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Verdier (w) condition implies the $w_f$ condition when the restriction of $f$ in each stratum is a submersion?

Let $X\subset\mathbb{R}^n$ be and let $\Theta=(X_\beta)_{\beta\in I}$ a Verdier stratification for X. Let $f:X\rightarrow\mathbb{R}$ be a polynomial function, such that $f_{|_{X_\beta}}$ is submersion ...
Giovanny Snaider Barrera Ramos's user avatar
11 votes
1 answer
605 views

If the universal cover has three boundary components, does it have infinitely many?

Suppose that $M$ is a compact, connected three-manifold with boundary. Suppose that $\pi_1(M)$ is infinite. Suppose that $\tilde{M}$, the universal cover of $M$, has at least three boundary components....
Sam Nead's user avatar
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0 votes
1 answer
142 views

Integral inner product with exponential function

Suppose on some unknown interval $[0, I]$ we have non-negative functions $f, g : [0, I] \rightarrow \mathbb{R}^{\geq 0}$. If we know that \begin{aligned} \int_0^I f & = c \\ \int_0^I e^f & = e^...
Lewwwer's user avatar
  • 129
1 vote
0 answers
118 views

Degrees of trigonometric numbers

For a rational number $r\in(0,1)$ the number $z=\sin(r\pi)$ is an algebraic number — such numbers appear to be called trigonometric numbers. What is its degree? That is, what is the minimal degree of ...
Joonas Ilmavirta's user avatar
13 votes
1 answer
596 views

A symmetric function related to sums of square roots

Let $x_1,x_2,\dots,x_n$ be indeterminates (say over $\mathbb{Q}$). For every sequence $\epsilon=(\epsilon_1, \dots,\epsilon_n)\in\{-1,1\}^n$ define $$ y_\epsilon = \sum_i \epsilon_i \sqrt{x_i}. $$ Let ...
Richard Stanley's user avatar
4 votes
1 answer
217 views

Is there a Bailey–Borwein–Plouffe (BBP) formula for $\gamma$ (euler-mascheroni constant)?

I was reading about BBP type formulas and there was a lot about $\pi$ and some $\log$'s. I started searching for some other constants and could find $2$ formulas for the catalan constant and learned ...
Pinteco's user avatar
  • 521
4 votes
1 answer
174 views

Maximal intersecting families on $\omega$ that are not ultrafilters

A family ${\cal S}\subseteq{\cal P}(\omega)$ is intersecting if any two members of ${\cal S}$ have non-empty intersection. Zorn's Lemma implies that every intersecting family is contained in a maximal ...
Dominic van der Zypen's user avatar
8 votes
1 answer
319 views

Why does this combinatorial sum vanish?

We define the coefficients $c_{k,k-i}$ of ${n \choose k}$ by the following easy expansion: \begin{align*} & {n \choose k} = \frac{1}{k!} n(n-1) \dots (n-k+1) = \frac{1}{k!} \prod\limits_{t=...
Ben Deitmar's user avatar
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1 vote
0 answers
36 views

Weighted least squares with matrices as unknowns

Let $M \in \mathbb{R}^{m \times n}$. Let $S \in \mathbb{R}^{m \times N_t}, U \in \mathbb{R}^{n \times N_t}$, with $ N_t \gg m,n$. Moreover, $\epsilon = S - M U$, with $\epsilon$ zero mean white noise ...
baptiste's user avatar
  • 123
1 vote
1 answer
362 views

Tensor product and semisimplicity of perverse sheaves

Let $X/\mathbb{C}$ be a smooth algebraic variety. Let $D_c^b(X,\bar{\mathbb{Q}}_{\ell})$ be the category defined in 2.2.18, p.74 of "Faisceaux pervers" (by Beilinson, Bernstein and Deligne). ...
Doug Liu's user avatar
  • 615
1 vote
0 answers
57 views

CP maps obeying an equality

Start with a completely positive unital map $\psi:A\to B$ between $C^*$ algebras with identity. The equality $\psi(a^*a)=\psi(a)^*\psi(a)$ is true for all $a\in A$ in the case where $\psi$ is a star ...
Edwin Beggs's user avatar
  • 1,143
2 votes
0 answers
84 views

Another variant of the Malfatti problem

We try to add to A Variant of the Malfatti Problem As stated in the Wikipedia entry on Malfatti circles, it is an open problem to decide, given a number $n$ and any triangle, whether a greedy method ...
Nandakumar R's user avatar
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4 votes
2 answers
219 views

Algorithm for grouping tetrahedra from Voronoi diagram

I have a set of 3D Voronoi generator points and their neighbouring points, which, when connected, should result in a Delaunay tetrahedralization. However, I'm having a hard time implementing this. My ...
catmousedog's user avatar
1 vote
0 answers
132 views

A generalisation of residual finiteness?

