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

**4**

votes

**1**answer

207 views

### Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property

ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ ...

**27**

votes

**4**answers

900 views

### Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?

A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{22} & a_{23} & a_{24} \\
a_{13} & a_{23} & a_{33} & ...

**2**

votes

**1**answer

146 views

### Elliptic regularity of second order pseudos

Let $M$ be a compact manifold. Choose a point $q \in M$. Let $P$ be a second order positive self-adjoint pseudodifferential operator such that $\text{Spec}(P) \subset (0, \infty)$. Also, we know that ...

**0**

votes

**1**answer

48 views

### Do the support sets of subspaces give the representable matroids?

Fact: Start with $V$ a subspace of $\mathbb R^n$. Take the set of all supports of vectors in $V$. Throw out $\emptyset$. You now have the dependent sets of some matroid.
Not sure you ...

**10**

votes

**1**answer

347 views

### Origin of group theory problem (bound on number of Sylow subgroups)

This problem (prove that the number of Sylow subgroups of a finite group $G$ is bounded by $\frac{2}{3}|G|$) posted on MSE proved rather difficult to solve. The OP has been silent about where the ...

**4**

votes

**0**answers

46 views

### Structure of Lagrangian algebras in the center of a Fusion algebra

(1) Let $\mathcal F$ be a spherical fusion tensor category. Then Müger showed that
$R=\bigoplus_{H\in\mathrm{Irr}(\mathcal F)} H\boxtimes H^\mathrm{op}$ canonically has the structure of a Frobenius ...

**6**

votes

**1**answer

92 views

### Reference for a PL flat torus embedding in $\mathbb{R}^3$

A piecewise linear flat torus embedded in $\mathbb{R}^3$ is shown at http://www.mathcurve.com/polyedres/toreplat/toreplat.shtml.
It is flat in the sense that the angle defect at the vertices is zero.
...

**1**

vote

**1**answer

88 views

### Conditions for the consistency of a system of affine polynomials

Let $f_1, f_2,\ldots,f_N$ be some affine polynomials. We consider the question if these polynomials have a common (affine) root. By homogenizing these polynomials, we can associate a projective ...

**4**

votes

**3**answers

618 views

### Looking for techniques of How to measure the Similarity/Dissimilarity between two images?

I would like to compute the similarity/dissimilarity between two images L and R.
I know one way which is : computing the histogram of blocks of each image, and then using Bhattacharyya measure I ...

**0**

votes

**1**answer

91 views

### Graph automorphisms that preserve independent sets [closed]

Let $G=(V,E)$ be a graph and $\mathrm{Ind}(G)$ be the collection of its independent sets.
We call a graph automorphism $f:V \to V$ of $G$ good if it is non-trivial and ...

**3**

votes

**1**answer

147 views

### Generalisation of “tangent space” to not-necessarily connected sets

I vaguely recall having read somewhere a definition similar to (but probably not exactly the same as) the following.
Definition (Blob) Let $S\subset \mathbb{R}^n$ be a set, and $p \in S$. The ...

**1**

vote

**0**answers

89 views

### Vector valued Sobolev spaces

My question is in reference to this question previously asked here. As asked there, consider a function $f \in H^s_x(L^2_y) \cap L^2_x(H^s_y)$. In the notation of Lions and Magenes (Chapter 4, Vol 2), ...

**4**

votes

**0**answers

175 views

### Primitive Closure Arithmetic

I am researching a system that was designed by myself and what I call PCA (Primitive Closure Arithmetic), because it looks like PRA.
The differences are:
- PRA uses recursive definition with a ...

**6**

votes

**1**answer

125 views

### Homology class of a Lagrangian Klein bottle

Nemirovskii's 2008 paper, by the same title in this question, claims that any Lagrangian Klein bottle in a closed symplectic 4-manifold $M$ must realize a nontrivial homology class in $H_2(M; \mathbb ...

**0**

votes

**0**answers

70 views

### Finding a set of generators of an ideal with certain property in $k[x_1, …, x_n]$

I am interested in the following problem, and I would appreciate any comments, inputs, answers, references! Let $k$ be a field. For each $1 \leq j \leq n$, let
$$
I_j = (x_j, u_{2}^{(j)}, ..., ...

**1**

vote

**0**answers

93 views

### cohomlogy of Diagonal ring

Let $S=\bigoplus_{\underline n\in\mathbb N^r } S_{\underline n}$ be a standard multigraded ring over a local ring and M be a finitely generated $\mathbb N^r $-graded $S$-module. Let ...

