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-4
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0answers
30 views

Finding Laplace inverse transformation of a product series [closed]

compute the inverse Laplace transformation of the following equation. \begin{align*} f(s)&=\frac{A}{\prod_{i=1}^{L}(s+a_i)^m} \\ &=\frac{A}{(s+a_1)^m\,(s+a_2)^m\cdots (s+a_L)^m}. \end{align*}...
4
votes
1answer
310 views

The range of the Euler totient function and multiplication by 28

If $n$ is in the range of the Euler totient function, certain multiples of $n$ are likewise guaranteed to be totient values. The simplest nontrivial example of this is that, if $n$ is in the range of ...
0
votes
0answers
89 views

Do three inequalities as follows holds $\sum_{k=1}^n \frac{{a_k(x)}}{b_k^\alpha} >0$ and $\sum_{k=1}^n \left( \frac{{a_k(x)}}{b_k} \right)^k >0$?

This topic are some generalizations of my previous question: Let series $b_1, b_2,\cdots, b_n$ and $a_1(x), a_2(x), \cdots , a_n(x)$ so that $$b_n > b_{n-1} > \cdots > b_1 \ge 1 \; \text{...
1
vote
1answer
135 views

Modular arithmetic and elementary symmetric functions

Denote the elementary symmetric functions in $n$ variables by $e_k(x_1, x_2,\dots, x_n)$. In the special case $x_j=j$, simply write $e_k(n)$ for $e_k(1, 2, \dots, n)$. Next, define the sequence $$a_{+}...
5
votes
2answers
190 views

Asymptotic rate for $\sum\binom{n}k^{-1}$

This MO question prompted me to ask: What is the second order asymptotic growth/decay rate for the sum $$\sum_{k=0}^n\frac1{\binom{n}k}$$ as $n\rightarrow\infty$?
0
votes
1answer
143 views

Is this a Borel summable $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ with $ a_k$ alternating sequence?

let $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ a divergent series such that $b_k=(-1)^k (k!)a_k >0 $ for $k>1$ , and $b_k$ signed this from $k=1$ to $20$ ,The asymptotic of the titled series ...
4
votes
2answers
121 views

Different notions of computable binary sequence

The standard definition of computability, for a sequence $s\in\{0,1\}^\omega$, is that there is a Turing machine outputting $s[i]$ on input $i$. I'm looking for strengthenings of this notion; for ...
2
votes
0answers
407 views

Is there an infinite product like this for $\cos x$?

There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example $$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
0
votes
0answers
115 views

A provably infinite infinitary aliquot sequence?

A divisor of n is called infinitary if it is a product of divisors of the form $p^{y_a 2^a}$, where $p^y$ is a prime power dividing n and $\sum_a y_a 2^a$ is the binary representation of y. [from OEIS ...
2
votes
2answers
179 views

Reference request for function by which to compute coefficients of continued fraction of algebaic number

The simple continued fraction is in the form $$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance. Obviously,the coefficients $x_i$can be computed by computable function $x_i=f(i),...
1
vote
0answers
128 views

Is there any irrational algebraic number among the set? [closed]

Suppose $S$ is set of numbers such that every number in it expands in decimal digits,every digit is 0 or 1,and $\lim_{n\rightarrow\infty}\frac{C_{n}(0)}{n}=\frac{1}{2}$ where ${C_{n}(0)}$ and ${C_{n}(...
4
votes
0answers
115 views

Irreducibility of polynomials corresponding to sequences

I have no experience with this, so I dont know if this is too easy for MO. Let $(a_n)$ be a strictly monotone sequence of natural numbers, then define the set of nice numbers of $(a_n)$ as $X(a_n):=\{...
2
votes
2answers
268 views

Algorithm to check if vertex belong to infinite path in Graph theory

My purpose is to understand if in a graph $G = \langle V, E\rangle$ given 4 vertices in input (a, b ,c and d) they belong to an infinite path. With infinite path I mean a vertex succession that has a ...
2
votes
2answers
178 views

What's “serialization” really called, and is there any theory surrounding it?

