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http://wok.oblomov.eu/ars/mathematical-anarchist/ | Apparently graffiti things such as this and this have appeared in Bruxelles (and who knows where else).
Note: this article makes use of MathML, the standard XML markup for math formulas. Sadly, this is not properly supported on some allegedly ‘modern’ and ‘feature-rich’ browsers. If the formulas don't make sense in your browser, consider reporting the issue to the respective developers and/or switching to a standard-compliant browser.
I must say it's not too common seeing mathematical graffiti, so let's have a look at them a little bit closer. Let's start with a transcription:
$f(x) = 2+ -(x-2)2 + 1 g(x) = 2- -(x-2)2 + 1 h(x) = 3x-3 i(x) = -3x+9 j(x) = 0,2x+1,7$
There's a few things I don't like about some choices made (such as the choice of decimal separator —which could be avoided altogether, as we'll see later), but let's first try to understand what we have, as-is.
What we're looking at is the definition of five distinct functions of a single variables. One of the nice things about real-valued functions of real-valued variables (which is the assumption we make here) is that, thanks to the brilliant intuition of Réne Descartes to associate algebra and geometry, we can visualize these functions.
Thus, another way to look at this is that we have an algebraic description of five curves. The obvious implication here would be that, if we were to plot these curves, we'd get another picture, the actual, hidden, graffiti.
Can we reason about the functions to get an idea about what to expect from the visualization? Indeed, we can.
For example, the functions $f$ and $g$ are obviously closely related, as they differ just for the sign of radical. We can thus expect them to be the two possible solution for a second-order equation, which we're going to discover soon. Additionally, the form these functions have also give as a domain of existence: since the argument to the radical must be non-negative for the functions to have real values (and thus be plottable), we must have
$-(x-2)2+1≥0,$
that is
$1≥(x-2)2,$
which would more commonly written as
$(x-2)2≤1,$
whose solution is
$-1≤x-2≤1,$
or
$1≤x≤3.$
Now we know that, for the functions $f$ and $g$ to exist, $x$ must be in the interval between 1 and 3. Since the other functions do not have restrictions on the domain of existence, we can take this as domain for the whole plot.
The next step is to find out what are $f$ and $g$ the two halves of. To do this, we can “combine” them using $y$ as placeholder for either, and writing:
$y=2±-(x-2)2+1,$
which can be rearranged to
$y-2=±-(x-2)2+1.$
Since we're taking both the positive and negative solutions for the square root, we can square both sides without worrying about introducing spurious solutions, obtaining:
$(y-2)2=-(x-2)2+1,$
which we can again rearrange to get
$(x-2)2+(y-2)2=1.$
This is the equation of a circle, centered at the point with coordinates $\left(2,2\right)$, and with radius 1. $f$ represents the upper half, $g$ the lower half.
The circle represented by the first two equations
## The lines
The next three functions ($h$, $i$, $j$) are much simpler, and anybody that remembers their analytical geometry from high school should recognize them for the equations of straight lines.
If we look at the first two of these more closely
$h(x) = -3x-3 i(x) = -3x+9$
we notice that they are rather steep (the absolute value of the $x$ coefficient is $3>1$), and symmetrical with respect to the vertical axis (the $x$ coefficient has opposite sign). They intersect for $x=2$ which is conveniently placed halfway through our domain (derived from the circle equations). Note that the resulting ordinate $y=3$ places the intersection point on the circle we've seen, as:
$(2-2)2+(3-2)2=1$
is satisfied.
The last equation, as I mentioned at the beginning meets my full displeasure due to the choice of decimal separator —in fact, the most annoying thing about is that it uses one at all, as the same values could be written in a more universal way by using fractions:
$j(x)=x5+1710$
or inline using the solidus:
$j(x)=x/5+17/10.$
This straight line much less steep (in fact, one could say it's barely sloping at all). It also intersects the other two lines in places with some funky values which I'm not even going to bother computing, as we are only interested in the visualization:
The straight lines
So, an ‘A’ shape.
## Putting it all together
If we put it all together now, we obtain the quite famous anarchist logo:
Anarchy | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 33, "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": 0.9385676980018616, "perplexity": 361.55483855960296}, "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-2020-05/segments/1579250606975.49/warc/CC-MAIN-20200122101729-20200122130729-00333.warc.gz"} |
http://math.stackexchange.com/questions/52113/probability-calculations-on-highway/52117 | # Probability Calculations on Highway
I read this Google Interview Question.
Q:If the probability of observing a car in 30 minutes on a highway is 0.95, what is the probability of observing a car in 10 minutes (assuming constant default probability)?
A:The trick here is that .95 is the probability for 1 or more cars, not the probability of seeing just one car. The prob. of NO cars in 30 minutes is 0.05, so the prob of no cars in 10 minutes is the cube root of that, so the prob of seeing a car in 10 minutes is one minus that, or ~63%
My question is, if The probability of NO cars in 30 minutes is 0.05, why the probability of no cars in 10 minutes is the cube root of that ??
Which algorithm used in this question?
-
Do you know the distribution from which you got the probability? I don't see how you could answer the question otherwise. – gary Jul 18 '11 at 8:29
## 3 Answers
The unstated (or, rather, very vaguely stated) assumption in the problem is that the probabilities of observing a car during any given non-overlapping time intervals of equal length are equal and independent.
(Of course, this assumption can't really be true in practice, even if "observing a car" is taken to be a point event — for example, if the road has $n$ lanes and you observe a different car within each of $n$ consecutive 1 millisecond intervals, you're not going to observe another one within the next millisecond — but it can be a fairly good approximation if the intervals are of moderate length and the road not very busy.)
This assumption (almost; see comments) implies that the arrival of cars is (assumed to be) a Poisson process. More specifically, it implies that the probability of no cars arriving within any given 10 minute interval is the same. Since we know that the probability of no cars arriving within a 30 minute interval equals the product of the probabilities of no cars arriving in each of the three consecutive 10 minute intervals within it, the answer follows.
To be specific, let $A$, $B$ and $C$ denote the events "no cars are observed within the first / second / third 10 minutes" respectively. Then we have
$$\mathrm{Pr}[A \text{ and } B \text{ and } C] = \mathrm{Pr}[A] \cdot \mathrm{Pr}[B \text{ if } A] \cdot \mathrm{Pr}[C \text{ if } A \text{ and } B].$$
Since the events $A$, $B$ and $C$ are independent by assumption, we get
$$\mathrm{Pr}[A \text{ and } B \text{ and } C] = \mathrm{Pr}[A] \cdot \mathrm{Pr}[B] \cdot \mathrm{Pr}[C],$$
and, since by assumption $\mathrm{Pr}[A] = \mathrm{Pr}[B] = \mathrm{Pr}[C]$,
$$\mathrm{Pr}[A \text{ and } B \text{ and } C] = \mathrm{Pr}[A]^3.$$
We know that $\mathrm{Pr}[A \text{ and } B \text{ and } C] = 0.05$, and we want to solve for $\mathrm{Pr}[A]$ (which, by assumption, equals the a priori probability of observing no cars within any given 10 minute interval), so we take the cube root of both sides and get
$$\mathrm{Pr}[A] = \sqrt[3]{\mathrm{Pr}[A \text{ and } B \text{ and } C]} = \sqrt[3]{0.05} \approx 0.3684.$$
Subtract that from one to get $\mathrm{Pr}[\text{not } A] \approx 0.6316$.
-
After posting this, I happened to come across a remark elsewhere saying that the assumption I gave may not actually be strong enough to imply that the process is Poisson; apparently there are "quasi-Poisson" processes which have independent Poisson distributed event counts over disjoint intervals, with mean proportional to the interval length, but which are not actually Poisson processes. In any case, this shouldn't affect the rest of the answer, for which the weaker assumption is sufficient. – Ilmari Karonen Jul 18 '11 at 14:13
Just to be sure, I edited the assumption to make sure it actually implies exactly the conditions I need (independence and equality). – Ilmari Karonen Jul 18 '11 at 14:25
The question seems rather ambiguous, but let's assume cars arrive as a Poisson process rate $\lambda$. If this is the case, the distribution of the time (from now) to the first arrival of the car is exponential with parameter $\lambda$. Therefore probability of no cars arriving is $P(T>t) = exp(-\lambda t)$. Thus
$P(T>10min) = \exp(-\lambda \times 10) = \exp(-\lambda*30/3) = \exp(-\lambda*30))^{\frac{1}{3}} = \sqrt[3]{P(T>30min)}$
Alternatively you could approximate by a binomial model. Suppose in 10 minutes the chance of no car arriving is $p$. Then in thirty minutes (assuming each period is independent) the probability of no cars passing is $p^3$. Whence, $p=\sqrt[3]{0.05}$.
-
Let's say that the probability of no cars in 30 minutes can be decomposed as (you assume constant probability)
P10 = probability of no cars in 10 minutes
P30 = P10 * P10 * P10 = P10^3 = 0.05
Thus P10 = cuberoot(0.05)
-
Note that this was possible only because of the constant assumption over probability. – Mauro Jul 18 '11 at 8:36
I already asking why P30 =P10^3 .. – Soner Gönül Jul 18 '11 at 8:46
@Soner, because probability of (A and B) equals (probability of A) times (probability of B), provided A and B are independent events - and the hidden-but-plausible assumption is that what happens in any one 10-minute interval is independent of what happens in any other (non-overlapping) 10-minute interval. – Gerry Myerson Jul 18 '11 at 12:37 | {"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": 0.8965063095092773, "perplexity": 357.5659171193055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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/1440645311026.75/warc/CC-MAIN-20150827031511-00349-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.ill.eu/users/instruments/instruments-list/pn1/examples/oxidation-and-diffusion-in-fuel-cladding-tubes-of-power-reactors | PN1
Fission-product spectrometer
Oxidation and Diffusion in Fuel Cladding Tubes of Power Reactors
During the operation of power reactors the fuel elements are subject to alterations due to fuel burn-up, oxidation of the cladding material, and diffusion of actinide material in the Zr-tubes, which contain the fuel.
Increased oxidation rates of the inner part of the cladding tubes are due to the impact of fission products, and due to the increased temperature. The same holds for the diffusion process where uranium and plutonium are deposited due to sputtering processes on to the inner surface of the tubes, which then mix with the Zr-oxide layer.
Oxidation and diffusion, under reactor conditions are subjects under investigation at the LOHENGRIN spectrometer (PN1). | {"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": 0.9594915509223938, "perplexity": 2070.6972212662477}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296944452.97/warc/CC-MAIN-20230322211955-20230323001955-00174.warc.gz"} |
https://spmchemistry.blog.onlinetuition.com.my/2012/03/relative-atomic-mass.html | # Relative Atomic Mass
### Relative Mass
The relative mass of an object is the comparison of the mass of the object to the mass of a standard object.
### Relative Atomic Mass
The relative atomic mass (Ar) of an element is the average mass of one atom of the element when compared with 1/12 of the mass of an atom of carbon-12, which taken as 12 units.
1. The mass of an atom when compared to another is known as the relative atomic mass (Ar).
2. The relative atomic mass (Ar) of an element is the average mass of one atom of the element when compared with 1/12 of the mass of an atom of carbon-12, which taken as 12 units.
3. 1/12 of the mass of an atom of carbon-12 is named as 1 atomic mass unit (amu).
4. The mass of one carbon atom is 12 amu.
5. 1/12 of the mass of an atom of carbon-12 is named as 1 atomic mass unit (amu).
6. The mass of one carbon atom is 12 amu.
Example 1
The mass of a sodium atom is 23 times greater than 1/12 of the mass of carbon-12 atom. What is the relative atomic mass of sodium?
23
Example 2:
The mass of element A is twice of the mass of carbon, therefore its relative atomic mass is __________. (Relative atomic mass of carbon = 12)
Relative Atomic Mass of Element A = 2 x 12 = 24
Example 3:
An atom of element X is 13 times heavier than one atom of helium. Calculate the relative atomic mass of X.( Ar: He = 4 )
Relative Atomic Mass of X = 13 x 4 = 52
Example 4:
How many times that the mass of 2 bromine atoms are greater than 4 neon atoms? (Ar: Ne = 20; Br = 80 ) | {"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": 0.8958725929260254, "perplexity": 791.85787989163}, "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-2021-17/segments/1618038084765.46/warc/CC-MAIN-20210415095505-20210415125505-00377.warc.gz"} |
http://mathhelpforum.com/differential-geometry/87677-continuous-functions-proof.html | # Math Help - Continuous Functions Proof
1. ## Continuous Functions Proof
Define f : R R by f(x) = 5x if x is rational and f(x) = x² + 6 if x is irrational. Prove that f is discontinuous at 1 and continuous at 2. Are there any other point besides 2 at which f is continuous?
2. at x=1 any open neighborhood about it will contain both rational and irrational numbers, but the irrational part is converging to 7 while the rational part is converging to 5 hear.
however at 2 both parts converge to 10.
to find the other point consider where
$x^2+6=5x \Rightarrow x^2-5x + 6 =0 \Rightarrow (x-3)(x-2)=0$
3 might be another good place to look.
3. Originally Posted by bearej50
Define f : R R by f(x) = 5x if x is rational and f(x) = x² + 6 if x is irrational. Prove that f is discontinuous at 1 and continuous at 2. Are there any other point besides 2 at which f is continuous?
Hi bearej50.
To prove that $f$ is discontinuous at 1, let $\epsilon=1.$ Then for any $\delta>0,$ the interval $(1-\delta,\,1+\delta)$ contains irrational numbers. If $y$ is one of them, then $|f(y)-f(1)|=|y^2+6-5|=|y^2+1|\geq1=\epsilon.$
To prove that $f$ is continuous at 2, let $\epsilon>0$ be given. Then if $\delta=\min\left\{1,\frac\epsilon5\right\},$ consider $y$ in the interval $(2-\delta,\,2+\delta).$ If $y$ is rational, we have $|f(y)-f(2)|=|5y-10|=5|y-2|<5\delta\leq\epsilon.$ If $y$ is irrational, then $|f(y)-f(2)|=|y^2-4|=|y+2||y-2|<3\delta<\epsilon.$
There is one other point at which $f$ is continuous, and that is found by solving the equation $5x=x^2+6$ for $x.$
4. Or consider $(a_n) = 1+ \frac{\sqrt{2}}{n}$. Then $a_n \to 1$. But $f(a_n) \not \to f(1)$. Thus $f$ is discontinuous at $x = 1$. It seems that if both rational parts and the irrational parts are equal at $x$ then $f$ is continuous at $x$ (e.g. in the case of $x = 2$). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 28, "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": 0.9891626834869385, "perplexity": 133.13730091568144}, "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-2015-22/segments/1432207930895.88/warc/CC-MAIN-20150521113210-00116-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/473732/problem-with-varwidth-package?noredirect=1 | # problem with varwidth package
I had a book that contains the varwidth package. Until now everything was OK, but finally this error appeared:
Sorry, but "C:\Program Files\MiKTeX 2.9\miktex\bin\x64\pdflatex.exe" did not succeed. The log file hopefully contains the information to get MiKTeX going again: C:\Users\HP 250\AppData\Local\MiKTeX\2.9\miktex\log\pdflatex.log
This problem appeared after that problem. I don't know if they are related. What do you believe? How can I solve it?
Here is the log-file and this is my code:
\documentclass[a4paper,11pt,twoside]{book}
\usepackage[a4paper,left=2.5cm,right=2.5cm,top=2.5cm,bottom=2.5cm]{geometry}
\usepackage{varwidth}
\begin{document}
text
\end{document}
• @UlrikeFischer , yes it compiles if I remove varwidth from the code I posted. I 'm saying "from the code I posted", because if I delete it from the book the error appears again. So, I suppose that an other package has problem too. – Kώστας Κούδας Feb 7 at 9:54 | {"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": 0.8741180896759033, "perplexity": 2725.08320195548}, "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-2019-18/segments/1555578806528.96/warc/CC-MAIN-20190426133444-20190426155444-00094.warc.gz"} |
http://www.edurite.com/kbase/find-the-measure-of-angle-abc | • Class 11 Physics Demo
Explore Related Concepts
find the measure of angle abc
Question:Ray BD bisects angle ABC. Find the measure of angle ABC if the measure of angle ABD is represented by 7x 2 and the measure of angle DBC is represented by 5x + 10.
Answers:Ray BD bisects angle ABC. Find the measure of angle ABC if the measure of angle ABD is represented by 7x 2 and the measure of angle DBC is represented by 5x + 10. 7x-2=5x+10 2x=12 x=6 5x+10=40 angle ABC=80
Question:Given: Segment BH is the angle bisector of angle ABC. The measure of angle ABC = 105. Find the value of x. http://i167.photobucket.com/albums/u142/XOJustaGurlOX/15.jpg
Question:In each Triangle ABC, find the measures for angle B and angle C that satisfy the given conditions. Draw diagrams to help you decide whether two triangles are possible. Remember that a triangle can have only one obtuse angle. 1. m angle A - 62 , a = 30, and b = 32 2. m angle A - 16 , a = 12, and b = 37.5 3. m angle A = 48 , a = 93, and b = 125 4. m angle A = 112 , a = 16.5, and b =5.4 5. m angle A = 23.6 , a = 9.8, and b = 17 6. m angle A = 155 , a = 12.5, and b = 8.4
Answers:Let's see... I will assume that side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Hmm... For these problems, I'd use the Law of Cosines and the Law of Sines. Law of Cosines: c^2 = a^2+b^2-2ab cos C Law of Sines: sin A/a = sin B/b = sin C/c In each one of these problems, we have a, b and angle A. We can rewrite the Law of Cosines so that it reads: a^2 = b^2+ c^2 - 2bc cos A. Rewriting this as a quadratic would give: c^2 -(2b cos A)c + (a^2+b^2) = 0. If you use the quadratic formula, you'll have your solution(s) for c. The number of positive solutions determines the number of possible triangles that can be made with the given information (i.e., 1 or 2). Any less, and the triangle is impossible. Once you have side c solved for, you can then use the Law of Sines to find angle C: sin C = c/a * sin A. Using the inverse of sine on both sides will give you a value for C. 180 - A - C = B (since, for every triangle ABC, A+B+C=180). This will probably involve a lot of tedious work; however, if you have no diagram (like myself), this is the way to do it.
Question:Find the measure of angle DBC. 19 76 104 180 2-Angle ABC is a straight angle. The measure of angle ABD=6x-5 and the measure of angle DBC=2x+1. Find the measure of angle ABD. 23 47 121 133 3-Ray SU bisects angle RST. The measure of angle RSU=6x-3 and the measure of angle UST=5x+4. Find the measure of RSU. 7 39 41 78 4-Ray SU bisects angle RST. The measure of angle RSU=3x-1, the measure of angle UST=2x+13 and the measure of angle RST=6x-2. Find x. 12 14 41 82 5-Use the information from the previous question to find the measure of angle RSU. 12 14 41 82 Really could use some help, I don't have a book for any help,so I'm going for yahoo!! cause yall are the best!=)
Answers:1. ABD + DBC = ABC 4x + 5x + 9 = 180 9x = 171 x = 19 DBC = 5x + 9 = 95 + 9 = 104 ------- option c 2.ABD + DBC = ABC 6x 5 + 2x + 1 = 180 8x = 184 x = 23 ABD = 6x 5 = 133 ------- option d 3-Ray SU bisects angle RST. The measure of angle RSU=6x-3 and the measure of angle UST=5x+4. Find the measure of RSU. RST = RSU + UST RSU = UST 6x 3 = 5x + 4 x = 7 RSU = 42 3 = 39 -------- option b 4 RSU = 3x-1 = UST=2x+13 3x 1 = 2x + 13 x = 14 --------- option b 5. RSU = 3x 1 = 42 1 = 41 --------- option c -------- | {"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": 0.9459103941917419, "perplexity": 694.4932252501541}, "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-2015-32/segments/1438042988922.24/warc/CC-MAIN-20150728002308-00082-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://mathonline.wikidot.com/the-weak-topology-induced-by-w-x | The Weak Topology Induced by W ⊆ X♯
# The Weak Topology Induced by W ⊆ X♯
Recall from The Weak Topology Induced by F page that if $X$ is a nonempty set and if $\mathcal F$ is a collection of functions defined on $X$ then for each $\epsilon > 0$, $F \subseteq \mathcal F$ finite, and $x \in X$ we defined:
(1)
\begin{align} \quad N_{\epsilon, F}(x) = \{ y \in X : |f(x) - f(y)| < \epsilon, \: \forall f \in F \} \end{align}
We then proved that the collection $\{ N_{\epsilon, F}(x) : \epsilon > 0, \: F \subseteq \mathcal F \: \mathrm{finite}, \: x \in X \}$ is a base for some topology $\tau$ on $X$ and we defined that topology to be the weak topology induced by $\mathcal F$.
Now let $X$ be a linear space. Recall that $X^{\sharp}$ is the collection of all linear operators on $X$. If $W \subseteq X^{\sharp}$ then we can consider the $W$-weak topology on $X$.
The following proposition tells us that the only $W$-weakly continuous functions are those functions in $W$.
Theorem 1: Let $X$ be a linear space and let $W \subseteq X^{\sharp}$. Then a linear functional $\varphi \in X^{\sharp}$ is continuous with respect to the $W$-weak topology on $X$ if and only if $\varphi \in W$.
• Proof: $\Rightarrow$ Let $\varphi : X \to \mathbb{C}$ be a linear functional that is continuous with respect to the $W$-weak topology on $X$. Consider the following set:
(2)
\begin{align} \quad \{ x \in X : |\varphi(x)| < 1 \} \end{align}
• Then the above set is the inverse image of the unit open ball in $\mathbb{C}$. Since $\varphi$ is continuous (with respect to the $W$-weak topology on $X$), the above set is open and also contains $0$. So there exists an element of the base that contains $0$ and is contained in this set. That is, there exists an $\epsilon > 0$ and a finite set $F \subseteq W$ for which:
(3)
\begin{align} \quad 0 \in N_{\epsilon, F} (0) = \{ x \in X : | \psi (x) - \psi(0) | < \epsilon, \forall \psi \in F \} = \{ x \in X : | \psi (x) | < \epsilon, \forall \psi \in F \} \subseteq \{ x \in X : |\varphi (x)| < 1 \} \end{align}
• Therefore, if $| \psi (x) | < \epsilon$ for all $\psi \in F$ we have that $| \varphi (x) | < 1$.
• Now let $x \in \bigcap_{\psi \in F} \ker \psi$. Then $\psi (x) = 0$ for every $\psi \in F$. Hence:
(4)
\begin{align} \quad | \psi (tx) | = 0 < \epsilon, \quad \forall t > 0, \forall \psi \in F \end{align}
• Hence $|\varphi (tx)| < 1$ for every $t > 0$ and:
(5)
\begin{align} \quad | \varphi (x) | < \frac{1}{t}, \quad \forall t > 0 \end{align}
• This shows that $| \varphi(x) | = 0$, so $\varphi (x) = 0$. Hence $x \in \ker \varphi$. So:
(6)
\begin{align} \quad \bigcap_{\psi \in F} \ker \psi \subseteq \ker \varphi \end{align}
• $\Leftarrow$ Suppose that $\varphi \in W$. Since the $W$-weak topology is the weakest topology which makes all of the functions in $W$ continuous, we have that $\varphi$ is continuous with respect to the $W$-weak topology. $\blacksquare$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 6, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999066591262817, "perplexity": 98.70802098248211}, "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-2019-13/segments/1552912203326.34/warc/CC-MAIN-20190324043400-20190324065400-00247.warc.gz"} |
https://www.whclasses.in/post/nearest-neighbour-analysis-nna | # Nearest Neighbour Analysis (NNA)
The study of settlements in order to discern any regularity in spacing by comparing the actual pattern of settlement with a theoretical random pattern.
The straight line distance from each settlement to its nearest neighbour is measured and this is divided by the total number of settlements to give the observed mean distance between nearest neighbours. The density of points is calculated as:
Where Rn is the nearest neighbour index. An index of 0 indicates a completely clustered situation. 1 shows a random pattern, 2 a uniform grid, and 2.5 a uniform triangular pattern. The interpretation of index values can be difficult since these values are not part of a continuum.
A technique for examining the spatial distribution of two-dimensional recorded points, for example settlement sites within a river catchment. Assuming that all the points to be examined are contemporary and that all relevant examples are known, a series of statistics can be calculated by measuring the linear distance between sites. This can very easily be done using a GIS system. A nearest-neighbour index (usually denoted by the symbol R), is calculated from the ratio of the average observed distance from each point in the pattern to its nearest neighbour, to the average distance expected if the pattern were randomly distributed, which depends solely on the density of the pattern being studied. The index R varies from 0.00 for a totally clustered pattern through 1.00 for a random distribution to a maximum of 2.15 for a completely regularly spaced pattern. | {"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": 0.9536158442497253, "perplexity": 511.80411157257237}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710962.65/warc/CC-MAIN-20221204040114-20221204070114-00852.warc.gz"} |
https://indico.mitp.uni-mainz.de/event/84/timetable/?view=standard | # 55. International Winter Meeting on Nuclear Physics
Europe/Berlin
Bormio, Italy
#### Bormio, Italy
, ,
Description
Long-standing conference bringing together researchers and students from various fields of subatomic physics. The conference location is Bormio, a beautiful mountain resort in the Italian Alps.
• Monday, 23 January
• 06:00 07:00
Pre-Conference School
• 06:00
Selected Topic in Hadron Physics 10m
Speaker: Dr Diego Bettoni (Istituto Nazione di Fisica Nucleare)
• 06:10
Selected Topics in Heavy Ion Physics 10m
Speaker: Torsten Dahms (Excellence Cluster Universe - Technische Universität München)
• 06:20
Selected Topic in Nuclear Structure and Nuclear Astrophysics 10m
Speaker: Dr Pierre Capel (Université Libre de Bruxelles (ULB))
• 06:30
Selected Topics in Flavour Physics 10m
Speaker: Prof. Marcel Merk (Nikhef)
• 09:00 13:40
Monday Morning
• 09:00
Welcome 10m
• 09:10
Classical Novae and the Physics of Exploding Stars 45m
At the turn of the 21st Century, new tools and developments, at the crossroads of theoretical and computational astrophysics, observational astronomy, cosmochemistry, and nuclear physics, have revolutionized our understanding of the physics of stellar explosions. The use of space-borne observatories has opened new windows to study the cosmos through multifrequency observations. In parallel to the elemental stellar abundances inferred spectroscopically, cosmochemists are now providing isotopic abundance ratios from micron-sized presolar grains extracted from meteorites. Encapsulated in those grains is pristine information about the suite of nuclear processes that took place in their stellar progenitors. The dawn of supercomputing has also provided astrophysicists withe appropriate tools to study complex physical phenomena that require a multidimensional approach. Last but not least, nuclear physicists have developed new techniques to determine nuclear interactions close to stellar energies. In this talk, a number of breakthroughs from all these different disciplines will be presented, with emphasis on the physical mechanisms that operate during nova explosions.
Speaker: Prof. Jordi Jose (Univ. Politcnica de Catalunya)
• 09:55
Higgs and New Physics at ATLAS and CMS 45m
..
Speaker: Prof. Stefano Giagu (CERN)
• 10:40
Coffee Break 30m
• 11:10
News from BESIII 40m
BESIII is an experiment located at the Institute of High Energy Physics in Beijing, China. It consists of a state-of-the-art 4π magnetic spectrometer that surrounds the beam-beam intersection region of the BEPCII e+e- collider that is operated by an international collaboration of nearl 400 researchers from 13 countries. BEPCII provides BESIII with record-high luminosity e+e- data over a center-of-mass energy range that includes the thresholds for pair production of all of the stable strange baryons, charmed mesons, -leptons and c charmed baryons, and direct access to charmonium and many of the XYZ charmoniumlike mesons that are strong candidates for non-standard, four-quark mesons. This talk will focus on a subset of recent BESIII results, including some unexpected phenomena seen at different baryon-antibaryon thresholds and high statistics measurements of properties of the Y(4260) that provide some insight into its underlying structure.
Speaker: Prof. Stephen Lars Olsen (Institute of Basic Science, Korea)
• 11:50
Detector at HL/LHC and Future Colliders 40m
Detector at HL/LHC and Future Colliders
Speaker: Dr Werner Riegler (CERN)
• 17:00 19:20
Monday Afternoon
• 17:03
Silicon Tracking System of the future CBM experiment at FAIR 3m
The physics aim of the Compressed Baryonic Matter (CBM) experiment is to explore the phase diagram of strongly interacting matter at highest net baryon densities and moderate temperatures in the range reachable with AA collisions between 2-45 AGeV, initially 2-14 AGeV (SIS 100). The Silicon Tracking System (STS) is the central detector for charge particle tracking and momentum determination. It is designed to be operated at high occupancies at collision rates up to 10 MHz. It is planned to reach the track reconstruction efficiency of 95%. The momentum resolution is expected to be around 1.5%. To achieve these goals, double sided double metal silicon sensors with a pitch of 58 μm are used. The sensors are mounted on light-weight carbon ladders, forming 8 stations. The read-out electronics is kept outside of the detector acceptance. The self-triggering read-out electronics is connected with the silicon microstrip sensors through the multi-line microcables. The resulting material budget is about 1%X0 per station. The entire system is going to be operated in a thermal enclosure to maintain constant temperature of -5◦C. Before the mass production of the silicon sensors, several studies are performed with prototypes. They include electrical and optical inspection, measurements with the radioactive sources and beam tests of the sensors and the read-out electronics. Particularly, we study possible impact of the severe radiation environment (10^14 neq cm^−2 ) to the sensor performance. The STS project is realized in cooperation of institutes from Germany, Poland, Russia and Ukraine. This presentation is given on behalf of the CBM Collaboration.
Speaker: Dr Maksym Teklishyn (FAIR)
• 17:06
Timing in a FLASH 3m
Very precise timing below the 100ps-mark is gaining importance in modern detector designs. Many technology demonstrators achieving this goal were based on the Cherenkov effect exploiting its prompt light emission. One common requirement is the necessity to compensate inherent walk effects to reach the sub-100ps timing precison. In traditional approaches either the amplitude is measured in addition to the timestamp, which requires the signal to be split, or a constant fraction discriminator is used. More recent developments include sampling the signal and extracting the relevant features which requires powerful frontend electronics and is not suited for all experimental circumstances. With the advent of high-precision TDCs another method becomes feasible: measuring the signal amplitude via Time-over-Threshold. In this case the frontend electronics can be kept simple by only using a discriminator circuit and, optionally, a wide-band amplifier. A prototype detector, called FLASH (Fast Light Acquiring Start Hodoscope), was built based on QUARTIC's design ideas. The fused silica radiator bars were coupled to a 10 micron-pore type PLANACON MCP-PMT with 64 channels and readout with custom electronics based on the NINO ASIC. The TRB3 system, a high-precision TDC implemented in an FPGA, was used as data acquisition system. The performance of the system was investigated at a dedicated test experiment at the Mainz Microtron (MAMI) accelerator. The validity of the Time-over-Threshold approach could be established and an overall timing resolution of approx 70ps could be achieved. The intrinsic resolution of the frontend electronics including the TDC was measured to be less than 25ps.
Speaker: Dr Matthias Hoek (JGU Mainz)
• 17:12
The upgrade of the ALICE Inner Tracking System 3m
ALICE (A Large Ion Collider Experiment) aims at studying the nuclear matter at high densities and temperatures characterizing a particular state of matter called Quark-Gluon Plasma (QGP), using proton-proton, proton-nucleus and nucleus-nucleus collisions at the CERN Large Hadron Collider (LHC). The third run of LHC will start in 2021 after a shutdown of two years to allow the upgrade of both the accelerator and the experiments. In Run 3 Pb-Pb collisions will be performed at a centre of mass energy per nucleon of 5.5 TeV, with a luminosity L_int=6×10^27 cm^(-2) s^(-1). The interaction rate will be up to 50 kHz and 400 kHz for Pb-Pb and pp collisions, respectively. To fulfil the requirements of the ALICE physics program for Run 3 of the LHC, a major upgrade of the experimental apparatus is planned for installation in 2019-2020. A key element of the ALICE upgrade is the construction of a new, ultra-light, high-resolution Inner Tracking System (ITS). The new ITS will significantly enhance the determination of the distance of closest approach to the primary vertex, the tracking efficiency at low transverse momenta, and the read-out rate capabilities, with respect to what achieved with the current detector. It will consist of seven layers equipped with silicon Monolithic Active Pixel Sensors with a pixel size of the order of 28x28 µm^2. To transmit (receive) data to (from) the sensors, Flex Printed Circuits (FPC) will be wire-bonded to the sensors themselves. The goal is to improve the reconstruction capabilities of heavy flavour (c and b quarks) mesons and baryons. In addition, the new tracking detector will allow us to study low-mass dileptons and low-p_T charmonia at mid (e^+ e^-) rapidities. In the talk, the first part is dedicated to the description of the present ALICE apparatus focusing the attention on the ITS, it’s operational limits and the physics motivations of its upgrade. Then, the future ITS layout will be outlined entering in the details of the various hardware components. In the second part, an overview of the expected physics performance will be shown.
Speaker: Dr Ivan Ravasenga (DISAT, Politecnico di Torino)
• 17:15
Study of corrections to the eikonal approximation 3m
For the last decades, multiple international facilities have developed Radioactive-Ion Beams (RIB) to measure reaction processes including exotic nuclei. These measurements coupled with an accurate theoretical model of the reaction enable us to infer information about the structure of these nuclei. The partial-wave expansion and the Continuum-Discretised Coupled Channel method (CDCC) provide a precise description of two- and three-body collisions respectively. Unfortunately, these methods have one main drawback: their computational cost. To cope with this issue, the eikonal approximation is a powerful tool as it reduces the computational time and still describes the quantum effects observed in reaction observables. Nevertheless, its range of validity is restricted to high energy and to forward scattering angles. In this work, we analyze the extension of the eikonal approximation to lower energies and larger angles through the implementation of two kinds of corrections. These aim to improve the treatment of the nuclear and Coulomb interactions within the eikonal model. The first correction is based on an expansion of the T-matrix [1] while the second relies on a semi-classical approach [2,3]. They permit to better account for the deflection of the projectile by the target, which is neglected in the standard eikonal model. The gain in accuracy of each correction is evaluated through the analyses of angular cross sections computed with the standard eikonal model, its corrections and either the partial-wave expansion (two-body collisions) or CDCC (three-body collisions). These analyses have been performed for tightly bound projectiles ($^{10}\mathrm{Be}$) and halo nuclei ($^{11}\mathrm{Be}$) from intermediate energies ($70$ MeV/nucleon) down to energies of interest of future RIB facilities such as HIE-ISOLDE and ReA12 at MSU ($10$ MeV/nucleon). [1] S. J. Wallace, Ann. Phys. 78, 190 (1972). [2] T. Fukui, K. Ogata, and P. Capel. Phys. Rev. C 90, 034617 (2014). [3] C. E. Aguiar, F. Zardi and A. Vitturi, Phys. Rev. C 56, 1511 (1997).
Speaker: Ms Chloë Hebborn (Université libre de Bruxelles)
• 17:18
New strategies of the LHC experiments to meet the computing requirements of the HL-LHC era 3m
The performance of the Large Hadron Collider (LHC) during the ongoing Run 2 is above expectations both concerning the delivered luminosity and the LHC live time. This resulted in a volume of data much larger than originally anticipated. Based on the status of current data production levels and the structure of the LHC experiment computing models, the estimates of the data production rates and resource needs were re-evaluated for the era leading into the High Luminosity LHC (HL-LHC), the Run 3 and Run 4 phases of LHC operation. It turns out that the raw data volume will grow ~10 times by the HL-LHC era and the processing capacity needs will grow more than 60 times. While the growth of storage requirements might in principle be satisfied with a 20% budget increase and technology advancements, there is a gap of a factor 6 to 10 between the needed and the available computing resources. The threat of a lack of computing and storage resources was present already in the beginning of Run 2, but could still be mitigated e.g. by improvements in the experiment computing models and data processing software or utilization of various types of external computing resources. For the years to come, however, new strategies will be necessary to meet the huge increase in the resource requirements. In contrast with the early days of the LHC Computing Grid (WLCG), the field of High Energy Physics (HEP) is no longer among the biggest producers of data. Currently the HEP data and processing needs are ~1% the size of the largest industry problems. Also, HEP is no longer the only science with large computing requirements. In this contribution, we will present new strategies of the LHC experiments towards the era of the HL LHC, that aim to bring together the desired requirements of the experiments and the capacities available for delivering physics results.
Speaker: Dr Dagmar Adamova (NPI ASCR Prague/Rez)
• 17:21
K* resonance dynamics in heavy-ion collisions 3m
We study the dynamics of the strange vector meson resonance K* in heavy-ion collisions using the microscopic Parton-Hadron-String-Dynamics (PHSD) transport approach with hadronic and partonic degrees-of-freedom and dynamics hadronisation. We investigate the behaviour of the K* in the medium by using Breit-Wigner spectral functions with self-energies based on the G-Matrix approach. The results from PHSD are compared to data from STAR at RHIC and ALICE at the LHC for different collisions, energies and centralities. The analysis shows that the final K*s (which can be observed experimentally) are produced predominantly during the hadronic phase, dominated by inelastic kaon-pion collisions, whereas K*s from the QGP decay and rescatter and thus are lost during the reconstruction. Furthermore the in-medium effects are relatively small at high energies (RHIC and LHC) as compared to lower energies (FAIR). Therefore the influence of the in-medium effects on the K* is rather modest.
Speaker: Andrej Ilner (Frankfurt Institute for Advanced Studies)
• 17:27
Backbending in the pear shaped 223Th nucleus : First evidence of a high spin octupole to quadrupole shape transition in the actinides. 3m
An experiment as been realized using the EUROBALL IV array at Strasbourg to study the fission products produced in the reaction 208Pb(180; F ). Nevertheless, the experiment also lead to the production of 223Th produced in the fusion-evaporation channel of this reaction with three neutrons evaporated. The structure of thorium isotopes is known to present features of octupole collectivity. The aim of the present talk is to report new results about the structure of 223Th, with possible consequences on the characterization of its octupole properties. Indeed, the quality of the data allowed us to establish more than 25 new levels and extend the yrast band up to 49/2+ . This observation has brought to light a sharp backbending occuring at the highest spins promoting the 223Th as the heavier thorium isotope having an accident observed in its moment of inertia. The interpretation of this phenomenon in terms of band crossing will be discussed by comparing experimental evidences with various calculations. This backbending is the first expected evidence of a quadrupole-octupole shape transition in the actinides at high spin. Moreover, a new non-yrast structure was discovered showing very dierent features and further details will be presented.
Speaker: Mr Guillaume Maquart (IPNl)
• 17:30
Coherent pi0 photoproduction on spin-zero nuclei 3m
The method of coherent pi0 photo production (gamma + Ag.s.-> pi0 +Ag.s., where Ag.s. is a nucleus in its ground state) provides an ecient tool to study the neutron skin of various nuclei. We will investigate the case of nuclei with zero spin and isospin from theoretical point of view in the framework of a distorted wave impulse approximation in momentum space. For the pion-nucleus nal-state interaction we will employ phenomenological pion-nucleus optical potential, which involves analysis of pion-pion elastic scattering as the solution of a Lippmann-Schwinger integral equation. As a rst application, we will show results for 12C.
Speaker: Dr Slava Tsaran (Uni. Mainz)
• 17:40
Poster Discussion 1h 20m
• Tuesday, 24 January
• 09:00 13:30
Tuesday Morning
• 09:00
Electric Dipole Moment Searches 45m
Electric Dipole Moment Searches
Speaker: Prof. Peter Fierlinger (TUM)
• 09:45
Status and perspectives of nuclear chiral EFT" 45m
I will present the status of nuclear forces and electroweak exchange currents in chiral effective field theory and discuss selected applications to few-nucleon systems.
Speaker: Prof. Evgeny Eppelbaum (Uni. Bochum)
• 10:30
Coffee Break 30m
• 11:00
recent results from ALICE on heavy flavor probes of the quark gluon plasma 45m
recent results on open charm and beauty observables as well as quarkonia will be reported
Speaker: Prof. Johanna Stachel (Universitaet Heidelberg)
• 11:45
Subthreshold charm and strangeness production at FAIR energies 30m
We present results on the sub- and near threshold production of multi-strange hadrons and of charmed hadrons. These particle allow to explore multi-step processes in dense hadronic matter. We provide an alternative explanation for the observed transparency ratios and show how the observed enhanced Cascade production at HADES energies can be explained. For the near and subtrehsold production of charmed hadrons, we present the first estimates for the SIS-100 energies, indicating that charm studies might still be feasable, even without SIS-300.
Speaker: Prof. Marcus Bleicher (Univ. Frankfurt)
• 12:15
Blessings of a phantom: what remains from the 750 GeV diphoton resonance 30m
Blessings of a phantom: what remains from the 750 GeV diphoton resonance
Speaker: Prof. Matthias Neubert (JGU)
• 17:00 19:40
Tuesday Afternoon
• 17:00
The FOOT (FragmentatiOn Of Target) Experiment 20m
Particle therapy uses proton or 12-C beams for the treatment of deep-seated solid tumors. Due to the features of energy deposition of charged particles a small amount of dose is released to the healthy tissue in the beam entrance region, while the maximum of the dose is released to the tumor at the end of the beam range, in the Bragg peak region. Dose deposition is dominated by electromagnetic interactions but nuclear interactions between beam and patient tissues inducing fragmentation processes must be carefully taken into account. In proton treatment the target fragmentation produces low energy, short range fragments along all the beam range. In 12-C treatments the main concern are long range fragments due to projectile fragmentation that release dose in the healthy tissue after the tumor. The FOOT experiment (FragmentatiOn Of Target) of INFN (Istituto Nazionale di Fisica Nucleare) is a new project designed to study these processes. Target (16-O,12-C) fragmentation induced by 150-250 MeV proton beam will be studied via inverse kinematic approach, where 16-O and 12-C beams, in the 150-200 AMeV energy range, collide on graphite and hydrocarbons target to allow the extraction of the cross section on Hydrogen. This configuration explores also the projectile fragmentation of these beams. The detector includes a magnetic spectrometer based on silicon pixel detectors, a scintillating crystal calorimeter with TOF capabilities, able to stop the heavier fragments produced, and a ΔE detector to achieve the needed energy resolution and particle identification. The detector, the physical program and the timetable of the experiment will be presented
Speaker: Dr Giuseppe Battistoni (INFN)
• 17:20
First measurement of the charge form factor of the proton at very low Q2 with Initial State Radiation 20m
In this talk, a new experimental method based on initial-state radiation (ISR) in e-p scattering will be presented, in which the radiative tail of the elastic e-p peak is used to extract information on the proton charge form factor (G_E^p) at extremely small Q^2. The ISR technique was validated in a dedicated experiment using the spectrometers of the A1-Collaboration at the Mainz Microtron (MAMI). This provided first measurements of G_E^{p} for 0.001
Speaker: Dr Harald Merkel (Institut für Kernphysik, Johannes Gutenberg-Universität Mainz)
• 17:40
Open heavy-flavour measurements in Pb-Pb collisions with ALICE at the LHC 20m
Strongly interacting matter at high densities and temperatures can be created and carefully studied under laboratory conditions in high-energy collisions of heavy atomic nuclei. Heavy quarks (charm and beauty) provide particular good probes to study the so-called Quark-Gluon Plasma state and its evolution since they are predominantly produced in initial hard partonic scattering processes in the early stages of the collision. Since 2010 the Large Hadron Collider at CERN delivers lead-lead collisions at an unprecedented energy in the TeV range. The measurement of open heavy-flavour production in heavy-ion collisions allows studies of the dynamical properties of the plasma phase. The ALICE experiment has measured charm and beauty production in Pb-Pb collisions at $\sqrt{\rm s_{NN}}$ = 2.76 TeV and 5.02 TeV, via the exclusive reconstruction of hadronic D-meson decays and semi-leptonic D and B-meson decays. In this contribution, I will give an overview of current open heavy-flavour results from ALICE ranging from the nuclear modification factor to elliptic flow measurements and of the interpretation of the data by comparing with different model calculations of in-medium energy loss.
Speaker: Andre Mischke (Utrecht University)
• 18:00
Looking for new Physics with Pion Decays 20m
In the Standard Model, electrons, muons, and tau leptons have identical electroweak gauge interactions, a hypothesis known as lepton universality. PIENU is a high precision measurement of the ratio of the rate of the pion decay to electron plus neutrino compared to pion decay to muon plus neutrino, including radiative processes. The SM theoretical prediction of this ratio is one of the most accurately calculated weak interaction observables involving quarks with uncertainty of <0.01%. The experimental value is an order of magnitude less precise. Testing lepton universality can constrain many non-Standard Model scenarios. However, pseudoscalar or scalar interactions induced by leptoquarks, supersymmetric particles, extra dimensions etc. can cause measurable deviations from the theoretical prediction. Experimental observation of such a deviation would be clear evidence of new physics sensitive to mass scales up to 1000 TeV. In this talk we present the latest results from the PIENU experiment which provide the most stringent test of lepton universality as well as limits on the presence of massive neutrino states coupled to electrons.
Speaker: Dr Luca Doria (TRIUMF)
• 18:20
Lifetime of the eta-prime meson at low temperature 20m
This work constitutes one part of an investigation of the low-temperature changes of the properties of the eta-prime meson. In turn these properties are strongly tied to the U(1) axial anomaly of Quantum Chromodynamics. The final aim is to explore the interplay of the chiral anomaly and in-medium effects. We determine the lifetime of an eta-prime meson being at rest in a strongly interacting medium as a function of the temperature. To have a formally well-defined low-energy limit we use in a first step Chiral Perturbation Theory for a large number of colors. We determine the pertinent scattering amplitudes in leading and next-to-leading order. In a second step we include resonances that appear in the same mass range as the eta-prime meson. The resonances are introduced such that the low-energy limit remains unchanged and that they saturate the corresponding low-energy constants. This requirement fixes all coupling constants. We find that the width of the eta-prime meson is significantly increased from about 200 keV in vacuum to about 10 MeV at a temperature of 120 MeV.
Speaker: Mrs Elisabetta Perotti (Uppsala University)
• 18:40
Nuclear medium modifications of properties of kaons measured around threshold with FOPI 20m
Modifications of basic properties of kaons, like mass and decay constants, in hot and dense nuclear medium, often parameterized by the in-medium potentials, are an intensely studied topic in the last two decades. However, until recently the experimental samples obtained from the heavy-ion collisions at 1-2A GeV, used to draw conclusions on the scale of these potentials were limited to narrow windows of the momentum space [1-4]. An installation of the new RPC-based ToF detector within the FOPI setup at GSI allowed for a considerable enhancement of the acceptance range of charged kaons. Recent measurement of the directed and elliptic flow of charged kaons [5], has been followed by the extraction of the K-/K+ ratio across phase space, and was supplemented by the determination of contribution of Phi meson decays to the K- meson yield at freeze-out. This new experimental data set may help to sharpen information on the scale of in-medium modifications of properties of charged kaons via comparisons to the predictions of the transport models. __________ [1] K.Wiśniewski et al., Eur. Phys. J. A 9, 515 (2000). [2] F.Laue et al., Eur. Phys. J. A 9, 397 (2000). [3] W.Scheinast et al., Phys. Rev. Lett. 96, 072301 (2006). [4] P.Gasik et al., Eur. Phys. J. A 52, 177 (2016). [5] V.Zinyuk et al., Phys. Rev. C 90, 025210 (2014).
Speaker: Dr Krzysztof Piasecki (University of Warsaw - Faculty of Physics)
• 19:00
Advances on micro-RWELL gaseous detector 20m
The R&D project on the micro-Resistive-WELL (μ-RWELL) detector technology aims in developing a new scalable, compact, spark-protected, single amplification stage Micro-Pattern Gas Detectors (MPGD) for large area HEP applications as tracking and calorimeter device as well as for industrial and medical applications as X-ray and neutron imaging gas pixel detector. The novel micro-structure, exploiting several solutions and improvements achieved in the last years for MPGDs, in particular for GEMs and Micromegas, is an extremely simple detector allowing an easy engineering with consequent technological transfer toward the photolithography industry. Large area detectors (up 1x2 m$^2$) can be realized splicing μ-RWELL_PCB tiles of smaller size (about 0.5x1 mm$^2$ – typical PCB industrial size). The detector, composed by few basic elements such as the readout-PCB embedded with the amplification stage (through the resistive layer) and the cathode defining the gas drift-conversion gap has been largely characterized on test bench with X-ray and with beam test.
Speaker: Dr Gianfranco Morello (LNF-INFN)
• Wednesday, 25 January
• 09:00 14:00
Wednesday Morning
• 09:00
Beam Energy Scan results from STAR 40m
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Speaker: Dr Lijuan Ruan (BNL)
• 09:40
Standard Model and Heavy Ion Physics with CMS 40m
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Speaker: Dr Elisabetta Gallo (DESY)
• 10:20
Coffee Break 30m
• 10:50
News about Experiments at JPARC 35m
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Speaker: Prof. Masa Iwasaki (Riken)
• 11:25
Nuclear Astrophysics at the Low-Energy Frontiers: Updates from the Laboratory 35m
Unlike powerful explosive scenarios such as supernovae and novae, quiescent stellar evolution is characterised by nuclear reactions at the lowest thermal energies, well below the Coulomb barrier of interacting nuclei. When trying to reproduce these processes in terrestrial laboratories, we are faced with tremendous experimental challenges that require the use of dedicated setups ideally in an underground environment to maximise the possibility for rare event detection. In this talk I will present some updates from recent investigations of key nuclear reactions that take place in various astrophysical sites and during different stages of stellar evolution. I will underline the challenges faced in each and review upcoming opportunities for future scientific advances.
Speaker: Prof. Maria Luisa Aliotta (Univ. Edinburgh)
• 12:00
New results on the Be-8 anomaly 25m
Recently, we measured the e+e− angular correlation in internal pair creation for the M1 transition depopulating the 18.15 MeV 1+ state in 8Be, and observed a peak-like deviation from the predicted IPC [1]. To the best of our knowledge no nuclear physics related description of such deviation can be made. The deviation between the experimental and theoretical angular correlations is significant and can be described by assuming the creation and subsequent decay of a boson with mass: m0c2= 16.70 ± 0. 35(stat) ± 0. 5(sys) MeV. The branching ratio of the e+e− decay of such a boson to the γ decay of the 18.15 MeV level of 8Be is found to be 5. 8x10-6 for the best fit [1]. The data can be explained by a 17 MeV vector gauge boson X that is produced in the decay of the excited state to the ground state, and then decays to e+e− pairs [2]. The X boson would mediate a fifth force with a characteristic range of 12 fm and would have millicharged couplings to up and down quarks and electrons, and a proton coupling that is suppressed relative to neutrons [2]. Recently we reinvestigated the anomaly observed previously by using a new Tandetron accelerator of our Institute. The multi-wire proportional counters were replaced with silicon DSSD detectors, as well as the complete electronics and data acquisition system was changed from CAMAC to VME. We have measured the e+e− angular correlation in internal pair creation for the M1 transition depopulating the 17.64 MeV 1+ state in 8Be, and observed a peak-like deviation from the predicted IPC as well. It is a smaller deviation than we observed from the decay of the 18.15 MeV 1+ state, and appeared at larger angles corresponding to the mass of: m0c2 = 17.0± 0. 3(stat) ± 0. 5(sys) MeV. The branching ratio of the e+e− decay of such a boson to the γ-decay agrees well with the prediction of Feng et al. [2]. [1] A.J. Krasznahorkay et al., Phys. Rev. Lett. 1 16 042501 (2016) [2] J. Feng et al., Phys. Rev. Lett. 1 17, 071803 (2016)
Speaker: Dr Attila Krasznahorkay (MTA Atomki, Hungary)
• 17:00 19:00
Wednesday Afternoon
• 17:00
Ab-initio calculation of neutrino-carbon scattering in the quasi-elastic region 20m
Several upcoming experiments have the ambitious goal to understand neutrino mixing, including the mass hierarchy and CP violation, to search for physics beyond the standard model. These experiments aim to reach a precision at the per-cent level, and, in order to accurately interpret these measurements, the knowledge of the neutrino-nucleus interaction is critical. In this talk we will present recent Green's Function Monte Carlo calculations of the euclidean correlation functions that are relevant for the neutrino-12C scattering in the quasi-elastic region. These non-perturbative calculations fully include long- and short-range correlations in the nuclear wave function, and give an excellent description of properties of light nuclei. We will show that the inclusion of two-body operators consistent with the nuclear Hamiltonian is crucial and their contribution is quite sizable, as already predicted by similar calculations and experimental measurements of electron-scattering. These contributions are necessary to understand electron scattering and are also very important in neutrino-nucleus scattering.
Speaker: Dr Stefano Gandolfi (Los Alamos National Laboratory)
• 17:20
First observation of $\Sigma^{0}$ production in proton induced reactions on a nuclear target* 20m
We have studied the production of neutral $\Sigma^{0}$ baryons in the nuclear reaction p + Nb at an incident proton energy $E_{kin}$ = 3.5 GeV. The measurement has been performed with the HADES experiment setup at GSI, Darmstadt. $\Sigma^{0} \rightarrow \Lambda^{0} \gamma$ decays were identified via the charged decay $\Lambda^{0} \rightarrow p \pi^{-}$ coincident to $e^{+}e^{-}$ pairs from external gamma conversion. Experimental details, analysis procedures and background determination are presented. An observed total of about 250 candidate events is used to determine the $\frac{\Lambda + \Sigma}{\Sigma}$ production ratio. The obtained numbers and spectra are compared to predictions from transport model calculations and are discussed in the context of thermal particle production in nuclear fireballs. *Supported by the Excellence Cluster Universe
Speaker: Mr Tobias Kunz (Technische Universität München)
• 17:40
Baryon Form Factors at BESIII 20m
The BESIII detector at the BEPCII $e^+e^-$ collider has been taken data since 2008. With a small amount of data, we have measured the cross section of $e^+e^-\to p\bar{p}$ at 12 center-of-mass energies from 2.2324 to 3.6710 GeV. Much larger data samples have been collected, for examples, energy scan in 2-3.1 GeV for 650/pb and in 3.85-4.6 GeV for 500/pb, in addition to other data samples. The prospect of baryon form factors measurements will be discussed, including $p\bar{p}$, $n\bar{n}$, $\Lambda\bar\Lambda$, etc. The preliminary but unexpected result of form factors of $\Lambda\bar\Lambda$ measurement is shown, as well as the expectation from the data collected at the threshold of $\Lambda_c\bar\Lambda_c$. Initial State Radiation (ISR) technique is also used, in both tagged and untagged methods, for $p\bar{p}$ study.
Speaker: Dr Cristina Morales
• 18:00
Recent results on soft QCD topics from ATLAS 20m
The ATLAS collaboration has performed several measurements in special data sets with low LHC beam currents, recorded at a center-of-mass energy of 13 TeV: Measurements of the inclusive charged-particle multiplicity and its dependence on transverse momentum and pseudorapidity are presented and compared with predictions of various MC generators. The collaboration has also performed measurements of the number and transverse-momentum sum of charged particles as a function of properties of the leading high pT track in the event at a center-of-mass energy of 13 TeV. The results are compared to predictions of several MC generators. In addition, the total inelastic proton-proton cross section and the diffractive part of the inelastic cross section was measured, using special forward scintillators or the calorimeters. The latter result completes the measurement of the elastic pp cross section in a dedicated run with high beta* optics at 8 TeV centre-of-mass energy with the ALFA Roman Pot detector. From the extrapolation of the differential elastic cross section to t=0, using the optical theorem, the total cross section is extracted with the luminosity-dependent method with high precision. Furthermore, the nuclear slope of the elastic t-spectrum and the total elastic and inelastic cross sections are determined. Finally, the collaboration has studied the hard double parton interactions (DPI) in events with 4 hadronic jets and translated into a measurement of the effective DPI cross section. Several DPI-sensitive variables are unfolded to particle level and compared to predictions of different MC models.
Speaker: Dr Andrey Minaenko (Protvino IHEP)
• 18:20
Open heavy-flavour measurements in pp and p-Pb collisions with ALICE at the LHC 20m
Heavy quarks (charm and beauty), produced in ultra-relativistic heavy-ion collisions, are formed in hard partonic scattering processes in the early stage of the collision, and therefore offer a unique opportunity to probe the properties of the strongly-interacting medium created. An interpretation of measurements in heavy-ion collisions requires measurements in pp and p-Pb collisions, which can themselves offer important information about heavy-flavour production. The study of heavy-flavour production in pp collisions offers a baseline measurement to understand in-medium modification in Pb-Pb collisions as well as a test of pQCD predictions, and p-Pb measurements can give crucial information on cold nuclear matter effects, such as nuclear modification of parton distribution functions, $k_T$ broadening or energy loss in cold nuclear matter, as well as address the possibility of collective behaviour in smaller systems. More differential measurements can give further insight into heavy-flavour production. The measurement of D-meson yields as a function of multiplicity in pp and p-Pb collisions can offer unique insight into particle production mechanisms, including the interplay between hard and soft mechanisms, and the role of multi-parton interactions in heavy-flavour production. In addition the study of angular correlations between D-mesons and charged particles can give interesting insight into particle production mechanisms and jet properties. The ALICE detector is well suited to measure charmed meson decays via hadronic channels and semi-leptonic heavy-flavour decays. This talk will give an overview of heavy-flavour measurements made with the ALICE detector during Run 1 in pp and p-Pb collisions, as well as outline current and future measurements for Run 2.
Speaker: Mr Jaime Norman (University of Liverpool)
• 18:40
From the deuteron to Pc(4450) 20m
Heavy hadrons interacting via pion exchange can form bound states (hadronic molecules), analogous to conventional nuclei. The potential between charmed hyperons and charmed mesons, with parameters constrained by the deuteron, leads naturally to a bound state, whose mass and quantum numbers are consistent with the LHCb "pentaquark" $P_c(4450)$. An appealing feature of the model is that it does not imply a proliferation of unobserved partner states. Within a significant (and constrained) parameter range, and independently of the poorly-known short-distance potential, only one additional partner with the same flavour is expected, and its experimental absence (so far) has a possible explanation. A further partner with different flavour is also predicted, and could be discovered in $\Lambda_b$ decays. A characteristic feature of the molecular interpretation is isospin mixing, resulting in striking signatures in production and decay.
Speaker: Dr Timothy Burns (Swansea University)
• Thursday, 26 January
• 09:00 13:20
Thursday Morning
• 09:00
Inferences on the specific heat and neutrino emissivity of dense matter from accreting neutron stars 45m
Many neutron stars are in binaries and accrete matter transferred from their companion. Often, this accretion is intermittent: the neutron star accretes rapidly for a time, and then is quiescent for a long time. In this talk, I will discuss recent efforts to constrain the core heat capacity and neutrino emissivity of matter at densities above saturation from observations of the surface temperatures of quiescent neutron star transients immediately following an accretion outburst.
Speaker: Prof. Edward Brown (MSU)
• 09:45
Optical Lattice Clocks: Reading the 18th decimal place of frequency 45m
Optical lattice clocks [1] benefit from a low quantum-projection noise by simultaneously interrogating a large number of atoms trapped in the standing wave of light (optical lattice) tuned to the “magic frequency” that mostly cancels out the light shift perturbation in the clock transition [2]. About a thousand atoms enable such clocks to achieve 10-18 instability in a few hours of operation [3-5], allowing intensive investigation and control of systematic uncertainties, such as multipolar and higher order light shifts [6] and the blackbody radiation shift [5]. It is now the uncertainty of the SI (International System of Units) second (~10-16) itself that restricts the absolute frequency measurements of such optical clocks [7, 8]. Direct comparisons of optical clocks are, therefore, the only way to demonstrate and utilize their superb performance beyond the SI second. In this presentation, we report on frequency comparisons of optical lattice clocks with neutral strontium (87Sr), ytterbium (171Yb) and mercury (199Hg) atoms. By referencing cryogenic Sr clocks [5], we have determined the frequency ratios, R = νYb/νSr and νHg/νSr of a Yb clock and a Hg clock with uncertainty at the mid 10-17 [9]. Such ratios provide an access to search for temporal variation of the fundamental constants [10]. We also present remote comparisons of cryogenic Sr clocks located at RIKEN and the University of Tokyo over a 30-km-long phase-stabilized fiber link. The gravitational red shift Δν/ν0 ≈ 1.1×10-18 Δh cm-1 reads out the height difference of Δh~15 m between the two clocks with uncertainty of 5 cm [11], which demonstrates a step towards relativistic geodesy [12]. Finally, we mention our ongoing experiments that reduce clock uncertainty to 10-19 by applying “operational magic frequency” [13] that effectively cancels out higher-order light shifts arising from the dipole, multipolar, and hyper-polarizability effects for a certain range of lattice intensity. References: [1] H. Katori, Optical lattice clocks and quantum metrology, Nature Photon. 5, 203 (2011). [2] H. Katori et al., Ultrastable optical clock with neutral atoms in an engineered light shift trap, Phys. Rev. Lett. 91, 173005 (2003). [3] N. Hinkley et al., An atomic clock with 10-18 instability, Science 341, 1215 (2013). [4] T. L. Nicholson et al., Systematic evaluation of an atomic clock at 2×10-18 total uncertainty, Nature Commun. 6, 6896 (2015). [5] I. Ushijima et al., Cryogenic optical lattice clocks, Nature Photon. 9, 185 (2015). [6] P. G. Westergaard et al., Lattice-Induced Frequency Shifts in Sr Optical Lattice Clocks at the 10-17 Level, Phys. Rev. Lett. 106, 210801 (2011). [7] R. Le Targat et al., Experimental realization of an optical second with strontium lattice clocks, Nature Commun. 4, 2109 (2013). [8] C. Grebing et al., Realization of a timescale with an accurate optical lattice clock, Optica 3, 563 (2016). [9] N. Nemitz et al., Frequency ratio of Yb and Sr clocks with 5 × 10−17 uncertainty at 150 seconds averaging time, Nature Photon. 10, 258 (2016). [10] J.-P. Uzan, The fundamental constants and their variation: observational and theoretical status, Rev. Mod. Phys. 75, 403 (2003). [11] T. Takano et al., Geopotential measurements with synchronously linked optical lattice clocks, Nature Photon. 10, 662 (2016). [12] C. Lisdat et al., A clock network for geodesy and fundamental science, Nature Commun. 7, 12443 (2016). [13] H. Katori et al., Strategies for reducing the light shift in atomic clocks, Phys. Rev. A 91, 052503 (2015).
Speaker: Prof. Hidetoshi Katori (Uni TOkyo)
• 10:30
Coffee Break 30m
• 11:00
Recent developments in nuclear structure theory 30m
Atomic nuclei constitute the heart of matter. They drive the synthesis of chemical elements, serve as star fuel and as laboratories to test fundamental interactions and the Standard Model. Predictions of nuclear properties that start from forces among nucleons and their interactions with external probes as described by chiral effective field theory are arguably the doorway to a solid connection between observations and the underlying fundamental theory of quantum chromo-dynamics. Today, thanks to advances in many-body theory and high performance computing, we can calculate nuclear structure and reactions in a unified way for increasingly large systems and estimate theoretical uncertainties. Recent highlights will be presented, that portrait the role of ab-initio calculations to tackle contemporary issues, such as the investigation of neutron-rich nuclei and the proton-radius puzzle.