A group $\Gamma$ is Residually Finite (RF) if $\forall g \neq e \in \Gamma$ there is a homomorphism $h: \Gamma \to G$ where $G$ is a finite group such that $h(g) \neq e$. Free groups are known to be ...
mathstudent42's user avatar
1 vote
1 answer
149 views

Is there a necessary and sufficient condition to determine whether a number sequence can serve as the first few moments of a Radon measure?

Given a few positive numbers $(M_1, M_2,\cdots, M_K)$, they are the moments of a measure if \begin{equation} M_k = \int d\mu(x) x^k,\quad k = 1,2,\cdots,K. \end{equation} This is related to the ...
Yi Changhao's user avatar
3 votes
0 answers
124 views

Branching problem of representation of Lie groups and orbit method

Branching problem asks how a restriction of an irreducible representation of $G$ to a subgroup $H$ decomposes. In case of (real) Lie groups, after labeling irreducible representations via highest ...
Seewoo Lee's user avatar
  • 2,215
2 votes
0 answers
146 views

What are the name and inverse of an interesting integer matrix?

It is practicable to compute the matrix inverses \begin{align*} \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 2 & 2^2 \\ \end{pmatrix}^{-1} &=\begin{pmatrix} 1 & 0 &...
qifeng618's user avatar
  • 1,101
5 votes
0 answers
249 views

Aspherical space whose fundamental group is subgroup of the Euclidean isometry group

Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
Chicken feed's user avatar
1 vote
1 answer
350 views

The distribution of product of normal matrices

Assuming we have three matrices: $\mathbf{X} \in \mathbb{R}^{4096 \times 1024}$, $\mathbf{Q} \in \mathbb{R}^{1024 \times 2048}$, $\mathbf{K} \in \mathbb{R}^{2048 \times 1024}$. Where all elements of ...
eternity's user avatar
2 votes
1 answer
198 views

Eigenvectors and eigenvalues of a symmetric matrix and its entry-wise absolute value

The modulus of matrices is meant componentwise in the following. Let $H$ be a sqaure matrix that satisfies the following assumptions: $H$ is real-valued, symmetric, and positive-definite.. $H$ is ...
keisuke murota's user avatar
1 vote
1 answer
339 views

Analysis of functions over Galois fields

I'm trying to understand how harmonic analysis generalises to functions over finite (Galois) fields. In particular I'm trying to understand - how to meaningfully map the function to somehow "work ...
GWB's user avatar
  • 301
0 votes
1 answer
152 views

Name for a monoid on the basis of a vector space?

Is there a name for the structure of a vector space with a monoid defined on its basis? Given a vector space V over a field F, we can choose a basis and define a monoid on it. Now we can use each ...
Spencer Woolfson's user avatar
2 votes
1 answer
40 views

Questions on the "generalized" min. singular value of $A$ given $B$: $\min_{L \in \mathbb{R}^{n \times m}} \{\|BL\|_F: \det(A + BL) = 0\}$

Let $A \in \mathbb{R}^{n \times n}$ be a matrix. Recall $\sigma_{\min}(A)$ is the Frobenius distance between $A$ and the set of singular matrices: $$\sigma_\min(A) = \min_{E \in \mathbb{R}^{n \times n}...
Spencer Kraisler's user avatar
0 votes
1 answer
327 views

Can we generalise groupoids to monoid-oids? [closed]

Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories. Groupoids correspond to small categories where every morphism is an ...
Diego de la Paz's user avatar
2 votes
0 answers
148 views

Abstract definition of hypertoric varieties

I'm reading Proudfoot's survey on hypertoric varieties. In Section 1.4 he mentioned such a conjecture: Conjecture 1.4.2 Any connected, symplectic, algebraic variety which is projective over its ...
TheWildCat's user avatar
2 votes
0 answers
110 views

Defining a metric on $\mathbb Z^n$ using Green's function for the simple random walk

Let $G$ be Green's function for the simple random walk on $\mathbb Z^n$ for $n\ge 3$, i.e., $G(x)$ is the expected number of visits to $x$ when the walk starts at the origin. Define $d(x,y)=G(x-y)^{1/(...
Alexander Pruss's user avatar
2 votes
0 answers
137 views

$p$-adic Banach group algebra

Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
Luiz Felipe Garcia's user avatar
2 votes
1 answer
385 views

Continuity equation $\partial_t \mu_t+\operatorname{div} (v_t \mu_t)=0$: are these two notions of weak solution equivalent?

