Highest scored questions
159,037 questions
-2
votes
1
answer
1k
views
When multiply absolutely convergent seires and conditionally convergent series [closed]
Is this series absolutely converges
$$
\sum_{n=1}^{\infty}C_nU_n
$$
when
$\sum_{n=1}^{\infty}C_n$ is absolutely convergent series and
$\sum_{n=1}^{\infty}U_n$ is conditionally convergent series?
- 49
-2
votes
1
answer
150
views
Combinatoric Problem [closed]
Let $2\leq k\leq r\leq n$ are positive integers and $r=kt$.
I construct sets such that $\cup_{i=1}^n A_i=\{1,2,3,\dots,n\}=X$, this union is disjoint and if $x\in A_i$ and $y\in A_j$ for all $i\leq j$...
- 355
-2
votes
1
answer
181
views
Polynomials of minimum degree that interpolate primes in intervals
Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of ...
- 1,826
-2
votes
1
answer
64
views
Interpretation of the word Random [closed]
I have previous knowledge of what a random experiment is, but sometimes I get confused by the use of the word Random.
I can express my doubts as the following questions: if something is random them it ...
-2
votes
1
answer
84
views
Ensemble averaging in a random graph (or network) in the large $N$ limit [closed]
I have a random graph/network described by the adjacency matrix $(a_{ij})_{N\times N}$ where $a_{ij}=1$ with probability $p$. Each node in the graph is associated with a continuous quantity $\eta_i=\...
- 137
-2
votes
1
answer
80
views
Density property for Sobolev spaces
My question is as follows: is the space $ C_c^{\infty}(\mathbb{R}^3 \setminus \mathcal{C}) $ dense in $ H^1( \mathbb{R}^3) $ where $ \mathcal{C} $ is the circle $ \{(x,y,z) \in \mathbb{R}^3 \mid x^2 +...
- 1
-2
votes
1
answer
307
views
If a sequence $X_n$ of RVs converges in probability to $X$, does the sequence $\mathbb{E}(X_n)$ also converge to $\mathbb{E}(X)$? [closed]
I couldn't find the answer in literature so any idea would be helpful.
-2
votes
1
answer
314
views
Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$
(Question is short and straight-forward. )
What is/are "nice and non-trivial" series representation/s of $\log(|\zeta(\frac{1}{2}+it)|)$ ??
By "nice and non-trivial" I mean contains no ...
- 375
-2
votes
1
answer
283
views
On the series of the product of the terms of two sequences whose respective series are one convergent and the other not [closed]
Let us consider two sequences of real numbers $a_n$ and $b_n$, about which we only know that:
$$\sum_{1}^{\infty}a_n = 0$$
and that all $b_n > 0$, with $b_{n+1} > b_n$. Can it be proved that ...
- 362
-2
votes
1
answer
133
views
Undecidable definition of mathematical expressions?
I am arguing a bit on Facebook regarding the definition of a mathematical expression. Some argue that equations are not expressions (and there are a few possibly dubious online sources which states ...
- 15.8k
-2
votes
1
answer
62
views
If $X$ is discrete and $Z,W$ are discrete or continuous, is it always the case that $P(X=x\mid Z) \geq P(X=x\mid Z,W)$? [closed]
Suppose $X$ is discrete and $Z,W$ are discrete or continuous, I am wondering if it is always the case (or at least non-trivially) that
$$
P(X=x\mid Z) \geq P(X=x\mid Z,W)
$$
for all $x\in X$.
It ...
- 97
-2
votes
1
answer
103
views
Estimating expectation of a slightly strange sum
Let $X$ be a random variable with support on the positive integers (you can assume $\mathbb{E}[X^2] <\infty$ if needed, or even higher moments if needed), and let $S(i)=\mathbb{P}(X\geq I)$. ...
- 526
-2
votes
1
answer
89
views
Alternating property of H_2(T, Z)
Let us consider the torus $T = S^1_X \times S^1_Y$, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. We have an alternate property $dX \wedge dY = - dY \wedge dX \...
- 563
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votes
1
answer
92
views
Existence or impossibility of Gaussian factory
Gaussian factory problem: given an iid sequence $x_i \sim \mathcal{N}(\mu,\sigma^2)$, $i=1,2,\dots$, with $\mu$ and $\sigma^2$ both unknown, construct a realization $y \sim \mathcal{N}(0,1)$.
-2
votes
1
answer
538
views
Is an SDE really equal to an integral equation, or is it rather "its integral" that is?
Ive been told and been reading in some textbooks on SDE's that an SDE really is an integral equation. In other words, that
$ dX= \beta dt + \sigma dW$ $\,$ "really means" $\,$ $X_{t}= X_{0} +\int_{0}...
- 243
-2
votes
1
answer
476
views
sum of positive definite matrix
sum of positive definite matrix $A+B $is positive definite. I want to look at the spectrum of $C=A+B$
can we say the ith largest eigenvalue of $C$ is no less than the ith largest eigenvalue of $A$ i....
