1
vote
1answer
27 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi ...
0
votes
1answer
35 views

Maximum value of function involving factorials

Define $$g_{(k,j)} = \frac{a^{n-k}b^k(k+n)!x^{k+n-j}}{k!(n-k)!(k+n-j)!}$$, where $n,k,j \in \Bbb{N}$ are fixed such that $(0 \leq x \leq a/b ),(b<a),(0 \leq k \leq n ),(2 \leq j \leq 2n),(0 \leq ...
1
vote
1answer
46 views

I'm searching for the formula of the series $ \sum_{n=0}^{\infty}a^{n^l} $

I'm searching for the sum-formula (if exists) of the following power series: $$ \sum_{n=0}^{\infty}a^{n^l} $$ where $l=2,3,....$, and $|a|<1$.
2
votes
0answers
37 views

an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...
1
vote
1answer
37 views

A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$ with Dirac delta function $\delta$. ...
0
votes
2answers
18 views

Show that for any $s \in S$ where $S = $ {$s \in \mathbb{N} | \exists m,n \in \mathbb{Z}$, we have $d|s$ where $d$ is the smallest element

Explain why the set $S = $ {$s \in \mathbb{N} | \exists m,n \in \mathbb{Z}, s = ma+ nb $} has a smallest element, call it d, so we know there exists $x,y \in \mathbb{Z}$ such that $d = ax + by$, and ...
0
votes
0answers
31 views

Prove that $s \not= \emptyset $ by showing that at least one of $|a|$ or $|b|$ is an element of $S$.

. I'm trying to show an alternative proof to Bezout's Lemma (let $a, b \in \mathbb{Z}$, then there exists $x,y \in \mathbb{Z}$ such that $gcd(a, b) = ax + by$). Heres one of the steps in proving it: ...
2
votes
1answer
103 views

Possible marks in a MCQ test

Just out of curiosity(we have a similar test) not homework : A multiple choice questions test has $100$ questions. The marking scheme is $+4$ for correct answer, $-1$ for wrong answer and $0$ marks ...
0
votes
2answers
275 views

Continuous functions from real numbers to discrete space

One of my homework problems is this: "Let X = $\mathbb{R} \!\,$ with the usual metric and let X′ be a discrete metric space. Describe all continuous functions from X to X′." A function f : X ...
0
votes
1answer
33 views

conversion of discrete to continuous

Given $N_{j+1}-N_j=kN_j$ How can I substitute some time variable in to make $delta(t)$ small? Meaning change in time. I need to show $N_j=e^{(j\ln(1+k))}$ How can I rewrite the given in terms of ...
1
vote
4answers
114 views

Two form of derivative $ f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

Why I can write formula derivative $$ f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$$ in this form: $$ f'(x)=\frac{f(x+h)-f(h)}{h}+O(h)$$ I know, that it's easy but unfortunately I forgot.
2
votes
0answers
121 views

What is Algorithmic Graph Theory? [closed]

I'm an undergraduate and I signed up for a course next semester called Algorithmic Graph Theory. The course description doesn't give any details on the contents of the class, and there's no listing of ...
1
vote
1answer
29 views

Convergence of Discrete Poisson equation

Are there any sources that show the convergence of the discrete poisson equation? To be clear, by convergence I mean: given the poisson equation in a domain $ M \subset R^2 $, $\Delta \psi = f $, one ...
3
votes
1answer
144 views

A combinatorial identity with Pochhammer's symbol

Let $m,k$ be an positive integers with $k\le m$. I am trying to prove $$\sum_{j=0}^k{\frac{1}{2}\choose k-j}\frac{2^{2j}(m+j)!}{(m-j)!(2j)!}=\frac{P(n,k)}{(2k)!}$$ where $n=2m+1$ and ...
2
votes
1answer
76 views

A combinatorial identity related to Chebyshev differential equation

Let $m,k$ be an positive integers with $k\le m$. Does anyone have a proof that $$\sum_{j=k}^m {2m+1\choose 2j+1}{j\choose k}=\frac{2^{2(m-k)}(2m-k)!}{(2m-2k)!k!}?$$ This is related to Chebyshev ...
6
votes
2answers
410 views

Elementary bound of binomial coefficient

I'm working my way through an Erdős paper from the sixties and some of the elementary bounds he claims seem to be just beyond my reach. The expression looks horrendous but maybe there is a clever ...
2
votes
3answers
206 views

What is the expression for this summation?

Known that $\sum_{n=0}^{\infty}{x^n}{z^n}=\frac{1}{1-xz}$. If we have $\sum_{n=0}^{\infty}\frac{{x^n}{z^n}}{n\beta + \alpha}$ where $\beta, \alpha $ are element of real numbers but not equal $0$. ...
23
votes
4answers
1k views

A stronger version of discrete “Liouville's theorem”

If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition $$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, ...