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What's the sign of $\det\left(\sqrt{i^2+j^2}\right)_{1\le i,j\le n}$? https://ift.tt/eA8V8J
Suppose $A=(a_{ij})$ is a n×n matrix by $a_{ij}=\sqrt{i^2+j^2}$. I have tried to check its sign by matlab. l find that the determinant is positive when n is odd and negative when n is even. How to prove it？
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A reason for $ 64\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt =-\pi^4$ ... https://ift.tt/eA8V8J
Question: How to show the relation $$ J:=\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt =-\frac 1{64}\pi^4 $$ (using a "minimal industry" of relations, possibly remaining inside the real analysis)?
So i have found a solution to the problem, it is part of my solution for math.stackexchange.com - questions - 3854736, but not a satisfactory solution. "There should be more", explaining why there is a "clean result" for the integral.
Here, i am not strictly interested in a computational approach. I just want to share this with the community in these days of isolation. Any idea to attack this, or a related integral involving "three log factors" is welcome. (Well, the $\arctan$ is a sort of $\log$ in a sense that i don't want to define closer, see below.) Computations may be safely done "modulo integrals involving two or one log factor". But an illuminating, short way to show the above formula for $J$ would be wonderful.
Motivation: The above relation appeared as i tried to solve the integral posted at the above link:
Calculate $\displaystyle\int_0^{2\pi} x^2\; \cos x \cdot\operatorname{Li}_2(\cos x)\; dx$ .
After several simplifications and substitutions, it turns out that the above integral is related to integrals of the shape
$\int_0^1\log t\; R(t)\; dt$ , and
$\int_0^1\arctan t\cdot \log t\; R(t)\; dt$ , and
$\int_0^1\arctan^2 t\cdot \log t\; R(t)\; dt$ ,
and "similar" expressions.
Here $R$ is in each case a (rather simple) rational function. (The more log and/or arctangent factors, the higher the computational complexity.)
I could compute more or less algorithmically most of the the needed integrals to solve the linked problem, all of them but the integral $$ K=\int_0^1\arctan^2 t\cdot\log t\cdot\frac2{1-t^2}\; dt\ , $$ which turned out to be very hard to attack with the methods of real analysis. Computing this integral is more or less equivalent to computing $J$, and the question wants $J$ instead, since we have a "clean formula", so that some speculation about a "clever substitution" may be accepted.
My solution (for $K$) works in complex analysis, the first step is to write $$ \int_0^1 =\int_0^i+\int_i^1\ , $$ then parametrize the first integral using a linear path, the second one using a path on the unit circle.
Some comments: I will say some more words, because the situation is rich in coincidences. Since a numerical evidence is the simplest and shortes way to present (instead of showing how to show), i will use this method to at least list the coincidences. Many equalities below are "equivalent" (modulo computation of integrals of lower complexity) to the formula for $J$.
First of all, a numerical experiment using pari/gp delivers some connection between $K$ and a "cousin" of $J$:
? 2 * intnum( t=0, 1, atan(t)^2 * log(t) / (1-t^2) ) %88 = -0.357038604620289042902893412499686912781214141574556097366337 ? real(intnum( t=0, I, (pi/4 - atan(t))^2 * log(t) / (1-t^2) )) %89 = -0.357038604620289042902893412499686912781214141574556097366337 ? intnum( t=0, 1, (pi/4 - atan(t))^2 * log(t) / (1-t^2) ) %90 = -0.357038604620289042902893412499686912781214141574556097366337
In words: $$ \begin{aligned} K &= \int_0^1\arctan^2 t\cdot\log t\;\frac{2}{1-t^2}\; dt \\ &= \Re \int_0^i\left(\frac \pi 4-\arctan t\right)^2 \cdot\log t\;\frac 1{1-t^2}\; dt \\ &= \int_0^1\left(\frac \pi 4-\arctan t\right)^2 \cdot\log t\;\frac 1{1-t^2}\; dt \ . \end{aligned} $$ Note the integration margins. What happens if we take the integral on $[0,i]$ instead of $[0,1]$ in the $K$-integral? Numerically:
? 2 * real(intnum( t=0, i, atan(t)^2 * log(t) / (1-t^2) )) %98 = 1.52201704740628808181938019826101736327699352613570971392919 ? pi^4/64 %99 = 1.52201704740628808181938019826101736327699352613570971392919
In words: $$ \begin{aligned} K^* &:= \Re\int_0^i\arctan^2 t\cdot\log t\;\frac{2}{1-t^2}\; dt \\ &=\frac 1{64}\pi^4 \\ &= -\int_0^1\left(\frac \pi 4+\arctan t\right)^2 \cdot\log t\;\frac 1{1-t^2}\; dt \\ &=-J\ . \end{aligned} $$
(These observations were leading to the formula for $K$ in loc. cit. .)