Speaker: Dr Sonia Bacca (TRIUMF)
• 11:30
LHCb Results on Flavour Physics 30m
LHCb Results on Flavour Physics
Speaker: Dr Stefania Ricciardi (STFC)
• 12:00
Theoretical Prediction of the Be8 anomaly 25m
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Speaker: Dr Tim Tait (UCI)
• 17:00 19:05
Thursday Afternoon
• 17:00
Nuclear Reactions for Neutrinoless Double Beta Decay 25m
Nuclear Reactions for Neutrinoless Double Beta Decay
Speaker: Dr Manuela Cavallaro (LNS)
• 17:25
Calculations of kaonic nuclei based on chiral meson-baryon coupled channel interaction models 20m
We review our latest calculations of $K^-$-nuclear quasi-bound states. We apply a self-consistent scheme for constructing $K^-$-nuclear potentials from subthreshold chirally inspired $K^-$$N$ scattering amplitudes, which are derived within several chirally motivated meson-baryon coupled-channel interaction models: Prague [1], Kyoto- Munich [2], Murcia [3], and Bonn [4]. They capture the physics of the Λ(1405) and reproduce low energy $K^-N$ observables, including the 1s level shift and width in the $K^-$ hydrogen atom from the SIDDHARTA experiment [5]. We consider the in-medium versions of the scattering amplitudes taking into account Pauli blocking in the intermediate states. We calculate $K^-$ binding energies and widths in various nuclei and study the effects of 2N absorption, core polarization and the P-wave interaction on the considered observables. The chirally inspired $K^-N$ potentials supplemented with phenomenological terms representing $K^-$ multinucleon processes were recently confronted [6] with kaonic atom and $K^-$ absorption data to provide another test of considered interaction models. [1] A. Cieply, J. Smejkal, Nucl. Phys. A 881 (2012) 115 [2] Y. Ikeda, T. Hyodo and W. Weise, Nucl. Phys. A 881 (2012) 98 [3] Z. H. Guo and J. A. Oller, Phys. Rev. C 87, no. 3,(2013) 035202 [4] M. Mai and U.-G. Meißner, Nucl. Phys. A 900, (2013) 51 [5] M. Bazzi et al (SIDDHARTA Collaboration), Phys. Lett. B 704 (2011) 113 [6] E. Friedman, A. Gal, to be published, arXiv:1610.04004 [nucl-th]
Speaker: Mrs Jaroslava Hrtankova (Nuclear Physics Institute, 25068 Rez, Czech Republic)
• 17:45
Recent results from the ATLAS heavy ion program 20m
The heavy ion program in the ATLAS experiment at the Large Hadron Collider aims to probe and characterize the hot, dense matter created in relativistic lead-lead collisions, in the context of smaller collision systems involving nuclei and hadrons. This talk presents recent results based on LHC Run 1 and Run 2 data, including measurements of bulk collectivity, electroweak bosons, jet modifications, and quarkonium suppression. Results will also be presented on electromagnetic processes in ultra-peripheral collisions, including forward dilepton production and light-by-light scattering.
Speaker: Dr Radim Slovak (Prague CU)
• 18:05
Constraining the symmetry energy at high density with the first SπRIT experiments 20m
The nuclear Equation of State (EoS) is a fundamental property of nuclear matter that describes relationships between energy, pressure, temperature, density, and isospin asymmetry in a nuclear system. The asymmetric part of EoS, which is originated by the isospin asymmetry, has not been well constrained yet above the saturation density, contrary to the symmetric part of EoS. Transport model calculations predict that pions emitted from the heavy-ion collisions are sensitive probe to constrain the symmetry energy above the saturation density. The SπRIT Time Projection Chamber (TPC) and ancillary trigger detectors were specifically designed and constructed to constraint the symmetry energy above the saturation density using the radioactive isotope beams produced by the Radioactive Isotope Beam Factory (RIBF) at RIKEN by measuring pions as well as light ions. In this talk, the SπRIT TPC and the first experimental campaign, completed in 2016, are described and preliminary results are presented. Data was collected for the four collision systems: 132Sn+124Sn, 112Sn+124Sn, 124Sn+112Sn, and 108Sn+112Sn with beam energy of 270 AMeV.
Speaker: Dr Giordano Cerizza (Michigan State University/NSCL)
• 18:25
Two-neutron removal from $^{11}$Li in a (p,t) reaction at low incident energy 20m
The structure of halo nuclei, such as $^{11}$Li is of considerable interest [1]. The two-neutron transfer reaction $^{11}$Li(*p,t*) $^{9}$Li promises to provide valuable information on the halo-neutron correlation of this exotic Borromean nuclear species. Unfortunately a relatively sophisticated theoretical prediction does not reproduce the first experimental cross section angular distribution of this reaction at an incident energy of 3 MeV [2] particularly well. The hope that a more refined calculation would be successful seems to be somewhat elusive, because a fairly sophisticated theoretical attempt fails spectacularly [3] to reproduce the angular distribution for the same reaction at a slightly higher incident energy of 4.4 MeV. On the other hand, as will be shown in this work, a very simplistic simultaneous transfer, zero-range distorted-wave Born Approximation gives an excellent reproduction of the angular-distribution shape. The implications of this surprising result will be discussed. [1] I. Tanihata, H. Savajols, and R. Kanungo, Progress in Particle and Nuclear Physics **68** (2013) 215. [2] I. Tanihata et al., Phys. Rev. Lett. **100** (2008) 192502. [3] Ian Thompson, 2011 Theory and Calculation of Two-Nucleon Transfer Reactions Lawrence Livermore National Laboratory, Report LLNL-PRES-492069 (unpublished).
Speaker: Prof. Anthony Cowley (Stellenbosch University)
• 18:45
Selected CMS Results in Higgs Physics 20m
In this talk a selection of the most recent results on the measurement of Higgs boson production and properties from the CMS experiment is presented
Speaker: Dr Milos Dordevic (Uni Belgrade)
• Friday, 27 January
• 09:00 13:10
Friday Morning
• 09:00
Sterile Neutrinos 45m
Sterile neutrinos
Speaker: Prof. Susanne Mertens (TUM, MPA)
• 09:45
Hadron Spectroscopy and Heavy Ion Results at LHCb 45m
Hadron Spectroscopy and Heavy Ion Results at LHCb
Speaker: Dr Giovanni Passaleva (INFN FI)
• 10:30
Coffee Break 30m
• 11:00
Challenging the Standard Model: SuperKEKB and the Belle II Experiment 40m
With SuperKEKB and the Belle II experiment a new era of high statistics flavor physics at the Upsilon (4S) resonance is at the horizon, providing almost two orders of magnitude more luminosity compared to the eminently successful B factories PEP II and KEKB. In this presentation we first give a short introduction into the B-meson system and CP violation within the Standard Model, summarize the present experimental tensions, and then present SuperKEKB’s potential searching for physics beyond the Standard Model. Concurrently with the construction of SuperKEKB, a massive detector upgrade of the Belle detector (“Belle II”) is ongoing. Most importantly, the tracking and particle ID systems are in the focus. Due to the largely increased background close to the beam pipe traditional Silicon strip detector will no longer work. We report on the design and construction of a unique pixel vertex detector for Belle II, coined “PXD”. The PXD sensors are based on the DEPFET-technology, with which an extremely small material budget and a high signal to noise ratio can be reached. The principles of the DEPFET technology will be explained as well as the construction of large self-supporting pixel matrices, making up the PXD. Finally, we show the various steps, tests and commissioning phases for the accelerator and the Belle II detector which are scheduled for first nano-beam collisions by early 2018.
Speaker: Dr Christian Kiesling (Max Planck Institute for Physics)
• 11:40
Mesons in the medium - experimental probes for chiral symmetry restoration 30m
The in-medium modications of hadron properties have been identied as one of the key problems in understanding the non-perturbative sector of QCD. Several theoretical papers discuss the possibility of a partial restora- tion of chiral symmetry in a strongly interacting environment. However, is it possible to nd experimental evidence for partial symmetry restoration by studying the in-medium behaviour of mesons, in particular the meson- nucleus interaction? Is this interaction suciently strong to allow even the formation of mesic states only bound by the strong interaction? The an- swers can be given by studying the meson-nucleus optical potential. What are the experimental approaches to deduce this potential? In this presen- tation experimental results from CBELSA/TAPS on the determination of the meson-nucleus optical potential will be presented and discussed in view of these questions. Data taken on a C and Nb target at CB/TAPS@ELSA have been analyzed to deduce the 0- and !-nucleus optical potential. The data for both mesons are consistent with a weakly attractive potential. The formation and population of !-nucleus and 0-nucleus bound states will be discussed. In case of the ! meson the in-medium width is found to be larger than the potential depth which hampers a successful identication of !-mesic states. The relatively small in-medium width of the 0 meson encourages ongoing experiments to search for 0-nucleus bound states. ?Funded by DFG(SFB/TR-16
Speaker: Dr Mariana Nanova (Uni. Giessen)
• 12:10
Speaker: Prof. Marco Vanderhaeghen (Univ. Mainz)
• 17:00 19:00
Friday Afternoon
• 17:00
Signals for dynamical process form IMF-IMF correlation function 20m
In Heavy Ion Collisions (HIC) at Fermi energies (10 MeV/nucleon ≤ E/A ≤ 100 MeV/nucleon) hot nuclear systems are produced and they may disassemble by a variety of dynamical and statistical mechanisms with vastly different time scales (neck emission, fission, multifragmentation, fusion-evaporation, ecc.). The space-time sensitivity of the fragment-fragment Correlation Function to the emission of Intermediate Mass Fragments (IMFs 3≤ Z ≤ 25) has been investigated in order to pin down size and time characteristics of their emission region. In particular, IMF-IMF correlation functions have been applied to 124Sn+64Ni at E/A= 35 MeV reverse kinematics reactions where strong competition between dynamical and statistical production mechanisms of heavy fragments has been clearly demonstrated [1]. Preliminary comparisons between data and theoretical simulations, will be also discussed. [1]P. Russotto et al. PHYSICAL REVIEW C 91, 014610 (2015)
Speaker: Dr Emanuele Vincenzo Pagano (LNS-INFN)
• 17:20
Kaon-Production in Pion-Induced Reactions at 1.7 GeV/c 20m
The production and properties of $K^0_S$, $K^+$, $\Phi$ and $K^-$ in cold nuclear matter generated in pion-nucleon reactions ($\pi^- + A$, $A = C, W$) at $p_{\pi^-}= 1.7$ GeV/c has been investigated with the HADES detector at GSI. Similar to the $K^0$ production at the surface of the nuclei ($\sigma \sim A^{b}, b=2/3$), as already verified by the FOPI collaboration, it was assumed to apply also for the $K^+$ and $K^-$ production allowing for the study of in-medium effects. In this context the $K^-$ absorption in nuclear matter which should be apparent through strangeness exchange processes ($K^- N\rightarrow Y\pi$) is investigated, contrary to the $K^+$ with no conventional absorption mechanism known. In this talk we are presenting the $K^-$ absorption on the basis of the $K^-/K^+$ ratios in both nuclear environments and obtained cross-sections. In addition the $K^-$ production is shown in terms of the $\phi$ feed-down. * supported by the DFG cluster of excellence "Origin and Structure of the Universe"
Speaker: Ms Joana Wirth (TU München)
• 18:00
The continuing story of two-photon exchange: results from the OLYMPUS experiment 20m
Over the past two decades, a discrepancy has emerged between two different techniques for measuring the proton's electromagnetic form factors. Unpolarized electron-proton cross section measurements paint a picture of the proton's internal structure that is incompatible with measurements from polarization transfer experiments. The leading hypothesis is that the discrepancy is caused by a typically neglected radiative correction, hard two-photon exchange (TPE), which would affect the two measurement techniques in different ways. There is no model independent way to calculate hard TPE, but it can be measured experimentally by looking for an asymmetry between the positron-proton and electron-proton elastic cross sections. Three recent experiments have attempted to quantify this asymmetry, and, just last month, the third of these, called OLYMPUS, released its results (arXiv:1611.04685). The OLYMPUS experiment collected data in 2012 at DESY, alternating between 2 GeV electron and positron beams, directed through a hydrogen gas target. The scattered lepton and recoiling proton were detected in coincidence with a large acceptance toroidal spectrometer. The relative luminosity between the two beam species was monitored with three independent systems, and the results comprise 3 inverse fb of integrated luminosity, exceeding by a factor of three the other two TPE experiments combined. In this talk, the case for the TPE hypothesis will be presented, the OLYMPUS experiment will be described, and the results of all three experiments will be compared.
Speaker: Dr Axel Schmidt (MIT)
• 18:20
Bottomia physics at RHIC and LHC energies 20m
The suppression of Y mesons in the hot quark-gluon medium (QGP) versus reduced feed-down is investigated in UU collisions at RHIC energies and PbPb collisions at LHC energies. Our centrality- and p_T-dependent model encompasses screening, collisional damping and gluodissociation in the QGP. For Y(1S) it is in agreement with both STAR and CMS data provided the relativistic Doppler effect and the reduced feed-down from the Y(nS) and chi_b(nP) states are properly considered. At both energies, most of the suppression for the Y(1S) state is found to be due to reduced feed-down, whereas most of the Y(2S) suppression is caused by the hot-medium effects. The role of the in-medium effects relative to reduced feed-down increases with energy. The p_T-dependence is flat due to the relativistic Doppler effect. We predict the Y(1S)-suppression in PbPb at sqrt(s_NN) = 5.02 TeV and consider the hot-medium vs. cold nuclear matter (CNM) contribution to the suppression in the asymmetric pPb system at the same energy. [1] J. Hoelck, F. Nendzig and G. Wolschin, arXiv:1602.00019
Speaker: Prof. Georg Wolschin (U Heidelberg)
• 18:40
Measurement of polarization transfer to a bound proton at large virtuality 20m
A measurement of the ratio $P_x/P_z$ of polarization transfer components in the $\mathrm{D}(\vec{e},e'\vec{p})n$ reaction at different missing momenta will be reported. The $P_x/P_z$ ratio observed for H, D, and $\mathrm{^{4}He}$, respectively, indicates a dependency on the proton's virtuality and the missing momentum direction, but not the average nuclear density.
Speaker: Dr Ulrich Müller (Inst. für Kernphysik, Univ. Mainz) | {"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": 0.8971582651138306, "perplexity": 2931.2212155910283}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347439928.61/warc/CC-MAIN-20200604094848-20200604124848-00332.warc.gz"} |
https://math.stackexchange.com/users/524175/art?tab=summary | Art
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Mathematics 326 rep 18 Stack Overflow 161 rep 7 Physics 104 rep 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": 0.9662762880325317, "perplexity": 3382.650042385904}, "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/1558232260358.69/warc/CC-MAIN-20190527005538-20190527031538-00089.warc.gz"} |
https://tex.stackexchange.com/questions/495378/the-equations-do-not-align-with-respect-to | # The equations do not align with respect to +
I have the following code to align a system of equations
\begin{aligned}
p &= q r &+ r_0 \\
r &= q_0 r_0 &+ r_1 \\
r_0 &= q_1 r_1 &+ r_2 \\
r_k &= q_{k+1} r_{k+1} &+ r_{k+2} \\
&\mathrel{\vdots} \\
r_{k+1} &= q_{k+2} r_{k+2}
\end{aligned}
I can not understand why these equations do not align w.r.t +.
• A general explanation of how elements of equations are treated by the align environment is given in this answer: Aligning equations – barbara beeton Jun 13 '19 at 3:12
You need one more & to have stuff left-aligned after it (if you want to work with aligned here).
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{aligned} p &= q r &&+ r_0 \\ r &= q_0 r_0 &&+ r_1 \\ r_0 &= q_1 r_1 &&+ r_2 \\ r_k &= q_{k+1} r_{k+1} &&+ r_{k+2} \\ &\mathrel{\vdots} \\ r_{k+1} &= q_{k+2} r_{k+2} \\ \end{aligned}
\end{document}
If you want to center the \vdots, you can use the mathtools package.
\documentclass{article}
\usepackage{mathtools}
\begin{document}
\begin{aligned} p &= q r &&+ r_0 \\ r &= q_0 r_0 &&+ r_1 \\ r_0 &= q_1 r_1 &&+ r_2 \\ r_k &= q_{k+1} r_{k+1} &&+ r_{k+2} \\ &\vdotswithin{=} \\ r_{k+1} &= q_{k+2} r_{k+2} \\ \end{aligned}
\end{document}
• Thank you so much! I got it. Could you please instruct me to put the \cdots right below the center of =? – Navier_Stokes Jun 12 '19 at 4:21
• @LeAnhDung I added something that does it. (Andrew Swann used this trick in an answer to a previous question of yours: tex.stackexchange.com/a/494424/121799.) – user121799 Jun 12 '19 at 4:26
• I admire your great memory :) – Navier_Stokes Jun 12 '19 at 4:28
• @LeAnhDung Those who live in burrows need it otherwise they cannot find out. ;-) – user121799 Jun 12 '19 at 4:30
• @marmot such a nice suggestion.... – MadyYuvi Jun 12 '19 at 4:41
You need alignedat, that doesn't add horizontal spacing, but with one & more.
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{alignedat}{2} p &= q r &&+ r_0 \\ r &= q_0 r_0 &&+ r_1 \\ r_0 &= q_1 r_1 &&+ r_2 \\ &\mathrel{\;\vdots} \\ r_k &= q_{k+1} r_{k+1} &&+ r_{k+2} \\ r_{k+1} &= q_{k+2} r_{k+2} \end{alignedat}
\end{document}
• I'm happy to see you on TexExchange ;) – Akira Jun 12 '19 at 9:28
• @Akira Happy to see you, too! – egreg Jun 12 '19 at 9:29 | {"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": 1.0000100135803223, "perplexity": 4001.899159996637}, "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-2020-10/segments/1581875145981.35/warc/CC-MAIN-20200224193815-20200224223815-00056.warc.gz"} |
http://mathhelpforum.com/calculus/130170-implicit-differentiation.html | 1. ## implicit differentiation
Hi:
If 8y(x+y)=4x, find an expression for dy/dx
8y(x+y)=4x
u=8y
v=(x+y)
du/dx=8
dv/dx=1
dy/dx=8y.1dy/dx+(x+y).1=9
dy/dx=8ydy/dx+(x+y)=9
I'm not sure if my method is solid here. Would somebody please check this for me. Thanks
2. $8y(x+y) = 4x$
$8xy + 8y^2 = 4x$
Using implicit differentiation and the product rule:
$8xy' + 8y + 16yy' = 4$
$8xy' + 16yy' = 4 - 8y$
$y'(8x + 16y) = 4 - 8y$
$y' = \frac{4 - 8y}{8x + 16y} = \frac{1 - 2y}{2(x + 2y)}$
3. Originally Posted by stealthmaths
Hi:
If 8y(x+y)=4x, find an expression for dy/dx
8y(x+y)=4x uv=8y(x+y)=4x
u=8y
v=(x+y)
du/dx=8 typo....du/dy=8
dv/dx=1 dv/dx=1+dy/dx
dy/dx=8y.1dy/dx+(x+y).1=9 no!
dy/dx=8ydy/dx+(x+y)=9
I'm not sure if my method is solid here. Would somebody please check this for me. Thanks
Hi stealthmaths,
you need to reconsider this...
$8y(x+y)=4x$
Both sides are equal.
Differentiate both sides, the derivatives are equal...
$\frac{d}{dx}[8y(x+y)]=\frac{d}{dx}[4x]$
$v\frac{du}{dx}+u\frac{dv}{dx}=4$
$u=8y,\ \frac{du}{dx}=\frac{d}{dx}[8y]=8\frac{dy}{dx}$
$v=x+y,\ \frac{dv}{dx}=\frac{d}{dx}x+\frac{d}{dx}y=1+\frac{ dy}{dx}$
$(x+y)8\frac{dy}{dx}+8y\left(1+\frac{dy}{dx}\right) =4$
$\frac{dy}{dx}\left(8(x+y)+8y\right)+8y=4$
$\frac{dy}{dx}\left(8(x+y)+8y\right)=4-8y$
$\frac{dy}{dx}=\frac{4-8y}{8x+16y}=\frac{1-2y}{2x+4y}$
4. I'm really missed some fundamentals from my lectures on this. I have a video tutorial I think I will watch before I fire some questions back, if that's ok.
In the meantime: I just bought mathtype so I can put my workings in Latex for easy comprehension, and managed to create the problem in the program. But how do I get it into this window without uploading an image file?
Attached Thumbnails
5. Hello, stealthmaths!
If $8y(x+y)\:=\:4x,\$ find an expression for $\frac{dy}{dx}$
First, I would divide by 4: . $2y(x+y) \:=\:x$
Then I would expand: . $2xy + 2y^2 \:=\:x$
Differentiate implicitly: . $2x\frac{dy}{dx} + 2y + 4y\frac{dy}{dx} \:=\:1$
. . . .Rearrange terms: . . . . $2x\frac{dy}{dx} + 4y\frac{dy}{dx} \;=\;1 - 2y$
. . . . . . . . . . .Factor: . . . . . $2(x+2y)\frac{dy}{dx} \;=\;1-2y$
. . . . . . . . Therefore: . . . . . . . . . . . $\frac{dy}{dx} \;=\;\frac{1-2y}{2(x+2y)}$
6. Great!
I have studied and can now understand.
Thanks to Archie Mead for the full version, as this is how we are being taught.
I have also enjoyed to learn the shorter versions thanks to Soroban and Icemanfan. | {"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": 23, "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": 0.8334360718727112, "perplexity": 1116.5073136884769}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662430.97/warc/CC-MAIN-20160924173742-00247-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://papers.neurips.cc/paper/2015/hash/c8c41c4a18675a74e01c8a20e8a0f662-Abstract.html | #### Authors
Sorathan Chaturapruek, John C. Duchi, Christopher Ré
#### Abstract
We show that asymptotically, completely asynchronous stochastic gradient procedures achieve optimal (even to constant factors) convergence rates for the solution of convex optimization problems under nearly the same conditions required for asymptotic optimality of standard stochastic gradient procedures. Roughly, the noise inherent to the stochastic approximation scheme dominates any noise from asynchrony. We also give empirical evidence demonstrating the strong performance of asynchronous, parallel stochastic optimization schemes, demonstrating that the robustness inherent to stochastic approximation problems allows substantially faster parallel and asynchronous solution methods. In short, we show that for many stochastic approximation problems, as Freddie Mercury sings in Queen's \emph{Bohemian Rhapsody}, Nothing really matters.'' | {"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": 0.8996226191520691, "perplexity": 2690.0888825208026}, "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-2021-21/segments/1620243991870.70/warc/CC-MAIN-20210517211550-20210518001550-00299.warc.gz"} |
https://search.datacite.org/works/10.4231/r70z715m | ### 32-digit values of the first 100 recurrence coefficients relative to the weight function w(x)=x^{-1/2}(1-x)^{-1/2}log(1/x) on (0,1) computed by the SOPQ routine sr_jacobilog1(100,-1/2,-1/2,32)
Walter Gautschi
32-digit values of the first 100 recurrence coefficients relative to the weight function w(x)=x^{-1/2}(1-x)^{-1/2}log(1/x) on (0,1) computed by the SOPQ routine sr_jacobilog1(100,-1/2,-1/2,32)
This data repository is not currently reporting usage information. For information on how your repository can submit usage information, please see our documentation. | {"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": 0.9238747954368591, "perplexity": 3834.4784438082925}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104597905.85/warc/CC-MAIN-20220705174927-20220705204927-00436.warc.gz"} |
https://tutorial.math.lamar.edu/Classes/CalcII/ArcLength_SurfaceArea.aspx | Paul's Online Notes
Paul's Online Notes
Home / Calculus II / Parametric Equations and Polar Coordinates / Arc Length and Surface Area Revisited
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Section 3-11 : Arc Length and Surface Area Revisited
We won’t be working any examples in this section. This section is here solely for the purpose of summarizing up all the arc length and surface area problems.
Over the course of the last two chapters the topic of arc length and surface area has arisen many times and each time we got a new formula out of the mix. Students often get a little overwhelmed with all the formulas.
However, there really aren’t as many formulas as it might seem at first glance. There is exactly one arc length formula and exactly two surface area formulas. These are,
\begin{align*}L & = \int{{ds}}\\ S & = \int{{2\pi y\,ds}}\hspace{0.5in}{\mbox{rotation about }}x - {\mbox{axis}}\\ S & = \int{{2\pi x\,ds}}\hspace{0.5in}{\mbox{rotation about }}y - {\mbox{axis}}\end{align*}
The problems arise because we have quite a few $$ds$$’s that we can use. Again, students often have trouble deciding which one to use. The examples/problems usually suggest the correct one to use however. Here is a complete listing of all the $$ds$$’s that we’ve seen and when they are used.
\begin{align*}ds & = \sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \,dx & \hspace{0.5in}& {\mbox{if }}y = f\left( x \right),\,\,a \le x \le b\\ ds & = \sqrt {1 + {{\left( {\frac{{dx}}{{dy}}} \right)}^2}} \,dy & \hspace{0.5in} & {\mbox{if }}x = h\left( y \right),\,\,c \le y \le d\\ ds & = \sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2}} \,\,dt & \hspace{0.5in} & {\mbox{if }}x = f\left( t \right),y = g\left( t \right),\,\,\alpha \le t \le \beta \\ ds & = \sqrt {{r^2} + {{\left( {\frac{{dr}}{{d\theta }}} \right)}^2}} \,d\theta & \hspace{0.5in} & {\mbox{if }}r = f\left( \theta \right),\,\,\alpha \le \theta \le \beta \end{align*}
Depending on the form of the function we can quickly tell which $$ds$$ to use.
There is only one other thing to worry about in terms of the surface area formula. The $$ds$$ will introduce a new differential to the integral. Before integrating make sure all the variables are in terms of this new differential. For example, if we have parametric equations we’ll use the third $$ds$$ and then we’ll need to make sure and substitute for the x or y depending on which axis we rotate about to get everything in terms of t.
Likewise, if we have a function in the form $$x = h\left( y \right)$$ then we’ll use the second $$ds$$ and if the rotation is about the y-axis we’ll need to substitute for the x in the integral. On the other hand, if we rotate about the x-axis we won’t need to do a substitution for the y.
Keep these rules in mind and you’ll always be able to determine which formula to use and how to correctly do the 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": 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": 0.9123900532722473, "perplexity": 644.6286244130924}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347388012.14/warc/CC-MAIN-20200525063708-20200525093708-00282.warc.gz"} |
https://www.gradesaver.com/textbooks/math/calculus/thomas-calculus-13th-edition/chapter-1-functions-section-1-3-trigonometric-functions-exercises-1-3-page-27/15 | Thomas' Calculus 13th Edition
$T=2$
When we are presented with a function such as $cos(bx)$ and are asked to find the period, it is useful to make use of the following formula: $\frac{2\pi}{T}=b$ The $T$ in the equation represents the period of the function, as in the amount of $x$ units before the function repeats itself. In this question, we have $cos(\pi x)$. Thus, $\frac{2\pi}{\pi}=T=2$ . Keep in mind that the formula above allows us to easily swap the $T$ and the $b$ in the formula, thus, the period of this function is $2$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8760606646537781, "perplexity": 57.74651124646476}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827281.64/warc/CC-MAIN-20181216051636-20181216073636-00038.warc.gz"} |
https://de.scribd.com/document/266107690/CFD-applications-for-latent-heat-th-pdf | Sie sind auf Seite 1von 11
# Renewable and Sustainable Energy Reviews 20 (2013) 353363
## Renewable and Sustainable Energy Reviews
journal homepage: www.elsevier.com/locate/rser
## CFD applications for latent heat thermal energy storage: a review
Abduljalil A. Al-abidi n, Sohif Bin Mat, K. Sopian, M.Y. Sulaiman, Abdulrahman Th. Mohammed
Solar Energy Research Institute, University Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia
a r t i c l e i n f o
abstract
Article history:
Accepted 19 November 2012
Available online 9 January 2013
Thermal energy storage is needed to improve the efciency of solar thermal energy applications (STEA)
and to eliminate the mismatch between energy supply and energy demand. Among the thermal energy
storages, the latent heat thermal energy storage (LHTES) has gained much attention because of its highenergy densities per unit mass/volume at nearly constant temperatures. This review presents previous
studies on the numerical modeling of phase change materials (PCMs) through a commercial computational uid dynamic (CFD) software and self-developed programming to study the heat transfer
phenomena in PCMs. The CFD (Fluent) software is successively used to simulate the application of
PCMs in different engineering applications, including electronic cooling technology, building thermal
storage, and heating, ventilation, air conditioning (HVAC). Using CFD software to design LHTES is
believed to be an effective way to save money and time and to deliver optimization tools for maximum
efciency of STEAs.
Keywords:
Phase change material
Thermal energy storage
CFD
Fluent
Contents
1.
2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Numerical solution of PCMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
2.1.
Fluent program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
2.2.
Mathematical formulation for Fluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
2.3.
Fluent solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
3. Numerical simulation of PCM in thermal storage geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
3.1.
Spherical capsule geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
3.2.
Square and rectangular geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
3.3.
Cylindrical capsule geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
4. Numerical simulation of the PCM applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
4.1.
Numerical simulation of PCM for electronic devices applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
4.2.
Numerical simulation of PCM for HVAC equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
4.3.
Numerical simulation of PCM for building applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
4.4.
Numerical simulation for other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
1. Introduction
Researchers intensively studied the thermal energy storage
of PCMs for the last three decades because of the latters high
thermal energy densities per unit volume/mass and their applicability to different engineering elds using wide temperature
## Corresponding author. Tel.: 60 14 7169 139; fax: 60 38 9214 593.
http://dx.doi.org/10.1016/j.rser.2012.11.079
## ranges. PCM thermal storage plays a key role in improving energy
efciency. It limits the discrepancy between the energy supply
and the energy demand of solar thermal energy applications
(STEAs), particularly when the STEA operation strategy depends
solely on solar energy as a main source. PCM thermal storage
indicates high performance and dependability with the advantages of high storage capacity and nearly constant thermal energy
[1]. Most STEAs need constant or near-constant temperature for
high-efciency strategies. Using latent heat thermal energy storage (LHTES) as thermal energy storage can provide the required
354
A.A. Al-abidi et al. / Renewable and Sustainable Energy Reviews 20 (2013) 353363
Nomenclature
g
C
Cp
g
h
H
L
k
T
u
## Mushy zone constant (kg/m3 s)
Specic heat of PCM, (J/kg)
gravity acceleration, (m/s2)
sensible enthalpy(J/kg)
enthalpy (J/kg)
latent heat fusion (J/kg)
thermal conductivity (W/m K)
Temperature (1C or K)
velocity (m/s)
m
a
## uid density (kg/m3)
liquid fraction
volumetric expansion coefcient (1/K)
Dynamic viscosity (kg/m s)
Phase volume fraction
Subscripts
ref
s
l
Reference
Solidus of the phase change materials
Liquidus of the phase change material
Greek letters
## constant temperature that matches the melting temperature of
the PCMs. PCMs are used in different engineering elds, such as in
the following: the thermal storage of building structures [37];
building equipment, including domestic hot water; heating and
cooling systems [8,9]; electronic products [1012]; drying technology [13], waste heat recovery [14]; refrigeration and cold
storage [1518]; solar air collectors[19]; and solar cookers [20].
Using a CFD software to design a LHTES is expected to be an
effective way to save money and time and to deliver optimization
tools for maximum efciency of STEAs.
## 2. Numerical solution of PCMs
The mathematical formulation of a phase transient known
as a phase change or moving boundary is governed by a partial
deferential equation that can be solved either analytically or
numerically. The analytical solution of PCMs is problematic
because of the nonlinear phase front interfaces, complex geometries, and nonstandard boundary condition; the few analytical
studies available are on 1D cases with regular geometries and
zisik [21] reported that the
a standard boundary condition. O
numerical method for solving PCMs can be categorized as xedgrid, variable grid, front-xing, adaptive grid generation, and
enthalpy methods.
Predicting the behavior of phase change systems is difcult
because of its inherent non-linear nature at moving interfaces, for
which the displacement rate is controlled by latent heat lost or
absorbed at the boundary [22]. The heat transfer phenomena in
solidliquid PCMs can be analyzed using two main methods: the
temperature-based and enthalpy-based methods. In the rst
method, temperature is considered a sole dependent variable.
The energy conservation equations for the solid and liquid are
written separately; thus, the solidliquid interface position can be
tracked explicitly to achieve an accurate solution for the problems.
@T s
@T l
ks
k rLkvn ,
@n
@n l
## to be satised at the solidliquid interface; (3) the enthalpy
formulation involves the solution within a mushy zone, involving
both solid and liquid materials, between the two standard phases;
and (4) the phase change problem can be solved more easily [23].
@rH
rU rvH rUkDT S
@t
## r is the density of the PCM, v is the uid velocity, and H is the
enthalpy, S is the source term.
Dutil et al. [22] presented an intensive mathematical and
numerical review of the PCM application based on the rst and
second laws of thermodynamics. Using 252 references, they
determined the mathematical and numerical methods applied
to solve heat transfer problems involving PCMs for thermal
energy storage, the mathematical fundamentals of PCMs, and
the different application geometries and applications. Verma et al.
[24] also introduced other PCM mathematical reviews.
Recently, researchers used the Fluent software by ANSYS to
simulate melting and solidication in engineering problems.
Other software that can be used to simulate the PCM process
include COMSOL Multiphysics and Star-CMM . However, Fluent
is preferred by most researchers for melting and solidifying PCMs.
Conversely, some researchers self-developed a program using
computational language (C, Fortran, Matlab) to study the heat
transfer phenomena in PCMs. Table 1 summarizes some of the
self-developed programs for different PCM geometries.
2.1. Fluent program
The Fluent software by ANSYS is a computational uid
dynamic (CFD) program used successfully to simulate different
engineering problems. This software has a specic model that can
simulate a range of different melting and solidication problems
in engineering, including casting, melting, crystal growth, and
## where Ts denotes the temperature in the solid phase, Tl denotes
the temperature in the liquid phase, ks is the thermal conductivity
of the solid phase, kl is the thermal conductivity of the liquid
phase, n is the unit normal vector to the interface, and vn is the
normal component of the velocity of the interface. L is the latent
heat of freezing, as shown in Fig. 1.
In the second method, the solidliquid interface position need
not be tracked. Researchers often use the enthalpy formulation
because of the following advantages: (1) the governing equations
are similar to the single-phase Eq.; (2) no explicit conditions need
Liquid
n
Solid
Ts (x, y, z, t)
Tl (x, y, z, t)
P
v
## Fig. 1. Solidliquid interface for a multidimensional situation [21].
A.A. Al-abidi et al. / Renewable and Sustainable Energy Reviews 20 (2013) 353363
355
Table 1
Self-developed numerical model for different PCM geometries.
Ref
PCM
Geometry
Dimension
Discretization
methods
## Numerical solution methods
Validation
FDM
Enthalpy method-based control volume approach; the power law used for
discretization and semi-implicit method for pressure-linked equations (SIMPLE) was
adopted for the pressurevelocity couple and for the tridiagonal matrix algorithm
(TDMA) solver to solve the algebraic equations
An enthalpy formation-based control volume approach, the fully implicit method, and
TDMA were used to solve the algebraic equations
The interpolating cubic spline function method was used to determine the specic
heat in each time step of the temperature calculations; the control volume approach
and the backward Euler scheme were chosen for the temperature discretization in
time
The power law used for discretization and SIMPLE were adopted for pressurevelocity
coupling
Enthalpy methods and the TDMA solver were used to solve the equations
Fixed control volume and enthalpy methods
A control volume-based implicit form was used. All equations were solved
simultaneously using the GaussSeidel iterative method
The enthalpy formulation approach, the control volume method, and the alternating
direction scheme were used to discretize the basic equations
The control volume approach, the power law used for discretization, and the SIMPLE
algorithm were adopted for pressurevelocity coupling
A temperature and thermal resistance iteration method was developed to analyze
PCM solidication to obtain the solutions
Enthalpy methods were used; the equations were solved simultaneously using the
GaussSeidel iterative method
The enthalpy method and the TDMA were used to solve the resulting algebraic
equations solved using an iteration procedure
Standard Galerkin nite element method was used
Using the enthalpy method, the equations were solved by adopting the alternating
direction implicit (ADI) method and the TDMA
An implicit FDM approach and a moving-grid scheme were used. The problem domain
was divided into two regions, solid and liquid, both of which were separated by the
interface. All algebraic equations were solved simultaneously
The heat delivered by the capsule could also be calculated by the thermal resistance
circuit. The resulting equation and associated boundary conditions were treated using
a moving grid method
Experimental
[66] Gallium
Rectangular 2D (X, Y)
[70] Parafn wax
RII-56
(X, Y)
PCM bags
3D (X, Y, Z) FDM
In duct
Cylindrical
2D (R, X)
FVM
[74] 4 PCM
[75] RT60
Cylindrical
Cylindrical
Cylindrical
2D (R, X)
2D (R, X)
2D (R, U)
FDM
FDM
FDM
## [76] Parafn 130/
135 Type 1
[77] Parafn wax
Cylindrical
2D (R, U)
FDM
Cylindrical
2D (R, U)
FDM
[80] n-Hexacosane
Cylindrical
1D (R)
[81] 4PCM
Cylindrical
2D (R, X)
FDM
[82] 2PCM
Cylindrical
2D (R, X)
FDM
[83] 5 PCM
[84] 3 PCM
Cylindrical
Cylindrical
2D (R, X)
2D (R, X)
FEM
FDM
[86] Water
Spherical
1D (R)
FDM
## [87] Water, water
Glycol
mixtures
Spherical
1D (R)
FDM
[68,69]
[71]
Experimental
Experimental
Experimental
Experimental
[78,79]
Experimental
[73]
[85]
Experimental
Experimental
FDM; Finite difference method, FEM; Finite elements method, FVM; Finite volume method.
## solidication. The program can be used to solve the phase change
that occurs at a single temperature (pure metals) or over a range
of temperatures (mixture, alloy, and so on). The applications and
limitations of Fluent can be found in Ref. [25].
To begin the Fluent software simulation, the physical engineering problem is drawn and meshed in a specic geometric
modeling using mesh generation tools that include the Fluent
software (Gambit, workbench). The mesh generation tools can be
imported from other programs like Auto CAD. After the physical
conguration is drawn and meshed, the boundary layers and
zone types are dened, and the mesh is exported to the Fluent
software.
Different grid sizes and time steps should be applied to the
numerical model to ensure that the numerical results are independent of the parameters. Small grid sizes and time steps are
preferred for a short simulation time in the computer.
## 1 to 0 as the material solidies. When the material has fully
solidied in a cell, the porosity becomes zero, resulting in the
drop of velocities to zero. In this section, an overview of the
solidication/melting theory is given. For details on the enthalpy
porosity method, refer to Voller and Prakash [26].
The energy equation is expressed as
@t rH r rvH rkrT S
3
## where href is the reference enthalpy at the reference temperature
Tref, cp is the specic heat, DH is the latent heat content that may
change between zero (solid) and 1 (liquid), L is the latent heat of
the PCM, and g is the liquid fraction that occurs during the phase
change between the solid and liquid state when the temperature
is T l 4 T 4 T s . Thus, g may be written as
## The mathematical equations used to solve the solidication
and melting models in Fluent depend on the enthalpyporosity
technique [2628] and on the nite volume methods. In the
former, the melt interface is not tracked explicitly. A quantity
called liquid fraction, which indicates the fraction of the cell
volume in liquid form, is associated with each cell in the domain.
The liquid fraction is computed at each iteration based on
enthalpy balance. The mushy zone is a region wherein the liquid
fraction lies between 0 and 1. The mushy zone is modeled as a
pseudo porous medium in which the porosity decreases from
## where r is the density of the PCM, v is the uid velocity, k is the
thermal conductivity, H is the enthalpy, and S is the source term.
The sensible enthalpy can be expressed as
Z T
h href
cpDT,
4
T ref
## and H can be dened as
H h DH,
g DH=L,
8
>
<0
g 1
>
: TT =T T
s
s
l
6
if
T oT s
if
T 4T l
if
Tl 4 T 4 Ts
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A.A. Al-abidi et al. / Renewable and Sustainable Energy Reviews 20 (2013) 353363
104 and 107.
2
S C 1g
g3 A
## 2.3. Fluent solver
It is the Darcys law damping terms (as source term) that are
added to the momentum equation due to phase change effect on
convection. Where A 0.001 is a small computational constant
used to avoid division by zero, the mushy zone constant C
PressureBased Segregated Algorithm
## The Fluent software has two main solvers: the pressure-based
solver and the density-based coupled solver (DBCS). Only the rst
method can be used to simulate the melting and solidication
PressureBased Coupled Algorithm
Update properties
Update properties
Solve sequentially:
Solve simultaneously:
equations and pressure -based
continuity equation
## U vel V vel W vel
Solve pressurecorrection
(continuity) equation
Update mass flux
Update mass flux,
pressure, and velocity
scalar equations
scalar equations
No
Yes
Converged?
Stop
No
Yes
Converged?
Stop
## Set the solution parameters
Solution parameters
Choosing the solver
Discretization schemes
## Initialize the solution
Initialization
Enable the solution monitors of
interest
Convergence
Monitoring convergence
Stability
Calculate a solution
Calculate a solution
1. Setting Under-relaxation
2. Setting Courant number
Check for convergence
Monitoring convergence
Accuracy
No
Yes
Grid Independence
No
Yes
Stop
## Fig. 3. Solution procedure overview for Fluent [30].
A.A. Al-abidi et al. / Renewable and Sustainable Energy Reviews 20 (2013) 353363
## problems. The pressure-based solver employs an algorithm that
belongs to a general class of methods called the projection
method [29] wherein the constraint of the mass conservation
(continuity) of the velocity eld is achieved by solving a pressure
(or pressure correction) equation. The pressure equation is
derived from the continuity and the momentum equations in
such a way that the velocity eld, corrected by the pressure,
satises the continuity. The governing equations are nonlinear
and coupled to one another; thus, the solution process involves
iterations where the entire set of governing equations is solved
repeatedly until the solution converges. Available in Fluent are
two pressure-based solver algorithms, namely, a segregated
algorithm and a coupled algorithm Fig. 2.
Different discretization (interpolation methods) schemes
are available for the convection terms in Fluent, including the
rst-order upwind scheme, power law scheme, second-order
upwind scheme, central-differencing scheme, and the quadratic
upwind interpolation scheme. The rst-order upwind, power law,
and second-order upwind schemes are mostly used with solidication and melting problems. The last two methods are more
accurate than the rst methods. In addition, the interpolation
method of the face pressure for the momentum equations are the
semi-implicit pressure-linked equation (SIMPLE), the SIMPLEconsistent (SIMPLEC), the pressure-implicit with splitting of operators (PISO), and the fractional step method (FSM). Fig. 3 shows the
solution procedure overview. More details about the solution,
initialization, and discretization methods can be found in Ref. [25].
The physical properties of materials, such as density,
thermal conductivity, heat capacity, and viscosity, may be temperature-dependent and/or composition-dependent. The temperature
dependence is based on a polynomial, piecewise-linear, or
Fig. 4. Melt fraction vs. time for various values of the mushy zone constant with
different models [31].
357
## piecewise-polynomial function. Individual component properties
are either dened by the user or computed via kinetic theory.
Nevertheless, these physical properties can be dened as a
constant value, a temperature-dependent function, or a userdened function (UDF) that can be written in a specic programming language to dene the temperature-dependence of the
thermophysical properties.
Some researchers deduced that the thermophysical properties
of PCMs, such as density and viscosity, are dependent on temperature changes and are determined by specic correlations.
r rl =bTT l 1
m exp A B=T
9
10
## b is the thermal expansion coefcient, and A, B are constant
coefcients. The relationship between PCM and air is described by
a specic model called the volume of uid (VOF). This model
denes the PCMair system with a moving internal interface, but
without inter-penetration of the two-phase uid. Three conditions reect the uids volume fraction in the computational cells,
denoted as an: if an 0, then the cell is empty of the nth uid; if
an 1, then the cell is full of the nth uid; and if 0 o an o1, then
the cell contains the interface between the nth uid and one or
more other uids.
Shmueli et al. [31] numerically investigated the melting of
PCM in a vertical cylinder tube with a diameter of 3 and 4 cm,
exposed to air above, and insulated at the bottom. The wall
temperature was between 10 and 30 1C above the melting
## Fig. 5. Comparison between pressure discretization schemes and experimental
results with different models [31].
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## temperature of the PCM. They studied the effect of various
numerical solution parameters such as the pressure velocity
scheme (PISO and SIMPLE) and the pressure discretization regime
(e.g., PRESTO and the weighted force method). They analyzed the
inuence of the mushy zone constant to the melting process,
as shown in Fig. 4 and Fig. 5. They also reported no difference in
the pressure velocity scheme (PISO and SIMPLE) as well as a
considerable difference in the result for pressure discretization.
## 3. Numerical simulation of PCM in thermal storage
geometries
3.1. Spherical capsule geometry
LHTES in a spherical container represents an important case
for thermal storage to be used in different engineering applications such as in packed-bed storage. Moreover, most commercial
companies produce the spherical capsule for this application.
Assis et al. [32] presented a numerical and an experimental
investigation of the melting process of RT27 with a PCM in a
spherical geometry lled with 98.5% solid PCM as an initial
condition of the simulation. One percent was left to account for
the increase of the PCM volume during the phase change transition. The variable density was dened during the liquid state with
linear variation in the mushy state, and the VOF was used for
the PCMair system. The numerical model was based on the axial
symmetry of the physical model using Fluent 6.0. Different design
and operation parameters were studied, and the results indicated
that the melting process of the PCM is affected by its thermal and
geometrical parameters, including the Stefan number and the
shell diameter. Assis et al. [33] numerically and experimentally
studied the solidication process of RT27 as a PCM in a spherical
shell lled with 98.5% liquid PCM. Different shell diameters were
examined, and a transient numerical model was developed using
Fluent 6.2; the numerical model was the same one used in the
melting process of PCM [32].
Hosseinizadeh et al. [34] numerically studied the effect of
various volume fractions of nanopractical copper on an unconstrained melting rate in a spherical container. Different volumes
of nanopractical copper (0, 0.02, 0.04) per volume were used, and
85% of the PCM was lled. They used Fluent to develop an axisymmetric numerical model and added the Darcy law to the
momentum equation to account for the effect of phase change on
convection. The VOF used for the PCMair system and the power
law scheme and the PISO method used for the pressurevelocity
coupling were adopted to solve the momentum and energy
equations, respectively. The PRESTO scheme was used for the
pressure correction equation. Three different Stefan numbers
were analyzed; the validation of the model was based on the
experimental work by Assis et al. [32]. The results indicated that
the nanopractical copper caused an increase in the thermal
conductivity of the nano-enhanced PCMs compared with the
conventional PCM. Unfortunately, the latent heat of fusions
decreased.
Tan et al. [35] reported an experimental and numerical work
on the constrained melting of PCM inside a transparent spherical
glass capsule. The Darcy law was added to the momentum
equation to account for the effect of phase change on convection.
They used Fluent 6.2.16 for the numerical simulation and the
SIMPLE method for solving the governing equations. The power
law differencing scheme was used to solve the momentum and
energy equations, whereas the PRESTO scheme was adopted for
the pressure correction equation. They reported that waviness
and excessive melting of the bottom of the PCM was underestimated by the experimental observation. This discrepancy
## was linked to the use of a support structure to hold the sphere
that could have inhibited heat from reaching the bottom of the
sphere.
Xia et al. [36] developed an effective packed-bed thermal
energy storage that contains a spherical PCM capsule. They
studied the effect of the arrangement of the PCM spheres and
the encapsulation of the PCM on the heat transfer performance of
the LTES bed. They developed a 2D numerical model via Fluent
6.2 using the standard kx model for turbulent ow; a SIMPLE
algorithm was adopted for pressure and velocity coupling. They
reported that random packing was more favorable than special
packing for heat storage and retrieval; both the material and the
thickness of the encapsulation had apparent effects on the heat
transfer performance of the LTES bed.
3.2. Square and rectangular geometry
Arasu and Mujumdar [37] presented a numerical investigation
on the melting of parafn wax dispersed with aluminum (Al2O3)
as a nano-PCM in a square enclosure heated from the bottom side
and from a vertical side with different volume percentages
(2% and 5%). They developed a numerical model using Fluent
6.3.26. The computational domain was resolved with a ne mesh
near the hot and cold wall to resolve the boundary layer; an
increasingly coarser mesh was used in the rest of the domain to
reduce computational time. They used a UDF to dene the
temperature-dependence of the thermophysical properties of
PCM, wherein the rst-order upwind difference scheme was used
to solve the momentum and energy equations, and the PRESTO
scheme was adopted for the pressure correction equation. They
reported that the effective thermal conductivity of a parafn wax
latent heat storage medium could be increased signicantly
by smaller volumetric concentrations of alumina particles in the
parafn wax.
Pinelli and Piva [38] developed a 2D numerical model using
Fluent to study the effects of natural convection on the PCM
process in terms of temperature distributions, interface displacement, and energy storage, and to enhance the agreement between
the experimental work with the numerical one when the conduction dominated is considered only for the heat transfer of the
PCM. The mushy zone constant for the model was approximately
1.6n103, and the cylinder cavity was lled with n-octadecane as
PCM in the solid state and heated from above with an electrical
heater. The overall heat transfer coefcient U considered conduction through the polystyrene layer as well as the convection and
radiation heat exchanged with the environment. The model was
validated with an analytical solution wherein the conductiondominated freezing and numerical methods were considered in
the convectionconduction. They reported that the agreement
between numerical and experimental results signicantly improved
when the presence of circulation in the liquid phase, instead of in
the conduction-dominated process only, was considered.
Researchers reported that the heat transfer in PCM could
be enhanced signicantly by adding nanopractical copper to the
PCM (nanouid). Such addition improves the PCMs thermal
conductivity. Wu et al. [39] numerically investigated the melting
processes of Cu/parafn nanouid PCMs. They developed a 2D
numerical model using Fluent 6.2. The enclosed cavity was lled
with 2.5-cm high PCM and 1 cm air; the top of the cavity was
enclosed and taken by air. The VOF was adopted to describe the
relationship between the PCM and air with a moving interface,
but without inter-penetration of the two media. The QUICK
differencing scheme was used to solve the momentum and energy
equations, whereas the PRESTO scheme was adopted for the
pressure correction equation. They reported a 13.1% decrease in
the parafns melting time.
A.A. Al-abidi et al. / Renewable and Sustainable Energy Reviews 20 (2013) 353363
## Li et al. [40] numerically studied the melting process of a PCM
in concentric horizontal annuli of a square external shell. A 2D
numerical model was adopted using Fluent 6.0. They reported
that the top part of the PCM melts faster than the bottom part as a
result of the convection-dominated mode, which accelerates the
melting of the top part of the annulus.
conductivity in contrast to the base material. They used a 2D
numerical model within the commercial code of Fluent 6.2.1. The
SIMPLE algorithm and the QUICK differencing scheme were used
to solve the momentum and energy equations, whereas the
PRESTO scheme was adopted for the pressure correction equation.
The Darcy Law damping terms were added to the momentum
equation. The results indicated an increase in the thermal conductivity of the nano-enhanced PCMs in contrast to the conventional PCM. Unfortunately, the latent heat of fusion decreased.
359
## outlet temperature. Based on their analysis, the best options for
energy storage density were characterized by latent storage units
installed on the heating network of the building rather than on
the primary district heating network.
Buyruk et al. [44] studied the solidication process of water
around different staggered and inline cylinders placed in a
rectangular ice storage tank. They developed a 2D numerical
model using Fluent to depict the temperature distributions and
ice formation in the tank. They adopted Darcys law and the
KozenyCarman equation to model the ow and permeability of
the mushy zone, respectively. Based on a xed ice storage tank
volume, the solidied area of four inline cylinders was larger than
that for six inline cylinders.
## 4. Numerical simulation of the PCM applications
4.1. Numerical simulation of PCM for electronic devices applications
## 3.3. Cylindrical capsule geometry
Cylindrical geometries are considered most promising for
devices for commercial heat exchangers, such as the double pipe
heat exchanger and shell and the tube heat exchanger, because of
their high-efciency in a minimum volume. Most researchers use
this geometry for LHTES, lling the PCM in the tube side or in the
annulus (shell). Commercial products of PCMs are delivered in
these geometries.
Darzi et al. [42] used Fluent 6.2 to simulate the melting
process of the n-octadecane as a PCM in the concentric and
eccentric double pipe heat exchanger. They studied the effect of
the internal tube position on the melting rate. They developed a
2D numerical model (R, y) and adapted low power to solve the
momentum and energy equations. The SIMPLE method was used
for pressurevelocity coupling, and the PRESTO scheme was used
for the pressure correction equation. The results indicated that
the melting rate was the same in the early stage of melting, and
then decreased in the concentric model.
Gou and Zhang [43] reported that the solidication process
of KNO3NaNO3 as PCM in solar power generation using the
direct steam technology is controlled by pure conduction. They
neglected the inuences of natural convection when comparing it
with heat conduction. They developed a 2D numerical model via
Fluent 6.2 and studied the storage with and without foil based on
the same tube diameters and PCM. The results indicated how the
heat transfer and discharge time were affected by the changes in
the geometry of the aluminum foil and in the tube radius. They
reported that the discharge process signicantly accelerated with
the addition of the aluminum foil.
Colella et al. [2] designed a medium-scale LHTES unit for
district heating systems. The shell-and-tube LHTES that included
RT100 as a PCM in the shell side was used to transfer heat from
the district to the heating networks of the building; the heat
transfer uid was water. The thermal storage was charged during
the night using CHP plants and then discharged during the day at
the peak of the thermal request. A 2D numerical model (R, X) was
developed using the Fluent simulation software to study the
thermal behavior of the PCM and the HTF. Grid meshing of the
computational model was rened systematically. The SIMPLE
algorithm was adopted to solve the pressurevelocity coupling.
The second-order upwind scheme was used for the convective
uxes. A naturally expanded graphite matrix with 15% volume
was added to the parafn to enhance the thermal conductivity of
the composite (parafngraphite) to achieve the heat uxes
required for the medium scale application. Different operation
strategies and parameters were studied, including the time-wise
variation of the liquid fraction, PCM temperature, and the HTF
## Shatikan et al. [45] numerically presented the melting process
of a PCM with internal ns exposed to air from the top and heated
from the horizontal base. They investigated different n dimensions with a constant ratio between the n and the PCM thickness. They used Fluent 6.0 to develop 2D and 3D numerical
models and adopted the VOF model to describe the relationship
between the air and the PCM; the SIMPLE algorithm was used for
the pressurevelocity coupling. The 2D model was considered
because the 3D model was time consuming. They reported that
the results for the 2D and 3D models were equal in some
dimensions, and the transient phase change process of the PCM
depended on the operation and design parameters of the system.
Such parameters were related to the temperature difference
between the base and the mean melting temperature, as well as
to the thickness and height of the ns. Shatikian et al. [10]
analyzed the same model with constant heat ux applied from
the horizontal base.
Wang and Yang [11] numerically studied the cooling technology of portable handheld electronic devices using PCM. They
developed half computational grids with the 3D numerical model
using Fluent 12 to investigate the effect of a different amount of
ns and various powers heating level. They adopted the VOF
model to describe the relationship between the air and the PCM.
In the VOF model, the Geo-Reconstruct, which is the volume
fraction discretization scheme, was used to calculate the transient
moving boundary. If the transient temperature of the calculation
cell in the PCM domain was equal to or higher than the melting
temperature, the continuity and momentum equations were
calculated in the system; the SIMPLE algorithm was used for the
pressurevelocity coupling. They studied the orientation of PCM
with the heat sink; the model was validated using the method by
Fok et al. [46]. The numerical results indicated that the orientation had insufcient inuence on the transient thermal performance of the hybrid cooling system, and that the transient surface
temperature caused a discrepancy of around 8.3%10.7%. They
reported that PCM in aluminum heat sink with six ns could
provide stable temperature control and good cooling ability.
Yang and Wang [12] numerically investigated the cooling
technique using a hybrid PCM-based heat sink within a closed
system to absorb constant and uniform volumetric heat generated
from portable handheld electronic devices. They tested different
computation grids and time steps to validate the model with
the experimental work; various power levels (24 W), different
orientations (vertical, horizontal, slanted), as well as charging
and discharging times were conducted in the simulation model.
The melting rate increased when a higher power level was
transferred to the system. They reported that the maximum
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A.A. Al-abidi et al. / Renewable and Sustainable Energy Reviews 20 (2013) 353363
## temperature of the hybrid system using PCM could be well
controlled under 320 K if the melting ratio was under 0.9 or if
the melting time was shorter than 6801.2 s.
Ye et al. [47] gured out the correlation between inuential
variables, such as the liquid fraction and thermal storage time, the
transient heat ux and melting period, and the solid fraction and
the solidication time by numerically studying the heat transfer
and uid ow in a plate n lled with PCM for rapid heat
absorbed/released techniques. They developed a 2D numerical
model through Fluent. The cavity was lled with 85% PCM and
15% air to consider thermal expansion during the solidliquid
phase change; nely structured meshing grids were installed near
the heating and cooling plate wall to resolve the boundary layers.
The VOF model was adopted to describe the relationship between
the air and the PCM. The UDF was written in C language to
account for the temperature dependence of the thermophysical
properties. The SIMPLE algorithm was considered for the pressure
velocity coupling, and different time steps were used for the
energy storage and energy release. Exploring via a 3D numerical
model, they reported no difference in the results between the 2D
and 3D models. The correlations they obtained will be useful for
future component design and system optimization.
Ye et al. [48] performed further studies on the effect of PCM
cavity volume fraction on the heat transfer behavior and uid
ow in LHTES. Different parameters of LHTES performance, such
as the volume expansion ratio, the time of complete thermal
storage, heat ux, liquid fraction, and velocity and temperature
elds, were investigated. They used the same model in Ref. [47],
except that the time step was chosen in relation to the grid size
via the CourantFriedrichesLewy number. They reported that
the volume expansion ratio decreased as cavity volume fraction
x increased. By contrast, the time for complete energy storage
increased.
Kandasamy et al. [49] numerically and experimentally investigated the use of a PCM-based heat sink for quad at package
(QAP) electronic devices. They developed one quarter of the
physical model as well as a 3D numerical model using the
commercial Fluent 6.2. The VOF was used to describe the relationship between the air and the PCM. The PISO scheme was used for
the pressurevelocity coupling. The cooling performance of the
PCM-based heat sinks improved, compared with the case without
PCM. The full melting of the PCM and the melting rate also
increased as the input power increased.
Sabbah et al. [50] presented the thermal and hydraulic
behavior of a micro-channel heat exchanger with MCPCM water
slurry for heat dissipation of a high-power electronic device. They
developed a 3D numerical model using Fluent 6.2, and the SIMPLE
algorithm was used for velocitypressure coupling. The heat
transfer in both the cooling uid and the metal heat sink on the
overall performance of the system was considered. The results
showed a signicant increase in the heat transfer coefcient that
depended on the channel inlet and outlet and on the selected
MCPCM melting temperatures. Lower and more uniform temperatures throughout the electronic device could be achieved
using less pumping power compared with the case using only
water as the cooling uid.
Hosseinizadeh et al. [51] numerically and experimentally
presented the thermal management of electronic products using
PCM. They compared the effect of the PCM cooling technology in
contrast to the normal ones. Different design parameters were
studied to determine their effects on the temperature of the heat
sink and on the melting of PCM inside the heat sink. 85% of the
heat sink height was lled with PCM, and 15% of the height was
modeled in the air region for the solid-to-liquid conversion in the
PCM expansion. A 2D numerical model was adopted via Fluent.
Density was considered constant in the solid state, whereas liquid
## density was a function of the temperature and the thermal
expansion coefcient. The VOF was adopted to describe the
PCMair system. The PISO algorithm was used for the pressure
velocity coupling. They set the time step at 0.0001 s at the early
stage of simulation and changed it to 0.001 s after 3 min of
simulation. The increased n thickness showed only a slight
improvement in thermal performance, whereas an increase in
the number of ns and n height resulted in an appreciable
increase in the overall thermal performance.
## 4.2. Numerical simulation of PCM for HVAC equipment
Tay et al. [52] modeled different tube congurations in a shell
and tube heat exchanger with PCM in the shell side. They
developed a 3D CFD model using ANSYS code to analyze the
transient heat transfer during the phase change process. Three
grid resolutions were evaluated for the tube, HTF, and PCM.
ANSYS CFX 12.1 was used to complicate the physical properties
of the mixture. Three domains were created using CFX.PRE
version 12.1, namely, liquid for HTF, liquid for PCM, and solid
for the tube. An experimental apparatus was developed to
compare the predicted simulation results of both freezing and
melting processes. They found that the results of the CFD melting
model were generally longer than the experimental results. The
thermal behavior of the freezing and melting processes of PCM
was different from those in the experimental results when the
natural convection was ignored.
Antony and Velraj [53] studied the heat transfer and uid ow
through a tube in a regenerative PCM heat exchanger (shell and
tube) module incorporated into a ventilation system. The PCM
was encapsulated in the shell side of the module for a free cooling
system. Fluent software was used to analyze the heat transfer and
ow through the model, which consisted of an air ow passenger
and two air spacers on top and at the bottom. PISO algorithm was
adopted for the pressurevelocity coupling. The second-order
upwind scheme was used for the approximation of the convective
terms. The kx model was used for the turbulence modeling of
the heat transfer uid. This module was validated experimentally.
The results from the experiment and the calculated results were
compared.
Xia and Zhang [54] prepared an acetamide/expanded graphite
composite (AC/EG) with 10% (mass fraction) as PCM to be used in
active solar applications, such as solar-driven solid/liquid desiccant dehumidication systems and solar-driven adsorption/
absorption refrigeration systems that require a higher heat source
temperature of over 60 1C. They measured the thermal properties,
including melting/freezing points, thermal conductivities, and the
heat of fusion, using the transient hot-wire method and a
differential scanning calorimeter. Two-thirds of the test model
was lled with solid PCM for the thermal expansion during the
solidliquid phase change. They developed a 2D numerical model
using Fluent 6.2 to study the release and absorption of heat
in LHTES. The numerical model was validated experimentally.
The thermal conductivity of the composite AC/EG (2.61) was six
times that of pure metal; 0.043 for solid at 60 1C and 0.2 for liquid
at 90 1C. The phase change temperature shifted and the latent
heat decreased compared with that of pure AC. The heat storage
and retrieval durations of the LTES unit with the AC/EG composite
showed 45% and 78% reductions, respectively, compared with
those for pure AC.
New internal ns inside spherical and cylindrical PCM encapsulation in the HVAC system for buildings were numerically
investigated by Siva et al. [55]. They developed a numerical
model using Fluent 6.3 to study the charging and discharging
process of PCMs for the two congurations. Incorporating ns for
A.A. Al-abidi et al. / Renewable and Sustainable Energy Reviews 20 (2013) 353363
## different congurations reduced the total solidication time by
65%72%.
4.3. Numerical simulation of PCM for building applications
LHTES is used extensively in building structures to remove
thermal load and to improve the thermal comfort of occupants.
PCM can be found in different items of the building structure,
such as the walls, roof, oor, ceiling, window shutters, plasterboard, and tiles. Most researchers mentioned in the present study
used CFD to simulate the process, reporting a good agreement
between the simulation and the experimental results.
Silva et al. [56] incorporated a PCM into a typical clay brick
masonry enclosure wall in Portugal as a passive technology to
enhance the buildings energy efciency. They experimentally and
numerically evaluated how the PCM reduced internal temperature uctuations and increased the time delay between internal
and external conditions. They generated the mesh grid in CFD and
developed a 2D numerical model in ANSYS Fluent 12. The SIMPLE
algorithm was used to solve the governing equations. The secondorder upwind scheme was used to solve the pressure, momentum, and energy equations. They reported that incorporating the
PCM into the wall contributed to the attenuation of the indoor
space temperature swing, which reduced from 10 to 5 1C, and to
the thermal amplitude. The time delay also increased by approximately 3 h.
