Newest Questions
1,693,661 questions
1
vote
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How to find the spectrum of $T+T^*$ for this compact operator $T$?
The question is related to this stackexchange question
We fix a parameter $0<q<1$. Let $l^2=l^2_{\geq 1}$. We define an operator $T$ on $l^2$ by
$$
T(e_n)=q^n\sqrt{1-q^{2n}}e_{n+1}.
$$
Its ...
- 1,134
2
votes
0
answers
17
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When do $\sum x_i=\frac{n(n+1)}2$ and $\prod x_i=n!$ imply $\{x_i\}=[n]$?
Let $(x_1,\dots,x_n)\in\{1,2,\dots,n\}^n$ such that $\sum\limits_{i=1}^nx_i=\frac{n(n+1)}2$ and $\prod\limits_{i=1}^nx_i=n!$. Find all $n$ such that it is necessary that $x_1,\dots$, $x_n$ is a ...
- 3,701
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0
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18
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Reasoning about M/M/1 queueing, but where arrival rate is dependent on service rate
M/M/1 queues make sense to me; if your arrival rate λ is greater than the service rate μ, the queue grows unbounded, and if λ < μ I can compute lots of nice properties like server utilization. My ...
0
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0
answers
13
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Is there a hypergeometric representation for this double-factorial sum?
I'm interested in the following finite sum:
\begin{align}
S(N, q_1, q_2) = \sum_{j=0}^{N-1} \frac{(j!)^2}{(j - q_1)! \, (j - q_2)!}
\end{align}
where $N \in \mathbb{Z}_{>0}$, and $q_1, q_2 \in \...
0
votes
0
answers
17
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On the equation $b_0^n = \sum_{k=1}^n b_k^n$ with $(b_k)$ an arithmetic sequence
On the equation $b_0^n = \sum_{k=1}^n b_k^n$ with $(b_k)$ an arithmetic sequence
I'm currently investigating the following problem and would be grateful for any feedback or insights. I would also ...
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2
votes
0
answers
19
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Smallest $i$ such that $n \Bigm| |\mathrm{GL}(i, p)|$
Let $p$ be a fixed prime, and define the function
$$
f(n) = \min \{\, i \in \mathbb{N} \text{ such that } n \Bigm| |\mathrm{GL}(i, p)| \,\},
$$
where $\mathrm{GL}(i, p)$ is the general linear group of ...
- 2,097
0
votes
0
answers
27
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Given symmetric equations, find $S = a+b+c$ and $Q = a^2+b^2+c^2$
Let $a, b, c$ be distinct reals such that $\frac{1}{a} + \frac{1}{a-b} + \frac{1}{a-c} = \frac{1}{b} + \frac{1}{b-a} + \frac{1}{b-c} = \frac{1}{c} + \frac{1}{c-a} + \frac{1}{c-b} = 3$. Find $S = a+b+c$...
0
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0
answers
17
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Asymptotic bound on modified binomial sum
Problem statement
For all $x>0$, $n, m \in \mathbb{N}$, let
$$
S(x, n, m) \triangleq \sum_{j=0}^n \binom{n}{j} (1 - x)^{n-j} x^j j^3 \frac{1}{1 + \frac{j}{m} }
$$
Assuming $n = o(m)$, for all $c&...
0
votes
0
answers
13
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Closure of probability distributions on an open set vs probability distributions on the closure of an open set
Consider the space of probability distributions on $(0,\infty)$ denoted through their cdf $F$. Let $F_0$ be a probability distribution having a point mass at $0$, so that $F_0$ has a step at $0$ (say ...
- 1,366
-2
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34
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Computing $\int \ln(\sin x)\,\mathrm dx$ using polylogarithm [closed]
I want to compute the integral
$$\int \ln(\sin x)\,\mathrm dx$$
using the polylogarithmic function .
1
vote
1
answer
16
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Tensor Products of $C^*$-algebras
I'm reading the Brown-Ozawa book and I'm looking at Exercise $3.3.6$ which asks you to prove that under the obvious embedding, the image of $B(\ell^2)\otimes_{\text{alg}}B(\ell^2)$ is not norm-dense ...
- 689
0
votes
0
answers
20
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l2 norm inequality
I am working on a proof
Given the matrix:
$$
U =
\begin{pmatrix}
u_1 & u_2 & 0 & 0 & \cdots & 0 \\
0 & 0 & 1 & 0 & \cdots & 0 \...
