0
votes
2answers
208 views
+100

How find all positive real $\beta$ such A finite number of $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

Define sequence $\{a_{n}\}$,such $$a_{1}=1,a_{2}=2,a_{k+2}=2a_{k+1}+a_{k},k\ge 1$$ Find all positive real number $\beta$,such only have a finite number of relatively prime integers $(p,q)$ ...
20
votes
4answers
525 views
+300

Closed-forms for several tough integrals

These integrals came up in the process of finding solution to Vladimir Reshetnikov's problem. I wonder if there are closed-forms for the following integrals: \begin{array}{1,1} &[\text{1}] ...
4
votes
1answer
89 views
+50

Where can I learn more about the Galois connection induced by a graph on its own powerset?

Given a binary relation $R \subseteq X \times Y$, we get an antitone Galois connection $(F,U) : \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ in the usual way: The function $U : \mathcal{P}(X) ...
1
vote
3answers
110 views
+50

Runge Kutta 8(5,3)

This is actually three small very related questions about Runge Kutta methods. 1) I have programmed a RK 7(8) method also RK 4(5). At the beginning I was assuming that the RK 7(8) uses two ...
6
votes
1answer
186 views
+50

If the set of values , for which a function has positive derivative , is dense then is the function increasing?

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $A:=${ $x \in \mathbb R :f'(x)>0$ } is dense in $\mathbb R$ , then is it true that $f$ is an increasing function ? What ...
13
votes
4answers
533 views
+300

Is there an elementary proof for Euler's product for Sine?

I've been looking at proofs of this equation: $$\displaystyle \frac {\sin \pi x} {\pi x} = \displaystyle \prod_{k \mathop = 1}^\infty \left({1 - \frac{x^2}{k^2} }\right)$$ All the proofs seem to ...
1
vote
2answers
82 views
+50

How find this $a$ such $(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$

Question: find all the complex $a$,such for every complex $z_{1},z_{2}(|z_{1}|,|z_{2}|<1,z_{1}\neq z_{2})$,such $$(z_{1}+a)^2+a\overline{z_{1}}\neq (z_{2}+a)^2+a\overline{z_{2}}$$ My idea: ...
0
votes
1answer
39 views
+50

Factorization of Compact Lie Algebras into Irreducible Ideals

I have read in some lecture notes on Lie theory that any compact Lie algebra $\mathfrak{g}$ can be factored as a direct sum of of irreducible ideals for the $\mathrm{ad}$ representation. That is, ...
3
votes
3answers
136 views
+50

How to solve limits?

The above limit was solved by making a seemingly arbitrary substitution. The previous limit was solved by making a linear substitution $y=mx$. Which again seemed a bit out of the blue. For another ...
5
votes
1answer
80 views
+50

How to calculate floating point numbers?

Here are two locations in small memory: 0110 | 1111 1110 1101 0011 0111 | 0000 0110 1101 1001 Interpret locations 6 and 7 as an IEEE floating point number. ...
1
vote
0answers
106 views
+50

Vector space basis change: is this “index-free” notation correct?

There are already quite a number of questions on basis change in a vector space. Nevertheless, to fully grasp the underlying idea I made up the following notation and I have some doubts on it (note: ...
1
vote
1answer
49 views
+150

Linear programming: Basic solution

Let $P = \{ x \in R^n : Ax \leq b, x\geq 0\}$ and $Q= \{ (x, r)' \in R^{n+m}: Ax + r= b, x \geq 0, r\geq 0\}.$ Show that $$x \text{ is basic solution of } P \iff (x, b-Ax) \text{ is basic solution of ...
2
votes
0answers
119 views
+50

What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and is contained in a parallel square with side-length 2 (which is its smallest containing square). What is the smallest possible angle of the ...
14
votes
3answers
562 views
+100

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
1
vote
1answer
60 views
+50

counting occurence of subgraphs by counting their occurence in larger subgraphs

I have a mental block in fully understanding the following notion. Let $G$ be a graph of order $n$ and $H$ a fixed small graph of order $k \le n$. Suppose that there are $d$ copies of $H$ as an ...
0
votes
2answers
76 views
+50

Functional of Einstein tensor

What does this equal to, and how do I actually calculate this correctly? $$ \frac{\delta G_{ab}}{\delta g_{cd}}=? $$
3
votes
2answers
245 views
+50