- 553
-2
votes
1
answer
970
views
What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [closed]
Suppose we have the following symmetric matrix.
$$A = \sigma^2 I + u u^T$$
What can we say about the eigendecomposition of $A$?
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-2
votes
1
answer
125
views
Notation on wreath product [closed]
While reading some (relatively old) papers on group-theory I encountered the following notations whose meanings I cannot understand:
If $W= G \wr H$ is the (unrestricted) wreath product of $G$ and $H$...
-2
votes
1
answer
75
views
What is an algorithm for generating a set of null (spacetime) vectors that add to zero? [closed]
I am interested in generating a list of n 4-vectors (t,x,y,z) such that -t^2+x^2+y^2+z^2=0 for each vector and the sum of the n 4-vectors equals zero. All of the t,x,y,z are real. I, particular, I am ...
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-2
votes
1
answer
162
views
Number of subgroups of a group of orders $p^3$ [closed]
Let $p$ be a prime number. Is there a formula for the number of subgroups of
$$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p^2\mathbb{Z}$$
$$\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}\times\...
- 39
-2
votes
1
answer
113
views
Demonstrations on an $L^1$ martingale [closed]
If $(X_n,\mathcal{F_n})_{n\in \mathbb{N}}$ is a martingale such that $\forall$ n $\in \mathbb{N}, \frac{X_{n+1}}{X_n}\in L^1$ How can be demonstrated that:
$\mathbb{E}[\frac{X_{n+1}}{X_n}]=1$ and ...
- 9
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votes
1
answer
142
views
The maximum known number n of non-attacking queens on the board n × n [closed]
Please inform.
For which maximum number n of non-attacking queens (located on a flat board with dimensions n × n), at least one solution is known?
Yours sincerely,
Alexander
- 13
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votes
1
answer
134
views
Multiplicative group of countable fields [closed]
Is there a countable field such that the multiplicative group of that field is isomorphic to $(\mathbb{Z},+)$?
- 48.9k
-2
votes
1
answer
209
views
Modulus of continuity an exponential type function [closed]
Fixed $0<a<1$, define $f(x):=(1-x)^{a}$ for every $x\in [0,1]$. Recalling that the modulus of continuity of $f$ of order $\varepsilon$ is given by
$\omega(f,\varepsilon):=\sup\{|f(x)-f(y)|:|x-y|\...
- 101
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votes
1
answer
158
views
About local maxima of multivariable polynomials
Lets say I have a real valued function which is writable as a polynomial in terms of Frobenius norms of a pair of matrices as in it is of the form, $f_B(A) = f(||A||_F^2, ||AB||_F^2, ||A^TAB||_F^2)$ ...
- 2,246
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votes
2
answers
446
views
Can ZFC be interpreted in a set theory having finitely many ranks?
One of the most notable features of $ZFC$ is that it builds up sets in recursively defined $V_i$ stages (where $i$ is an ordinal), however the usual formulation of $ZFC$ has infinitely many stages. ...
- 11.3k
-2
votes
1
answer
201
views
Solutions to a diophantine system
What is the smallest $\gamma_1,\gamma_2,\gamma_3>0$ such that given coprime $p,q=\Theta(\ell)$ and integer $t\geq3$ there are coprime $m,n=\Theta(\ell^{t-1})$ with $(mn,pq)=1$, $\alpha_i\in\Bbb Z$ ...
- 13.9k
-2
votes
1
answer
111
views
Is this space discrete? [closed]
Let X be a Tychonoff space such that for any closed set A there exist a continuous function f: X to R such that A=cl(X-Z(f)). Is this space X discrete?
- 5
-2
votes
1
answer
457
views
Expectation of random integral of deterministic function
Suppose I have some random variable $W$ along with its expectation $\mathbb{E}[W]$. My goal it to compute the integral
\begin{equation}
\mathbb{E}\left[\int_{0}^{W}f(t)dt\right] = \int_{0}^{\mathbb{E}...
- 48
-2
votes
1
answer
253
views
why do the Computability theory choose the natural number as the object of study? [closed]
I am wondering why the computable function is defined in the natural number set. Can people give me the answer or some resources that can solve my puzzle.
- 23
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votes
1
answer
86
views
Complete and saturated linear hypergraphs
A linear hypergraph is a hypergraph $H=(V,E)$ such that
$|e|\geq 2$ for all $e\in E$,
$|e_1\cap e_2|\leq 1$ for all $e_1, e_2\in E$ with $e_1\neq e_2$.
We call a linear hypergraph complete if there ...
- 48.9k
-2
votes
1
answer
106
views
Combinatorics- Equal partitions [closed]
Found this question on a list of past USAMO problems- and the one of the few that I couldn't solve. At a quick glance, seems like a deep problem.
P(n) is to partition an integer n greater than 1 so ...
- 1
-2
votes
1
answer
215
views
Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ [closed]
I am reading from the book Topics in Galois theory by Serre.