Joulin et al. [6] analyzed the energy conservation of a building
with PCM conditioned in a parallel epipedic polyn envelope to
be used in passive solar walls. They developed a 1D Fortran
numerical model and a 2D model using commercial Fluent to
study the thermal behavior of the PCM to validate the experimental work. The self-developed numerical model was based on
the enthalpy formulation and on the fully implicit nite difference solution method used for discretization. The equations were
solved using TDMA. The liquid fraction was updated in each
iteration until convergence was achieved. For the commercial
Fluent, the SIMPLE method was used to solve the governing
equations. The power law differencing scheme was used to solve
the momentum and energy equations, whereas the PRESTO
scheme was adopted for the pressure correction equation. They
reported that the 1D and 2D numerical simulations did not satisfy
because the results were very far from the experimental ones: the
codes did not represent the supercooling phenomenon. Numerical
modeling showed that the supercooling phenomenon must be
taken into account correctly to predict the PCM thermal behavior.
Susman et al. [57] constructed four 150 mm2 prototype PCM
sails from parafn/low-density-polyethylene composites installed
below the ceiling of an occupied ofce space in London in the
summer. During daytime, the modules were hung on an internal
rig 2.5 m above the ground; after 8 p.m., the modules were
transferred to the outside rig to discharge the energy, and were
moved to the inside rig at 7 a.m. They developed a semi-empirical
model of the modules in Fluent using an enthalpyporosity
formulation to model phase change. The modules were validated
well using the temperature measurements, including a notable
divergence when the maximum liquid fraction was reached.
4.4. Numerical simulation for other applications
Bo et al. [58] introduced a metal PCM energy storage in a
high-temperature heat pipe steam boiler used for direct steam
generation in a solar thermal power generation system. The AlSi
eutectic alloy contained 12.07% Si used as a PCM at a melting
temperature of 577 1C The temperature distributions of the heat
reservoir tank and the heat storage medium under high solar heat
ux were analyzed using a 2D numerical model via Fluent. The
361
## newly developed heat storage boiler could withstand a focused
solar energy ux of 400 kW/m2 and was compatible with the heat
storage medium.
Tan et al. [16] used water as a PCM to recover and store cold
energy from a cryogenic gas in a cryogenic cold energy system.
They developed a 2D numerical model using Fluent to analyze the
effect of the thermal boundary and thermal resistance on the
freezing process in an LHTES system. Their numerical model was
based on half the domain because the geometrical model was
considered symmetric. The standard kx model was adopted to
calculate the internal forced convection of the gaseous coolant.
The CFD model was validated experimentally and showed good
agreement with the experimental results. They reported that the
dimensionless numbers, such as the Biot number and the Stefan
number of the PCM, and the Stanton number of the coolant
owing in the tube had a noticeable inuence on the PCM
freezing characteristics. A larger Biot number was benecial to
the heat transfer between the PCM and the coolant, promoting a
higher freezing rate. A higher Stefan number appeared to result in
larger freezing rates in a xed axial position. Moreover, the frozen
layer slope along the tube was steeper at larger Stanton numbers.
Xiaohong et al. [59] numerically studied a PCM at high melting
solar dynamic power generation system. A 2D numerical model
was adopted via Fluent 6.2 to simulate the heat transfer of the
PCM canister of the heat pipe receiver. UDF was used in the
simulation. The numerical results were compared with NASAs
numerical results. The temperature of the heat pipe wall and
the outer wall of the PCM canister in the numerical model and in
the NASA simulation differed only slightly. These results can be
used to guide the design and optimization of PCM canisters for
Koizumi and Jin [60] proposed a new compact slab container with
an arc outer conguration to promote direct contact melting. They
studied two congurations of the close contact melting process,
namely, at slab container and curved slab container. The 2D
numerical model was performed via Fluent 6.3. The VOF model was
adopted to describe the PCMair system. A rst-order accurate
upwind difference scheme was adopted for the convection terms
because the convection velocity in the liquid PCM was extremely
small (within several mm/s). The simulated results quantitatively
elucidated the experimental melting process from beginning to end.
MacPhee and Dincer [61] numerically investigated and thermodynamically analyzed the melting and solidication process of
a de-ionized water as PCM in a spherical capsule geometry. They
developed a 3D model using Fluent 6.0 and studied the effect of
the HTF inlet temperature and the ow rate. The numerical model
was validated and was found to be in good agreement with the
experimental data. The variations of the incoming HTF temperature had a much larger effect on the charging times and on the
energy and exergy efciencies compared with the changing of the
HTF ow rate.
Reddy [62] presented a solar-integrated collector storage
water heating system with PCM to optimize the solar gain and
heat loss characteristics through a 2D numerical model using
commercial Fluent and UDF to express the variable heat ux
condition on the absorber plate. Different numbers of ns were
attached to the PCM water solar system to enhance the heat
transfer. The latent heat storage with nine ns was optimal at
maximum water temperature and had minimum heat loss to the
surroundings.
Rao et al. [63] discussed the use of PCM with an aging
commercial LiFePO4 power battery to study the thermal behavior
of the PCM and the cell for thermal energy management.
They developed a 3D computation model for a single cell and a
battery package using the commercial computational Fluent. They
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A.A. Al-abidi et al. / Renewable and Sustainable Energy Reviews 20 (2013) 353363
## reported that the design for battery thermal management should
include a proper selection of the thermal conductivities for the
PCM and the cell because the thermal conductivity increased,
whereas the latent heat of fusion decreased.
Darkwa and Su [64] studied the effect of particle distribution on
the thermal performance of micro-encapsulated PCMs in different
congurations of a composite high conductivity laminated MCPCM
board. They developed a 3D numerical model using Fluent 6.3. The
SIMPLE algorithm was used for the pressurevelocity coupling, and
the second-order upwind scheme was adopted for both momentum
and energy equations. The PRESTO pressure discretization scheme
was used for the pressure correction equation. The thermal response
times for the rectangular and triangular geometries were approximately half of that for the pyramidal geometry during the cooling
and heating processes of the board.
MacPhee et al. [65] numerically studied the effect of different
capsule geometries, HTF inlet temperatures, and HTF ow rates
on energy and exergy efciencies for the solidication process in
encapsulated ice thermal energy storage (EITES). They used Fluent
6.3 to simulate different geometries (slab, cylindrical, spherical),
three HFT ow rates, and three HTF inlet temperatures to achieve
the highest efcient charge method for the thermal storage. They
recommended the exergy efciency methods to study system
performance, stating that these methods provide better insight
into system losses. They also advised EITES designers to increase
both the ow rate and inlet HTF temperature to achieve full
system charging in an acceptable amount of time.
5. Conclusion
The sustainable thermal energy storages for different engineering
applications, such as building structures and HVAC equipment, are
indispensable to reducing greenhouse-gas emission. LHTES is an
important component of thermal solar engineering systems, playing
a key role in improving the energy efciency of these systems.
Thermal energy storage systems, especially LHTES, have gained
widespread attention in relation to global environmental problems
and energy-efciency improvement. The following conclusions can be
summarized for CFD applications in latent heat thermal storage.
1. The numerical solution for the PCM phenomena is more
accurate than the analytical solution and can be used in
different engineering conditions.
2. The CFD software is used successively to simulate the
application of PCM in different engineering applications,
such as electronic cooling technology, building thermal
storages, and HVAC.
3. Different CFD software are used in PCM engineering, including
ANSYS Fluent, Comsol Multiphysics, and Star-CCM. The
most commonly used software is ANSYS Fluent.
4. The numerical results of the 2D numerical model are generally
the same as those of the 3D numerical model. Researchers
reported that the time required for simulation was reduced.
5. The application of CFD in designing PCM thermal storages
is a feasible method because of the highly accurate results
of the experimental work. CFD also delivers optimization
tools to help users achieve maximum efciency while
saving time and money.
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applications. International Journal of Heat and Mass Transfer 2009;52:4195202. | {"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": 0.8739382028579712, "perplexity": 2897.5157219678285}, "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-2020-45/segments/1603107890273.42/warc/CC-MAIN-20201026031408-20201026061408-00417.warc.gz"} |
https://www.physicsforums.com/threads/particle-motion.590018/ | # Homework Help: Particle Motion
1. Mar 24, 2012
### Cbray
1. The problem statement, all variables and given/known data
A particle moves in a straight line with acceleration which is inversely proportional to t3 , where t is the time. The particle has a velocity of 3ms-1 when t=1 and its velocity approaches a limiting value of 5ms-1 . Find an expression for its velocity at time t.
2. Relevant equations
a=-kt^3
3. The attempt at a solution
*****ROOKIE MISTAKE SHOULD BE -2/t^2 + d (last equation)***
Last edited: Mar 25, 2012
2. Mar 24, 2012
### agent_509
I won't answer the question directly for you, but this is just a basic algebra problem, remember that when y is proportional to x, you use the formula y=kx where k is some constant, inversly proportional to x and you have y = k/x. Once you know how y is proportional to x, and you have test values for y and x, you can plug them in and solve for k.
3. Mar 25, 2012
### Cbray
I did some more and got stuck again, do you mind looking at it?
4. Mar 25, 2012
### agent_509
I'm not entirely sure what you did there here's what would be a good idea:
∫ dv/dt dt = k∫1/t^2 dt
v= -k/2t^2 + d
now you know that when t=1 v=3, and you also know that the limiting velocity (when t→∞) is 5, so when t→∞, d=v, and v= 5, so d = 5, so now you have the formula
v=-k/2t^2 + 5, and you know that when t=1, v=3.
so just plug it in and solve for k
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https://wir-sind-dann-mal-wech.de/db-to-ratio-calculator.html | db to ratio calculator. To calculate Signal to noise ratio, you need Number of bits (n). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. This applies to the field quantity voltage or sound pressure. Further, dBW is the abbreviation for dB referenced to one watt. Motorcycle speed and sprocket calculator with bike database for gearing, sprockets, tires and chains of over 1600 bikes. x \space dB x dB is in linear scale. 293 where resistors and ratio tolerances are equal. The unit expresses a relative change or an absolute. For converting the ratio of two voltage or current values in dB, we use ans(dB)=20*log10(ratio) and ratio=10(ans(dB)/20). This difference can be defined as sound intensity is the rate of sound moving through a region while sound intensity level is a ratio. golden ratio room calculator. In other words, SNR is the ratio of signal power to the noise power, and its unit of expression is typically decibels (dB) is calculated using Signal to noise ratio = (6. decibel (dB) - A unit of relative amplitude defined on a logarithmic scale. So with a terrible sample size, about 98% of people have a ratio between and 0. If there are two or more amplifiers connected in cascade, you can compute their overall gain. This dB calculator will help you find the sound pressure level (SPL) and intensity level in decibels. To use it enter only ONE dimension in the "Desired" column and. For example, if a linear amplifier produces 2 volts (V) . This value is usually presented in the percentage of the requests or hits to the applicable cache. What if you are given a ratio in dB and asked to calculate the power or voltage ratio? Here are the formulas:. Basically it shows how heavy the boat is in comparison to the waterline length. It is often convenient to compare two quantities in an audio system using a proportionality ratio. You can use it for dB conversion or for . The procedure to use the signal to noise ratio calculator is as follows: Step 1: Enter the inputs separated by a comma in the input field. Equations: dB= 20log (V1/V2)= 10log (P1/P2) The dBm is a logarithmic measure of power compared to 1mW. The voltage ratio in dB is defined as. This calculator can also be used to express differences between two distances or pressures. The heavy math behind decibels. If we increase r by a ratio of 10, we decrease the level by 20 dB. Just fill in one field and the calculator will convert the other two fields. A decibel-milliwatt is a unit of level used to indicate that a power ratio is expressed in decibels (dB) with reference to one milliwatt (mW). dBm is defined as power ratio in decibel (dB) referenced to one milliwatt (mW). Find the logarithm of the power ratio. With the Scaling menu you can choose between power calculation (10db / decade), or switch over. dBW value in dBm is always 30 more because 1 watt is 1000 milliwatts, and ratio of 1000 (in power) is 30 dB, eg 10 . The decibel is the conventional relative power ratio, rather than the bel, for expressing relative powers because the decibel is smaller and therefore more convenient than the bel. gain-multiplier → dB: d = 20 * log10(g)for example: +6dB = 20 * log10(2)dB → gain-multiplier: g = 10^(d / 20)for example: 2 = 10^(6dB / 20). Equation 1: Tone-to-Noise Ratio (TTNR) is dB difference between Tone level (T) In the calculation of Prominence Ratio, the critical band . Decibels are all about power ratios. 10, To convert Milliwatts To dB, Enter the . A unit of measurement used to express the ratio of one value of a power or field quantity to another . In the case where the wave is linearly polarized, r_T tends to infinity and the above formula cannot be used. Also, a ratio greater than 0 dB or higher than 1:1, signifies more signal than noise. The deciBel formula or equation for power is given below: NdB=10log10(P2P1). single mode link at 1310nm with 2 connector pairs and 5 splices. Decibels convert multiplication and division calculations into . Decibels are defined in terms of Power ratios. In decibels, this would be just 60 dB. Enter the values in one or two of the text boxes and press the corresponding Convert button: Quantity type: Power Voltage Current Frequency Sound pressure. We must realize, that there are relative and absolute decibel levels. Receiver C/N ratio in dBHz = 85. Thus the formula becomes Decibels (dB) = 10 log(P 2/P 1) [2] The power ratio need not be greater than unity as shown in the previous examples. Step #2: Click the calculate button ("Simplify Ratio," "Solve Ratio," or "Check Equality"), which will perform the desired calculation and generate a. Calculate the ratio of background noise (unwanted signal) on a channel using signal to noise ratio (SNR) calculator online. The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). If a polarization is circular, enter 0 for the dB axial ratio and 0 for the tilt angle. A decibel is a ratio of the observed amplitude, or intensity level to a . 26) or root-power ratio of 10 1 ⁄ 20 (approximately 1. For example, if you enter the value -6 for the scaling "Power (10db / decade)", the result is 0. For more information or to do calculations involving each of them, please visit the 401 (k) Calculator, IRA Calculator, or Roth IRA Calculator. This formula is used to calculate the VSWR from the given return loss. The way it is calculated is through a division of the boat's displacement measured in long. It is usually referred to sailing boats and it is also seen with the symbol D/L. Sojamod, a 20 dB RF Amplifier up to more than 1. Calculates the input and output power and voltage ratios from the decibel (gain). See also root mean square (RMS) and total harmonic distortion. Decibels convert multiplication and division calculations into simple addition and subtraction operations. 3 dB mean power (energy) related to a ratio with the factor 2 (doubling) . Signal to noise Ratio (SNR ) Calculator. In your question you say you want to calculate the voltage (or current) db ratio. System Bandwidth (MHz) : Signal : Noise : Signal To Noise Ratio: The SNR is typically measured in units of decibels (dB)s. Where: Ndb is the ratio of the two power expressed in deciBels, dB. dB = 20log10(V1/V2) = 10log10(P1/P2) You can also use this calculator to find out the Power Gain and Voltage Gain from the Decibel value. com A collection of really good online calculators for use in every day domestic and commercial use!. integration on a home computer or programmable calculator. In electronics, the decibel is most often used to express the ratio between two powers, two voltages, or two currents. We are concerned about the ratio of P1/P2 or vice versa. The negative value means that the common mode voltage was reduced by 142 dB. Besides conversion of Voltage and Power ratio to dB and vice-versa, dBm, V, A, W, dBµA, . signal-to-noise ratio (SNR) - The RMS value . If a polarization is linear, enter 60 for the dB axial ratio and R for the rotation sense. FORMING THE ACOUSTIC EQUATIONS. For example, you can calculate the difference in . 120 or 130 dB is the threshold of pain - for example, a jet aircraft taking off in your. If the impedances of the two circuits are same, then the Power ratio can be expressed as, Power ratio in dB = 20 log 10 [v1/ v1] Voltage Ratio in dB. When dB is referenced to a carrier signal, it is termed dBc; likewise, dB referenced to 1mW is termed dBm. −6 dB is equivalent to 50% (factor = 1/2 = 0. When SPL is given in decibels, we can estimate the pressure of everyday sounds usually in the 20-100 dB range. Calculate! Where do these calculations come from? Well, start with the formula for VSWR: If we invert this formula, we can calculate the reflection coefficient (, or the return loss. Supposing the ratio of IR/I0 = 1/1000, i. Enter the values 50 (L2), 100 (d2) and 1. 0 will result in a much smaller tolerance. Calculate the sound attenuation over distance based on a starting sound the sound pressure level decreases by approximately 6 dB. 2, dB - power and power ratio converter. Decibels express the ratio between identical quantities. The MACRS Depreciation Calculator allows you to calculate depreciation schedule for depreciable property using Modified Accelerated Cost Recovery System (MACRS) GoodCalculators. dBm is referenced to one milliwatt (0. Note well that the voltage-ratio equations are valid only if the two voltages appear across equal impedances. The defining equation for decibels is. 5 (d1) into the calculator and it will calculate L1=86. Decibels are defined as ten times the log of a power ratio. Power ratio in dB = 10 log 10 [ (v12/ z1)/ (v12/ z2)] The z1 and z2 are the magnitudes of the impedances of the two circuits. If you want to compare two ratios, enter values for A, B, C, and D. For example, calculate the amplitude scaling factor if you want to adjust a sound waveform by -3dB or +6dB, or find out what the decibel difference would be if you scaled the waveform by 0. Enter Attenuation and Zo to solve for R1 and R2. Golden ratio also known as the divine proportion, golden mean, or golden section is a term for dividing a segment so the long/short ratio is equal to all/long ratio. The dBCalculator software comprises five independent calculation tools: dBm Calculator: This tools helps to add and subtract an arbitrary number of power . Return Loss to VSWR Calculator. dB and ratios dBW and Watts W dBm and milliwatts mW. Convert dB, dBm, dBW, dBV, dBmV, dBμV, dBu, dBμA, dBHz, dBSPL, dBA to watts, volts, ampers, hertz, sound pressure. INSTRUCTIONS: Enter the following. Enter a value for VSWR (remember: VSWR should be a number larger than 1. Enter any value and then press "Calculate" to find out the equivalent values in volts, dBV, or dBu. (notice the 3 in 30 dB corresponds to the number of zeros in the power ratio) Conversely, a ratio of less than 1. The formula for converting a linear . Relative level – a ratio of two voltage, current, power, sound intensity values etc. Two signals whose levels differ by one decibel have a power ratio of 10 1/10 (approximately 1. RF Power Ratio to dB Converter. This page of converters and calculators section covers receiver C/N ratio calculator. Definition: dBW means dB relative to 1 watt, so 0 dBW = 1 watt, -3 dBW = half watt. Formula used by the above calculator i2/i1= (d1/d2) 2 ΔL=10*Log (i2/i1) where ΔL is the difference in sound levels (L1-L2) in dB (or dBA). Choose the number of amounts (in dB) that you want to add (2 as a minimum) with the + button. The quotient i2/i1 is the sound intensity ratio. Decibel is a dimensionless value of relative ratios. Air Change Rate Calculator Air Mixing Calculator. The values of L1 Ln can be modified. log (100) = log (102) = 2 Multiply this result by 10 to find the number of decibels. Step 3: Finally, the signal to noise ratio will be displayed in the output field. The Bel is a logarithmic measure of a ratio. Another example is the ratio 0. **Note: All of our calculators allow SI prefix input. The sound intensity level and the sound pressure levels in dB have the same. The Decibel Calculator shows the addition and subtraction of dB values in the usual acoustic range of 0 to 200 dB. k = 10^ {\frac {x} {10}} k = 1010x. decibel calculation dB calculator voltage power ratio sound pressure. EXAMPLE: G/T ratio = 10, satellite EIRP=50, Propagation loss =200, margins = 3. ** When you divide watts by the reference watts, for example, the result is a simple ratio (no units). How to Calculate the Displacement to Length Ratio of a Boat. 4, Use this spreadsheet to convert dB and power. For example, instead of saying that the reflection coefficient in the line is 0. Criteria Labs - Criteria Labs Partners with CAES to Support US Navy's Advanced Offboard Electronic Warfare Program - May 04, 2022; Movandi Corporation - Ubicquia and Movandi Partner to Develop mmWave Streetlight Repeaters for 5G Deployment - May 04, 2022; Arbe Introduces New RF Chipset with Ultra-High Resolution Radar Image Quality and Increased Range - May 04, 2022. The tipping point is around 1-sqrt(0. 0 is a loss, a negative gain, and will be expressed as a negative dB value. Here's a simple VSWR calculator. What is the ratio of the full sound intensity to the reduced sound intensity the worker hears? Answer: The decibel scale is logarithmic, and so a small drop in . You can use this voltage divider calculator to determine any one of the four variables associated with a simple two-resistor voltage divider when the values of the other three variables are available. Pasternacks's Power Ratio Conversion Calculator converts from a power output-input ratio to a dB ratio measurement. A decibel is one-tenth of a Bel - named after Alexander Graham Bell, the inventor of the telephone. The calculator also plots the circuit diagram and generates the component values. We have used the decibel to express power ratios (watts), but it can also be used to express voltage ratios (volts). V o l t a g e: V o V i = 10 G 10 P o. Convert RF microvolts or millivolts to dBuV, dBmV, or dBV. VSWR Converter This tool converts reflection quantities such as Voltage Standing Wave Ratio VSWR, reflection coefficient, reflected power, return loss and mismatch loss. The formula for converting logarithmic (db) to linear is: $$\displaystyle a=10^{\left(\displaystyle \frac{x[db]}{10}\right)}$$ a is the factor (P1 / P2) here Values to remember 0 db ≡ factor 1 3 db ≡ factor 2 6 db ≡ factor 4 10 db ≡ factor 10 Convert voltage ratio to db. The power ratio is proportional to the square of the voltages. Decibels (dB) Calculator Decibels are defined as ten times the log of a power ratio. The formula for converting logarithmic (db) to the linear voltage ratio is: a = 10⎛ ⎝x[db] 20 ⎞ ⎠ a = 10 ( x [ d b] 20) Values to remember 0 db ≡ factor 1 6 db ≡ factor2 12 db ≡ factor 4 20 db ≡ factor 10 Other basic functions Ohm's law and power Wire resistance Voltage drop Temperature drift of resistance Table of temperature coefficients. The SNR, measured in dB, is twenty times the ratio of the base-10 logarithm of the amplitude of the received signal to the amplitude of the noise. It can be converted to a voltage, if the load is known. , today very rarely is the term "DC plan" used to refer to pension plans. Signal to noise ratio Calculator. Set the quantity type and decibel unit. Note! - the decibel value of a signal increases with 3 dB if the signal is doubled (L = 10 log (2) = 3). MCAT Content / Sound / Intensity Of Sound Decibel Units Log Scale. Home Go to Gearing Commander main page. - New Sensible Heat Ratio Line Plotting - New Humidification Delta-Enthalpy / Delta-Humidity Ratio Line Plotting - New Partial Mixing of Airstreams Allows for Component Mixing Bypass - New Cooling Coil Leaving Air Calculator / Auto-Plotting - New Organized Toolbar Menu Setup NEW TOOLS - New Air Collection Calculator with Auto-Plotting. Aside from the technical definition of SNR, the way I define it in other terms is by using a comparative. Fill in the boxes L1, L2, Ln with the amounts (in dB). A change in voltage, from V 1 to V 2 can be expressed as a ratio in decibels with the equation RV = 20*log (V 2 / V 1 ). Golden Ratio Formulas: For this calculator we use phi = ( 1 + sqrt (5)) / 2, which is rounded to 1. Calculate power & voltage ratio from a +dB or -dB value: Gain or Loss (±) : dB. of Connectors Power Ratio, Result. Convert dBuV, dBmV, or dBV to microvolts or millivolts. Ratio to Decibel Calcultor. Enter any two values and press "Calculate" for the remaining value. VSWR = Voltage standing wave ratio. Signal To Noise Ratio Calculator. It is an abbreviation for dB with respect to 1 mW and the "m" in dBm stands . Converting Decibels to Power Ratios. The signal units depends on the nature of the signal - can be W for power. dB to Power Ratio Power Ratio to dB dB to Voltage Ratio Voltage Ratio to dB. For power, it is 10log(P A /P B). This formula works as long as you have spherical wavefronts. The Decibel (dB) calculator calculates the decibel gain (or loss) value from either a power ratio or a voltage/current ratio, according to the above formulas. Convert dB gain or loss to a Power Ratio and Voltage Ratio. Now let us dive into the wonderful world of pascals, decibels, formula. To convert dB to ratio, divide by 10 and then do ten to the x, like 10 x Example: dB = +12. Using the logarithm allows very large or very small ratios to be represented with a small number for convenience. SiliSoftware » Tools » deciBel-Amplitude converter. A = 10*log10 (P2/P1) (dB) where P1 is the power being measured, and P1 is the reference to which P2 is being compared. Thus the formula becomes: (2) \begin{equation*} Decibels (dB) = 10\ . For example, if you wish to input"25000000", just type "25M" instead. 5 GHz; Opamp Circuits • Knowledge. How To Calculate Common Mode Rejection Ratio. To improve this 'Ratio of the amplitude from the decibel Calculator', please fill in questionnaire. This calculator converts between decibels, voltage gain (or current), and power gain. Resistive divider calculator. SiliSoftware » Tools » deciBel-Amplitude converter Useful for converting decibals into amplitude scale factors. Take your calculator to verify it if you wish. Remember these three important values. 8, an engineer would rather say that the signal suffered a return loss of 1. The amplitude ratio in decibels (dB) is 20 times base 10 logarithm of the ratio of V 1 and V 0: RatiodB = 10⋅log 10 ( V12 / V02) = 20⋅log 10 ( V1 / V0). You can choose different calculations from the drop down menu. So, if we double the distance, we reduce the sound pressure by a ratio of 2 and the sound intensity by a ratio of 4. , the information being transmitted) compared to the noise. The term 'forward power is synonymous with powered deliver to the load, or thru power. The formula to memorize and take to your . Enter dry bulb temperature, plus either wet bulb temp, %RH or dewpoint temp. 0 or the results don't mean anything!). Appendix C: Learning Decibels and Their Applications. This calculator is the result of a few years' worth of fooling around, off and on, with performing calculations using Factorio's recipe graph. RF Power Ratio Conversion Calculator Pasternacks's Power Ratio Conversion Calculator converts from a power output-input ratio to a dB ratio measurement. This calculator converts decibels to percentages and vice versa. Voltage Divider Calculator. The calculation of decibels uses a. Simply put, a decibel is a way to express a ratio. How to calculate decibels – dBu dBv dBm dB SPL. The decibel (dB) is 10 times the decimal logarithm of the ratio between two values of a variable. 0, and its source may be found on github, here. RatiodB = 10⋅log 10 ( P1 / P0) Amplitude ratio The ratio of quantities like voltage, current and sound pressure level are calculated as ratio of squares. Free Online Engineering Calculator to quickly convert a Voltage Standing Wave Ratio into Mismatch / Return Loss. In equations [1] and [2], P 1 is usu ally the reference power. This calculation is simply the sum of all worst-case loss variables in the link: Link Loss = [fiber length (km) × fiber attenuation per km] +. You might now remember dB is a reference number, and it shows you the ratio between two forces. Commercial Kitchen Exhaust Hood (ASHRAE) Condensate Rate Calculator. This function converts a decibel value into the linear ratio between two voltages or powers. The decibel is a relative quantity and it is expressed as ratio of either two . In other words, SNR is the ratio of signal power to the noise power, and its unit of expression is typically decibels (dB). For example, the ratio 100000000/100 would be presented as 1000000. Blood urea nitrogen and creatinine are two metabolites constantly produced by the body but whilst the BUN is filtered in the nephrons, in the kidney, then reabsorbed in the blood. To convert from decibel measure back to power ratio: P2/P1 = 10^ (A/10) Voltage is more easily measured than power, making it generally more convenient to use: A = 20*log10 (V2/V1) (Z2. Signal to noise ratio(SNR) is a measurement parameter that compares the level of the desired signal to the level of background noise. Bridged Tee Attenuator Calculator. Definition: dBm means dB relative to 1 milliwatt, so 0 dBm = 1 milliwatt (one thousandth of 1 watt or 0. The formula for calculating a signal-to-noise ratio in dB is: SNR = 20 x log ( . If you want to solve for a missing value, enter any 3 of the 4 terms. Knowing the intensity of a sound wave allows one to calculate the deciBel (dB) . They are more likely to be referred to by their programs, such as "401 (k)," the "457 plan," or IRA, etc. C/N in dB (input1) : Bit rate in Mbps (input2) : Bandwidth in MHz (input3) : Eb/N0 ratio (Output): Eb/No calculator example: INPUTS: C/N = 14. Decibel (dB) Calculator Decibels are defined as ten times the log of a power ratio or twenty times the log of Voltage Ratio. calculate the ratio of 20 watts to 10 watts in decibels:. 0 dB is a ratio of 1, 3 dB is a ratio of approximately 2, and 10 dB is a ratio of 10. at a known distance from the center of the pump, then using the formula: dB . For example, calculate the amplitude . The level of the output voltage level is 0 dB, that is 100% (factor or ratio = 1). Request yours to be added as well. We will first consider a ratio of powers. The level of −3 dB is equivalent to 50% (factor = 0. It is licensed under the Apache License 2. The level of −3 dB is equivalent to 70. The decibel (dB) scale is logarithmic, meaning that an increase of roughly 3 dB is equivalent to doubling the pressure if expressed in Pascals. In other words, we reduce the sound level by 6 dB. Figure 4 – A decibel is ten times the log of a power ratio. dB Voltage Ratio Calculator Formula used: dB = 20 • log 10 ( Voltage 1 / Voltage 2 ) 1 Bel = 10 decibels (dB) Decibel The decibel (dB) is a unitless measurement for expressing ratios. At the LT box the addition result will appear (in dB). This calculation will give you the ratio, in decibels, between two power values. VSWR / Return Loss Calculator. The displacement to length ratio of a boat is a good comparing parameter for most boats. The ratio of a sinusoidal signal reflected back from the load to the formward signal absorbed by the load in dB is called return loss. Algebraically we express the decibel by this formula: dB value = 10 log(P1/P2) or 10 log(P2/P1). dBm possesses a fixed value, POi is consistently 1. For example, you can calculate the difference in dB between two amplifiers with different power output specifications. If you want to simplify a ratio, enter values for A and B. Step 2: Now click the button "Solve" to get the ratio value. This calculator is copyright 2015-2020 Kirk McDonald. Before you apply the demonstration and . Using these formulas we can calculate three important identities that you need to remember, only three: important identities for conversion of dB to linear units. A 10:1 power ratio, 1 bel, is 10 dB; a 100:1 ratio, 2 bels, is 20 dB. Signal-to-noise ratio is also frequently stated in decibels (dB). Well, dBm represents the opposite. This Eb/No calculator takes C/N, Bit rate and Bandwidth and calculates Eb/No. To sum up, decibels do not represent an exact number, it only tells you the ratio between two forces. Decibels are handy in presenting very small or very large ratios. 25 , So a performance ratio of 1/4. If you make a mistake entering the values, you can enter them again. For example: 30 dBm = 0 dBW, which is another way of saying that 1000 milliwatts equals one Watt. It expresses the ratio of two values of a power or root-power . The best way to calculate a cache hit ratio is to divide the total number of cache hits by the sum of the total number of cache hits, and the number of cache misses. The level of the output power level is 0 dB. This CalcTown calculator converts ratio of power and intensity into decibel (dB) units. A doubling in the voltage level translates to a power ratio of 6 dB. Conversion Ratio to dB · dB calculation problem: What is the output power level in the following situation and will the transmitter output . A voltage ratio of 1:10 therefore corresponds to 20 db. 363 (So if you use 50s for DB this would convert to ~138 for BB) Standard Deviation: 0. It is typically expressed in decibels. Calculate the power gain of the amplifier in decibels. To use the calculator below, input a value into the field and hit the enter key. Sound Pressure Level (SPL) = 20·log (p/po) dB, where p is . On the other hand, ratios close to 1. An explanation of decibel addition and subtraction is included. [splice loss × # of splices] [connector loss × # of connectors] + [safety margin] For example: Assume a 40 km. Decibel (dB) definition; Decibels to watts, volts, . An adjustable 20 dB range attenuator, initially set half way, by you, at the 10 dB setting: 10 dB attenuation (10 times reduction) Long coax cable: Loss 3 dB attenuation (reduction to half the level) High power 10 watt amplifier: Gain=30 dB (increase in signal level by a factor of 1000 times. To calculate the intensity level in decibels, find the ratio of the intensity of sound to . Useful for converting decibals into amplitude scale factors. Decibel (dB) definition, how to convert, calculator and dB to ratio table. You can verify the concept of . 7071), and the level of −6 dB is equivalent to 50% (factor = 1/2 = 0. A signal-to-noise ratio over 0 dB indicates that the signal level is greater than the . The level of the output power level is 0 dB, that is 100% (factor or ratio = 1). Bookmark or "Favorite" this page by pressing CTRL + D. In other words: small ratios are bad for tolerance, large ratios are good. The Signal-to-Noise Ratio (SNR) calculator computes a relative measure of the strength of the received signal (i. Decibels are defined as ten times the log of a power ratio or twenty times the log of Voltage Ratio. Engineers are used to seeing power losses in dB, thus having parameters presented in dB is more convenient than ratios (like what the VSWR and reflection coefficient are). Historically, decibels were defined strictly for use with power ratios. For voltage values, dB is given by 20log(V A /V B). dB Converter This tool converts linear power and voltage ratios to dB and vice versa. Equation used to calculate the data: dB = 10 * Log (Pout / Pin). One tenth of the common logarithm of the ratio of relative powers, equal to 0. Just fill in one field and the calculator will convert the other two. The macOS version can be downloaded for free from the Mac App Store. Similarly, halving the power equals a -6 dB voltage ratio. Application note tutorial: Definition & use of Decibel, dBm, dB units in optical communications. The BUN Creatinine Ratio is one of the kidney function indicators most often determined, especially if needed to determine the exact cause of renal malfunction. | {"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": 0.9352903366088867, "perplexity": 1772.0609212930472}, "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-2022-21/segments/1652662522741.25/warc/CC-MAIN-20220519010618-20220519040618-00606.warc.gz"} |
https://www.physicsforums.com/threads/how-to-calculate-moles.7596/ | # How to calculate moles
1. Oct 21, 2003
### repugno
Hello everyone. I have a slight problem with my chemistry and was wondering if there is someone that could help me. Chemistry is not my strong point so please go easy. Thank you.
---------------------------------------------------------------------
The problem is that I have gotten mixed up with how to calculate moles. Now, I know that I should use the formula:
n= Mass (g)/ Mr
The problem is that this doesn’t seem to apply everywhere and I don’t know why. If I have 1.5g of Na2CO3 in 250cm^3 of solution. Then I know that,
n= 1.5g/ 106 = 0.014150943mol
So, there is 0.014mol of Na2CO3 in 250cm^3 of solution.
---------------------------------------------------------------------
Now, I did a question in my textbook where this method of calculating moles did not work.
“A solution of Na2CO3 contains 12.5g of anhydrous salt in 1000cm^3 of solution. When 25cm^3 of this solution was titrated with a solution of HCl using methyl orange indicator, 23.45cm^3 of the acid was required.”
Naturally I would then use the formula to work out the moles…
n= 12.5g/ 106 = 0.117924528mol
However, the book states that this gives the concentration, and to calculate the moles I will have to multiply it by the volume (i.e. 25cm^3).
What am I not understanding? Any help would be greatly appreciated, Thank you.:)
Last edited by a moderator: Mar 7, 2013
2. Oct 22, 2003
### Chemicalsuperfreak
What is the question asking for? You calculated the number of moles correctly. But if the question is asking for the Molarity of the HCl solution than the volume of the HCl solution is essential.
3. Oct 22, 2003
### repugno
The question is asking me to calculate the molarity of HCl. If I calculated to moles right, I should be able to then calculate the molarity.
Na2CO3 + 2HCl ==> 2NaCl + H2O + CO3
Ratio (1:2)
So, 0.117924528mol * 2 = 0.235849056mol of HCl reacting with Na2CO3
Knowing
c= n/ V
then,
c= 0.235849056mol/ 23.45cm^3
c= 0.0101mol cm^-3
This doesn’t seem to be the concentraton, the answer in the book says it is 2.516mol cm^-3
I’m completely mystified ....
4. Oct 22, 2003
### lavalamp
I kind of have a standard way of setting these things out. It helps me remember what comes next.
"A solution of Na2CO3 contains 12.5g of anhydrous salt in 1000cm^3 of solution. When 25cm^3 of this solution was titrated with a solution of HCl using methyl orange indicator, 23.45cm^3 of the acid was required."
Na2CO3 + 2HCl ==> 2NaCl + H2O + CO3
Code (Text):
Number of moles of Na2CO3 = mass/Mr
= 12.5/106
= 0.118 mol
Molarity of Na2CO3 = moles/volume
= 0.118/1
= 0.118 mol dm^-3
Moles of Na2CO3 in 25 cm^3 = concentration*volume
= 0.118 * 25 * 10^-3
= 0.00295 mol
Number of moles of HCl = 0.00295 * 2
= 0.00590 mol
Molarity of HCl = moles/volume
= 0.00590/(23.45*10^-3)
= 0.251 mol dm^-3
5. Oct 23, 2003
### repugno
Thank you very much, that certainly made it much clearer.
6. Oct 23, 2003
### lavalamp
Happy to help.
7. Oct 23, 2003
### Chemicalsuperfreak
So methyl orange will indicate in the neutral solution but not the sodium bicarbonate solution?
Last edited: Oct 23, 2003
8. Oct 24, 2003
### lavalamp
The solution starts off neutral, and the indicator will colour the solution yellow. As the acid is added, the pH lowers and when it reaches pH=4.4 the methyl orange will start to colour the solution orange, as the pH lowers the colour becomes more and more orange until it gets to pH=3.1 when the soloution will become red.
This means that you should stop the titration at the first hint of orange in the solution. The chances are that it will go straight from yellow to red on one drop, since the acid is a strong acid.
So to answer you question, it depends what you mean by "indicate".
9. Oct 24, 2003
### Chemicalsuperfreak
A solution of sodium carbonate is neutral? I didn't know that.
10. Oct 24, 2003
### lavalamp
There are two ions that affect the pH, they are H+ ions (cause acidity) and OH- ions (cause alkalinity). Since sodium carbonate contains neither of these, the pH of the solution depends on the H+ ions provided by the water. Since water is neutral, the solution will be neutral.
Even though the solution is neutral, the pH will be slightly higher than 7 (at s.t.p.). This is because the Na2CO3 (s) that is added will actually increase the volume of the solution just very slightly, which causes a drop in the concentration of H+ ions, and so, although the pH will be above 7, it will probably be in the be of the order 7.0001.
So for all practicality, the solution will have a pH of 7 and overall it will be neutral, unless H+ or OH- ions are added (or removed).
11. Oct 24, 2003
### Chemicalsuperfreak
So then in order to make a solution basic you need to add something with a hydroxide, like NaOH of K(OH)2? I didn't know that.
12. Oct 24, 2003
### lavalamp
To make solution less acidic or more alkaline then, yes, you could add something like those two alkalis.
There are also things called buffer solutions, these resist changes to their pH, so although when you add acid or alkali to them their pH does change, it only changes a little. These may be a little complicated to explain, but if you want I'll try.
To put it simply, they contain both the un-dissociated acid/alkali molecules and the acidic/alkaline salt ions in roughly equal proportions. They can only occur as a result of weak acids and bases because for strong acids and bases, all of the molecules dissociate.
13. Oct 24, 2003
### Chemicalsuperfreak
Let's not worry about buffers for now. What I want to know is, if you add something with hydroxide to water (say Ca(OH)2) it will become basic, but if it does not have hydroxide ions, such as KCO3 or butyl lithium, then it will remain neutral?
14. Oct 24, 2003
### lavalamp
Yep, you've got it twigged.
15. Oct 24, 2003
### Chemicalsuperfreak
That's interesting. Because I thought butyl lithum was highly basic and would rip the proton off of water if it looked at it funny.
16. Oct 24, 2003
### lavalamp
Lol, very funny. OK, you laid a very good trap and I walked straight into it. Since you didn't want to know about buffer solutions, I assume that you also didn't want to know about Lowry-Bronsted acids and bases.
The definition of an LB acid is a proton donar, or a lone pair acceptor such as an H+ ion.
The definition of an LB alkali is a proton acceptor, or a lone pair donor such as an OH- ion.
However not all LB acids and alkalis affect the pH of a solution as some do not contain any H+ or OH- ions.
It seems that you've had a slightly higher chemistry education than you let on.
17. Oct 24, 2003
### Chemicalsuperfreak
So that butyllithium or Na2CO3 will not effect the pH of water?
18. Oct 24, 2003
### lavalamp
Not that I know of, I just know that it isn't just H+ and OH- ions that cause acidity and alkalinity.
I don't think that Na2CO3 will affect the pH of the solution. I suppose that it could cause the water molecules to dissociate and that would affect the pH, (even though the solution would be neutral overall while at the same time being an acid or an alkali). But we haven't done how LB acids and bases affect pH.
19. Oct 24, 2003
### Chemicalsuperfreak
HCl is a Bronsted-Lowry acid. NaOH is a BL base. Butyllithium is a BL, it accepts a proton from H2O. Now, if you accept a proton from H2O, what have you got?
20. Oct 24, 2003
### lavalamp
Well water minus a proton leaves OH- ions. So I suppose the pH would change:
pH = 14 + log10[OH-]
Although I must say, I'm not familiar wuth butyl lithum, could you provide a structural formula for it please.
I'd never really connected those two dots about water losing a proton and the solution becomming alkaline before just now.
21. Oct 24, 2003
### Chemicalsuperfreak
n-Butyl lithium: CH3CH2CH2CH2- Li+
Reacts with water to form butane (C4H10) and LiOH.
So what is the structure of sodium carbonate? What happens when you stick it in water?
22. Oct 24, 2003
### lavalamp
Over here we call that Lithium-1-Butane (or 1-Lithium-Butane, I can't quite remember).
I suppose that the CO3 ion could act as a proton acceptor but I don't know. The conversation has moved into an area that I don't have much experience in.
23. Oct 24, 2003
### Chemicalsuperfreak
Well that would be the IUPAC name, but I'm pretty sure most brits still use n-butyllithium. Or en-buely for short.
OK, so carbonate ion excepts a proton. What do you get.
24. Oct 24, 2003
### lavalamp
Ok, I see where you're going with this, I was wrong. I offered duff advice, sorry. What more do you want me to say? | {"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": 0.8514308929443359, "perplexity": 3302.0125097668038}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591718.31/warc/CC-MAIN-20180720154756-20180720174756-00210.warc.gz"} |
https://tex.stackexchange.com/questions/409139/bar-is-barely-visible | # Bar is barely visible
I have two objects I need to clearly identify in my equations, tilde and bar as operators on variables. However, these are barely legible and not really differentiable, in particular if they appear in the denominator of a fraction.
I'd like to keep referring to the variables as \tilde and \bar in my text, and not introduce a new command. If this implies "too much" space for the bars and tildes outside of fractions, I'm happy to take that cost.
Is there something I can do in the preamble to make these more legible?
% !TEX encoding = UTF-8 Unicode
% !TEX TS-program = pdflatexmk
\documentclass[11pt]{article}
\usepackage{geometry} % See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper} % ... or a4paper or a5paper or ...
\usepackage{dcolumn}
\usepackage{booktabs}
%\usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}
\newcommand{\mbeq}{\overset{!}{=}}
\usepackage{epstopdf}
\usepackage[utf8]{inputenc}
\usepackage{marginnote}
\renewcommand*{\marginfont}{\color{red}\sffamily}
\usepackage[backgroundcolor=green, linecolor=green]{todonotes}
\usepackage{placeins}
\usepackage{filecontents}
\usepackage{amsthm}
\usepackage[round]{natbib}
\usepackage[notref]{showkeys}
\bibliographystyle{chicago}
\renewcommand{\topfraction}{0.85}
\renewcommand{\textfraction}{0.1}
\renewcommand{\floatpagefraction}{0.85}
\renewcommand\thesubsection{\thesection.\alph{subsection}}
\usepackage[
urlcolor=black, %no colors
citecolor=black,
bookmarks=true, %tree-like TOC
bookmarksopen=true, %expanded when starting
hyperfootnotes=false, %no referencing of footnotes, does not compile
pdfpagemode=UseOutlines %show the bookmarks when starting the pdf viewer
]{hyperref}
\urlstyle{same}
\date{}
\begin{document}
\begin{align*}
\bar p_x &=\begin{cases} a^\frac{1}{\bar\theta} & i < \gamma \\
a^\frac{1}{\tilde\theta} & i \geq \gamma \\
\end{cases}
\end{align*}
\end{document}
• It could be informative to include also the result that you get from the example file. – mickep Jan 6 '18 at 18:38
• @mickep I thought that was universal given the latex code - and I can't upload a pdf here it appears.. – FooBar Jan 6 '18 at 18:46
• Well, you are correct it will most likely show the same here. But what you then ask for people is to first copy your code, and compile. Instead of just looking at your output and suggesting a solution. You can convert your output to png and upload. (I suggest this in all well meaning, just for you to increase the chance that someone takes a look.) – mickep Jan 6 '18 at 18:48
I assume the bar over the \theta in the denominator is the problem. Some variations:
\documentclass[11pt]{article}
\usepackage{amssymb}
\usepackage{amsmath}
\begin{document}
\begin{align*}
\bar p_x & =
\begin{cases}
a^\frac{1}{\bar\theta} & i < \gamma \\
a^\frac{1}{\tilde\theta} & i \geq \gamma \\
\end{cases}\\
\bar p_x & =
\begin{cases}
a^{1/\bar\theta} & i < \gamma \\
a^{1/\tilde\theta} & i \geq \gamma \\
\end{cases}\\
\bar p_x & =
\begin{cases}
a^{\bar\theta^{-1}} & i < \gamma \\
a^{\tilde\theta^{-1}} & i \geq \gamma \\
\end{cases}\\
\bar p_x & =
\begin{cases}
a^\frac{1}{
\vbox{\kern.2ex\hbox{$\scriptscriptstyle\bar\theta$}}
} & i < \gamma \\
a^\frac{1}{\tilde\theta} & i \geq \gamma \\
\end{cases}
\end{align*}
\end{document}
The latter variation in macro form. \barfix adds a space about the math symbol in the mandatory argument. The optional argument before allows fine-tuning.
\documentclass[11pt]{article}
\usepackage{amssymb}
\usepackage{amsmath}
\makeatletter
\newcommand*{\barfix}[2][.175ex]{%
\mathpalette{\@barfix{#1}}{#2}%
}
\newcommand*{\@barfix}[3]{%
% #1: space
% #2: math style
% #3: symbol
\vbox{%
\kern#1\relax
\hbox{$#2#3\m@th$}%
}%
}
\makeatother
\begin{document}
\begin{align*}
\bar p_x & =
\begin{cases}
a^\frac{1}{\barfix{\bar\theta}} & i < \gamma \\
a^\frac{1}{\barfix[.1ex]{\tilde\theta}} & i \geq \gamma \\
\end{cases}
\end{align*}
\end{document}
Other macros can be defined on top of \barfix, e.g.:
\newcommand*{\barF}[1]{\barfix{\bar#1}}
\newcommand*{\tildeF}[1]{\barfix[.15ex]{\tilde#1}}
...
$\barF\theta, \tildeF\theta$
• I like the last version most. Is there any way to redefine \tilde and \bar such that the \barfix{} wrapper is not necessary? – FooBar Jan 6 '18 at 19:09
• @FooBar There is no easy way. The accents do not know the context, the object above them. If the space is always be set, then it might lead to problems in other contexts (asymmetric line spacing in matrices, ...). LuaTeX allows to traversal the node lists, thus the situations that the accent is too close to a line could be detected. But the programming amount would be huge. – Heiko Oberdiek Jan 6 '18 at 19:49 | {"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": 0.9265861511230469, "perplexity": 2544.1485265729834}, "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/1558232255251.1/warc/CC-MAIN-20190520001706-20190520023706-00226.warc.gz"} |
http://math.stackexchange.com/questions/259781/proof-correctness-eigenvalues-and-isomorphisms | # Proof correctness Eigenvalues and Isomorphisms
Prove that $\lambda$ is an eigenvalue of $T \iff$ the map represented by $T-\lambda 1$ is not an isomorphism.
Proof:
$\rightarrow$
Suppose $\lambda$ is an eigenvalue of $T$, then we have $(T-\lambda 1)v = 0$ where $v\neq 0$. It is enough to show that $T$ is not one-to-one. By contradiction if $T$ were one-to-one then $(T-\lambda 1) = 0 \implies v = 0 \rightarrow \leftarrow$.
$\leftarrow$
By contraposition, Suppose $T$ is an isomorphism. We must show that $\lambda$ is not an eigenvalue of $T$ where $(T-\lambda 1)v = 0$. Since $T$ is an isomorphism then $T$ is one-to-one and we have $(T-\lambda 1)v = 0 \implies v = 0 \implies v$ is not an eigenvector and hence $\lambda$ is not an eigenvalue of $T$.
1. Is the above "proof" correct?
-
You prove the same thing twice: $P \Rightarrow Q$ and ~$Q \Rightarrow$~$P$ (which are equivalent statements), so you need either a "Suppose $T- \lambda 1$ is not an isomorphism" ($Q$), or a "Suppose $\lambda$ is an eigenvalue of $T$" (~$P$). – andybenji Dec 16 '12 at 6:02
I think that my proof would be useful for you.
Theorem:
$T$ is endomorphism of finite-dimentional vector space $V$. $\lambda$ is eigenvalue of $T$ if and only if the endomorphism $$T-\lambda E:V\mapsto V$$ is not an isomorphism.
Proof: In proof I'm going to use three lemmas (I think you know them well)
1. $f:V\mapsto W$ is injective if and only if $Ker(f)$ is zero subspace of $V$.
2. $f:V\mapsto W$ is surjective if and only if $Im(f)=W$.
3. $dim(Ker(f))+dim(Im(f))=n$ where n is dimension of space.
$\rightarrow$
Suppose $\lambda$ is eigenvalue of $T$ and $v\neq0$,$v\in V$, then $$Tv=\lambda v \implies (T-\lambda E)v=0$$ So, $v\in Ker(f)\implies Ker(f)\neq \{0\}$. Therefore, from the first lemma we have that $f=T-\lambda E$ is not an isomorphism.
$\leftarrow$ $f=T-\lambda E$ is not an isomorphism then $Ker(f)\neq \{0\}$ or $Im(f) \neq V$ (from first and second lemmas)
But we have lemma 3, so at any rate $Ker(f)\neq \{0\}$. And so we can choose $v\in Ker(f)$. Then
$$(T-\lambda E)v=0 \implies Tv=\lambda v ,$$
so $\lambda$ is eigenvalue of $T$
$\blacksquare$
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9976114630699158, "perplexity": 125.42318813556366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246655589.82/warc/CC-MAIN-20150417045735-00045-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://math.chapman.edu/~jipsen/structures/doku.php?id=semilattices | ## Semilattices
Abbreviation: Slat
### Definition
A \emph{semilattice} is a structure $\mathbf{S}=\langle S,\cdot \rangle$, where $\cdot$ is an infix binary operation, called the \emph{semilattice operation}, such that
$\cdot$ is associative: $(xy)z=x(yz)$
$\cdot$ is commutative: $xy=yx$
$\cdot$ is idempotent: $xx=x$
Remark: This definition shows that semilattices form a variety.
##### Morphisms
Let $\mathbf{S}$ and $\mathbf{T}$ be semilattices. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism:
$h(xy)=h(x)h(y)$
### Definition
A \emph{join-semilattice} is a structure $\mathbf{S}=\langle S,\leq,\vee\rangle$, where $\vee$ is an infix binary operation, called the \emph{join}, such that
$\leq$ is a partial order,
$x\leq y\implies x\vee z\leq y\vee z$ and $z\vee x\leq z\vee y$,
$x\le x\vee y$ and $y\leq x\vee y$,
$x\vee x\leq x$.
This definition shows that semilattices form a partially-ordered variety.
### Definition
A \emph{join-semilattice} is a structure $\mathbf{S}=\langle S,\vee \rangle$, where $\vee$ is an infix binary operation, called the \emph{join}, such that
$\leq$ is a partial order, where $x\leq y\Longleftrightarrow x\vee y=y$
$x\vee y$ is the least upper bound of $\{x,y\}$.
### Definition
A \emph{meet-semilattice} is a structure $\mathbf{S}=\langle S,\wedge \rangle$, where $\wedge$ is an infix binary operation, called the \emph{meet}, such that
$\leq$ is a partial order, where $x\leq y\Longleftrightarrow x\wedge y=x$
$x\wedge y$ is the greatest lower bound of $\{x,y\}$.
### Examples
Example 1: $\langle \mathcal{P}_\omega(X)-\{\emptyset\},\cup\rangle$, the set of finite nonempty subsets of a set $X$, with union, is the free join-semilattice with singleton subsets of $X$ as generators.
### Properties
Classtype variety decidable in polynomial time decidable undecidable yes 2 no no yes no no no yes yes yes yes yes
\end{table}
### Finite members
$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &2 f(4)= &5 f(5)= &15 f(6)= &53 f(7)= &222 f(8)= &1078 f(9)= &5994 f(10)= &37622 f(11)= &262776 f(12)= &2018305 f(13)= &16873364 f(14)= &152233518 f(15)= &1471613387 f(16)= &15150569446 f(17)= &165269824761 \end{array}$
These results follow from the paper below and the observation that semilattices with $n$ elements are in 1-1 correspondence to lattices with $n+1$ elements.
Jobst Heitzig,J\“urgen Reinhold,\emph{Counting finite lattices}, Algebra Universalis, \textbf{48}2002,43–53MRreview | {"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": 0.991327166557312, "perplexity": 1011.7968093262597}, "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-2022-21/segments/1652662526009.35/warc/CC-MAIN-20220519074217-20220519104217-00447.warc.gz"} |
https://chemistry.stackexchange.com/questions/55213/range-of-distance-for-van-der-waals-force?noredirect=1 | # Range of distance for Van der Waals force
Is there any range of distance between the nuclei of the atoms (in Angstrom, say between 4 to 12 Angstrom) within which there will be an occurrence of Van der Waals force (attraction) between them?
For example, say within a distance of 4 A between two atoms of same type, there will van der Waals attraction present. Can such a certainty be obtained? If two atoms are very far apart, then there will be not Van der Waals attraction between them, this is known. So my question is how close should the nuclei of two atoms be so that one can say with certainty that there will be Van der Waals attraction? | {"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": 0.8923856019973755, "perplexity": 278.3755651416542}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578742415.81/warc/CC-MAIN-20190425213812-20190425235812-00520.warc.gz"} |
https://www.physicsforums.com/threads/lower-mass-more-elliptical-orbit.97777/ | # Lower mass = more elliptical orbit?
1. Nov 1, 2005
Does it?
2. Nov 1, 2005
### Labguy
It shouldn't; orbits have many other factors that determine thier path. But, do you mean lower mass of the primary body or "orbiting" body? The mass can determine the orbit's decay, change, tidal locking, etc. but shouldn't have much to do with initial eccentricity.
3. Nov 1, 2005
### vincentm
i mean the orbiting body. Let's say there is a system that contains just two bodies orbiting a star, one with a higher mass than the other. which one would have a more elliptical orbit? Or does this depend on the mass of the star it's orbiting?
4. Nov 1, 2005
### Garth
In a binary system with unequal masses each orbit, as seen from the Centre of Mass of the combined system, is the mirror image of the other with dimensions inversely proportional to the mass of each body.
The eccentricities are equal.
Garth
5. Nov 1, 2005
### SpaceTiger
Staff Emeritus
The ellipticity of an orbit is not determined uniquely by the values of the masses in the system -- it depends also on its history and initial conditions. In other words, I could take the three-body system you describe above and give a kick to either one of the planets, making its orbit more elliptical.
One reason you might find that lower mass objects tend to have more elliptical orbits is that it takes less energy to disturb them. For example, I can send a satellite into an elliptical orbit with a small amount of fuel, while doing the same with the earth would require an enormous quantity of energy.
Last edited: Nov 2, 2005
6. Nov 1, 2005
### vincentm
Thank you, that helps alot! | {"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": 0.9269099235534668, "perplexity": 571.4999123502074}, "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-2017-04/segments/1484560279933.49/warc/CC-MAIN-20170116095119-00237-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://www.gamedev.net/index.php?app=forums&module=extras§ion=postHistory&pid=5068042 | • Sign In
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### #ActualJuliean
Posted 07 June 2013 - 10:45 AM
See what I did there......?
Judging from your formatting, I'd say you forgot to put parenthesis after the if, so that the third line was executed regardless of the selected value? If so, it would be interesting what the results of this was
### #1Juliean
Posted 07 June 2013 - 10:42 AM
See what I did there......?
Judging from your formatting, I'd say you forgot to put parenthesis after the if? If so, it would be interesting what the results of this was
PARTNERS | {"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": 0.8688516020774841, "perplexity": 1564.3918276321108}, "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-2014-42/segments/1414637900019.55/warc/CC-MAIN-20141030025820-00165-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/probability-problem.65890/ | # Probability problem
1. Mar 4, 2005
### Niels
I'm trying to learn probability on my own and I'm stuck.
My multiple-variable-calculus is not so strong so the following problem got me stuck.
I have density function
f(x,y) = x^2 + xy/3 for 0<x<1; 0<y<2 otherwise 0
And I need to calculate Prob(X > Y). X and Y are random variables.
I know how to do Prob(X <= 0.5) etc.
Also would be nice if someone could explain Prob(Y < 1/2 and X < 1/2) and
Prob(X+Y < 1)
Thanks
/Niels
2. Mar 4, 2005
### hypermorphism
Just to make sure, this is just integrating over all values of Y, such that X <= 0.5, which is a rectangle.
Similarly, this is an integration of the density function over the region where Y < 1/2 and X < 1/2. This is a square of sidelength 1/2 with one corner at the origin. It is a double integral that can be written as:
$$\int_0^{\frac{1}{2}} \int_0^{\frac{1}{2}} f(x,y) dy dx$$
Do you know how to find P(X<1/2 OR Y<1/2) ?
If you cannot picture the region, rewrite it in a friendlier form; ie., Y < -X + 1. You're then integrating the density function for all points below the line y = -x + 1. This integral can be written:
$$\int_0^1 \int_0^{-x+1} f(x,y) dy dx$$
So your original problem, P(Y<X) is just the set of points below the line y=x.
Last edited: Mar 4, 2005
3. Mar 4, 2005
thank you! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.9350719451904297, "perplexity": 901.121609142465}, "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-2018-22/segments/1526794866107.79/warc/CC-MAIN-20180524073324-20180524093324-00534.warc.gz"} |
http://math.stackexchange.com/questions/99037/k%c3%97k-grid-has-tree-width-at-least-k | # k×k grid has tree-width at least k
I am looking for ideas how to solve the problem from Diestel's textbook Graph Theory.
Chapter 12. Minors, Trees, and WQO.
Problem 16. Apply Theorem 12.3.9 to show that the $k \times k$ grid has tree-width at least $k$, and find a tree-decomposition of width exactly $k$.
Theorem 12.3.9. (Seymour & Thomas 1993). Let $k \geq 0$ be an integer. A graph has tree-width $\geq k$ if and only if it contains a bramble of order $> k$. (A bramble is a set of mutually touching connected vertex sets in $G$; its order is the minimum number of vertices in a set meeting each member of the bramble.)
The problem is the proof of the theorem 12.3.9 was given in the terms of bramble, which is a bit confusing, at present I don't really see the way to solve the problem by using this theorem.
If you familiar with the topic, please, help me out.
Addendum:
In Graphs & Algorithms: Advanced Topics on the slide 5.
The $n\times n$ - grid on $\left \{(i,k) | 1 \leq i, j \leq n\right \}$ has treewidth $\leq n$: Consider the path on
$X_{n(i-1)+j}=\left \{(i,k)|j\leq k\leq n\right \}\cup\left \{(i+1,k)|1\leq k\leq j\right \}, 1\leq i\leq n-1, 1\leq j\leq n$
How this is supposed to help me?
-
The $X_{n(i-1)+j}$ are the vertex sets (Wagner’s bags in slide 3) in a tree decomposition of the $n\times n$ grid of treewidth $n$. – Brian M. Scott Jan 14 '12 at 17:29
## 2 Answers
Here are a couple of big hints.
First, you need a bramble of order $k+1$. Let $G_k$ be the $k\times k$ grid, with vertices $\langle i,j\rangle$ for $1\le i,j\le k$. Thus, $G_{k-1}$ is the $(k-1)\times(k-1)$ grid in the upper lefthand corner of $G_k$. Start with the $(k-1)^2$ crosses in $G_{k-1}$; they form a bramble $\mathscr{B}$ of order $k-1$ for $G_{k-1}$. Of course these crosses don’t meet the bottom row and last column of $G_k$ at all. It’s possible to add just two sets to $\mathscr{B}$ to make a bramble of order $k+1$ for $G_k$. A good place to look for these sets is the part of $G_k$ that isn’t touched by $\mathscr{B}$ at all.
Then you need a tree decomposition of $G_k$ of width $k$. Uli Wagner’s slide 5 gives you one, though you still have to figure out the tree on which it’s based in order to prove that it is a tree decomposition. Note that he indexed the sets in the decomposition by consecutive integers; this is a big hint for the shape of the tree in question, and once you make the right guess, it isn’t hard to verify.
-
No we both have same mistake, crosses doesn't cause to bramble of order k in k*k grid, you just need to select intersection of two cross to make a hitting set which causes to Floor(k/2) not k, i thought about it and I see it's wrong (Still I didn't edit my answer because I couldn't find any way). – Lrrr Jan 14 '12 at 19:30
@Ali: If you use all $k^2$ crosses ($\operatorname{row}_i\cup\operatorname{col}_j$ for $1\le i,j\le k$, you do get a bramble of order $k$: any set of fewer than $k$ points misses at least one row and column and therefore misses at least one cross. – Brian M. Scott Jan 14 '12 at 19:36
You take all the crosses right :) it was simple mistake i made :) – Lrrr Jan 14 '12 at 19:42
+1 for correct answer – Lrrr Jan 14 '12 at 19:47
@BrianM.Scott: Thank you very much for the answer, I very appreciate it. My problem is I cannot get the idea behind the bramble, it's not actually a cover, it contains joined and disjoined connected by edge subgraphs. What the intuition of the bramble? I don't get the proof from Diestel's book too, because of the same reason - bramble. And how they get the path on the 5th slide, there is also should be something behind it. – com Jan 15 '12 at 20:33
I didn't read Diestel's book, but simple observation may be help is, if $G$ is a grid then we know $\operatorname{tw}(G) \ge BN(G)-1$ and as you know if you select $row_i \cup col_j$ as a bramble set, its hitting set size is $n$, so treewidth is at least $n-1$. Also you can simply construct a tree decomposition with size of $n$. So treewidth is $n$ or $n-1$. (By $BN(G)$ I mean bramble number of $G$.)
Edit: As Braian mentioned you can simply find good brambles and answer of sample tree decomposition is in your question. (my mistake was I thinking about $cross_{i,i}$ not $cross_{i,j}$
-
Your bramble of order $n$ shows only that the treewidth of the $n\times n$ grid is at least $n-1$. – Brian M. Scott Jan 14 '12 at 17:34
That’s irrelevant to the point that I was making. You said that the bramble of crosses shows that the treewidth is at least $n$, but it doesn’t: it shows only that the treewidth is at least $n-1$. Thus, finding a tree decomposition of width $n$ leaves open the possibility that the treewidth is actually $n-1$. Your argument would be incomplete even if you had included the tree decomposition of width $n$. – Brian M. Scott Jan 14 '12 at 18:03
@BrianM.Scott, you are right i should find better bramble, i'd edited my answer respect to your comment. – Lrrr Jan 14 '12 at 18:29 | {"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": 0.8475785851478577, "perplexity": 271.01772799396235}, "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/1440646249598.96/warc/CC-MAIN-20150827033049-00035-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://cs.stackexchange.com/questions/100357/in-isgci-unit-interval-graphs-are-denoted-as-c-n4-s-3-claw-net-free-i/100365 | # In ISGCI, unit interval graphs are denoted as ($C_{n+4}$,$S_3$,claw,net)-free. Is this an accurate notation?