- 41
1
vote
0
answers
31
views
$\mathbb{N}\setminus\{0\}$ is not an elementary submodel of $\mathbb{N}$? [duplicate]
As the title suggests, I am trying to show that $\mathbb{N}\setminus\{0\}$ is not an elementary submodel of $\mathbb{N}$, together with the usual ordering $<$.
(I am not sure that it is true). my ...
- 391
1
vote
0
answers
17
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Find solution for the integral: $\int_{d}^{\infty} \frac{1}{\sqrt{x}}\exp\left(- bx^2 - c x \right) dx,$ with $d>0,c>b>0$
I encountered the following integration in my probabilistic research
$\int_{d}^{\infty} \frac{1}{\sqrt{x}}\exp\left(- bx^2 - c x \right) dx,$
with $d>0,c>b>0$
I tried with some method by ...
- 83
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0
answers
30
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Number of real roots of a polynomial of degree $7$
Define $$P(x)=\prod_{k=0}^7 (x-a_k)-\prod_{k=0}^7 (x-a_0+a_k)$$
where $a_k\in\mathbb{Z}^{+} \forall \ 0\leq k\leq 7$ satisfying $a_1\leq a_2\leq...\leq a_7<a_0/2$ and $\sum_{k=1}^{7} a_k<3a_0$.
...
- 879
0
votes
0
answers
19
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A condition for bounded algebraic surface? [closed]
I'm interested in surfaces in $\mathbb{R^3}$ :
I will focus on real algebraic surfaces for this question :
$S_P = \{(x,y,z) \in \mathbb{R^3} ~~|~~ P(x,y,z)=0 \}$ with $P\in \mathbb{R}[X]$
Do you know ...
- 3
0
votes
1
answer
15
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torsion free sheaf on surface has filtration with torsion free factors of rank 1
Let X be a nonsingular complete surface over $\mathbb{C}$. Suppose $F$ is a torsion free sheaf of arbitrary positive rank. It is claimed in section 5.1 of Huybrechts and Lehns' "The Geometry of ...
- 329
1
vote
0
answers
26
views
Asympotics of $\sum_{p \leq x} \log^k(p)$ for any rational $k \in \mathbb{Q}$
I'm interested in finding generic asymptotics for the following sum:
$$S_k(x) = \sum_{p \leq x} \log^k(p)$$
with any $k \in \mathbb{Q}$. It might not be possible to get something like this for all ...
- 31
2
votes
0
answers
12
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Cohomology of a cyclic group: dependence on a generator
Let $G$ be a finite cyclic group of order $n$.
Choose a generator $\sigma$ of $G$.
Let $A$ be a $G$-module.
We have an isomorphism
$$\phi_\sigma\colon H^2(G,A)\overset\sim\longrightarrow A^G/N(A),$$
...
- 1,378
0
votes
1
answer
23
views
Closed-form or simplification for a triple sum involving binomial coefficients and factorials
I would like to find a closed-form or simplified expression for the following triple sum, where $N \in \mathbb{Z}_{>0}$, and $k_1, k_2 \in \mathbb{Z}_{>0}$:
\begin{align}
\sum_{i=0}^{N-1} \sum_{...
1
vote
2
answers
50
views
Suppose $t\in\Bbb F_{q^2}\setminus \Bbb F_q$ with $w=t+t^{-1}\in \Bbb F_q$. Is $t^3-t\in\Bbb F_q$?
The Question:
Suppose $t\in\Bbb F_{q^2}\setminus \Bbb F_q$ with $w=t+t^{-1}\in \Bbb F_q$. Is $t^3-t\in\Bbb F_q$?
Here $\Bbb F_{q}$ is the field of $q=p^r$ elements, where $p$ is prime and $r\in\Bbb N$...
- 47.8k
0
votes
0
answers
19
views
Fusion dynamics on an infinite graph: does every configuration stabilize uniquely?
Description of the graph:
We consider an infinite directed graph with a triangular structure:
The graph is composed by 2 different rows of nodes: the upper rows and the lower rows.
Each node on the ...
0
votes
1
answer
26
views
Is provability of existential arithmetic sentences/halting of Turing machines independent with theory
Let $D(x_1, x_2, …, x_n) = 0$ be a Diophantine equation of which coefficients are computable integers via Turing machines. Suppose we can prove in theory like ZFC+large cardinal that $D$ has a zero, ...