Applications of integrals of rational functions of sine and cosine

I earlier asked this question about conformal equivalence of flat tori with embedded tori. In the ensuing thread the integral $\displaystyle\int\frac{dx}{R+\cos x}$ occurred. If I'm not mistaken, it ...
5
votes
0answers
94 views
+100

On distributivity of lattice of group topologies

Let $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Q$ (the additive group of all rational numbers) such that $(\Bbb Q,\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ is a ...
3
votes
0answers
70 views
+100

Compositions of filters on finite unions of Cartesian products

Let $\Gamma$ be the lattice of all finite unions of Cartesian products $A\times B$ of two arbitrary sets $A,B\subseteq U$ for some set $U$. See this note for other equivalent ways to describe the set ...
1
vote
0answers
71 views
+100

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
1
vote
1answer
51 views
+100

Framing a travelling salesman problem

I have an optimization(optimisation) problem, I think it is travelling salesman, where I want to find an answer to the question: "What is the best coffee shop for person x within a 50km radius?" The ...
6
votes
0answers
66 views
+100

Why is there no theory of $G$-ic varieties, for linear algebraic groups $G$?

A toric variety is an algebraic variety $X$ with an embedding $T \hookrightarrow X$ of an algebraic torus $T$ as a dense open set, such that $T$ acts on $X$ and the embedding is equivariant. It ...
1
vote
5answers
958 views
+50

Proving unique weak solution.

I am attempting to solve some problems from Evans , I need some help with the following question. If $u\in H^2_0\Omega$, where $\Omega$ is open, bounded subset of $R^n$. How can i solve biharmonic ...
14
votes
1answer
188 views
+200

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
5
votes
9answers
276 views
+100

Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$

Let $\bf u$ be any column vector and $\bf v$ be any row vector, each with $n \geq 2$ arbitrary entries from a field. Then it is well known that ${\bf u} {\bf v}$ is an $n \times n$ matrix such ...
1
vote
0answers
32 views
+50

Resources for understanding game trees?

I am trying to make an AI to solve the popular game $2048$, and I think that the theory of game trees would help me quite a bit in this endeavor. The only issue is that most of the results I've found ...
8
votes
3answers
122 views
+300

Simplification of an expression containing $\operatorname{Li}_3(x)$ terms

In my computations I ended up with this result: $$\mathcal{K}=78\operatorname{Li}_3\left(\frac13\right)+15\operatorname{Li}_3\left(\frac23\right)-64\operatorname{Li}_3\left(\frac15\right)-102 ...
3
votes
1answer
53 views
+50

Monte Carlo p-test and early stopping

Say you have a coin with some probability $p$ of falling on heads. You would like to determine if this probability is less than or equal to $0.05$ with some reasonable degree of confidence and stop ...
3
votes
1answer
80 views
+50

On a property of polylogarithm

I have an observation, and I don't know that the following statement is true or not. If not give a counterexample, if it is true prove it, or give a reference about it. Let $n \in \mathbb{R}$, $z \in ...
5
votes
1answer
89 views
+100

Find a smooth function with prescribed moments

In several contexts I’ve encountered variants of the following problem : let $m_0,m_1,m_2$ be real numbers such that $0 < m_1 < m_0$ and $\frac{m_1^2}{m_0} <m_2 < m_1$. Then, show that ...
1
vote
1answer
26 views
+100

Kernel of an integral operator with Gaussian kernel function

Suppose we have the integral operator $T$ defined by $$Tf(y) = \int_{-\infty}^{\infty} e^{-\frac{x^2}{2}}f(xy)\,dx,$$ where $f$ is assumed to be continuous and of polynomial growth at most (just to ...
27
votes
0answers
301 views
+100

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
0
votes
1answer
47 views
+100

How do I solve an inhomogeneous Helmholtz boundary value problem in 2-D with a rectangular boundary?

I need to solve the following BVP: $\Delta u - 1/\delta * u = R(x,y)$, where $\Delta$ is the laplacian operator. and boundary conditions: $u(0,y)=u(L,y)=u(x,0)=u(x,L)=0$ where $L=1$, ...
5
votes
0answers
92 views
+200

Closed form for integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral. $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$ From Vladimir ...
3
votes
1answer
78 views
+100

How to represent Fermat number $F_n$ as a sum of three squares?