When I search unit interval graphs in ISGCI, it says that the unit interval graphs (UIG) are equivalent to ($$C_{n+4}$$,$$S_3$$,claw,net)-free graphs.
I am confused about the definition of an $$S_3$$ graph. I am aware that in some resources, $$S_n$$ means $$K_{1,n}$$, and in others $$S_{n-1}$$ means $$K_{1,n}$$.
However, in both cases, ($$C_{n+4}$$,$$S_3$$,claw,net)-free $$\equiv$$ UIG seems like a wrong notation.
If $$S_3$$ is $$K_{1,3}$$, then $$S_3$$ is also a claw graph. Thus, ($$S_3$$,claw)-free is a redundant notation. On the other hand, if $$S_3$$ is $$K_{1,2}$$, then it is wrong because a path of length 3 is realizable as a unit interval graph.
If this notation is true, then can I also write UIG $$\equiv$$ ($$C_{n+4}$$,$$S_3$$,$$K_{1,2}$$,$$P_2$$,claw,net)-free?
If not, then is it correct to write that UIG $$\equiv$$ ($$C_{n+4}$$,claw,net)-free?
You can look up the small graphs that ISGCI uses to define classes. Specifically, $$S_3$$ is the following "sun" graph:
a chordal graph on $$2n$$ nodes ($$n\geq3$$) whose vertex set can be partitioned into $$W = \{w_1,\dots,w_n\}$$ and $$U = \{u_1, \dots,u_n\}$$ such that $$W$$ is independent and $$u_i$$ is adjacent to $$w_j$$ iff $$i=j$$ or $$i=j+1\pmod{n}$$.
Implicitly, the subgraph induced by $$U$$ can be anything. Wolfram Mathworld also talks about "complete sun graphs", which is the special case where $$U$$ is a clique. | {"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": 35, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9773046970367432, "perplexity": 447.2639504966167}, "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-2019-43/segments/1570986657949.34/warc/CC-MAIN-20191015082202-20191015105702-00006.warc.gz"} |
https://documen.tv/question/how-to-find-a-planet-s-gravitational-field-strength-using-its-radius-solve-the-radius-of-earth-i-15175005-35/ | ## how to find a planet’s gravitational field strength using its radius? solve: the radius of earth is 6.4 x 10^6 m and its mass is
Question
how to find a planet’s gravitational field strength using its radius?
solve:
the radius of earth is 6.4 x 10^6 m and its mass is 6.0 x 10^24. calculate earth’s gravitational field strength.
in progress 0
3 months 2021-08-22T11:37:30+00:00 1 Answers 1 views 0
1. The gravitational field strength is approximately equal to 10 N.
Explanation:
Gravitational field strength is the measure of gravitational force acting on any object placed on the surface of the planet. Generally, the mass of the object is considered as 1 kg.
So the gravitational field strength will be equal to the gravitational force acting on the object.
The formula for gravitational field strength is
Here g is the gravitational field strength, m is the mass of the object placed on the surface and F is the gravitational force acting on the object.
Since, the mass of any object placed on the surface of earth will be negligible compared to the mass of Earth, so the mass of the object is considered as 1 kg.
Then the g = F
And
Here G is the gravitational constant, M is the mass of Earth and m is the mass of the object placed on the surface, while r is the radius of the Earth.
So, the gravitational field strength is approximately equal to 10 N. | {"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": 0.9688677787780762, "perplexity": 163.1206545202926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363189.92/warc/CC-MAIN-20211205130619-20211205160619-00389.warc.gz"} |
http://mathhelpforum.com/calculus/187255-integrals-leading-log-function.html | 1. ## Integrals leading to a Log function
Hi all,
I am working on a problem at the moment which i have the answer for but i don't understand one step of the solution.
Evaluate "integration sign" (2x-6)dx/[(x^2)-6x+10]
The answer is given as Ln2.
I can see how the answer comes about but a step of it is confusing me.
Once i substitue and solve i get Ln[1x^2)-6(x)+10]/??? as per the rule for integrating 1/ax+b=ln(ax+b)/a
In otherwords 1/ax+b=(1[ln(ax+b)]/a) +c
However which value do i pic for a? It seems there are 2 coefficients of x in this case, 1 for the x squared, and -6 for the x.
It looks like the a chosen in the solution is the coefficient of the X^2 which is 1.
Why is the coefficient of the -6x not used for the a ?
It seems a pretty fundemental issue that i need solved.
When given a quadratic expression & when integrating is the a coefficient for the solution always the coefficient associated with the x^2?
Apologies if this is a ridiculous question
John
2. ## Re: Integrals leading to a Log function
You're right that:
$\int \frac{2x-6}{x^2-6x+10}dx= \ln|x^2-6x+10|+C$
I think it's strange there're no integration limits given.
3. ## Re: Integrals leading to a Log function
I think we have to take t = [(x^2)-6x+10]. Then dt = 2x-6. If you substitute this, your problem will become integration of dt/t.
4. ## Re: Integrals leading to a Log function
sorry guys the limits are b=4 a=3
5. ## Re: Integrals leading to a Log function
That changes the cases, that means we have:
$[\ln|x^2-6x+10|]_{3}^{4}=\ln|4^2-24+10|-\ln|3^2-18+10|=\ln|2|-\ln|1|=\ln|2|-0=\ln(2)$
6. ## Re: Integrals leading to a Log function
i should have just solved the expression -tut!
thanks all
John
7. ## Re: Integrals leading to a Log function
But if you are going to make the substitution $t= x^2- 6x+ 10$, $dx= (2x- 6)dx$, you might as well change the limits of integration also: when x= 3, $t= 9- 18+ 10= 1$ and when x= 4 $t= 16- 24+ 10= 2$ so
$\int_3^4 \frac{2x- 6}{x^2- 6x+ 10}dx= \int_1^2\frac{dt}{t}= \left[ln(t)\right]_1^2= ln(2)- ln(1)= ln(2)$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 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": 0.9408628344535828, "perplexity": 1195.606933575825}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607620.78/warc/CC-MAIN-20170523103136-20170523123136-00425.warc.gz"} |
https://www.questarter.com/q/sum-of-dirichlet-distributions-21_3387715.html | # Sum of Dirichlet distributions
by Rubertos Last Updated October 10, 2019 03:20 AM - source
Let $$\alpha\in (0,\infty)^n$$ and $$\{e_1,...,e_n\}$$ be the standard ordered basis for $$\mathbb{R}^n$$, and define $$\alpha_0:=\sum_{i=1}^n \alpha_i$$.
Let $$Dir(\alpha+e_k)$$ be Dirichlet distributions and $$\mu_k$$ be its distribution measures.
Then, how do I show that $$\sum_{i=1}^n \frac{\alpha_i}{\alpha_0} \mu_k$$ is the distribution measure of $$Dir(\alpha)$$? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.987703800201416, "perplexity": 357.5635755634742}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987803441.95/warc/CC-MAIN-20191022053647-20191022081147-00557.warc.gz"} |
https://www.physicsforums.com/threads/discriminant-is-a-symmetric-polynomial.100612/ | # Homework Help: Discriminant is a symmetric polynomial
1. Nov 19, 2005
### Pietjuh
I've got to proof the following:
Let f be a monic polynomial in Q[X] with deg(f) = n different complex zeroes. Show that the sign of the discriminant of f is equal to (-1)^s, with 2s the number of non real zeroes of f.
I know the statement makes sense, because the discriminant is a symmetric polynomial over Q, so it can be written as a polynomial in elementary symmetric polynomials.
The question seems to suggest that the complex zeroes always come in pairs of a zero and its conjugate. But even if this is true, i still don't know how to proceed :(
Can anyone give me a hint?
2. Nov 19, 2005
### AKG
Prove that if z is a root of the polynomial, then so is $\bar{z}$. Use the fact that rational numbers are self-conjugate, and that
$$\overline{ab + c} = \overline{a}\overline{b} + \overline{c}$$
Next, go to the definition of the descriminant as a product of squares of differences of roots. You know that if ri is a non-real root and if rj is a real root, then both $(r_i - r_j)^2$ and $(\overline{r_i} - r_j)^2$ will occur in the product, so consider what the product of these two factors will be (well, just consider the sign). Consider the case when both roots under consideration are real, when both are complex and not conjugate to one another, and when they are complex and conjugate to one another. | {"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": 0.9288536310195923, "perplexity": 207.25218512375986}, "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-2018-30/segments/1531676589350.19/warc/CC-MAIN-20180716135037-20180716155037-00312.warc.gz"} |
https://wiki.cs.byu.edu/cs-677sp2010/slice-sampling?do=diff&rev2%5B0%5D=1418341502&rev2%5B1%5D=1418341688&difftype=sidebyside | ##### Differences
This shows you the differences between two versions of the page.
cs-677sp2010:slice-sampling [2014/12/11 16:45]ryancha cs-677sp2010:slice-sampling [2014/12/11 16:48]ryancha Both sides previous revision Previous revision 2014/12/11 16:49 ryancha 2014/12/11 16:48 ryancha 2014/12/11 16:47 ryancha 2014/12/11 16:47 ryancha 2014/12/11 16:46 ryancha 2014/12/11 16:45 ryancha 2014/12/11 16:44 ryancha 2014/12/09 09:52 ryancha created Go Next revision Previous revision 2014/12/11 16:49 ryancha 2014/12/11 16:48 ryancha 2014/12/11 16:47 ryancha 2014/12/11 16:47 ryancha 2014/12/11 16:46 ryancha 2014/12/11 16:45 ryancha 2014/12/11 16:44 ryancha 2014/12/09 09:52 ryancha created Go Last revision Both sides next revision Line 4: Line 4: Suppose you want to sample some random variable ''X'' with distribution f(x). Suppose that the following is the graph of f(x). The height of f(x) corresponds to the likelihood at that point. Suppose you want to sample some random variable ''X'' with distribution f(x). Suppose that the following is the graph of f(x). The height of f(x) corresponds to the likelihood at that point. - [[media:cs-677sp10:fx_dist.png|350px]] + [[media:cs-677sp10:350px-fx_dist.png|350px]] If you were to uniformly sample ''X'', each value would have the same likelihood of being sampled, and your distribution would be of the form f(x)=y for some ''y'' value instead of some non-uniform function f(x). Instead of the original black line, your new distribution would look more like the blue line. If you were to uniformly sample ''X'', each value would have the same likelihood of being sampled, and your distribution would be of the form f(x)=y for some ''y'' value instead of some non-uniform function f(x). Instead of the original black line, your new distribution would look more like the blue line. - [[media:cs-677sp10:uniform_dist.png|350px]] + [[media:cs-677sp10:350px-uniform_dist.png|350px]] In order to sample ''X'' in a manner which will retain the distribution f(x), some sampling technique must be used which takes into account the varied likelihoods for each range of f(x). In order to sample ''X'' in a manner which will retain the distribution f(x), some sampling technique must be used which takes into account the varied likelihoods for each range of f(x). Line 24: Line 24: ==Univariate Case== ==Univariate Case== - [[media:cs-677sp10:x_y.png|thumb|350pxFor a given sample x, a value for y is chosen from [0, f(x)], which defines a "slice" of the distribution (shown by the solid horizontal line). In this case, there are two slices separated by an area outside the range of the distribution.]] + [[media:cs-677sp10:350px-x_y.png|thumb|350pxFor a given sample x, a value for y is chosen from [0, f(x)], which defines a "slice" of the distribution (shown by the solid horizontal line). In this case, there are two slices separated by an area outside the range of the distribution.]] To sample a random variable ''X'' with density f(x) we introduce an auxiliary variable ''Y'' and iterate as follows: To sample a random variable ''X'' with density f(x) we introduce an auxiliary variable ''Y'' and iterate as follows: Line 39: Line 39: *A candidate sample is selected uniformly from within this region. If the candidate sample lies inside of the slice, then it is accepted as the new sample. If it lies outside of the slice, the candidate point becomes the new boundary for the region. A new candidate sample is taken uniformly. The process repeats until the candidate sample is within the slice. (See diagram for a visual example). *A candidate sample is selected uniformly from within this region. If the candidate sample lies inside of the slice, then it is accepted as the new sample. If it lies outside of the slice, the candidate point becomes the new boundary for the region. A new candidate sample is taken uniformly. The process repeats until the candidate sample is within the slice. (See diagram for a visual example). - [[Image:slice.png|thumb|center|500pxFinding a sample given a set of slices (the slices are represented here as blue lines and correspond to the solid line slices in the previous graph of f(x) ). a) A width parameter ''w'' is set. b) A region of width ''w'' is identified around a given point $x_0$. c) The region is expanded by ''w'' until both endpoints are outside of the considered slice. d) $x_1$ is selected uniformly from the region. e) Since $x_1$ lies outside the considered slice, the region's left bound is adjusted to $x_1$. f) Another uniform sample $x$ is taken and accepted as the sample since it lies within the considered slice.]] + [[media:cs-677sp10:500px-slice.png|500pxFinding a sample given a set of slices (the slices are represented here as blue lines and correspond to the solid line slices in the previous graph of f(x) ). a) A width parameter ''w'' is set. b) A region of width ''w'' is identified around a given point $x_0$. c) The region is expanded by ''w'' until both endpoints are outside of the considered slice. d) $x_1$ is selected uniformly from the region. e) Since $x_1$ lies outside the considered slice, the region's left bound is adjusted to $x_1$. f) Another uniform sample $x$ is taken and accepted as the sample since it lies within the considered slice.]] ==Multivariate Methods== ==Multivariate Methods== Line 55: Line 55: In this graphical representation of reflective sampling, the shape indicates the bounds of a sampling slice. The dots indicate start and stopping points of a sampling walk. When the samples hit the bounds of the slice, the direction of sampling is "reflected" back into the slice. In this graphical representation of reflective sampling, the shape indicates the bounds of a sampling slice. The dots indicate start and stopping points of a sampling walk. When the samples hit the bounds of the slice, the direction of sampling is "reflected" back into the slice. - [[media:cs-677sp10:reflection_sampling.png|350px|alt=alt text]] + [[media:cs-677sp10:350px-reflection_sampling.png|350px|alt=alt text]] ==Example== ==Example== | {"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": 0.9016992449760437, "perplexity": 1431.041302455571}, "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-2021-04/segments/1610704847953.98/warc/CC-MAIN-20210128134124-20210128164124-00538.warc.gz"} |
https://arxiv.org/abs/1404.5565 | cs.CC
(what is this?)
# Title: On the Satisfiability of Quantum Circuits of Small Treewidth
Abstract: It has been known for almost three decades that many $\mathrm{NP}$-hard optimization problems can be solved in polynomial time when restricted to structures of constant treewidth. In this work we provide the first extension of such results to the quantum setting. We show that given a quantum circuit $C$ with $n$ uninitialized inputs, $\mathit{poly}(n)$ gates, and treewidth $t$, one can compute in time $(\frac{n}{\delta})^{\exp(O(t))}$ a classical assignment $y\in \{0,1\}^n$ that maximizes the acceptance probability of $C$ up to a $\delta$ additive factor. In particular, our algorithm runs in polynomial time if $t$ is constant and $1/poly(n) < \delta < 1$. For unrestricted values of $t$, this problem is known to be complete for the complexity class $\mathrm{QCMA}$, a quantum generalization of MA. In contrast, we show that the same problem is $\mathrm{NP}$-complete if $t=O(\log n)$ even when $\delta$ is constant.
On the other hand, we show that given a $n$-input quantum circuit $C$ of treewidth $t=O(\log n)$, and a constant $\delta<1/2$, it is $\mathrm{QMA}$-complete to determine whether there exists a quantum state $\mid\!\varphi\rangle \in (\mathbb{C}^d)^{\otimes n}$ such that the acceptance probability of $C\mid\!\varphi\rangle$ is greater than $1-\delta$, or whether for every such state $\mid\!\varphi\rangle$, the acceptance probability of $C\mid\!\varphi\rangle$ is less than $\delta$. As a consequence, under the widely believed assumption that $\mathrm{QMA} \neq \mathrm{NP}$, we have that quantum witnesses are strictly more powerful than classical witnesses with respect to Merlin-Arthur protocols in which the verifier is a quantum circuit of logarithmic treewidth.
Comments: 30 Pages. A preliminary version of this paper appeared at the 10th International Computer Science Symposium in Russia (CSR 2015). This version has been submitted to a journal and is currently under review Subjects: Computational Complexity (cs.CC); Quantum Physics (quant-ph) Cite as: arXiv:1404.5565 [cs.CC] (or arXiv:1404.5565v2 [cs.CC] for this version)
## Submission history
From: Mateus de Oliveira Oliveira [view email]
[v1] Tue, 22 Apr 2014 17:31:56 GMT (197kb)
[v2] Sun, 7 Feb 2016 21:47:30 GMT (157kb) | {"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": 0.9087596535682678, "perplexity": 371.11162331143066}, "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/1498128320003.94/warc/CC-MAIN-20170623045423-20170623065423-00468.warc.gz"} |
https://www.splashlearn.com/math-vocabulary/fractions/dividing-fractions-with-whole-numbers | # Dividing Fractions With Whole Numbers - Definition with Examples
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## Dividing Fractions with Whole Numbers
A fraction is a part of a whole. The given pizza is cut into 5 equal slices and 3 of slices are left. That is, 3 out of 5 slices of pizza are there. The fraction shown is 35
Now, if we divide this three-fifths of the pizza into 3 equal parts, each part will have one part out of the 5 parts as shown.
That is 35÷3= 1⁄5.
Dividing 35 by 3, will give us one-third of 35
That is 3115.
We can also verify this as 1x 3= 35.
Consider dividing the fraction 46 by 2.
The portion shaded in pink is divided equally into two parts – shaded in green and the one in blue respectively. The part in green is 36 of the rectangle, and so is the part shaded in blue.
That is 462=26.
We can verify this using multiplication as 2x 2 = 46.
Here again, on dividing 46 by 2, we precisely find the one-half of 46
That is 41226.
In both the examples, in the procedure, the division symbol is replaced with multiplication, and its multiplicative inverse or reciprocal replaces the divisor.
The rule is, to divide a fraction by a whole number, multiply the given fraction by the reciprocal of whole numbers.
Example: Find 14÷3.
The reciprocal of 3 is 13
14÷3 = 11112
Conceptually this can be shown as:
Example: If 5 out of 12 pieces of an apple pie were shared among 3 people, what fraction of apple pie does each person gets?
We know that 512 of the apple pie is equally shared between 3 people.
So, we have to find 512 ÷3.
The reciprocal of 3 is 13
512 ÷ 3 = 512 1536
Therefore, each person gets 536 of the apple pie.
Fun Facts:
What if the divisor is a fraction? The rule remains the same!
3÷ 14 = 3x 4 = 3
How many ¼ s are there in a ¾?
There are 3 one-fourth in a three-fourth!
• If the dividend is a mixed fraction, first convert the mixed fraction into an improper fraction and apply the rule.
Example: 51÷ 8
First, convert 513 into an improper fraction.
51= (5x3)+1163
Now, 51÷ 8 = 163 ÷ 8 = 163 x 18 = 23 .
Won Numerous Awards & Honors | {"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": 0.840130090713501, "perplexity": 1448.169529945202}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178355937.26/warc/CC-MAIN-20210225211435-20210226001435-00248.warc.gz"} |
http://home.cern/about/updates/2015/05/low-energy-collisions-tune-lhc-experiments | # Low-energy collisions tune LHC experiments
Two beams of protons at 450 GeV collide in the the CMS detector for a total collision energy of 900 GeV. The LHC experiments are using these collisions to tune and align their detectors (Image: CMS/CERN)
At about half past nine CET this morning, for the first time since the Large Hadron Collider (LHC) started up after two years of maintenance and repairs, the accelerator delivered proton-proton collisions to the LHC experiments ALICE, ATLAS, CMS and LHCb at an energy of 450 gigaelectronvolts (GeV) per beam.
These collisions, which take place with each beam at the so-called injection energy, that is, the energy at which proton beams are injected into the LHC from the Super Proton Synchrotron, enable the LHC experiments to tune their detectors. This process is also an important step towards readying the accelerator to deliver beams at 6.5 teraelectronvolts (TeV) for collisions at 13 TeV.
Each low-energy collision sends showers of particles flying through an experiment's many layers. The experimental teams can use this data to check their subdetectors and ensure they fire in the correct place at the precise instant that a particle passes. Reconstructing flight paths of the particles from many parts of the detector at once helps the experiments to check the alignment and synchronization of various subdetector elements.
So just as the LHC team tests each component, system, and algorithm one after the other, the experiments go through checklists that confirm that everything is fully functional and no mistakes, bugs or failures are present when collisions are delivered at 13 TeV.
Meanwhile the LHC Operations team is halfway through its eight weeks of scheduled beam commissioning, during which the accelerator's many subsystems are checked to ensure that beams will circulate stably and in the correct orbit. Sensors and collimators around the accelerator's full 27 kilometres send information to the CERN Control Centre, from where the operators can remotely adjust the beam by fine-tuning the positions and field strengths of hundreds of electromagnets.
Though the first beam at 6.5 TeV circulated successfully in the LHC last month, there are many more steps before the accelerator will deliver high-energy collisions for physics to the LHC experiments. Well before the full physics programme begins, the LHC operations team will collide beams at 13 TeV to check the beam orbit, quality and stability. | {"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": 0.8981549143791199, "perplexity": 1801.2566664831004}, "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-2017-04/segments/1484560284270.95/warc/CC-MAIN-20170116095124-00114-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/article-higgs-boson-in-superconductors.651805/ | # Article: Higgs Boson in Superconductors
1. Nov 13, 2012
### DrDu
I ran into the following article by Varma, Higgs Boson in Superconductors
http://arxiv.org/abs/cond-mat/0109409
Varma compares the Gross-Pitaevskii equation with the Higgs Lagrangian and calculates the elementary excitations. He shows that although symmetry is broken in both cases, Higgs type excitations are absent in the G-P equation although it looks very similar to the Higgs equation, the main difference being that the former is only first order in t.
He then shows why in Superconductors one nevertheless obtains Higgs type excitations which he identified in experimental spectra together with Littlewood in 1982.
It is interesting how easily -at least my- intuition gets lost in QFT.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook
Can you offer guidance or do you also need help?
Draft saved Draft deleted | {"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": 0.9410990476608276, "perplexity": 2292.149296181815}, "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-2018-30/segments/1531676593010.88/warc/CC-MAIN-20180722041752-20180722061752-00351.warc.gz"} |
https://freakonometrics.hypotheses.org/tag/odd | # Playing with fire (or water)
A few days ago,http://www.futilitycloset.com/published a short post based on the fourth problem of the 1987 Canadian Mathematical Olympiad (from on a problem from the 6th All Soviet Union Mathematical Competition in Voronezh, 1966). The problem is simple (as always). It is about water pistol duels (with an odd number of players)
What puzzled me in this problem is the following: if we know, for sure, that at least one player won’t get wet, we don’t know exactly how many of them won’t get wet (assuming that if they shoot at the closest, they hit him for sure) ? It is simple to run simulations, e.g. assuming that players are uniformly distributed over a square,
```NOTWET=function(n){
x=runif(n)
y=runif(n)
(d=as.matrix(dist(cbind(x,y), method = "euclidean",upper=TRUE)))
diag(d)=999999
dmin=apply(d,2,which.min)
notwet=n-length(table(dmin))
return(notwet)}```
It is then rather simple to get the distribution of the number of player that did not get wet,
```N25=Vectorize(NOTWET)(n=rep(25,NSim))
T=table(N25)
plot(as.numeric(names(T)),T/NSim,type="b")```
The graph for different values for the total number of players is the following (based on 25,000 simulations)
If we investigate further, say with 51 players, we have a distribution for the total number of players that did not get wet which looks exactly like the Gaussian distribution,
```NSim=25000
N51=Vectorize(NOTWET)(n=rep(51,NSim))
T=table(N51)
plot(as.numeric(names(T)),T/NSim,type="b",col="blue")
u=seq(0,51,by=.1)
lines(u,dnorm(u,mean(N51),sd(N51)),col="red",lty=2)```
If anyone has an intuition (not to say a proof) for that, I’d be glad to hear it… | {"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": 0.9173243045806885, "perplexity": 464.4439385791937}, "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-2022-05/segments/1642320305317.17/warc/CC-MAIN-20220127223432-20220128013432-00633.warc.gz"} |
https://www.physicsforums.com/threads/electricity-formula.739913/ | # Electricity formula
1. Feb 23, 2014
### Astronomy
Hi
I want to know what is the diffrences between E=σ/2ε0 & E=σ/ε0
of course I know we use the first for a very huge page but what about the second formula?
I saw in some solving of question when we have 2 pages infront of each others we use both of them...
I want to understand the deep meaning and usage...
pls tell me simply beacuse my profession english is not good!
thank u
2. Feb 23, 2014
### dauto
The first formula is used for a thin sheet where the field lines split half in each direction. The second formula is used for the surface of a metal where all the field lines are in one side of the surface (outward), and there is no field in the other side (inward). | {"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": 0.8006230592727661, "perplexity": 878.2146926166532}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589179.32/warc/CC-MAIN-20180716041348-20180716061348-00157.warc.gz"} |
https://economics.stackexchange.com/questions/25127/showing-marginal-product-of-capital-is-independent-of-the-scale-of-production | # Showing marginal product of capital is independent of the scale of production
The image is pretty much self-explanatory. To add some context, I'm learning Solow-Swan Growth Theory and my professor said that the marginal product will not change if both capital and labor increase at the same scale.
I can intuitively understand that it makes sense but trying to apply a simple equation (the blue one, the definition of constant marginal product) is just not working.
It's either I'm not partial differentiating correctly or the whole theory is wrong.
I don't see anything wrong with what I've done but why are they not the same?
• The trick is to show the aggregate production function is homogenous of degree 1 – Pedro Cavalcante Oliveira Oct 21 '18 at 20:01
To compute the MPK, we must differentiate the production function with respect to the current level of capital: $$\partial F/\partial K$$.
But in your final line, you are not differentiating with respect to the current level of capital (which is $$\tilde{K}=\lambda K$$). You are instead differentiating with respect to $$K$$, which is a fraction $$1/\lambda$$ of the current amount of capital.
If we compute the derivative with respect to $$\tilde{K}\equiv \lambda K$$ instead of just $$K$$ then everything works as it should:
$$\frac{\partial F(\lambda K,\lambda L)}{\partial \lambda K}=\frac{\partial (8(\lambda K)^{1/2}(\lambda L)^{1/2})}{\partial \lambda K}=4(\lambda K)^{-1/2}(\lambda L)^{1/2}=4\frac{\sqrt{L}}{\sqrt{K}}.$$
This does not depend on $$\lambda$$ so MPK is indeed independent of the scale of the economy. QED | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9572557806968689, "perplexity": 283.55387618224813}, "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-2019-04/segments/1547583857993.67/warc/CC-MAIN-20190122161345-20190122183345-00016.warc.gz"} |
http://www.msri.org/seminars/16506 | # Mathematical Sciences Research Institute
Home » MSRI Colloquium: "The orbifold vertex: computing the Donaldson-Thomas invariants of toric orbifolds by counting colored boxes"
# Seminar
MSRI Colloquium: "The orbifold vertex: computing the Donaldson-Thomas invariants of toric orbifolds by counting colored boxes" March 13, 2009
Location: MSRI: Simons Auditorium
Speaker(s) Professor Jim Bryan
Description No Description
Video
Abstract/Media
The topological vertex is a powerful formalism first
discovered in physics for computing the Gromov-Witten theory of any
toric Calabi-Yau threefold in terms of a universal power series (the
vertex). Maulik, Nekrasov, Okounkov and Pandharipande found an
equivalent formalism for Donaldson-Thomas invariants in which the
vertex has a very concrete combinatorial interpretation --- it is a
generating function for counting boxes piled in a corner. We present
an orbifold version of the vertex formalism which computes the
Donaldson-Thomas invariants of a toric orbifold. The orbifold vertex
counts boxes which are colored by representations of a finite Abelian
group. As an application, we prove the Donaldson-Thomas Crepant
Resolution Conjecture in the toric case. | {"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": 0.8538044691085815, "perplexity": 3264.233719457074}, "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/1414119645329.44/warc/CC-MAIN-20141024030045-00027-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/85120-normal-subgroups-factor-groups-print.html | # Normal Subgroups and Factor Groups
• April 22nd 2009, 03:44 PM
o&apartyrock
Normal Subgroups and Factor Groups
Let G = GL(2,R) and let K be a subgroup of R*. Prove that H = {A ∈ G| det A ∈ K} is a normal subgroup of G.
• April 22nd 2009, 04:00 PM
Gamma
Normal
A subgroup $H\leq G$ is said to be normal iff $\forall g \in G$ we have $gHg^{-1} \subset H$
So take $A\in G=GL(2, \mathbb{R})$ it is a group and therefore invertible so $A^{-1} \in G$ but consider for any $X\in H$ we would have:
$det(AXA^{-1})=det(A)det(X)det(A^{-1})=det(A)det(X)\frac{1}{det(A)}=det(X)$ so we see that for any $X\in H$ and for all $A\in G$ we get $AXA^{-1}\in H$.
This shows $AHA^{-1} \subset H$ thus it is normal. | {"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": 11, "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": 0.9910293221473694, "perplexity": 560.7127819216562}, "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-2014-41/segments/1410657132646.40/warc/CC-MAIN-20140914011212-00200-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/156414-expected-value-print.html | # Expected Value
• September 16th 2010, 09:04 AM
cuteylion
Expected Value
Hi,
Suppose E(U|X) = X^2. Suppose that E(X) = 0, Var(X) = 1 and E(X^3) = 0
What is the E(UX)?
Thanks alot for helping!
• September 16th 2010, 09:48 AM
Moo
Hello,
$E[UX]=E[E[UX|X]]=E[XE[U|X]]=E[X^3]$
:)
• September 16th 2010, 11:17 PM
aman_cc
@ Moo - Thanks for your response.
But I am not confortable with all the theorms/or rather properties of Expectation (in case of joint distributions). I basically get confused and at times rely on intuition.
Will it be possible for you to reccomend me
1. A source/book I can read to understand this better
2. For e.g. Why is
E(Y) = E(E(Y|X))
And then how you got
E[E[UX|X]]=E[XE[U|X]] - I understand it intuitively but not rigorously.
Thanks
• September 17th 2010, 12:50 AM
matheagle
E(Y) = E(E(Y|X))
is easy to prove, just write the conditional expectation and integrate/sum a second time
• September 17th 2010, 01:08 AM
aman_cc
Thanks.
But where I'm weak is a better understanding of Joint Distributions. I tried referring some online resources but couldn't get a very clear idea. And, I'm not attending any university so little hard for me to find out where can you read up on it. Thanks
• September 17th 2010, 02:06 AM
cuteylion
• September 17th 2010, 08:26 AM
matheagle
In the continuous setting....
$E(Y|X)=\int yf_{Y|X}(y)dy$
$=\int y{f_{X,Y}(x,y)\over f_X(x)}dy$
now take the expectation wrt x...
$E(E(Y|X))=\int E(Y|X)f_X(x)dx$
and wave the Fubini wand...
$E(E(Y|X))=\int \int y{f_{X,Y}(x,y)\over f_X(x)}dyf_X(x)dx$
$=\int \int yf_{X,Y}(x,y)dydx=E(Y)$
• September 17th 2010, 12:31 PM
Moo
aman_cc,
I have a very formal proof of the fact that $E[E[X|Y]]=E[X]$ but it's kind of long, and very theoretical. You also need measure theory knowledge. Well I don't think it's a relevant one...
The general idea is to say that in a probability space $\Omega,\mathcal A,\mathbb P)$ and given a sigma-algebra $\mathcal B$, and X a random variable $\in L^1(\Omega,\mathcal A,\mathbb P)$, then there exists a unique random variable $Z=E[X|\mathcal B$ such that for any $B\in \mathcal B,\int_B Z d\mathbb P=\int_B Xd\mathbb P$.
In particular, for $B=\Omega$, we get the desired equality.
Note that when we write $E[X|Y]$, it's in fact $E[X|\sigma(Y)$, where $\sigma(Y)$ is the sigma-algebra generated by $Y$. So it's in place of $\mathcal B$ in the above paragraph.
Also note that with the Aussie singer's method, matheagle, we have to assume that these random variables have a pdf.
For the second point, the proof doesn't look that difficult, it's just long and painful... I don't know how to explain it with words...
• September 19th 2010, 10:36 PM
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http://www.purplemath.com/learning/viewtopic.php?p=22 | ## Solve 2^(2x) - 7(2^x) - 8 = 0 (Hint: Let y = 2^x)
Simplificatation, evaluation, linear equations, linear graphs, linear inequalities, basic word problems, etc.
little_dragon
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### Solve 2^(2x) - 7(2^x) - 8 = 0 (Hint: Let y = 2^x)
I know some about powers and about solving by factoring. One of my "challenge" problems is to solve 2^(2x) - 7(2^x) - 8 = 0, and the hint in the book says "Let y = 2^x". I think this gives me y^2 - 7y - 8 = 0. Is that right? Because then I'm pretty sure I can finish this. Thanks.
stapel_eliz
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little_dragon wrote:I think this gives me y^2 - 7y - 8 = 0. Is that right?
That's exactly right!
Now factor, and set each factor equal to zero. Once you've done that, plug the $2^x$ back in for the y, and use what you know about powers to find the answers.
If you get stuck, please reply showing how far you got. Thank you!
Eliz.
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http://cms.math.ca/cjm/kw/generalized%20Frobenius%20partition | location: Publications → journals
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Search: All articles in the CJM digital archive with keyword generalized Frobenius partition
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1. CJM 2011 (vol 63 pp. 1284)
Dewar, Michael
Non-Existence of Ramanujan Congruences in Modular Forms of Level Four Ramanujan famously found congruences like $p(5n+4)\equiv 0 \operatorname{mod} 5$ for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on $\Gamma_{1}(4)$ that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored $F$-partitions. Keywords:modular form, Ramanujan congruence, generalized Frobenius partition, overpartition, crankCategories:11F33, 11P83 | {"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": 0.9076259732246399, "perplexity": 2613.6804203930988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119646554.36/warc/CC-MAIN-20141024030046-00042-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/6842/is-there-a-simple-formulaic-way-to-construct-a-modular-exponentiation-circuit/6877 | # Is there a simple, formulaic way to construct a modular exponentiation circuit?
I'm a newcomer to quantum computing and circuit construction, and I've been struggling to understand how to make a modular exponentiation circuit. From what I know, there are several papers on the matter (like Pavlidis, van Meter, Markov and Saeedi, etc.) but they are all so complicated and involve a lot of efficiency and optimization scheme that make it impossible for me to understand. When I read it in Nielsen and Chuang, specifically in Box 5.2 the author wrote them without any example, as if it is very easy to make (it probably is, but not for me).
Anyway, I've learned about the algorithm to do modular exponentiation using binary representation (it's simple enough at least this thing), but I don't know how to make a circuit out of it. Here's the picture I believe describing the process:
So how do I build those $$U$$ circuit? Can anyone, for example, tell me how do things changed when say I went from $$11^x (\mod{15})$$ to $$7^x (\mod{21})$$? I don't care if the circuit is not gonna be optimized and contain thousands of gates, but I want to at least understand the first step before going into more advanced stuffs like optimization.
Thank you very much!
• This question and answer about an implementation of the order finding circuit and modular exponation circuit gives some insight.quantumcomputing.stackexchange.com/questions/4852/… – Bram Jul 22 '19 at 20:26
• Thank you! I've also seen these modular multiplication circuits in Markov and Saeedi in the form of SWAP gate, and even though I did try the numbers in and it works, I still couldn't see how; it's like something we make when we've already know the answer, and just make a circuit that gives out the correct answer. Why do we use the SWAP gates, and in that order for C = 2 , 4 or 7, 11? How do things changed when say M = 21 for example? I couldn't understand those things... – Kim Dong Jul 23 '19 at 3:56
Here's, surely, a very non-optimal way of doing it. Imagine we have a unitary $$V$$ which performs the operation $$V|x\rangle|y\rangle=|x\rangle|xy\text{ mod }N\rangle.$$ We can deal with how $$V$$ might work separately, but if you have that, we want to see how we can use it to calculate any $$|x^{2^i}\text{ mod }N\rangle$$. The trick is that if both inputs are $$x^{2^j}\text{ mod }N$$, then the output is $$x^{2^{j+1}}\text{ mod }N$$, so we only have to repeat this construction $$i$$ times. For example, in the circuit below: Here, I've used the control-not to denote transversal application to achieve copying of one (effectively classical) register to another. This lets you do any of the $$U$$ operations that you need, assuming that you know how to implement $$V$$. Don't forget that, as part of a larger circuit, you have to 'uncompute' the data on any auxiliary registers.
So, how do we implement $$V$$? Let me give some of the components. Let $$x=x_1x_2x_3\ldots x_n$$ and $$y=y_1y_2\ldots y_n$$ be the binary representations of $$x$$ and $$y$$. The product $$xy$$ is easy to calculate by long multiplication. For example, $$x_iy_j$$ is a bit value (so there are never any carries from the multiplication steps) that is equivalent to applying the Toffoli (controlled-controlled-not) with $$x_i$$ and $$y_j$$ as the two inputs. So you can calculate $$y_1x$$, $$y_2x$$, $$y_3x\ldots$$ on separate registers and then add them up.
Addition is another standard circuit. Imagine you want to add $$x_1x_2\ldots x_n$$ and $$y_1y_2\ldots y_n$$. We need to have additional two registers: one for the output, one for the carry bit. The least significant bit of output is $$x_n\oplus y_n$$, which can be calculated with controlled-nots. The carry bit has value $$z_n=x_ny_n$$. The next output is $$x_{n-1}\oplus y_{n-1}\oplus z_{n}$$, which we can again do with controlled nots. The carry bit is a majority vote - are two or more of $$x_{n-1},y_{n-1},z_n$$ value 1? One way to implement this is: You can keep repeating this process bit by bit to compute the sum. Then, again, don't forget to uncompute all the ancillas.
• Thank you for your answer. I still have 2 problems though:$\\$ 1. I'm still not sure how we can use multiplication and addition to make a circuit that calculate modulus. I'm a physics student and I don't really know about circuit. Can you elaborate for me? $\\$ 2. This method looks like it's gonna take a massive number of qubits since we effectively just multiply everything out instead of doing memory-efficient modular arithmetic right? I estimated using 30 qubits just to store the $x^j$ answer if we were to do $19^7 \mod(21) = 1$. So this is definitely unusable as of right now right? – Kim Dong Jul 24 '19 at 12:47
• @D.Tran You didn't ask for usable. You asked for a general method that could get you started. All the methods that have been used have involved a very deliberate choice of $a$ that makes things easy, and some very specific improvements for those cases. – DaftWullie Jul 24 '19 at 12:59
• Yes, the 2nd question isn't too important, but as in the 1st one I'm still not sure how to use it to do modulo... – Kim Dong Jul 24 '19 at 13:18
• @D.Tran Functionally, what is it you would do if you were asked to calculate $x\text{ mod }N$, given $x$ and $N$? – DaftWullie Jul 25 '19 at 7:17
• If I were to do it by hand, I'll try to divide x to N, then multiply N by the rounded down result, then take x minus the multiplied result. – Kim Dong Jul 25 '19 at 8:55
The method of transversal copying by a CNOT is solid and you can stack up the building blocks for the quantum part of shor algorithm. However ad-hoc circuit synthesis based at a pattern of the function truth table could be efficiënt in some cases as described in arxiv 1310.6446v2. First case is for factoring N=15 and base a = 2 with period r = 4. In exponential formulation we have $$f(x)=a^{x}\text{ mod }15$$ With values
Setup a truth table for input x between 0 and 3. Input x is represented by 2 qubits x2 and x1. Output y is represented by 4 qubits y4,y3,y2,y1 For example if x = 2 then x2=1 and x1=0 then only y3 =1 so put a NOT on this line.
Furthermore underlined entries in table 1 are the ones that are modified by a toffoli gate to get the right output in the circuit according the table 1. We can use this module in the overall algorithm according to https://arxiv.org/pdf/0705.1398.pdf | {"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": 31, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8247562050819397, "perplexity": 319.3677557525414}, "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/1593655901509.58/warc/CC-MAIN-20200709193741-20200709223741-00248.warc.gz"} |
https://math.stackexchange.com/questions/245237/extension-of-a-uniformly-continuous-function-between-metric-spaces | # Extension of a Uniformly Continuous Function between Metric Spaces
Let $(X,d)$ and $(Y,d')$ be metric spaces with $(Y,d')$ complete. Let $A\subseteq X$. I need to show that if $f:A\to Y$ is uniformly continuous, then $f$ can be uniquely extended to $\bar{A}$ maintaining the uniform continuity.
My attempt at this has involved taking each point $a\in \bar{A}-A$ and forming a Cauchy sequence to it by considering open balls $B_{\frac{1}{n}}(a)-B_{\frac{1}{n+1}}(a)$ beginning with $n$ large enough so there is such a sequence, and defining $g(a)$ to be the limit in $Y$. The uniqueness seems to be obvious just by thinking about the uniqueness of limits (referring to the sequence in $Y$), but I have to admit I don't know how to rigorously show it. The uniform continuity seems natural, but I don't know how to show it, either.
This seems to be correct, but I'm not entirely sure... Any help would be very appreciated!
You may want to see the answers for this question, which answer yours, Extending a function by continuity from a dense subset of a space.
I built the proof myself based on Srivatsan's answer for that question. If anybody still needs it, here it goes:
Theorem
If $X$ and $Y$ are metric spaces and $f:S \to Y$ is uniformly continuous with $S$ dense in $X$, and $Y$ is complete, then there exists a unique continuous extension of $f$ in $\overline{S}$ which by the way is uniformly continuous.
Proof
Let $d$ and $D$ be the metrics of $X$ and $Y$ respectively.
Let $g:\overline{S} \to Y$ be given by $g(a) = \lim f(x_n)$, where $(x_n)$ is any sequence of points in $S$, with $x_n \to a$.
$g$ is well defined:
• $\lim f(x_n)$ exists:
Let $\varepsilon > 0$. Because of the uniform continuity of $f$, there exists $\delta>0$ such that for every $a,b \in S$, if $d(a,b) < \delta$, then $D(f(a),f(b)) < \varepsilon$.
Since $x_n \to a$, $(x_n)$ is Cauchy, there exists $N \in \mathbb{Z}^{+}$ such that if $n,m \geq N$, $d(x_n,x_m)<\delta$.
Hence, if $n,m \geq N$, $D(f(x_n),f(x_m))<\varepsilon$. Then $(f(x_n))$ is Cauchy, and since $Y$ is complete, $\lim f(x_n)$ exists.
• If $x_n \to a$ and $y_n \to a$ then $\lim f(x_n) = \lim f(y_n)$:
Let $(z_n) = (x_1,y_1,x_2,y_2,...)$. If $\varepsilon>0$, there exists $N \in \mathbb{Z}^{+}$ with $d(x_n,a) < \varepsilon$ and $d(y_n,a) < \varepsilon$ for each $n \geq N$.
Consequently, if $n \geq 2N$, then $n/2,(n+1)/2 \geq N$ and so, if $n$ is even, $d(z_n,a) = d(y_{n/2},a) < \varepsilon$, and if $n$ is odd, $d(z_n,a) = d(y_{(n+1)/2},a) < \varepsilon$. Therefore $z_n \to a$.
So, $\lim f(z_n)$ exists and since $(f(x_n))$ and $(f(y_n))$ are subsequences of $(f(z_n))$, $\lim f(x_n) = \lim f(z_n) = \lim f(y_n)$.
$g$ is an extension of $f$:
• If $a \in S$, $a \to a$, therefore $g(a) = \lim f(a) = f(a)$.
$g$ is uniformly continuous:
• Let $\varepsilon > 0$. Since $f$ is uniformly continuous, there exists $\delta > 0$ such that $D(f(a),f(b))<\varepsilon/3$ for every $a,b \in S$ with $d(a,b)<\delta$.
Let $a,b \in \overline{S}$ with $d(a,b)<\delta/3$.
There exist sequences in $S$, $(x_n)$ and $(y_n)$ with $x_n \to a$ and $y_n \to b$. Since $x_n \to a$ and $y_n \to b$, there exists $N_1 \in \mathbb{Z}^{+}$ with $d(x_n,a)<\delta/3$ and $d(y_n,b)<\delta/3$ for every $n\geq N_1$.
If $n \geq N_1$, $d(x_n,y_n) \leq d(x_n,a) + d(a,b) + d(b,y_n) < \delta$ and so, $D(f(x_n),f(y_n)) < \varepsilon/3$.
Also, since $f(x_n) \to g(a)$ and $f(y_n) \to g(b)$, there exists $N_2 \in \mathbb{Z}^{+}$ with $D(f(x_n),g(a))<\varepsilon/3$ and $D(f(y_n),g(b))<\varepsilon/3$ for every $n\geq N_2$.
Then, if $N=max\{N_1,N_2\}$, $D(g(a),g(b)) \leq D(g(a),f(x_N)) + D(f(x_N),f(y_N)) + D(f(y_N),g(b)) < \varepsilon.$
$g$ is unique:
• If $h$ is a continuous extension of $f$ in $\overline{S}$ and $a\in \overline{S}$, there exists a sequence $(x_n)$ in $S$ with $x_n \to a$. Since $h$ is continuous, $h(x_n) \to h(a)$. But $(h(x_n)) = (f(x_n))$ and $f(x_n) \to g(a)$, then $h(a) = g(a)$ must hold.
• When proving that $\lim f(x_n)$ exists, you need a sequence $(x_n)$ converging to $a$. AFAIK, constructing it requires $a\in\overline{S}$ and $d$. May 13, 2016 at 14:53
If $$a \in \overline{A}$$ then $$a = \lim_n a_n$$ where $$a _n \in A$$. Then, $$a_n$$ is Cauchy and as $$f$$ is uniformily continuous n $$A$$, $$f(a_n)$$ is Cauchy and as $$(Y,d´)$$ is complete you can define $$f(a) : = \lim_n f(a_n)$$. For example, if $$b \in A$$ you have that $$\begin{eqnarray} d(f(a),f(b)) &\le& d(f(a_n),f(b)) + d(f(a),f(a_n)) \end{eqnarray}$$ and by the definition of uniform continuity it is clear that the extension of $$f$$ is uniformily continuous. Analogously, if $$b \in \overline{A}, b =\lim_n b_n$$ where $$b_n \in \overline{A}$$ and $$\begin{eqnarray} d(f(a),f(b)) &\le& d(f(a_n),f(a)) + d(f(b_n),f(a_n)) + d(f(b_n), f(b)) \end{eqnarray}$$
[This is, except for a few details, same as Carlos Pinzon's answer above]
Let $${ X,Y }$$ be metric spaces and $${ A (\subseteq X ) \overset{f}{\to} Y }$$ a continuous map. Both $${ d _X, d _Y }$$ will be denoted by $${ d }$$ for brevity.
The following question arises. Can we impose a "general enough" constraint such that : There is a unique continuous map $${ \overline{A} \overset{\overline{f}}{\to} Y }$$ with the property $${ \overline{f} \vert _{A} = f }$$ ?
Turns out imposing that $${ f }$$ is uniformly continuous and $${ Y }$$ is complete will do. Further, in this case $${ \overline{f} }$$ is uniformly continuous.
$${ \underline{ \textbf{Defining} \text{ } \overline{f} } }$$
For every $${ p \in \overline{A}, }$$ there exists a seq $${ (x _n) \subseteq A }$$ with $${ d(x _n, p) \to 0 }.$$
Let $${ p \in \overline{A} }.$$ For every $${ n \in \mathbb{Z} _{\gt 0} },$$ pick an $${ x _n \in A }$$ with $${ d(x _n, p) \lt \frac{1}{n} }.$$
So if we ensure that
$${ {\color{purple}{(1)}} }$$ If $${ (x _n) \subseteq A }$$ and $${ d(x _n, p) \to 0 }$$ for some $${ p \in X }$$ then $${ \lim _{n \to \infty} f(x _n) }$$ exists
$${ {\color{purple}{(2)} } }$$ If $${ \lbrace (x _n) \subseteq A; d(x _n, p) \to 0 \rbrace }$$ and $${ \lbrace (y _n) \subseteq A; d(y _n, p) \to 0 \rbrace }$$ for some $${ p \in X }$$ then $${ \lim _{n \to \infty} f(x _n) = \lim _{n \to \infty} f(y _n) }$$
then we can define a map $${ \overline{A} \overset{\overline{f}}{\to} Y }$$ naturally by : Let $${ p \in \overline{A} }.$$ Pick an $${ (x _n) \subseteq A }$$ with $${ d(x _n, p) \to 0 },$$ and set $${ \overline{f}(p) := \lim _{n \to \infty} f(x _n) }.$$
Ensuring $${ {\color{purple}{(1)} } }$$: Let $${ (x _n) \subseteq A }$$ and $${ d(x _n, p) \to 0 }$$ for some $${ p \in X }.$$ So $${ (x _n) }$$ is Cauchy. So imposing that $${ f }$$ is uniformly continuous ensures $${ (f(x _n)) }$$ is Cauchy.
Let $${ \epsilon \gt 0 }.$$ Pick $${ \delta \gt 0 }$$ such that $${ x,y \in A },$$ $${ d(x,y) \lt \delta }$$ implies $${ d(f(x), f(y)) \lt \epsilon }.$$ Pick $${ N }$$ such that $${ d(x _m, x _n) \lt \delta }$$ whenever $${ m, n \geq N }.$$ Now $${ d(f(x _m), f(x _n)) \lt \epsilon }$$ whenever $${ m, n \geq N }.$$
Further imposing that $${ Y }$$ is complete ensures $${ (f (x _n)) }$$ is convergent.
Ensuring $${ {\color{purple}{(2)} } }$$: Say $${ f }$$ is uniformly continuous and $${ Y }$$ is complete, which ensures $${ {\color{purple}{(1)} } }.$$ Let $${ (x _n), (y _n) \subseteq A }$$ with $${ d(x _n, p) \to 0 },$$ $${ d(y _n, p) \to 0 }$$ for some $${ p \in X }.$$ So $${ \lim _{n \to \infty} f(x _n) = \ell _1 }$$ and $${ \lim _{n \to \infty} f(y _n) = \ell _2 }$$ exist. We'll show $${ \ell _1 = \ell _2 }.$$
As $${ d(\ell _1, \ell _2) }$$ $${ \leq \underbrace{ d(\ell _1, f(x _n)) } _{\to 0} }$$ $${ + d(f (x _n), f(y _n)) }$$ $${ + \underbrace{ d(f(y _n), \ell _2) } _{\to 0} },$$ it suffices to show $${ d(f(x _n), f(y _n)) \to 0 }.$$
Let $${ \epsilon \gt 0 }.$$ Pick $${ \delta \gt 0 }$$ such that $${ x,y \in A, }$$ $${ d(x,y) \lt \delta }$$ implies $${ d(f(x), f(y)) \lt \epsilon }.$$
From $${ d(x _n, y _n) }$$ $${ \leq d(x _n, p) + d(p, y _n) \to 0 },$$ we have $${ d(x _n, y _n) \to 0 }.$$ So pick $${ N }$$ such that $${ d(x _n, y _n ) \lt \delta }$$ for all $${ n \geq N }.$$ Now $${ d(f(x _n), f(y _n)) \lt \epsilon }$$ for all $${ n \geq N },$$ as needed.
$${ \underline{ \textbf{Properties of} \text{ } \overline{f} } }$$
From now, $${ A \overset{f}{\to} Y }$$ is uniformly continuous and $${ Y }$$ is complete, and $${ \overline{A} \overset{\overline{f}}{\to} Y }$$ is as defined above.
Note $${ \overline{f} \vert _{A} = f }.$$ We'll show $${ \overline{f} }$$ is uniformly continuous.
Let $${ \epsilon \gt 0 },$$ and $${ p, q \in \overline{A} }.$$ We want a $${ \delta \gt 0 }$$ independent of $${ p,q },$$ such that $${ d(p,q) \lt \delta }$$ implies $${ d(\overline{f}(p), \overline{f}(q) ) \lt \epsilon }.$$
Pick $${ \eta \gt 0 }$$ such that $${ x,y \in A, d(x,y) \lt \eta }$$ implies $${ d(f(x), f(y)) \lt \frac{\epsilon}{10} }.$$
We'll show $${ \delta := \frac{1}{10} \min\lbrace \epsilon, \eta \rbrace }$$ will work.
Suppose $${ {\color{green}{d(p,q) \lt \delta}} }.$$
Pick seqs $${ (x _n), (y _n) \subseteq A }$$ with $${ d(x _n, p) \to 0 },$$ $${ d(y _n, q) \to 0 }.$$ Now $${ d(f(x _n), \overline{f}(p)) \to 0 }$$ and $${ d(f(y _n), \overline{f}(q)) \to 0 }$$ too.
So pick an $${ m }$$ such that $${ {\color{green}{d(x _m, p)}} },$$ $${ {\color{green}{d(y _m, q)}} },$$ $${ {\color{green}{d(f(x _m), \overline{f}(p))}} }$$ and $${ {\color{green}{d(f(y _m), \overline{f}(q))}} }$$ are all $${ {\color{green}{\lt \delta}} }.$$
Now $${ d(\overline{f}(p), \overline{f}(q)) }$$ $${ \leq d(\overline{f}(p), f(x _m)) }$$ $${ + d(f(x _m), f(y _m)) }$$ $${ + d(f (y _m), \overline{f}(q)) }$$ $${ \leq 2\delta + {\color{red}{d(f(x _m), f(y _m))}} . }$$
But as $${ d(x _m, y _m) }$$ $${ \leq d(x _m, p) }$$ $${ + d(p,q) }$$ $${ + d(q, y _m) }$$ $${ \lt 3 \delta \lt \eta },$$ we have $${ {\color{red}{d(f (x _m), f(y _m)) \lt \frac{\epsilon}{10}}} }.$$ This gives $${ d(\overline{f}(p), \overline{f}(q)) }$$ $${ \leq 2\delta + \frac{\epsilon}{10} }$$ $${ \lt \epsilon }.$$
Finally $${ d(\overline{f}(p), \overline{f}(q)) \lt \epsilon },$$ as needed.
[This shows we could've set $${ \delta := \frac{1}{3} \min \lbrace \epsilon, \eta \rbrace }$$ to begin with. But $${ \frac{1}{10} }$$ was a "safer factor" to carry out the estimates]
[Uniqueness] Suppose $${ \overline{A} \overset{g}{\to} Y }$$ is continuous and $${ g \vert _{A} = f }.$$ We see $${ g = \overline{f} }$$ :
Let $${ p \in \overline{A}. }$$ Pick a seq $${ (x _n) \subseteq A }$$ with $${ d(x _n, p) \to 0 }.$$ As $${ g }$$ is continuous, $${ g(p) = \lim _{n \to \infty} g(x _n) }.$$ But $${ g(x _n) = f(x _n) }$$ and $${ \overline{f}(p) = \lim _{n \to \infty} f(x _n). }$$ So $${ \overline{f}(p) = g(p) },$$ as needed. | {"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": 151, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9952040314674377, "perplexity": 66.0005495296823}, "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-2022-27/segments/1656103271864.14/warc/CC-MAIN-20220626192142-20220626222142-00209.warc.gz"} |
https://physics.stackexchange.com/questions/73763/what-is-the-electric-charge-of-the-sun-and-its-corona?noredirect=1 | # What is the electric charge of the Sun and its corona?
What is the net electric charge (in magnitude and sign) of the Sun and its corona?
According to this post and references there, the charge of the Sun is positive, the magnitude is estimated as 77 Coulombs, or about 1 electron per million tons of matter.
The reason for this is that
The net global charge on the Sun comes about because electrons, being rather less massive than protons, are more able to escape the sun as part of the solar wind. The net charge achieved is a result of the balance between the forces that eject the solar wind, which push electrons more efficiently then protons, and the attractive force on the electrons of the net positive charge that results. Equilibrium of these forces establishes the allowed net charge.
• @honeste_vivere: I respectfully disagree with your estimate. An electric field of a point charge $q$ at a distance $r$ is $q/(4\pi\epsilon_0 r^2)$. When I substitute $q=77$, $\epsilon_0=8.85\times 10^{-12}$, $r=7\times 10^8$ (the Sun radius, SI units everywhere), I get $1.4 \mu V/m$. – akhmeteli May 3 '16 at 1:33
• Yes, you are correct... I forgot to square the $8 \ R_{s}$ factor. My colleagues eventually corrected my mistake. – honeste_vivere May 3 '16 at 12:20 | {"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": 0.9235122799873352, "perplexity": 416.00651624499505}, "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-2019-35/segments/1566027321351.87/warc/CC-MAIN-20190824172818-20190824194818-00230.warc.gz"} |
https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_ChemPRIME_(Moore_et_al.)/06%3A_Chemical_Bonding_-_Electron_Pairs_and_Octets/6.17%3A_Polyatomic_Ions | # 6.17: Polyatomic Ions
Our discussion of ionic compounds was confined to monatomic ions. However, more complex ions, containing several atoms covalently bonded to one another, but having a positive or negative charge, occur quite frequently in chemistry. The charge arises because the total number of valence electrons from the atoms cannot produce a stable structure. With one or more electrons added or removed, a stable structure results. Well-known examples of such polyatomic ions are the sulfate ion (SO42–),
the hydroxide ion (OH),
the hydronium ion (H3O+),
and the ammonium ion (NH4+).
The atoms in these ions are joined together by covalent electron-pair bonds, and we can draw Lewis structures for the ions just as we can for molecules. The only difference is that the number of electrons in the ion does not exactly balance the sum of the nuclear charges. Either there are too many electrons, in which case we have an anion, or too few, in which case we have a cation.
Consider, for example, the hydroxide ion (OH) for which the Lewis structure is
A neutral molecule containing one O and one H atom would contain only seven electrons, six from O and one from H. The hydroxide ion, though, contains an octet of electrons, one more than the neutral molecule. The hydroxide ion must thus carry a single negative charge. In order to draw the Lewis structure for a given ion, we must first determine how many valence electrons are involved. Suppose the structure of H3O+ is required. The total number of electrons is obtained by adding the valence electrons for each atom, 6 + 1 + 1 + 1 = 9 electrons. We must now subtract 1 electron since the species under consideration is not H3O but H3O+. The total number of electrons is thus 9 – 1 = 8. Since this is an octet of electrons, we can place them all around the O atom. The final structure then follows very easily:
In more complicated cases it is often useful to calculate the number of shared electron pairs before drawing a Lewis structure. This is particularly true when the ion in question is an oxyanion (i.e., a central atom is surrounded by several O atoms). A well-known oxyanion is the carbonate ion, which has the formula CO32–. (Note that the central atom C is written first, as was done earlier for molecules.) The total number of valence electrons available in CO32– is
$$4 \text{(for C)} + 3 \times 6 \text{(for O)} + 2 \text{(for the –2 charge)} = 24$$
We must distribute these electrons over 4 atoms, giving each an octet, a requirement of 4 × 8 = 32 electrons. This means that 32 – 24 = 8 electrons need to he counted twice for octet purposes; i.e., 8 electrons are shared. The a ion thus contains four electron-pair bonds. Presumably the C atom is double-bonded to one of the O’s and singly bonded to the other two:
In this diagram the 4C electrons have been represented by dots, the 18 O electrons by ×’s, and the 2 extra electrons by colored dots, for purposes of easy reference. Real electrons do not carry labels like this; they are all the same.
There is a serious objection to the Lewis structure just drawn. How do the electrons know which oxygen atom to single out and form a double bond with, since there is otherwise nothing to differentiate the oxygens? The answer is that they do not. To explain the bonding in the CO32– ion and some other molecules requires an extension of the Lewis theory. We pursue this matter further when we discuss resonance. Now we end with an example.
Example $$\PageIndex{1}$$ : Lewis Structure
Draw a Lewis structure for the sulfite ion, SO32–.
Solution The safest method here is to count electrons. The total number of valence electrons available is
6(for S) + 3 × 6(for O) + 2(for the charge) = 26 To make four octets for the four atoms would require 32 electrons, and so the difference, 32 – 26 = 6, gives the number of shared electrons. There are thus only three electron-pair bonds in the ion. The central S atom must be linked by a single bond to each O atom.
Note that each of the S—O bonds is coordinate covalent. | {"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": 0.849679172039032, "perplexity": 913.5912212097204}, "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-2021-39/segments/1631780057913.34/warc/CC-MAIN-20210926175051-20210926205051-00286.warc.gz"} |
https://scholars.ttu.edu/en/publications/measurement-of-the-differential-cross-sections-for-the-associated | # Measurement of the differential cross sections for the associated production of a W boson and jets in proton-proton collisions at s =13 TeV
(CMS Collaboration)
Research output: Contribution to journalArticlepeer-review
22 Scopus citations
## Abstract
A measurement of the differential cross sections for a W boson produced in association with jets in the muon decay channel is presented. The measurement is based on 13 TeV proton-proton collision data corresponding to an integrated luminosity of 2.2 fb-1, recorded by the CMS detector at the LHC. The cross sections are reported as functions of jet multiplicity, jet transverse momentum pT, jet rapidity, the scalar pT sum of the jets, and angular correlations between the muon and each jet for different jet multiplicities. The measured cross sections are in agreement with predictions that include multileg leading-order (LO) and next-to-LO matrix element calculations interfaced with parton showers, as well as a next-to-next-to-LO calculation for the W boson and one jet production.
Original language English 072005 Physical Review D 96 7 https://doi.org/10.1103/PhysRevD.96.072005 Published - Oct 27 2017
## Fingerprint
Dive into the research topics of 'Measurement of the differential cross sections for the associated production of a W boson and jets in proton-proton collisions at s =13 TeV'. Together they form a unique fingerprint. | {"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": 0.9940710663795471, "perplexity": 1631.2124030240966}, "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/1627046154321.31/warc/CC-MAIN-20210802141221-20210802171221-00450.warc.gz"} |
http://jcore-reference.highwire.org/highwire/markup/4976/expansion?width=1000&height=500&iframe=true&postprocessors=highwire_tables%2Chighwire_reclass%2Chighwire_figures%2Chighwire_math%2Chighwire_inline_linked_media%2Chighwire_embed | Table 2.
Examples of Goals Set for 3 Different Participants in the Intervention Group Using Goal Attainment Scaling (GAS)a
• a The current level of skill attainment (determined within the first 3 intervention sessions) was set at a score of −1; the expected outcome at the end of the intervention was set at a score of 0; a somewhat better outcome than expected was set at 1; a much better outcome than expected was set at a score of 2; a worse outcome than expected was set at a score of −2. The scores at the end of the intervention were determined by the intervention therapist, either based on direct observation (all 3 examples here) or by participant interview (if self-report). | {"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": 0.8182873129844666, "perplexity": 1647.6611881974065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764501555.34/warc/CC-MAIN-20230209081052-20230209111052-00386.warc.gz"} |
http://mathhelpforum.com/number-theory/148085-solved-modified-wallis-example-print.html | [SOLVED] Modified Wallis Example
• Jun 7th 2010, 03:23 AM
Samson
[SOLVED] Modified Wallis Example
Hello all, I have a quick question concering a Wallis Example (click here for background information on Wallis problems). My book states that positive integer solutions exist within many Wallis type examples, followed by a list of them. It does not however list the solutions to them.
I've google searched a good bit and I was able to find how a few problems matched up, but I can't seem to find the solution to (x^3)+(y^3)=35 using methodology that others used to solve their problems.
