# Tagged Questions

This tag is used if a reference is needed in a paper or textbook on a specific result.

**2**

votes

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29 views

### When is a linear operator on $C^{0,\alpha}(\overline{\Omega})$ a multiplication?

The title says it all, really. Suppose that I have a linear operator $T$ from $C^{0,\alpha}(\bar{\Omega})$ into itself, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^d$ (e.g. the unit ball ...

**3**

votes

**0**answers

59 views

### On the principal eigenvector of an elliptic operator

Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$:
\...

**0**

votes

**0**answers

33 views

### Generalizing Integration by parts for general bounded continous measure

Consider a probability measure $d\mu = w(t) dt$ with $w(t)\in L^1(I)$, $I =\left[ 0,1\right]$. What are the minimal assumption I can take on two functions $f,g:I\ \to \mathbb{R}$ so that an ...

**3**

votes

**0**answers

89 views

### Universal enveloping algebra functor preserves quasi-isomorphism

Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}$ denote the category of DG algebras and $\mathtt{DGLA}_{k}$ denote the category of DG Lie algebras. It is well known that there are model ...

**1**

vote

**1**answer

71 views

### Derivations of polynomial identity rings

Does anyone have any references to general theory of derivations of PI rings? I have had a quick look around without much luck.

**4**

votes

**1**answer

88 views

### Does the Nash inequality hold on manifolds with Lipschitz boundary?

Let $N$ be a smooth manifold without boundary of dimension $n$. $M$ is a manifold with Lipschitz boundary if $M \subset N$, $M$ and $N$ are of the same dimension, and in the charts of $N$, the ...

**6**

votes

**4**answers

511 views

### Minimum negative eigenvalue of zero-one matrices

The following question must have been answered decades ago.
For $n$ fixed, what is the most negative eigenvalue among all trace zero zero-one matrices (that is, all entries are either zero or one, ...

**7**

votes

**1**answer

188 views

### When is a mapping the proximity operator of some convex function?

Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ?
That is, given $p : ...

**1**

vote

**2**answers

261 views

### Maximal subgroups of $\mathrm{SL}(n,\mathbb{R})$

I would like to find a list (or at least a description) of the maximal closed connected subgroups of $\mathrm{SL}(n, \mathbb{R})$ , and also of $\mathrm{SU}(p,q)$.
In the following MO discussion is ...

**1**

vote

**2**answers

311 views

### Expository articles on Algebraic Number Theory

I am about to start learning Algebraic Number Theory and thus was looking for some expository articles on this subject. So far I have found two such articles:
Dickson, L. E.. (1917). Fermat's Last ...

**7**

votes

**1**answer

226 views

### Thorough reference on regular homotopy

I would like to learn this topic of algebraic topology but I cannot find a relevant reference to answer my basic questions on the subject (for example, is there a Hurewicz theorem for regular homotopy ...

**1**

vote

**0**answers

47 views

### Dual equivalence for multioperators

This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...

**2**

votes

**1**answer

161 views

### Regarding a conjecture Fogarty proposed

In a paper by Fogarty titled "Algebraic Family On An Algebraic Surface,"
he conjectured that $\bf Hilb^n(\mathbb P^N)$ is always variety-- reduced and irreducible.
Is this still a conjecture; any ...

**3**

votes

**0**answers

75 views

### Donnelly-Fefferman growth of eigenfunctions

Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...

**1**

vote

**0**answers

43 views

### The integral of $\Gamma\left(\zeta\right) \, W_{-\zeta,\mu}(z) $

Someone has a reference that addresses an integral of the followns type
$$I_{a,b,x} = \int_{0}^{+\infty} \zeta^{-a} \, \Gamma\left(\zeta+b\right) \, W_{-\zeta-b,\tfrac{-1}{2}}(x) \, d\zeta$$
where ...

**2**

votes

**2**answers

230 views

### Attaching an ideal whose square is zero: does this operation have a name and a notation?

I know I met the following construction somewhere, but I cannot remember where. Let $A$ be
a (unital associative) ring, and let $N$ be an $A$-$A$ bimodule. On the product set $A\times N$ we define ...