One idea is to use partial integration in $J$ or $K$. Well, we have for $K$: $$ \begin{aligned} K &= \int_0^1\arctan^2 t\cdot\log t\;\left(-\log\frac {1-t}{1+t}\right)'\; dt \\ &= \underbrace{\int_0^1\arctan^2 t\cdot\frac 1t\cdot \log\frac {1-t}{1+t}\; dt}_{=2K\text{ (why?)}} \\ &\qquad\qquad+ \underbrace{\int_0^12\arctan^2 t\cdot\frac 1{t+t^2}\cdot \log t\cdot \log\frac {1-t}{1+t}\; dt}_{=-K\text{ (why?)}} \ . \end{aligned} $$
Note that $\arctan$ is related to the logarithm (over $\Bbb C$), we have the relation (around $0$) $$ \arctan t=\frac 1{2i}\log\frac {1+it}{1-it}\ . $$ The substitution $t=\frac{1-u}{1+u}$ and the formula for $\tan(\arctan 1-\arctan u)$ are giving: $$ \begin{aligned} K &= \int_0^1\arctan^2 t\cdot\log t\;\frac{2}{1-t^2}\; dt \\ &=\int_0^1 \left(\frac\pi2-\arctan u\right)^2\cdot\log\frac {1-u}{1+u}\cdot \frac {du}u\ . \\ &=\int_1^\infty \left(\frac\pi2-\arctan u\right)^2\cdot\log\frac {u-1}{1+u}\cdot \frac {du}u\ . \end{aligned} $$ (Write $\log t=\frac 12\log t^2$ to have the same expression under the integral on $(0,1)$ and on $(1,\infty)$.)
Note the fact that the factor $\frac 2{1-t^2}$ is not "random". It is the right one to make things feasible. It is the derivative of $\displaystyle -\log\frac{1-t}{1+t}$, and plugging in $t=iu$ into $\displaystyle \log\frac{1-t}{1+t}$ leads to an expression related to $\arctan u$. And conversely, $\arctan(iu)$ is related to such a logarithmic expression in $u$.
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Nov 13, 4 Jobs You Can Do With a Major in Software Engineering https://ift.tt/eA8V8J
4 Jobs You Can Do With a Major in Software Engineering
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Expressing "does not imply'' https://ift.tt/eA8V8J
In ordinary discourse, when we say that $A$ implies $B$, we shall formalize it by writing the following: $$A\rightarrow B$$ But when we say that $A$ does not imply $B$, we cannot formalize it as the following: $$\neg(A\rightarrow B)$$ Because by material implication, we have the following equivalences for the second formula: $$\neg(A\rightarrow B)\Longleftrightarrow\neg(\neg A\vee B)\Longleftrightarrow A\wedge\neg B$$ But the ordinary meaning of the sentence "$A$ does not imply $B$" is that $B$ does not follow from $A$: if $A$ is true, $B$ is either false or undecidable. I wonder how "$A$ does not imply $B$" is formally expressed in the object language (not at the meta-level). Thanks!
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Check if an integer is present in a linear recurrence https://ift.tt/eA8V8J
Given the following recurrence relation :
$f(n) = 5f(n-1) - 2f(n-2)$ where $f(0) = 0, f(1) = 1$
I need to find out if an integer $F_n$ is present in the sequence in $O(1)$ time and space.
Solving the equation, there are two distinct real roots.