Can anyone explain how we find all the positive integers solutions to that problem ( x^3 + y^3 = 35 ) ?
All help is appreciated!
• Jun 7th 2010, 03:57 AM
undefined
Quote:
Originally Posted by Samson
Hello all, I have a quick question concering a Wallis Example (click here for background information on Wallis problems). My book states that positive integer solutions exist within many Wallis type examples, followed by a list of them. It does not however list the solutions to them.
I've google searched a good bit and I was able to find how a few problems matched up, but I can't seem to find the solution to (x^3)+(y^3)=35 using methodology that others used to solve their problems.
Can anyone explain how we find all the positive integers solutions to that problem ( x^3 + y^3 = 35 ) ?
All help is appreciated!
I don't understand how number theory really plays into this particular problem. There are very few possibilities. It can be seen with very little work that (x,y) = (2,3) is the only solution for which $x \le y$.
• Jun 7th 2010, 04:00 AM
Samson
Quote:
Originally Posted by undefined
I don't understand how number theory really plays into this particular problem. There are very few possibilities. It can be seen with very little work that (x,y) = (2,3) is the only solution for which $x \le y$.
Is it required that $x \le y$ ? That is the only solution possible? How did you reach that conclusion?
• Jun 7th 2010, 04:08 AM
undefined
Quote:
Originally Posted by Samson
Is it required that math]x \le y[/tex] ? That is the only solution possible? How did you reach that conclusion?
I let $x \le y$ to avoid having to count (x,y) = (2,3) and (x,y) = (3,2) as two separate solutions.
Just plug in numbers. There are very few choices of (x,y) that do not exceed 35.
For example, if (x,y) = (3,4) then x^3 + y^3 is already quite a bit greater than 35.
• Jun 7th 2010, 04:10 AM
Samson
Quote:
Originally Posted by undefined
I let $x \le y$ to avoid having to count (x,y) = (2,3) and (x,y) = (3,2) as two separate solutions.
Just plug in numbers. There are very few choices of (x,y) that do not exceed 35.
For example, if (x,y) = (3,4) then x^3 + y^3 is already quite a bit greater than 35.
Okay, thank you! So if I was to write a formal proof of this, I would write (x,y,)=(2,3) and (x,y,)=(3,2) or would it be more proper to write (x,y)=(2,3) where $x \le y$ ? (This is just for formality purposes)
• Jun 7th 2010, 04:16 AM
undefined
Quote:
Originally Posted by Samson
Okay, thank you! So if I was to write a formal proof of this, I would write (x,y,)=(2,3) and (x,y,)=(3,2) or would it be more proper to write (x,y)=(2,3) where $x \le y$ ? (This is just for formality purposes)
This is up to preference. Letting $x \le y$ is a mathematical shorthand of sorts, see: Without loss of generality.
• Jun 7th 2010, 04:25 AM
Samson
Thank you undefined! | {"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": 8, "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": 0.9048469066619873, "perplexity": 637.672783808444}, "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-2017-39/segments/1505818690307.45/warc/CC-MAIN-20170925021633-20170925041633-00466.warc.gz"} |
http://mathhelpforum.com/calculus/131806-average-value-function-help.html | # Math Help - Average value function help
1. ## Average value function help
Hey
Can you please tell me if the integration in my photo is correct
Thank you
Muchas gracias
Attached Thumbnails
Welcome to Math Help Forum!
Hey
Can you please tell me if the integration in my photo is correct
Thank you
Muchas gracias
The lower limit of the integral in each case is $0$, and the upper limit is $a$. This will give a positive sign in the answer.
Also, each result can be simplified, by dividing by $a$.
Thank you for welcome
g(x) functions answer would be a^2/(3), and the first part f(x) works out a/(2) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "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": 0.8942787051200867, "perplexity": 1232.9972048817822}, "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/1454701160822.87/warc/CC-MAIN-20160205193920-00101-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://wsw.academickids.com/encyclopedia/index.php/Conditional_probability | # Conditional probability
This article defines some terms which characterize probability distributions of two or more variables.
Conditional probability is the probability of some event A, assuming event B. Conditional probability is written P(A|B), and is read "the probability of A, given B".
Joint probability is the probability of two events in conjunction. That is, it is the probability of both events together. The joint probability of A and B is written [itex]P(A \cap B)[itex] or [itex]P(A, \ B).[itex]
Marginal probability is the probability of one event, ignoring any information about the other event. Marginal probability is obtained by summing (or integrating, more generally) the joint probability over the ignored event. The marginal probability of A is written P(A), and the marginal probability of B is written P(B).
In these definitions, note that there need not be a causal or temporal relation between A and B. A may precede B, or vice versa, or they may happen at the same time. A may cause B, or vice versa, or they may have no causal relation at all.
## Relations
If A and B are events, and P(B) > 0, then
[itex]P(A\mid B)=\frac{P(A \cap B)}{P(B)}.[itex]
Equivalently, we have
[itex]P(A \cap B)=P(A\mid B)\cdot P(B).[itex]
If [itex]P(A \cap B) = P(A)P(B)[itex], or equivalently, [itex]P(A|B) = P(A)[itex], then we say that [itex]A[itex] and [itex]B[itex] are independent.
If [itex]B[itex] is an event and [itex]P(B) > 0[itex], then the function [itex]Q[itex] defined by [itex]Q(A) = P(A|B)[itex] for all events [itex]A[itex] is a probability measure.
If [itex]P(B)=0[itex], then [itex]P(A|B)[itex] is left undefined.
Conditional probability is more easily calculated with a decision tree.
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• Clip Art (http://classroomclipart.com) | {"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": 0.9643217921257019, "perplexity": 1684.024095234796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989856.11/warc/CC-MAIN-20210511184216-20210511214216-00415.warc.gz"} |
http://mathhelpforum.com/algebra/279701-integral-dx-2-bx-x-2-1-5-a.html | # Thread: integral of dx/(a^2 + bx + x^2)^1.5
1. ## integral of dx/(a^2 + bx + x^2)^1.5
Hi
Can anybody help me with this one : integral of dx/(a^2 + bx + x^2)^1.5
Thanks
2. ## Re: integral of dx/(a^2 + bx + x^2)^1.5
Originally Posted by Roger44
Hi
Can anybody help me with this one : integral of dx/(a^2 + bx + x^2)^1.5
Have a look HERE.
3. ## Re: integral of dx/(a^2 + bx + x^2)^1.5
You would complete the square in the polynomial and then look for a trigonometric substitution, I think.
4. ## Re: integral of dx/(a^2 + bx + x^2)^1.5
Plato, the link you gave me was perfect. Two other sites I had tried before didn't find a solution. Many thanks
The formula I submitted gives the lumens/sq m on a point on the floor coming from a bulb fixed to the wall, and this integral will give me lumens/sq m on a point on the floor coming from a horizontal strip of light fixed to the wall.
5. ## Re: integral of dx/(a^2 + bx + x^2)^1.5
Hello again
Wolfram gives this solution for another integral but as you can see only if a>b. Or this is a real world situation where a will be rarely less than b. Any workaround? | {"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": 0.9960699677467346, "perplexity": 1577.247927819674}, "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-2018-51/segments/1544376824338.6/warc/CC-MAIN-20181213010653-20181213032153-00211.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=4242050 | Solving heat equation for heat-pulse in a point on the surface
by Jbari
Tags: heat equation, pde, semi-infinite solid
P: 8 Hi everybody, I'm trying to find a solution for the 3D heat equation for pulsed surface heating of a semi-infinte solid with insulated surface. I know the method of reflection is required, and that a point source in an infinite solid gives the following solution: $U(x,y,z,t)= \frac{Q}{8\sqrt{(πκt)^3}}*e^{-\frac{x^2+y^2+z^2}{4κt}}$ Where κ is thermal conductivity and Q is a measure for the strength of the heat source. However, I have only found a solution for a semi-infinite solid with surface temperature zero and a heat source inside the solid. In my case however, the heat source is on the surface, (let's say in point (0,0,0)), hence surface temperature cannot be zero, yet to make matters (a little less) complicated, let's assume a perfectly insulated surface with no heat transfer.. Thanks in advance for help, or tips for usefull literature
HW Helper
Thanks
PF Gold
P: 4,394
Quote by Jbari Hi everybody, I'm trying to find a solution for the 3D heat equation for pulsed surface heating of a semi-infinte solid with insulated surface. I know the method of reflection is required, and that a point source in an infinite solid gives the following solution: $U(x,y,z,t)= \frac{Q}{8\sqrt{(πκt)^3}}*e^{-\frac{x^2+y^2+z^2}{4κt}}$ Where κ is thermal conductivity and Q is a measure for the strength of the heat source. However, I have only found a solution for a semi-infinite solid with surface temperature zero and a heat source inside the solid. In my case however, the heat source is on the surface, (let's say in point (0,0,0)), hence surface temperature cannot be zero, yet to make matters (a little less) complicated, let's assume a perfectly insulated surface with no heat transfer.. Thanks in advance for help, or tips for usefull literature
Your solution looks OK to me. The partial derivative of U with respect to z is zero, so the surface heat flux (normal to the x-y plane) is zero. Your semi-infinite solid lies above the x-y plane (z > = 0).
P: 8
Quote by Chestermiller Your solution looks OK to me. The partial derivative of U with respect to z is zero, so the surface heat flux (normal to the x-y plane) is zero. Your semi-infinite solid lies above the x-y plane (z > = 0).
Thanks for the quick response, but are you sure this is correct? Because this is the solution for a point source inside an infinite solid, and it looks quite odd to me that a surface heat source in a semi-infinite solid has te same solution. I mean, intuitively, one would expect a different heat propagation in this semi-infinite solid since there is only one way the heat is dissipated (in the part where z>=0) and the heat is 'blocked' in the other direction (where z<0)? Or is this assumption not correct?
HW Helper
Thanks
PF Gold
P: 4,394
Solving heat equation for heat-pulse in a point on the surface
Think of it first as an infinite solid. You have a sudden instantaneous spherically symmetric injection of heat at the origin. Then you let the heat diffuse away. Half the previously injected heat goes up, and half the previously injected heat goes down. So there is symmetry of the temperature distribution with respect to the x-y plane (z = 0). No heat crosses this boundary (after t = 0). This is exactly what your solution tells you. The upward heat flux at z = 0 is -kdU/dz, but dU/dz is zero at all times after t = 0.
P: 8 Thanks a lot for the extra information, your explanation looks correct indeed, and I understand that in this case my equation covers the problem since I cannot see a theoretical error in your explanation . However, I still find it odd that there is no effect of the insulated boundary (although I might of course be mistaken): Isn't the heat that normally goes down in the z<0 area (when the solid is infinite), 'trapped' due to thermal insulation of the surface, resulting in the fact that it is dissipated to the other side, leading to a higher heat input in the z>=0 area? And if this is the case, does this only affect the factor Q (heat input) in the equation?
Engineering
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Thanks
P: 6,339
Quote by Jbari However, I still find it odd that there is no effect of the insulated boundary
The point is that, because of the symmetry of the heat flow, there would be no heat flow across the "boundary" plane even if the solid was infinite, not semi-infinite.
If there is no heat flow through a surface, replacing that surface with an insulated boundary doesn't change the heat flow.
P: 8 Ok, thanks a lot! I didn't realise I had the correct solution lying around all this time :).
HW Helper
Thanks
PF Gold
P: 4,394
Quote by AlephZero The point is that, because of the symmetry of the heat flow, there would be no heat flow across the "boundary" plane even if the solid was infinite, not semi-infinite. If there is no heat flow through a surface, replacing that surface with an insulated boundary doesn't change the heat flow.
I like your explanation much better than mine. I was struggling to say just this, but was unable to articulate it as simply and concisely.
Chet
P: 305
Quote by Chestermiller I like your explanation much better than mine. I was struggling to say just this, but was unable to articulate it as simply and concisely. Chet
Very counter-intuitive. So adding a thin insulating plane to an infinite solid at z=0 would have no influence at all on the T distribution?
P: 8 About the insulation: in my question I assumed a perfect insulation with no heat transfer across the boundary, but what if we try to make the situation more realistic and insulation is not perfect.. So instead of a semi-infinite solid, we now actually have 'two semi-infinite solids with their surfaces in thermal contact' (or something like that). Does this change anything for the heat transfer? Again, intuitively, one would assume it does, but I cannot rely on my intuition :). Furthermore, small differences are quite important here, because I'm also intrested in the heat propagation on the boundary plane. Hope this isn't too much to ask?
Sci Advisor HW Helper Thanks PF Gold P: 4,394 Let me say this back so that I understand correctly. You now have 2 semi-infinite solids with an imperfect insulating material of thickness ?? between then, and you release the heat at a point on the surface of one of the solids, but not at the mirror image point on the other solid. And the thermal properties of the insulating material is different from that of the two semi-infinite solids. Correct? Chet
P: 8 Ah no, sorry for the confusion, there is 1 semi-infinite solid, and the other semi infinite solid IS the insulating material.. So 2 materials in contact with eachother, with different thermal properties, and with a point heat source on the boundary surface between them. I hope this makes it more clear to you?
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PF Gold
P: 4,394
Quote by Jbari Ah no, sorry for the confusion, there is 1 semi-infinite solid, and the other semi infinite solid IS the insulating material.. So 2 materials in contact with eachother, with different thermal properties, and with a point heat source on the boundary surface between them. I hope this makes it more clear to you?
Is the other semi-infinite solid a perfect insulator?
P: 8
Quote by Chestermiller Is the other semi-infinite solid a perfect insulator?
No, it is a 'realistic' insulator, so just a material with a lower thermal conductivity than the other semi-infinite solid, so from the mathematical point of view it could be any material..
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PF Gold
P: 4,394
Quote by Jbari No, it is a 'realistic' insulator, so just a material with a lower thermal conductivity than the other semi-infinite solid, so from the mathematical point of view it could be any material..
If the thermal conductivities are not equal, you can still have zero heat flux at the interface if the thermal diffusivities are equal. The heat pulse will just initially partition between the two slabs in proportion to the thermal conductivities. After that, no heat flow will occur across the interface.
If the thermal diffusivities are unequal, there will be heat flow across the interface. I'm not sure whether this problem has an analytic solution. Of course, it can always be solved numerically.
P: 305 What confuses me is how come the OP's original equation does not have a diffusion term at all? Shouldn't k, Cp and density always occur in combination (i.e. Diffusivity) in the solutions of the heat equation?
P: 8
Quote by rollingstein What confuses me is how come the OP's original equation does not have a diffusion term at all? Shouldn't k, Cp and density always occur in combination (i.e. Diffusivity) in the solutions of the heat equation?
You are correct, I ment κ to be thermal diffusivity (= K/(ρ*Cp)) but I defined it incorrectly in my explanation as thermal conductivity..
And to return to the problem: the thermal diffusivities are NOT equal, but I would still like to find an analytical solution (if possible of course). Any suggestions on how/where to find it (maybe in literature, but my search has been fruitless untill now).
Two reasons. First of all, in the OP's relationship, κ (kappa) is the thermal diffusivity, not, as he stated, the thermal conductivity k. Secondly, the term involving Q is not quite right. The units don't properly give temperature. I'm too lazy to look up what the correct expression for what this term should be, but, at the very least, there should be a k (thermal conductivity) in the denominator (if U has units of temperature and Q has units of energy). I'm guessing that the denominator should be a constant times $kt\sqrt{\kappa t})$. This would give the correct units for temperature. | {"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": 0.8282049894332886, "perplexity": 496.81525385596836}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00455-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://cvgmt.sns.it/paper/3240/ | # Minimization of anisotropic energies in classes of rectifiable varifolds
created by derosa on 24 Nov 2016
[BibTeX]
Preprint
Inserted: 24 nov 2016
Last Updated: 24 nov 2016
Year: 2016
We consider the minimization problem of an anisotropic energy in classes of $d$-rectifiable varifolds in $\R^n$, closed under Lipschitz deformations and encoding a suitable notion of boundary. We prove that any minimizing sequence with density uniformly bounded from below converges (up to subsequences) to a $d$-rectifiable varifold. Moreover, the limiting varifold is integral, provided the minimizing sequence is made of integral varifolds with uniformly locally bounded anisotropic first variation. | {"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": 0.96578049659729, "perplexity": 1700.828325680887}, "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-2017-22/segments/1495463608936.94/warc/CC-MAIN-20170527094439-20170527114439-00071.warc.gz"} |
https://mathoverflow.net/questions/218078/stable-homotopy-groups-and-zeta-function/218204 | # stable homotopy groups and zeta function
I have heard during a discussion that there is a well known relation between the stable homotopy groups of a sphere (more precisely the order of stable homotopy groups of localized sphere spectrum with respect to some homology theory $E$) and the values of the zeta function at some integers.
$$|\pi_{i}^{s}L_{E}\mathbb{S}| =^{?} \zeta(-n)$$ I'm not sure that I understood well, I will be glad if someone can explain this relation.
• Perhaps this is what you mean. Take $E = K(1)$, the first Morava $K$-theory. You can find the homotopy groups $\pi_*L_{K(1)}S^0$, at least when $p$ is odd, in, for example, Lurie's course notes (math.harvard.edu/~lurie/252xnotes/Lecture35.pdf). The order of the cyclic summand that appears can be expressed as the denominator of a certain expression involving Bernoulli numbers. This is related to the image of $J$, see en.wikipedia.org/wiki/J-homomorphism – Drew Heard Sep 13 '15 at 9:58
• @DrewHeard: as this question has been highly upvoted, you should probably promote your comment to an answer. – Neil Strickland Sep 13 '15 at 12:29
Here is a slightly more fleshed out version of my comment. Let $K(1)$ be the first Morava $K$-theory. When $p$ is odd one can calculate the homotopy groups of the $K(1)$-localised sphere spectrum to be $$\pi_nL_{K(1)}\mathbb{S} = \begin{cases} \mathbb{Z}_p, &n=0,1\\ \mathbb{Z}/p^{\nu_p(t')+1} & n=2(p-1)t'-1, t' \in \mathbb{Z}. \end{cases}$$ Here $\nu_p(x)$ is the $p$-adic valuation of $x$.
Following Adams define a function $m(l)$ by $$\nu_p(m(l)) = \begin{cases} 0 & l \not \equiv 0 \mod (2(p-1)) \\ 1+ \nu_p(l) & l \equiv 0 \mod (2(p-1)). \end{cases}$$ Adams shows (following Milnor and Kervaire) that $m(2s)$ is the denominator of $\beta_{2s}/4s$, where $\beta_s$ is the $s$-th Bernoulli number, and the fraction is expressed in the lowest possible form.
Using standard properties of $\nu_p(x)$ there is an equivalence $\nu_p(t')+1 = \nu_p((n+1)/2)+1$. Since $(n+1)/2 \equiv 0 \mod (2(p-1))$ we see $$\nu_p(m\left(2\cdot \frac{n+1}{4}\right)) = \nu_p((n+1)/2)+1$$ and so the order of $\pi_nL_{K(1)}S^0$ is the denominator of $\beta_{(n+1)/2}/(n+1)$.
Edit: Let me try and say something about the image of $J$ then. This is a homomoprhism $J:\pi_nSO \to \pi_n\mathbb{S}$. When $n=4k-1$ the order of the image of $J$ is cyclic of order the denominator of $\beta_{2k}/4k$. Let $\text{Im}(J_n)_p$ denote the image of the composite $\pi_nSO \to \pi_n\mathbb{S} \to \pi_n \mathbb{S}_{(p)}$. I believe this is meant to be isomorphic to $\pi_nL_{K(1)}\mathbb{S}$ for $n>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": 0.9866039752960205, "perplexity": 131.6083025434895}, "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-2020-34/segments/1596439737168.97/warc/CC-MAIN-20200810175614-20200810205614-00545.warc.gz"} |
https://math.stackexchange.com/questions/2172473/generalisation-of-integers-for-infinite-length/2172491 | # Generalisation of integers for infinite length?
As most people know, an integer can only be finite length when expressed in the form of a series of digits in some base. However, real numbers in general can be infinite length, so long as there is a "first digit".
That is, if we express an "infinite integer" in the form $$\sum_{n=0}^\infty a_n 10^n$$ where $a_n\in\mathbb{Z}_{10}$, then addition and multiplication should both be well-defined on this sequence. Indeed, I suspect that such a system may actually form a field, as additive and multiplicative inverses should exist (consider that $\bar{9}+1=0$ where $\bar{9}$ is the number for which $a_n=9$ for all $n$, and $\overline{285714}3\times7=1$, where $\overline{285714}3$ has $a_0=3$ and then $a_{6n+1}=4$, $a_{6n+2}=1$, and so on).
Has such a set with addition and multiplication been investigated? Does it have a name? Does it actually form a field, or does it fail one of the required properties of a field? Is the system functionally the same irrespective of chosen base, or do different bases change some properties of the system?
You've rediscovered the $10$-adic integers. Such "infinite decimal expansions of integers" form a ring using the usual rules for adding and multiplying decimals. However, they do not form a field. A quick and easy way to see this is that $10$ has no inverse (since if you multiply $10$ by anything, then the last digit of the product will be $0$).
Even worse, however, this ring has zero divisors: there are nonzero elements $x$ and $y$ such that $xy=0$. These take a bit of work to construct, but here's the idea. We'll construct the digits of $x$ and $y$ one at a time, starting from the end. Start by saying the last digit of $x$ is $2$ and the last digit of $y$ is $5$, so the last digit of $xy$ is $0$. Then choose the preceding digit of $x$ so that $x$ is divisible by $4$, and the the preceding digit of $y$ so that $y$ is divisible by $25$, so their product will be divisible by $100$ and end in two zeroes (for instance, $x$ might end in $12$ and $y$ might end in $25$). Continue choosing digits of $x$ and $y$ one at a time so that $x$ is divisible by every power of $2$ and $y$ is divisible by every power of $5$. Every digit of the product $xy$ will end up being $0$.
You can, of course, do the same thing with a different base instead of $10$, giving you the $b$-adic integers for any integer $b>1$. Unlike the case of finite base expansions, however, you get a genuinely different number system for different values of $b$! (To be precise, it turns out that the number system you get depends only on the set of prime factors of $b$, so for instance the $10$-adic integers are isomorphic to the $50$-adic integers.)
It turns out that whenever $b$ is composite, the $b$-adic integers will have zero divisors, as in the case $b=10$. When $b$ is a prime (say $b=p$), however, the $p$-adic integers are an integral domain (they have no zero divisors). The $p$-adic integers aren't quite a field, though, because $p$ has no inverse. If you adjoin an inverse to $p$, you do get a field, called the $p$-adic numbers and written $\mathbb{Q}_p$. Elements of $\mathbb{Q}_p$ are "base $p$ expansions that can be infinite on the left": that is, formal sums $$\sum_{n=k}^\infty a_np^n$$ where each $a_n$ is an integer between $0$ and $p-1$ and $k\in\mathbb{Z}$ (so $k$ can be negative, allowing finitely many negative powers of $p$ in the sum).
There is a much larger story here: $p$-adic numbers play a huge role in modern number theory, and have a very rich structure. Since it is easily possible to write an entire book on the topic, I'll end my answer here though. You can read a bit more about them on the Wikipedia page I linked at the beginning.
• Thanks. This basically addresses all of what I was hoping to check. I hadn't considered the case of the final digit being a zero (I did suspect that the base would matter, though). I had seen the "p-adic numbers" referred to in other questions/answers, but had never really looked into them (as my expertise is in calculus and mathematical physics). – Glen O Mar 5 '17 at 5:04
Yes. What you've defined is called the 10-adic numbers. They don't form a field for two reasons, one of which is that $10$ is not invertible; the other reason is left as an exercise.
Most books will only talk about the $p$-adic numbers for $p$ prime, and this is because the general case turns out to reduce to the prime case, essentially because of the Chinese remainder theorem.
• Thanks. Your answer addressed the most important part of my question, and did so very quickly. I've upvoted your answer. However, Eric's answer provides much more detail and addresses a number of the other questions I asked as well, so I'm afraid I have to accept his answer over yours. – Glen O Mar 5 '17 at 5:00
Another fun way that the rings differ depending on the base used is that in base 5, $-1$ (i.e. $\bar{4}$) has a square root while $2$ doesn't, but in base 7 it's the other way around (and $-1$ is of course $\bar{6}$ in that base).
I should note that I haven't proved those, but my messing around with the idea pointed strongly that way. If you like programming it's a fun exercise to write up a class that implements these numbers. I did it in Python, as its yield/generator system made it easy to model these numbers as a potentially infinite list that produced the digits from units place up. | {"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": 0.9299675822257996, "perplexity": 124.94681496556116}, "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/1563195525863.49/warc/CC-MAIN-20190718231656-20190719013656-00180.warc.gz"} |
http://scholarpedia.org/article/Talk:Galactic_dynamics | Notice: Undefined offset: 6285 in /var/www/scholarpedia.org/mediawiki/includes/parser/Parser.php on line 5961
Talk:Galactic dynamics - Scholarpedia
# Talk:Galactic dynamics
## (New) Reviewer A
Comments as of 28 Mar 2011
Galactic Dynamics is a vast subject involving several areas of mathematics and physics. The authors have chosen to illustrate these matters from a particular perspective, that of stellar dynamics, which, even if not embracing every aspects involved in the field, is surely wide enough to shed light on many of the building blocks of the physics of large self-gravitating systems. On this basis, the article provides a thorough illustration of the main results so far obtained and of the ideas and tools to obtain them. These ideas and results are also the starting point of many research programs concerning issues that are still to be understood.
The authors faced a difficult task because the body of mathematical gear needed to deepen the subject is vast and complex. On this side I think that they should do an extra effort in order to clarify some statements and to fill some gaps in the presentation of the results. In the following I provide some examples to illustrate what I think about:
1. Introduction and basic concepts: the generic notion of "disc galaxy" appears somewhere but is not explained its relation with the Hubble classification. This remark is useful also to justify the distinction between the contents of section 2.3, 4 and 5. In introducing relaxation, it is useful to refer to the discussion in section 6.1.
2. Orbits and integrals: the Hamiltonian pops out from nothing and the same for the notion of formal integral and that of Poisson brackets. The hints on "KAM theory", "overlapping of resonance" and "Lyapunov exponents" is so sketchy that it is difficult to think it can be useful to the inexpert reader. Since it is clear this is not the place to fully introduce these issues, appropriate references seem mandatory.
3. Construction of equilibria: Concerning the Jeans equation, some words of warning could be useful about problems with possibly unphysical required distribution functions. Some illustration of the construction of more refined analytic self-consistent models can be useful, for example models with distribution function depending also on the angular momentum.
4. Gas dynamics: In equation 38 concerning collisional Boltzmann equation, the index "c" in the right hand side is probably referred to the "collisions"; it seems useful to give a hint of these effects, the time-scale and a comparison with the stellar relaxation time.
5. Spiral structure: This section is quite detailed and clear.
6. N-body systems: Some warning concerning the present status of the theory of "Violent Relaxation" is mandatory, since it is not clear how to reconcile almost trivial and rough theoretical results with generic numeric simulations of collapses.
We thank the new reviewer for his comments, which were all taken into account. In particular:
1. We added a sentence on the meaning of "disc galaxies" in the second paragraph of the introduction. We also added a sentence at the end of the fifth paragraph were we refer to section 6 for a more detailed discussion of relaxation.
2. We extended the introductory paragraph of section 2 so as to go more gently into the definitions of concepts like formal integrals, KAM theory, Lyapunov characteristic numbers etc. We then suggest the book of Contopoulos (2004) for further reading. In the first paragraph of section 2.1 we better explain how the Hamiltonian (8) is derived. Finally, we rephrased the sentence before Eq.(13) so as to render clear that the formal integrals series are actually defined by the requirement of having a vanishing Poisson bracket with the Hamiltonian.
3. In the second paragraph of section 3 we give a warning on possibly unphysical distribution functions arising by solutions of Jeans' equation, mentioning explicitly the cases where the d.f. may turn to be negative in some parts of the modelled galaxy or the velocity dispersion curve may have an unphysical form. We think that these cases are the most characteristic. Finally, we wrote a new paragraph on inversion formulae for spherical models depending on the angular momentum or axisymmetric models depending on the z-component of the angular momentum.
4. We explained the meaning of the subscript "c" in Eq.(38) and we speak about the relevant phenomena immediately after Eq.(38).
5. No change
6. We revised the whole subsection 6.2 in order i) to give a more fair account, and ii) better explain our own point on this matter. Briefly, this can be summarized as follows: while we certainly agree that Lynden-Bell's formula and the whole theory around it are far from describing realistic galactic systems, we think that this theory continues to play in important role today in our understanding of the structure of galaxies. This is because violent relaxation has offered a unique up to the present "paradigm" of theoretical framework showing how can statistical mechanics be applied even in principle in collisionless systems. In fact, we think that the main contribution of this theory is to explain how can relaxation be at all possible in galaxies over a timescale as short as the dynamical time. We hope that the new text conveys this message while providing a fair account.
## Authors replies to earlier comments by reviewers
Report on "Galactic dynamics", by G. Contopoulos and C. Efthymiopoulos
edited by reviewers "A" and "B": reply by the authors
A: I have read the article "Galactic dynamics" and found it useful and interesting, but I have a number of comments that I hope the authors will be able to consider.
B: Given the title "Galactic Dynamics", it's obligatory to compare this article with the excellent textbook of the same title by Binney and Tremaine (2008, Princeton University Press. Essentially, a html version of that book would be a VERY useful resource, for everybody who could possibly be interested in reading this scholarpedia article. I suspect that this is a general feature/problem for many scholarpedia articles). Clearly, this article has no chance in a such comparison: it fails in width (too specialised on a few subjects), clarity, and up-to-dateness; its only advantage naturally lies in brevity and availability.
While we thank both reviewers for their constructive comments, we have to stress that the spirit of a scholarpedia article is to present only very basic notions on a subject within an extent of ~5000 words. It is not reasonable to compare this to the scope of a textbook like Binney and Tremaine, which is larger than the present article by a factor 100, or other books listed in our reference section. The present article has not even the form of a review article. In fact, our choice of topics is based on the sole criterion of outlining a set of notions which are an absolute must for a first step in the study of galactic dynamics. Even so, we had to limit ourselves to a a severely restricted list of topics which is obviously far from exhaustive. On the other hand, we intend to continuously create new electronic links to some article keywords, as an on-going "curator" task that will render the article more useful.
Major points
(1) A: The title appears to be discordant with the content of the paper. Galactic dynamics encompasses a number of astrophysical issues (such as processes of galaxy formation and evolution, and, in particular, the dynamics of the dissipative components) that are basically ignored in this review, which covers mostly the dynamics of stellar orbits. In my opinion, a more appropriate title would be "Stellar dynamics" or "Orbits in stellar systems". The use of the word "galaxies" is sometimes unjustified. The sections "Orbits in axisymmetric galaxies" and the following "Orbits in triaxial galaxies" are actually a discussion of orbits in axisymmetric or triaxial models inspired by galaxies; the potential described by Eq. 18 is interesting and useful, but should not be considered an approximation of the potential present in elliptical galaxies, because it has been found to be too shallow and associated with unrealistic density distributions.
B: I sort-of agree with the first reviewer. This article is meant to be theoretical and traditionally galactic dynamics has been concerned with the motion of stars (and dark-matter) only. Galactic gas dynamics is still not very well developed theoretically and hence hard to review. I think it helpful if this article were accompanied by or appended with an observational review about galaxies and their properties, similar to the book "Galactic Astronomy" by Binney & Merrifield.
B: However, I also think that even in the limited realm of stellar dynamics the article is too narrow, mainly dealing with orbits in static potentials and density-wave theory in disc galaxies (which are, I think, the main areas of expertise of the authors). However, galaxy dynamics is MUCH richer than this, just look at the table of contents of "Galactic Dynamics" by Binney & Tremaine.
We emphasize stellar dynamics because it is our basic thesis that stellar dynamics is the most important ingredient of galactic dynamics. We believe that this viewpoint is shared by most leading experts in the field. One referee invokes Binney and Tremaine's book. But in the first paragraph of the "Preface" of this book it is clearly stated that "the principal tool for answering these questions" (regarding the formation, structure, and evolution of galaxies) "is stellar dynamics". This is repeated in the first two paragraphs of the "Introduction" of the same book.
Dissipative processes due to gas play, however, also a role in galaxies, in particular as regards phenomena of secular evolution. In order not to leave this aspect completely ignored, we added a new section 3 entitled "gas dynamics", where we briefly discuss how stellar dynamics should be modified and what sort of new phenomena could arise due to the gas.
In section 2, we prefer to keep the word "galaxies" rather than "models inspired by galaxies" in all subsection titles. However, we mention in the text that this is an idealization. Regarding Staeckel potentials (Eq.21), their discussion is necessary, because the nomeclature for orbits found in such potentials (Box, ILAT, OLAT SAT) has become standard in the characterization of orbits in generic triaxial potentials. In the new version we stress that box orbits are unlikely to survive in cuspy potentials, being replaced, instead, by chaotic orbits.
Reviewer B characterizes this article as "too narrow". We can hardly see how to substantially widen the article's content without exceeding by far the length standards of a scholarpedia article (which have already been exceeded in the first article's version). Besides the new section 3, the only marginal additions that we found possible to accomodate were: i) a reference to virial equilibria and to the concept of rotationally vs. pressure supported systems, ii) possible forms of distribution functions and how these affect the form of velocity ellipsoids in galaxies, and iii) add more to the section on density wave theory (see below).
We finally agree that a separate lemma on "Galaxies: structure and evolution" would be useful.
(2) A: (Possibly related to the previous point.) Some statements do not do justice to the subject. In particular, "The shapes of the galaxies are governed by their stars, that form their main bodies. The gas forms a relatively small proportion..." is incorrect and misleading. As proved by near-infrared (K-band) observations, and as expected from the analysis of Jeans instability, gas plays an important role in determining the morphology of spiral galaxies.
B: I don't think that gas-dynamics is very important. Even in spiral galaxies, the gas contributes only about ~10% (in the Milky Way) to the baryonic mass. Spirals are seen in the old stars (K-band) of most disc galaxies, are the natural modes of discs, and do not require gas dynamics, though dissipative processes may play some role.
While we now discuss the role of gas (section 3), we should point out that the argument based on K-band observations is not correct. These observations yield the distribution of matter as deduced mainly by old stars (red giants) rather than the gas. Such a distribution is in general characterized by smooth and well-defined spiral arms. In fact, the reason why such observations have been used extensively in recent years in reconstructions of the gravitational potential of disc galaxies lies, precisely, on the consideration that old stars are better tracers of the overall mass distribution because they have more time to respond' to the total potential, hence settling to a smooth matter distribution. On the contrary, the distribution of young stars, molecular clouds, or OB associations exhibits large and non-smooth variations. This remark notwithstanding, we followed the reviewer's suggestion to rephrase our sentence in a more balanced way regarding the role of all components (dark matter, stars, gas) in the shapes of galaxies.
(3) A: The general set-up of the basic equations is inadequate. The set of Eqs 1- 4 is practically inapplicable, even when we assume that the interstellar medium can be neglected. The main reason is that it is now established that galaxies are embedded in halos of dark matter and that dark matter generally has a density distribution different from that of visible matter. This requires the inclusion of an external component in Eq. 4, the symmetry of which has to be discussed separately, with important consequences on dynamics, and in particular on the structure of stellar orbits.
B: Essentially, only equation (4) needs to be amended by adding the dark-matter potential. Alternatively, one may extend the formalism to include dark-matter and stars on an equal footing. However, I think one should try to keep these equations as simple as possible. An important point is that the dark-matter potential must not be considered as a god-given external potential, but as an independent player. From a dynamical point of view, stars and dark-matter are governed by the same equations: perturbations affect the dark-matter as well as stras, dark-matter exchanges energy and angular momentum with spiral and bar modes (rather than just providing an additional static background potential).
We essentially agree with reviewer B. The set of equations (1-4) is valid to a very good approximation if $f$ represents the distribution function of all collisionless matter, i.e. the stars and the dark matter. Also, it is valid for describing the luminous matter alone, if the dark matter is considered to yield a fixed external potential. On the other hand, we may consider the dark halo as live', i.e. responsive to the luminous matter, in which case the study of the problem becomes more complex. These facts are mentioned in the revised text.
(4) A: The section on "Orbits in disc (spiral - barred) galaxies" is confused, not only because of the general difficulties noted in point 1 above. The reader is not informed of the assumptions made for the description of the non-axisymmetric part of the potential. In particular, in applications to real galaxies one has to justify to what extent such non-axisymmetric part should be treated as stationary (time independent) and bisymmetric.
B: I think the simplified picture presented is useful and helps understanding the situation with real galaxies. As usual in science (especially in physics), simplifying assumptions allow for analytic insight and hence understanding of much more complicated systems. It is important, though, to emphasise this and clearly state which assumptions are made and how good they are.
We essentially agree with the comment of reviewer B. We added some sentences on the importance of both even and odd components of the Fourier trasform of the potential, but we emphasize that the most basic theory of orbits in disc galaxies stems from considering that the m=2 component of the non-axisymmetric perturbation is the most important one.
(5) A: The section "Density wave theory" is poor, because (i) key issues at the basis of the density wave theory (basic issues raised by the observations, such as the winding dilemma or the preference, but non-universality, of trailing bisymmetric structures; dynamical issues such as collective instabilities, from Jeans instability and beyond, wave propagation and dynamics, self-regulation,overreflection, light disks vs. heavy disks, role of dark halos; observational tests, i.e. what do the observations tell us in favor or against the alternative scenarios that may be proposed) are ignored; (ii) the description of the theory is improperly reduced to a statement of Fourier analysis (Eq. 31); and (iii) the impression left by the presentation and by sentences such as "The density wave theory of spiral arms was developed first by Lindblad, and later by Lin, Shu, Kalnajs, Toomre, Lynden-Bell, etc." is that the theory has remained frozen at the stage of its initial formulation in the '60s. (The justification of the "Nonlinear theory" given in this review is misleading: as for many other problems in hydrodynamics or plasma physics, although it is fair to say that nonlinear effects are interesting and likely to be important at resonances, the linear theory can make and has made adequate predictions on the role of resonances.) The book "Spiral structure in galaxies: a density wave theory, Publisher: Cambridge, MA MIT Press, 1996 (http://adsabs.harvard.edu/abs/1996ssgd.book.....B)" by G. Bertin & C.C. Lin is fairly complete (in terms of references and topics covered) up to the mid '90s; additional interesting work has been done and published in the last 15 years, after the publication of that book.
B: I am not an expert on density-wave theory, but thought that this theory is dead: it cannot explain spirals galaxies. Why is swing amplification not mentioned? As I understand it, this is thought to be essential for understanding the formation of spirals and resolving the anti-spiral theorem. I think that chapter 6 of "Galactic Dynamics" (Binney & Tremaine) is a much better account of our current knowledge in this area.
It is hard to compromise the two reviewers' opinions on this point. One reviewer wants us to add more on new developments of spiral density wave theory beyond Lindblad, Lin, Shu, Kalnajs and Toomre, while the other thinks the theory is "dead". Our own viewpoint is that the most important, up to the present, contribution of the density wave theory lies in what was the central idea of its founders, namely, that this theory provides a "paradigm" of dynamical mechanism which describes spiral structure as a wave propagating in the galactic disc with angular speed different than that of most stars or gas going in and out of it. This paradigm is not "dead" but it remains today the most basic description of rotating spiral arms in galaxies. The main open questions are rather different, focusing mainly on the degree of quasi-stationarity and/or secular evolution of such a structure. Namely, it is still unknown whether it is more consistent to approximate spiral structure i) by quasi-stationary solutions to Boltzmann's equation supporting a unique pattern speed, or ii) by relatively short-lived, recurrently generated spiral arms formed by instabilities in the disc, which support multiple pattern speeds (e.g. of the bar and spiral arms).
In the new version we i) discuss to some extent the winding dilemma and the problem of preference of trailing waves, ii) only marginally refer to most other topics mentioned by the reviewers, since their treatment in reasonable length is practically impossible, iii) added the book of Lin and Bertin in the bibliography, iv) mention alternative scenaria of spiral structure considered as a reccurent disc instability.
(6) A: Given the importance of "N-body systems" and "Violent relaxation - collective instabilities" in galactic dynamics, the description given at the end of this review is too short and out-of-balance with respect to the details on stellar orbits provided in the main body of the article; in addition, the statements given on "N-body simulations" (the text around Eqs. 32-34) appear to be restricted to stellar systems with little or no rotation (except for the reference to instabilities in disks, at the very end).
B: I completely agree with the first reviewer.
We enriched the text, adding some reference to N-Body simulations of galactic discs, and of other interesting phenomena like interacting or merging galaxies. But there is no real need for further additions at this point, since we found a separate scholarpedia article specialising on "N-Body simulations (gravitational): curators: M. Trenti and P. Hut", which is very specific and informative. We thus added a link to that article.
Minor points (reviewer A)
7. The asymptotic expansion at the basis of Eq. 11 and the physical meaning of the expansion parameter in the context of galaxy models should be explained better.
8. The distinction between "smooth core" and "core with a central point mass" (see the discussion illustrated by Fig. 4) should be explained better. What about a cuspy core, without a central point mass?
9. After Eqs. 18-21, the attention moves to a polynomial potential (22). Could a classification diagram like Fig. 7 be drawn also for orbits in Staeckel potentials?
10. (see also point 4 above) Before Eq. 23, "Disc galaxies and their patterns rotate in general with large angular velocities \Omega_s" is a very strange and ambiguous statement. The assumptions on the potential, the question of drawing orbits in an inertial or rotating frame, the symmetry break, and the location of L_1, L_2, L_4, L_5 with respect to potential minima and maxima could be explained much better. The current text is confused and some figures are difficult to understand (in particular, Fig. 11). The assumption of stationarity of the symmetry-breaking perturbation in a suitable rotating frame is crucial and not given in a sufficiently explicit way (especially if the authors believe that the models studied are to be applicable to observed galaxies).
11. The factor (1/2) in the definition of the radial wavenumber (after Eq. 31) should be dropped.
12. It would be nice to explain better how and why in general the Fourier analysis associated with Eq. 31 leads to an integral equation with eigenvalues \omega. Then to explain how in the initial stages (until the mid '70s) the theory could only handle, through an approximate dispersion relation, the local properties of density waves. And finally to show what mechanisms are thought to be at the basis of the excitation of a discrete spectrum of global modes and how the problem has been addressed analytically and numerically.
13. The infinities at turning points and resonances that appear in algebraic dispersion relations reflect a wrong choice of representation of the perturbation and are removed in more general analyses of the linear theory.
14. The use of "collision time" as a synonym of "dynamical time" (after Eq. 32) is misleading.
7) We added a sentence clarifying the physical meaning of small parameter in galaxies.
8) We now clarify that the distinction is between i) "smooth core" and ii) "cuspy core or core with a central point mass".
9) The answer is no. The periodic orbits in Staeckel potentials can only have neutral stability since the potentials are integrable. However, Fig.7 is important because it represents the generic case, i.e. when periodic orbits of all four types of (in-)stability co-exist.
10) We thank the reviewer for this point. We rephrased the statement in a hopefully satisfactory way. As regards Fig.11 we think that it is important to give some examples of characteristics of periodic orbits.
11) We dropped 1/2.
12) The mathematical derivation of the dispersion formula is too lengthy to present in all detail, thus we just sketch it by words. We furthermore clarify that this is a local theory and mention also modal theory.
13) We agree with the reviewer's comment that there are methods to take care of the problem of infinities at resonances. In fact, one of the authors (G.C.) has been a main proponent of resonant theory in discs along the lines mentioned by the reviewer. We doubt, however, that we should call such a theory linear'. At least it is not linear in the same way as the Lin and Shu theory, because the formal scheme yielding the corrections for the distribution function near resonances is different from the formal scheme used by Lin and Shu. Thus, we prefer to keep this point unaltered, i.e. as part of nonlinear theory.
14) We removed "collision time".
01-11-2010
remark. The email alert triggered by me was errorneous (bug in scholarpedia, I suppose), as it said "Reviewer A alerts ...", rather than reviewer B. Also, I have not been informed about the decision by reviewer A --- has he accepted, so that now there is only one of us, and hence I default to be "A" ??? this is very confusing.
Reviewer A:
I have read the revised article and the authors replies. The article is somewhat improved, but on the whole the changes are minor and rather disappointing.
My main criticism is that the article is still quite badly balanced with way too much emphasis on orbits and density waves--perhaps the authors should consider to write an article about these subjects instead. Other important topics are almost entirely missing, such as equilibriums systems other than discs and their stability properties (a lot has been learned from spherical models); methods to construct equilibria; interactions, dynamical friction, tides, and mergers; dynamics in the vicinity of a dominating point mass (supermassive black hole); formation/dynamics of bars and warps to name a few.
It is certainly impossible to write a scholarpedia article (with the imposed limit on its size) on galactic dynamics to the same depth as the existing section on density waves. However, the solution then is NOT to pick just one or two (density waves and orbits) of many subjects and expand on them, rather all subjects should be mentioned at the same level. Naturally, there may be further articles about each of these more specialised subjects. Moreover, the article merely describes methods and tools, but I am missing the greater picture: what is the goal/purpose of galactic dynamics? what are the most pressing open questions? what are its successes? How does it fit into astrophysics as a whole?
We have already agreed with the reviewer that this article cannot present all aspects of galactic dynamics. The section on density wave theory was expanded because the other reviewer requested a number of additions in his/her original report. On the other hand, with all these additions this section deals now with the general problem of spiral structure rather than pure density wave theory. We thus changed its title to "Spiral Structure", while we did some reduction by merging some subsections (doing some re-ordering) and by removing some paragraphs. It now occupies about 15% of the article.
On the other hand, we think that the section on orbits' should be kept essentially as it now stands (apart from one paragraph on the dynamics in the corotation region, which we removed as too technical). This section occupies about 30% of the article, and we think that this percentage should not be considered as too high. In fact, our experience is that some topics like e.g. the form and bifurcations of periodic orbits in rotating galaxies are less known than they should be, and thus it is useful and informative to present them in some depth. Finally, it should be noted that the article appears presently under the category "Celestial Mechanics" , where orbital dynamics is a key issue (although we believe that the article should eventually be moved to an independent category on "Galactic Dynamics).
Apart from the previous changes:
i) We added a sentence in the introduction stating what is the focus of the article.
ii) We added a section on construction of equilibria' where we speak about fixed models and methods of construction (Schwarszchild).
iii) we expanded the last subsection on collective instabilities, which is now an independent sub-section.
iv) Responding to some comments of the other reviewer (who has now accepted the article), we added a reference to Jeans instability in "Gas dynamics", and we discuss some differences between gas and stellar dynamics.
v) We give replies to the specific comments of Reviewer A (see below).
It has to be understood that by this process the article is now well beyond the limits of an encyclopedic introduction. In particular, there is no way to seriously go into questions like 'how does' galactic dynamics 'fit into astrophysics as a whole'. Even the question what are 'the most pressing questions' of galactic dynamics is rather subjective. We thus prefer to be concise rather than going into such subjective discussions. %-------------------------------------------------------------------------
Reviewer A:
In addition, I identify the following issues with the existing text (but this list is very likely incomplete).
(1) The idea/concept of a "rigid halo" is not a useful one, even though it has been used a lot in the past. Essentially, this concept is at best dangerous as the effect of a rigid halo can be opposite to that of an active halo, e.g. a halo may suppress the bar instability by reducing the importance of self-gravity in the disc, but a life halo can also absorb angular-momentum and thus promote the growth of a bar once it has become non-linear.
While it is true that a rigid halo can suppress the bar instability and lead to other spurious phenomena, we do not agree that the concept "is not a useful one". In fact, whenever one is interested in the orbits on the disc plane, it is useful to assume a fixed halo potential, at least temporarily. For example, when we attempt to estimate the total gravitational potential from K-band observations, our only way to fix the mass-to-light ratio is by adopting an ansatz on how much the halo contributes to the total rotation curve in the inner or outer parts of the disc (e.g. to choose a maximal or sub-maximal disc solution). The same thing we practically do with the bulge (this was noted by reviewer B). The correct attitude is then that a rigid halo model can be useful in some contexts while it is not so in other contexts. The rephrasal we did in the text is, we think, fair. %-------------------------------------------------------------------------
Reviewer A:
(2) the concept of "approximate" third integrals is confusing and should be avoided. An orbit is either regular or not, it either admits a third integral or not. Perhaps what the authors meant is whether the third integral is universal and can be given in closed form for all orbits or whether it only holds for an orbital family at a time and no closed form can be given. In the latter case, the third integral is not approximate. Another option is to give a closed form (e.g. derived using perturbation theory) which is not exactly but approximately conserved and can thus be used effectively as a third integral in some approximate modelling--in this sense the angular momentum w.r.t. the short axis is an approximate integral for short-axis tubes in triaxial potentials.
The referee is not right at this point and this remark actually reveals a common confusion about the concepts "regular orbit" and integrals that can be "local", "formal" or "approximative". The confusion stems from the following point: the word "integral" in general means a function $I(q,p)$ of the phase space coordinates which has zero Poisson bracket with the Hamiltonian i.e. ${I,H}=0$ (in authonomous systems). But in order that such a bracket can be defined in the first place, we should be able to define the function $I$ in an {\it open domain} of the phase-space. To give an example: if we have a nearly-integrable system, most motions lie on invariant tori whose existence is guaranteed by the Kolmogorov - Arnold - Moser theorem. However, a careful study of the proof of the KAM theorem reveals that there is no function $I$, either global, or local on whatever small open domain, that represents such tori, because these tori are defined on a Cantor set of values in the action space which is not an open set. Of course, this does not mean that the orbits on these tori are not regular. They are regular because they are quasi-periodic and they have zero Lyapunov exponents. But contrary to the reviewer's claim, these completely regular orbits do NOT 'admit' any exact third integral'.
On the other hand, a 'formal integral' is a formal series whose Poisson bracket with the Hamiltonian vanishes. A series does not represent in general a quantity in 'closed form'. Thus integrals in 'closed form' are not in general 'given by perturbation theory'. in fact, the formal series are not convergent in general. However, a proper truncation creates an 'approximate integral'. This quantity is defined in open domains of the phase space and it is therefore useful in practice, since the remainder can be exponentially small according to Nekhoroshev theorem. In fact, this seems to be a very useful quantity in Galactic Dynamics, because it bounds the motions of {\it both regular and weakly chaotic orbits} for times surppassing by far the Hubble time.
Finally, the fact that the formal integrals (and their truncations) take different forms at different resonances is well known, but it is irrelevant in this discussion.
These facts are well known to nonlinear dynamists, but not so well known to dynamical astronomers and one often sees a confusion in the literature stemming from inappropriate use of the various concepts. In our text we have been very careful in giving the right definitions and we think the text should be left as it now stands.
%------------------------------------------------------------------------
Reviewer A:
(3) in the virial theorem section, both T and K are called kinetic energy. I think T should not be called kinetic energy, as it is only the component of ordered motion to the kinetic energy.
We changed the terminology by properly defining the kinetic energy tensor. %------------------------------------------------------------------------
Reviewer A:
(4) N-body force solvers that scale as O(N) are not available for "rough models" only, but are as general as the tree method (see Trenti & Hut).
To our knowledge, there are two types of O(N) N-Body algorithms: (i) the so-called "self-consistent field" method, pioneered by Clutton-Brock, uses basis functions to develop the galactic potential. This has led to useful applications (e.g. the Hernquist-Ostriker approach), which, however, are limited to isolated systems and seem to hardly be able to treat more complex systems such as merging galaxies, spiral structure etc. Furthermore, recent research by many groups (including some works of ours) shows that the choice of a basis set must by very finely tailored to the specific morphological details of a system under study in order to correctly represent the dynamics. (ii) Improvements of the tree method (e.g. Dehnen 2000) claim to scale linearly with N rather than O(NlogN). However, Trenti & Hut (recalled by the referee) are clearly very sceptic about that. In our article, we rephrased this sentence mentioning in more detail the self-consistent field method. %-------------------------------------------------------------------------
Reviewer A:
(5) The Lynden-Bell statistic is presumably incorrect (in a more recent paper co-authored by Lynden-Bell, the authors admit that Lynden-Bell's original method was inconsistent), and is, as far as I am aware, hardly used at all. Moreover, there is more recent work on violent relaxation (e.g. Tremaine et al 1986 and Dehnen 2005).
This is a highly controversial issue. The referee may wish to consult section 11.3 of a tutorial article of one of us (C.E. et al. Lect. Notes in Physics 729, 297 (2007)), where the questions raised by the referee are reviewed in detail (in particular see subsection 11.3.6). Briefly, the work of Tremaine et al. has itself been largely criticised on various grounds (i.e. on that the choice of entropy functional cannot be arbitrary or that even Tolman's theorem may not be consistent with their approach). Clearly, we cannot here present all the new developments. Lynden-Bell's original distribution deserves, in our opinion, particular reference because it opened the way to a whole new branch of statistical mechanics, namely the statistical mechanics of collisionless gravitating systems. We added, however, one paragraph speaking about the various criticisms and deferring to the tutorial of C.E. et al for more information.
%-------------------------------------------------------------------------
Reviewer A (on 13-Jan-2011):
Sorry for this delay (I missed the email on 25-Nov and only got a second one today). I have read the article and the authors reply. I am not happy with this article, as detailed below. However, I have the feeling the authors are not prepared to meet with me on this and therefore decided to quit without accepting. For the benefit of any future reviewer, I leave some comments as follows.
1 The main criticism is that this article does not encompass "Galactic dynamics" in its whole width. Some subjects are laid out it great detail (the section on Orbits and Integrals takes about 40% of the whole article), while others are left out altogether (see my previous review). Perhaps a solution is to split this into two scholarpedia articles on "Spiral Structure" and "Stellar Orbits", respectively.
2 In answering many of my minor comments the authors have sometimes demonstrated a rather narrow and opinionated view: they present their own occasionally ill-informed opinion rather than seeking to present a fair account. An example is their account of violent relaxation, which is certainly one-sided (the authors admit that this is a controversial issue).
3 The reply to my previous point (2) about regular orbits should be incorporated into the article either directly or via a suitable reference.
4 There is only one type of O(N) force solvers: the FMM (fast-multipole method). The so-called "SCF" methods are not O(N) if one increases the force resolution with N, as one should and does with any other method (in fact, in this case they become O(N^2). The FMM, on the other hand are O(N), though most implementations have failed to reach this goal with the exception of Dehnen's (2002) algorithm, which he demonstrated to have complexity O(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": 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": 0.8992946743965149, "perplexity": 782.2585922881952}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499831.97/warc/CC-MAIN-20230130232547-20230131022547-00091.warc.gz"} |
https://questions.examside.com/past-years/jee/question/a-2-kg-block-slides-on-a-horizontal-floor-with-a-spe-2007-marks-4-cyklcxddi80lgcxx.htm | ### JEE Mains Previous Years Questions with Solutions
4.5
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1
### AIEEE 2007
A $2$ $kg$ block slides on a horizontal floor with a speed of $4m/s.$ It strikes a uncompressed spring, and compress it till the block is motionless. The kinetic friction force is $15N$ and spring constant is $10, 000$ $N/m.$ The spring compresses by
A
$8.5cm$
B
$5.5cm$
C
$2.5cm$
D
$11.0cm$
## Explanation
Let the block compress the spring by $x$ before coming to rest.
Initial kinetic energy of the block $=$ (potential energy of compressed spring) $+$ work done due to friction.
${1 \over 2} \times 2 \times {\left( 4 \right)^2} = {1 \over 2} \times 10000 \times {x^2} + 15 \times x$
$10,000{x^2} + 30x - 32 = 0$
$\Rightarrow 5000{x^2} + 15x - 16 = 0$
$\therefore$ $x = {{ - 15 \pm \sqrt {{{\left( {15} \right)}^2} - 4 \times \left( {5000} \right)\left( { - 16} \right)} } \over {2 \times 5000}}$
$\,\,\,\,\, = 0.055m = 5.5cm.$
2
### AIEEE 2006
The potential energy of a $1$ $kg$ particle free to move along the $x$-axis is given by $V\left( x \right) = \left( {{{{x^4}} \over 4} - {{{x^2}} \over 2}} \right)J$.
The total mechanical energy of the particle is $2J.$ Then, the maximum speed (in $m/s$) is
A
${3 \over {\sqrt 2 }}$
B
${\sqrt 2 }$
C
${1 \over {\sqrt 2 }}$
D
$2$
## Explanation
Velocity is maximum when kinetic energy is maximum and when kinetic energy is maximum then potential energy should be minimum
For minimum potential energy,
${{dV} \over {dx}} = 0$
$\Rightarrow {x^3} - x = 0$
$\Rightarrow x = \pm 1$
$\Rightarrow$ Min. Potential energy (P.E.) =${1 \over 4} - {1 \over 2} = - {1 \over 4}J$
$K.E{._{\left( {\max .} \right)}} + P.E{._{\left( {\min .} \right)}} = 2\,$ (Given)
$\therefore$ $K.E{._{\left( {\max .} \right)}} = 2 + {1 \over 4} = {9 \over 4}$
$\therefore$ ${1 \over 2}mv_{\max }^2$ = ${9 \over 4}$
$\Rightarrow {1 \over 2} \times 1 \times {v^2}_{\max .} = {9 \over 4}$
$\Rightarrow {v_{\max }} = {3 \over {\sqrt 2 }}$ m/s
3
### AIEEE 2006
A particle of mass $100g$ is thrown vertically upwards with a speed of $5$ $m/s$. The work done by the force of gravity during the time the particle goes up is
A
$-0.5J$
B
$-1.25J$
C
$1.25J$
D
$0.5J$
## Explanation
Kinetic energy at point of throwing is converted into potential energy of the particle during rise.
$K.E = {1 \over 2}m{v^2} = {1 \over 2} \times 0.1 \times 25 = 1.25\,J$
$W = - mgh = - \left( {{1 \over 2}m{v^2}} \right) = - 1.25\,J$
$\left[ \, \right.$ As we know, $mgh = {1 \over 2}m{v^2}$ by energy conservation $\left. \, \right]$
4
### AIEEE 2006
A ball of mass $0.2$ $kg$ is thrown vertically upwards by applying a force by hand. If the hand moves $0.2$ $m$ while applying the force and the ball goes upto $2$ $m$ height further, find the magnitude of the force. (consider $g = 10\,m/{s^2}$).
A
$4N$
B
$16$ $N$
C
$20$ $N$
D
$22$ $N$
## Explanation
According to energy conservation law,
Work done by the hand and due to gravity = total change in the kinetic energy
Initially the the ball is at rest and finally at top its velocity become zero so total change in kinetic energy $\Delta K$ = 0
${W_{hand}} + {W_{gravity}} = \Delta K$
[Here distance covered would be 0.2 meter for force by hand as force is applied while ball is in contact with hand.
And gravity will still work while ball is in contact with hand so total distance due to gravity would be 2 + 0.2 = 2.2 meter.]
$\Rightarrow F\left( {0.2} \right) - \left( {0.2} \right)\left( {10} \right)\left( {2.2} \right)$ $= 0 \Rightarrow F = 22\,N$
$\therefore$ Option (D) is correct. | {"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": 0.9423888921737671, "perplexity": 685.729408156082}, "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-2022-05/segments/1642320304287.0/warc/CC-MAIN-20220123141754-20220123171754-00092.warc.gz"} |
https://nips.cc/Conferences/2022/ScheduleMultitrack?event=63156 | Timezone: »
Fanny Yang: Surprising failures of standard practices in ML when the sample size is small.
Fanny Yang
Sat Dec 03 01:30 PM -- 01:55 PM (PST) @
In this talk, we discuss two failure cases of common practices that are typically believed to improve on vanilla methods: (i) adversarial training can lead to worse robust accuracy than standard training (ii) active learning can lead to a worse classifier than a model trained using uniform samples. In particular, we can prove both mathematically and empirically, that such failures can happen in the small-sample regime. We discuss high-level explanations derived from the theory, that shed light on the causes of these phenomena in practice. | {"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": 0.8755531907081604, "perplexity": 1516.8324496587209}, "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/1674764500671.13/warc/CC-MAIN-20230208024856-20230208054856-00726.warc.gz"} |
https://www.physicsforums.com/threads/is-this-use-of-the-grad-function-correct.670489/ | # Is this use of the grad function correct?
1. Feb 9, 2013
### CraigH
Hi,
I am currently studying vector calculus, and to try and understand it I have created a question for myself, and it would be greatly appreciated if someone could verify that I have answered my question correctly, and I have done everything right.
Okay here it is; Point Q in the room is a heat source, how does the temperature vary as you move away from the origin (0,0,0)
http://img571.imageshack.us/img571/4250/gascube.png [Broken]
http://imageshack.us/a/img841/4250/gascube.png [Broken]
.The further away from point Q you are the less heat there is. This can be modelled by:
$dt = \nabla t .dr$
.(dt) is a scalar, it is a change in temperature
.(dr) is a vector, it is a change in distance of x, y and z $(dr=dx\hat{x} +dy\hat{y} +dz\hat{z} )$
.(del t) is a vector, it is the amount t changes for x, y, and z, for a change in distance of x,y,and z
For the purposes of this example I am going to say that the heat at any point equals 1 divided by the square of the distance from q.
In the x direction
$t = \frac{1}{(xq-x)^{2}}$
In the y direction
$t = \frac{1}{(yq-y)^{2}}$
In the z direction
$t = \frac{1}{(zq-z)^{2}}$
Where (xq,yq,zq)=(0,2.5,2.5) And (x,y,z) is the point you are at
So, $\frac{dt}{dx} = 2(xq-x)^{-1}, \frac{dt}{dy} = 2(yq-y)^{-1}, \frac{dt}{dz} = 2(zq-z)^{-1}$
$dt =(2(xq-x)^{-1}\hat{x} + 2(yq-y)^{-1}\hat{y} + 2(zq-z)^{-1}\hat{z}) . dr$
Is this correct?
Thank you
Last edited by a moderator: May 6, 2017
2. Feb 9, 2013
### Mmm_Pasta
The gradient points in the direction of greatest increase. Since temperature is decreasing, you will want to take the negative gradient of your function. You also differentiated incorrectly. The derivative of x^(-2) is not 2^(-1). It's -2^(-3). Remember: d/dx(x^n) = nx^(n-1).
3. Feb 9, 2013
### CraigH
Ah yes I can see I have differentiated incorrectly, although I think it would be 2(xq-x)^-3 and not -2(xq-x)^-3 because of the chain rule:
derivative of (xq-x) with repect to x = -1, so the whole thing will be multiplied by -1.
And can you explain exactly what I did wrong apart from this, I do not really understand why I should have to take the negative gradient, or what this actually even means.
When you say:
I've always thought that this just meant that if you plot the gradient vector it would point in the direction of greatest increase, but that was just a property of the gradient function, and the true reason you would use the gradient function is to see how much change one variable would change (dt) as three other variables are changing (dr)
$dt = \nabla .dr$
4. Feb 9, 2013
### Mmm_Pasta
I totally missed that -x part. Thanks!
The fact that the gradient points in the direction of greatest increase is the reasons for the negative sign in your case. The temperature is decreasing as you go away from the source. Without this negative sign, the resulting vector field would be pointing away from the source which implies that temperature is increasing.
By multiplying the gradient by -1, the resulting vector field would be pointing towards the source which means that temperature increases towards the source. Gradients are not only used for seeing *how much* a variable changes, but we are also interested in *which direction* the greatest increase/decrease occurs (for other directions we use directional derivatives). In your case, you specified that temperature drops away from the source.