Define an operator $\mathop{\vec{\bigcup}}$ as follows: Definition. Whenever $A$ is an $I$-indexed family of sets, where $I$ is a totally-ordered set, we have $$\mathop{\vec{\bigcup}}_{i \in I} A_i ...
0
votes
2answers
137 views

Closed form of $\sum_{i=k}^\infty i h {i \choose {k-1}} h^{k-1} (1-h)^{i - (k-1)}$?

Is there a closed form solution to the expression below? Or, if there is no closed form solution but the series converges, is there some upper bound on this expression? $$\mathbb E_{i \sim Q}[i] = \...
4
votes
1answer
364 views

Put 10 balls in the jar then randomly take 1 out. Do it infinitely many times. Find the probability of resulting in an empty jar [closed]

The original discussion (in Chinese): https://www.zhihu.com/question/58702489 The original problem was from an probability theory exam. The problem is translated as: Assume an infinitely large jar ...
5
votes
1answer
376 views

The need for nets in topology

I remember when I first heard about nets in topology (called also Moore-Smith sequences). I was told that most of useful topological properties which can be exressed in terms of sequences in the ...
3
votes
1answer
103 views

Adding the harmonic sequence and a permutation of it

Let $\pi:\mathbb{N}\to\mathbb{N}$ be a bijection. Then does there exist another bijection $\nu:\mathbb{N}\to\mathbb{N}$ and a constant $C$ such that $$ \frac{1}{n} + \frac{1}{\pi(n)} \leq \frac{C}{\...
5
votes
0answers
115 views

Sum of squared hypergeometric polynomials

$\sum_{m=1}^\infty \frac{1}{m} \bigg[{}_2F_1(-m,m,2,u)\bigg]^2 = \frac 1 4 -\frac 1 2 \log u$ I have very strong numerical support that this is true when $0<u\le1$. Can anyone help proving or ...
3
votes
1answer
257 views

Solving recurrent relation

I have the following recurrent relation and I want to find a close form of it if it exists at all. $$ P_n = (1-p)^{n-1}P_{n-1} + \sum\limits_{k=2}^{n} \binom{n-1}{k-1} p^{\binom{k}{2}} (1-p)^{k(n-k)} ...
1
vote
2answers
241 views

An elementary proof for a limit? [closed]

This question is motivated by pedagogical reason, not research. I will provide a simple proof for contrast, but I would like to see another approach that does not involve integrals, instead even more ...
1
vote
1answer
118 views

boundedness of a nonlinear recursive sequence

Consider a real sequence $(x_k)$ for $k=0,1,2,\dots,N$ as $x_0=1$ and for $k>0$ $$ x_k=x_{k-1}+\frac{\gamma}{N}x_{k-1}^2,\qquad (\gamma>0).$$ I wonder to show that the sequence is bounded as $N\...
4
votes
2answers
282 views

Moment problem on [-1,1]: necessary and sufficient conditions

Consider a sequence of real numbers $s=(s_0,s_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that $$ s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;? $$ ...
2
votes
2answers
125 views

Sum of subsequence, over index set of non-zero density, of monotone divergent sum also divergent?

For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, $$0&...
2
votes
2answers
319 views

Non-Formal Applications: Higman and Kruskal

After looking through many papers, I noticed that most of the discussions and proofs for Higman's Lemma and Kruskal's Tree Theorem only have formal applications in set theory, logic, and type theory. ...
3
votes
1answer
179 views

Hermite-Kakeya Theorem for entire functions

In a question asked by Bobby Ocean, the following theorem is cited: Hermite-Kakeya Theorem(for polynomials) - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) r$ has only real zeros ...
2
votes
1answer
120 views

Asymptotic expansion of a sequence given by an integral with reciprocal Gamma function

I would like to know the asymptotic expansion of the sequence of positive numbers given by $$I_{n}:=-\int_{0}^{1}\frac{n^{x-1}}{\Gamma(x-1)}dx,$$ for $n\rightarrow\infty$. One can easily derive an ...
-2
votes
1answer
123 views

Looking for the name of an infinite sequence [closed]

I am looking for information about a sequence that seems like it should converge. The sequence is textually described as: ...
3
votes
1answer
52 views

Complete classification of complexity classes / infinite approaching sequences

http://en.wikipedia.org/wiki/Time_complexity#Table_of_common_time_complexities For complexity as seen in the above link, complexity classes can be log, polynomial, exp, or composition of any of these ...
2
votes
1answer
143 views

Is there a dense rational sequence of positive separation?

