Newest Questions
159,036 questions
1
vote
1
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162
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Generating function of the stopped simple random walk
Let $\varepsilon_i$ be independent random variables such that $\mathbb{P}(\varepsilon_i = \pm 1)= 1/2$ and denote $W_n = \sum_{i=1}^{n}\varepsilon_i$. That is, $W_n$ is the simple random walk on $\...
- 13
3
votes
1
answer
351
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In which ways did geometric flows and variational methods from Riemannian geometry enter the symplectic world?
I am interested to learn about the role of geometric analytic methods for solving problems in symplectic geometry, In particular, I would like to know what results heavily rely on this machinery (incl....
- 219
4
votes
1
answer
372
views
Does Hermite-Einstein imply Kähler-Einstein?
Let $M$ be a compact Kähler manifold and let $\nabla$ be its Levi-Civita, or equivalently its Chern, connection. Denoting the vector bundle of complexified one forms of $M$ by $\Omega^1_{\mathbb{C}}$, ...
4
votes
3
answers
726
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Does there exist a topological space $X$ such that $X^2$ and $[0,1]$ are homeomorphic?
I have proved that if $X$ is not connected then $X^2$ is not connected either. So my idea was to prove that if $X$ is connected then $X^2$ blown up any point is also connected. But I don't know ...
- 515
2
votes
1
answer
136
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On equal area planar sections of 3D convex bodies
This is an extension of On segments of equal area cut from planar convex regions by chords.
While the 3D analog of the above question would be about 3D pieces cut from a convex body $C$ by planes, ...
- 5,979
7
votes
2
answers
1k
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Polynomials having all their zeros on the unit circle
Let $P(z)=\sum_{k=0}^na_kz^k$ be a polynomial of degree $n$ having all its zeros on the unit circle. Let $M=\max_{0\leq k\leq n}\lvert a_k\rvert$. The polynomial $P(z)=z^n+1$ has $\max_{\lvert z\...
- 559
1
vote
1
answer
298
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Calculation of an integral from BCS superconducting theory
How do I calculate the following integral?
$$\int_0^{\infty} d z ~z^{1 / 2}\left[\frac{\tanh ((z-1) / 2 t)}{2(z-1)}-\frac{1}{2 z}\right]=\ln \left(\frac{8 C}{\pi e^2 t}\right),$$
where $C=e^\gamma$ ...
- 419
4
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0
answers
235
views
A combinatorial proof for equality of two $q$-series
Consider the following two $q$-series
\begin{align*}
f(q):&=\sum_{k=1}^{\infty} \frac{(-1)^{k-1}(1 + q^k)\,q^{\binom{k + 1}2}}{(1 - q^k)^2} \qquad \text{and} \\
g(q):&=\frac1{\prod_{j=1}^{\...
- 43.2k
2
votes
1
answer
106
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A modified Paley–Wiener theorem with weaker condition
Let's consider the following argument: let $f$ be a function in $L^2(\mathbb R)$ such that $\hat f$ extends to an entire function on $\mathbb C.$ Assume that for each $t>0$ and $x \in \mathbb R$
$$
...
- 1,755
2
votes
0
answers
170
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finite dimensionality of a subspace of a Banach space
Let $H$ be the space of measurable functions on $(0,1)$ such that
$$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$
Let $C>0$ be a constant. Suppose that $W \...
- 4,115
1
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0
answers
121
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Does a gauge-invariant Caccioppoli inequality hold?
(I previously asked this question on Math.SE but got no responses after two weeks.)
Let $V \Subset U$ be domains in a Riemannian manifold $M$, and $W := U \setminus \overline V$. If $u: U \to \mathbb ...
- 708
3
votes
1
answer
134
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Connected components of a spherical subgroup from spherical data?
This question is in a similar spirit to this one by Mikhail Borovoi.
Let $G$ be a reductive group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety.
Losev proved that the spherical $...
