Highest scored questions
159,037 questions
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Bounded metric spaces with non-surjective self-isometry
A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$.
A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
- 48.9k
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votes
1
answer
247
views
Minimum number of people such that 2 can be expected to sit next to each other [closed]
We are given a large, round table with $n$ seats. It is easy to see that whenever $p\geq \text{int}(\frac{n}{2}) + 1$ people are seated, at least $2$ people will sit next to each other (here $\text{...
- 48.9k
-2
votes
1
answer
147
views
Combinatorical meaning of such expression [closed]
Any combinatorical meaning or interpretation of
$$1^{\alpha_1}2^{\alpha_2}3^{\alpha_3}...s^{\alpha_s}\alpha_1!\alpha_2!...\alpha_s!$$
for partition $(1^{\alpha_1},2^{\alpha_2},3^{\alpha_3},...,s^{\...
- 443
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votes
1
answer
1k
views
Degree of a rational function [closed]
I would like to have a simple proof for the following result:
Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). ...
- 2,091
-2
votes
1
answer
486
views
finitely generated subgroups of SO(3) [closed]
Is it known whether there is any example of a pair of rotations in $SO(3)$ about orthogonal axes such that the group that they generate is not a free product of the two cyclic groups generated by each ...
- 2,125
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votes
2
answers
119
views
Systems of ODEs that fulfill a matrix relationship at steady state [closed]
It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$
with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, ...
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votes
2
answers
226
views
Any duality between different real forms of a complex Lie group? [closed]
A complex Lie group may have several real forms.
Are there any duality/trinity... between them?
Maybe a trivial question to ask, is $SL(3,\mathbb{C})$ a real form of $SL(3,\mathbb{C})\times SL(3,\...
- 783
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votes
1
answer
147
views
What are the formula of representation of quasicrystals and the law or mechanism of the formation [closed]
I vaguely recall that formula of representation of quasicrystals is relevant to tiling plane,and tiling plane without period is relevant to recursiveness, and do not know the mechanism or physics ...
- 3,104
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votes
1
answer
183
views
($^{\omega}2$,<) is not well-order. [closed]
Let < be a lexicographic order on $^{\omega}2$ or in other words given distinct functions $f,g$ from $\omega$ to 2, let $f<g$ if and only if $f(n)=0$ and $g(n)=1$, where $n$ is the lease $m<\...
-2
votes
1
answer
172
views
Action of automorphism group on Lie algebra [closed]
I want to know whether an automorphism group of a simple Lie algebra over $GF(2)$, acts transitively on non-zero elements of Lie algebra or not? How can I check this property?
- 367
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votes
1
answer
634
views
Doubt in this proof of Horrocks theorem
I'm beginning to study some research papers and I need right now to understand the solution of Vaseršteĭn of Serre's theorem (simplest proof of this theorem), to do so, I'm beginning to understand ...
- 195
-2
votes
1
answer
213
views
Solving a difficult equation for a variable?
I'm trying to obtain the maximum likelihood estimate of the parameters for a model I'm building. I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ ...
-2
votes
1
answer
208
views
Can the isometry group of the set of zeros of an L-function $F$ be used to make $F$ automorphic?
I'm still trying to understand the notion of automorphic (L-)function. Due to my lack of knowledge of the subject, this question may appear pretty vague and therefore may not be suitable for MO. I ...
- 6,982
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votes
1
answer
3k
views
How to obtain the determinant of the difference of two matrices? [closed]
I am trying to obtain the determinant of the difference between the identity matrix and an A matrix. The question is such:
...
-2
votes
1
answer
257
views
A question on parallelizable manifolds [closed]
Let $M$ be a manifold with the property that $f^{*}(TM)$ is isomorphic to TM, for every diffeomorphism $f$ on $M$. Does this imply that $M$ is parallelizable?
- 366
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votes
2
answers
503
views
spanning tree of a graph of minimum degree three
Does each graph of minimum degree three admit a spanning tree whose vertices have degree three (exactly) except the leaves (degree one)?
- 1
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votes
1
answer
190
views
Dixon's Theorem [closed]
I am going through a sketch of the proof of Dixon's Theorem (the probability that two randomly chosen elements of A_n generate A_n -> 1 as n -> infinity) due to M. Liebeck and its underlying idea is ...
- 65
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votes
1
answer
146
views
a measure convolution equation
My question is:
Given a function $f$ in the Schwartz class, we are looking for a measure $\mu$ which is a solution of the convolution equation: $f = e^{-|.|^2/2} \ast \mu$, where $e^{-|.|^2/2}$ is ...
