Questions tagged [infinite-sequences]
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77
questions
15
votes
1
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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 $ ...
11
votes
1
answer
613
views
Integrals of power towers
Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...
- 6,709
9
votes
1
answer
1k
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The justifiable universe
Over the years of my study of set theory, I have encountered several sentences of the form V = X: V = L, V = HOD, V = WF (the exclusive assertion of the cumulative hierarchy), and (if I understand ...
9
votes
1
answer
700
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 ...
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9
votes
0
answers
397
views
Can this infinite sum for the Riemann zeta function be generalised?
I recently derived the following identity (which is probably a rediscovery of something well-known to experts).
$$\sum_{k=1}^\infty{\frac{(-1)^{k+1}k^{4n+1}}{e^{k \pi}-(-1)^k}}=\frac{1}{2}\zeta\left(-...
- 2,852
7
votes
0
answers
135
views
Characterization of tempered distributions from tempered sequences
Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported and infinitely smooth functions for its usual topology. Let
$\mathcal{D}'(\mathbb{R})$ be the topological dual of $\mathcal{D}(\...
- 2,174
6
votes
2
answers
2k
views
References for the result that $\sqrt{n}$ is equidistributed mod 1
It is not difficult to show (even without Weyl criterion) that the sequence $\sqrt{n}$, $n=1,2,\ldots$ is equidistributed mod 1. However, I need a reference to this result. Can you help me? Thanks.
- 463
6
votes
2
answers
525
views
Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped
I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...
user504691
6
votes
1
answer
810
views
The need for nets in topology
I remember when I first heard about nets in topology (called also Moore-Smith sequences). I was told that most of useful topological properties which can be exressed in terms of sequences in the ...
- 9,150
6
votes
2
answers
422
views
Moment problem on [-1,1]: necessary and sufficient conditions
Consider a sequence of real numbers $s=(s_0,s_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that
$$
s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;?
$$
...
- 505
6
votes
0
answers
183
views
Sum of squared hypergeometric polynomials
$\sum_{m=1}^\infty \frac{1}{m} \bigg[{}_2F_1(-m,m,2,u)\bigg]^2 = \frac 1 4 -\frac 1 2 \log u$
I have very strong numerical support that this is true when
$0<u\le1$.
Can anyone help proving or ...
- 181
5
votes
2
answers
1k
views
Ends of topological spaces. Why independent of choice of ascending sequence of compact subsets?
Quoting from http://en.wikipedia.org/wiki/End_(topology):
"Let X be a topological space, and suppose that
K1 ⊂ K2 ⊂ K3 ⊂ · · ·
is an ascending sequence of compact ...
- 53
5
votes
2
answers
372
views
Asymptotic rate for $\sum\binom{n}k^{-1}$
This MO question prompted me to ask:
What is the second order asymptotic growth/decay rate for the sum
$$\sum_{k=0}^n\frac1{\binom{n}k}$$
as $n\rightarrow\infty$?
- 41.8k
5
votes
2
answers
227
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 ...
- 2,489
5
votes
1
answer
191
views
Does every integer appear in the modular sum sequence?
$\newcommand{\N}{\mathbb{N}}$Let $\N$ denote the set of non-negative integers. We inductively define a sequence $a:\N\to\N$ by:
$a(0) = 0, a(1) = 1$ and
$a(n) = \big(\sum_{k=0}^{n-1}a(k)\big)\text{ ...
- 45.5k
4
votes
1
answer
1k
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 ...
- 3,585
4
votes
0
answers
67
views
Draw an arbitrary line on a Penrose tiling. Determine a sequence of tiles can it intersect
Let us consider a Penrose tiling of $\mathbb R^2$. Starting with an arbitrary point on the tiling, draw an arbitrary straight line. Assume that this straight line never overlaps perfectly with a ...
- 775
4
votes
0
answers
131
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):=\{...
- 26.1k
3
votes
1
answer
266
views
When is an upper bound on the longest irreducible program outputting something computable?
Given some way to to encode programs to strings with a finite alphabet, which we assume has a computable translation to/from Turing machines, a program is irreducible if no subsequence of it has the ...
- 2,979
3
votes
1
answer
89
views
Meetability of $\pm 1$-functions on $\omega$
If ${\cal S}$ is a collection of functions $f:\omega\to\omega$ we say that ${\cal S}$ is meetable if there is a "global function" $g:\omega\to \omega$ such that for every $f\in {\cal S}$ there is $n\...
- 45.5k
3
votes
1
answer
321
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)} ...
- 332
3
votes
1
answer
454
views
Is there a summation method where the divergent series $S = U_0+U_1+U_2+\dots$ converges to a finite value?