$\phi = \frac{5 + \sqrt17}2$
$\psi = \frac{5 - \sqrt17}2$
Therefore, $F_n = \frac{\phi^n - \psi^n}{\sqrt17}$
Similar to Binet's rearranged formula, I want to solve for $n$ in terms of $F_n$.
Since, $\psi = \frac{2}{\phi}$
$\sqrt17F_n = \phi^n - \frac{2^n}{\phi^n}$
$Or,$
$\phi^{2n} - \sqrt17F_n\phi^n-2^n = 0$
Here I'm not able to find out a solution to express $n$ purely in terms of $F_n$ so that I can calculate the perfect square just like in Binet's formula.
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How to show that if two Platonic solids have the same number of edges, vertices, and faces, then they are similar in $\mathbb{R}^{3}$? https://ift.tt/eA8V8J
I was looking into the proof that there are only five Platonic solids in Basic Concepts of Algebraic Topology by F.H. Croom at page 29, Theorem 2.7. To clarify,
We define a Platonic solid as a simple, regular polyhedron.
We define a simple polyhedron to be a polyhedron homeomorphic to the $2$-sphere.
We define a regular polyhedron to be a polyhedron whose faces are regular polygons all congruent to each other and whose local regions near the vertices are all congruent to each other.
Using homology theory, one can prove that the Euler formula $V-E+F=2$ must hold for Platonic solids. Then by using Euler's formula and invoking a counting argument, we find that there are five possible tuples $(V, E, F)$. This is a beautiful proof, but I am unsatisfied with a question: How do we know there can't be two non-similar Platonic solids that have the same $(V, E, F)$-tuple?
Almost all sources I've looked at seem to assume it is obvious that two Platonic solids with the same $(V, E, F)$-tuple are similar, and it is not obvious to me.
Does anyone have any suggestions for how to prove this? Alternatively, does anyone know of a reference where this is proved rigorously?
Edit 1: It's not completely clear, but it seems like the definition I used for "regular polyhedra" is different than the one commonly used. Note that I am not assuming any global symmetry, so if any global symmetry is to be invoked, it needs to be proven.
Edit 2: I've been made aware of Cauchy's rigidity theorem, which is proven in, e.g., Proofs From the BOOK by Aigner & Zeigler. One can show that any two Platonic solids that have the same $(V, E, F)$-tuple must be combinatorically equivalent. However, in order for the theorem to apply, we need to show that our Platonic solids are convex. I can't seem to think of any rigorous argument for why the Platonic solids have to be convex.
And actually, you don't need to show that the entire polyhedron is convex. If I'm not mistaken, the proof for Cauchy's rigidity theorem only relies on the vertices of the polyhedron being locally convex. So really it suffices to show the vertices are convex.
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The mathematics of how universities are giving misleading Covid-19 numbers
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Let $S=\{AB-BA| A,B \in M_n(K)\}$ where $K$ is a field. Prove that $S$ is closed under matrix addition. https://ift.tt/eA8V8J
I know there is a result that $S=$ collection of all trace $0$ matrices, and that collection forms a vector space. But I want to prove it independently i.e.
For any $A,B,C,D\in M_n(K)$ we have to find $E,F\in M_n(K)$ such that $(AB-BA)+(CD-DC)=EF-FE$.
But I don't know how to solve this. But the thing I can observe that the above equation gives rise to $n^2$ equation (equating each entries of matrices of both the sides) with $2n^2$ variables (Total number of entries of $E,F$ is $2n^2$).Can this problem be simplified if we choose special kind of $E$ say diagonal matrix.
Edit-(This is valid only for $\Bbb{R}$ or $\Bbb{C}$)
$AB-BA=(A+aI)(B+bI)-(B+bI)(A+aI)$ for all $a,b\in\Bbb{R}$. And there is $a,b$ in $\Bbb{R}$ such that $A+aI,B+bI$ are invertible.
Hence, we can assume $A,B$ to be invertible matrices i.e. $S=\{AB-BA|A,B\in GL_n(K)\}$
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Why is the military interested in geometric representation theory?
In 2004, mathematicians Edward Frenkel, Dennis Gaitsgory, Kari Vilonen, and Mark Goresky were awarded millions of dollars in the form of a grant by DARPA (Defense Advanced Research Projects Agency). The project being funded was to relate the geometric Langlands program to quantum field theory.