**18**

votes

**1**answer

950 views

### Anti-Mandelbrot set

I clearly remember seeing a paper where the dynamic of the anti-conformal map
$f(z)=\overline{z}^2+c$ was studied (the bar means complex conjugation). There was a picture of the analog of the ...

**14**

votes

**1**answer

254 views

### Geometric Mean of $L(1,\chi)$ for quadratic Dirichlet characters

Let $S = \{D_1, D_2, D_3, \ldots \}$ be the set of all prime discriminants
(or positive prime discriminants) of quadratic number fields. For such a
discriminant let $\chi_j(n) = (\frac{D_j}n)$ be ...

**1**

vote

**0**answers

107 views

### Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function:
$$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$
or to know where I can find some techniques to ...

**13**

votes

**0**answers

567 views

### Derived functors - homotopical vs homological approach

This question is a crosspost of the second part of this MSE question.
In my first course in homological algebra, derived functors were defined in terms of universal $\delta$-functors. In the text ...

**2**

votes

**0**answers

104 views

### Nonnegative integers represented by $\prod_{i=1}^m \sum_{j=1}^n a_{i,j} x_j $, where the $a_{i,j}$ are positive integers

Fix $m, n \in \mathbf N^+$ with $m+n \ge 3$, and let $A = (a_{i,j})_{1 \le i \le m, 1 \le j \le n}$ be an $m$-by-$n$ matrix of positive integers. What is known about the asymptotic behavior of the ...

**1**

vote

**2**answers

166 views

### What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?

Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the ...

**47**

votes

**3**answers

4k views

### Is this differential identity known?

Recently I discovered the differential identity
$$ \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}$$
valid for any odd natural number $k$; for ...

**28**

votes

**5**answers

1k views

### The unpublished papers in reference to the published papers

Sometimes it happens that a published paper refers to an unpublished paper for a result used.
In this case, if we want to check this result by ourselves, we need to access to this unpublished paper.
...

**1**

vote

**0**answers

62 views

### The dimension of the space of automorphic forms with multiplier system

Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{Z})$ and $\vartheta$ a multiplier system of weight $k$ for $\Gamma$, by which we mean a function $\vartheta:\Gamma \rightarrow \mathbb{C}$ ...

**1**

vote

**1**answer

91 views

### Norm equivalent to Sobolev norm? [closed]

On the hyperbolic space $\mathbb{H}^n$, it is known that the spectrum of the Laplacian satisfies $\text{Spec}(-\Delta) \subset [\frac{(n - 1)^2}{4}, \infty)$. Consider the operator $P = -\Delta + a$, ...

**0**

votes

**0**answers

107 views

### What is the symplectic manifold whose Delzant polytope is a trapezoid?

What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ...

**3**

votes

**0**answers

101 views

### How can one “extend scalars” for (motivic) ring spectra and for modules over it?

Let $S$ be a (motivic symmetric) ring spectrum (more generally, one can possibly consider a ring object in a symmetric stable model category); let $R$ be a flat associative commutative unital algebra ...

**6**

votes

**1**answer

242 views

### Ordinals in constructive mathematics ? (references)

I'm looking for references presenting a constructive treatment of the theory of ordinals. By constructive I mean valid in the internal logic of a topos (so no axiom of choice and no law of excluded ...

**2**

votes

**0**answers

93 views

### relations in (\mathbb P^1)^n

What is a minimal set of relations of the image of $(\mathbb P^1)^n$ in the Segre embedding? For $n=2$ its just the determinant $x_1x_2.y_1y_2=x_1y_2.x_2y_1$. Is it written explicitly somewhere?

**1**

vote

**1**answer

75 views

### Rank of Cartesian product of well-partial-orders

We are interested in the ordinal rank $h(P)$ of a wpo (a well-partial-order). It is known that it coincides with the order type of its longest chain.
When we consider the cartesian product $P\times ...

**6**

votes

**1**answer

306 views

### Introductory article of knot Heegaard Floer Homology

I am looking for some article that gives an introduction to Heegaard Floer homology of knot.
I heard that it is very useful to determine the unknotting number of a knot, but I couldn't find any ...

**3**

votes

**2**answers

226 views

### A galois group is topologically finitely generated or finitely generated as a $\mathbb{Z}_p$-module

The basic set up is the following:
Let $K$ be a number field. let $p$ be an odd prime. Let $\Sigma_p$ be the set of primes of $K$ lying above $p$. Let $M$ be the composite of all finite $p$-extensions ...