- 367
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votes
1
answer
347
views
Forms of multivariate CLT [closed]
I am looking for a good reference for differnt kinds of multivariate central limit theorems. I was wondering how far the i.i.d. condition of the standard multivariate clt can be relaxed, as in can the ...
-2
votes
1
answer
219
views
Howto plot a specific complex function [closed]
We need to plot the real and imaginary parts of a complex function $k(\omega)$, and cannot find a good way to do this without using "ad hoc tricks."
Definitions
$k$ is a complex-valued function ...
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votes
1
answer
2k
views
Upper bound of a series
Given $N$ and $a$ positive integers, with $a\ge 2$ is it possible to prove the inequality:
$$\sum_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}\le\frac{N}{2}$$
- 399
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votes
1
answer
272
views
representation of teichmuller space Teichmuller space [closed]
I want to study representation of teichmuller space of surface of genus g in psl(2,R).
can you suggest any good references.
- 217
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votes
2
answers
500
views
Is this divisor ample on the product of two curves [closed]
Let $X$ and $Y$ be complete curves over a field $k$ of characteristic zero. Let $S = X \times_k Y$. Assume that Y has a $k$-rational point and use this point to consider $X$ as a divisor (also denoted ...
- 95
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votes
3
answers
1k
views
Turing-Shannon connection
From Alan Turing we know what we can expect from a computer and from Claude Shannon what we can expect from a communication channel.
Does anyone know any connection between these two theories (namely,...
- 342
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votes
2
answers
893
views
Graphic representation of an antisymmetric relation on a set [closed]
Hi,
I'm learning about relations on sets, and I'm trying to figure out what exactly antisymmetric means.
The way we represent a relation is like a adjacency matrix. In my textbook I see that ...
- 139
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votes
2
answers
5k
views
Water jug puzzle [closed]
There are n red & n blue jugs of different sizes and shapes. All
red jugs hold different amounts of water as the blue ones. For every red jug,
there is a blue jug that holds the same amount of ...
-2
votes
1
answer
840
views
Generic coordinate system representations [closed]
Please excuse the verboseness which follows, as the question is rather basic, so I would like to state it carefully so that it will not be accidentally neglected as automatically trivial. If, after my ...
-2
votes
1
answer
890
views
Determine noise distribution [closed]
I'm trying to solve the following least squares problem:
$\underset{x}{\text{min}} ||Ax - \tilde{b}||_2$
where $Ax = b$ and $\tilde{b} = b + w$
Question:
How do I determine which probability ...
- 35
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votes
2
answers
2k
views
probability of subset sum after rolling dice 4 times [closed]
If we roll 4 dices (fair), what is the probability of "sum of subset" being 5. e.g. 1432,1121, 2344, 2354 have a subset sum of 5. Can you illustrate how to calculate this.
- 1
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votes
1
answer
2k
views
Sufficient Conditions for Graph "Non-Isomorphisms" [closed]
Suppose we have two graphs $G_1$ and $G_2$. To check whether these two graphs are not isomorphic, is it sufficient to find a $k$-cycle in $G_1$ but can't find a $k$-cycle in $G_2$ (or vice versa)?
- 7
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votes
2
answers
1k
views
Similarity of Ellipsoids
Suppose I have two ellipsoids $A$ and $B$ respectively represented as $(x-C_A)^T M_A(x-C_A)$ and $(x-C_B)^T M_B(x-C_B)$ in the matrix representation.
What's the best way to find a a function that ...
- 35
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votes
1
answer
162
views
What is the weakest condition on the matrices A_k that guarantees v_k->0 => A_kv_k->0 ? [closed]
What is the weakest condition on the sequence of real matrices A_k that guarantees that whenever a sequence of real vectores v_k converges to zero, the product A_kv_k also converges to zero?
Edit: ...
- 1,638
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votes
1
answer
205
views
Can this theory interpret Peano arithmetic?
Logic: Bi-sorted first order logic with equality, first sort written in lower case range over natural numbers, the second sort written in upper case range over sets of naturals, "$=$" has no ...
- 11.3k
-2
votes
1
answer
181
views
What is the computational complexity to verify a P solution with a deterministic Turing machine? [closed]
As we know, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is &...
- 3,104
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votes
1
answer
414
views
Defining the set of natural numbers in the first order Peano arithmetic [closed]
The question seems simple, but I'm not sure:
let's consider a first order Peano Arithmetic and its standard model $N = \{ 0,1,2,3,... \}$ of natural numbers.
A question: how can we define the whole ...