According to DARPA’s Wikipedia page, the agency is “a research and development agency of the United States Department of Defense responsible for the development of emerging technologies for use by the military.” In other words, it seems unlikely that an agency dedicated to researching military technology would fund anything with no chance of being applicable to its goal.
Now anyone who has spent any time trying to understand the mathematics related to the geometric Langlands program knows that it’s a highly abstract field. Although the program has roots in number theory, its central conjectures are certain equivalences of DG categories whose definitions take literal books to write out fully. Even the program’s connections to physics (via S-duality a la Witten) seem too deep to have any real impact on “real world” issues.
My question is as stated in the title. Why is the US military interested in funding research related to highly abstract subfields of math and physics?
Two possibilities spring to mind:
1) DARPA has no understanding of the project; it just knows that it’s a hot topic and hopes that some practical application will come of it in the future
2) There are actual concrete applications the military has in mind but they’re classified.
Any thoughts?
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Biggest circle under a polynomial curve https://ift.tt/36qWcjq
What is the biggest circle that is contained in the region bounded by the graph of the polynomial $f(x) = x(1-x)(2x+1)$ and the x-axis interval $[0, 1]$?
(Here's the thing in Desmos)
Here's what I have tried
Let's denote the center of the circle by $(c_x, c_y)$ and the points of tangency by $(x_k, y_k)$, $k=1,2$. From distance to the x-axis, we know that the radius of the circle is then $c_y$. We can express the tangency by the dot product of the vectors $(x_k-c_x, y_k-c_y)$ and $(1, f'(x_k))$ being zero. So we get the group of equations (all for $k=1,2$)
$$\begin{cases} (x_k-c_x)^2 + (y_k-c_y)^2 &=& c_y^2 \\ x_k-c_x + (y_k-c_y)f'(x_k) &=& 0 \\ f(x_k) &=& y_k \end{cases} $$
Then I tried to solve this group by using Gröbner basis and elimination ideals. I wrote this SageMath code:
f(x) = x*(1-x)(2*x+1) fd = f.derivative() R.<x1, y1, x2, y2, cx, cy> = PolynomialRing(QQ, order='lex') polys1 = [(cx-xk)**2+(cy-yk)**2 - cy**2 for (xk, yk) in [(x1,y1), (x2, y2)]] polys2 = [(xk-cx)+(yk-cy)*fd(xk) for (xk, yk) in [(x1,y1), (x2, y2)]] polys3 = [f(xk)-yk for (xk, yk) in [(x1,y1), (x2, y2)]] I = R.ideal(polys1+polys2+polys3) gb = I.groebner_basis() for poly in gb: print (poly) print (I.dimension())
It gave me a long list of polynomials, the last one being
$$c_x^6 - 2c_x^4c_y^2 + \frac{5}{4}c_x^4c_y + \frac{1}{64}c_x^4 + c_x^2c_y^4 + \frac{3}{4}c_x^2c_y^3 + \frac{3}{16}c_x^2c_y^2 + \frac{1}{64}c_x^2c_y$$
and said the dimension is $1$. I now realize the reason we don't get dimension $0$ has probably to do with the fact that the points $(x_1, y_1)$ and $(x_2, y_2)$ could be the same point(?) So for a solution, the point $(c_x, c_y)$ should be a zero of that polynomial? But is even that correct, since the points could be collapsed and the other side go over the curve??
How to solve this even numerically? I tried with Sage by minimizing the square sum of all those polynomials that should be zero. I also tried adding constraints for the variables to be near the solution that can be seen from the picture but that didn't work either.
