Newest Questions
165,357 questions
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Projection formula for D-modules
I am currently reading Claude Sabbah’s classic paper “Monodromy at Infinity and Fourier Transform” (the electronic version is available at https://perso.pages.math.cnrs.fr/users/claude.sabbah/articles/...
- 31
1
vote
0
answers
8
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Is every function to a lower dimensional Euclidean space a contraction on a large set?
Let $f: \mathbb R^n \to \mathbb R^m$ be a measurable function with $m < n$.
Question: Is it true that there exists a subset of $\mathbb R^n$ of Hausdorff dimension at least $n-m$ on which $f$ is a ...
- 9,990
1
vote
0
answers
16
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Criteria for continuity of bounded linear functionals which are continuous on a dense, non-normable subspace of a complete locally convex vector space
In what follows, all vector spaces are over the field of scalars $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. One has as basic facts in functional analysis that:
Any bounded linear operator $T:E\...
- 9,118
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15
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Geometric structures on a Lie-groupoid as flat bundles
Let $\Gamma$ be an étale connected Lie-groupoid. Let $G$ be a Lie-group. Let $X$ be a smooth manifold equipped with a smooth $G$-action. Assume more about the action if needed. People say that an $(X,...
- 131
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54
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Counterexample request: kernel of coproduct is not coproduct of kernels
Let $\mathcal{A}$ be an abelian category. Let $\{A_\alpha\xrightarrow{f_\alpha}B_\alpha\}$ be an infinite family of maps. Then there is an induced map:
$$\bigoplus_\alpha A_\alpha\xrightarrow{f=\...
- 2,413
-4
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71
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Axioms of Soft Logic
Someone here bumped into the papers by Moshe Klein & Oded Maimon on Soft Logic? I try to understand whether their axioms actually exclude the zero-product property.
Here are the axioms from one of ...
- 115
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19
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Single source-path, Inverse shortest path problem with bounds, complexity and algorithms
Given a network $G=(V,E,w)$, two vertices $s$ and $t$ as source and sink, and a designated path $P_0$ from $s$ to $t$, the inverse shortest path problem (ISP) asks to adjust new weights $w^*$ at ...
- 25
2
votes
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answers
97
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Does there exist some result about vector-valued non-commutative measure spaces?
It is well-known that for any Banach space $X$, and a probability space $(\Omega,\Sigma,P)$, we may define a $X$-valued measurable function space $(\Omega,\Sigma,X)$. And a von-Neumann algebra $\...
- 29
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32
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Polyhedral complexes and shellings
By a polyhedral complex, I mean a collection of polytopes $\mathscr{P}$ in $\mathbb{R}^n$ such that if $P\in \mathscr{P}$ then all of the faces of $P$ are in $\mathscr{P}$, and if $P,Q \in \mathscr{P}...
- 731
3
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1
answer
127
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Diophantine equation $a^2+b^4+c^6=d^8$ (parameterization)
How to get parametrization for the diophantine equation $a^2+b^4+c^6=d^8$ There is an infinite set of solutions to this equation (example $79^2+4^4+2^6=3^8$), but it is not easy to reduce it to a ...
- 33
5
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1
answer
112
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Invertible perturbations of matrices
Let $P \in M_n(\mathbb{C})$ be a rank $k$ orthogonal projection, $k \geq 2$, and let $A_1, \ldots, A_r \in M_n(\mathbb{C})$ be matrices. Suppose that for every rank $k - 1$ orthogonal projection $Q &...
- 44.3k
3
votes
1
answer
161
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Does anyone know how to solve $f(n+1,m+1) = (m+2)(f(n,m) - f(n,m+1))$?
Does anyone know how to solve $$f(n+1,m+1) = (m+2)(f(n,m) - f(n,m+1))$$
I'm not too sure how to approach it. I've found some specific results like:
$$f(n+1,n) = (n+1)!$$
$$f(n,0) = (-1)^{n+1}$$
$$f(n,...
- 31
3
votes
1
answer
155
views
Fast and simple recurrence for sums over powers and factorials
Let
$f(n,m)$ be a function such that $$ f(n,m) = mf(n-1,m) + 1, \\ f(0,m) = 1. $$
$T(n,m)$ be a coefficients such that $$ T(n,m) = n! \sum\limits_{k=1}^{n} \frac{m^{k-1} n^{n-k-1}}{(n-k)!}. $$
See ...
5
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1
answer
157
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Stochastic dominance $ \tanh Y\tanh Z \succeq \tanh X $
Let $X,Y,Z$ independent Gaussian r.v.'s with mean=variance. Let's denote these mean/variance parameters by $g_X,g_Y,g_Z>0$ respectively.
Set $T_1:=\tanh X$, $T_2:=\tanh Y\tanh Z$.
My question. ...
- 361
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answers
19
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Nonlinear Hammerstein integral equation: existence and bifurcation of solutions
Let
𝐹
be a family of subsets of
[
𝑛
]
such that for any two distinct sets
𝐴
,
𝐵
∈
𝐹
, we have
∣
𝐴
∩
𝐵
∣
≤
𝑘
.
For fixed
𝑘
, what is known about the maximum possible size of such a family as
𝑛...
- 63