Let $F_n=2^{2^n}+1$ be the Fermat number. How to represent the Fermat number $F_n$ for $n \geq 3$ as a sum of three squares of different natural numbers? For example for $n=3$ we have $$ ...
8
votes
2answers
133 views
+100

Closed form of $I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$ where $\operatorname{Li}_2$ is the dilogarithm function. A numerical ...
1
vote
1answer
63 views
+50

A possible mistake in Hartshorne chapter 2 proposition 2.6

Here is the context of this question. Hartshorne claim that $O_X(U)\cong \beta_*(O_V)(U)=O_V(\beta^{-1}(U))$ for any open $U\subset X=\operatorname{Spec}A$,but it is possible that ...
0
votes
0answers
56 views
+50

Example about the difference between Morse and degree theory

i found this example but i don't understand how we applyed Morse theory and why we can't applyed degree theory. if the functional $f$ behaves like $<lu,u>$ at infinity where the symmetric ...
0
votes
1answer
48 views
+50

A question on Kronecker Index

I am reading A book by Milnor (Lectures on characteristic classes) and I can across this section on Stiefel-Whitney numbers (page 16) and he uses the Kronecker index but never defines it (He says to ...
1
vote
1answer
41 views
+50

Codimension 1 homology represented by Embedded Submanifold

I'm looking for a reference for the following statement: Given an oriented manifold $M$ and a class $\xi \in H_{n-1}(M)$, there is some embedded oriented submanifold $F^+ \hookrightarrow M$ such that ...
5
votes
0answers
48 views
+50

Question about B. Host paper 'Nombres, normaux entropie, translations'

I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let $X = \mathbb{R}/\mathbb{Z}$, $p > 1$ a integer, $D_n = \{kp^{-n}\colon 0 \leq k < p^n ...
1
vote
1answer
130 views
+50

Prove complex function goes counterclockwise around the unit circle at least once

Suppose $f$ is non-constant and holomorphic in a neighborhood of the closed unit disk, s.t. $|f(z)| = 1$ for all $|z| = 1$. Then show that as $f(e^{i\theta})$ traverses the unit circle and makes at ...
0
votes
0answers
34 views
+50

Given $A\in Tn(R)$ show thast A is a scalar matrix if $e_{ij}=Ar_{ij}$, where $a\le i\le j\le n$

Given $A\in Tn(R)$ show thast A is a scalar matrix if $e_{ij}=Ar_{ij}$, where $a\le i\le j\le n$ Prove for $1: e_{ii}A=Ae_{ii}$ and for $2: e_{ij}A=Ae_{ij}$ where $i\le j$ Now for 1, I understand ...
0
votes
2answers
47 views
+50

Is it true that $\biggl\|I-\frac{vv^T}{v^Tv}\biggr\|=1$?

I am following a proof in the text OPTIMIZATION THEORY AND METHODS a springer series by WENYU SUN and YA-XIANG YUAN. I come across what seems obvious that for a column vector $v$, with dimension ...
3
votes
1answer
34 views
+50

Geometry question pertaining to $4$ points in the plane where $90$ degree projectors are on each point and we must illuminate the whole plane.

Suppose we have $4$ points that can be positioned anywhere in $\mathbb{R}^2$. Now imagine each point has $90$ degree projectors coming out of them and you can rotate these projectors any way you would ...
4
votes
0answers
60 views
+50

How prove there exist $(a,b)$ such $f'^2_{x}(a,b)+f'^2_{y}(a,b)-4(a^2+b^2)=0$

Question: let $D=\{(x,y):x^2+y^2<1\}$,and $f\in C^{1}(D)$,if $$|f(x,y)|\le 1 ,((x,y)\in D)$$ show that: $\exists (a,b)\in D,$$$f'^2_{x}(a,b)+f'^2_{y}(a,b)-4(a^2+b^2)=0$$ I only solve ...
11
votes
6answers
242 views
+100

Understanding the solution of a riddle about lions and sheep.

I heard a riddle once, which goes like this: There are N lions and 1 sheep in a field. All the lions really want to eat the sheep, but the problem is that if a lion eats a sheep, it becomes a sheep. ...
2
votes
0answers
75 views
+50

numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$ In this linear PDE: \begin{cases} B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...
4
votes
1answer
64 views
+500

How many directed graphs of size n are there where each vertex is the tail of exactly one edge?

In a research problem in an unrelated area, me and a student found it necessary to count the number of directed graphs with every vertex having one outward-pointing edge, with no restrictions on the ...
-34
votes
8answers
3k views
+100

Unique Representation and The Fundamental Theorem of Arithmetic

While reading this thread Is 1 a prime number?, I recalled that The Fundamental Theorem of Arithmetic (FTA) which says that every positive integer greater than 1 can get written uniquely as a product ...

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