Let $\Omega$ be an open connected convex subset of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of Borel probability measures on $\Omega$. Let $C_0 (\Omega)$ be the space of real-valued ...
Analyst's user avatar
  • 657
9 votes
3 answers
553 views

Bounding the $n$-th derivatives of $\frac{1-\cos(x)}{x^2}$

Define the smooth map $f : \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) := \frac{1-\cos(x)}{x^2} = -\sum\limits_{k=1}^\infty \frac{(-1)^k}{(2k)!} x^{2k-2}$. I am looking for a nice bound on $|f^{(n)}(x)...
Ben Deitmar's user avatar
  • 1,295
0 votes
1 answer
143 views

From superoperator of unitary to the unitary itself

I have a question about super operators. Let's say I have the super operator of some unitary matrix $u$ called $SU$ where $SU = u^\ast\otimes u$ (here $u^\ast$ is the complex conjugate of $u$). If I ...
Mushahid Khan's user avatar
1 vote
1 answer
205 views

Degree three, codimension one subvarieties lying on a quadratic hypersurface

Let $H$ be an irreducible hypersurface in $\mathbb P^n$ of large-ish degree, say 14. This question is about subvarieties $V$ of $H$ such that $V$ has codimension 1 in $H$ (i.e. $V$ has dimension $n-2$...
Simon L Rydin Myerson's user avatar
2 votes
1 answer
624 views

On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$

Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$. I have found in several research ...
Arghya kundu's user avatar
2 votes
0 answers
62 views

Localized estimate for divergence free vector field

Suppose $\Omega \subset \mathbb{R}^3$ is a simply connected Lipchitz domain. For a divergence free field $w\in [L^2(\Omega)]^d$, it is well known that there exists a vector field $v\in [W^{1,2}(\Omega)...
Ryan Li's user avatar
  • 31
1 vote
1 answer
256 views

Moser iteration in dimension $6$

Let $M$ be a closed Riemannian manifold of dimension $6$. We have a function $f\geq 0$ on $M$ satisfying \begin{align*} \Delta f \leq gf-\frac{3}{4}f^2 \end{align*} Where $g$ is another smooth ...
Partha's user avatar
  • 954
5 votes
1 answer
313 views

Can the free module in the representation ring be characterised this way?

Let $F$ be a field and $G$ be a finite group. Let $V,W$ be two $G$-representations of the same dimension. I am interested to find all modules $U$ with the property that $V\otimes U$ and $W\otimes U$ ...
HenrikRüping's user avatar
0 votes
1 answer
109 views

Extending maps from a discrete set to a Stone-Čech compactification while retaining an injectivity condition

For $S$ a set, let $\beta_{\bf2}(S)$ be a compact, totally disconnected space containing $S$ where $S$ in the subspace topology is discrete and $S$ is a dense subspace, and $\beta_{\bf2}(S)$ has the ...
Tri's user avatar
  • 1,644
0 votes
0 answers
69 views

What should we call the area for which the lower border is a Motzkin path?

We can draw a Motzkin path from $(0,0)$ to $(n/2,n/2)$ using steps $(0,1)$, $(1,0)$ and $(1/2,1/2)$, such that the path never goes below the line $y=x$. Consider the area bounded by the Motzkin path, ...
xmchenhit's user avatar
  • 135
3 votes
0 answers
219 views

Do these cousins of permanents satisfy the following inequality?

Let $H$ denote an $n$ by $n$ hermitian positive semidefinite matrix. Let $G$ and $K$ be two subgroups of the symmetric group $\Sigma_n$. Define $$ f_{G, K}(H) = \sum_{(\sigma, \tau) \in G \times K} \...
Malkoun's user avatar
  • 5,215
2 votes
0 answers
147 views

Homotopy equivalence of chain complexes from subcomplexes and quotient complexes

Let $C_1$ be a finite-dimensional chain complex over $\mathbb{C}$ coefficients. Let $S_i$ be a subcomplex of $C_1$ and let $Q_1$ be the quotient complex. Suppose $S_1$ and $Q_1$ are chain homotopy ...
Faniel's user avatar
  • 673
3 votes
0 answers
171 views

Grothendieck-Messing in characteristic 0?

