5
votes
0answers
118 views
+50

Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
14
votes
1answer
477 views
+100

Big-Daddy-Conjectures and Hierarchy of Mathematical Conjectures

I am interested in the Hierarchy and Connections between various different open problems in Mathematics, and the most general conjectures in various fields of Mathematics. Examples of Hierachy ...
3
votes
0answers
105 views
+100

$Z_n \backslash \{0\}$ splits into octets

Let $n=8m+1, m\in\mathbb{N}$. Does the set of nonzero elements of $\mathbb{Z}_n$ split into disjoint octets of the form $8_k=\{\pm a_k,\pm b_k,\pm a_k\pm b_k\}$? The computer tells me it is possible ...
3
votes
1answer
205 views
+100

Optimal Tic Tac Toe algorithm without lookahead

Is there any algorithm for tic tac toe that does not rely on a lookahead algorithm that is perfect for any sized boards?
9
votes
2answers
401 views
+150

Understanding this ODE

(I did not change anything, I just rewrote the ODE in a simpler form): I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - \left( \alpha x + \gamma x^2 \right) y(x) = \lambda y(x),$ ...
14
votes
4answers
529 views
+50

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
8
votes
3answers
194 views
+200

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
4
votes
1answer
71 views
+100

Is there a finite non-solvable group which is $p$-solvable for every odd $p\in\pi(G)$?

Let $G$ be a finite non-solvable group and let $\pi(G)$ be the set of prime divisors of order of $G$. Can we say that there is $r \in \pi(G)-\{2\}$ such that $G$ is not a $r$-solvable group?
14
votes
1answer
120 views
+50

Is there a geometric meaning of a prime power not being primary?

I guess that the standard example of a prime power that is not a primary ideal is $$\mathfrak p^2 :=(x,z)^2\subset k[x,y,z]/(xy-z^2):=A.$$ Because $\mathfrak p^2 = (x^2,xz,xy)$, we see that $x\not ...
9
votes
4answers
340 views
4
votes
2answers
487 views
+100

Trends in the distribution of reordered digits of Pi (OEIS A096566)

First let me try to describe in more details below the approach of "reordering" digits of Pi, which is used in OEIS A096566 https://oeis.org/A096566 and what I have done analyzing it so far. I am ...
2
votes
1answer
127 views
+50

Time to extinction

Suppose that a population has a rate of extinction equal to $k \exp({-\epsilon N})$ where $\epsilon$ and $k$ are constant and $N$ is the population size. This also means that the time to extinction is ...
2
votes
0answers
70 views
+50

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
0
votes
0answers
52 views
+50

Partial Differential equations and applications- Reference request

I will be taking up a PDEs course next semester and would like to find some good references. The topics covered in the syllabus is given below. Partial differential equations: Conservation laws, ...
1
vote
0answers
27 views
+50

About the integral forms of Appell Hypergeometric Function and Lauricella Functions

According to http://en.wikipedia.org/wiki/Appell_series#Integral_representations and http://en.wikipedia.org/wiki/Lauricella_hypergeometric_series#Integral_representation_of_FD, it is known that ...
0
votes
0answers
393 views
+100

Fitting Expanding Spheres in the Irregular Surface

This is a kind of a puzzle: I have a convex hull filled with $N$ small spheres(with radius $r$). Now, the balls grow in size till it hit any side of the Cavity (Convex hull) or any other ball. ...
0
votes
0answers
53 views
+50

congruence relation between 2 primes and possible equivalent relation in polynomial ring over $GF(2)$

Let $p, q$ be primes. Then linear congruence equation, $ap \equiv r(\mod q)$ can be solved for $a$ and will have unique solution for each value of $r$ such that $a < q$. Is this right ...
5
votes
0answers
63 views
+100

Convergence of Euler-transformed zeta series

I am trying to prove that the expression $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^\infty\sum_{m=0}^n\frac{(-1)^m}{2^{n+1}}{n\choose m}(m+1)^{-s}$$ converges to an analytic continuation of the alternating ...
12
votes
3answers
199 views
+500

An inequality on sequences with each term dividing sum of two neighbouring terms

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is $$\frac{x_{i-1}+x_{i+1}}{x_i} = k_i $$ is an integer for ...
3
votes
1answer
440 views
+50

Derivation of formula for finding median for grouped data

I know the formula of formula for finding median for grouped data that is $$Median = L_m + \left [ \frac { \frac{n}{2} - F_{m-1} }{f_m} \right ] \times c$$ and I know what all the letters stand for. ...
31
votes
3answers
485 views
+500

When does a real function together with its derivatives form a basis?

