-KPP type of nonlinearity (for example $f(u) = u(1-u)$). In the local diffusion case, the stable state $u=1$ invades the unstable state $u=0$ at a constant speed. In the nonlocal case (any $s<1$), the invasion holds at an exponential rate.
The explanation of the difference can be understood intuitively from the fact that the fat tails in the fractional heat kernels make diffusion happen at a much faster rate [1][2].
### The optimal regularity for the fractional obstacle problem exceeds the scaling of the equation
Given a function $\varphi$, the obstacle problem consists in the solution to an equation of the form $\min((-\Delta)^s u , u-\varphi) = 0.$
If $\varphi$ is smooth enough, the solution $u$ to the obstacle problem will be $C^{1,s}$ and no better. There is a big difference between the case $s=1$ and $s<1$ which makes the proof fundamentally different. In the classical case $s=1$, the optimal regularity matches the scaling of the equation. The classical proof of optimal regularity is to show an upper bound in the separation of $u$ from the obstacle in the unit ball and then just scale it. In the fractional case $s<1$, this method only gives $C^{2s}$ regularity, which matches the scaling of the equation. It is somewhat surprising that a better regularity result holds and it requires a different method for the proof.
The intuitive explanation is that $(-\Delta)^s u$ satisfies an extra elliptic equation in terms of its Laplacian to the power $1-s$, and that equation provides the extra regularity [3].
### Solutions to nonlocal elliptic equations can have interior maximums
Solutions to linear (and nonlinear) integro-differential equations satisfy a nonlocal maximum principle: they cannot have a global maximum or minumum in the interior of the domain of the equation. Local extrema are possible.
This is related to the fact that Dirichlet boundary conditions have to be given in the whole complement of the Domain and not only on its boundary. It is also related to the failure in general of the classical Harnack inequality unless the positivity of the function is assumed in the full space [4].
In fact, any function $f\in C^k(\overline{B_1})$ can by approximated with a solution to $(-\Delta)^su$ in $B_1$ that vanishes outside a compact set [5]. That is, s-harmonic functions are dense in $C^k_{loc}$. This is clearly in contrast with the rigidity of harmonic functions, and is a purely nonlocal feature.
### Boundary regularity of solutions is different from the interior
For second order equations, the boundary regularity of solutions to $\Delta u=0$ is the same as in the interior. For example, a solution to $\Delta u=0$ in $B_1^+$, with $u=0$ in $\{x_n=0\}$, can be extended (by odd reflection) to a solution of $\Delta u=0$ in $B_1$. Thus, in this case the boundary regularity of $u$ just follows from the interior regularity --one has $u\in C^\infty(\overline{B_{1/2}^+})$.
In a general smooth domain $\Omega$ one can flatten the boundary and repeat the previous argument to get that $u\in C^\infty(\overline\Omega)$.
This is not the case for nonlocal equations. Indeed, the function $u(x)=(x_+)^s$ satisfies $(-\Delta)^su=0$ in $(0,\infty)$, and $u=0$ in $(-\infty,0)$. However, $u$ is not even Lipschitz up to the boundary, while all solutions are $C^\infty$ in the interior. This is related to the fact that the odd reflection of $u$ is not anymore a solution to the same equation.
More generally, solutions to $(-\Delta)^su=f$ in $\Omega$, with $u=0$ in $\mathbb R^n\setminus\Omega$, are smooth in the interior of $\Omega$, but not up to the boundary. The optimal Holder regularity is $u\in C^s(\overline\Omega)$. See boundary regularity for integro-differential equations for more details.
### For some equations, the weak Harnack inequality may hold while the full Harnack inequality does not
The weak Harnack inequality relates the minimum of a positive supersolution to an elliptic equation to its $L^p$ norm. It is an important step used to derive Hölder estimates and also the usual Harnack inequality. However, there are examples of non local elliptic equation for which the weak Harnack inequality and Hölder estimates hold, whereas the classical Harnack inequality does not. There is a discussion about this fact in an article by Moritz Kassmann, Marcus Rang and Russell Schwab [6].
### Solutions to elliptic linear and translation invariant equations may not be smooth
For second order equations, any solution to an elliptic linear and translation invariant equation $Lu=f$ in $\Omega$ is smooth in the interior whenever $f$ is smooth. For second order equations, $L$ must be of the form $Lu=a_{ij}\partial_{ij}u$, and hence after an affine change of variables it is just the Laplacian $\Delta$.
For nonlocal equations, solutions to $Lu=f$ in $\Omega$, with $f$ smooth, may not be smooth inside $\Omega$, even if $L$ is an elliptic linear and translation invariant operator like $Lu(x)=\int_{\mathbb R^n}\bigl(u(x+y)-u(x)\bigr)K(y)dy,$ with $K(y)=K(-y)$ and satisfying $\frac{\lambda}{|y|^{n+2s}}\leq K(y)\leq \frac{\Lambda}{|y|^{n+2s}}.$ It was proved in [7] that there exist a solution to $Lu=0$ in $B_1$, with $u\in L^\infty(\R^n)$, which is not $C^{2s+\epsilon}(B_{1/2})$ for any $\epsilon>0$. The counterexample can be constructed even in dimension 1, and it is very related to the regularity of the kernel $K$.