**4**

votes

**0**answers

121 views

### Categories where every Mono Splits

When every epi splits a category is said to satisfy the Axiom of Choice.
When every idempotent splits a category is called Cauchy Complete or Idempotent complete.
These look to be well-studied ...

**2**

votes

**1**answer

137 views

### Want more details about the image of a Maass form in the AIM press release concerning LMFDB

Actually I came upon this through MO a couple of days ago: in here
(http://aimath.org/aimnews/lmfdb/) there is a mesmerizing image
The caption reads
A Maass form, one of the 20 different types ...

**5**

votes

**0**answers

158 views

### Where can I find Andre's “Cinq exposés sur la désingularisation”?

Many expositions of Popescu's desingularization theorem indicate that an other proof of this theorem can be found in
"Cinq exposés sur la désingularisation" by M. Andre, Ecole Polytechnique ...

**3**

votes

**2**answers

128 views

### base point in Heegaard Floer homology

It is stated in Mcduff's overview paper "FLOER THEORY AND LOW DIMENSIONAL TOPOLOGY", end of page 9, that the Heegaard Floer homology without considering a base point depends only on the homology of ...

**7**

votes

**2**answers

257 views

### Surfaces contained in a ball

In this Paper there is a proof that a closed plane curve of length
$L$ and curvature bounded by $K$ can be contained inside a circle of radius
$L/4 - (\pi - 2)/2K$. Are there similar results for ...

**15**

votes

**1**answer

332 views

### Combinatorics problem about sum of natural numbers

Following combinatorics problem is claimed to be an open problem in "The Princeton Companion to Mathematics" (pp. 6)
Let $a_1,a_2,a_3,...$ be a sequence of positive integers, and suppose that each ...

**0**

votes

**0**answers

98 views

### What to do about Secondary Results that are of Independent Interest?

I am thinking about writing an online article about a simplified algorithm for a problem $P$, for which efficient solutions are already known, whose implementation is however challenging.
My ...

**3**

votes

**0**answers

120 views

### Reference for the Hodge polynomial or the Hodge Characteristic

What is the first work that studies, refers to, or mentions the Hodge characteristic?
The Hodge polynomial is the unique ring homomorphism
$$
P_{hdg}:K_0(\mathbf{Var}/\mathbb{C)}\to \mathbb{Z}[u,v,u^{...

**4**

votes

**1**answer

92 views

### Pfaffian of several skew-linear transformations / matrices

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...

**15**

votes

**1**answer

306 views

### Classification of fake (quaternionic, octonionic) projective spaces

If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a ...

**10**

votes

**2**answers

245 views

### Finite objects for which isomorphism is NP-hard or harder?

Are there finite objects for which deciding isomorphism
is NP-hard or harder?
Graphs and groups are not solutions.
Searching the web didn't return answer for me.
Partial result based on Chow's ...

**1**

vote

**1**answer

183 views

### Deformation Quantization

I am a beginner and I want to learn about deformation quantization. Please suggest me with which book or notes, I should start?

**3**

votes

**1**answer

134 views

### Bounded solutions for Schrödinger equation at the edge of the essential spectrum

Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...

**1**

vote

**0**answers

56 views

### History of Cauchy-Euler Equations

As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of -
$\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall k=0,\...

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vote

**0**answers

48 views

### Reference: Heat Kernel for Siegel Upper Half plane

Is there a ready reference for explicit computation of the heat kernel for Siegel upper half space $\mathbb{H}_n=\{Z=X+iY\in \mathrm{Mat}_n(\mathbb{C}) \vert Y>0\} $? I could find it for general ...

**6**

votes

**1**answer

230 views

### coefficient-wise powers of matrices. Reference wanted

Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...

**0**

votes

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164 views

### Matrix with roots of unity entries

For given prime p, i am interested in the norms of $p \times p$ matrices which have roots of unity entries, $M_{k,l} ∈ \{1,ζ,…,ζ^{p−1}\}$ with $ζ=\exp(2πI/p)$. Are there any studies of the norms of ...

**5**

votes

**0**answers

150 views

### When a proper smooth fibration is isotrivial?

Let $\pi : X \to Y$ be a proper smooth morphism between complex quasi-projective varieties. Assume there exists a connected and Zariski dense analytic subvariety $Z$ of $Y$ such that for any two ...

