Trending questions
159,029 questions
3
votes
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158
views
How to maximize the variance of a subset of integers?
$\DeclareMathOperator{\Var}{Var}$Given the set of numbers $\Omega := \{1, \ldots, n\}, n \in \mathbb{Z}^+$, how can I choose a subset, $A$ of $\Omega$ , such that $\min(\Var(A), \Var(\Omega \setminus ...
- 41
-4
votes
0
answers
62
views
Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs?
To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$
$$
\eta(s) = \sum_{n=...
1
vote
1
answer
170
views
On the condition of preadditive categories being locally small
The theory of categories is more flexible when not adding the (quite common) condition of being locally small. So the general notion of a category is the following (assuming we have a suitable ...
- 63.1k
8
votes
1
answer
318
views
Fibers of generic smooth maps between manifolds of equal dimension
I have heard that the following is a "well-known"
Claim. Let $M$ and $N$ be smooth manifolds with equal dimensions and $M$ compact. Then a generic smooth map $f\colon M\to N$ has finite ...
- 501
-1
votes
0
answers
45
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Linear and non-linear intersection to solve ODE
Consider a linear operator $$L(u(t)) = \dfrac{d}{dt}u(t)+p(t)u(t)$$ for known function $p(t)$. It is well known the homogeneous equation $$L(u) = 0 ~~\text{or}~~\dfrac{d}{dt}u(t)+p(t)u(t)= 0$$ has ...
1
vote
0
answers
66
views
Construction of the smallest nucleus above a prenucleus: what does this proof tell us?
While reading Hyland's paper on the effective topos [retyped version here] in the L. E. J. Brouwer Centenary Symposium, specifically prop. 16.3, I realized that the following proposition is implicit:
...
- 32.5k
3
votes
0
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62
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What upper bounds are known, for the number of divisors of Mersenne numbers?
Short version. What upper bounds are known, for the number of divisors of Mersenne numbers?
Long version.
Studying the structure of the factors of $M_n = 2^n - 1$ appears to be an active and difficult ...
- 1,444
1
vote
0
answers
26
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On $N$-partition of some common subsets $\Omega\subset\mathbb R^d$
Let $\Omega\subset\mathbb R^d$ be compact and convex, and denote by $\ell$ the normalised Lebesgue measure such that $\ell(\Omega)=1$. Let $N$ be an arbitrary but fixed integer.
In this post we set $d=...
- 1,399
3
votes
2
answers
187
views
Algorithms (or packages) to find recurrence relations for given sequence of q-polynomials?
Assume we have sequence of polynomials : $P_i(q)$ - each term is polynomial in $q$. (With integer coefficients, but hopefully it is not important).
We expect that there exists recurrence relation a ...
- 24.9k
1
vote
0
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59
views
Minimum number of sets for a union-closed sets conjecture counterexample
Let $\mathcal{F}$ be a family of $n$ finite sets. In this case, the family can be regarded as a multiset, since it is allowed to contain multiple instances of the same set. Let $U(\mathcal{F})$ be the ...
- 1,000
2
votes
0
answers
52
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Can we bound the degree of a one dimensional smooth compact leaf of a holomorphic foliation in terms of its genus?
Let $X$ be a smooth projective variety over the complex numbers with a fixed ample line bundle $H$. Suppose that $\cal F$ is a foliation in curves over $X$ (which may be singular).
Can you find a ...
- 388
0
votes
0
answers
116
views
How near are a groupoid and its 'preorderification'?
As remarks, a groupoid is a category with only (categorical) isomorphisms as its morphisms and a preorder is a category only having one morphism between each object. If we choose one isomorphism by ...
0
votes
0
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53
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Special determinant formula
Consider two column vectors $\textbf{a}$ and $\textbf{b}$ of length $k$ and $m$ respectively, $km$ variables denoted $y_{i,j}$ (i=1 to k, j=1 to m), and a quadratic form $\textbf{y}^{T}\mathbb{M}\...
- 419
6
votes
2
answers
739
views
Shifting an irrational binary sequence
Let $\newcommand{\tn}{\{0,1\}^\mathbb{N}}\tn$ be the collection of all infinite binary sequences. For $s\in\tn$ and $k\in\mathbb{N}$ let the left-shift of $s$ by $k$ positions, $\ell_k(s)\in \tn$, be ...
- 48.9k
7
votes
1
answer
498
views
Invertibility of a matrix defined using inner product
Let $n,m \geq 1$. We fix $n$ distinct vectors $x_1, ... , x_n \in \mathbb{R}^m$. We define $A \in \mathbb{R}^{n\times n}$ as
\begin{equation}
A_{ij} = x_i^T \left(n x_j - \sum_{1 \leq k \leq n} x_k \...