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Finding $\lim_{x \to 0+}(2\sin \sqrt x + \sqrt x \sin \frac{1}{x})^x$ https://ift.tt/eA8V8J
I need to compute a limit:
$$\lim_{x \to 0+}(2\sin \sqrt x + \sqrt x \sin \frac{1}{x})^x$$
I tried to apply the L'Hôpital rule, but the emerging terms become too complicated and doesn't seem to simplify.
$$ \lim_{x \to 0+}(2\sin \sqrt x + \sqrt x \sin \frac{1}{x})^x \\ = \exp (\lim_{x \to 0+} x \ln (2\sin \sqrt x + \sqrt x \sin \frac{1}{x})) \\ = \exp (\lim_{x \to 0+} \frac {\ln (2\sin \sqrt x + \sqrt x \sin \frac{1}{x})} {\frac 1 x}) \\ = \exp \lim_{x \to 0+} \dfrac {\dfrac {\cos \sqrt x} {x} + \dfrac {\sin \dfrac 1 x} {2 \sqrt x} - \dfrac {\cos \dfrac 1 x} {x^{3/2}}} {- \dfrac {1} {x^2} \left(2\sin \sqrt x + \sqrt x \sin \frac{1}{x} \right)} $$
I've calculated several values of this function, and it seems to have a limit of $1$.
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Engaging students: Law of Sines https://ift.tt/eA8V8J
In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to […]
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Study maths as second degree?
Good morning.
TL; DR: For a long time I convinced myself that computing science (and in particular AI/data science) was the way, until I realised I dislike how everything is monopolised by the industry, and that CS has become solving very 1st world problems (in a bad way). Would it make sense to try taking another degree/doing a PhD in mathematics?
Here is some (long) background about me to explain how did I arrive at such point:
I am a computing science student who has been dealing with identity crises for a long time. I enrolled in computer science because I thought it was an easy way to find a job and make decent salaries (which is partially true) but I came to realise coding is more and more off-putting.
Back in the days of high school, I used to take part to mathematics competitions and studied some advanced stuff like maths for winter camps in preparation to IMO, despite never reaching that level. During my last year of high school I considered whether to continue computing science (Italian high schools have fixed subjects you can take according to the type of your school, so mine had a lot of cs/swe classes) - for which I started to lose my initial fascination after doing cs-ish stuff for 5 years -, mathematics because I like problem solving a lot, and physics. After some considerations I opted for CS as I had clear ideas about what I could do and learn.
At first I was sceptical about joint degrees, thus I did not apply for a joint CS+Maths degree, and I am regretting it now. I guess it's a cultural thing: in my country we always believed that you should study one thing and do it well, so I could not initially accept the Anglosaxon idea of joint degrees. I thought that, despite interesting, one would have ended up having incomplete education about both.
Fast forward to last year: I went down the rabbit hole of data science and for a while I was quite happy of my decision. Until... well, until I realised data science is clearly applied science. And with that a number of factors come in, for example that much of the stack is proprietary (i.e. runs on non-free software), is being monopolised by a small number of companies and that you cannot really do actual data science without both data and computing resources. Data-wise, you need to perform massive (and unethical) data mining on users to stay on top of the concurrence. Computing-wise, the university has some resources, yes, but they can be nowhere as near as the ones that - say - Google, Nvidia, OpenAI etc. have when they train language models like BERT and GPT-2/3 from scratch. Which in practise means your research will not be truly free until you work for a company that wants to make profit out of your work.
I have worked with people coming from one of the aforementioned companies and it sounds like my worst fears were true: extreme pressure to get things done, intellectually dead applied research (e.g. writing a lot of parsers just to get the Google Assistant to 'understand' the intent of your utterance), actually no freedom of self-defining your work until you become a senior researcher etc.
And, last but not least, the actual 'usefulness'. At first I thought "I am going to do CS to solve real-world problems" like treatments for cancer, optimizing aqueduct planning in developing countries etc. I thought of things like combinatorial optimizations, bioinformatics etc.; however, I ended up doing NLP - the 'easiest' field of ML one can get into at the moment - and realised how little it actually "help" end users. Yes, you can make good information retrieval systems but at the end of the day it mainly serves the purpose of making better and better advertisements. Even if you put DS aside, you quickly realise much of software engineering is mainly about making internal tools, dashboards, websites for the nth corporate etc... Why not other classical CS? Because:
AI is clearly "the future". For example, I considered doing formal verification for a while. FV is a tool humans use to proof check their code by - to simply put it - "convert it" into a theorem and run a proof assistant on it, but if an AI can program (yes, current AI models can program, even if at a toy level) and debug itself then what's the point? It would end up being a technique for Good Old-Fashioned AI (GOFAI) - which, to plainly put it, fails to be AI.