It's a subtle point, but it's important. | {"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": 0.8158902525901794, "perplexity": 471.1174939479189}, "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-2017-34/segments/1502886107720.63/warc/CC-MAIN-20170821060924-20170821080924-00415.warc.gz"} |
https://www.physicsforums.com/threads/4-momentum-problem.90142/ | # 4-momentum problem
1. Sep 21, 2005
### coregis
I'm having trouble understanding conservation of 4-momentum. My problem is about dertermining the threshold for triplet production from a photon and an electron using 4-momentum conservation. The answer is 4m0c^2. So far, I say the initial energy is:
E1^2=(pc)^2 + (m0c^2)^2=(hv)^2+(m0c^2)^2
At threshold energy after collision, three particles with no momentum are produced, so
E2^2=(3m0c^2)
E1=E2, but substituting and solving for hv does not work out correctly, and I have not used 4-momentum at all anyhow. What am I doing wrong/forgetting? Help!
2. Sep 21, 2005
### David
It might help to know what the question actually asks.
3. Sep 21, 2005
### coregis
verbatim- "Show the threshold for triplet production is 4m0c^2 using the concept of conservation of 4-momentum." Sorry for the paraphrase.
Last edited: Sep 21, 2005 | {"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": 0.8077260255813599, "perplexity": 3205.679973150261}, "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-2017-43/segments/1508187823630.63/warc/CC-MAIN-20171020025810-20171020045810-00238.warc.gz"} |
http://mathhelpforum.com/calculus/126467-volumes-revolution-print.html | # Volumes of Revolution
• Jan 31st 2010, 12:51 PM
dalbir4444
Volumes of Revolution
Calculate the volume of the solid obtained when the triangle with vertices (2,5), (6,1) , (4,4) is rotated about the line x=-3 using either cylindrical shells or disks.
I found the slope for each line, and found an equation fro each. Then, using cylindrical shells with radius x+3, I found the volume generated by each line and then subtracted the results. However, I got a negative value.
• Jan 31st 2010, 04:45 PM
TKHunny
Couple of things wrong with this:
1) Lines don't tend to generate much volume.
2) You have three lines. Which did you subract from which?
3) What were you integration limits? What did your integrals look like?
4) Have you ever heard of Pappas' Theorem?
5) Please show your work. Someone will guide you once you demonstrate where it is that you need guidance.
• Jan 31st 2010, 05:29 PM
dalbir4444
My first integral was 2pi(3+x)(7-x) in the interval [2,6].
My second integral was 2pi(3+x)(-0.5x+6) over the interval [2,4].
My third integral was 2pi(3+x)(-1.5x+10) over the interval [4,6].
I subtracted the second and third integral from the first one. We haven't learned Pappas' Theorem yet, so we can only use cylindrical shells or disks method.
• Jan 31st 2010, 08:38 PM
TKHunny
You're upside down. y = 7-x is less than the other two on [2,6].
That's a lot of nice calculus to go down with an analytic geometry error.
• Feb 1st 2010, 09:16 AM
dalbir4444
That's why I was getting a negative answer, but I can't seem to visualize the volume of revolution. Am I on the right path, and if so, which ones am I supposed to subtract?
• Feb 1st 2010, 10:21 AM
TKHunny
1) Visualization should become unnecessary. If you learn to rely on it with so few variables, what will you do when the number increases?
2) Part of your difficulty visualizing, I think, is your unusual construction. You probably cannot visualise subtraction of volumes. If you were to define the two pieces that you want as single entities, it may help. For one thing, it will get rid of all the unnecessary distractions down to the x-axis.
In this example, using just the domains of your integrals, rather than [2,4]+[4,6]-[2,6], as you have it, try ([2,4]-[2,4])+([4,6]-[4,6]) - as in, also split up the subtraction piece into rational chunks. It's much more palatable for your eyes.
• Feb 1st 2010, 05:26 PM
dalbir4444
I don't completely understand what you mean. I know that the limits of my integrals are incorrect, but what I'm not sure of is why they are incorrect. You said " try ([2,4]-[2,4])+([4,6]-[4,6]) - as in, also split up the subtraction piece into rational chunks". Which integrals am I supposed to take over these limits?
• Feb 1st 2010, 07:23 PM
TKHunny
I knew that description with just the limits would not be clear. I regretted it when I hit the [submit] button.
You have these
$2\pi\cdot\int_{2}^{4}(x+3)\left(6-\frac{x}{2}\right)\;dx$
$2\pi\cdot\int_{4}^{6}(x+3)\left(10-\frac{3}{2}x\right)\;dx$
$2\pi\cdot\int_{2}^{6}(x+3)(7-x)\;dx$
These three chunks are "right" and the do solve the problem. They do not contribute to your visualization. This next version should contribute to your visualization.
$2\pi\cdot\int_{2}^{4}(x+3)\left[\left(6-\frac{x}{2}\right)-(7-x)\right]\;dx$
$2\pi\cdot\int_{4}^{6}(x+3)\left[\left(10-\frac{3}{2}x\right)-(7-x)\right]\;dx$
It's more a picture of the pieces you want. Same thing, just a different presentation. | {"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": 0.8847072720527649, "perplexity": 1053.2299095439296}, "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-2017-09/segments/1487501172017.60/warc/CC-MAIN-20170219104612-00583-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://www.researchgate.net/researcher/47870530_M_Takashina | # M. Takashina
Osaka University, Suita, Osaka-fu, Japan
Are you M. Takashina?
## Publications (36)84.98 Total impact
• ##### Article: Effect of tensor interactions in 16O studied via (p,d) reaction
[Hide abstract]
ABSTRACT: The differential cross sections of the 16O(p,d) reaction populating the ground state and several low-lying excited states in 15O were measured using 198-, 295- and 392-MeV proton beams at the Research Center for Nuclear Physics (RCNP), Osaka University, to study the effect of the tensor interactions in 16O. Dividing the cross sections for each excited state by the one for the ground state and comparing the ratios over a wide range of momentum transfer, we found a marked enhancement of the ratio for the positive-parity state(s). The observation is consistent with large components of high-momentum neutrons in the ground-state configurations of 16O due possibly to the tensor interactions.
Journal of Physics Conference Series 12/2014; 569(1).
• ##### Article: Probing effect of tensor interactions in 16O via (p, d) reaction
[Hide abstract]
ABSTRACT: We have measured the 16O(p, d) reaction using 198-, 295- and 392-MeV proton beams to search for a direct evidence on an effect of the tensor interactions in light nucleus. Differential cross sections of the one-neutron transfer reaction populating the ground states and several low-lying excited states in 15O were measured. Comparing the ratios of the cross sections for each excited state to the one for the ground state over a wide range of momentum transfer, we found a marked enhancement of the ratio for the positive-parity state(s). The observation is consistent with large components of high-momentum neutrons in the initial ground-state configurations due to the tensor interactions.
Physics Letters B 10/2013; 725(4-5):277-281. · 6.02 Impact Factor
• ##### Article: Erratum: Global optical potential for nucleus-nucleus systems from 50 MeV/u to 400 MeV/u [Phys. Rev. C 85, 044607 (2012)]
Physical Review C 06/2012; · 3.88 Impact Factor
• Source
##### Article: Possible evidence of tensor interactions in 16O observed via (p,d) reaction
[Hide abstract]
ABSTRACT: We have measured 16O(p,d) reaction using 198-, 295- and 392-MeV proton beams to search for a direct evidence on the effect of the tensor interactions in light nucleus. Differential cross sections of the one-neutron transfer reactions populating the ground states and several low-lying excited states in 15O were measured. Comparing the ratios of the cross sections for each excited state to the one for the ground state over a wide range of momentum transfer, we found a marked enhancement for the positive-parity state(s). The observation indicates large components of high-momentum neutrons in the initial ground-state configurations, due possibly to the tensor interactions.
The European Physical Journal Conferences 05/2012; 66.
• Source
##### Article: Global optical potential for nucleus-nucleus systems from 50 MeV/u to 400 MeV/u
[Hide abstract]
ABSTRACT: We present a new global optical potential (GOP) for nucleus-nucleus systems, including neutron-rich and proton-rich isotopes, in the energy range of \$50 \sim 400\$ MeV/u. The GOP is derived from the microscopic folding model with the complex \$G\$-matrix interaction CEG07 and the global density presented by S{\~ a}o Paulo group. The folding model well accounts for realistic complex optical potentials of nucleus-nucleus systems and reproduces the existing elastic scattering data for stable heavy-ion projectiles at incident energies above 50 MeV/u. We then calculate the folding-model potentials (FMPs) for projectiles of even-even isotopes, \$^{8-22}\$C, \$^{12-24}\$O, \$^{16-38}\$Ne, \$^{20-40}\$Mg, \$^{22-48}\$Si, \$^{26-52}\$S, \$^{30-62}\$Ar, and \$^{34-70}\$Ca, scattered by stable target nuclei of \$^{12}\$C, \$^{16}\$O, \$^{28}\$Si, \$^{40}\$Ca \$^{58}\$Ni, \$^{90}\$Zr, \$^{120}\$Sn, and \$^{208}\$Pb at the incident energy of 50, 60, 70, 80, 100, 120, 140, 160, 180, 200, 250, 300, 350, and 400 MeV/u. The calculated FMP is represented, with a sufficient accuracy, by a linear combination of 10-range Gaussian functions. The expansion coefficients depend on the incident energy, the projectile and target mass numbers and the projectile atomic number, while the range parameters are taken to depend only on the projectile and target mass numbers. The adequate mass region of the present GOP by the global density is inspected in comparison with FMP by realistic density. The full set of the range parameters and the coefficients for all the projectile-target combinations at each incident energy are provided on a permanent open-access website together with a Fortran program for calculating the microscopic-basis GOP (MGOP) for a desired projectile nucleus by the spline interpolation over the incident energy and the target mass number.
Physical Review C 04/2012; 85(4). · 3.88 Impact Factor
• ##### Article: Three-body model analysis of α+ d elastic scattering and the^{2} H (α, γ)^{6} Li reaction in complex-scaled solutions of the Lippmann-Schwinger equation
[Hide abstract]
ABSTRACT: We investigate the α+d elastic scattering and the radiative capture reaction of 2H(α, γ)6Li based on the α + p + n three-body model. The α+d scattering states are described by using the complex-scaled solutions of the Lippmann-Schwinger equation. We calculate the elastic phase shifts for the α+d scattering and the radiative capture cross section of 6Li. We evaluate the contributions of the α + p + n structures in those observables. It is found that in the α+d scattering process, the deuteron breakup and the rearrangement to the 5He + p and 5Li + n channels play prominent roles in reproducing the observed phase shifts and radiative capture cross section.
Physical Review C 12/2011; 84(6). · 3.88 Impact Factor
• ##### Article: Two-Neutron Correlations in Halo Nuclei via Coulomb Breakup Reactions
[Hide abstract]
ABSTRACT: We investigate the three-body Coulomb breakup of a two-neutron halo nucleus, 6He. We calculate the breakup cross section and the invariant mass spectra by using the complex-scaled solutions of the Lippmann-Schwinger equation, and discuss the relations between the structures in these observables and the n-n and alpha-n correlations in 6He.
International Journal of Modern Physics E 04/2011; 20:843-846. · 0.84 Impact Factor
• Source
##### Article: Enhanced collectivity in 74Ni
[Hide abstract]
ABSTRACT: The neutron-rich nucleus 74Ni was studied with inverse-kinematics inelastic proton scattering using a 74Ni radioactive beam incident on a liquid hydrogen targetat a center-of-mass energy of 80 MeV. From the measured de-excitation gamma-rays, the population of the first 2+ state was quantified. The angle-integrated excitation cross section was determined to be 14(4) mb. A deformation length of delta = 1.04(16) fm was extracted in comparison with distorted wave theory, which suggests that the enhancement of collectivity established for 70Ni continues up to 74Ni. A comparison with results of shell model and quasi-particle random phase approximation calculations indicates that the magic character of Z = 28 or N = 50 is weakened in 74Ni.
Physics Letters B 08/2010; · 6.02 Impact Factor
• ##### Article: DiNeutron Correlation in 6He Through Coulomb Breakup Reactions
[Hide abstract]
ABSTRACT: We investigate the three-body Coulomb breakup of a two-neutron halo nucleus, 6He. The three-body scattering states of 6He are described by using the Complex-scaled solutions of the Lippmann-Schwinger equation. We calculate the breakup cross section and the invariant mass spectra, and discuss the relations between the structures in these observables and the n-n and alpha-n correlations of 6He.
Modern Physics Letters A 07/2010; 25:1907-1910. · 1.34 Impact Factor
• ##### Article: Elastic scattering of B from ¹²C with internal three-cluster structure of B
[Hide abstract]
ABSTRACT: We study theoretically the elastic scattering of B from ¹²C at E{sub lab}=95 MeV. The B nucleus consists of weakly bound Be and proton, while the Be nucleus has an internal cluster structure of alpha+³He. We treat the last proton in B in the adiabatic recoil approximation and also take into account the excitation of Be including resonance states by a coupled-channel method with consideration of the cluster structure. It turns out that the excitation to the resonance state of Be in B is important for the B elastic scattering.
Physical Review C 06/2010; 81(6):061602-061602. · 3.88 Impact Factor
• ##### Article: ^{16} O+^{16} O inelastic scattering studied by a complex G-matrix interaction
M. Takashina, T. Furumoto, Y. Sakuragi
[Hide abstract]
ABSTRACT: We reanalyze the 16O+16O inelastic scattering to the single 21+ and 31- excitation channels at an incident energy of Elab=1120 MeV by a microscopic coupled-channel (CC) calculation with complex G-matrix interaction. The imaginary part of the folded potential is renormalized so as to reproduce the elastic angular distribution. It is found that the results of full CC calculations are still inconsistent with the experimental data for both 21+ and 31- channels. It is also found that when the imaginary coupling terms are omitted, the calculated 21+-channel cross sections are improved, although the 31--channel cross sections are slightly overestimated.
Physical Review C 04/2010; 81(4). · 3.88 Impact Factor
• ##### Article: Two-neutron correlations in He in a Coulomb breakup reaction
[Hide abstract]
ABSTRACT: We investigate the three-body Coulomb breakup of a two-neutron halo nucleus, He. Based on the alpha+n+n model, the three-body scattering states of He are described by using the combined methods of the complex scaling and the Lippmann-Schwinger equation. We calculate the breakup cross section, the two-dimensional energy distributions, and the invariant mass spectra for the E1 transition of He. We discuss the relations between the structures in these strengths and the n-n and alpha-n correlations of He. It is found that the He resonance in the final states contributes to make a low-energy enhancement of the strength. The n-n final-state interaction also contributes to enhance the strength globally. However, the ground-state correlations of He, such as a dineutron, are difficult to recognize in the strength because of the dominant effect of the final-state interaction.
Physical Review C 04/2010; 81(4):044308-044308. · 3.88 Impact Factor
• ##### Article: Complex-Scaled CDCC method for nuclear breakup reactions
[Hide abstract]
ABSTRACT: Nuclear breakup process is very important for light unstable nuclei (typically halo nuclei) induced reactions because of their weak-binding nature. The continuum-discretized coupled-channel (CDCC) method is known to be one of the powerful method to describe the nuclear breakup reaction. Indeed, CDCC has been applied to a number of analyses for the breakup reactions of both the stable and unstable nuclei, and the successful results have been obtained. In the present study, we propose complex-scaled CDCC (CS-CDCC) method, in which only the internal coordinate and momentum of the projectile are complex-scaled. The expected advantages of CS-CDCC are (1) in spite of the discretization, we can obtain the continuous S matrix elements without the smoothing function, because the continuum level density is correctly obtained, (2) in the framework of the complex scaling method, three-body scattering state can be solved properly, (3) the resonance state is strictly separated from the continuum states, and this fact is more advantageous for investigation of reaction mechanism than the ordinary CDCC method. We apply CS-CDCC to the d ->p+n breakup reaction on a ^58Ni target at Ed=80 MeV to confirm the availability of CS-CDCC. We also plan to apply it to breakup reactions of light unstable nuclei.
10/2009;
• ##### Article: Subsystem correlations in soft E1 excitation of ^11Li
[Hide abstract]
ABSTRACT: The ^11Li nucleus has characteristc features of neutron-rich nuclei such as two-neutron halo structure and large s-wave mixing in the ground state, and has been studied with keen interest from both theoretical and experimental sides. Experimentally, the Coulomb breakup reactions have been performed to investigate the exotic features of ^11Li, and significant E1 strength was measured at low excitation energy. However, the nature of this soft E1 excitation for ^11Li is not clearly understood. To understand the nature of the soft E1 excitation, it is necessary to understand the complicated structure of ^11Li, which contains both ^9Li-n and n- n correlations. In the present study, we investigate soft E1 excitation for ^11Li based on the core+n+n three-body model. We analyze the E1 strength as a function of relative energies in binary subsystems in ^11Li, and discuss the correlations of ^9Li-n and n-n subsystems through the soft E1 excitation.
10/2009;
• ##### Article: Multistep effect in^{16} O+^{16} O inelastic scattering
M. Takashina, Y. Sakuragi
[Hide abstract]
ABSTRACT: We have analyzed 16O+16O inelastic scattering to the single 21+ and 31- channels at incident energies of Elab=350 and 1120 MeV by a microscopic coupled-channel calculation. We have found that the strong rotational coupling among the channels of the α+12Cg.s. type cluster states plays an important role in reproducing the angular distribution of the 21+ channel for both energies. We have also found that the coupling with the 02+, 21+, and 41+ channels as well as those among the shell channels has a large effect on the 31- channel cross sections at Elab=350 MeV. However, their absolute values could not be reproduced by our calculation, unless we assumed that the strengths of the 31-→02+,21+,41+ transitions were unphysically strong.
Physical Review C 07/2009; 80(1). · 3.88 Impact Factor
• ##### Article: Low-lying states in 32Mg studied by proton inelastic scattering
[Hide abstract]
ABSTRACT: Proton inelastic scattering on the neutron-rich nucleus 32Mg has been studied at 46.5 MeV/nucleon in inverse kinematics. Populated states were identified by measuring de-excitation γ rays, in which five new states were found by γ-γ coincidence analyses. By analyzing the angular differential cross sections via coupled-channel calculations, their spins and parities were constrained and the amplitudes for each transition were extracted. The spin and parity of the 2321-keV state was assigned as 41+. The ratio between the energies of the 21+ and 41+ states indicates that 32Mg is a transitional nucleus rather than an axially deformed rigid rotor. The collectivities in the nucleus 32Mg with N=20 are discussed based on the results obtained in the present experiment.
Phys. Rev. C. 05/2009; 79(5).
• Source
##### Article: Coulomb Breakup Reactions in Complex-Scaled Solutions of the Lippmann-Schwinger Equation
[Hide abstract]
ABSTRACT: We propose a new method to describe three-body breakups of nuclei, in which the Lippmann-Schwinger equation is solved combining with the complex scaling method. The complex-scaled solutions of the Lippmann-Schwinger equation (CSLS) enables us to treat boundary conditions of many-body open channels correctly and to describe a many-body breakup amplitude from the ground state. The Coulomb breakup cross section from the 6He ground state into 4He+n+n three-body decaying states as a function of the total excitation energy is calculated by using CSLS, and the result well reproduces the experimental data. Furthermore, the two-dimensional energy distribution of the E1 transition strength is obtained and an importance of the 5He(3/2-) resonance is confirmed. It is shown that CSLS is a promising method to investigate correlations of subsystems in three-body breakup reactions of the weakly-bound nuclei. Comment: 12 pages, 6 figures, submitted to Progress of Theoretical Physics; section 2.4 added, 2 equations added, 1 equation replaced
Progress of Theoretical Physics 05/2009; · 2.06 Impact Factor
• ##### Article: Persistent decoupling of valence neutrons toward the dripline: Study of 20C by γ spectroscopy
[Hide abstract]
ABSTRACT: The very neutron-rich nucleus 20C has been investigated by inelastic scattering on 208Pb and liquid hydrogen targets. Through distorted wave analysis, the reduced electric quadrupole transition probability B(E2;0g.s.+→21+)<18.4 (stat) e2 fm4 and the neutron transition probability Mn2=292±52 (stat) fm4 have been derived. A simple shell model calculation has shown a need for a factor of about 0.4 decrease of the normal polarization charges to elucidate the results. This is interpreted as a decoupling of the valence neutrons from the nuclear core in carbon isotopes heavier than 14C.
Physical Review C 01/2009; 79(1). · 3.88 Impact Factor
• ##### Article: Analysis on Two-Neutron Correlation Through Dalitz Plot of Coulomb Breakup Reaction for 6He
[Hide abstract]
ABSTRACT: A new method to describe the three-body decaying states is developed. In this method, the Lippmann-Schwinger equation is combined with the complex scaling method to take the correct boundary condition into account. For the application, the E1 transition strength of 6He is investigated. The energy and Dalitz distributions of the E1 transition are calculated, and the internal correlation of 6He is discussed. As results, it is found that the 6He --> 5He(3/2-)+n --> 4He+n+n sequential decay process is dominant in the E1 transition reaction of 6He.
International Journal of Modern Physics E 11/2008; 17(10):2368-2373. · 0.84 Impact Factor
• ##### Article: Interpretation of a diffraction phenomenon observed in the angular distribution of α inelastic scattering on^{12} C exciting the 0_ {2}^{+} state
M. Takashina
[Hide abstract]
ABSTRACT: We analyze α inelastic scattering on 12C exciting the 02+ state at the incident energy of Eα=240 MeV using the eikonal wave Born approximation and investigate what determines the oscillation pattern appearing in the inelastic angular distribution. We present an interpretation for the results obtained by the present author and his collaborator [Phys. Rev. C 74, 054606 (2006)] that the oscillation pattern in inelastic angular distribution is almost independent of the nuclear radius of 12C(02+).
Physical Review C 07/2008; 78(1). · 3.88 Impact Factor
#### Publication Stats
177 Citations 84.98 Total Impact Points
#### Institutions
• ###### Osaka University
• • Graduate School of Medicine
• • Research Center for Nuclear Physics
Suita, Osaka-fu, Japan
• ###### Kyoto University
• Yukawa Institute for Theoretical Physics
Kyoto, Kyoto-fu, Japan
• ###### RIKEN
Вако, Saitama, Japan
• ###### Osaka City University
• Department of Physics
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http://www.nature.com/articles/s41598-017-00962-7?error=cookies_not_supported&code=98f7f3b0-dd58-42cd-bfe8-c19d45908387 | Article | Open
# Investigation of spin-orbit torque using current-induced magnetization curve
• Scientific Reportsvolume 7, Article number: 790 (2017)
• doi:10.1038/s41598-017-00962-7
Accepted:
Published online:
## Abstract
Manipulation of magnetization using current-induced torque is crucial for magnetic recording devices. Recently, the spin-orbit torque (SOT) that emerges in a ferromagnetic thin film on a heavy metal is focused as a new scheme for magnetization switching in perpendicularly magnetized systems. Since the SOT provides a perpendicular effective field to the system, the formation of a magnetic multiple domain state because of Joule heating is supressed in the magnetization reversal process. This means that high reliable switching is possible using the SOT. Here, by utilizing the SOT induced domain stability, we show that an electrical current directly injected to a perpendicularly magnetized Pt/Co/Pd system can magnetize itself, that is, current-induced magnetization process from multi to single domain state. A quantitative determination of the SOT is performed using the current-induced magnetization curve. The present results are of great importance as another approach to evaluate the SOT effect, as well as a demonstration of domain state switching caused by the SOT.
## Introduction
In magnetic recording devices, the information of a bit is retained as a magnetization direction. Spin-torque-induced magnetization switching1,2,3,4,5,6 in magnetic tunnel junctions (MTJs) and current-induced domain wall (DW) displacement7,8,9,10,11,12,13 in magnetic wires have been widely investigated as information writing methods in magnetic memory. Recently, a current-induced spin-orbit torque (SOT) emerging in a thin ferromagnetic film deposited on a heavy metal layer has been recognized as a new scheme for magnetization switching in perpendicularly magnetized materials14,15,16,17,18,19. Moreover, it is well known that current-induced DW dynamics is strongly affected by the SOT20,21,22,23,24.
The SOT is known to act on magnetizations as an effective field. This contributes to the magnetization switching and DW motion. More importantly, the SOT effective field also plays a role in stabilizing the domain state against current-induced Joule heating25, 26, which is often a severe problem in devices based on current-induced torque27,28,29. Owing to this characteristic of the SOT, random multiple domain formation is suppressed even under high current density as much as 1011–1012 A/m2 during the magnetization reversal. In this letter, we show that in a perpendicularly magnetized Pt/Co/Pd structure with a multi domain (MD) state in thermal equilibrium (having no net magnetization), the magnetization of the system increases with injected current, and finally, a single domain (SD) state is created (see Fig. 1a). This “current-induced” magnetization process is observed only when an external in-plane magnetic field parallel to the current exists, indicating that the SOT is responsible for the effect. A quantitative determination of the SOT effective field from the current-induced magnetization curve is performed.
Figure 1b shows an optical microscope image of our device. The Hall bar component consists of asymmetrical Pt/Co/Pd layers deposited on a SiO2/intrinsic Si substrate (see also Methods). The width of the channel is 5 μm. Owing to the interface magnetic anisotropy at the Pt/Co and Co/Pd interfaces, the system exhibits perpendicular magnetic anisotropy PMA. The high PMA realizes an MD state with perpendicular magnetization near the Curie temperature (~370 K in our device). Four Cr/Cu electrodes are formed to apply the current and to detect the Hall resistance RHall. The definitions of each external magnetic field (μ0H x , μ0H y , and μ0H z ) and the current flow along the x-axis are indicated in Fig. 1b. The temperature of the stage Ts, which is in thermal contact with the fabricated device, is controlled using a heater (see Methods).
Figure 2a shows the results of the RHall measurement when a dc current density Jdc of +2.8 × 109 A/m2 is injected at room temperature (Ts = 304 K). μ0H z is swept to obtain the curves. Jdc is determined by simply dividing the current by the cross-sectional area of the metallic layers. RHall is proportional to the perpendicular component of the magnetization because of the anomalous Hall effect. A clear hysteresis loop with a coercivity μ0Hc of 2.4 mT is observed. The remanent value of RHall (RHallr) is almost equal to the saturation value, indicating that the SD state is stable at fields near zero.
Next, RHall measurements with Jdc sweeping are performed under constant in-plane magnetic fields. The procedure is as follows. First, the magnetization direction of the entire device is set upward by applying a μ0H z of +30 mT. After returning to ~0 T, a constant in-plane magnetic field (μ0H x or μ0H y ) of +38 mT is applied. Then, RHall is monitored with a sweeping Jdc. The positive (negative) sweep corresponds to a Jdc change from 0 to +1.7 (−1.7) × 1011 A/m2. Figure 2b and c show RHall as a function of Jdc measured under the application of μ0H y and μ0H x , respectively. As shown in the positive sweep (PS) in Fig. 2b, RHall abruptly decreases when a Jdc of +0.8 × 1011 A/m2 is applied, and drops toward zero in the region of Jdc > +0.8 × 1011 A/m2, indicating that the MD state is formed at Jdc ≥ 0.8 × 1011 A/m2 due to the Joule heating. A rapid decrease in RHall also appears symmetrically in the negative sweep (NS) case (see NS curve in Fig. 2b), suggesting that the Joule heating effect is independent of the Jdc direction.
By contrast, the situation is completely different when μ0H x , which is parallel to Jdc, is applied. For the PS curve in Fig. 2c, a full RHall switching from positive to negative, which corresponds to the abrupt reversal of the magnetization direction from up to down, is observed at Jdc = +0.5 × 1011 A/m2. In the NS curve, no switching occurs up to Jdc = −1.7 × 1011 A/m2. When the sign of μ0H x is reversed, switching is observed only for a negative Jdc (not shown). The switching direction in the present configuration is consistent with the SOT induced switching previously reported in the Pt/Co structure2, 26. Although the sign of the spin Hall angle of the top Pd is the same as that of the Pt, the SOT in the present device is dominated by the spin current injection from the bottom Pt because the magnitude of the spin Hall angle of Pd is one order smaller than that of Pt30, 31. We also checked that in a similar device structure, the top Pd effect on the SOT is negligibly small32. In order to evaluate the magnitude of SOT effective fields, i.e. the Slonczewski-like torque μ0HSL and field-like torque μ0HFL, the harmonic Hall measurement, which is widely used for the quantitative determination of the SOT33, was conducted for a similar device structure. The procedure is shown in the Supplementary Information. μ0HSL and μ0HFL are determined to be 6.55 ± 0.20 mT/1011 A/m2 and 1.57 ± 0.13 mT/1011 A/m2, respectively. These values are close to those previously reported for Pt/ferromagnet bilayer structure22. Another important point here is that under μ0H x application, |RHall| always represents a value close to saturation even at |Jdc| ≥ 0.8 × 1011 A/m2, indicating that the MD formation is completely suppressed against Joule heating. In the present case, the magnetization experiences a finite perpendicular effective field μ0Heff derived from μ0HSL because the magnetization tilts slightly toward the x-direction. When the current direction is the same as the μ0H x direction, the sign of μ0Heff becomes negative in the Pt/Co system. As a result, the SD state becomes stable because of the gain by the Zeeman energy reduction. Thus, the SOT plays an important role in stabilizing the SD state.
In the above experiments, Td was increased by injecting a current. In the following, the MD state at thermal equilibrium is prepared by simply increasing Td using a heater. Figure 3a shows the results of the Hall measurements performed at Ts = 343 K. Jdc for this measurement is 2.8 × 109 A/m2, and the Joule heating effect is negligibly small. In this case, an anhysteric RHall curve with the almost zero remanent is observed. This indicates that the MD state is realized and the system is demagnetized. Figure 3b shows RHall as a function of Jdc obtained under μ0H z of ~0 T. RHall is almost independent of Jdc in the range of ±0.5 × 1011 A/m2. Similarly, no change in RHall is observed for a μ0H y of +38 mT, as shown in Fig. 3c. The slight RHall deviation from zero is probably due to the small z-component of the field. The slight Jdc dependence on RHall shown in Fig. 3b and c might be a result of the Oersted field. Based on these results, it can be concluded that in both cases, the MD state is kept under current injection. The result obtained under μ0H x = +38 mT is completely different, as shown in Fig. 3d. For a small Jdc, RHall showed an intermediate value, indicating that the MD state remains. However, a clear increase and decrease in RHall with increasing current toward the negative and positive magnetization directions, respectively, are observed. For both current directions, RHall saturates above |Jdc| = 0.1–0.2 × 1011 A/m2 and the RHall saturation values are consistent with those for the SD state. This result indicates that magnetization process of the system is caused by the SOT.
In order to check whether the device is fully in the SD state, a cooling experiment under current is carried out. The procedure is as follows. After making RHall saturation using current under μ0H x = +38 mT, Td is gradually decreased from 343 K to room temperature with remaining the current and x-field application. Jdc of −0.25 × 1011 A/m2 is continuously applied during the cooling. During the Td decrease, RHall is monitored. The result is shown in Fig. 3e. One can see that RHall gradually increases with decreasing Td and at room temperature it shows ~0.4 Ω, which is the value for the SD state. This indicates that the SD state is created by the current at 343 K and is maintained during cooling. On the other hand, when the above experiment is performed with an injection of Jdc = −2.8 × 109 A/m2, RHall shows a small value of ~0.1 Ω even at room temperature, suggesting that the MD state formed at 343 K is retained even at room temperature. Therefore, the RHall saturation in Fig. 3(d) corresponds to the complete SD state, i.e. a situation where no domain with an opposite magnetization exists in the device is realized against thermal agitation. The result presented here demonstrates that the magnetization curve of Pt/Co/Pd system can be obtained by only sweeping electrical current owing to the stability caused by the SOT. In addition, the magnetization direction can be reversibly controlled by simply changing the current polarity.
The magnetization processes shown in Fig. 3(a) and (d) are caused by μ0H z and μ0Heff, which is proportional to Jdc, respectively. The current-induced magnetization process is expected to develop with the DW motion in the current direction, while the magnetic domains isotropically expands when the magnetization process is caused by the external field. Here, we focus on the gains of the Zeeman energy for each case and determine μ0Heff, and consequently μ0HSL, by comparing them. First, EZ obtained by μ0H z (EZ_ H ) is calculated for the up magnetization case. Figure 4(a) shows the normalized RHall (RHalln) as a function of μ0H z in the range from 0 to +4.0 mT. EZ_ H can be calculated using the following equation:
$E Z _ H = M s ∫ 0 1 d R Hall n μ 0 H z ,$
(1)
where Ms is the saturation magnetization of the system. The integral term of (1) is defined by the coloured area of Fig. 4(a), and EZ_ H /Ms is determined to be 0.180 mT. Subsequently, using the current-induced magnetization curve shown in Fig. 3(d), EZ resulting from μ0Heff (EZ_ J ) is calculated. RHalln as a function of Jdc in the negative Jdc sweep, where the positive μ0Heff is applied to the system, is shown in Fig. 4(b). The small Jdc offset is corrected in this figure. Since μ0Heff is expected to be proportional to Jdc, EZ_ J can be determined from the following:
$E Z _ J =− M s ∫ 0 1 d R Hall n α J dc ,$
(2)
where α is the constant value defined as μ0Heff/Jdc. The calculation of EZ_ J /Ms is done in the same manner and is determined to be 0.662α × 1010 A/m2. The same calculations are performed for the down magnetization case. Assuming EZ_ H /Ms = EZ_ J /MS, α = 2.51 ± 0.01 mT/1011 A/m2 is finally obtained.
Next, we calculated μ0HSL (per 1011 A/m2) using μ0Heff determined above. Figure 4(c) shows a schematic where the direction of μ0HSL under the x-field is denoted. μ0Heff is the perpendicular component of μ0HSL and θ is the tilting angle of the magnetization from the z-axis. Thus, μ0HSL is expressed as μ0Heff/sinθ. From the x-field dependence of the anomalous Hall resistance, θ at μ0H x = +38 mT is approximately 19° in our Pt/Co/Pd structure. Therefore, μ0HSL of 7.67 ± 0.04 mT/1011 A/m2 is obtained. This value shows good agreement with that obtained by the harmonic measurement in our Pt/Co/Pd system. μ0Heff was also calculated using a Langevin fit and the result is consistent with the direct Zeeman energy calculation (see Supplementary Information). The calculation results presented here indicate that the SOT effective field can be quantitatively determined from the current-induced magnetization curve.
Finally, we demonstrate current-induced alternate switching between the MD and SD states. Figure 5(a) shows the sequence of Jdc injection into the device. A pulsed Jdc with three values of +0.4 × 1011, +2.7 × 109, and −0.4 × 1011 A/m2 is injected in series to create the SD state with up and down magnetization. The duration of each Jdc pulse is 1.0 s, and a μ0H x of +38 mT is applied during the measurement. The RHall value is measured during the pulse injection, and the result is shown in Fig. 5(b). Maximum and minimum RHall values corresponding to the up and down SD states appear alternately with the injections of +0.4 and −0.4 × 1011 A/m2. At Jdc = 2.7 × 109 A/m2, RHall always exhibits an intermediate value, that is, the MD state is restored. We checked that the current-induced SD state returns to the MD state within 1 ms after pulse off. This result indicates that an arbitrary switching of domain states between MD and SD can be achieved by injecting a current. Although in the present case, 1-ns-long pulses were used for the convenience of the measurement, sub-ns domain state switching is expected to be possible because the SOT induced magnetization switching occurs in this time scale34.
When the SOT effectively acts on the magnetization, the electrical current flowing in the system enhances the stability of the single domain state. In this study, owing to this domain stability, we show that the magnetization curve of the nonmagnetic/ferromagnetic metal structure can be obtained by sweeping electrical current, while it is conventionally obtained by sweeping external magnetic field. This effect would be marked in smaller size applicable to current IT devices because the SOT field is proportional to the current density flowing in the heavy metal layer. Anisotropy-wedged film16, an interlayer exchange coupled system17, and an antiferromagnet/ferromagnet layered structure18, where in-plane field free magnetization switching by the SOT was achieved, may also enable domain state switching without an in-plane field. In addition, this work offers a novel method to determine the SOT effective field from the current-induced magnetization curve.
## Methods
### Film deposition and device fabrication
Multilayered Ta (2.7 nm)/Pt (3.0)/Co(0.36)/Pd (0.8) film was deposited on a thermally oxidized Si substrate using rf sputtering, and a 0.5-nm-thick Ta layer was formed on the film as a cap. The base pressure of the sputter chamber was below 1.0 × 10−6 Pa, and Xe process gas was used for the deposition. The X-ray diffraction profile indicates that the Pt layer has an fcc (111) texture. The film was patterned into a Hall bar structure by photolithography and Ar ion milling. Cr (1.0)/Cu (100) electrodes were deposited by thermal evaporation and defined by a lift-off process using photolithography.
### Measurement setup
Measurements were performed using a prober system in which a vector magnetic field can be applied. The device temperature was controlled by a plate-shaped heater placed under the device and monitored using a Pt thermometer (Pt-100). A current source (Yokogawa 7651) and nano-voltmeter (Keithley 2182A) were used for anomalous Hall measurements.
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## Acknowledgements
The authors thank S. Ono, A. Tsukazaki, F. Matsukura, and H. Ohno for their technical help. This work was partly supported by the Grants-in-aid for Young Scientists (A) (No. 15H05419), Scientific Research (S) (No. 25220604), Specially promoted Research (No. 15H05702) from JSPS, the Murata Science Foundation, and Spintronics Research Network of Japan.
## Author information
### Affiliations
1. #### Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo, 113-8656, Japan
• Tomohiro Koyama
• , Yicheng Guan
• & Daichi Chiba
### Contributions
T.K. and D.C. planned the study. T.K. deposited the film and fabricated the device. T.K. and Y.G. performed the measurement and data analysis. T.K. wrote the manuscript with input from D.C. All authors discussed the results.
### Competing Interests
The authors declare that they have no competing interests.
### Corresponding authors
Correspondence to Tomohiro Koyama or Daichi Chiba. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "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": 0.8138449192047119, "perplexity": 2698.27384672987}, "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/1521257647768.45/warc/CC-MAIN-20180322034041-20180322054041-00263.warc.gz"} |
http://mathonline.wikidot.com/the-intersection-and-union-of-subspaces | The Intersection and Union of Subspaces
# The Intersection and Union of Subspaces
We will now look at a couple of theorems regarding the intersection of subspaces and the union of subspaces.
Theorem 1: Let $V$ be a vector space over the field $\mathbb{F}$ and let $U_1, U_2, ..., U_m$ be any collection of subspaces of $V$. Let $I$ be a set of indices denote subspaces. Then $\bigcap_{i \in I} U_i$ is a subspace.
• Proof: Consider the collection of subspaces $\{ U_i : i \in I$ where $I = \{ 1, 2, ..., m \}$, that is the set $\{ U_i : i \in I \}$ is a set any set of subspaces of $V$.
• Now let $x, y \in \bigcap_{i \in I} U_i$, and so $x, y \in U_i$ for all $i \in I$. Since $U_i$ is a subspace for all $i \in I$, then we know that for all scalars $a, b \in \mathbb{F}$, $(ax + by) \in U_i$ for all $i \in I$ and so $(ax + by) \in \bigcap_{i \in I} U_i$.
• Therefore the intersection of the collection of subspaces $\{ U_i : i \in I \}$ is a subspace. $\blacksquare$
Theorem 2: Let $V$ be a vector space over the field $\mathbb{F}$ and let $U_1, U_2$ be any subspaces of $V$. Then $U_1 \bigcup U_2$ is a subspace of $V$ if and only if $U_1 \subseteq U_2$ or $U_1 \supseteq U_2$.
• Proof: Let $U_1$ and $U_2$ be any subspaces to the vector set $V$.
• $\Rightarrow$ We first want to show that if $U_1 \subseteq U_2$ or $U_2 \subseteq U_1$ then $U_1 \cup U_2$ is a subspace of $V$. Without loss of generality suppose that $U_1 \subseteq U_2$. Then it follows that $U_1 \cup U_2 = U_2$. But $U_2$ is defined to be a subspace of $V$ and so $U_1 \cup U_2$ is a subspace of $V$.
• $\Leftarrow$ We now want to show that if $U_1 \cup U_2$ is a subspace of $V$ then either $U_1 \subseteq U_2$ or $U_1 \supseteq U_2$. Suppose that $U_1 \not \subseteq U_2$ and $U_1 \not \supseteq U_2$. We will derive a contradiction with this.
• Let $U_1 \cup U_2$ be a subspace of $V$. Thus there exists elements $u_1 \in U_1 \setminus U_2$ and $u_2 \in U_2 \setminus U_1$. We have that $u_1, u_2 \in U_1 \cup U_2$, and since $U_1 \cup U_2$ is a subspace, then the vector $x = u_1 + u_2 \in U_1 \cup U_2$. Therefore $x \in U_1$ or $x \in U_2$.
• First consider the case where $x \in U_1$. Since $x = u_1 + u_2$ then $u_2 = x - u_1$ and so $u_2 \in U_1$ which is a contradiction since $u_2 \in U_2 \setminus U_1$.
• Now consider the case where $x \in U_2$. Since $x = u_1 + u_2$ then $u_1 = x - u_2$ and so $u_1 \in U_2$ which is a contradiction since $u_1 \in U_1 \setminus U_2$.
• Therefore either $U_1 \subseteq U_2$ or $U_1 \supseteq U_2$. $\blacksquare$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9997467398643494, "perplexity": 32.60008582804692}, "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-2021-04/segments/1610704795033.65/warc/CC-MAIN-20210126011645-20210126041645-00558.warc.gz"} |
http://mathhelpforum.com/calculus/91609-vectors-parametric-problem-urgent-print.html | # Vectors (Parametric) Problem Urgent.
• Jun 2nd 2009, 09:41 PM
037810
Vectors (Parametric) Problem Urgent.
find parametric eq of a linethat intersects both lines, [x,y,z]=[4,8,-1]+t[2,3,-4]
[x,y,z]=[7,2,-1]+s[-6,1,2] at 90 degrees.
So with direction vectors, i cross multiplied to get [1,2,2].
I just need to know the point that the line goes through to complete the equation.
I also did...
x=4+2t=6s
y=8+3t=2+s
z=-1-4t=-1+2s
and got t = -6/5 and s=12/5
so since no interception and not parallel, it's a skew line.
What would be the intercept point if the two screw lines were in the same plane. (i know they are not. but how would you find the point where the two lines make an X (top-view))
• Jun 5th 2009, 05:30 PM
Media_Man
You're on it so far. Keep going. For notation's sake, let the two vector equations represent lines: $X(t)=(4+2t,8+3t,-1-4t)$ and $Y(s)=(7-6s,2+s,-1+2s)$ . You correctly determined that the line sought will be of the form $Z(u)=(a,b,c)+u(1,2,2)$ for some "starting vector" $(a,b,c)$. Notice that the choice for the starting vector is not unique, it can be any point on the line $Z$. Since we know $Z$ intersects $X$, without losing generality, let $Z(0)=(a,b,c)=X(t)$ for some $t$. In other words, line $Z$ "starts" at its intersection with $X$ and progresses in the direction $(1,2,2)$ towards $Y$. At some value of $u$, $Z(u)$ will intersect line $Y$. We now have the ingredients to set up a system of equations: There exists integers $t,s,u$ such that $X(t)+u(1,2,2)=Y(s)$ . This yields a 3x4 matrix with $t,s,u$ variables that can be reduced to find a unique solution. I'll let you do the grunt work for yourself, but the answer, so you can check it, is $Z(u)=(2,5,3)+u(1,2,2)$. (By the way, what would it mean if said matrix were singular? (Thinking)) | {"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": 20, "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": 0.9421729445457458, "perplexity": 299.76419563854097}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948608836.84/warc/CC-MAIN-20171218044514-20171218070514-00480.warc.gz"} |
https://math.stackexchange.com/questions/1076179/series-expansion-of-frac11x1%E2%88%92x1x21%E2%88%92x21x31%E2%88%92x3-cdots | Series expansion of $\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$?
How would I find the series expansion $\displaystyle\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$ so that it will turn into an infinite power series again??
• This infinite power series looks like it is counting some partitions of integers. What I would do to find the answer is to calculate the coefficient of the first few terms then use OEIS to find the combinatorial meaning. – clark Dec 21 '14 at 2:27
• @clark: I'm not exactly sure how to find OEIS, but I did post the original question here: text – Mathy Person Dec 21 '14 at 2:28
• @vadim123 typo, i meant to say i don't know how to use* OEIS – Mathy Person Dec 21 '14 at 2:46
• here is the OEIS results: oeis.org/A000041 – Math-fun Dec 22 '14 at 11:16
If you set
$$f(x)=\prod_{n=0}^\infty (1-x^n)^{-1}=\sum_{n=0}^\infty p(n)x^n$$
we see that yours is just
$$f(x^2)=\sum_{n=0}^\infty p(n)x^{2n}.$$
$$\frac{1}{1-x^k}=1+x^k+x^{2k}+x^{3k}\cdots$$ $$\frac{1}{1+x^k}=1-x^k+x^{2k}-x^{3k}\cdots$$
• Would that result in....$(1-x+x^2-x^3...)(1+x+x^2+x^3+....)(1-x^2+x^4-x^6....)(1+x^2+x^4+x^6....)(1-x^3+x^6-x^9....)(1+x^3+x^6+x^9+....)....$? – Mathy Person Dec 21 '14 at 2:53
• Is there an alternative to find the truncation of $x^{10}$ from that without doing messy expansions? – Mathy Person Dec 21 '14 at 3:09
• Yes, and yes. Use the other product I gave you in the solution to your other question. – vadim123 Dec 21 '14 at 3:13
• Which product? In the comment, or in your answer? – Mathy Person Dec 21 '14 at 3:19
• Are you referring to $$\prod_{k>0}1-x^{2k-1}=\prod_{k>0}\frac{1}{1+x^k}$$? – Mathy Person Dec 21 '14 at 3:30
I suppose that you can make the expresion shorter using $(1+a)(1-a)=1-a^2$ so $$A=\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}=\frac{1}{(1−x^2)(1−x^4)(1-x^6)\cdots}$$ and now use the fact that $$\frac{1}{1-y}=\sum_{i=0}^{\infty}y^i$$ and replace successively $y$ by $x^2$, $x^4\cdots$,$x^{2n}$ before computing the overall product.
Doing so, for a large number of terms, you should arrive to $$A=1+x^2+2 x^4+3 x^6+5 x^8+7 x^{10}+11 x^{12}+15 x^{14}+22 x^{16}+30 x^{18}+42 x^{20}+56 x^{22}+77 x^{24}+101 x^{26}+135 x^{28}+176 x^{30}+O\left(x^{31}\right)$$ and, as mentioned earlier, the coefficients correspond to sequence $\text{A00041}$ of $\text{EOIS}$ where $a_n$ is the number of partitions of $n$ (the partition numbers).
Probably off-topic, for an infinite number of terms, $$A=\frac{1}{\left(x^2;x^2\right){}_{\infty }}$$ where appears the q-Pochhammer symbol and which, for sure, leads to the same expansion. | {"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": 0.8848246335983276, "perplexity": 414.11415036266646}, "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-2019-43/segments/1570986659097.10/warc/CC-MAIN-20191015131723-20191015155223-00171.warc.gz"} |
http://www.jiskha.com/display.cgi?id=1329171535 | Tuesday
April 28, 2015
# Homework Help: physics
Posted by kas on Monday, February 13, 2012 at 5:18pm.
Long, long ago, on a planet far, far away, a physics experiment was carried out. First, a 0.250 kg ball with zero net charge was dropped from rest at a height of 1.00 m. The ball landed 0.350 s later. Next, the ball was given a net charge of 7.60 muC and dropped in the same way from the same height. This time the ball fell for 0.645 s before landing.
What is the electric potential at a height of 1.00 {\rm m} above the ground on this planet, given that the electric potential at ground level is zero? (Air resistance can be ignored.)
No one has answered this question yet. | {"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": 0.947115421295166, "perplexity": 735.540220234414}, "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-2015-18/segments/1429246661095.66/warc/CC-MAIN-20150417045741-00108-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.answers.com/Q/What_is_the_probability_of_exactly_three_heads_in_four_flips_of_a_coin_given_at_least_two_are_heads | Math and Arithmetic
Statistics
Probability
# What is the probability of exactly three heads in four flips of a coin given at least two are heads?
###### Answered 2013-06-09 20:47:41
If you know that two of the four are already heads, then all you need to find is
the probability of exactly one heads in the last two flips.
Number of possible outcomes of one flip of one coin = 2
Number of possible outcomes in two flips = 4
Number of the four outcomes that include a single heads = 2.
Probability of a single heads in the last two flips = 2/4 = 50%.
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## Related Questions
The probability of obtaining 7 heads in eight flips of a coin is:P(7H) = 8(1/2)8 = 0.03125 = 3.1%
The requirement that one coin is a head is superfluous and does not matter. The simplified question is "what is the probability of obtaining exactly six heads in seven flips of a coin?"... There are 128 permutations (27) of seven coins, or seven flips of one coin. Of these, there are seven permutations where there are exactly six heads, i.e. where there is only one tail. The probability, then, of tossing six heads in seven coin tosses is 7 in 128, or 0.0546875.
We can simplify the question by putting it this way: what is the probability that exactly one out of two coin flips is a head? Our options are HH, HT, TH, TT. Two of these four have exactly one head. So 2/4=.5 is the answer.
The probability of obtaining exactly two heads in three flips of a coin is 0.5x0.5x0.5 (for the probabilities) x3 (for the number of ways it could happen). This is 0.375. However, we are told that at least one is a head, so the probability that we got 3 tails was impossible. This probability is 0.53 or 0.125. To deduct this we need to divide the probability we have by 1-0.125 0.375/(1-0.125) = approximately 0.4286
you toss 3 coins what is the probability that you get exactly 2 heads given that you get at least one head?
Pr(3H given >= 2H) = Pr(3H and >= 2H)/Pr(>=2H) = Pr(3H)/Pr(>=2H) = (1/4)/(11/16) = 4/11.
Three in eight are the odds of getting exactly two heads in three coin flips. There are eight ways the three flips can end up, and you can get two heads and a tail, a head and a tail and a head, or a tail and two heads to get exactly two heads.
The probability that you will toss five heads in six coin tosses given that at least one is a head is the same as the probability of tossing four heads in five coin tosses1. There are 32 permutations of five coins. Five of them have four heads2. This is a probability of 5 in 32, or 0.15625. ----------------------------------------------------------------------------------- 1Simplify the problem. It asked about five heads but said that at least one was a head. That is redundant, and can be ignored. 2This problem was solved by simple inspection. If there are four heads in five coins, this means that there is one tail in five coins. That fact simplifies the calculation to five permutations exactly.
50-50. each toss is independent of any previous toss. if all tosses are to be heads/tails then each toss you multiply by the number of chances. i,e. 2, starting with 1. three heads in a row is 1x2x2
The probability of throwing exactly 2 heads in three flips of a coin is 3 in 8, or 0.375. There are 8 outcomes of flipping a coin 3 times, HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. Of those outcomes, 3 contain two heads, so the answer is 3 in 8.
We have no way of knowing the probability of any given person flipping any given coin at any given time. But for any two flips of an honest coin, the probability that both are tails is 25% . (1/4, or 3 to 1 against)
The probability is 25%. The probability of flipping a coin once and getting heads is 50%. In your example, you get heads twice -- over the course of 2 flips. So there are two 50% probabilities that you need to combine to get the probability for getting two heads in two flips. So turn 50% into a decimal --> 0.5 Multiply the two 50% probabilities together --> 0.5 x 0.5 = 0.25. Therefore, 0.25 or 25% is the probability of flipping a coin twice and getting heads both times.
Five coin flips. Any outcome on a six-sided die has a probability of 1 in 6. If I assume that the order of the outcome does not matter, the same probability can be achieved with five flips of the coin. The possible outcomes of five flips of a coin are as follows: 5 Heads 5 Tails 4 Heads and 1 Tails 4 Tails and 1 Heads 3 Heads and 2 Tails 3 Tails and 2 Heads For six possible outcomes.
The probability of getting 3 or more heads in a row, one or more times is 520/1024 = 0.508 Of these, the probability of getting exactly 3 heads in a row, exactly once is 244/1024 = 0.238
The answer depends on how many times the coin is tossed. The probability is zero if the coin is tossed only once! Making some assumptions and rewording your question as "If I toss a fair coin twice, what is the probability it comes up heads both times" then the probability of it being heads on any given toss is 0.5, and the probability of it being heads on both tosses is 0.5 x 0.5 = 0.25. If you toss it three times and want to know what the probability of it being heads exactly twice is, then the calculation is more complicated, but it comes out to 0.375.
The probability of a fair coin to land head is 1/2. Since for 4 flips it must land heads each time, the probability of 4 heads is 1/2 * 1/2 * 1/2 * 1/2 = 1/16.
Pr(3 flips at least one H) = 1 - Pr(3 flips, NO heads) = 1 - Pr(3 flips, TTT) = 1 - (1/2)3 = 1 - 1/8 = 7/8
###### Math and ArithmeticProbabilityStatisticsAlgebra
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https://www.physicsforums.com/threads/magnetic-field-problem-help-needed.229394/ | # Magnetic Field Problem! HELP NEEDED
1. Apr 16, 2008
### physicsbhelp
[SOLVED] Magnetic Field Problem! HELP NEEDED!!!!!!!
1. The problem statement, all variables and given/known data
A proton of mass mp and charge e is in a box that contains an electric field E, and the box is located in Earth's magnetic field B. The proton moves with an initial velocity vertically upward from the surface of Earth. Assume gravity is negligible.
(a) On the diagram above, indicate the direction of the electric field inside the box so that there is no change in the trajectory of the proton while it moves upward in the box. Explain your reasoning.
(b) Determine the speed of the proton while in the box if it continues to move vertically upward. Express your answer in terms of the fields and the given quantities.
The proton now exits the box through the opening at the top.
(c) On the figure above, sketch the path of the proton after it leaves the box.
(d) Determine the magnitude of the acceleration a of the proton just after it leaves the box, in terms of the given quantities and fundamental constants.
2. Relevant equations
F=IL*B = qv*B
B= uI / 2piR u=4pi*10-7
I don't have the diagram but i will try to find a picture of it on the internet.
Last edited: Apr 16, 2008
2. Apr 18, 2008
### Kurdt
Staff Emeritus
3. Apr 18, 2008
### physicsbhelp
well that is becasue i am very confused.
i don't even know where to start, that is why i haven't posted anything, usually i do, but i just am lost!
help? :(
4. Apr 18, 2008
### Kurdt
Staff Emeritus
Well what have you been doing? I'm sure you have some sort of notes or a book that can give you a slight clue as to what the answer to part a) should be.
5. Apr 19, 2008
### physicsbhelp
i should use the curved hand rule right?
Last edited: Apr 19, 2008
6. Apr 19, 2008
### EngageEngage
If the magnetic field is also in the box, you need to balance the force by the magnetic field with the force of the electric field, so the trajectory within the box is straight.To do this, you need to think about which way the force from the magnetic points initially, and how to place an electric field so the force on the charge is in the opposite direction. Once you have figured this out, you can start working with some formulas. Also, In such a problem its reasonable to neglect gravity i would think.
7. Apr 20, 2008
### physicsbhelp
thank you engageengage!
okay so for part a) the electircal field is in the postive z direction (out of the page) right?
and for part b) do i use the formula Fm=qv*B ? Where v is the velocity or in other words, the speed? (by the way, Fm=Force magnetic)
PLEASE help! thanks to all who have helped me till now. buti still need to understand how to do the rest of it.
thanks!
8. Apr 20, 2008
### Staff: Mentor
First figure out the direction of the magnetic force on the proton. (You'll need to use the right hand rule.)
9. Apr 20, 2008
### physicsbhelp
well if i use the right hand rule, the magnetic force is in the -x direction (to the left).
is that correct?
10. Apr 20, 2008
### Staff: Mentor
Good! So which way must the electric force act in order to cancel out the magnetic force? So which way must the electric field point?
11. Apr 20, 2008
### physicsbhelp
in order to cancel the magnetic force, the electric field must act in the positive x direction.
so the elctric field should point in the positive z direction?
12. Apr 20, 2008
### Hootenanny
Staff Emeritus
Now why would you say that? How does the electric field relate to the force experienced by a charged particle?
13. Apr 20, 2008
### physicsbhelp
do they attract each other?
14. Apr 20, 2008
### Staff: Mentor
You need to understand how to find the force on a charge due to both electric and magnetic fields. Look it up in your textbook.
15. Apr 20, 2008
### physicsbhelp
we don't have a textbook.
16. Apr 20, 2008
### physicsbhelp
can someone explain to me "how to find the force on a charge due to both electric and magnetic fields. " ????? Please?
17. Apr 20, 2008
### Staff: Mentor
Say what? What do you have?
18. Apr 20, 2008
### Hootenanny
Staff Emeritus
Edit: Sorry Doc, didn't see you there.
19. Apr 20, 2008
### physicsbhelp
Please don't be rude to me.
Our teacher said that physics is better learned w/out a textbook. he justs gives us problems to work.
Does this have anything to do with induction?
Currently, i am studying my class notes. i have been, i just cna't seem to find anything about electic fields.
in my class notes it says taht E= Fe/q
Last edited: Apr 20, 2008
20. Apr 20, 2008
### Staff: Mentor
This is what textbooks are for. If you don't have one, get one!
A good resource is hyperphysics. For example: Lorentz Force Law | {"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": 0.9311906695365906, "perplexity": 980.2637603534632}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721347.98/warc/CC-MAIN-20161020183841-00443-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/216224 | ## A Tight Linear Time (1/2)-Approximation For Unconstrained Submodular Maximization
We consider the Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function f : 2(N) -> R+, and the objective is to find a subset S subset of N maximizing f(S). This is one of the most basic submodular optimization problems, having a wide range of applications. Some well-known problems captured by Unconstrained Submodular Maximization include Max-Cut, Max-DiCut, and variants of Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige, Mirrokni, and Vondrak [SIAM J. Comput., 40 (2011), pp. 1133-1153]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem.
Published in:
Siam Journal On Computing, 44, 5, 1384-1402
Year:
2015
Publisher:
ISSN:
0097-5397
Keywords:
Laboratories: | {"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": 0.91802579164505, "perplexity": 812.783979518932}, "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-2018-09/segments/1518891813322.19/warc/CC-MAIN-20180221024420-20180221044420-00742.warc.gz"} |
https://proxies-free.com/probability-how-do-i-show-that-the-mean-recurrence-time-for-transient-states-is-infinity/ | # probability – How do I show that the mean recurrence time for transient states is infinity?
The random variable $$T_i$$, the “Hitting Time of $$i$$” is defined to be the first $$n$$ such that $$X_n=i$$ given that $$X_0=i$$.
By the mean recurrence time of $$T_i$$, I mean the expected value of this random variable.
I wish to show that if $$i$$ is transient, then the expectation does not converge to any finite real number. While this, intuitively makes sense, I do not know how to formally prove this and any help is appreciated. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9707031846046448, "perplexity": 81.0378452737647}, "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/1623488556482.89/warc/CC-MAIN-20210624171713-20210624201713-00033.warc.gz"} |
https://www.physicsforums.com/threads/question-about-quarks.860566/ | Tags:
1. Mar 4, 2016
### Fizica7
Hi. I'm wondering if anyone has any info on "quark seeding" like:
Is it possible to dope the crystal lattice of a solid material by replacing electrons with quarks ?
2. Mar 4, 2016
### Orodruin
Staff Emeritus
No.
3. Mar 4, 2016
### Fizica7
Is the mass of the antiquark negative ?
Are the antiquark charges in the neutron 2, -1, -1 ?
4. Mar 4, 2016
### Orodruin
Staff Emeritus
No, these questions appear to me as wild speculation only. Is ere a point to this?
5. Mar 4, 2016
### Fizica7
They shouldn't appear as wild speculation... they were written by Dr F Winterberg in 1975...but maybe quark knowledge was incorrect back then so his work incorrect by today's standards?
Do you have a PhD or doctorate in this field?
6. Mar 4, 2016
### Orodruin
Staff Emeritus
Do you really think nothing has happened in theoretical particle physics in 40 years? I suggest you pick up a modern textbook instead of reading outdated stuff.
7. Mar 4, 2016
### Fizica7
I would read physics manuals or even go to university and study physics if I could afford it... For the moment I can only ask a question on a forum and hope for an educated answer to satisfy my curiosity.
Suppose the magnetic field between quarks is 10^17 gauss...a laser of 10^13 erg, 10^-9 second pulse and focused on an area of 10^-20 sqcm should be enough to achieve the fission of a proton, right?
As in 10^42 erg / sqcm should overcome the 10^17 gauss field, right?
8. Mar 4, 2016
### Staff: Mentor
Below are a few articles to get you started on the subject.
No. Quarks are not held together through the EM force, but the color force (also known as the strong force). Also, I'm not sure fission is the right term for what would happen to the proton. But I'm also not sure what the right term is...
https://en.wikipedia.org/wiki/Quark
https://en.wikipedia.org/wiki/Color_confinement
https://en.wikipedia.org/wiki/Strong_interaction
9. Mar 5, 2016
### Staff: Mentor
Free quarks do not exist.
To focus a laser to a picometer, you would need a gamma ray laser. Good luck building that. High-energetic gamma rays can react with protons independent of the energy, but classical physics does not give a proper description of that interaction, and the results are always hadrons, not free quarks.
10. Mar 5, 2016
### Fizica7
So by what means did a Dr. in 1970s manage to calculate the force between quarks in a neutron to be equal to 10^17 gauss?
Anyway.. his paper suggests a few main ideas:
1) breakdown of neutron to obtain quarks with such a powerful laser ( BTW can "erg" be converted into layman like watt or something?)
2) he presumes the spacing of mass in an electric force field is determined by the Heisenberg uncertainty principle depending on the magnitude of the masses regardless if the masses are positive or negative. So a crystal's lattice spacing would be reduced when electrons are replaced with antiquark thanks to the greater, although negative, mass of the quark vs mass of the electron.
He then gives the theoretical strength as:
e^2/r^4
where e is electron charge and r is Bohr radius which varies inversely with particle mass.
So he says for a replacement of 0.3% of electrons with quarks, the lattice spacing should decrease by a factor of 10 and the strength increases by a factor of 10000.
Also because melting point depends on e^2/r^4 then again a increase in melting point by a factor of 10000.
3) the fission would release double the binding energy per unit mass of proton-antiproton annihilation as calculated with these equations:
Legend: Eb( binding energy), mp(nucleon mass), 3|Maq|(mass of 3 antiquark)
Eb=(Mp-3Maq)c^2
because for antiquarks Maq=-|Maq|
Eb=(Mp+3|Maq|)
So the huge energy release(double that of matter annihilation) comes from
Mp=3|Maq| , and Eb=2Mp c^2
Have to mention that Dr. Winterberg's work appears to be based on the quark theory of matter from P.A.M. Dirac.
Would really appreciate a sort of layman explanation on why this might or might not work.
Last edited: Mar 5, 2016
11. Mar 5, 2016
### Fizica7
There is mention of a "hard X-ray" laser.
12. Mar 5, 2016
### Staff: Mentor
Gauß is not a unit of force.
1970 is long ago. The gluon was not even experimentally observed back then.
107 erg = 1 J
That does not make sense at all.
Antiquarks have positive masses, by the way.
That does not make sense either.
Forget what Winterberg wrote in 1970. I doubt it was reasonable physics back then, but it is certainly not reasonable today.
It does not work, because there is no content that would remotely make sense.
13. Mar 5, 2016
### Fizica7
Right...ok.. thank you... By the way it was 10^17 erg not ^7.
Edit: just one last question. Is there anything discovered so far that has negative mass?
Edit2: last edit I promise: what are leptons made of, and what are quarks made of? And what's the size ratio of the two? Could there be something hiding inside either of them(sub-quarks particles and sub-lepton particles) ?
Last edited: Mar 5, 2016
14. Mar 5, 2016
### phinds
quarks are elementary (aka fundamental) particles.