- 353
-1
votes
0
answers
38
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Rational points on this elliptic curve? [closed]
For a "wurzelnightrider" (fairy chess) tour, it would be interesting to know if $i^4+j^4-i^2\cdot j^2$ can be a (rational would suffice!) square. This is easy to transform to an elliptic ...
- 829
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0
answers
11
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Codimension of singular points of hypersurface $Y\subset X$
Let $X$ be a complex manifold, and let $Y\subset X$ be an irreducible hypersurface (i.e. analytic subset of codimension 1). Lemma 2.3.22 in Huybrechts' Complex Geometry shows the sheaves $\mathcal{O}(-...
-3
votes
0
answers
39
views
nCr and perfect squares [closed]
Find the largest two digit $n$ such that, $$\binom{n}{3}\binom{n}{4}\binom{n}{5}\binom{n}{6}$$ is a perfect square.
- 68
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0
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10
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Deriving the Jacobi equation as linearization of the geodesic equation
Consider the geodesic equation $D_{\gamma'}\gamma'=0$. In coordinates this reads as the non-linear ODE:
$$(1)\hspace{2cm}(\gamma^i)''+(\gamma^j)'(\gamma^k)'\Gamma_{jk}^i=0.\hspace{5cm}$$
In Eschenburg'...
- 1,175
1
vote
0
answers
22
views
Proof check: $\hat{\mathbb{C}}$-valued limit of holomorphic functions is holomorphic or identically $\infty$
Question: Let $d$ be the distance function induced on the Riemann sphere $\hat{\mathbb{C}}$ via stereographic projection,
$$ d(z,w) = \frac{|z-w|}{ \sqrt{1+|z|^2} \sqrt{1+|w|^2}}, \quad d(z,\infty) = \...
- 1,426
0
votes
1
answer
18
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Deriving the inequality $|R(1,w)|\geq k$ via wall-crossing parity in Davis's Coxeter Groups book
Suppose $\mathbf{s}=(s_1,\dots,s_k)$ is a word in $S$. Define $w_i\in W$ by $w_0=1$ and $w_i=s_1\cdots s_i$, and $r_i\in R$ by $r_i=w_{i-1}s_iw_{i-1}^{-1}$. Set $\Phi(\mathbf{s}):=(r_1,\dots,r_k)$. ...
- 624
0
votes
0
answers
16
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Positive solution of nonlinear elliptic equation
We consider the following problem
$$
\begin{cases}
- \Delta u = \lambda f(u), \quad u>0 \quad &\text{in} \ \Omega, \\
u=0\quad&\text{on }\partial\Omega.
\end{cases}
$$
where $\Omega \...
- 387
3
votes
1
answer
45
views
Expected value on a fair 6 sided die with coin flip
You are given a fair
6−sided die and play the following game: You receive the value of the face-up side on each roll. If you roll an odd number, the game ends. If you roll an even number, you flip a ...
- 108
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0
answers
26
views
Explanation of an assertion by Gauss on an embedding of a surface in space
[I ask this question after a previous question of mine, which deals with a related theme, was closed (because it was too broad and unfocused). I will try now to ask several much narrower questions ...
- 1,236
2
votes
0
answers
20
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Identity in MCMC autocorrelation
I am reading an article on metrics regarding the samples of MCMC methods. It is this one: https://arxiv.org/abs/1903.08008 . On pg. 7 the authors define the autocorrelation of lag $t$ of a chain of ...
- 192
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0
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29
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$f:U\to\mathbb{R}$ is $C^k$ at $p\in U$. Is this definition really correct? ("An Introduction to Manifolds Second Edition" by Loring W. Tu.)
I am reading "An Introduction to Manifolds Second Edition" by Loring W. Tu.
Definition 1.1. Let $k$ be a nonnegative integer. A real-valued function $f:U\to\mathbb{R}$ is said to be $C^k$ ...
- 9,896
0
votes
0
answers
21
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How to endow a fiber of smooth map at a point with manifold structure? [closed]
Let $f: M \to N$ be a smooth map between smooth manifolds and $q \in N$ be a point.
I wonder when and how I can endow the fiber $f^{-1}(q) \subset M$ with structure of smooth manifold? What type of ...
- 381
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0
answers
12
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Proof of a characterization of quasiconcave functions.
A function f is quasiconcave at x iff it is concave on the tangent to the level set of f through x. I am looking for a proof of this but can't find any. Can someone please point me to the right source?...