Consider the function
$$
M(v) = \frac{m_0}{\sqrt{1 - \left(\frac{v}{c}\right)^2}},
$$ where $v \in \left]-c;c\right[$, $m_0\in\mathbb{R}^{*+}$, and $c=3\cdot10^8$.
Let $(U_n)$ be a sequence with ...
- 33
3
votes
1
answer
128
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
$$
\frac{1}{n} + \frac{1}{\pi(n)} \leq \frac{C}{\...
- 5,795
3
votes
1
answer
182
views
"Approximating" linear recursion with homogenous polynomial coefficients by linear recursion with constant coefficients
In a lecture I once attended, I remember the speaker using a result of the following nature:
$``$Let $\{A_n\}_{n=1}^\infty \subset \mathbb R$ be a sequence satisfying a recursion of the form
$$P(n) ...
- 719
3
votes
1
answer
217
views
Is there a two-dimensional Higman's lemma?
A string over a finite alphabet $A$ can be thought of as a function
$f:\{1,2,...,m\} \rightarrow A$ for some natural number $m$.
A 2-Dim string over $A$ is a function $f$ where
$f:\{1,2,\ldots,m\}\...
3
votes
1
answer
329
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 ...
- 603
3
votes
1
answer
63
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 ...
- 31
3
votes
0
answers
179
views
Shift Operators and the Weyl Algebra
I have a question about the action of a shift operator $E$ on polynomials $Ep(x) = p(x+1)$ in the context of linear differential operators in one variable with polynomial coefficients, i.e. ...
3
votes
0
answers
62
views
Some exercise on the regularity of a summability method
I was reading the book of Johann Boos "Classical and modern method in summability theory" and I came across an exercise from the Chapter 2 (page 50, exercise 2.3.15). Here is the statement ...
- 31
3
votes
1
answer
452
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 ...
- 39
2
votes
2
answers
253
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),...
- 3,000
2
votes
2
answers
959
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?
- 23
2
votes
2
answers
354
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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. ...
2
votes
2
answers
1k
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 ...
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2
votes
2
answers
224
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 $$\mathop{\vec{\bigcup}}_{i \in I} A_i ...
- 3,693
2
votes
1
answer
118
views
A second attempt at a two-dimensional Higman's Lemma
Let $A$ be a fixed finite alphabet.
If $s$ and $t$ are finite strings over $A$, define $s\leq t$ if $s$ can be obtained by deleting zero or more characters from $t$. Higman's Lemma states that if $s_1,...
2
votes
1
answer
134
views
Mellin transform (of sequences)
Is it possible to define the Mellin transform for sequences of real numbers or even for tuples? Is there any book treating this argument?
Any idea or suggestion will be greatly appreciated
Since the ...
- 131
2
votes
2
answers
213
views
Sum of subsequence, over index set of non-zero density, of monotone divergent sum also divergent?
For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, $$0&...
- 357
2
votes
1
answer
152
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 $\varepsilon>0$, ...
- 2,427
2
votes
1
answer
109
views
Decreasing sequences in a finitely generated closure algebra
I am interested in finitely generated closure algebras (as a special case of Heyting algebras), and in decreasing sequences of elements within such an algebra that have no lower bound.
Call two ...
- 21
2
votes
1
answer
198
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 ...
- 2,188
2
votes
1
answer
357
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 $...
- 1,355
2
votes
0
answers
59
views
Factor group of all the sequences by the subgroup of bounded sequences
Consider the group G of the sequences of real numbers (the group operation is addition). It contains a subgroup H of bounded sequences.
Is there any nice description of the factor group G/H ?
It is ...
- 4,761
2
votes
0
answers
196
views
Uniform distribution of log(log((n!)!)) mod 1
Can it be shown that the sequence $\log(\log((n!)!))$ is uniformly distributed mod 1? I believe it should be, but I'm not certain if Stirling's approximation repeated twice combined with the ...
2
votes
0
answers
235
views
Cardinal numbers and the Bolzano-Weierstrass theorem
Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows:
$\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...
- 551
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votes
0
answers
1k
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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
$$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
1
vote
4
answers
8k
views
Does Cauchy continuity imply uniform continuity? [No.] [closed]
It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff
$$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X ...
- 3,710
1
vote
1
answer
114
views
Given a real $x>1$, construct an aperiodic substitution sequence whose complexity functions grow like $xn$
The Fibonacci word is a binary sequence defined as follows.
We use a substitution rule $0\to 01$, $1\to 0$. Then, starting with the binary string $0$, apply the substitution rules successively. So we ...
- 775
1
vote
1
answer
163
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_{+}...
- 41.8k
1
vote
2
answers
265
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 ...
- 41.8k