Consider real functions defined on an interval, let's say $[-\pi,\pi]$. Let $s(x)$ be an analytic, periodic function such that all of its Fourier components are non-zero. We build a set of functions ...
0
votes
1answer
150 views
+50

Question on differentiate under integral

First we have the following theorem: Then we apply it to a concrete problem: Finally how to obtain the second rectangle?
19
votes
3answers
357 views
+100

Is this graph connected

Define the following equivalence relation $\sim$: We have $a \ \mathcal{R} \ b$ if and only if $a+b \ | \ ab-1$. $a \sim b$ if and only if there exist a sequence of integers $a_1, \ldots, a_n$ such ...
10
votes
1answer
194 views
+50

Can we prove that the solutions of $\int_0^y \sin(\sin(x)) dx =1$ are irrational?

Can we prove that the solutions of $$\int_0^y \sin(\sin(x)) dx =1$$ are irrational? Wolfram Alpha gives two approximate sets of solutions as $\{4.58+2\pi k|k\in\mathbb{Z}\}$ and $\{1.69+2\pi ...
22
votes
1answer
316 views
+50

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
2
votes
1answer
50 views
+50

Rankine Hugoniot Jump Condition Derivation

I follow the majority of this derivation I just do not understand where the two terms highlighted in green come from.
6
votes
0answers
42 views
+50

Invariant tensors in adjoint representation

Suppose we have a simple Lie group $G$ with algebra $\mathfrak{g}=\{X_a\}$, where the generators $X_a$ are in some matrix representation. Is it true that the only invariant rank $n$ tensor in the ...
0
votes
0answers
46 views
+50

Traverse resultant 2d array after integer partition

I have used the solution of integer partitioning using dynamic programming explained in this post and in this article. Following is the resultant matrix when N is equal to 6: $$\begin{bmatrix} 1 ...
3
votes
2answers
39 views
+100

How to work out miles between Longitude values based on a Latitude value.

We know that when Latitude is 0, the distance between Longitude values is roughly 69 miles. When the Latitude is +/-90, Longitude values are 0 miles. At 0 Latitude, the earths circumference is ...
3
votes
0answers
109 views
+50

A reflective subcategory of the category of inverse semigroups.

The Question. I'm reading Lawson's "Inverse Semigroups: The Theory of Partial Symmetries" and I've hit something I don't understand. It's claimed on page 34 of my copy that The category of ...
4
votes
0answers
24 views
+50

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
0
votes
1answer
24 views
+50

periodicity of an interval exchange transformation(IET)

Let $T$ be an IET. That is, $T:[0,1] \rightarrow [0,1]$ is a piecewise orientation-preserving isometry. Let $D$ be the set of points whose entire forward iterates are well-defined. I have the ...
2
votes
6answers
220 views
+50

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2…$.

Assume that $ f ∈ L([a, b])$ and $\int x^nf(x)dx=0$ for $n=0,1,2...$. Prove $f=0$ a.e. Since there exist polynomials going to f almost everywhere all I would need to do is bring the limit in to ...
2
votes
1answer
46 views
+100

Extensions of universal measures

Let $(\Omega,\mathcal F)$ be a measurable space, and let $\mathcal P$ be the set of all probability measures no this space. Let $\mathcal F^p$ denote a completion of $\mathcal F$ w.r.t. $p\in P$ and ...
0
votes
0answers
53 views
+50

Creating a Topological group from modulo multiplication Group.