Let $A$ is an abelian scheme over a base scheme $S$. Let $S \rightarrow S'$ be a thickening defined by an ideal of square zero (for example). If $p$ is locally nilpotent on $S$, then Serre-Tate and ...
351910953's user avatar
  • 261
2 votes
1 answer
206 views

Bound for zero-crossings of heat equation

I am considering the following problem. Let $\mathcal{P}$ the classical heat-diffusion problem: $$\mathcal{P} : \left(\partial_t u (t,x)=\frac{1}{2}\partial_{xx}^2u(t,x)\text{ with }u(0,\cdot) = f(x)\...
NancyBoy's user avatar
  • 393
4 votes
1 answer
321 views

Sub-Gaussian random variables and convex ordering

Suppose that $X$ is a $1$-sub-Gaussian real-valued random variable, i.e. for all $t \in \mathbf{R}$, it holds that $\log \mathbf{E} \exp \left( t X \right) \leqslant \frac{1}{2} t^2 $. Does there ...
πr8's user avatar
  • 801
0 votes
1 answer
137 views

Is there a canonical mapping between countable transfinite ordinals and $\omega$? What about recursive ordinals?

Consider $\omega^2$. We can build a simple bijection between the ordinal and $\omega$ similarly to how the bijection between $\mathbb{Q}$ and $\mathbb{N}$ can be built. I was wondering if there is a ...
Guillermo Mosse's user avatar
4 votes
2 answers
414 views

A measure assigning values in $\{0,1\}$ must be a Dirac measure?

Let $\mu$ be a measure on some measurable space $(\Omega, \mathcal F)$ such that $$\mu(B)\in \{0,1\},\quad \forall B\in \mathcal F.$$ Can we show that $\mu$ must be a Dirac measure (under suitable ...
Fawen90's user avatar
  • 1,399
7 votes
0 answers
296 views

A system with distinct infinite cardinalities but no "best" version of $\mathbb{N}$

Let $\mathfrak{S}=(M_z,U_z)_{z\in\mathbb{Z}}$ be a sequence such that for each $z\in\mathbb{Z}$ we have $M_z\models\mathsf{ZFC}+$ "$U_z$ is a nonprincipal ultrafilter on $\omega$" (so in ...
Noah Schweber's user avatar
2 votes
1 answer
238 views

Need some clarification to understand an inequality involving exponential sums

I was looking into Montgomery's proof of the large sieve inequality in his book on Topics in Multiplicative Number Theory, and on page $18$, we have $$B(x)=\sum_{k=-\infty}^{\infty}b_ke(kx),$$ for $x\...
Anish Ray's user avatar
  • 309
3 votes
0 answers
112 views

Solving an underdetermined system of linear equations among the rationals, in the neighbourhood of an approximate solution

I have a system of linear equations $Ax=b$. Extremely underdetermined, for concreteness $x \in \mathbb{R}^{17,000}, b \in \mathbb{Z}^{156}$. $A$ is sparse, integer, full rank. I have a very precise ...
Dániel Varga's user avatar
5 votes
1 answer
889 views

Showing that decay results of Fourier coefficients are sharp

I originally post the question here but the it seems too advanced for undergraduate level. At least, an example is very hard to find, so I am putting it here. It is well-known that if $f$ is ...
Ma Joad's user avatar
  • 1,755
2 votes
1 answer
206 views

Conjecture about families of subsets of $\{1,\ldots,2n+1\}$ of size $n+1$

Let $\mathcal{A}$ be the family of all subsets of $U = [2n+1] = \{1,2,\ldots,2n,2n+1\}$ with size $n+1$, $n \ge 1$. The size of $\mathcal{A}$ is therefore $\binom{2n+1}{n+1}$. For any family $\mathcal{...
Fabius Wiesner's user avatar
7 votes
0 answers
185 views

Interest in the size of ultrapowers in model theory

It seems that in the 60s (at least), there was interest in computing the size of ultrapowers by countably incomplete ultrafilters. For example, given an ultrafilter $U$ on some relatively small set ...
Monroe Eskew's user avatar
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