15. Mar 5, 2016
### Staff: Mentor
I posted the conversion factor, so you can convert whatever energy value you have.
No.
At least according to current knowledge, they are elementary particles, and do not have a size. It is possible that they are composite particles but that would require really weird physics to explain all the precision experiments that did not find any hint of an internal structure.
16. Mar 5, 2016
### Fizica7
So if 10^7= 1 Joule, then 10^17= 10000000 joules? An extra 10 zeroes? So watt=j/s, then 10mil joules/s ...1millionth of a second, 1 million joules.... 1 million watts... So that laser given in the example at the beginning would be a hard X-ray of 1 MW pulse for 1 nanosecond?
edit: No wait ...10mil j/s and only 1 millionth of s...10 joules...10watts for 1 nanosecond... I'm lost.
17. Mar 5, 2016
### Staff: Mentor
10, not 7.
1017 erg = 1010*107 erg = 1010 J = 10 GJ.
Post 3 mentions 1013 erg in 10-9 s, that is 1013 erg / (10-9 s) = 1022 erg/s = 1015 J/s = 1015 W.
18. Mar 5, 2016
### Fizica7
Oh right... yes it was only 10¹³ for erg.. so you added the -9 to 13 at the powers and got 10²²... Cool.
So that's 10^15 watts if the laser pulse is 1 second.. and if it's only 1ns pulse then the laser needs only be 10^9 Watts= 1bn Watts=1GW.
edit: so hard xray 1 GW pulsed for 1 ns. Is that achievable with q switching? Nd:glass? Nd:yag? Or it requires free electron laser?
19. Mar 5, 2016
### Staff: Mentor
It is 1015 W (1PW) for 1 nanosecond. The energy is 106 J.
Those numbers have nothing to do with proton-light interactions however. Those happen frequently at high-energetic particle accelerators, for example, with individual high-energetic photons of negligible intensity.
20. Mar 5, 2016
### Fizica7
Right... one last question... Might be a bit of a leap... is it possible that the Sun is actually a fission reactor at the core and a fusion reactor at the surface... so that the fusion is only a secondary recycling reaction only present at the surface but the main power comes from core fission? | {"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": 0.8587281703948975, "perplexity": 3300.0619508814502}, "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/1508187825147.83/warc/CC-MAIN-20171022060353-20171022080353-00874.warc.gz"} |
http://psjd.icm.edu.pl/psjd/element/bwmeta1.element.bwnjournal-article-appv88z103kz | PL EN
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## Acta Physica Polonica A
1995 | 88 | 1 | 21-27
Article title
### Liquid Crystalline Polyurethanes Studied by Positron Annihilation Method
Authors
Content
Title variants
Languages of publication
EN
Abstracts
EN
A series of liquid crystalline polyurethanes of different both flexible spacer length and mesogenic group content was studied by means of positron annihilation method. The investigated polymers were targeted on the separation of aromatic hydrocarbons from their mixtures with aliphatic ones. The effect of modification of the liquid crystalline polyurethane chain on the polyurethane free volume size and free volume distribution was determined on the basis of the positron annihilation lifetime spectra. In the positron annihilation lifetime spectra measured for the samples under study two long components of several nanoseconds, characteristic of o-Ps decaying by pick-off, occurred. The correlation of the values of the o-Ps lifetime with the size of the free volume region allowed to recover the free volume distributions with the use of the method employing the numerical Laplace inversion technique. The obtained results were compared with the diffusion data on the mobility of liquid hydrocarbons in the liquid crystalline polyurethanes enabling the correlations between polymer structure and its transport properties to be evaluated.
Keywords
EN
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21-27
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published
1995-07
References
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Publication order reference | {"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": 0.8395252227783203, "perplexity": 3974.003932679045}, "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-2018-22/segments/1526794863410.22/warc/CC-MAIN-20180520112233-20180520132233-00470.warc.gz"} |
https://www.physicsforums.com/threads/what-value-will-make-these-two-vectors-parallel.642934/ | # What value will make these two vectors parallel?
1. Oct 10, 2012
### kasda-1
1. The problem statement, all variables and given/known data
What value of c will make these two vectors parallel?
2. Relevant equations
3i + 2j + 9k = w
5i - j + ck = v
3. The attempt at a solution
I tried too find a common factor to multiply by the w vector to get the components of the v vector, but no luck.
3i (5/3) + 2j (5/3) +9k (5/3) = 5i + 10/3 j + 45/3 k ≠ 5i - j +ck
2. Oct 10, 2012
### Dick
Would that lead you to conclude that there is no value of c that will make them parallel? Because that would be correct.
3. Oct 10, 2012
### kasda-1
I thought and thought about this problem, wondering if this is a trick question. It probably is that simple :). | {"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": 0.9281079769134521, "perplexity": 955.1193301899166}, "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/1521257648113.87/warc/CC-MAIN-20180323004957-20180323024957-00643.warc.gz"} |
https://www.physicsforums.com/threads/understanding-probability-is-probability-defined.704217/ | # Understanding probability, is probability defined?
1. Aug 4, 2013
### bobby2k
I have taken a course in probability and statistics, and did well, but still I feel that I do not grasp the core of what holds the theory together. It is a little weird that I should use a lot of theory when I do not get the simple building block of the theory.
I am basically wondering if probability is defined in some way?
In the statistics books I have looked in, probability is not defined, but at the beginning of the book, they give a describtion of how we can look at probability, and this is usually the relative frequency model, but they never define it to be this?
These steps is what I seem to see in a statistics books, do they seem fair?
1. Probability is described in terms of events, outcomes and relative frequency, but never defined.
2. A lot of theory is then built regarding probability.
3. Then with the help of Chebychevs inequality, we are able to show that the relative frequency model is correct. That is, if the probability for an event is p, and X is a bernoulli random variable, then mean(X) will converge to p.
Do you see my problem? If we say that the probability for an event is p, then we can show that the relative frequency of the of the event in the long run is p. In order to show this, we used all the theory of linear combinations, variance etc.. But this means that the relative frequency model is a consequence of our theory, correct?
I mean, we can not say that the probability is the relative frequency, then develeop a lot of theory, and then prove that p equals the relative frequency, then we are going in a circle?
2. Aug 4, 2013
### Stephen Tashi
If you look closely about what theorems in probably say about relative frequency, they only talk about the probability of relative frequency taking on a certain value. They may have wording such as "the probability approaches 1 as the number of trials approaches infinity", but this is still not a guarantee that relative frequency will behave in a certain manner - it just probably will.
Probability theory not circular in the way that you describe, but it is circular in that results of probability theory are results about probabilities of things, not guarantees of actual outcomes.
In the axiomatic statement of probability theory, probability is not defined in terms of relative frequency. It is defined abstractly as a "measure". If you look closely at the axiomatic development of probability theory ( the high class approach, not the approach taken in elementary texts) you will find that there isn't any discussion of whether an event actually happens or not. There isn't even any assumption that you can take a random sample - there are only statements that random variables of various kinds (which people think of as representing random samples) have certain distributions.
The mathematical theory of probability does not describe any way to measure "probability" in the same way that physical theories describe how to measure a quantity like "mass" or "force". It is not clear whether probability has any physical reality. If it does then it is rather mysterious. Consider how the probability of an event changes. If a prize is placed "at random" behind one of 3 doors, the probability of it being behind the second door is 1/3. If we open the first door and the prize is not there then the probability of it being behind the second door changes to 1/2. Does this involve a physical change in the doors? Does the probability change from 1/3 to 1/2 instantaneously or does it go from 1/3 up to 1/2 in a finite amount of time? The mathematical theory of probability does not deal with such questions. A person who applies probability theory may tackle them, but mathematically he is on his own.
3. Aug 4, 2013
### economicsnerd
^ This.
The most popular formalism for probability consists of (i) states of the world, (ii) events (where an event is just a collection of states), and (iii) a number attached to each event, which is just called the "probability" of said event.
The formalism exists even without any interpretation.
Indeed, one popular interpretation of probability is the frequentist one. The (strong) law of large numbers---the theorem to which you alluded here---suggests that the formalism somehow agrees with the frequentist interpretation. It suggests that, if somebody really wants to think of probability in terms of long-run frequencies, then then it usually won't lead them astray in doing rigorous study of probability theory.
4. Aug 5, 2013
### bobby2k
So, probability in it's core, is just a measure of likelihood?, with 0 not happening, 1 certain to happen, and if P(A) > P(B), then it is more likey that the event A happens over B? We can say no more of probability as it is defined at the bottom of all the theory?
It may sound stupid, but I still feel that there is a gap between saying that probability is a measure of something, to us beeing able to calculate probabilities, make confindence-intervals and all that stuff.
From what I see we can do is this:
1. definie probability as a measure of likelihood as you said
2. define events, outcomes etc
3. define random variables, both continous and discrete etc.
4. define probability distribution functions for the random variables
5. define expected values and variance
6. calculate expected values of linear combinations, show that the law of large numbers etc. holds(chebychev)
If I do the things in this list, I run into a problem at step 5. If I do not allready have the relative frequency model in the back of my mind, step 5 does not make any sence. I mean when I learned to understand the expected value, I thought of the probability as relative frequencies for expected values to make sense(it was the avarage in the long run, for this to work, we have to look at probability as frequencies). But I can not really do this, because this comes in step 6, after expected value have been defined. How is this explained?
Thanks for your time guys, this is really important for me to understand.
Last edited: Aug 5, 2013
5. Aug 5, 2013
### Stephen Tashi
I did not say probability is defined as a "measure of liklihood". I just said it was defined as a "measure". A "measure" in mathematics is an abstraction of ideas connected with the physical ideas of length, area, volume etc. When we apply probability theory, we think of probability as a tendency for a thing to happen - but that thought is not expressed in the axioms of probability theory.
An attempt to define probability as "tendency for something to happen" or a "liklihood" merely offers undefined words such as "tendency" or "liklihood" for the undefined word "probability". Such a definition has no mathematical content. (As a matter of fact, the word "liklihood" has a technical definition in probability and statistics that is different than the man-in-the-street's idea of what liklihood means.)
You apparently are seeking a formulation of probability theory that somehow guarantees some connection between the mathematics of probability and applications to real world problems. There is no such mathematical theory. Applications of any sort of math to the real world involve assuming certain math is a correct model. There is no mathematical proof that mathematics applies to the real world. There is no mathematical proof or definition that says probability is a frequency of occurrence. The only connection between probability theory and observed frequency is that probability theory tells you about the probability of various frequencies.
The expectation of a random variable can be thought of as the average of taking infinitely many independent samples of the random variable, but such a thought is a way of thinking about how to apply probability theory. It isn't part of the mathematical theory of probablity.
6. Aug 5, 2013
### bobby2k
Thanks, I still have some follow-ups, I hope that's ok, I am getting closer to the end though.
Does this step by step seem fair then:
1. Probability is a "measure" but undifined. However we say that it is a measure of how likely something will happen.
2. We define the basic probability axioms, these are mathematical.(P(S)=1 etc.)
3. We define dependent and indepent variables. We define that the measure of two independents events are going to happen, to be the product of those two individual measures.
4. We define expected value and variance mathematically, we don't give them any other meaning.
5. Since we know have defined the measure of indepentent events, if X is a bernoulli random variable, we get that the measure of mean(x) beeing close to p, approces one as the number of events goes to infinity. All this is still only matehmatical, and all it means is that the measure goes to 1.
Then we start assuming things:
6. Lets say there is a price between one of three doors. Since we assume that it is equally likely that each door has the price, P(door 1 has price)=1/3. Still, this is just a meassure of how likely the price is there.
7. Then we choose a door many times, and count how many times we are correct. Now in our real physical world, we assume that the it is equally likely to get the price each time, no matter what we get the previous times. Now we adopt the mathematical model, we say that since they are physically indepentent, we assume that their probabilities can me multiplied. Then we get that the probability that we will guess correct 1/3 of the times approces 1, as the number of trials goes to infinity.
What more do I need to do/assume, to be able to say that the relative frequency of the number of correct guesses will approch 1/3? Is it ok to say that since one axiom defines probability to be maximum 1, then we can say that it is extremely likely that the relative frequency will approch 1/3?
7. Aug 5, 2013
### jostpuur
One of the most important lessons in philosophy is that nearly nothing can be defined properly. Everytime you define something, you use some other concepts in the definition, and the definitions of the used concepts become new problems.
The concept of probability is one of those eternal philosophical problems. It seems intuitive, but cannot be defined.
Mathematicians have rigor definitions for measures and random variables, but these definitions don't give answers to what probability is. In the mathematical approach, the intuitive idea of probability is assumed accepted in the beginning, and the theory is then developed with rigor mathematical definitions into which some intuitive interpretations are attached.
Science is not only about knowing as much as possible, but also about knowing what you don't know.
8. Aug 5, 2013
### jostpuur
Keep in mind that likelihood has its own meaning in statistical inference. Likelihood and probability are different things, and in fact probability is needed in the definition of likelihood.
http://en.wikipedia.org/wiki/Likelihood_function
So likelihood should not be used rhetorically when attempting to define probability.
9. Aug 5, 2013
### Stephen Tashi
Can you explain what the goal of these steps is supposed to be?
You aren't paying attention to the previous posts. It doesn't do any good, mathematically, to say that "probabllity" is a measure of "how likely" something is to happen. The idea of "how likely" contains no more information that the word "probability".
You are correct that the basics of probability theory are implemented as definitions.
I think you want to phrase that in terms of N independent realizations of X and in terms of the mean of those realizations, not in terms of the single random variable X.
Limits of things involving probabilities are complicated to state exactly. They are more complicated that the limits used in ordinary calculus. To make your statement precise, you'll have to study the various kinds of limits involved in probability theory.
Again, limits of probabilities are complicated. If the number of trials is not a multiple of 3, the fraction that are bernoulli "successes" can't be exactly 1/3. So the limit of the probability of getting exactly 1/3 successes doesn't approach 1 as the number of trials approaches infinity. To express the general idea that you have in mind takes more complicated language.
You can't say that it "will" by any standard assumptions of probability theory. If you express your idea precisely, you can say "it probably will".
Realize that when you say "extremely likely", you aren't saying anything that has mathematical consequences. You are just using words that make you feel psychologically more comfortable. There is no mathematical definition for "extremely likely" except in terms of "probability".
Look at the formal statement of the weak and strong laws of large numbers and look at the sophisticated concepts of limits that are used ("convergence in probability" and "almost sure convergence").
You aren't going to get around the fact that probability theory provides no guarantees about the observed frequency of events, or about the limits of observed frequencies except for those theorems that say something about the probability of those frequencies. You are presenting your series of steps as if the goal is to say something non-probabilistic about observed frequencies or to prove that "probability" amounts to some kind of observed frequency. This is not the goal of probability theory.
10. Aug 5, 2013
### bobby2k
Thanks for still beeing in the thread, really appreciate it!
My goal in the steps is to have a pathway from the basic building-blocks to the more complex usage, and results. For instance I really liked that the theory of integration and derivation, can be built from the 10 basic axioms+the axiom of the least upper bound. It is very interesting when you see the complex theorems beeing built with starting with these axioms and then making the exterme value theorem, intermediate value theorem etc. and then going on. I want to see something similar in probability theory, but it is difficult.
Ok, I get that we can say that the probability of those frequencies goes to 1. But what does this mean then? That it is "probable" that the frequencies will behave like this?
11. Aug 5, 2013
### Stephen Tashi
As I said, if you want to know what it means, you have to deal with the various ways that limits involving probabilities are defined. To say "the probability of those frequencies goes to 1" is not a precise statement. (In fact the probability of observing a frequency of successes exactly equal to he probability of success for a bernoulli random variable that is the subject of a large number of independent trials goes to zero as the number of trials increases.) If you want to understand what probability theory says about the limiting probability of observed frequencies, you have to be willing to deal with the details of how the various limits are defined.
12. Aug 5, 2013
### bobby2k
Can you reccomend a good book so that I will be able to learn what I need, to understand what I want?
I'd like it to be not so long, and easy to read if possible. I have not taken real analysis yet(or measure theory), but I have read about logic and set theory on my own, so I can del with that if the book contains it.
13. Aug 5, 2013
### Stephen Tashi
I didn't encounter the various types of limits used in probability theory until I took graduate courses, so I can't recommend a book. I'll keep my eyes out for something online that explains the various types of limits ( usually referred to as types of "convergence of sequences of functions").
Perhaps some other forum member knows a good book.
14. Aug 6, 2013
### jostpuur
bobby2k, it looks like you feel like looking for something, that cannot be found (at our time at least). IMO you have already understood the essential. You only need to calm down, take a step back, and try to see the big picture.
Yes, you are seeing a real problem with a circular thinking. If you define (or attempt to define) probability with frequencies, and then use probability concept to prove some basic frequency results, you are going in a circle.
Some of the basic results related to the frequencies are important, so I wouldn't speak bad about them. But if we are discussing attempts to define probability (which is philosophy IMO), the circular thinking should be recognized.
(The choice of words is confusing, but you can tell what's the point in the quote)
It means that we have assumed the probability concept defined and accepted, and then we have proven something technical / mathematical about probabilities of some sequences.
15. Aug 6, 2013
### Stephen Tashi
Probability theory is more of a tangle than single variable calculus. (In fact, I've read that developing probability theory is one of the main reasons that calculus was extended to include ideas like Stieltjes integration and the more general idea of "measures".)
In my opinion, mathematical topics have a "flat" and simple character when they involve the interaction between one kind of thing and a distinct kind of thing. For example, in introductory calculus, you study the limit of a function in the situation where the limit is a number. When a mathematical subject begins to study the interaction of a thing with the same kind of thing, it takes on the complexities of the "snake eating its tail" sort. For example, in real analysis you study the situation where the limit of a sequence of functions is another function. It turns out that this type of limit can be defined in several different non-equivalent ways, so even the definition of limits becomes complicated.
In probability theory, if you think of the object of study as a single "random variable" then the situation appears "flat". However, as soon as you begin to study anything involving several samples from that random variable, you introduce other random variables. Typically you have one random variable (with it's associated mean, variance etc.) and you have some sampling procedure for it. The sample value is itself a random variable. (Technically you aren't supposed to say things like "the mean of the sample 'is' 2.35" since "2.35" is only a "realization" of the sample mean. Of course both non-statisticians and statisticians say such things!) Since the sample mean is a random variable, it has its own mean and variance. The variance of the sample is also a random variable and has its own mean and variance. There is even ambiguity about how the quantity "the sample variance" is defined.
justpuur says to be calm. I'll put it this way. As you study math, you will find calm quiet areas where complex things are developed from simple things. However, there are also many turbulent places where things are developed from the same kind of thing. Don't get upset when this happens. Don't get upset because theorems in probability theory only tell you about probabilities.
16. Aug 6, 2013
### jostpuur
For example, in calculus, beginner students often feel like there is something about infinitesimals that they have not understood, but which could be understood (which is true). Then they also observe that they are unable to solve some technical calculation problems. Then, from the point of view of the beginner student, it might make sense to contemplate on the infinitesimals, because it seems like it could be, that better understanding of infinitesimals could lead to improved capability to solve technical calculation problems.
Then it takes some time for the student to learn that actually better understanding of infinitesimals will not improve capability to solve technical calculation problems, but anyway, it seemed a reasonable idea from the point of view of a beginner.
In this thread bobby2k began with quite philosophical touch (IMO), asking about circular definitions with sequences and so on. But then...
Ok so now bobby2k is only wanting to learn the definitions for sake of pratical applications?
These are elements of the thread:
There are some philosophical problems related the definition of probability.
There are complicated probability problems, whose mathematical treatment isn't obvious (at least not to everyone, beginners, us...)
Perhaps better understanding of the definitions would lead to better capability to solve technical problems?
Well there's no way to know in advance what turns out to be useful and what useless. You have to keep open mind, and remember not to get stuck in ideas to don't seem to lead anywhere.
17. Aug 6, 2013
### atyy
The idea that probability is relative frequency is not part of the mathematical structure of probability theory. The mathematical theory just defines abstract mathematical concepts with names like measure and expectation. When we say that probability is relative frequency, we are interpreting the mathematics and giving the abstract concepts operational meaning, so that the mathematics has the possibility of being used to describe and predict the results of experiments.
It is the same with geometry. Points and lines are abstract concepts. When you think of a point as the mark you make with a pencil on paper, then you are interpreting the mathematics.
In both cases, the mathematics exists without the science. Probability theory exists without relative frequency, and geometry exists without pencil and paper. The idea of probability as relative frequency, or that a point is something you draw with a pencil on paper are additional things you add so that you can pass from mathematics to science.
In Bayesian interpretations of probability, probability is not necessarily relative frequency. Some Bayesian interpretations, like de Finetti's are beautiful, if impractical to carry out exactly. Others are very practical and powerful, for example in providing evidence for a positive cosmological constant like in http://arxiv.org/abs/astro-ph/9812133 and http://arxiv.org/abs/astro-ph/9805201.
18. Aug 8, 2013
### bobby2k
Thanks for your patience guys. I was maybe not clear enough in my question. Maybe a better formulation would have been, "why does probability theory work, when the intuiative(relative frequency) part of probability, is not defined in the axioms?".
You may say that, it was not meant as a deep question. But it isn't really to do better in practical applications. Because just by having the intuitive explanation in the back of my mind, I can solve all the problems. But it is more rewarding to understand why we can use this to solve the problems.
I think my main problem is/was that I struggle to see where we go from the math to making assumptions and making a model. It seems alot easier in physics, there you may model a car with friction, it is clear what is the mathematical model, and what is the real thing. I mean, I thought that flipping coins hypothetically could be part of the mathematics, but it seems as though we have moved out of the probability world, even though it is just hypothetical.
I think I have finally wrapped my head around what many of you say. That the CLT only says something about what the probability of an event is, nothing more about what probability really is.
I made a picture to try and communicate how I view it now. Have I understood it?:
Last edited: Aug 8, 2013
19. Aug 8, 2013
### bahamagreen
A strange thing about probability is that it is not like other fundamental theories - it is not time reversible even at the smallest scale.
An event (to happen at a particular time) is said to have a probability before that time, but afterwards, what happens to that probability? It disappears or changes into a certainty?
That is, one can only use probabilities to describe things in the unknown future, not the certain past. One does not continue to say that the probability of a past event is 1/3 or 1/5; after the event the historical probability would have to be 1 or 0... but what if you don't know yet?
It gets more mysterious when a past event would seem to have either occurred or not, but we have not checked it yet to know which way it came out... can there be a probability for the person who has not checked, and a certainly for one who has?
20. Aug 10, 2013
### bobby2k
That is some very interesting points bahamagreen. :)
But do you guys think that at the basic level, my picture above describes the interaction with probability and using probability in an adequate way? I am eager to get closure. :)
21. Aug 10, 2013
### Stephen Tashi
No. People who apply probability theory (correctly know that probabilities do not represent relative frequencies. And you haven't yet dealt with the meaning of "mean(x) -> p".
To apply math, you need "understanding", not "closure".
22. Aug 10, 2013
### bobby2k
But is the text I have marked in red in my statistics book here then wrong? It deals with interpreting a confidence interval, but it may as well be interpreting the probability of a confidence interval.
23. Aug 10, 2013
### Stephen Tashi
Yes. It's wrong. The problem is the the statement that "A will occur 95% of the time". If you want to say something like "I'd be willing to bet that A occurs about 95% of the time" or "We will do calculations assuming A occurs 95%" of the time, those statements could be called an "interpretation".
As I pointed out before, if an event has probability .95 of occurring and you conduct a large number of independent trials, the probability that the event happens with a relative frequency of exactly .95 in the trials approaches zero as the number of trials approaches infinity.
By the way, from that page, it looks like your textbook is about to make an important point regarding confidence intervals. Do you understand how to interpret confidence intervals correctly?
24. Aug 10, 2013
### atyy
I'm not sure whether it is "exactly" ok, but it looks fine to me. My feeling is that one shouldn't lose sleep over this.
I don't think the problem is related only to probability theory. What is an electron? It is a particle that is deflected by an electric field. What is an electric field? It is a thing that deflects electrons. There is no problem if we consider electron and electric field as mathematical objects, but what happens if I give you an unidentified particle and ask you to show me that it is an electron?
So to connect mathematics with physics, it seems we always need some circularity. We accept as useful the mathematics and the interpretation as long as their predictions are consistent with observation.
From David O. Siegmund's Britannica article:
"Insofar as an event which has probability very close to 1 is practically certain to happen, this result justifies the relative frequency interpretation of probability. Strictly speaking, however, the justification is circular because the probability in the above equation, which is very close to but not equal to 1, requires its own relative frequency interpretation. Perhaps it is better to say that the weak law of large numbers is consistent with the relative frequency interpretation of probability."
Last edited: Aug 10, 2013
25. Aug 16, 2013
### bobby2k
My main problem when I started the thread was that I thought that the law of large numbers somehow validated that we could look at probabilities as relative frequencies(since both contained relative frequencies). But as we can read in the link you give says that the law of large numbers is consistent with the relative frequency interpretation. So this means that if we choose to use the relative frequency interpretation, then the mathematical theory seems "fair" to use?
But how would you connect the relative frequency interpretation to the axiomatic mathematical theory? I do not mean a specific connection in the sense of a theorem that guarantees something. But some connection there must surely be? And the only connection I see is that when people assume probabilities represent relative frequencies, then the axiomatic theory seems fair. Do you agree that the connection comes only when you choose how to view a probability?
I have to admit that I interpret them as the book writes, that if the confidence level is 0.95, then about 95 percent of confidence intervals in the long run will contain the parameter, because of this we can be "confident" that the paramter is in the interval we made, even if we just make one.
I have done some research about the subjects you have talked about and the thing is that they are part of the upper courses in the bachelor, and a master in mathematics in for example stochastic analysis. At my school atleast people taking even a master in statistics does not learn about these subjects(measure theory etc.) And think about how many people learn statistics(echonomists, social sciences etc.), surely not all of these will have learn about the advanced mathematical theory. This must mean that there is an adequate way to understand probability, without a master in mathematics? | {"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": 0.933696448802948, "perplexity": 409.62220317827536}, "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-2018-26/segments/1529267859766.6/warc/CC-MAIN-20180618105733-20180618125733-00099.warc.gz"} |
https://tex.stackexchange.com/questions/168913/left-aligning-equations-without-align-character | # Left aligning equations without align character
I realise that this has been asked many times before, but no answers I've found seem to provide a solution to my question.
I'd like to left align all text in an align block, in:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align}
b := (a \oplus s_1) \oplus s_2 \\
e := 0 \\
\end{align}
\end{document}
This presents me with:
If I change the first line to \documentclass[fleqn]{article} , then instead I get:
What I'm really looking for is the ability to align the first character on every line in the align block, to the far left i.e. so that b and a are vertically aligned. I would much rather not have to use a character such as & to guide the alignment, I'd prefer to always be the first character of each word after a \\.
• i don't think you want the \\ after the last line; that gives you an empty (but numbered) extra line in the output. – barbara beeton Apr 1 '14 at 14:50
Don't use align. Use gather instead.
\documentclass[fleqn]{article} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9737797975540161, "perplexity": 1053.774063273777}, "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/1563195526536.46/warc/CC-MAIN-20190720153215-20190720175215-00255.warc.gz"} |
http://physics.stackexchange.com/questions/33885/does-the-large-red-shift-value-of-galaxies-mean-they-are-far-away | # Does the large red shift value of galaxies mean they are far away?
When the red shifts of galaxies are large, why do we think that they are far away?
I know about Hubble's law, Tully-Fisher relation of spiral galaxies, Faber-Jackson relation of elliptical galaxies, and so on , but they seem to the empirical rules.
I think that the value of red shift has only the information on speed, so they might be the objects which move with the high speed near by our Galaxy.
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Comment to the question(v2): OP's underlying implicit question seems to be How do we experimentally measure radial distances from Earth to various galaxies? Answer: By using so-called standard candles. – Qmechanic Aug 10 '12 at 13:00
The reason why think a large red shift corresponds to large distance is that this is what is predicted by General Relativity.
If you make a few simplifying assumptions about the universe you can solve Einstein's equation for the universe to give a result called the FLRW metric. This predicts the universe is expanding and predicts the red shift increases with distance.
However we have experimental evidence for the red shift-distance relation. For example there is a type of supernova called SN1a for which we can calculate the brightness. Because we know what the brightness should be we can measure the brightness as seen from Earth and use this to calculate how far away the supernova is. Then we can measure the red shift and test the red shift-distance relationship.
The measurement of SN1a red shifts is the way dark energy was detected, because if you do the experiment you find the supernovae are actually slightly farther away than the red-shift predicts they should be.
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Kind of unrelated, but can you explain how SN1a red shift has anything to do with dark matter? I was under the impression that dark energy was "detected" when they had trouble explaining the orbital velocities of stars in galaxies. – Captain Claptrap Aug 10 '12 at 11:55
@CaptainClaptrap I think you're mixing up terms. One of the early signs of dark matter did come from rotation curves of galaxies. Dark energy and the accelerating expansion of the universe are signaled by SNe Ia, which have had no bearing on our understanding of dark matter. Despite the common use of the word "dark," these things have nothing to do with one another. If there's still any confusion, perhaps you can open another question. – Chris White Aug 10 '12 at 12:07
I think that clears it up. Thanks. – Captain Claptrap Aug 10 '12 at 12:43
@John Rennie: The most important reason to think they are far away is that they are small and faint, but otherwise similar to close objects. So they are obviously far away. If I see a tiny airplane outside my window, I don't assume it's actually a tiny airplane very close. – Ron Maimon Aug 10 '12 at 19:17 | {"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": 0.8383665680885315, "perplexity": 541.4643208251506}, "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/1435375096156.35/warc/CC-MAIN-20150627031816-00085-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/156072-continuity.html | 1. ## Continuity
I must say if the function is continuous in the point (0,0). Which is $\displaystyle \displaystyle\lim_{(x,y) \to{(0,0)}}{f(x,y)}=f(0,0)$
The function:
$\displaystyle f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x^2+y^2}) & \mbox{ if }& y\neq{-x}\\1 & \mbox{if}& y=-x\end{matrix}$
I think its not continuous at any point, cause for any point I would ever have a disk of discontinuous points, but I must prove it. And I wanted to do so using limits, which I think is the only way to do it.
$\displaystyle \displaystyle\lim_{(x,y) \to{(0,0)}}{(x+y)^2\sin(\displaystyle\frac{\pi}{x^ 2+y^2})}$
What should I do? should I use trajectories? the limit seems to exist, as the sin oscilates between -1 and 1, and the other part tends to zero.
2. Originally Posted by Ulysses
I must say if the function is continuous in the point (0,0). Which is $\displaystyle \displaystyle\lim_{(x,y) \to{(0,0)}}{f(x,y)}=f(0,0)$
The function:
$\displaystyle f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x^2+y^2}) & \mbox{ if }& y\neq{-x}\\1 & \mbox{if}& y=-x\end{matrix}$
I think its not continuous at any point, cause for any point I would ever have a disk of discontinuous points, but I must prove it. And I wanted to do so using limits, which I think is the only way to do it.
$\displaystyle \displaystyle\lim_{(x,y) \to{(0,0)}}{(x+y)^2\sin(\displaystyle\frac{\pi}{x^ 2+y^2})}$
What should I do? should I use trajectories? the limit seems to exist, as the sin oscilates between -1 and 1, and the other part tends to zero.
$\displaystyle \lim\limits_{(x,y)\to (0,0)} (x+y)^2=0\,\,\,and\,\,\,\sin\left(\frac{\pi}{x^2+y ^2}\right)$ is bounded, so the limit does exists and equals..., and thus the function isn't continuous at the origen.
Tonio | {"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": 0.9509186744689941, "perplexity": 248.3074764326832}, "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-2018-17/segments/1524125948617.86/warc/CC-MAIN-20180426222608-20180427002608-00613.warc.gz"} |
https://www.tau.ac.il/~tsirel/dump/Static/knowino.org/wiki/Electromagnetic_spectrum.html | # Electromagnetic spectrum
The electromagnetic spectrum is the name given to the range of electromagnetic waves covering all frequencies and wavelengths. It includes radio and television transmission, Microwaves, Infrared, visible light, Ultraviolet, X-rays, and Gamma rays.
## Contents
All electromagnetic radiation can be described in terms of its energy (E), frequency (f), or wavelength (λ). These properties are related by the following equations:
$\lambda = \frac{c}{f} \,\!$ ,
$E=hf \,\!$ ,
where c = 299 792 458 m/s (the speed of light) and h = 6.626 x 10-34 Js (Planck's constant)
Electromagnetic radiation is generally considered to consist of waves. However, in the description of several effects, especially those related to the emission, absorption, and scattering of light, EM radiation seems to consist of particles rather than of waves. That is, the energy carried by EM waves is packaged in discrete bundles called photons or light quanta. The particle character of EM radiation, such as light, is described in quantum electrodynamics, a theory which began with Dirac's work of 1927.[1]
## Electromagnetic spectrum as a resource
Especially in radio and radar frequencies, the electromagnetic spectrum can be a scarce and valuable resource, since a given frequency in a given geographic area can often be used only for one purpose. This is a classic requisite of a technical monopoly. If a frequency is in use for commercial television, than it cannot be used for cellular telephony. In electronic warfare, there may be difficult tradeoffs between letting an enemy use a radio frequency for his own purposes while one's own side is gaining signals intelligence from it, or jamming to deny its use for command and control.
See electromagnetic spectrum management for the policy, management, and commercialization of parts of the spectrum.
## Reference
1. P.A.M. Dirac, Proc. Royal Society (London), The Quantum Theory of the Emission and Absorption of Radiation, vol. A114, p. 243 (1927) | {"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": 2, "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": 0.8890573382377625, "perplexity": 1045.8578792679536}, "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-2022-05/segments/1642320303729.69/warc/CC-MAIN-20220122012907-20220122042907-00473.warc.gz"} |
http://math.stackexchange.com/questions/69488/is-this-determinant-equal-to-1 | # Is this determinant equal to 1
Let $V$ be a finite dimensional vector space over $\mathbf{C}$ with a hermitian inner product. Let $e=(e_1,\ldots,e_n)^t$ and $f=(f_1,\ldots,f_n)^t$ be orthonormal bases for $V$.
There is a matrix $A$ such that $e =A f$.
Is $\det A = 1$?
-
Do you mean $e_i=Af_i,\;i=1,2,\dots,n$? – anon Oct 3 '11 at 8:34
No. $e_i = \sum_{j=1}^n a_{ij} f_j$ and $A= (a_{ij})$. – shaye Oct 3 '11 at 11:36
Homan: how are they each an orthonormal basis for $V$ (which I assume is $n$-dimensional) if they're each only a single vector? – anon Oct 3 '11 at 11:40
For real vector spaces and two bases of the same orientation this will be true. – Mark Oct 3 '11 at 14:38
No, as a counterexample, take the matrix
$$A = \left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right) \; .$$
And take for $f$ the standard basis
$$f_1=\left(\begin{array}{c} 1 \\ 0 \end{array}\right) \; , \; f_2=\left(\begin{array}{c} 0 \\ 1 \end{array}\right) \; .$$
Clearly, the determinant of $A$ is $-1$.
-
It may be worth noting that, however, $|\det A|=1$, that is, $A$ is a unitary matrix. – joriki Oct 3 '11 at 9:00
Indeed. Basically, the matrices Homan defines are the unitary matrices, i.e. $U(n)$ . – Raskolnikov Oct 3 '11 at 9:01
So in general the determinant of $A$ is a complex number of modulus $1$, right? – shaye Oct 3 '11 at 11:18
That's indeed right. – Raskolnikov Oct 3 '11 at 11:28 | {"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": 0.9831427931785583, "perplexity": 292.2975424885505}, "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/1469257824185.14/warc/CC-MAIN-20160723071024-00246-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://mathhothouse.me/2014/09/12/pythagorean-triples/ | ## Pythagorean Triples
The Pythagorean Theorem, that “beloved” formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse. In symbols,
$a^{2}+b^{2}=c^{2}$ where a and b are the sides and c is the hypotenuse.
Since we are interested in number theory, that is, the theory of the natural numbers, we will ask whether there are any Pythagorean triangles all of whose sides are natural numbers. There are many such triangles. The most famous has side 3,4 and 5. Here are the first few examples:
$3^{2}+4^{2}=5^{2}$
$5^{2}+12^{2}=13^{2}$
$8^{2}+15^{2}=17^{2}$
$28^{2}+45^{2}=53^{2}$
Our first naive question is whether there are infinitely many Pythagorean triples, that is, triples of natural numbers $(a,b,c)$ satisfying the equation $a^{2}+b^{2}=c^{2}$
The answer is “YES” for a silly reason. If we take a Pythagorean triple $(a,b,c)$ and multiply it by some other number d, then we obtain a new Pythagorean triple
$(ad,bd,cd)$. This is true because
$(da)^{2}+(db)^{2}=d^{2}(a^{2}+b^{2})=d^{2}c^{2}=(dc)^{2}$
Clearly, these new Pythagorean triples are not very interesting. So we will concentrate our attention on triples with no common factors. We will even give them a name:
A primitive Pythagorean triple (or PPT for short) is a triple of numbers $(a,b,c)$ so that a,b, and c have no common factors and satisfy
$a^{2}+b^{2}=c^{2}$.
To investigate whether are infinitely many PPT’s is the same as asking whether there is a formula to find as many PPT’s as we want — the formula would contain relationship(s) between a, b and c.
As explained in the previous blog, the first step is to accumulate some data. I used a computer to substitute in values for a and b and checked if $a^{2}+b^{2}$ is a square. Here are some PPT’s that I found:
$(3,4,5)$; $(5,12,13)$; $(8,15,17)$; $(7,24,25)$;
$(20,21,29)$; $(9,40,41)$; $(12,35,37)$; $(11,60,6)$;
$(28,45,53)$; $(33,56,65)$; $(16,63,65$.
A few conclusions can be easily drawn even from such a short list. For example, it certainly looks like one of a and b is odd and the other is even. It also seems that c is always odd.
It is not hard to prove that these conjectures are correct. First, if a and b are both even, then c would also be even. This means that a,b and c would have a common factor of 2, so the triple would not be primitive. Next, suppose that a and b are both odd, which means that c would have to be even. This means that there are numbers x,y and z so that
$a=2x+1$ and $b=2y+1$ and $c=2z$
We can substitute this in the equation $a^{2}+b^{2}=c^{2}$ to get
$(2x+1)^{2}+(2y+1)^{2}=(2z)^{2}$
Hence, $2x^{2}+2x+2y^{2}+2y+1=2z^{2}$
This last equation says that an odd number is equal to an even number, which is impossible, so a and b cannot both be odd. Since, we have just checked that they cannot both be even and cannot both be odd, it must be true that one is even and the other is odd. It’s then obvious from the equation $a^{2}+b^{2}=c^{2}$ that c is also odd.
We can always switch a and b, so our problem now is to find all solutions in natural numbers to the equation
$a^{2}+b^{2}=c^{2}$ with a odd, b even and a,b,c, having no common factors.
The tools we will use are factorization and divisibility.
Our first observation is that if $(a,b,c)$ is a primitive PPT, then we can factor
$a^{2}=c^{2}-b^{2}=(c-b)(c+b)$
Here are a few examples from the list given earlier, where note that we always take n to be odd and b to be even:
$3^{2}=5^{2}-4^{2}=(5-4)(5+4)=1.9$
$15^{2}=17^{2}-8^{2}=(17-8)(17+8)=9.25$
$35^{2}=37^{2}-12^{2}=(37-12)(37+12)=25.49$
$33^{2}=65^{2}-56^{2}=(65-56)(65+56)=9.121$
It looks like $c-b$ and $c+b$ are themselves always squares. We check this observation with a couple more examples:
$21^{2}=29^{2}-20^{2}=(29-20)(29+20)=9.49$
$63^{2}=65^{2}-16^{2}=(65-16)(65+16)=49.81$
How can we prove that $c-b$ and $c+b$ are squares? Another observation apparent from our list of examples is that $c-b$ and $c+b$ seem to have no common factors. We can prove this last assertion as follows. Suppose that d is a common factor of $c-b$ and $c+b$, that is, d divides both $c-b$ and $c+b$. Then, d also divides
$(c+b)+(c-b)=2c$ and $(c+b)-(c-b)=2b$
Thus, d divides $2b$ and $2c$. But, b and c have no common factor because we are assuming that $(a,b,c)$ is a primitive Pythagorean triple. So, d must equal 1 or 2. But, d also divides $(c-b)(c+b)=a^{2}$ and a is odd, so d must be 1. In other words, the only number dividing both $c-b$ and
$c+b$ is 1, so $c-b$ and $c+b$ have no common factor.
We now know that $c-b$ and $c+b$ have no common factor, and that their product is a square since $(c-b)(c+b)=a^{2}$. The only way that this can happen is if $c-b$ and $c+b$ are themselves squares. So, we can write
$c+b=s^{2}$ and $c-b=t^{2}$ where $s>t \geq 1$ are odd integers with no common factors. Solving these two equations for b and c yields
$c=(s^{2}+t^{2})/2$ and $b=(s^{2}-t^{2})/2$ and then
$a=\sqrt{(c-b)(c+b)}=st$
We have finished our first proof of elementary number theory! The following theorem records our accomplishment:
Theorem (Pythagorean Triple Theorem) You will get every primitive Pythagorean triple $(a,b,c)$ with a odd and b even by using the formulas
$a=st$ and $b=(s^{2}-t^{2})/2$ and $c=(s^{2}+t^{2})/2$ where
$s>t \geq 1$ are chosen to be any odd integers with no common factors.
More later…
Nalin Pithwa
1. Sachit Shanbhag
Note that the pythogorean triplet is 11,60,61(missed a digit).
2. Posted December 11, 2015 at 1:53 am | Permalink | Reply
Do you not think you should honour your source with a reference when you quote verbatim from somebody else’s work? In this case “A Friendly Introduction to Number Theory” by Joseph H. Silverman. This text is take from chapter 2. (http://www.math.brown.edu/~jhs/frint.html).
• Posted December 13, 2015 at 4:39 am | Permalink | Reply
Hello Dr. Childs:
Many heartfelt thanks for pointing out my mistake at an early stage of my blogging activity. Yes, not only do I acknowledge what you said, but I strongly recommend Dr. Silverman’s book, “A Friendly Introduction to Number Theory” as the first book of number theory to be studied 🙂 Dr Silverman is an active, real research mathematician and I am only a lecturer in Math.
I just wrote “The purpose of my blog and references I used” so that all real mathematicians get their due credit.
Regards,
Nalin Pithwa
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 76, "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": 0.8171364068984985, "perplexity": 217.05834094590386}, "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/1579250597230.18/warc/CC-MAIN-20200120023523-20200120051523-00109.warc.gz"} |
http://mathoverflow.net/questions/128231/is-the-connected-sum-of-knots-an-isometry | # Is the connected sum of knots an isometry?
Take $X$ as the set of knots in the 3-sphere (i.e. smooth embeddings of $S^1$ in $S^3$ up to smooth isotopy), endowed with the Gordian distance $d$.
For a fixed knot $K$ we can define the map $\varphi_K : X \rightarrow X$ as $\varphi_K (K^\prime) = K^\prime \sharp K$ .
It is easy to show that $\forall K \in X$ we have $d(K_1, K_2) \ge d(\varphi_K (K_1) , \varphi_K (K_2))$.
My question is: does the equality hold? Or in other words is the map $\varphi_K$ and isometry on $X$ with respect to $d$?
-
What is this distance between knots ? (pure curiosity) – Thomas Richard Apr 21 '13 at 16:45
@Thomas: Distance between knots is the least number of crossings (in a knot diagram) one needs to change in order to transform one knot to the other. For the question itself, it is not even known if crossing number is additive (see arxiv.org/abs/0805.4706 for the best result so far); distance-preservation would be much harder to prove. Maybe, though, a counter-example would be easier. On the other hand, Lackenby's result suggests to look for questions on "coarse geometry" of the "knot space X" e.g., if X is coarsely Euclidean or hyperbolic. – Misha Apr 21 '13 at 17:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9199460744857788, "perplexity": 334.08962461936864}, "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/1440646249598.96/warc/CC-MAIN-20150827033049-00035-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/titrate-sulphuirc-acid-with-sodium-hydroxide.66152/ | # Titrate sulphuirc acid with sodium hydroxide
1. Mar 6, 2005
### The Bob
Coursework time has hit me at college and no matter how much I hit the books the answer is not hitting me.
It might be simple but I can't see it. Of course, the question is needed.
I had to titrate sulphuirc acid with sodium hydroxide. Not hard to write coursework on and I am sure that if I did it without what I am trying to find out (below) then it would be fine but I am not happy just to accept that there should be equal amounts of sulphuric acid and sodium hydroxide to create a neutral solution.
I read in these forums that the molar pH of sulphuirc acid is less than that of the pOH of the sodium hydroxide. For my hypothesis I want to be able to say why I think that there needs to be more sodium hydroixde to create a neutral pH overall.
For what I need help with is the maths, really. I have been studying for the last 4 hours trying to make sense of the dissonance(?) of acids and bases and how this affects the pH or the moles (not even sure which). I installed a system that was posted up here from the 'few questions' thread (called BATE) and it says that for an equal amount of $$0.01$$ $$mol$$ $$dm^{-3}$$ concentrations of each susbstance I should get an overall pH of 6.17. I am trying to find out how to get to this number and how I can explain it along the way.
Even if this wasn't coursework I would have ask the same question (actually, to be honest I don't need it in my coursework it is more for interest but I really want to understand it).
Cheers
P.S. After re-reading it I hope it makes sense. What I want to know is how to work out that the pH after mixing, say, 10cm³ of sodium hydroixde to 10cm³ of sulphuric acid (both concentrated to 0.01M) is going to be 6.17, and therefore how much sodium hydroxide is actually needed. Also I will need to explain why so if you know any websites that will do that, to save you writing it, then I would be grateful.
Last edited: Mar 6, 2005
2. Mar 6, 2005
### GCT
The pH of an equimolar sodium hydroxide and sulfuric acid is 7, both are considered strong acid/base; the product salt of the two is weak with respect acid or base and thus the pH will remain 7. In the case of a different combination of where one of them is a weak acid/base and the other a strong base/acid with equimolar values of both, you'll need to consider that the product salt component of the weak acid/base will actually influence the pH upon the complete neutralization.
Yes, the strong base acid/base will react to completion with the weak acid/base (that is completely neutralized)...the sole factor which affects the pH afterwards is the product salt.
3. Mar 6, 2005
### The Bob
I have done the titration and the results show that for 10.0cm³ of 0.01M sulphuric acid, approximately 11.0cm³ of 0.01M sodium hydroixde is needed. Are you trying to say that on of the substances is strong and the other is weak because with equal concentrations I can't see that making any difference.
I have been looking around and I found these sorts of equations:
$$K_w = [H^+][OH^-]$$
$$K_a = \frac{[H^+][HA^-]}{[H_2 A]}$$ coming from $$H_2A \rightarrow H^+ + HA^-$$
Last edited: Mar 6, 2005
4. Mar 6, 2005
### Staff: Mentor
It is not. Equimolar means 'Having an equal number of moles'. pH will be much lower, depending on the H+ concentration.
Chemical calculators for labs and education
BATE - pH calculations, titration curves, hydrolisis
Last edited: Mar 6, 2005
5. Mar 6, 2005
### Staff: Mentor
The problem is, you will not get this number, because it is wrong - and I have no idea how did you get it using BATE :( Can you give some more details?
Chemical calculators for labs and education
BATE - pH calculations, titration curves, hydrolisis
Last edited: Mar 6, 2005
6. Mar 7, 2005
### GCT
ok, first of all, as long as there is a strong acid/base component present all of the base/acid component will be neutralized (think, la chatelier). In all cases of titrations, assuming equimolar reaction, the final pH is actually pertains to the salt product obtained from the acid base reaction.
If you're referring to to the second ionization of sulfuric acid, it is negligible compared to the first. Look up the ionization constants yourself.
Last edited: Mar 7, 2005
7. Mar 7, 2005
### Gokul43201
Staff Emeritus
The pH comes from the concentration of the acid/base as well as the dissociation constant (or you can use Ka/Kb). If the dissociation constants are very close to 1 (Ka/Kb large), the pH is roughly a function of concentration alone.
8. Mar 7, 2005
### Staff: Mentor
pH of endpoint depends on the salt present at endpoint regardless of whether the reaction is equimolar or is not.
I remember them so I don't have to check. pKa1=-3, pKa2 = 2. Difference is large but pKa2 is far from being negligible. NaHSO4 solutions are highly acidic - 0.01M solution has pH 2.21, not 7 as you stated.
Chemical calculators for labs and education
BATE - pH calculations, titration curves, hydrolisis
9. Mar 7, 2005
### The Bob
I tried it again and I didn't get the same answer. To be honest because I do not understand about the dissonance of the acids/bases I can't use the system properly.
This is what I want to know. How to work it out and how to apply it.
Cheers.
10. Mar 7, 2005
### GCT
If it is equimolar than it will depend on the salt alone, if not we'll need to account for the excess of the acid/base.
I never said that a NaHS04 solution, an acidic solution, was neutral. The pH of HSO4- is not 2.21. The pH due to the first acid dissociation alone is 2.
An acid/base classified as strong is deemed for the most part to dissociate almost to completion in water, in this case $K_a$ will be very large (Gokul...way above 1) since $K_a= \frac{[H^+][anion]}{[H_SO_4]}$. $K_{a2}$ is actually $1.2 x 10^{-2}$. There's really no comparison.
Bob, solve this equation for x, assuming a original solution of sulfuric acid of .01M and sodium hydroxide .01M...
$$.012= \frac{[x]^2}{[.01M-x]}$$
Subtract from pH 7. What you can also do is to actually calculate the pH of sulfuric acid as a whole, including Ka for both its acids. After finding the total concentration of H+ produced, find the excess of the latter by subtracting from the concentration of hydroxide. This would be more accurate.
$$.012= \frac{[.01+x][x]}{[.01-x]}$$, solve for x. $[H_3O^+]=[x+.01]$.
I'm sure you can do the rest.
You'll find that the pH of an equimolar solution of sodium hydroxide and sulfuric acid is roughtly 6.9, perhaps a bit higher.
Last edited: Mar 7, 2005
11. Mar 7, 2005
### The Bob
Wouldn't be too sure about that. I am studying this one my own and this is the only place I can get help.
So how do you know that $$K _{a2}$$ is 0.012 and why is it that you are using $$K _{a2}$$???
12. Mar 7, 2005
### Staff: Mentor
On 03-06-2005 at 11:27 you stated:
Now you deny it. Further discussion makes no sense.
Chemical calculators for labs and education
BATE - pH calculations, titration curves, hydrolisis
13. Mar 7, 2005
### The Bob
Here is the problem. I understand nothing at all to do with working out pHs or pOHs. I only found out yesterday that there was a unit pOH.
The problem is I have 10 cm3 of Sodium Hydroxide and 10 cm3 of Sulphuric Acid. I mix them together and they, obviously, create an overall pH. I need to work out what that pH is. How do I do it, using the formulas that are needed and an explaination to what each bit is for and means, please.
I would be really, really grateful if someone could explain it to be and, more or less, drag me through it.
Thanks.
14. Mar 7, 2005
### Staff: Mentor
Dissociation equation:
$$AcH \leftrightarrow Ac^- + H^+$$
Equilibrium constant:
$$K_a = \frac{[Ac^-][H^+]}{[AcH]}$$
If you know total concentration - say it is $$C_a$$ - you may calculate $$H^+$$ concentration assuming - in accordance with reaction equation - that $$[Ac^-] = [H^+]$$. If so, undissociated acid concentration is $$[AcH] = C_a - [H^+]$$ and
$$K_a = \frac{[H^+]^2}{C_a - [H^+]}$$
which is a quadratic equation. Solve it for $$[H^+]$$:
$$[H^+] = \frac{-K_a + \sqrt{K_a^2 + 4 K_a C_a}}{2}$$
and you know how to perform pH calculation of the weak acid solution.
Assumption that $$[Ac^-] = [H^+]$$ is not always valid - to be precise you should take account of $$[H^+]$$ ions from the water autodissociation. For not very weak acids and not very diluted solutions assumption holds.
Now get back to the sulphuric acid and its equimolar solution with sodium hydroxide - which is the same as $$NaHSO_4$$ solution. You may treat $$HSO_4^-$$ the same way acetic acid was treated above - first dissociation step was already 'consumed' by neutralization and is not influencing the situation (that's not always the case - here is).
As I already wrote $$pK_{a2}$$ equals 2 - so $$K_{a2}$$ (which describes second dissociation equilibrium) equals 0.01. Put it into the equation for the $$[H^+]$$ together with the 0.01 concentration (in the real titration you will have to calculate the dilution factor) and you will get
$$[H^+] = \frac{-0.01 + \sqrt{0.01^2 + 4*0.01*0.01}}{2} = 0.00618$$
and pH = 2.21
Hope that helps.
Chemical calculators for labs and education
BATE - pH calculations, titration curves, hydrolisis
15. Mar 7, 2005
### GCT
Borek, I mentioned a solution of the strong acid base, however you were implying that I had made a statement where an pure acid solution had a pH of 7.
Bob, I'm not able to explain everything about acid/base chemistry right here, what you'll need to do is read up on your own. For now I'll explain a simple case...what is the pH of a solution of .01M sulfuric acid?
We'll assume that the first acid dissociates to completion, this will give .01M of [H+] and [HSO4-] upon dissociation. The pH at this point can be calculated by taking the negative log of the concentration of [H+]....$pH=-log[H^+]$, at this point the pH is 2. However, if you wish to be absolutely exact the you'll need to incorporate the second ionization which has its own dissociation constant $K_{a2}$.
This is where this equation comes in
$$K_{a2}= \frac{[H+][SO_4^{2-}]}{[HSO_4^-]}$$
the initial concentration of [H+] is .01M, 0 for the sulfate, and .01 for the acid. Thus when x M of the acid has dissociated,
$$.012= \frac{[.01+x][x]}{[.01-x]}$$
Solving for x and then finding [.01 + x] will give you the hydronium concentration of which the pH is approximately 1.9.
16. Mar 8, 2005
### The Bob
I can accept all of this. This is all fine.
This is where I get lost. I don't see what you have done and why.
The rest of this post I can, again, accept, so long as if I try another one I can run it through here.
And yes, BATE says that too.
Cheers for the help so far.
17. Mar 8, 2005
### The Bob
Understood. I am not looking for the easy road. Just the chance to understand.
What is the 0.12 from???
18. Mar 8, 2005
### GCT
It is Ka2, the equilibrium constant for the second dissociation of sulfuric acid. So again, the summary is that by neglecting this dissociation, a sulfuric acid solution will be deemed to have a pH of 2; by taking account of the second dissociation, it will roughly have a pH of 1.9. From this you can obtain a good idea of the pH due to equimolar solution pertaining to your initial proposal. For most calculations at most schools/universities, the second dissociation is neglected, so you won't have an opportunity to show off your skills.
19. Mar 8, 2005
### The Bob
I am sure I have more questions but I am going to do some independent learning first and see what I understand of this thread and of books.
As I have said I am learning this myself, I mean the teacher haven't set us a task I just want to know, but my teacher is saying I should wait until next year. The problem is if I wait until next year then I will have more to learn rather than just a little revision.
I am going to, now, set myself the task of finding the pH of 10cm³ of 0.03 M Acetic Acid and 10cm³ of 0.03 M of Sodium Hydroxide so be ready for some more explaining of a wrong answer.
Thanks for your help so far. Really do appreaciate it (even if I can't spell it).
I assume, GCT, that you have a degree in Chemistry or so, no???
Last edited: Mar 8, 2005
20. Mar 8, 2005
### The Bob
I got 1.43pH for the mixed solution of Acetic Acid and Sodium Hydroxide.
Similar Discussions: Titrate sulphuirc acid with sodium hydroxide | {"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": 0.8499516844749451, "perplexity": 1420.089039044592}, "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-2017-13/segments/1490218189031.88/warc/CC-MAIN-20170322212949-00233-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://people.maths.bris.ac.uk/~matyd/GroupNames/432/S3xC72.html | Copied to
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## G = S3×C72order 432 = 24·33
### Direct product of C72 and S3
Series: Derived Chief Lower central Upper central
Derived series C1 — C3 — S3×C72
Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — S3×C36 — S3×C72
Lower central C3 — S3×C72
Upper central C1 — C72
Generators and relations for S3×C72
G = < a,b,c | a72=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 124 in 74 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4, C22, S3 [×2], C6 [×2], C6 [×3], C8, C8, C2×C4, C9, C9, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C2×C8, C18, C18 [×3], C3×S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C2×C12, C3×C9, C36, C36 [×2], C2×C18, C3×Dic3, C3×C12, S3×C6, S3×C8, C2×C24, S3×C9 [×2], C3×C18, C72, C72 [×2], C2×C36, C3×C3⋊C8, C3×C24, S3×C12, C9×Dic3, C3×C36, S3×C18, C2×C72, S3×C24, C9×C3⋊C8, C3×C72, S3×C36, S3×C72
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, C9, C12 [×2], D6, C2×C6, C2×C8, C18 [×3], C3×S3, C24 [×2], C4×S3, C2×C12, C36 [×2], C2×C18, S3×C6, S3×C8, C2×C24, S3×C9, C72 [×2], C2×C36, S3×C12, S3×C18, C2×C72, S3×C24, S3×C36, S3×C72
Smallest permutation representation of S3×C72
On 144 points
Generators in S144
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 25 49)(2 26 50)(3 27 51)(4 28 52)(5 29 53)(6 30 54)(7 31 55)(8 32 56)(9 33 57)(10 34 58)(11 35 59)(12 36 60)(13 37 61)(14 38 62)(15 39 63)(16 40 64)(17 41 65)(18 42 66)(19 43 67)(20 44 68)(21 45 69)(22 46 70)(23 47 71)(24 48 72)(73 121 97)(74 122 98)(75 123 99)(76 124 100)(77 125 101)(78 126 102)(79 127 103)(80 128 104)(81 129 105)(82 130 106)(83 131 107)(84 132 108)(85 133 109)(86 134 110)(87 135 111)(88 136 112)(89 137 113)(90 138 114)(91 139 115)(92 140 116)(93 141 117)(94 142 118)(95 143 119)(96 144 120)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 129)(52 130)(53 131)(54 132)(55 133)(56 134)(57 135)(58 136)(59 137)(60 138)(61 139)(62 140)(63 141)(64 142)(65 143)(66 144)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
G:=sub<Sym(144)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,25,49)(2,26,50)(3,27,51)(4,28,52)(5,29,53)(6,30,54)(7,31,55)(8,32,56)(9,33,57)(10,34,58)(11,35,59)(12,36,60)(13,37,61)(14,38,62)(15,39,63)(16,40,64)(17,41,65)(18,42,66)(19,43,67)(20,44,68)(21,45,69)(22,46,70)(23,47,71)(24,48,72)(73,121,97)(74,122,98)(75,123,99)(76,124,100)(77,125,101)(78,126,102)(79,127,103)(80,128,104)(81,129,105)(82,130,106)(83,131,107)(84,132,108)(85,133,109)(86,134,110)(87,135,111)(88,136,112)(89,137,113)(90,138,114)(91,139,115)(92,140,116)(93,141,117)(94,142,118)(95,143,119)(96,144,120), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,133)(56,134)(57,135)(58,136)(59,137)(60,138)(61,139)(62,140)(63,141)(64,142)(65,143)(66,144)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,25,49)(2,26,50)(3,27,51)(4,28,52)(5,29,53)(6,30,54)(7,31,55)(8,32,56)(9,33,57)(10,34,58)(11,35,59)(12,36,60)(13,37,61)(14,38,62)(15,39,63)(16,40,64)(17,41,65)(18,42,66)(19,43,67)(20,44,68)(21,45,69)(22,46,70)(23,47,71)(24,48,72)(73,121,97)(74,122,98)(75,123,99)(76,124,100)(77,125,101)(78,126,102)(79,127,103)(80,128,104)(81,129,105)(82,130,106)(83,131,107)(84,132,108)(85,133,109)(86,134,110)(87,135,111)(88,136,112)(89,137,113)(90,138,114)(91,139,115)(92,140,116)(93,141,117)(94,142,118)(95,143,119)(96,144,120), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,99)(22,100)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,133)(56,134)(57,135)(58,136)(59,137)(60,138)(61,139)(62,140)(63,141)(64,142)(65,143)(66,144)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78) );
G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,25,49),(2,26,50),(3,27,51),(4,28,52),(5,29,53),(6,30,54),(7,31,55),(8,32,56),(9,33,57),(10,34,58),(11,35,59),(12,36,60),(13,37,61),(14,38,62),(15,39,63),(16,40,64),(17,41,65),(18,42,66),(19,43,67),(20,44,68),(21,45,69),(22,46,70),(23,47,71),(24,48,72),(73,121,97),(74,122,98),(75,123,99),(76,124,100),(77,125,101),(78,126,102),(79,127,103),(80,128,104),(81,129,105),(82,130,106),(83,131,107),(84,132,108),(85,133,109),(86,134,110),(87,135,111),(88,136,112),(89,137,113),(90,138,114),(91,139,115),(92,140,116),(93,141,117),(94,142,118),(95,143,119),(96,144,120)], [(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,99),(22,100),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,109),(32,110),(33,111),(34,112),(35,113),(36,114),(37,115),(38,116),(39,117),(40,118),(41,119),(42,120),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,127),(50,128),(51,129),(52,130),(53,131),(54,132),(55,133),(56,134),(57,135),(58,136),(59,137),(60,138),(61,139),(62,140),(63,141),(64,142),(65,143),(66,144),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78)])
216 conjugacy classes
class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A 8B 8C 8D 8E 8F 8G 8H 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 18A ··· 18F 18G ··· 18L 18M ··· 18X 24A ··· 24H 24I ··· 24T 24U ··· 24AB 36A ··· 36L 36M ··· 36X 36Y ··· 36AJ 72A ··· 72X 72Y ··· 72AV 72AW ··· 72BT order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 9 ··· 9 9 ··· 9 12 12 12 12 12 ··· 12 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 24 ··· 24 24 ··· 24 24 ··· 24 36 ··· 36 36 ··· 36 36 ··· 36 72 ··· 72 72 ··· 72 72 ··· 72 size 1 1 3 3 1 1 2 2 2 1 1 3 3 1 1 2 2 2 3 3 3 3 1 1 1 1 3 3 3 3 1 ··· 1 2 ··· 2 1 1 1 1 2 ··· 2 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3
216 irreducible representations
dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C8 C9 C12 C12 C18 C18 C18 C24 C36 C36 C72 S3 D6 C3×S3 C4×S3 S3×C6 S3×C8 S3×C9 S3×C12 S3×C18 S3×C24 S3×C36 S3×C72 kernel S3×C72 C9×C3⋊C8 C3×C72 S3×C36 S3×C24 C9×Dic3 S3×C18 C3×C3⋊C8 C3×C24 S3×C12 S3×C9 S3×C8 C3×Dic3 S3×C6 C3⋊C8 C24 C4×S3 C3×S3 Dic3 D6 S3 C72 C36 C24 C18 C12 C9 C8 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 2 2 8 6 4 4 6 6 6 16 12 12 48 1 1 2 2 2 4 6 4 6 8 12 24
Matrix representation of S3×C72 in GL2(𝔽73) generated by
5 0 0 5
,
8 8 0 64
,
1 0 7 72
G:=sub<GL(2,GF(73))| [5,0,0,5],[8,0,8,64],[1,7,0,72] >;
S3×C72 in GAP, Magma, Sage, TeX
S_3\times C_{72}
% in TeX
G:=Group("S3xC72");
// GroupNames label
G:=SmallGroup(432,109);
// by ID
G=gap.SmallGroup(432,109);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,92,142,192,14118]);
// Polycyclic
G:=Group<a,b,c|a^72=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
×
𝔽 | {"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": 0.9920534491539001, "perplexity": 4162.009276378154}, "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/1579250610919.33/warc/CC-MAIN-20200123131001-20200123160001-00276.warc.gz"} |
https://www.qb365.in/materials/stateboard/11th-standard-mathematics-half-yearly-model-question-paper-1-315.html | Mathematics Half yearly Model Question Paper 1
11th Standard
Reg.No. :
•
•
•
•
•
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Maths
Do not write anything on the question paper
Time : 02:30:00 Hrs
Total Marks : 80
Section A
20 x 1 = 20
(a)
A\B
(b)
B\A
(c)
AΔB
(d)
A'
2. Let X = {1, 2, 3, 4} and R = {(1, 1), (1, 2), (1, 3), (2, 2), (3, 3), (2, 1), (3, 1), (1, 4),(4, 1)}. Then R is
(a)
reflexive
(b)
symmetric
(c)
transitive
(d)
equivalence
3. If A = {x / x is an integer, x2 $\le$ 4} then elements of A are
(a)
A = {-1, 0, 1}
(b)
A = {-1, 0, 1, 2}
(c)
A = {0, 2, 4}
(d)
A = {- 2, - 1, 0, 1, 2}
4. Solve $\sqrt{7+6x-x^2}=x+1$
(a)
(1, -3)
(b)
(3, -1)
(c)
(1, -1)
(d)
(3, -3)
5. For the below figure of ax2 + bx + c = 0
(a)
a < 0, D > 0
(b)
a > 0, D > 0
(c)
a < 0, D < 0
(d)
a > 0, D = 0
6. cos2ፀ cos2ф+sin2(ፀ-ф)-sin2(ፀ+ф) is equal to
(a)
sin2(ፀ-$\phi$)
(b)
cos2(ፀ+$\phi$)
(c)
sin2(ፀ-$\phi$)
(d)
cos2(ፀ-$\phi$)
7. In 3 fingers, the number of ways four rings can be worn is ways.
(a)
43-1
(b)
34
(c)
68
(d)
64
8. If n+1 C3=2.nC21 then n =
(a)
3
(b)
4
(c)
5
(d)
6
9. The remainder when 3815 is divided by 13 is
(a)
12
(b)
1
(c)
11
(d)
5
10. In the series $\frac{1}{1+\sqrt 2}+\frac{1}{\sqrt 2+\sqrt 3}+\frac{1}{\sqrt 3+\sqrt 4}+...$ some of first 24 number is:
(a)
4
(b)
$\sqrt 24$
(c)
$\frac{1}{\sqrt 24}$
(d)
$\frac{1}{\sqrt 25-\sqrt 24}$
11. Which of the following point lie on the locus of 3x2+3y2-8x-12y+17 = 0
(a)
(0,0)
(b)
(-2,3)
(c)
(1,2)
(d)
(0,-1)
12. Distance between the lines 5x + 3y - 7 = 0 and 15x + 9y + 14 = 0 is
(a)
$\frac{35}{\sqrt{34}}$
(b)
$\frac{1}{3\sqrt{34}}$
(c)
$\frac{35}{2\sqrt{34}}$
(d)
$\frac{35}{3\sqrt{34}}$
13. The lines x + 2y - 3 = 0 and 3x - y + 7 = 0 are:
(a)
parallel
(b)
neither parallel nor perpendicular
(c)
perpendicular
(d)
parallel as wellas perpendicular
14. If A=$\begin{bmatrix} 1 & -1 \\ 2 &-1 \end{bmatrix}$,B=$\begin{bmatrix} a & 1 \\ b &-1 \end{bmatrix}$ and (A+B)2=A2+B2, then the values of a and b are
(a)
a = 4, b =1
(b)
a =1, b = 4
(c)
a = 0, b = 4
(d)
a = 2, b = 4
15. A vector $\overrightarrow{OP}$ makes 60° and 45° with the positive direction of the x and y axes respectively. Then the angle between $\overrightarrow{OP}$and the z-axis is
(a)
45°
(b)
60°
(c)
90°
(d)
30°
16. The vector in the direction of the vector$\hat{i}-2\hat{j}+2\hat{k}$ that has magnitude 9 is
(a)
$\hat{i}-2\hat{j}+2\hat{k}$
(b)
$\frac { \hat { i } -2\hat { j } +2\hat { k } }{ 3 }$
(c)
3($\hat{i}-2\hat{j}+2\hat{k}$)
(d)
9($\hat{i}-2\hat{j}+2\hat{k}$)
17. $lim_{\theta\rightarrow0}{Sin\sqrt{\theta}\over \sqrt{sin \theta}}$
(a)
1
(b)
-1
(c)
0
(d)
2
18. If y=cos (sin x2),then ${dy\over dx}$ at x= $\sqrt{\pi\over 2}$ is
(a)
-2
(b)
2
(c)
$-2\sqrt{\pi\over 2}$
(d)
0
19. If ,then the right hand derivative of f(x) at x = 2 is
(a)
0
(b)
2
(c)
3
(d)
4
20. Choose the correct or the most suitable answer from the given four alternatives.
$Iff\left( x \right) =4{ x }^{ 8 },\quad then$
(a)
$f^{ ' }\left( \frac { 1 }{ 2 } \right) =f^{ ' }\left( \frac { -1 }{ 2 } \right)$
(b)
$f\left( \frac { 1 }{ 2 } \right) =-f^{ ' }\left( \frac { -1 }{ 2 } \right)$
(c)
$f\left( \frac { 1 }{ 2 } \right) =f\left( \frac { -1 }{ 2 } \right)$
(d)
$f\left( \frac { 1 }{ 2 } \right) =f^{ ' }\left( \frac { -1 }{ 2 } \right)$
21. Section B
Answer any seven question in which question no. 30 is compulsory.