Let us consider the set $\ell_\neq$ of bounded sequences of unequal real terms. We use the following descriptions. A sequence $x=(x_0,x_1,...)\in\ell_\neq$ is dense if, for all $\varepsilon>0$, ...
0
votes
1answer
174 views

Lemma about infinite sequences we are hoping is true [closed]

Given N infinite sequences of non-negative integers, some of which diverge to infinity, must there exist two steps i, j in which $x_i \leq x_j$ for all sequences x?
2
votes
1answer
250 views

Iterated projections in Hilbert spaces

Let $E$ be an Hilbert space and $F, G$ two subspaces such that $F \cap G =\{0\}$. Let $(x_n)$ be the sequence of iterated orthogonal projections: $x_0 \in F$, $x_1$ is the orthogonal projection of $...
1
vote
0answers
393 views

Greedy sequences without k-term arithmetic progressions

If $S_k$ is the greedy sequence with no length-k arithmetic subsequence, (ie $S_3$ = A003278 , $S_4$ = A005837 , $S_5$ = A020655 ), is it guaranteed that any other sequence $a$ with no length-k ...
0
votes
2answers
217 views

sequence, such that sum of any combinations in the sequence does not equal another [closed]

Hi, Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence. ...
-2
votes
1answer
213 views

How to work with infinite random graph(s) ?

Hi, In the case where we are dealing with an infinite random graph (RG with infinite nodes). How do we model/work with notions like degrees, degree distribution ? How are they defined ? Thanks!
1
vote
2answers
350 views

sum of infinite series

Given the complex variable $x$, complex constant $c$, and integer number $r$. I want to solve the equation: $\sum_{k=1}^{\infty}\frac{e^{kx}}{k^r}=c$. I was thinking that if there is a formula or ...
2
votes
2answers
563 views

Is it necessary that gcd > 1 of an infinite set? [closed]

Consider an infinite set $S$, of positive integers. If all the finite subsets of $S$ have GCD $>$ $1$, is it necessary that the GCD of $S$ is greater than $1$ as well?
4
votes
2answers
818 views

References for the result that $\sqrt{n}$ is equidistributed mod 1

It is not difficult to show (even without Weyl criterion) that the sequence $\sqrt{n}$, $n=1,2,\ldots$ is equidistributed mod 1. However, I need a reference to this result. Can you help me? Thanks.
0
votes
0answers
838 views

In a network with N nodes, what is the general formula for computing the propagation of a set of numbers?

I am creating a circular neural network with N nodes. Each node is connected via a send pathway to every other node, and the connection between two nodes has a weight. Any number sent over the ...
15
votes
1answer
891 views

Self-avoiding walk on $\mathbb{Z}$

This one is an unanswered question in Math.SE. I've posted it here because I think it deserves more attention. How many sequences $\{a_n\}$ exist satisfying: a) $a_1=0$ b) $\forall k\ge1 $ ...
5
votes
2answers
712 views

Ends of topological spaces. Why independent of choice of ascending sequence of compact subsets?

Quoting from http://en.wikipedia.org/wiki/End_(topology): "Let X be a topological space, and suppose that K1 ⊂ K2 ⊂ K3 ⊂ · · · is an ascending sequence of compact ...
7
votes
1answer
551 views

Magic square on an infinite lattice

This question came to me while reading the discussion of magic square in the complex plane with equal integrals along every horizontal, vertical and diagonal "magic square in the complex plane with ...
0
votes
4answers
5k views

Does Cauchy continuity imply uniform continuity? [No.] [closed]

It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff $$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X ...