If I were to create a Topology out of the Modulo 3 Multiplication group $\mathbb{Z}_3$, what elements would it consist of and why? So $\mathbb{Z}_3 = \{0,1,2\}$ as a group over modulo 3. What are the ...
8
votes
1answer
83 views
+150

Blocking lines of length $5$ in a $7 \times 8$ matrix; how can we know the solutions have a specific form?

A friend shared with me the following puzzle he encountered in a Chinese math competition: In a $7 \times 8$ matrix, we place tokens so that any straight line of length $5$ (horizontal, vertical, ...
3
votes
3answers
109 views
+50

An equilateral triangle formed using points of tangency

P.S:I am looking for a hint and not the whole solution. BdMO 2012 nationals secondary: The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The ...
2
votes
1answer
58 views
+100

The definition of the right regular representation

I'm having difficulties understanding the definition of the right regular representation as it appears in Dummit & Foote's Abstract Algebra text. On page 132 it says Let $\pi:G \to S_G$ be the ...
5
votes
1answer
158 views
+50

Breaking chocolate bars game

About two weeks ago, a friend of mine taught me the following game without his knowing the answer. It may be famous, but I haven't known it. There are $N\ (\in\mathbb N)$ chocolate bars composed of ...
3
votes
0answers
57 views
+50

What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
4
votes
1answer
58 views
+50

Measures which cannot be uniquely written as the sum of a purely atomic measure and a nonatomic measure

Maharam's theorem says that every complete measure can be written as the sum of a purely atomic measure and a nonatomic measure. According to the paper "Atomic and Nonatomic Measures" by R.A. Johnson, ...
2
votes
1answer
153 views
+50

An arithmetic sequence whose members do not contain the digit ‘9’

There is a non-constant arithmetic progression made of natural numbers only; none of them contains the digit $9$. Prove that such an arithmetic progression has no more than $72$ terms.
3
votes
1answer
98 views
+50

Difficult infinite integral involving a Gaussian, Bessel function and complex singularities

I've come across the following integral in my work on flux noise in SQUIDs. $$\intop_{0}^{\infty}dk\, e^{-ak^{2}}J_{0}\left(bk\right)\frac{k^{3}}{c^{2}+k^{4}} $$ Where $a$,$b$,$c$ are all positive. ...
3
votes
1answer
58 views
+100

Convergence of ${\large\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx$

Consider $$I(a)={\int}_{-\infty}^\infty J_0(x)\,J_0(x+a)\,dx,$$ where $J_0(z)$ is the Bessel Function of the $1^{st}$ kind and $a>0$. Does this integral converge for any values of $a$? If so, is ...
0
votes
0answers
60 views
+50

quasi-convexity of a function

Can someone help me identify whether the following function is quasi-convex? Let $p>1$. For $x=(x_1,\dots,x_n),x_i>0,\sum_ix_i=1$, we define $$f(x) = -\log \sum_i (x_i/\|x\|_p)^{p-1}.$$ Plots ...
1
vote
1answer
22 views
+50

A question about conic sets of functionals.

The problem is the following. Let $(X,\|\cdot\|)$ be a normed space and let $C \subseteq X$ be a closed convex set with nonempty interior. Let be $x\in C$. I define to be a "normal cone to $C$ in ...
1
vote
0answers
29 views
+50

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin t}{t}J_m(t)\,dt$

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin(t)}{t}J_m(t)\,dt$ $$\int_R \frac{\sin(t)}{t}J_m(t)\,dt=\int_R \hat{\chi_{{(-1,1)}}}(t)J_m(t) \,dt =\int_R \hat{\chi_{{(-1,1)}}}(t)J_m(t) \,dt ...
1
vote
1answer
45 views
+50

Prove that $\frac{d}{dx}(z^kJ_k(z))=z^kJ_{k-1}(z)$ for every $ k \geq \frac{1}{2}$.

Prove that $\frac{d}{dx}(z^kJ_k(z))=z^kJ_{k-1}(z)$ for every $ k \geq \frac{1}{2}$. Then write an explicit expression for (a) $J_\frac{1}{2}(x)-J_\frac{5}{2}(x)$, and (b) an expression for $J_1+2J_3$ ...