7 x 2 = 14
22. If the equations x2 - ax + b = 0 and x2 - ex + f = 0 have one root in common and if the second equation has equal roots, then prove that ae = 2 (b + f).
23. Find cos(x - y), given that cos x = $-\frac{4}{5}$ with $\pi<x<{{3\pi}\over{2}}$ and sin y = $-\frac{24}{25}$ with$\pi<y<{{3\pi}\over{2}}$.
24. Find the value of $\frac { 12! }{ 9!\times 3! }$
25. Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
${ \left( x+2 \right) }^{ \frac { 2 }{ 3 } }$
26. lf P(2,-7) is a given point and Q is a point on (2x2 + 9y2 = 18), then find the equations of the locus of the mid-point of PQ.
27. Find the sum A + B + C if A, B, C are given by
A=$\begin{bmatrix} { sin }^{ 2 }\theta & 1 \\ { cos }^{ 2 }\theta & 0 \end{bmatrix}$B= $\begin{bmatrix} cos^{ 2 }\theta & 1 \\ { -cosec }^{ 2 }\theta & 0 \end{bmatrix}$ and C= $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$
28. Verify whether the following ratios are direction cosines of some vector or not${1\over\sqrt{2}},{1\over 2},{1\over 2}$
29. Complete the table using calculator and use the result to estimate the limit.
$lim_{x\rightarrow0}{cos x-1\over x}$
x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x)
30. Find the derivatives of the following$\sqrt{x^2+y^2}=tan^{-1}({y\over x})$
31. Section C
Answer any seven question in which question no. 40 is compulsory.
7 x 3 = 21
32. If R is the set of all real numbers, what do the cartesian products R x Rand R x R x R represent?
33. If $\left( { x }^{ +\frac { 1 }{ 2 } }+{ x }^{ -\frac { 1 }{ 2 } } \right) ^{ 2 }=\frac { 9 }{ 2 }$ ,then find the value of $\left( { x }^{ \frac { 1 }{ 2 } }-{ x }^{ -\frac { 1 }{ 2 } } \right)$for x>1
34. Prove that $cos\frac { B-C }{ 2 }= \frac { b+c }{ a } sin\frac { A }{ 2 }$
35. A spring was hung from a hook in the ceiling. A number of different weights were attached to the spring to make it stretch, and the total length of the spring was measured each time shown in the following table.
Weight, (kg) 2 4 5 8 Length, (cm) 3 4 4.5 6
(i) Draw a graph showing the results.
(ii) Find the equation relating the length of the spring to the weight on it.
(iii) What is the actual length of the spring.
(iv) If the spring has to stretch to 9 cm long, how much weight should be added?
(v) How long will the spring be when 6 kilograms of weight on it?
36. The line 2x - y = 5 turns about the point on it, whose ordinate and abscissae are equal, through an angle of 45° in the anti-clockwise direction. find the equation of the line in the new position.
37. If A=$\begin{bmatrix} 1&-1 &2 \\ -2 & 1 & 3 \\ 0 &-3 &4 \end{bmatrix}$ and B= $\begin{bmatrix} 1&-3 \\ -1 & 1 \\ 1 &2 \end{bmatrix}$ find AB and BA if they exist.
38. Use the graph to find the limits (if it exists). If the limit does not exist, explain why?
$lim_{x\rightarrow3}(4-x)$.
39. Differentiate : $y=(x^3-1)^{100}$
40. Show that the points whose position vectors are 2$\hat{i}$+ 3$\hat{j}$− 5$\hat{k}$, 3$\hat{i}$+ $\hat{j}$− 2$\hat{k}$ and, 6$\hat{i}$− 5 $\hat{j}$+ 7$\hat{k}$ are collinear
41. Section D
5 x 5 = 25
1. Write the values of f at -4, 1, -2, 7, 0 if $f(x)=\begin{cases}-x+4\quad if\ -\infty <x\le -3 \\ x^2+4\quad if-3<x<-2 \\ x^2-x\quad if-2\le x<1\\x-x^2\quad if\ 1\le x <7\\0\quad \quad \quad otherwise \end{cases}$
2. If $x={{\sqrt{3}-\sqrt{2}}\over{\sqrt{3}+\sqrt{2}}}$ and $y={{\sqrt{3}+\sqrt{2}}\over{\sqrt{3}-\sqrt{2}}}$ find the value of x2+xy+y2.
1. Prove that $cos({ 30 }^{ 0 }-A)cos({ 30 }^{ 0 }+A)cos({ 45 }^{ 0 }-A)cos(45^{ 0 }+A)=cos2A+\frac { 1 }{ 4 }$
2. Prove that $\left( 1+\frac { 1 }{ 1 } \right) \left( 1+\frac { 1 }{ 2 } \right) \left( 1+\frac { 1 }{ 3 } \right) ...\left( 1+\frac { 1 }{ n } \right) =\left( n+1 \right)$ for all $n\epsilon N$ by the principle of mathematical induction.
1. Compute the sum of first n terms of the following series 6 + 66 + 666 + .......
2. Consider a hollow cylindrical vessel, with circumference 24cm and height 10 cm. An ant is located on the outside of vessel 4cm from the bottom. There is a drop of honey at the diagrammatically opposite inside of the vessel, 3cm from the top. (i) What is the shortest distance the ant would need to crawl to get the honey drop? (ii) Equation of the path traced out by the ant. (iii) Where the ant enter in to the cylinder? Here is a picture that illustrates the position of the ant and the honey.
1. Show that the locus of the mid-point of the segment intercepted between the axes of the variable line x cos $\alpha$ + y sin $\alpha$ = p is $\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$ where p is a constant.
2. Compute all minors, cofactors of A and hence compute |A| if A=$\begin{bmatrix} 1& 3 &-2 \\4 & -5 &6 \\ -3 & 5 & 2 \end{bmatrix}$ .Also check that | A | remains unaltered by expanding along any row or any column.
1. Evaluate the following limits :
$lim_{x\rightarrow a}{\sqrt{x-b}-\sqrt{a-b}\over x^2-a^2}(a>b)$
2. If y = $(cos^{-1}x)^2$ ,prove that $(1-x^2){d^2y\over dx^2}-x{dy\over dx}-2=0.$ Hence find y2 when x=0 | {"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": 0.8465556502342224, "perplexity": 766.5449870318345}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986664662.15/warc/CC-MAIN-20191016041344-20191016064844-00250.warc.gz"} |
https://www.taylorfrancis.com/books/9780203854204/chapters/10.4324/9780203854204-9 | chapter 3
15 Pages
## The Analysis of Linguistic Variation
The previous chapter laid the groundwork for the analysis of linguistic variation. We defined linguistic variation as “differences in linguistic form without (apparent) changes in meaning”. We also introduced the analytic construct of the variable, which we defined as “different ways (variants) of saying the same thing (the variable context)”. We saw that variables can occur at a number of different linguistic levels: phonetics or phonology, morphology, syntax, the lexicon and discourse. We introduced the principle of accountability, which requires that we examine not only the variant of interest to us but also its relative frequency with respect to all of the other variants of the same variable. In this respect, the definition of the variable context, the place where the speaker has a choice between forms, assumes a central position in the analysis of linguistic variation. How we define the variable context determines which forms we include in the analysis, how we calculate relative frequencies and, ultimately, how we interpret the variation. | {"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": 0.9604638814926147, "perplexity": 925.3095658569443}, "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-2020-34/segments/1596439738944.95/warc/CC-MAIN-20200812200445-20200812230445-00546.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-a-combined-approach-4th-edition/chapter-12-section-12-7-common-logarithms-natural-logarithms-and-change-of-base-exercise-set-page-889/30 | ## Algebra: A Combined Approach (4th Edition)
$\frac{1}{2}$
Based on the definition of the common logarithm, we know that $log(x)=log_{10}x$. Furthermore, recall that $log_{b}x=y$ is equivalent to the statement $b^{y}=x$ (where $x\gt0$, $y$ is a real number, and $b\gt0$ and $b\ne1$). Therefore, $log(\sqrt 10)=log_{10}\sqrt 10=\frac{1}{2}$, because $10^{\frac{1}{2}}=\sqrt 10$. | {"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": 0.996968150138855, "perplexity": 126.37866029513881}, "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-2018-17/segments/1524125947931.59/warc/CC-MAIN-20180425154752-20180425174752-00247.warc.gz"} |
https://www.physicsforums.com/threads/simple-word-problem.96358/ | # Simple word problem.
1. Oct 23, 2005
### danizh
Question: A rectangle with a perimeter of 40cm is rotated around one of its sides, creating a right cylinder. What is the largest possible volume for this cylinder?
Here's what I have done so far:
Equation #1:
40 = 2h + 2r
r = 20 - h
Equation #2:
Volume = pi*r^2h
= pi(20-h)(h)
= 20pi*h - h^2*(pi)
Derivate of volume: 20pi-2pi*h
10=h
Therefore, r also equals 10.
Thus, the maximum volume is 3141.592cm^3, which is incorrect.
The actual answer is 3723.37cm^3. Any help would be great.
Sorry, but I think this should be in the "Calculus and Beyond" board.
I'm not too sure how to move it there, though.
2. Oct 23, 2005
### moose
I think you forgot to square the radius (20-h) when putting it in....
I may be wrong, I just looked at it quickly and that's what I saw.
EDIT: My way, my calculator now tells me 3723.3691 so yup, that was your mistake
Last edited: Oct 23, 2005
3. Oct 23, 2005
### danizh
Thanks for the help, I really appreciate it.
I'll be more careful next time.
4. Oct 23, 2005
### danizh
I have another question, wouldn't it be more appropriate if the "restraint equation" was 40 = 2h + 4r rather than 40 = 2h + 2r. It just seems to make more sense since if the cylinder is transformed into a rectangle, each side of the triangle would be a diameter (or two times the radius) rather than just the radius, which we are assuming right now. I'm just curious to know why I get the wrong answer if I do it the way that seems to be more logical to me.
5. Oct 24, 2005
### HallsofIvy
Staff Emeritus
I assume you meant "rectangle" where you wrote "triangle" above. The reason a side of the rectangle is a radius not a diameter is that the rectangle is rotated about one side, not about a center line of the rectangle.
6. Oct 24, 2005
### danizh
Ah, I understand it now! Thanks for clearing that up.
I think the key to the question is that it is rotated to create a right cylinder.
7. Feb 28, 2008
### chaosblack
what did you get for your "r" value? I'm doing somethign wrong...
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Similar Discussions: Simple word problem. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8711745738983154, "perplexity": 1410.9813923239144}, "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-2017-09/segments/1487501170864.16/warc/CC-MAIN-20170219104610-00161-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://davegiles.blogspot.com/2012/02/asymptotic-properties-of-estimators-and.html | ## Tuesday, February 14, 2012
### "Asymptotic" Properties of Estimators and Tests
We're so familiar with "large-sample" asymptotics as a way of characterizing the behaviour of our estimators and tests in econometrics, that we tend to forget that there are other, very interesting ways of evaluating their behaviour, and approximating small-sample behaviour.
I touched on this in an earlier earlier post when I discussed "small-sigma" (or "small error") asymptotics. However, that's by no means the end of the story.
There are at least two other types of "asymptotics" that have proven themselves to be very useful in econometric analysis. The first of these dates back at least as far as the work of Kunitomo (1980), Morimune (1983), and others, during the hey-day of work on simultaneous equations models.
It's what is nowadays called "many instruments" asymptotics. In this set-up, the number of instruments and the sample size are allowed to grow at the same rate. (Of course, the number of regressors, and hence the number of parameters to be estimated, is constant). This type of asymptotic behaviour has been shown to be especially useful in the construction of asymptotic covariance matrices, and hence asymptotic standard errors, for a number of problems. For example, see Hansen et al. (2008).
The other type of "asymptotics" is the so-called "small bandwidth" asymptotics, as exemplified by the work of Cattaneo et al. (2010, 2011). In this case, a non-parametric approach is taken, and the asymptotics are based on a sequence of bandwidths, rather than a sequence of sample sizes (as with the usual large-n asymptotics), or a sequence of values for the disturbance variance (as with "small-sigma" asymptotics).
Very recently, Cattaneo (2012) showed that actually there is a very close connection between "small bandwidth" asymptotics and "many instruments" asymptotics.
One of the many things that is interesting about these other types of asymptotic behaviour is that they enable us to construct asymptotically valid covariance matrix estimators in situations where estimators such as Hal White's "heteroskedasticity-consistent" estimator fails.
A great example of this is in the context of a linear panel-data model with fixed effects. If you thought that White's covariance matrix estimator is consistent in this context, then think again. Better yet, take a look at Stock and Watson (2007). It is inconsistent if T (> 2) is fixed, and the number of cross-section units increases without limit. Stock and Watson supply an alternative covariance matrix estimator that is (nT)1/2 consistent.
In short, the "asymptotic behaviour" of our estimators and tests can depend on what we mean by "asymptotics". There's more than one way to think about this type behaviour.
Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.
References
Cattaneo, M. D., R. K. Crump, and M. Jansson 2010. Robust data-driven inference for density-weighted average derivatives. Journal of the American Statistical Association, 105, 1070-1083.
Cattaneo, M. D., R. K. Crump, and M. Jansson 2011. Small bandwidth asymptotics for density-weighted average derivatives. Forthcoming in Econometric Theory.
Cattaneo, M. D., M. Jansson and W. K. Newey, 2012. Alternative asymptotics and the partially linear model with many regressors. CREATES Research Paper 2012-02, Department of Economics and Business, Aarhus University.
Hansen, C., J. Hausman, and W. K. Newey, 2008. Estimation with many instrumental variables. Journal of
Business and Economic Statistics, 26, 398-422.
Kunitomo, N., 1980. Asymptotic expansions of the distributions of estimators in a linear functional relationship and simultaneous equations. Journal of the American Statistical Association, 75, 693-700.
Morimune, K., 1983. Approximate distributions of k-class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica, 51, 821-841.
Stock, J. H., and M. W. Watson, 2007. Heteroskedasticity-robust standard errors for fixed effects panel data regression. Econometrica, 76, 155-174. | {"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": 0.8815481066703796, "perplexity": 1132.8671195783343}, "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/1508187822116.0/warc/CC-MAIN-20171017144041-20171017164041-00846.warc.gz"} |
http://www.nanoscalereslett.com/content/9/1/51/ | Nano Express
# Optical absorption of dilute nitride alloys using self-consistent Green’s function method
Masoud Seifikar12*, Eoin P O’Reilly12 and Stephen Fahy12
Author Affiliations
1 Tyndall National Institute, Lee Maltings, Dyke Parade, Cork, Ireland
2 Department of Physics, University College Cork, Cork, Ireland
For all author emails, please log on.
Nanoscale Research Letters 2014, 9:51 doi:10.1186/1556-276X-9-51
Received: 20 November 2013 Accepted: 14 January 2014 Published: 29 January 2014
This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
### Abstract
We have calculated the optical absorption for InGaNAs and GaNSb using the band anticrossing (BAC) model and a self-consistent Green’s function (SCGF) method. In the BAC model, we include the interaction of isolated and pair N levels with the host matrix conduction and valence bands. In the SCGF approach, we include a full distribution of N states, with non-parabolic conduction and light-hole bands, and parabolic heavy-hole and spin-split-off bands. The comparison with experiments shows that the first model accounts for many features of the absorption spectrum in InGaNAs; including the full distribution of N states improves this agreement. Our calculated absorption spectra for GaNSb alloys predict the band edges correctly but show more features than are seen experimentally. This suggests the presence of more disorder in GaNSb alloys in comparison with InGaNAs.
##### Keywords:
Dilute nitride semiconductors; Self-consistent Green’s function; Optical absorption; Band anticrossing model
### Background
The substitution of a small fraction x of nitrogen atoms, for group V elements in conventional III-V semiconductors such as GaAs and GaSb, strongly perturbs the conduction band (CB) of the host semiconductor. The band structure of dilute nitride alloys has been widely investigated [1]. We have recently developed [2] a SCGF approach to calculate the density of states (DOS) near the conduction band edge (CBE) in these alloys.
One way to test the accuracy of this model is to look at optical absorption spectra for dilute nitride samples, where we expect to see features related to the N states present in the samples. The absorption spectrum arises from transitions between valence and conduction band states. It provides knowledge of the energy gap in semiconductors, and also gives significant information about the band structure of materials. Experimental measurements of absorption spectra can be used to benchmark band structure calculations. In this paper, we investigate two different materials: InyGa1-yNxAs1-x, for which the band structure has been widely studied and many of the features are well established and GaN xSb1-x, for which much less information has been reported in the literature.
We consider two different models for the band structure of dilute nitride alloys, firstly a five-level band anticrossing (BAC) model, including the host semiconductor CB and valence bands, isolated N and pair N-N states and, secondly, the linear combination of isolated nitrogen states (LCINS) model [3,4], which allows for interaction between N states on nearby sites as well as inhomogeneous broadening and produces a distribution of N state energies. In the LCINS model, the band structure of the alloys is calculated using a SCGF approach [2].
For InyGa1-yNxAs1-x alloys, we find that the BAC model reproduces the main features in the absorption spectrum, in agreement with previous work [5,6]. However this model shows some additional features which are related to the N and N-N state energies, reflecting that in the BAC model, we have ignored the actual distribution of localised states. Including the LCINS distribution of N states in InyGa1-yNxAs1-x using the SCGF approach [2] removes the additional features found in the BAC calculations and gives absorption spectra that are in very good agreement with experimental data.
We then apply our methods to GaN xSb1-x, where much less information is known theoretically and experimentally. The overall width of the optical spectrum can be well fitted by our models for the absorption spectrum. Both the BAC and LCINS models account for the absorption edge of GaNSb alloys, supporting the presence of a band anti-crossing interaction in these alloys. However, the five-level BAC model gives more features than are seen experimentally in the absorption spectrum. Including a distribution of localised state energies, obtained by modifying those calculated for GaNAs, makes the calculated absorption spectra smoother and gives better agreement with experimental data but still shows some discrepancies around the localised state peak energies. These results suggest the presence of more disorder in GaNSb samples than in InGaNAs. This disorder may be due to sample inhomogeneities or due to an intrinsically broader distribution of N states in GaNSb than in InGaNAs.
The remainder of this paper is organised as follows. In the ‘Methods’ Section we first provide an overview of optical absorption calculation, followed by a description of the band structure models used for dilute nitride alloys. The theoretical results for InGaNAs and GaNSb are presented and compared with experiment in the ‘Results and discussion’ Section. Finally, we summarise our conclusions in the last section.
### Methods
The absorption spectrum α(E) describes the rate of absorption of photons with energy per unit distance and can be described using a ‘single-electron’ approximation. In this approach, the absorption spectrum for allowed transitions between valence band v and conduction band c states is given by [7-10]
(1)
where is the photon energy, e and m0 are the electron charge and mass, c is the speed of light and nr is the refractive index. f is the Fermi-Dirac distribution function. Here, we assume a filled valence and empty conduction band, so that fv-fc=1. The matrix element in Equation 1 can be written for transitions between valence p- and conduction s-like zone centre states as [9]
(2)
where is the interaction energy, and the momentum interband matrix element, p, can be estimated from experiment as
(3)
where is the conduction band effective mass, Eg is the band gap (between conduction and valence band) and Δso is the spin-orbit-splitting energy. Therefore, Equation 1 can be written as
(4)
where E=Eck-Evk is the transition energy between conduction (Eck) and valence (Evk) states with wavevector k, Jcv(E) is the joint density of states, and we ignore for now the energy dependence of Mb.
The hole-electron interaction can be included assuming Elliot’s theory, which applies to parabolic and nondegenerate bands. According to Elliot’s model [11], the absorption spectrum is modified because of the hole-electron interaction through a multiplicative function Fex given by [8]
(5)
where
(6)
and Ry is the exciton Rydberg energy, given by [12,13]
(7)
where κ is the static dielectric constant, h is the Planck constant, is the reduced mass, and , where mh and ml are the heavy-hole and light-hole effective masses.
Including the hole-electron interaction, the total absorption spectrum αtot(E) can then be obtained as
(8)
where αlh, αhh and αso are the absorption spectra for transitions from the light-hole (LH), heavy-hole (HH) and spin-orbit split-off (SO) bands to the conduction band, respectively.
The effect of the incorporation of N in (In)GaNAs alloys can be described in different ways. We investigate here how the model chosen influences the calculated alloy absorption spectrum. We first present a simple model including isolated and pair N states using the BAC model. This model includes the nonparabolicity of the conduction and light-hole and split-off bands and the interaction between the split-off and conduction bands. In the second model, we then include the full LCINS distribution of localised states using the SCGF model.
#### Optical absorption of dilute nitride alloys in five-level BAC model
Here we first consider a simpler model, including isolated and pair N states and their interaction with the host semiconductor conduction, valence and spin-orbit split-off bands. The conventional BAC model treats the host III-V conduction band dispersion as a parabolic band [14]. Test calculations that we have undertaken show that the inclusion of band non-parabolicity can strongly modify the calculated absorption spectra due to the change in the joint density of states caused by the non-parabolicity. In order to treat the host matrix conduction band nonparabolicity and the effect of N on the alloy conduction band dispersion, we construct a 5×5 Hamiltonian. This includes the Kane nonparabolicity of the host matrix conduction band, due to interactions with the light-hole and split-off bands, and treats the effects of N using a three-level BAC model [15], including isolated and pair N states. This Hamiltonian is given as
(9)
where Δso indicates the spin-orbit-splitting energy, and Ev0 is the energy of the valence band maximum. The energy of the isolated N levels (EN), N-N pair states (ENN) and the conduction band edge (Ec0) are assumed to vary with composition, x, and temperature, T, as [15]
(10)
(11)
and
(12)
The interaction parameters are assumed to vary as and , where the concentrations of single N and N-N pair states xN and xNN, respectively, are determined from the total N concentration x as xNN=6x2 and xN=x-2xNN. The chosen values of the above parameters for (In)GaNAs and GaNSb are given in Tables 1 and 2, respectively [16-18].
Table 1. BAC model parameters for InyGa1-yNxAs1-x
Table 2. GaNxSb1-x parameters at room temperature
Calculating the eigenvalues of the matrix in Equation 9 gives the dispersion for five bands, namely, the light-hole and split-off valence bands and three conduction bands denoted by El, Em and Eu for lower, middle and upper bands, respectively. However, the five-band model of Equation 9 overestimates the LH nonparabolicity and omits the heavy-hole band dispersion. Figure 1 displays the band dispersion for In 0.04Ga0.96N0.01As0.99 where we have included a parabolic heavy-hole band, and a nonparabolic light-hole band calculated using the six-band Luttinger-Kohn (LK) valence band Hamiltonian [21]. In this model, the band dispersion of the light-hole band is given by
(13)
Figure 1. The band dispersion for In 0.04Ga0.96N0.01As0.99 in five-level BAC model. Arrows show the possible optical transition from spin-orbit split-off band (solid arrows), light hole (dashed arrows) and heavy hole (dash-dotted arrows) to conduction sub-bands.
where and are the parabolic light-hole and spin-orbit bands, respectively, and
(14)
The fractional Γ character, defined as the contribution of the host matrix CB states to a given level, can be calculated by finding the eigenvectors of Equation 9 and is given by
(15)
Here, for each conduction sub-band, we use the appropriate energy E, as shown in Figure 1 by El, Em and Eu.
The joint density of states for transitions from the valence band vi to the conduction band cf, times the host matrix Γ character of the conduction state, determines the absorption strength. It is given by
(16)
where is the energy separation between the CB cf and the valence band vi states. The absorption spectrum for transitions between bands vi and cf is given, using Equation 4, by
(17)
Figure 1 represents the band dispersion and all possible optical transitions for In 0.04Ga0.96N0.01As0.99. The total absorption spectrum is given by the summation of nine individual absorption components, shown in Figure 1, for transitions from VBs vi (including LH, HH and SO bands) to CBs cf calculated using the 5-level BAC model (i.e.El, Em and Eu) as
(18)
where αSO-l, αSO-m and αSO-u are the absorption spectra from split-off band to lower, middle and upper sub-bands, respectively, and with a similar notation used for transitions from the HH and LH bands.
#### Optical absorption of dilute nitride alloys in the LCINS model
In order to include the full distribution of N states, we need to use the Green’s function method in the framework of the LCINS model. In our recent work [2], we showed that the conduction band Green’s function could be written as
(19)
where the (complex) energy shift of each localised state j is given by
(20)
We can solve Equations 19 and 20 self-consistently by an iterative method [2] to calculate ΔEj(E). The density of states per unit volume for the CB, projected onto the host matrix conduction states, is given by
(21)
The joint density of states between the CB host matrix components and valence band vi can then be obtained using
(22)
where Evi is the energy of the valence band vi which, as in the previous section, can be the LH, HH, or split-off (SO) band. We take into account the nonparabolicity of the LH band given by Equation 13, but assume parabolic heavy-hole and split-off bands, as we find that the parabolic split-off band dispersion in the relevant energy range is very close to that obtained when nonparabolicity effects are also included.
Having the joint DOS the optical absorption spectrum can be calculated using
(23)
The Green’s function given by Equation 19 ignores the nonparabolicity of the host semiconductor CB. In order to consider the Kane non-parabolicity, the Green’s function given by Equation 19 is modified to include the conduction-valence band interaction
(24)
### Results and discussion
Here, we present the absorption spectra calculated using the five-level BAC and LCINS model and compare them with experiments. Perlin et al. [5,6,22-24] measured the optical absorption spectra for In0.04Ga0.96N0.01As0.99 and In0.08Ga0.92N0.015As0.985 and compared them with GaAs absorption data. Turcotte et al. [16,25] recently measured the optical absorption spectrum of GaNxAs1-x and InyGa1-yNxAs1-x for several values of x and y. Here, we calculate the absorption spectra for In0.04Ga0.96N0.01As0.99 and compare them with Skierbiszewski measurements at different temperatures.
#### Five-level model for InyGa1-yNxAs1-x
The interaction between the InGaAs valence and conduction bands and isolated and pair N states in InyGa1-yNxAs1-x can be described using Equation 9. The band structure parameters for In yGa1-yAs are taken to vary with In composition, y, and temperature, T, as shown in Tables 1 and 3. Also, the energy and the interaction of isolated and pair N states are taken to vary with In composition as given in Table 1. Figure 1 shows the calculated band structure of In 0.04Ga0.96N0.01As0.99 where the three conduction sub-bands (Eu(k), Em(k) and El(k)) are determined as the eigenvalues of Equation 9. Also, we consider the lowest eigenvalue of Equation 9 as the split-off band energy (ESO). The non-parabolic light-hole (ELH) is given by Equation 13, and the heavy-hole band (EHH) has been taken to be parabolic.
Table 3. Electrical and optical parameters in GaAs
The fractional Γ character of the conduction sub-bands is also required in order to calculate the joint density of states between the Γ-like conduction band components and the valence bands. Figure 2 shows the calculated Γ character of the CB for In 0.04Ga0.96N0.01As0.99, obtained using Equation 15. It is observed that in the lower sub-band, fΓ has its maximum value at the CBE and decreases toward zero at the top of the lowest band. It increases again from zero to a maximum value of around 0.4 and goes back to zero in the middle band. Then in the upper band, it increases gradually from its minimum at the bottom of the upper band, approaching an approximately constant value around E=2.1 eV.
Figure 2. The fractionalΓ character for InyGa1-yNxAs1-x withy=4% andx=1%. The fractional Γ character calculated using the five-level BAC model for conduction sub-bands, at T=10 K.
Figure 3 shows the calculated contributions of the different transitions to the total absorption spectrum. The solid, dashed and dotted lines in this figure represent the contributions due to transitions from the LH, HH and SO bands, respectively, to the conduction sub-bands. The red, blue and green lines indicate transition to the upper, middle and lower conduction sub-bands, respectively. The summation of these nine transitions is shown by the black dash-dotted line in this figure. Multiplying this by Fex gives the total absorption spectrum shown by the brown circles in this figure.
Figure 3. The absorption spectrum for InyGa1-yNxAs1-x, withy=4% andx=1% calculated using the five-level model atT=10 K. The contribution of the transitions between three valence bands and three conduction sub-bands are shown. The solid arrows designate the transitions from HH and LH bands to lower (l), middle (m) and upper (u) conduction sub-bands. The dashed arrows indicate the transitions from spin-orbit split-off band to conduction sub-bands.
The usual BAC model predicts a gap in the DOS [2,27] of (In)GaNAs alloys. However, it is clear from Figure 3 that the joint DOS for different transitions overlaps and fills this gap. Therefore, no gap is found in the absorption spectrum in (In)GaNAs alloys when using the BAC model.
Figure 4 compares the calculated absorption spectrum using the five-level BAC model with that measured and calculated by Skierbiszewski [5]. The black dots in this figure are the experimental results for the absorption spectrum of In 0.04Ga0.96N0.01As0.99 at T=10 K. The solid black line shows the calculated absorption coefficient using the two-level BAC model [6] with constant VNc=2.7 eV and EN=1.65 eV. This line shows some discrepancies with the experimental data, especially around the transition to the upper sub-band of the BAC model. The arrows in this figure indicate the different transitions from the split-off, heavy- and light-hole band edges to the lower and upper sub-band edges, in the BAC model. The dashed red line in this figure displays the absorption spectrum calculated using the five-level BAC model, which shows much better agreement with the experimental measurements. This model still shows some steps corresponding to the transitions from the HH and LH band to the lower, middle and upper conduction sub-bands (see Figure 3), whereas the experiment shows a much smoother absorption spectrum and has only one pronounced step around E=1.85 eV. This is due to the fact that in the five-band model, we have considered isolated and pair N states. Considering the full distribution of N states makes the calculated absorption spectrum smoother, in better agreement with the experimental data.
Figure 4. The absorption spectrum for InyGa1-yNxAs1-x withy=4% andx=1%. The red dashed line and blue squares represent the absorption spectrum calculated using the five-level BAC model and the SCGF including the LCINS distribution of localised states (obtained for GaN 0.012As0.988), respectively. The black dots and line show the measured and calculated spectrum by Skierbiszewski [5] at temperature 10 K.
#### LCINS approach for InyGa1-yNxAs1-x
In order to calculate the absorption spectrum using the LCINS model, we first calculate the Green’s function for the CB, given by Equation 24. The inset in Figure 5 shows histograms of the distribution of localised states for GaN xAs1-x with x=0.84% and x=1.2%, calculated using the LCINS approach [3,4]. This figure shows that the LCINS distributions for x=0.84% and x=1.2% are very similar. This implies that the LCINS distribution for GaN xAs1-x with x=1.0% can be approximated by the one for x=1.2%. However, the calculated CBE at x=1.2% (indicated by E-) is about 30 meV lower than the CBE for x=0.84%. In the five-band BAC model, we assumed that the energy gap in In yGa1-yAs is given by Eg,GaAs-1.33y+0.27y2. Also, the BAC model parameters in Table 1 suggest that including 4% In in In 0.04Ga0.96N0.01As0.99 reduces the interaction VN(E) by about 5%, which we ignore in the LCINS model. Because we do not have the LCINS distribution for In 0.04Ga0.96N0.01As0.99, we approximate it here by the LCINS distribution calculated for GaN 0.012As0.988.
Figure 5. The density of state for InyGa1-yNxAs1-x, withy=4% andx=1%. Density of states calculated using the SCGF approach and LCINS distribution of N states at room temperature. Inset shows calculated distribution of N cluster state energies at low temperature, weighted by their interactions with the conduction band edge state for GaN xAs1-x with x=0.84% and x=1.2%.
We then solve Equations 20 and 24 self-consistently [2]. Figure 5 shows the CB DOS calculated by Equation 21 for In 0.04Ga0.96N0.01As0.99 alloy. We observe that use of the LCINS distribution of states inhibits the gap predicted by the BAC model in the DOS of (In)GaNAs alloys.
The blue squares in Figure 4 show the calculated absorption spectrum at T=10 K including the full LCINS distribution of N states, which is compared with the absorption spectrum measured by Skierbiszewski [5]. Clearly, the sharp steps that we saw in the five-level BAC model disappear due to the inclusion of the distribution of localised states. This gives a better overall agreement with the experimental data. The remaining discrepancies between the calculated and experimental data may be partly due to the fact that we have approximated the N distribution by the one that was previously calculated for GaN 0.012As0.988.
The room temperature absorption coefficient calculated from the SCGF method including the full LCINS distribution of N states is shown in Figure 6. The solid black line in this figure displays the absorption spectrum measured by Skierbiszewski [5] at T=300 K. The red circles in this figure show the absorption spectrum calculated in the LCINS model, where Fex is calculated using Equation 5. The blue diamonds here indicate the result when we consider Fex=1. The dashed blue and red lines in this figure show the optical absorption calculated using the five-level BAC model, assuming Fex=1 and given by Equation 5, respectively. Here, we again observe that the results calculated using the LCINS model have lower values in comparison with those calculated by the five-level BAC model. This is because of the differences in the band non-parabolicity that we have considered for the valence bands, in the LCINS and five-level LCINS models. Figures 4 and 6 suggest that Fex might have a stronger temperature dependence than what we have considered in our calculation. The temperature dependence is considered only in the static dielectric constant as shown in Table 3 in calculating exciton Rydberg energy in Equation 6.
Figure 6. Room temperature absorption spectrum for In 0.04Ga0.96N0.01As0.99 calculated using LCINS and five-level models. The absorption spectrum calculated using the LCINS (red circles and blue diamonds) and five-level BAC (dashed lines) models. The red and blue curves display the results with and without including the electron-hole interaction. The solid black line shows the experimental data.
#### The absorption spectrum for GaN xSb1-x
We can apply our method to calculate the band structure and absorption spectrum of other dilute nitride alloys. Here, we extend our calculations to investigate the absorption spectrum of GaNSb. The room temperature band gap of GaSb is about 725 meV, around half that of GaAs. Lindsay et al. [17,28] have reported that N-related defect levels lie close to the CBE in GaNSb and therefore strongly perturb the lowest conduction states in this alloy. The band gap and optical properties in GaN xSb1-x have been shown to be strongly affected and highly sensitive to the distribution of the nitrogen atoms. Lindsay et al. [28] found that there is a wide distribution of N levels lying close to and below the CBE. The higher-lying N states push the CBE down in energy, as in GaAs, but the large number of lower energy N states are calculated to mix in strongly with the conduction band edge states, severely disrupting the band edge dispersion in GaNSb.
Here, we first investigate the band structure and optical absorption spectra of GaN xSb1-x in the five-level BAC model and compare the results with the absorption spectra measured by Veal et al. [20] and Jefferson et al. [29]. We then apply the SCGF method to GaN xSb1-x in the Section ‘LCINS model for GaN xSb1-x’. As the LCINS distributions have not yet been calculated for these alloys, we modify those calculated for GaNAs alloys and use them in our calculations.
#### Five-level BAC model for GaN xSb1-x
When a single Sb atom is replaced by N in GaSb, the N atom introduces a localised state with energy EN. However, a GaNSb alloy can also contain clusters of N atoms, such as N-N nearest neighbour pairs as well as larger clusters that introduce states in the band gap of GaSb. Table 2 contains the band parameters that we use for GaN xSb1-x, including the isolated N state energies, EN and N pair state energies, ENN relative to the valence band maximum energy and the BAC interaction parameters βN and βNN. As shown in this table, isolated N states are calculated to be less than 0.1 eV above the conduction band minimum, while the N pair states have energies that lie in the GaSb band gap. The calculated energy gap of GaN xSb1-x depends strongly on the assumed N distribution, reflecting that N cluster states introduce a series of defect levels close to the CBE in this alloy. In addition, the interaction parameters (βN and βNN) in GaN xSb1-x are calculated to be about 20% larger than for GaNAs alloys.
Figure 7 shows the conduction and valence band dispersion, calculated using the five-level BAC model given by the Hamiltonian of Equation 9. The solid lines in this figure show the conduction sub-bands. Here, we include the isolated and pair N states and their interaction with the GaSb conduction and valence bands, as explained in the previous section. Since EN is very close to the GaSb CBE and ENN just below it, we observe that the lower sub-band (El) is almost flat and located within the GaSb band gap. The band edge minimum for this band is 0.39 eV and its maximum energy is 0.45 eV. This implies that substitution of only x=1.2% N by Sb in GaSb reduced the energy gap rapidly from 725 to 390 meV. This value for the band gap of GaN xSb1-x with x=1.2% is very close to that which was previously measured [20] and calculated using k.p[28] and ab initio pseudopotential [30] calculations.
Figure 7. The band dispersion of GaN 0.012Sb0.988 calculated by the five-level BAC model. The solid lines display the conduction bands including upper (Eu), middle (Em) and lower (El) sub-bands. The dashed lines show the spin-orbit split-off (ESO), light-hole (ELH) and heavy-hole (EHH) bands.
The middle sub-band (Em) lies between 0.55 and 0.78 eV, and the upper sub-band (Eu) minimum is close to 1.0 eV. The blue dashed line in Figure 7 shows the non-parabolic spin-orbit split-off band (ESO), calculated by the lowest eigenvalue of Equation 9. Also, the non-parabolicity of the light-hole band (ELH) has been taken into account using Equation 13, while we assumed that the heavy-hole (EHH) band has a parabolic dispersion.
Given the band dispersion, we can calculate the optical absorption as described earlier for InGaNAs. The red dashed line (with circles) in Figure 8 shows the absorption spectrum of GaN 0.012Sb0.988 calculated using the five-level BAC model. Green and brown solid lines in this figure show the absorption spectra measured by Mudd et al. [31] for GaN xSb1-x with x=1.18% and x=1.22%. Our calculated absorption edge is in good agreement with these experiments. However, there are two sharp steps in the calculated spectra corresponding to transitions from the light- and heavy-hole bands to the middle and upper sub-bands. We observe that the experimental absorption spectrum, α(E), increases from zero at the CBE, to about 3×103 cm -1 at energy E=0.55 eV, that is the calculated band edge of the middle sub-band. After this point the slope of the spectra decreases up to E=1 eV, the minimum of the upper conduction sub-band. Then, due to the transition from valence bands to the upper conduction sub-band, the magnitude of the calculated absorption spectra increases rapidly. The spin-orbit splitting energy, Δso, is 0.76 eV. Therefore, transitions from the split-off band commence at 1.15 eV, where we see a small increase in the calculated absorption spectrum due to transitions from the spin-orbit split off band to the lowest conduction sub-band El. The calculated optical absorption using a 5×5 k.p Hamiltonian, accounts well for the absorption edge. Wang et al. [32] have also measured the absorption edge of GaN xSb1-x with x=0.3%, x=0.7% and x=1.4%, with the measured band edge energies in very good agreement with those calculated by the five-level model of this paper.
Figure 8. The absorption spectrum of GaN 0.012Sb0.988. The absorption spectrum for Gan xSb1-x with x=1.2%, calculated using the SCGF model and including the distribution of localised states (blue diamonds), in comparison with experimental data measured for GaN xSb1-x with x=1.18% and x=1.22% (solid lines). The results calculated by the five-level BAC model are displayed by the red line with circles.
We note however that there are two sharp features in the calculated results that are not observed in experiments. This could be due to the fact that we included only isolated and pair N states in this model and ignored the distribution of N states and their inhomogeneous broadening. The results for the calculated absorption spectrum using a five-level BAC model suggest that we need to include the full distribution of N states in optical absorption calculations.
#### LCINS model for GaN xSb1-x
It has been shown that the calculated electronic structure of GaN xSb1-x strongly depends on the assumed distribution of N atoms [28]. Therefore, in order to calculate an accurate band dispersion for this alloy, we need to have the distribution of localised states. Unfortunately, such a distribution has not been calculated for GaN xSb1-x. However, we expect that the distribution of N states in GaN xSb1-x should have a similar general form to the LCINS distribution that Lindsay et al. [3] have calculated for GaN xAs1-x and for GaN xP1-x alloys [17]. Therefore, here, we consider the LCINS distribution of N states in GaN xAs1-x and, with some small modifications, use that for GaN xSb1-x alloys.
In the previous section, we have seen that the calculated energy of an isolated N state EN is at about 0.82 eV. So, we first need to shift the LCINS distribution of GaN 0.012As0.988 to locate the highest peak at this energy. The dashed red line in the inset of Figure 9 displays the LCINS distribution of N states, weighted by , calculated for GaN xAs1-x with x=1.2%, and shifted down in energy by 888 meV. This distribution can be approximated by three Gaussian distributions, each corresponding to different N environments. It is observed in this figure that if we align the main peak at E=0.82 eV, the lowest peak corresponding to pair N-N states is located at 0.55 eV, which is higher than the values that we considered for ENN in the BAC model. Therefore, we shift the Gaussian distributions corresponding to pairs and larger clusters of N states down by a further 70 meV. Moreover, the BAC model parameters in Table 2 suggest that in GaNSb, the interaction parameters, βN and βNN, are 20% stronger than in GaNAs. Therefore, we multiply the N LCINS values by 1.44 to account for this difference. The blue solid line in the inset of Figure 9 presents the distribution of N states that we consider for GaN 0.012Sb0.988 in our calculation.
Figure 9. The DOS of GaN 0.012Sb0.988, calculated using the SCGF method and including the LCINS distribution given in the inset. Inset displays the distribution of N states assumed for GaN 0.012Sb0.988 (blue line), in comparison with the LCINS distribution of GaN 0.012As0.988, shifted down in energy by 888 meV (dashed red line). The zero of energy is taken to be at the top of the GaSb valence band.
Having the distribution of N states, we are able to calculate the Green’s function for GaN 0.012Sb0.988 using Equations 19 and 20, self-consistently. Also, the density of CB states can be calculated using Equation 21. Figure 9 shows the DOS of GaN 0.012Sb0.988 calculated by the SCGF method and including the distribution of localised states shown by the solid blue line in the inset of Figure 9. The gaps corresponding to isolated and pair N states are clearly observed in this plot. Also, at energies around 0.65 eV, the DOS has a small gap that is related to the higher cluster of N states.
We can also calculate the absorption spectrum using the SCGF model. The blue line with diamonds in Figure 8 shows the calculated absorption coefficient using this method. As expected, this method shows a better agreement with experiments than the result of the five-level BAC model (shown by red circles in this plot).
For the considered N distribution, this calculation suggests more gaps in the DOS of GaN 0.012Sb0.988 compared to GaN 0.012As0.988[2]. However, experimental data indicate that there are fewer features in the GaNSb absorption spectra than in the GaNAs ones. This could be due to inhomogeneities in the samples investigated experimentally, either due to fluctuations in the N composition in the experimental samples or because of intrinsic differences between the short-range N ordering in GaNSb and in InGaNAs samples.
Recent work by Mudd et al. [31] has shown that the composition dependence of the energy gap in GaN xSb1-x is well described using a three-level model including interactions between the host matrix band edge and the N isolated states and N-N pair states. The energy gap calculated using the LCINS model is also determined primarily by these interactions. The energy gap calculated here using the SCGF and LCINS method is consistent with experiment for the N composition x=1.2% which we consider and should closely follow the theoretical energy gap results presented in [31] as a function of N composition x.
### Conclusions
In this paper, we presented an analysis of the optical absorption spectra of dilute nitride alloys, calculated using the band structure model presented in our earlier work [2]. We have considered two different models to calculate the absorption spectra in InGaNAs and GaNSb alloys and compared our results with experimental measurements. We note however that there are some discrepancies between experimental data in similar samples that make quantitative comparison difficult.
Two models have been considered to calculate the absorption spectrum in these materials: a five-level BAC model and a LCINS-based model. The five-level BAC model included isolated and pair N states and their interactions with the host semiconductor valence and conduction bands. The results of this model for InGaNAs alloys give an overall good agreement with experiments, and predict accurate absorption edge for these alloys. However, the results of the five-level BAC model include several additional features not seen experimentally, supporting the need to consider a full distribution of N state energies in the electronic structure calculations.
We therefore extended our calculations to include the LCINS distribution using the SCGF approach presented in [2]. The calculated absorption spectra using this approach for InGaNAs provide very good agreement with experiments, supporting the validity of the LCINS approach to describe dilute nitride conduction band structure.
Our calculated absorption spectra for GaNSb alloys fit well with experiments at the absorption edge [31], and predict the correct band gap in these alloys. However, the absorption spectrum calculated in the BAC model contains features associated with individual transitions to lower and upper sub-bands in the model that are not seen in the measured absorption spectra. Taking the distribution of localised states into account reduces the impact of these features and gives results more similar to experimental absorption. But we still see some dips in our calculated spectra that are not seen in any experiment. We conclude that the distribution of N states in the GaNSb alloys studied are different from that for InGaNAs samples. We conclude that further work is required to address and resolve why more structure is found in the calculated absorption spectra compared to what is observed in the experimentally measured spectra.
### Competing interests
The authors declare that they have no competing interests.
### Authors’ contributions
SF proposed the SCGF approach to study the band structure of dilute nitride alloys. EOR suggested to apply this method to calculate the absorption spectrum. All calculations have been carried out by MS. All authors helped in drafting the manuscript. All authors read and approved the final manuscript.
### Acknowledgements
This work was supported by the Science Foundation Ireland (06/IN.1/I90; 10/IN.1/I2994; 07/IN.1/I1810). The authors thank Tim Veal for providing measured value of absorption spectrum data for GaNSb samples.
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J Appl Phys 2009, 105(1):014904-014904. Publisher Full Text | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.906648576259613, "perplexity": 2610.4624521263495}, "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/1424936463444.94/warc/CC-MAIN-20150226074103-00086-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Talk:Elliptic_geometry | # Talk:Elliptic geometry
WikiProject Mathematics (Rated B-class, High-importance)
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## Elliptic vs. Spherical
In some texts these are topologically distinct but with the same local curvature. Elliptic geometry is the one where the poles in spherical geometry are identified. These are the only two globally isotropic spaces of constant positive curvature but there are other compact topologies which are locally isotropic. This distinction is relevant for cosmological world models and FAIK the terminology is not very standard. Would anyone like to comment before I dive in and add a section on the issue? PaddyLeahy 15:40, 22 May 2007 (UTC)
I've certainly seen the distinction elsewhere, indeed there is a little about in the article. The whole article could certainly do with a bit of fleshing out. --Salix alba (talk) 17:40, 22 May 2007 (UTC)
"Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p." This is not Euclid's parallel postulate, it's Playfair's axiom. The following is the parallel postulate:
"If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles." —Preceding unsigned comment added by 82.148.66.254 (talk) 13:41, 26 February 2008 (UTC)
Quite right, although Euclid's parallel postulate is equivalent to one direction of Playfair's axiom, namely that there is at most one line parallel to L passing through p. Hyperbolic geometry violates this direction but not elliptic geometry which has no parallel lines to begin with, e.g. on the sphere the counterpart of a straight line is a geodesic or great circle, and all great circles intersect, in two places in fact, which by Euclid's definition of parallel makes them not parallel. The reason Euclid omitted the other direction is that in proving Proposition 16 he thought he'd proved that direction, but his argument either was fallacious (the more likely reason in my view) or depended on a fast-talking interpretation of Postulate 2, that straight lines go on forever, which many have interpreted as ruling out great circles because they are bounded sets, i.e. of finite length, unlike the geodesics of the Euclidean plane and the hyperbolic plane which are unbounded sets, i.e. of infinite length.
Any objections to my fixing this? (Not in anywhere near that detail of course.) --Vaughan Pratt (talk) 04:51, 12 December 2008 (UTC)
I think both of these problems are fixed in the present version of the article.--76.167.77.165 (talk) 19:27, 1 March 2009 (UTC)
I haven't contributed to wikipedia before, and I guess I'm not now since I don't have time to edit the article. Apologies in advance for any protocol I am violating. The claim made in the article that the first 28 propositions of Euclid are true in elliptic geometry is false. For example, propositions 16, 17, and 27 are not true for elliptic geometry. Either Euclid was making assumptions beyond the scope of his second postulate, or his second postulate isn't satisfied by elliptic geometry, take your pick. —Preceding unsigned comment added by 75.33.252.98 (talk) 08:42, 17 March 2009 (UTC)
## re-revert
User Shawnpoo reverted a bunch of work I did on the article. He left me a message on my talk page, but I couldn't tell what he thought the problem was. I re-reverted, and left a response on his talk page inviting him to discuss it here.--76.167.77.165 (talk) 19:05, 1 March 2009 (UTC)
## Riemann Geometry?
The Wiki page on Non-Euclidean Geomotry states that "This kind of geometry, where the curvature changes from point to point, is called Riemannian geometry". I would take from this that Riemann Geometry is not Elliptic, although I may be wrong.
I'm going to remove that line, but feel free to put it back if it is in fact true. —Preceding unsigned comment added by 86.149.136.187 (talk) 07:10, 1 April 2009 (UTC)
I think it's fair to say: Elliptic geometry is Riemannian but not all Riemannian geometries are elliptic. Zaslav (talk) 04:01, 29 July 2009 (UTC)
## Models vs. types of geometry
I see a problem in the article. The notion of a "model" of an axiom system is different from having inequivalent examples of the axioms. The spherical model and the stereographic model are different "models" of the same abstract geometry. The projective "model" is a different system of elliptic geometry. I'm attempting to revise to clarify this very important point. I hope others will correct any oversights, errors, or poor writing. Zaslav (talk) 04:05, 29 July 2009 (UTC)
This article is misleading. Consider the elliptic plane, the simplest example of elliptic geometry. There is no such thing as "spherical elliptic geometry", only spherical geometry. Two antipodal points on the sphere are identified to form the elliptic plane. The article real projective plane describes the actual "elliptic plane". See Coxeter, Introduction to Geometry (1969), pages 92 to 95. In conventional geometry, one presumes that two lines intersect in a single point or not at all. Great circles intersect in two points. The content of this article must be changed; perhaps I will.Rgdboer (talk) 20:50, 28 September 2011 (UTC)
Have introduced "Definition" section. Pruning to follow.Rgdboer (talk) 21:41, 29 September 2011 (UTC)
The picture is good, but the inset of Cape Cod should be labelled with approximation signs before the numbers, since on any finite area the triangle angles do NOT add up to exactly 180 degrees. Erasmuse (talk) 01:53, 9 March 2012 (UTC)
## Alert: "Comparison with Euclidean Geometry Section"
There's a problem in the "Comparison with Euclidean Geometry Section":
"For example, the first and fourth of Euclid's postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry. Postulate 3, that one can construct a circle with any given center and radius, fails if "any radius" is taken to mean "any real number," but holds if it is taken to mean "the length of any given line segment." Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry..."
This is garbled since it skips over Postulate 2, the infinite line postulate and one which needs more discussion in the context of elliptical geometry since whether it holds depends on one's interpretation of it. I'm not expert enough on this topic to make the changes, tho. Erasmuse (talk) 01:56, 9 March 2012 (UTC)
You might have to think of an infinite line as a line segment that goes around the sphere and meets up with itself, and then keeps on doing that, looping infinitely. If you think of a (infinite) line as just a great circle, I don't think the second postulate would hold, seeing as the circumference would be the longest distance, unless distances were changed in a similar way to what is done with two dimensional representations of hyperbolic geometry. My interpretation of what was said about the third postulate is that there's an upper bound on the radius of a circle, but the way I constructed lines, you can have a line segment of any length, so excluding "any real number" doesn't work. This is resolved when the third postulate is defined as "between any two points, there is one and only one circle that goes through one of the points and has the other as its center". If the two points are more than a quarter of the circumference away from each other, you make a circle with one point and the antipode of the other (which is the same point). I think even without defining a line as I did, defining "any real number" as "the length of any given line segment" doesn't work because even if the upper bound of a line segment were half the circumference (it could be me more if we didn't consider antipodes, but that wouldn't change anything here), we can't make a circle with a radius of that length. The biggest circle you can make has a radius equal to a quarter of the circumference of the sphere (a great circle). I could be mistaken about some of this, and criticism is welcome.
## Elliptic Geometry vs. Projective Geometry
Hi.
I saw this:
"Mathematicians commonly refer to the elliptic plane as the real projective plane. Especially in spaces of higher dimension, elliptic geometry is called projective geometry."
But on the article for projective geometry, it says:
"It is not possible to talk about angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant under projective transformations, as is seen clearly in perspective drawing."
and
"Projective geometry is less restrictive than either Euclidean geometry or affine geometry. It is an intrinsically non-metrical geometry, whose facts are independent of any metric structure."
From Harold S.M. Coxeter's "Projective Geometry", in the intro it is said that projective geometry has "no circles, no distances, no angles, no intermediacy (or "betweenness"), and no parallelism".
Elliptic geometry, on the other hand, most certainly does deal with angles and distance (e.g. we can talk of the angle sum in a triangle and how it exceeds ${\displaystyle \pi }$). There are circles, and ordering (though it is like a cyclic order, rather than a linear order), as well. The only thing missing is parallelism. So elliptic geometry is not the same as projective geometry, even though the elliptic plane and the projective plane are topologically homeomorphic. mike4ty4 (talk) 01:24, 22 September 2013 (UTC)
Correct, thank you for noting this mistake. Changes have been made today.Rgdboer (talk) 22:47, 24 September 2013 (UTC) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.8383178114891052, "perplexity": 552.8585548707351}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718866.34/warc/CC-MAIN-20161020183838-00536-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://physmath.spbstu.ru/en/article/2013.20.10/ | # Triplet states of heavy alkali metal dimers: observation and analysis
Authors:
Abstract:
The latest results on the study of the triplet states of the heavy alkali metal dimers have been summed up. The investigations were aimed at an analysis of the experimental data on the perturbation facilitated optical-optical double resonance (PFOODR) spectroscopy. The technique of such data treatment was described and demonstrated. | {"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": 0.8485751152038574, "perplexity": 1990.97429726099}, "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/1585370493121.36/warc/CC-MAIN-20200328225036-20200329015036-00240.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/23579-metric-spaces-print.html | # Metric Spaces
• November 26th 2007, 08:52 PM
hannahs
Metric Spaces
Problem:
Let X = {the set of all continuous functions f: [a,b] to R}. Let d(f,g) = the integral from a to b of |f(x) - g(x)|dx. Show that d is a metric on X, and therefore, X,d is a metric space.
I also have a hint: Recall that if h(x) is greater than or equal to 0 on [a,b] then the integral from a to b of h(x)dx is greater than or equal to 0; also if the integral from a to b of h(x)dx = 0 for h(x) greater than or equal to 0 then h(x) = 0 for all x in [a,b]
Here are the three conditions for a metric space:
1. d(x,y) = 0 if and only if x=y
Meaning that the distance from function to itself is 0
2. d(x,y) = d(y,x)
Reflexive property
3. d(x,y) is less than or equal to d(x,z) + d(z,y)
Triangle Inequality
I have no clue where to even start on this problem so any help would be greatly appreciated!
• November 26th 2007, 09:04 PM
Jhevon
Quote:
Originally Posted by hannahs
Problem:
Let X = {the set of all continuous functions f: [a,b] to R}. Let d(f,g) = the integral from a to b of |f(x) - g(x)|dx. Show that d is a metric on X, and therefore, X,d is a metric space.
I also have a hint: Recall that if h(x) is greater than or equal to 0 on [a,b] then the integral from a to b of h(x)dx is greater than or equal to 0; also if the integral from a to b of h(x)dx = 0 for h(x) greater than or equal to 0 then h(x) = 0 for all x in [a,b]
Here are the three conditions for a metric space:
1. d(x,y) = 0 if and only if x=y
Meaning that the distance from function to itself is 0
we need to show that $\int_a^b |f(x) - g(x)|~dx = 0 \implies f(x) = g(x)$
and that $f(x) = g(x) \implies \int_a^b |f(x) - g(x)|~dx = 0$
these are trivial, since |a| = 0 iff a = 0. here we can take a = f(x) - g(x)
Quote:
2. d(x,y) = d(y,x)
Reflexive property
Again, by the definition of the absolute value function, it is obvious that $|f(x) - g(x)| = |g(x) - f(x)|$ hence $\int_a^b |f(x) - g(x)|~dx = \int_a^b |g(x) - f(x)|~dx$
Quote:
3. d(x,y) is less than or equal to d(x,z) + d(z,y)
Triangle Inequality
Let $h(x): [a,b] \to \mathbb{R}$ be continuous, then $h \in X$ and we have that
$\int_a^b |f(x) - g(x)|~dx = \int_a^b |f(x) - h(x) + h(x) - g(x)|~dx$ $\le \int_a^b|f(x) - h(x)|~dx + \int_a^b |h(x) - g(x)|~dx$ by the Triangle Inequality
thus we have shown d is a metric
• November 26th 2007, 09:11 PM
hannahs
Thank you so much! :) | {"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": 8, "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": 0.9489160180091858, "perplexity": 391.6548508273917}, "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/1435375098468.93/warc/CC-MAIN-20150627031818-00037-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/is-this-really-a-module.268175/ | # Is this really a module?
1. Oct 30, 2008
### Office_Shredder
Staff Emeritus
1. The problem statement, all variables and given/known data
K is a field with finite characteristic p, G is a finite group, and W is a set that G acts on transitively (so for all x,y in W, there exists g s.t. gx=y). It then says consider M=KW the permutation module.
What is KW supposed to mean? I know for a group G that KG is the group algebra, but we don't know that W is a group (in fact, it probably isn't). Furthermore, what ring is intended to be used for multiplication? I'm confused out of my mind. I've looked back in my lecture notes so far but haven't seen anything to resolve the issue
3. The attempt at a solution
2. Oct 30, 2008
### morphism
I think in this setting the permutation module is the KG-module you obtain by letting G act on KW = set of formal linear combinations of elements of W with coefficients in K (which is basically the free K-module generated by W).
3. Oct 30, 2008
### Office_Shredder
Staff Emeritus
But G isn't a ring. Unless we just use formal addition in G to make it one?
4. Oct 30, 2008
### morphism
G isn't; KG is.
This sort of stuff comes up when you talk about things like "G-modules". See planetmath article.
5. Oct 30, 2008
### Office_Shredder
Staff Emeritus
Oh, I misunderstood what you wrote originally. That makes sense now | {"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": 0.8442721366882324, "perplexity": 1103.4549336676146}, "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-2017-09/segments/1487501171171.24/warc/CC-MAIN-20170219104611-00048-ip-10-171-10-108.ec2.internal.warc.gz"} |
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