- 2,516
6
votes
1
answer
331
views
A combinatorial identity involving increasing functions from $\{1, \dots, n\}$ to itself
This is related to the post An order statistics problem with some interesting geometry. The following identity arose in the context of the problem.
Fix an integer $N \geq 2$. Let $\mathcal S_N^+$ ...
- 6,321
2
votes
2
answers
558
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Solution of a linear hyperbolic PDE
I trying to find the solution of the following Goursat problem for a second-order hyperbolic linear PDE
$$
\begin{cases}
u_{xy} + k(u_x+u_y) + (k^2 - \sigma^2 P(x-y))u = f(x,y) \\
u(x,0) = 0 \\
u(0,y) ...
- 71
23
votes
5
answers
2k
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PDEs and algebraic varieties
Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
- 8,998
0
votes
0
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183
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Why is the sign of the integration negative?
Let
\begin{aligned}
I=\int_0^1 B^{\frac{1}{1-\alpha + \alpha x}} x^{k - 1} \left(\frac{\alpha \log{\left(B \right)}}{(\alpha-k-1)^2} +\frac{1}{k} + \log{\left(x \right)} \right) dx,
\end{aligned}
...
- 9
1
vote
0
answers
72
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Infinite dimensional version of the Laplace transform and Gaussian integrals
This question is somehow related to my previous one Convergence of the Gaussian integral on $\mathcal{E}'$ for a mapping supported on $L^2$
Let $F : L^2(S^1) \to L^2(S^1)$ be a (nonlinear) Borel-...
- 3,477
5
votes
0
answers
259
views
Equations for conic del Pezzo surfaces of degree one
Let $X$ be a del Pezzo surface of degree one over a field $k$ of characteristic not $2$ equipped with a conic bundle $\pi: X \rightarrow \mathbb{P}^1$. By Theorem 5.6 of this paper, $X$ admits a ...
- 241
2
votes
3
answers
264
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Control of values of an entire function in a strip around the real line
Consider an entire function $f: \mathbb C \to \mathbb C$ such that $f|_{\mathbb R}(x)\to 0$ as $x \in \mathbb R \to \pm\infty.$ Does that imply that for each $T>0,$ we have $f(x+iy) \to 0$ as $x\to ...
- 1,755
0
votes
0
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178
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Finite monomorphism $A \to B$ with reduced $A$ and special fiber implies $B$ reduced
I have a question about correctness of following statement claimed here in $\boxed{2} \ $:
Let $k$ arbitrary field, let $f : X \longrightarrow Y$ be a finite dominant morphism between finite type $k$-...
- 6,038
7
votes
1
answer
509
views
An order statistics problem with some interesting geometry
Let $a_n$ be a given sequence of positive numbers, and $X_n$ a sequence of independent random variables with each $X_n$ uniformly distributed on $[0, a_n]$.
Question: Let $N \geq 2$ be an arbitrary ...
- 6,321
2
votes
1
answer
270
views
Least number of ways to color a map using at most 4 colors
Given all the maps with k regions, consider the map(s) that can be colored in the least amount of possible ways. In how many ways can it be colored (n)? (using at most 4 colors and not counting ...
- 89
4
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0
answers
121
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Is there a faithful functor from the freely generated bicartesian closed category to $\mathbf{Set}$?
Does there exist a faithful (bicartesian closed) functor $\operatorname F$ from the freely generated bicartesian closed category $\mathbf B$ to $\mathbf{Set}$? Preferably, $\mathbf B$ should contain ...
1
vote
0
answers
81
views
Pre-positive definite functions?
A function $f(x,y)$ is positive definite if matrices $( f(x_i, x_j) )_{i, j \in F}$ are positive definite for all finite index sets $F$. This is frequently hard or impossible to check given some ...
- 620
0
votes
1
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108
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Functional CLT with an asymptotically small time change
This question was posted to MathSE but it seems like MathOverflow might be the more appropriate place for it.