**4**

votes

**1**answer

171 views

### (Etale) fundamental group of quotient singularity $\mathbb{C}^n/G$

I don't know much about (algebraic/etale) fundamental groups, so sorry if this question sounds stupid. I am interested in quotient singularities (quotients $X$ of $\mathbb{C}^n$ by a finite subgroup $...

**3**

votes

**0**answers

52 views

### Reference for classification of positive involutions

An involution on a finite dimensional algebra $A$ over $\mathbb{Q}$ is an involutive anti-automorphism of $A$. If $\sigma$ is an involution on $A$, we say that $\sigma$ is positive if $\mathrm{Tr}_{A/\...

**0**

votes

**0**answers

165 views

### SDP based heuristics for graph coloring

This is a question about semidefinite programming heuristics for graph vertex coloring based on the Lovasz theta number such as "Approximate Graph Coloring" by Karger, Motwani and Sudan or "A ...

**7**

votes

**4**answers

471 views

### General Relativity and Differential Geometry intuitions of Second Bianchi Identity

In General Relativity, one uses the Riemann Tensor in its coordinate form $R_{abcd}$, and proves the Second Bianchi Identity-
$R_{abcd;e} + R_{abde;c} + R_{abec;d} = 0$
It is said that ...

**0**

votes

**0**answers

42 views

### Largest instance of highly nonlinear benchmark functions (e.g. Rastrigin function)

What is the largest instance size (number of variables) ever numerically solved for highly nonlinear (continuous, not combinatorial) optimization benchmarks functions, such as Rastrigin, Schwefel or ...

**7**

votes

**1**answer

150 views

### $C^{k,\alpha}$ diffeomorphisms and vector fields

This feels like something I should know, but I have a hard time finding a definite reference.
Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a $C^k$...

**3**

votes

**2**answers

287 views

### A possible norm on a subspace of $C^\infty([0,1])$?

I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.
Take the vector space of infinitely ...

**5**

votes

**3**answers

345 views

### Elementary reference for the isometry group of $\mathbb{RP}^2$

Endow the real projective plane with the distance defined by $d(L,L')$ := "the angle between the lines $L$ and $L'$ ".
It is the case that every isometry from $RP^2$ onto $RP^2$ is induced by an ...

**1**

vote

**0**answers

97 views

### Applications of Iss'sa's theorem on homomorphisms between algebras of meromorphc functions

In Remmert's book Funktionentheorie II, the following theorem, apparently due to Hironaka under the pseudonym Iss'sa, is proved:
Let $U,V \subseteq \mathbb{C}$ be open subsets. Every $\mathbb{...

**1**

vote

**1**answer

86 views

### Weighted counting of circular codes

Given a circular code $X$ (for example: $X=\{ w,b \}$) with generating function $u(z)=\sum\limits_{k=0}^{\infty}{u_k z^k}$ (in this example : $u(z)=2z$), the generating function $p(z)=\sum\limits_{k=0}...

**4**

votes

**2**answers

713 views

### Divergent Series as a topic of research

About a year ago, while studying real analysis, I got very much interested in divergent series. I discussed possible research topics related to divergent series with my teachers but couldn't find any. ...

**2**

votes

**0**answers

115 views

### $f(x)$-th largest number of prime factors

Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...

**1**

vote

**0**answers

60 views

### A question on uniformly corepresented functor

Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly ...

**0**

votes

**0**answers

50 views

### Convexity condition of matrices

In studying the viscoelastic theory of elastodynamics, I encounter a problem on the convexity condition of matrix functions. It has been known that for the energy function $E=E(v,F) = \frac{1}{2} v^2 +...

**1**

vote

**0**answers

135 views

### On the cardinality of the set of right-truncatable primes

We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime:
\begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ a_{n}a_{n-1}...

**4**

votes

**1**answer

80 views

### Groebner bases for differential operators with field coefficients (reference request)

Let $K$ be a field, $\partial_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial_1\ldots \partial_n]$ (it is implicitly assumed that the derivations do not commute with elements ...