Related to this, it was shown in [8] that there is a $C^\infty$ domain $\Omega$ and an operator of the form $Lu(x)=\int_{\mathbb R^n}\bigl(u(x+y)-u(x)\bigr)\frac{a(y/|y|)}{|y|^{n+2s}}\,dy,$ for which the solution to $Lu=1$ in $\Omega$, $u=0$ in $\mathbb R^n\setminus\Omega$, is not $C^{3s+\epsilon}$ inside $\Omega$ for any $\epsilon>0$. See the survey [9] for more details.
### Viscosity solutions can be evaluated at points
The concept of viscosity solutions is developed in order to make sense of an elliptic equation even for continuous functions for which the equation cannot be evaluated classically at points. The idea is to use test functions whose graphs are tangent to the graphs of the weak solution at some point, and then evaluate the equation on that test function. The ellipticity property tells us that the value of the equation for that test function at the point of contact must have certain sign, and this is the condition that a viscosity solution fulfill.
It turns out that for a large class of fully nonlinear integro-differential equations, every time a viscosity solution can be touched by a smooth test function at a point, then the equation can be evaluated classically for the original function at that point [10].
### Viscosity solutions to fully nonlinear integro-differential equations can be approximated with classical solutions
It is a very classical trick that if we have a weak solution to a linear PDE with constant coefficients, we can approximate it with a smooth solution via a simple mollification. For nonlinear equations this trick is no longer available and we are always forced to deal with the technical difficulties of viscosity solutions. This is an apparent difficulty for example when proving regularity estimates, since in general we cannot derive them an a priori estimate for a classical solution. On the other hand, viscosity solutions to fully nonlinear integro-differential equations can be approximated by $C^2$ solutions to approximate equations [11].
This procedure can also be carried out to approximate viscosity solutions to fully nonlinear elliptic equations with smooth solutions of an approximate equation. However, the approximated equation is integro-differential [12].
### Improved differentiability of solutions to integro-differential equations in divergence form
A classical theorem asserts that solution to uniformly elliptic equations in divergence form $\mathrm{div} \, ( A(x) \nabla u) = 0,$ where $\lambda I \leq A(x) \leq \Lambda I$ belong to the space $W^{1,2+\varepsilon}$ for some $\varepsilon > 0$. This is a nontrivial result, since the variational formulation of the problem only gives us a solution in $W^{1,2}$. The result provides an improvement in the integrability of $|\nabla u|$ from $L^2$ to $L^{2+\varepsilon}$.
The fractional version of the equation consists in a function $u$ so that $\int (u(x)-u(y)) (\eta(x)-\eta(y)) K(x,y) \, dx dy = 0,$ for all compactly supported, smooth enough, functions $\eta$. It turns out that under the uniform ellipticity assumption $\lambda |x-y|^{-d-2s} \leq K(x,y) \leq \Lambda |x-y|^{-d-2s}$, the solution $u$ turns out to belong to the space $W^{s+\varepsilon,2+\varepsilon}$ [13]. The surprising part of the result is that there is an improvement of differentiability. Not only is the power of integrability improved from $2$ to $2+\varepsilon$, but also the order of differentiability is improved from $s$ to $s+\varepsilon$.
## References
1. Cabré, Xavier; Roquejoffre, Jean-Michel (2009), "Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire", Comptes Rendus Mathématique. Académie des Sciences. Paris 347 (23): 1361–1366, doi:10.1016/j.crma.2009.10.012, ISSN 1631-073X
2. Cabré, Xavier; Roquejoffre, Jean-Michel (to appear), "The influence of fractional diffusion in Fisher-KPP equations", Comm. Math. Phys.
3.
4. Kassmann, Moritz (Preprint), The classical Harnack inequality fails for non-local operators
5. Dipierro, Serena; Savin, Ovidiu; Valdinoci, Enrico, "All functions are locally $s$-harmonic up to a small error", arXiv preprint arXiv:1404.3652
6. Rang, Marcus; Kassmann, Moritz; Schwab, Russell W, "H$\backslash$" older Regularity For Integro-Differential Equations With Nonlinear Directional Dependence", arXiv preprint arXiv:1306.0082
7. Serra, Joaquim, "$C^{2s+\alpha}$ regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels", arXiv preprint
8. Ros-Oton, Xavier; Valdinoci, Enrico, "The Dirichlet problem for nonlocal operators with singular kernels: convex and nonconvex domains", arXiv preprint
9. Ros-Oton, Xavier, "Nonlocal elliptic equations in bounded domains: a survey", arXiv preprint
10. Caffarelli, Luis; Silvestre, Luis (2009), "Regularity theory for fully nonlinear integro-differential equations", Communications on Pure and Applied Mathematics 62 (5): 597–638, doi:10.1002/cpa.20274, ISSN 0010-3640
11. Caffarelli, Luis; Silvestre, Luis (to appear), "The Evans-Krylov theorem for non local fully non linear equations", Annals of Mathematics, ISSN 0003-486X
12. Caffarelli, Luis; Silvestre, Luis (2010), "Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations", Nonlinear partial differential equations and related topics, Amer. Math. Soc. Transl. Ser. 2, 229, Providence, R.I.: American Mathematical Society, pp. 67–85
13. Kuusi, Tuomo; Mingione, Giuseppe; Sire, Yannick (2015), "Nonlocal self-improving properties", Anal. PDE 8: 57--114, doi:10.2140/apde.2015.8.57, ISSN 2157-5045 | {"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": 0.9564132690429688, "perplexity": 292.9352958389095}, "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-2018-13/segments/1521257651780.99/warc/CC-MAIN-20180325025050-20180325045050-00701.warc.gz"} |
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