- 19
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votes
1
answer
152
views
Branching process with varying offspring distribution at each step
Consider a simple branching process $Z_0,Z_1,Z_2...$ such that at every discrete step, a particle splits into $k\geq1$ particles where $k$ follows a discrete distribution with probability mass $p(k)$.
...
- 109
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votes
1
answer
81
views
attempt improve rational approximation against Pade aproximation
I try to improve the approximation of the exponential function, using orthogonal function as BesselI , it seems that it is better than Pade approximation with the same number of terms
$$e^z-\frac{2 ...
- 11
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votes
1
answer
217
views
Convergence and roots of alternating periodic infinite series
Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...
- 317
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votes
1
answer
108
views
applying the watson lemma to an integral [closed]
So i thought about applying the Watson lemma to determine the asymptotic behavior of the integral
$$
I(x)=\int_{0}^{\infty} \frac{e^{-x(t-\ln(t))}}{(1+t^2)} dt,
$$
as $x \rightarrow \infty$.
I think ...
- 7
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votes
1
answer
193
views
Finite normal extensions
Suppose that $K$ is a finite field extension of $F$. Is the following equivalent to the extension being normal?
If $L$ is an extension of $K$ and $\sigma:K\to L$ fixes $F$, then $\sigma(K) = K$.
I ...
- 147
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1
answer
369
views
Is this extension of the projectively extended real line, consistent?
This posting has been Edited. The edited material shall be noted.
The projectively extended real line $\hat {\mathbb R}= \mathbb R \cup \{\infty\}$ is one system which allows division by zero! Yet it ...
- 11.3k
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votes
1
answer
184
views
Conjecture about the equality : $f(y)=y\ln(y)+\sum_{n=2}^{\infty}\pm\frac{\left(y\ln y\right)^n}{2^{a_n}}$
I try here because I expect I cannot have any answer on MSE :
Problem :
Let :
$$f\left(x\right)=\frac{\left(1+\ln27\right)x!\ln x!}{x+1},g\left(x\right)=x\ln x$$
Then it seems $\exists y\in(0,1)$ and $...
-2
votes
1
answer
505
views
In Galois theory, why solvable groups must have their quotient groups be Abelian? [closed]
The definition of solvable groups can be regarded as two constraints, one is that there must be a sequence of normal subgroups, and the other is that the quotient groups between these sequences are ...
- 1
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votes
1
answer
149
views
How many positive roots can $\sum_{i}\frac{a_i}{x+b_i}$ have where $b_i$'s are all positive? [closed]
What is the maximum number of positive roots $\sum_{i}^N\frac{a_i}{x+b_i}$ can have where $b_i$'s are all positive? (everything here is a real number. To provide context, I encountered this problem ...
- 433
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votes
1
answer
217
views
The norm of the difference of two normal states
Let $M$ be a type III$_1$ factor and $\rho$ be a normal state on $M$. If $p$ is a projection in $M$, can we find another normal state $\rho'$ on $M$ such that $\rho'(p)=0$ and $\|\rho-\rho'\|=k\rho(...
- 427
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votes
1
answer
214
views
Can the same dataset be described as Chaotic & Pareto/ Power law distribution?
I'm trying to abstract the mathematical part of the problem as much as possible before the details follow,
There's this dynamic data set that's $O(2^{32})$, a recent result described it as a power-law ...
- 5
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votes
1
answer
294
views
What does the Concordant constructible universe model?
Define a ranking function $\cal R$ as:
$\mathcal{R}: V \to ON; \,\mathcal {R}(x)= \min \alpha \, \forall y \in x: \alpha > \mathcal {R}(y) $
Now the constructible rank $\mathcal R^c$ of a set $X$ ...
- 11.3k
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votes
1
answer
112
views
Relationship between intersections [closed]
Given $N$ sets $X_1, \dots, X_N$ and two definitions of intersection $\cap'$ and $\cap''$, is it possible to show that
$$
\vert X_i \cap' X_j \vert \le \vert X_i \cap'' X_j \vert,
\quad
\forall i,j \...
- 189
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votes
1
answer
182
views
Stationary distribution of a weighted directed acyclic graph
Is there any way to calculate the equilibrium (stationary) distribution for a weighted directed acyclic graph?
Some references emphasized adjacency matrix to be symmetric.
https://arxiv.org/abs/1012....
-2
votes
1
answer
109
views
If a sequence of measures is weakly convergent outside each compact ball, the sequence itself is weakly convergent
Let $E$ be a $\mathbb R$-Banach space and $\mathcal M_+(E)$ denote the space of finite nonnegative measures on $\mathcal B(E)$.
If $\lambda\in\mathcal M_+(E)$, let $$\left.\lambda\right|_\delta(B):=\...
- 167