- 2,306
-3
votes
0
answers
28
views
Convergence of measures in the Lévy–Prokhorov metric and weak convergence of measures
How to prove that over R the convergence of measures in the Levi-Prokhorov metric is equivalent to the weak convergence of measures
- 1
3
votes
1
answer
76
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Tangential Sobolev spaces
Let $Ω⊂R^n$ be a smooth domain, define $U_s=\{x∈Ω | d(x,∂Ω)<s\}$; let $f∈W^{1,p}(Ω)∩W_{\mathrm{loc}} ^{2,p}(Ω)$; let $v$ be the unit normal to $Ω$; consider $v$ to be smooth with bounded ...
4
votes
0
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150
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Lemma in Roth's Theorem for Primes
I am reading Ben Green's paper Roth's Theorem in the Primes and I don't follow the proof of Lemma 6.1. I am not sure where the fact there are no more than $n^{3/4}$ elements $x\in A_0$ with $x\leq n^{...
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83
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Research without context is no research? [closed]
I was wondering what is research in mathematics. Can it be considered research only the research in open problems or it can be considered research also the finding of new formulas, new classes of ...
4
votes
0
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159
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Are the natural powers of two conservatively embedded in $\mathbb{C}$?
This is a followup to this question.
Consider $\mathbb{C}$ as a structure - in the sense of first-order logic - with the graphs of addition and multiplication. Let $\mathcal{X}$ be the substructure ...
- 20.5k
0
votes
1
answer
77
views
Newton method for polynomials with random starting points
I know that this question exists, but unfortunately it doesn't cover my issue sufficiently.
Assume that we have a polynomial $p(x)$ of degree $n$ with real coefficients, we can assume that all its ...
- 1,171
-1
votes
0
answers
23
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A question on Ibragimov's theorem on strong unimodality
I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask ...
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1
vote
1
answer
49
views
Graph classes which have small edge k-cuts
I am interested in graph classes that have the following property: There exists a function $f(k)$ such that for every graph $G$ in the class, for every choice of $k$ vertices $v_1, \ldots, v_k$ in the ...
- 526
7
votes
1
answer
80
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A syntactic characterisation of morphisms of algebraic theories whose induced algebraic functors admit right adjoints
Let $f : S \to T$ be a morphism of algebraic theories. Such a morphism induces a monadic functor $f^* : \mathrm{Mod}(T) \to \mathrm{Mod}(S)$ (hence $f^*$ has a left adjoint). We may view $f$ ...
- 10.7k
12
votes
1
answer
590
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+200
Fundamental group of the complement of a codimension two submanifold
Let $M$ denote an arbitrary smooth, closed, connected, n-dimensional manifold for $n\geq 4$. For every such $M$, does there exist a closed (not necessarily connected!) codimension two submanifold $S \...
- 1,174
3
votes
1
answer
142
views
Forcing equivalence and equal generic extensions
Two forcing notions $\Bbb P$ and $\Bbb Q$ could be defined to be forcing equivalent if the associated complete Boolean algebras are isomorphic (so, the CBA's formed by considering the regular opens of ...
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5
votes
1
answer
153
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Does there exist a section of $\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ that is "nearly Boolean"?
The following might be a somewhat esoteric question:
Does there exist an infinite cardinal $\kappa$ and a section $f$ of the quotient map $\pi:\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ (...
- 2,830
-5
votes
0
answers
45
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New Framework for Handling Undefined Mathematical Operations: Seeking Comments and Suggestions [closed]
The framework addresses undefined mathematical operations like zero raised to the first power (0^1) and division by zero (1/0) by introducing definitions that include infinity as the inverse of zero. ...
- 1
47
votes
10
answers
6k
views
Algebraic theorems with no known algebraic proofs
What are some good examples of algebraic theorems that have no known algebraic proofs?
A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...
1
vote
1
answer
119
views
Rational functions on elliptic curves over global fields with given support
Let $E$ be an elliptic curve over a global field $k$. Let $x_1, \dots, x_r$ be a set of generators of $E(k) / E(k)_{tor}$ (or more generally, a $\mathbb Q$-basis of $E(k)_{\mathbb Q}$), and let $x_0$ ...
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2
votes
2
answers
127
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Existence of k-complete uniform ultrafilter over a regular cardinal, k is strongly compact
This is a question about set theory. Let $\kappa\leq \lambda$ be infinite cardinals such that $\kappa$ is strongly compact and $\lambda$ is regular. My question is: how to construct a $\kappa$-...
- 61
5
votes
0
answers
101
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Query about extender embeddings
This seems as though it should be a result which is possible to prove but I was just wondering if I have it right and also if there is a source for it.
Suppose that $j:V_{\alpha} \rightarrow V_{\beta}$...
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13
votes
1
answer
864
views
Mistake on article about Bohr compactification?
$\DeclareMathOperator\b{b}\newcommand\B{{\operatorname B}}$I wish to get help understanding the content of two theorems of [Iva] that seem mutually contradictory. First some context. Let $\b(\mathbb{R}...