- 1
0
votes
1
answer
34
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Conditions on linear independance of a family of vectors defined as an infinite linear combination
Let $(E,\left<.,.\right>)$ a vector space over $\mathbb{R}$ with an inner product. I consider $x:= (x_n)_{n \in \mathbb{N}} \in E^{\mathbb{N}}$ a sequence of elements of $E$ such that any finite ...
0
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0
answers
10
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Existence of rank 3 lattice of signature (1,2) containing two copies of $U$ intersecting in a positive vector
Let $U = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ denote the standard hyperbolic plane. I am trying to construct, for a given natural number $N$, an explicit example of a rank 3 lattice $...
- 573
2
votes
1
answer
78
views
Does there exist $1<x<2$ such that the sequence $\lfloor x\rfloor,\lfloor x^2\rfloor,\ldots,\lfloor x^k\rfloor$ is of the form $1,2,3,4,\ldots,n,n+4?$
There are many examples of $1<x<2$ such that the sequence $\left\lfloor x\right\rfloor, \left\lfloor x^2\right\rfloor, \left\lfloor x^3\right\rfloor, \ldots, \left\lfloor x^k\right\rfloor$ is of ...
- 24.3k
0
votes
1
answer
37
views
Is it really true that the (left-) inverse of a strictly increasing function is strictly increasing?
There is a question here on the site that claims that if a function $f:X\rightarrow Y$ is strictly increasing, then also its left inverse $f^{-1}$ is strictly increasing.
Inverse of Strictly Monotone ...
- 482
1
vote
2
answers
20
views
Average Daily Output of a Repair Shop with Limited Capacity
I've been puzzling over a probability problem and solved it using Markov chains, but I'm wondering if there's a more direct or intuitive way to arrive at the solution.
Imagine a business that repairs ...
- 133
3
votes
1
answer
41
views
What is the topology on the set of all plays of a game?
I read the wikipedia page on the "Angel Problem"(https://en.wikipedia.org/wiki/Angel_problem). It stated that there must be a winning strategy for either the angel or the devil because the &...
- 511
-3
votes
0
answers
62
views
What is the extension of the complex numbers that allows $(a+bi)(a-bi)^2=-1$? [closed]
So what is a simple extension of the complex numbers that allows $|z|^2=-1$? defining $|z|^2=z\bar{z}$ where $\bar{z}$ is the complex conjugate of $z$. Such an algebra can be constructed because one ...
- 449
0
votes
0
answers
28
views
Motivation for choosing generalized coordinates in classical mechanics
Background -
At the moment I am taking a (first) physics course named "Physics for mathematicians". I am pointing out the name to illustrate that it is intended for people with the "...
- 388
1
vote
2
answers
68
views
$S =\sum\limits_{n=0}^\infty\frac{(-1)^n}{(2n+1)^2}\left(\psi\left(\frac{3n}{2}+\frac{3}{2}\right)-\psi\left(\frac{3n}{2}+1\right)\right)$
I'm trying to find this sum
$$S = \sum\limits_{n = 0}^\infty {\frac{{{{\left( { - 1} \right)}^n}}}{{{{\left( {2n + 1} \right)}^2}}}} \left( {\psi \left( {\frac{{3n}}{2} + \frac{3}{2}} \right) - \psi \...
- 3,543
0
votes
0
answers
40
views
Questionable proof of the statement that connected components of a topological space are not necessarily open
I'm performing a close study of the Oxford Concise Dictionary of Mathematics (6th ed., 2021) edited by Richard Earl and James Nicholson.
I have reached the section connected component, where I see the ...
- 5,214
2
votes
0
answers
33
views
Terminology for limits in full subcategories
Let $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose $\mathcal{A}$ has, say, binary products; we might be interested in checking whether $\mathcal{B}$ has binary products as well. Perhaps ...
- 7,489
-5
votes
0
answers
41
views
Prime number solution of $p^2=4k-27$ [closed]
The equation $p^2=4k-27$ where $p$ and $k$ are prime numbers has one solution $(p,k)=(7,19)$.
Is this solution unique?
- 856
-3
votes
0
answers
63
views
What is the logical meaning of the word 'let' when used in mathematical theorems and definitions? [closed]
The word 'let' is ubiquitous in mathematical theorems and definitions.
For example:
Let g be differentiable on an open interval O and let c ∈ O . . .
What is the logical meaning of the word 'let' in ...
- 141
1
vote
1
answer
20
views
SageMath's Automaton method number_of_words error
Consider the language of those $w\in \{a,b,c,d\}^*$ that contain $ab$. Corresponding DFA is (written in SageMath):
...
- 12.4k