Suppose I know that $(\frac{1}{\sqrt{m}}X(mt))_{0\leq t\leq 1}\xrightarrow[m\to\infty]{\...
- 177
5
votes
1
answer
489
views
Does coefficient-wise limit preserve real-rootedness?
Let $P_n$, $n=1,2,\ldots$ be polynomials with real roots only (and real coefficients), and $P_n$ converge to a non-zero polynomial $Q$ coefficient-wise. Does it follow that $Q$ has real roots only?
...
- 109k
2
votes
0
answers
120
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Closure of Laplacian
Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator
$$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$
There are two ...
- 1,171
2
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0
answers
154
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Translation of an article of Littlewood
I want to read the English translation of an article of Littlwood titled "Quelques conséquences de l'hypothese que la fonction $ζ (s)$ de Riemann n'a pas de zéros dans le demi-plan $ℜs> 1/ 2$.&...
- 253
2
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0
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205
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Can anyone figure out the sign of the following definite integral? [duplicate]
$$
\int_0^1 B^{\frac{1}{1-\alpha + \alpha x}} x^{k - 1} \left(\frac{\alpha \log{\left(B \right)}}{(\alpha-k-1)^2} +\frac{1}{k} + \log{\left(x \right)} \right) dx,
$$
where $B\geq 2$, $\alpha \in (...
- 29
2
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1
answer
366
views
Filtered homotopy colimits of spectra
Let $\mathcal{I}: \mathbb{N} \to \operatorname{Sp}$ be a diagram in the infinity category of spectra. Let $\pi_0(\mathcal{I})$ denote the corresponding $1$-categorical diagram (i.e. compose $\mathcal{...
- 373
3
votes
1
answer
238
views
1D topological defects in $d>3$ spatial dimensions
I am trying to construct a 1D topological defect solution in 4 spatial dimensions, i.e., a solution to some PDE (likely the equations of motion of some Lagrangian) on $\mathbb{R}^{4}$ which is ...
- 147
3
votes
0
answers
50
views
Stability of indefinitely damped mechanical system with diagonal stiffness
I'm trying to find conditions for the asymptotic stability of the following linear system,
\begin{equation}
\mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0
\end{equation}
given the ...
2
votes
0
answers
102
views
Category O for (Yangian) toroidal Lie algebras?
Suppose throughout that $g$ is a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let us denote:
$$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$
$$g_{[2]}^+ := g \...
- 1,516
1
vote
1
answer
211
views
Tensor product of faithful normal states is faithful
I know that given C*-algebras $A, B$ with faithful states $\omega,\varpi$, the tensor product state $\omega\otimes\varpi$ on the minimal tensor product $A\otimes_{\text{min}}B$ is faithful.
I also ...
- 439
2
votes
2
answers
277
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Characterization of locality in Fourier multiplier
Obviously Laplacian $-\Delta$ is a local operator, and it can be written as a Fourier multiplier $|\xi|^2$. Similarly fractional Laplacian $(-\Delta)^s$ is with Fourier multiplier $|\xi|^{2s}$ and is ...
4
votes
1
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275
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Interesting Grothendieck topologies or coverages on the category Prob
I am currently trying to understand Grothendieck Topologies and coverages and want to endow the category Prob, consisting of finite probability spaces and measure preserving maps, with a Grothendieck ...
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0
votes
1
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134
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Existence of cyclic subspace decompositions for pairs of commuting matrices
Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...
4
votes
0
answers
78
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A functional equation: Functional families that are "weakly" closed under product?
Suppose that for any real number $a$, we have a function $f_a:\mathbb R \to \mathbb R$ or such that $f_a(x)$ is monotonically strictly increasing in $x$ and hence invertible on its image. We also ...
- 91
2
votes
1
answer
304
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Exact sequence for relative cohomology + normal crossing divisors
Let $X$ be smooth algebraic variety over $\mathbb C$ and $D_1, D_2$ are snc divisors such that $D_1\cup D_2$ is also snc.