**21**

votes

**7**answers

1k views

### What's that shape? Inferring a 3D shape from random shadows

Let $P$ be a bounded, simply connected region of $\mathbb{R}^3$.
$P$ could be a polyhedron, or a smooth shape, or an arbitrary shape;
I'll assume below that $P$ is a (non-degenerate, perhaps ...

**3**

votes

**2**answers

337 views

### Who needs a symmetric upper asymptotic density on the integers?

The upper asymptotic density on $\mathbf Z$, viz. the function
$$
{\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n},
$$
has a ''symmetric ...

**8**

votes

**2**answers

304 views

### Reference for an unbiased definition of a symmetric monoidal category

In the references I know, a symmetric monoidal category is defined as a monoidal category equipped with some additional data, so in particular the monoidal product is a functor
$$
\mathcal{C}\times ...

**7**

votes

**1**answer

177 views

### Poincaré inequality for curl-integrable functions

Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and
$$
u_B := \frac1{|B|}\int_B u \, dx.
$$
The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...

**1**

vote

**0**answers

41 views

### Symmetry conjecture for minimal dilatation pseudo Anosov mapping classes

The conjecture is something like the following:
The minimal dilatation among pseudo-Anosov mapping classes on a surface $S_{g,n}$ is realized by $\rho\circ\omega$ where $\omega$ is supported on a ...

**2**

votes

**1**answer

81 views

### References about spectral theory on Hyperbolic space

Can anyone suggest me some books or papers that include details about spectral theory on Hyperbolic spaces or related topics such as scattering theory on Hyperbolic spaces?
After some googling, I ...

**7**

votes

**1**answer

262 views

### Reference request, zeta function is rational function via Riemann-Roch?

I am looking for a reference to a proof that the zeta function of a function field in one variable over a finite field $\mathbb{F}_q$ is a rational function in $q^{-s}$ by using the Riemann-Roch ...

**1**

vote

**0**answers

51 views

### Multivariable polynomial interpolation via evaluations from entrywise powers of a point

I am interested in multivariate polynomial interpolation. Within computational complexity theory, I use it to create efficient reductions between counting problems. In the univariate case, there is ...

**2**

votes

**0**answers

63 views

### Generalized Hurwitz Spaces

In this question all the varieties are over $\mathbb{C}$. Classic Hurwitz spaces $\mathcal{H}_{g,r}$ are moduli spaces of simple branched coverings $f \colon X \to \mathbb{P}^1$ of degree $d$, where ...

**20**

votes

**2**answers

668 views

### Are symplectic methods used in (classical) Economics?

The tl;dr question is this: are economists using coordinate-free formulations in studying theory?
Borrowing from classical mechanics, the framework I have in mind for classical economics--involving ...

**3**

votes

**0**answers

148 views

### Where can I find a proof of this result on optimal tessellation of a unit square?

Here is an excerpt from the paper "The Hexagon Theorem" by Donald J.Newman
Does anyone know where I can find a proof of the underlined statement?
Newman states it without a proof, and I could get ...

**2**

votes

**2**answers

276 views

### Suggestions for dealing with the “timed” balls-into-bins model

Definitions: Let $T$ (for "time") be a random variable $T \sim \text{Exp}(\lambda)$ and $\Delta t$ is a realization (or called an observed value) of $T$. Let $D$ (for "delay") be a random variable $D ...

**4**

votes

**0**answers

111 views

### $H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...

**1**

vote

**0**answers

41 views

### Perturbation of a Fredholm sections which preserves compactness of 0-set

I am learning Morse-Bott-Floer theory and found the following cool paper
http://de.arxiv.org/abs/1310.5080
by P. Albers and D Hein. In order to prove a cup-length estimate on the number of critical ...

**5**

votes

**1**answer

117 views

### Example of a $G$-sphere that is not a $G$-representation sphere

Let $G$ be a finite group with the discrete topology. To set terminology:
a $G$-sphere is a sphere equipped with a continuous $G$-action
a $G$-representation sphere is a $G$-sphere obtained from an ...

**4**

votes

**1**answer

129 views

### Loss of derivative of subelliptic operator

Consider the differential operator $P$ on $\mathbb{R}^2$, given by $P = \frac{\partial^2}{\partial x^2} + x^2\frac{\partial^2}{\partial y^2}$. Clearly it is elliptic everywhere except on the $y$-axis. ...

**8**

votes

**3**answers

706 views

### Catalan numbers as sums of squares of numbers in the rows of the Catalan triangle - is there a combinatorial explanation?

This question arose from an answer to my recent question How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, ... algebras?
What I need from that ...