I have the following question ,
take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by
$$\sigma x\;=\;1/(1-x)$$
where $\sigma$ ...
- 601
-2
votes
1
answer
254
views
Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable? [closed]
Is there a proof (maybe formulated by Feferman) which says that a proof about the (in)consistency of ZFC is unachievable?
A professor said it to me a long time ago, but I don't have any references.
...
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votes
1
answer
126
views
fast way to get subextensions in magma? [closed]
If $l \equiv 1$ mod 3 then $\mathbb{Q}(\zeta_l)$ has a unique cubic subextension. I've been getting this field with the following magma code
F:=CyclotomicField(l);
S:=Subfields(F);
for ...
-2
votes
1
answer
138
views
How one can use a real math function on transaction in Hybrid Petri Net fundamental equation?
Say we have a simple HPN with 2 continuous places $A$ and $B$ and one transition. We want a transition not only add and substract $N$ marks from $A$ and add $M$ to $B$ but use mathematical function $...
- 59
-2
votes
1
answer
116
views
Is this intergral inequality valid? [closed]
Does the inequality $\int_2^{\infty} \dfrac{\sqrt x(\log x)^3 + (1+ \log x^2) x}{x(\log x)^2(x^2 - 1)} \,\mathrm {d}x > \ln \dfrac{17}{10}$ hold ?
- 53
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votes
1
answer
572
views
Tensor products of simple modules over algebras [closed]
Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules.
We can regard $M\otimes_{\mathbb C}N$ as $A\otimes_{\mathbb C} B$-modules in natural ...
-2
votes
1
answer
74
views
Behavior of "integer complex number" on computer [closed]
I want to provide software to compute with "integer complex numbers", that live in $\mathbb{Z}\times i \mathbb{Z}$, rather than the $\mathbb{C}$. Some operations are going to give results that are ...
- 97
-2
votes
1
answer
280
views
Deterministic Finite Automata question [closed]
I am very new to finite automata, and I came across an issue in my professors lecture slides which I think is wrong, and I'd wonder if any of you could confirm:
Alphabet: {1}
Automata
Surely the ...
- 7
-2
votes
1
answer
80
views
Suppose a real differentiable function with its derivative not infinity, it is sure that its second symmetric derivative should exist? [closed]
Suppose a real differentiable function $h(x)$ with its derivative not infinity, it is sure that its second symmetric derivative $\lim_{\epsilon->0}\frac{h(x+\epsilon)-2h(x)+h(x-\epsilon)}{\epsilon^...
- 39
-2
votes
1
answer
413
views
Covering space theory, category theory [closed]
Requiring covering spaces of a well-behaved connected topological space $X$ to be connected, let $\mathcal{Cov}(X)$ be the category of covering spaces of $X$ and maps over $X$ and maps over $X$. Can ...
-2
votes
1
answer
121
views
On Sylow subgroup of a finite group [closed]
Let $p\mid n$, then by $n_p$ we mean the $p$-part of $n$, i.e. $n_p = p^k$ if $p^k\mid n$ but $p^{k+1}\nmid n$. Let $G$ be a finite group, $M\leq G$ and $P\in Syl_p(G)$. Is It true that $|M\cap P|=|M|...
- 7
-2
votes
1
answer
2k
views
expected value of cosine wirh Gaussian phase
Is there a solution to the expected value/variance for a Gaussian with random phase:
$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$
?
For $t=0$, the solution is for example ...
- 167
-2
votes
1
answer
193
views
Analysis of Sobolev spaces [closed]
I just wanted to know wthether the following is OK or not.
Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
- 157
-2
votes
1
answer
224
views
using jensen's inequality
Suppose we have an expression
f(x, h(x,y)), for some function f and h, and x, y are random variables,
now we know that the function f(a, b) is concave w.r.t. a for given b. Can we use Jensen's ...
-2
votes
1
answer
472
views
how to reduce 3-colorable graph to this? [closed]
suppose we have a finite set X and a set S of subsets of X and we want to determine is there a subset S' of S such that all members of X belong to exactly one set in S' I think the best problem to ...
-2
votes
1
answer
212
views
A calculus question [closed]
Fix $q>1$. Define the function
$$
f_q(c):=\int_e^\infty \frac{e^{-c r^2}r}{\log(r)^q}d r.
$$
The problem is whether the following is true,
$$
\lim_{c\rightarrow 0} c \log(1/c)^q f_q(c) = C \in ...
- 1,649
-2
votes
2
answers
219
views
Lack of parabolicity of PDE due to invariancy under diffeomorphisms? [closed]
Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?
- 75
-2
votes
1
answer
180
views
Decimal digits multiplied by powers of 2: leads to mod 8? [closed]
This is more a puzzle than a research question,
a puzzle to me. Perhaps it is straightforward for others.
Imagine Repeatedly interpreting a number
expressed with the usual base-$10$ digits
as "digits"...
- 151k