Is it true that there is an exact sequence
$$H^*(X, D_1\cup D_2)\to H^*(X, D_1)...
- 219
5
votes
1
answer
342
views
Kirby diagrams of Mazur manifolds
In the 1980's, Fintushel-Stern and Fickle independently proved that Brieskorn spheres $\Sigma(2,3,25)$ and $\Sigma(3,5,19)$ bound some Mazur type contractible 4-manifolds with a single $0$-, $1$, and $...
- 51
11
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1
answer
1k
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Existence of skeletons in ZFC
Influenced by this question from a fellow lagomorph, I would like to get to the bottom of existence of a skeleton of a category. I want to stay in ZFC, so I do not assume the global axiom of choice. ...
- 12.4k
3
votes
1
answer
439
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Chinese remainder theorem for target interval
Given $n$ pairwise coprime natural numbers $m_{1}, \dots, m_{n}$ with remainders $y_{i}$, for all $i \leq n$. Furthermore, we have a target interval $I := \left[ a, b \right]$, with $1 \leq a < b \...
- 33
4
votes
0
answers
205
views
Notion of connected components for $\mathbb{Q}_p$-points of algebraic variety
Is there an interesting notion of connected components for the $\mathbb{Q}_p$-points of an algebraic variety over $\mathbb{Q}_p$? By "interesting" I mean a notion satisfying the following. ...
- 563
1
vote
0
answers
94
views
cycle types of all words in a permutation group
I have been working with permutation groups. For a given $G\subset S_n$, what I have been computing depends only on the conjugacy class of $G$.
Say all permutation groups in this question are ...
- 2,287
3
votes
2
answers
276
views
Continuous version of the union-closed sets conjecture
Let $F = \{f_1, \ldots, f_n\}$ be a set of continuous functions $f_i: [0,1] \rightarrow [0,1]$,
$i = 1, \ldots, n$, such that $f_i \in F \land f_j \in F \implies \max(f_i,f_j) \in F$.
I would like to ...
- 1,000
3
votes
1
answer
145
views
Topological amenability of actions - forgetting topology
Let $G$ be a (countable) discrete group and let $X$ be a locally compact Hausdorff space.
Assume that $G$ acts on $X$ by homeomorphisms. Recall that the action is (topologically) amenable if there ...
- 536
2
votes
1
answer
300
views
If $\mathcal{H}^{n-1}(E)=0$ then $\mathbb{R}^n\setminus E$ is connected
Let $E\subset \mathbb{R}^n$ be a (measurable) subset with $\mathcal{H}^{n-1}(E)=0$, where $\mathcal H^{n - 1}$ is the ($n - 1$)-dimensional Hausdorff measure. I want to know if $\mathbb{R}^n\setminus ...
- 1,149
1
vote
2
answers
213
views
If $\mathcal{H}^{n-1}(K)=0$ then $\mathcal{H}^n(K\times \mathbb{R})=0$
I am reading a paper Simon and Wickramasekera - A Frequency Function and Singular Set Bounds for Branched Minimal Immersions where the authors seem to claim that if $K\subset\mathbb{R}^n$ is a compact ...
- 1,149
6
votes
2
answers
1k
views
Prime gaps within which every "small" prime appears as a factor: Are there only finitely many? Is this the last one?
For a bounded range of positive integers $n,n+1,\ldots,m,$ call a prime number "small" if it does not exceed $\sqrt m,$ so that if one is trying to factor all of these numbers into primes, ...
3
votes
1
answer
121
views
When are homologous embedded surfaces in 3-manifolds related by embedded cobordisms?
Let $M$ be an orientable closed 3-manifold and suppose $A$ and $B$ are embedded incompressible closed orientable surfaces in $M$ with $[A] = [B]$ in $H_2(M,\mathbb{Z})$.
In general, there are a ...