Other fields of CS that are ML-free are at the moment mainly in combinatorics (e.g. stringology and graph theory) or numerical optimization or constraint programming. Now, I can't get to like the second one. About the first one... many cool problems in combinatorics are unfortunately very NP, so basically untreatable. Approximated algos are nice but I don't feel like they are 'challenging' enough. You are not cracking a problem but simply finetuning algorithms to better cover edge cases.
Yes, I am aware I can do Data science on medical data (which is something I am seriously trying to do at the moment). But the problems about availability of data, computing resources and, lastly, actual interests of your employer remain.
So, given that much of CS is about solving very 1st world problems... why not just agreeing on solving very artificial puzzles? But this time, nice problems like the ones I find in mathematics? The small issue here is that my knowledge of maths is a bit rusty: yes I know some bits of real analysis, linear algebra, general topology, number theory, category theory, ring theory etc. but I never fully really wondered what I would like to study if I were to do a PhD. Which is why I think it's unlikely I would apply for a PhD in maths even if I were eligible, as I do not have enough preparation to clearly define a research goal.
But then I question myself: would I be up for doing 3/4 more years of studies at a university for Mathematics? I am 21 now and I am starting to feel like I am wasting my youth. Will I do research as usual? How would I sustain myself during this period? What are the policies for second-degrees students (I am European and study in Scotland, but will likely go back to the mainland because of Brexit - thus some financial advice would be great too).
Is there anyone here who has a similar background, in particular wanted to migrate from the "cool" CS to mathematics? What made you do the switch?
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How to compute $\int_{0}^{+\infty}\frac{x^2\mathrm{d}x}{e^{x}-1}$ analytically? https://ift.tt/eA8V8J
Does anyone know how to compute analytically the following integral:
$$\int\limits_{0}^{+\infty}\dfrac{x^2\mathrm{d}x}{e^{x}-1}$$
It should be equal to $2\zeta(3)$ according to Maple. I tried the following using the binomial theorem for negative integer exponents:
$$I = \int\limits^{+\infty}_{0} e^{-x}(1-e^{-x})^{-1}x^2\mathrm{d}x = \int\limits^{+\infty}_{0}\left[\sum_{k=0}^{+\infty}(-1)^k(-1)^ke^{-(k+1)x}\right]x^2\mathrm{d}x=\int\limits^{+\infty}_{0}\left[\sum_{k=0}^{+\infty}(-1)^{2k}e^{-(k+1)x}\right]x^2\mathrm{d}x$$
After another change of variables, $y=(k+1)x$:
$$I = \sum_{k=0}^{+\infty}(-1)^{2k} \frac{1}{(k+1)^3}\int\limits_0^{+\infty} y^2e^{-y}\mathrm{d}y$$
The keen eye might recognize $\int\limits_0^{+\infty} y^2e^{-y}\mathrm{d}y$ as the gamma function, $\Gamma(3)=(3-1)!=2$. This, together with a slight nudge to the bottom limit of the summation we can rewrite things as:
$$I = \Gamma(3)\sum_{k=1}^{+\infty} \dfrac{(-1)^{2k}}{k^3}$$
And i see immediately (since the beginning in fact...) an infinite sum that makes me troubles and i can't get rid of. I tried to found if i did any trivial error but i'm focusing since to many hours to found it. That's why I need an external view to point me out my obvious error.
Thanks in advance for your help
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Nov 13, Solving System of Linear Equations Using Rank Method https://ift.tt/eA8V8J
Solving System of Linear Equations Using Rank Method
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Mathematical Objects: Ball of wool with Pat Ashforth https://ift.tt/3ee6Gpi
A conversation about mathematics inspired by a ball of wool (yarn). Presented by Katie Steckles and Peter Rowlett, with special guest Pat Ashforth.
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How good can you get at math through Youtube?
I minored in mathematics in undergrad and am thinking of pursuing a masters in math. I have studied up to calculus III, a foundations class, and prob & stats. I really enjoy math and learn a lot from Khan Academy, but how much will this actually prepare me for my future studies?
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