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4
votes
1
answer
283
views
Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
- 20.2k
6
votes
1
answer
563
views
Destroying scales
Suppose $\lambda$ is a cardinal with uncountable cofinality and $C\subseteq\lambda$ is a club of ordertype cf$(\lambda)$. Let $f=\langle f_\xi\mid\xi<\lambda^+\rangle$ be a scale consisting of ...
- 533
3
votes
1
answer
141
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Surjectivity of pushforward on image
Let $\mathcal X\subseteq\mathbb R^m$ be a Borel measurable set. $\Phi:\mathcal X\to\mathbb R^n$ be a continuous mapping and $\mathcal Y = \Phi(\mathcal X)\subseteq\mathbb R^n$ its image. Let $\mathcal ...
- 345
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votes
0
answers
38
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Examples of subharmonic functions
Let $A$ be a constant symmetric matrix with $\lambda < A < \Lambda$ and $0<\lambda < \Lambda$ are fixed constants. Let $u$ be a solution of $\text{div}A \nabla u = 0$. Is it true that $\...
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76
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Can one decompose quasi finite morphism as a composition of an open immersion and a finite morphism?
Can one decompose quasi-finite separated morphism of schemes as a composition of an open immersion and a finite morphism?
I am willing to assume that all the involved schemes are Noetherian.
- 2,649
2
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0
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116
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Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
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-3
votes
0
answers
48
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Do the domains of the two square roots of a positive (unbounded) operator coincide? [closed]
Let $H$ be a Hilbert space and $D:\mathrm{Dom}(D) \to H$ a densely defined operator on $H$. We further assume that $D$ is closed and self-adjoint. If we further assume that $D$ is positive, then we ...
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0
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93
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Generic representations of $\mathrm{GL}_n(\mathbb{R})$
Let $F$ be a local field of characteristic $0$, $G=\mathrm{GL}_n(F)$.
When $F$ is $p$-adic, Bernstein and Zelevinsky classified the irreducible generic representations. The statement is:
Let $\delta_{...
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17
votes
2
answers
2k
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Polynomials for natural numbers and irreducible polynomials for prime numbers?
Let $p$ be a prime and $n$ be a natural number.
Define inductively for prime numbers: $f_1(x) := 1$, $f_2(x):=x$, $f_p(x) := 1+\prod_{q\mid p-1} f_q(x)^{v_q(p-1)}$.
Is $f_p(x)$ always irreducible for ...
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11
votes
2
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193
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For what $n$ do there exist non-periodic tilings with rotational symmetry of order $n$?
More precisely, given an integer $n$, does there exist a non-periodic tiling, where there are infinitely many patches within the tiling, of indefinitely large area, with rotational symmetry of order $...
- 283
8
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0
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255
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+300
Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
- 501
0
votes
0
answers
41
views
Markov chain on the real line: Numerical methods for evaluating the stationary distribution
Consider a Markov chain on the real line with transition probabilities
$$
p(x_0,x)=\mathbf 1_{\{x\geq x_0+\alpha\,\cup\,x\leq x_0-\beta\}}\phi(x)+\delta(x-x_0)\left(\Phi(x_0+\alpha)-\Phi(x_0-\beta)\...
31
votes
2
answers
1k
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Why does this system of gcd equations have no solutions?
In February 2024 the following question was posed by user @Aig on Math.StackExchange:
Find the solutions to the system of equations $$\begin{cases} a + b = \gcd(a^3,b^3) \\ b+c = \gcd(b^3,c^3) \\ c+a =...
- 361
-2
votes
0
answers
72
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There is a typo in Stall's textbook on Set Theory: unable to prove the trichotomy of sets (m ∈ n or m = n, or n ∈ m) [migrated]
Here is the textbook, chapter 7, page 300. This lemma seems very of important, and I've spend about 8 hours trying to figure it out, but I'm unable to prove even the weaker version of the lemma (only ...
- 105
1
vote
0
answers
49
views
Square-integral involving Brownian bridge
Let $B(t)$ be a standard Brownian bridge on $[0,1]$. Let $x>0$ be a (small) parameter. What is the distribution of
$$
\int_0^{1-x} \left( B(t + x) - B(t) \right)^2 dt?
$$
As noted I am interested ...
- 2,217
2
votes
1
answer
145
views
Exotic Hopf algebra structures on the $p$-fold direct product in characteristic $p > 0$
Let $k$ be an algebraically closed field of characteristic $p > 0 $ and let $A$ be an algebra over $k$, which is a local ring.
There is an isomorphism of algebras $\prod_{i=1}^p A \cong A \otimes k[...
- 542
0
votes
0
answers
13
views
Chains with full range on a Boolean algebra with convex measure
Preliminaries. Let $X$ be a set and let $\mathcal A$ be a Boolean algebra of subsets of $X$ (i.e., $\mathcal A\subset 2^X$ such that $\mathcal A$ contains the empty set and is closed under finite ...
- 111