1
vote
0answers
58 views

Contact structures on circle cross plane

Can anyone provide an explicit contactomorphism between the following two contact structures on the circle cross the plane? 1) The standard contact structure on threespace, but with the line that ...
2
votes
0answers
54 views

Notions of/References for freely generated (symmetric) monoidal categories

We often describe a category by giving a (directed, multi-)graph and freely generating a category of paths. I would like to know to what degree this intuition generalizes to monoidal categories, and ...
1
vote
0answers
47 views

$\Gamma$ cohomology of principal series

Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup with Langlands ...
1
vote
1answer
46 views

Nested convex optimization

Suppose I have a convex optimization problem of the form $$\min_x f(x) ~~s.t.\\x\in X$$. Say that $f(x)$ and its (sub)gradient are not given in a closed form, but are determined by solving a convex ...
4
votes
1answer
197 views

Motives of a variety of type D4

Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For ...
1
vote
1answer
69 views

Expectation of Gaussian random vector & arbitrary function thereof?

I saw in a paper (https://www.princeton.edu/~wbialek/rome/refs/bialek+ruyter_05.pdf Eq.37) the following identity: where the <.> operator refers to a population average. No source or ...
2
votes
1answer
155 views

Normal subgroup of a totally ordered group

A totally ordered group is a group equipped with a compatible total order, that is, $x\leq y$ and $z\leq t$ imply $x+z\leq y+t$ for all $x,y,z,t$ in the group. Is it true that every totally ordered ...
0
votes
1answer
89 views

Continuity of Intersection Multiplicities

I’m looking for a correct technical version (and in the best case a reference) for a statement of the following type: Consider a complex algebraic variety $X\subset\mathbb{P}^n$ and a sequence of ...
0
votes
0answers
21 views

Normalizing Entries In Defining Random Matrices (Wigner Matrix)

In the definition of Wigner Matrix (a certain type of random Matrices) we take to independent family of i.i.d zero mean distributions $\{Z_{i,j}\}_{1<i<j}$ and $\{Y_{i}\}_{1\leq i}$ and then the ...
0
votes
0answers
83 views

Dissolution of Tensors

I have a question that might seem odd to linear logic experts (I am somewhat of a novice). I know that two items of the same type can be combined into one premise with a tensor (multiplicative ...
0
votes
1answer
75 views

Unipotent orbit in adjoint group over finite field

[Editted: The assertion is wrong; see Jay's answer] My apology if this question is too simple. I am reading Deligne-Lusztig "Reductive groups over finite fields" and at the beginning of Chap. 4, ...
5
votes
1answer
95 views

Abelian varieties with good reduction everywhere over function fields

There is a famous theorem due to J.-M. Fontaine, Il n'y a pas de variété abélienne sur Z (and independently by V.A. Abrashkin) that there are no abelian varieties over Z. I was wondering whether ...
3
votes
1answer
91 views

Prescribed spherical representations, symplectic group $Sp(n)$

An irreducible representation $(\pi,V_\pi)$ of a compact group $G$ is called spherical with respect to the pair $(G,K)$, $K$ is closed subgroup of $G$, if $V_\pi$ has a non-zero vector invariant by ...
3
votes
1answer
284 views

Notation: Categories of measur(abl)e spaces

Is there a common notation in the literature for the category of measurable spaces and measurable maps? the category of measure spaces and measure-preserving maps? The nlab suggests ...
3
votes
2answers
250 views

Adeles and twisted adeles

Let $\mu_n$ denote the group of $n$-th roots of unity in ${\mathbb{C}}$, i.e., $\mu_n=\ker[{\mathbb{C}}^*\overset{n}{\longrightarrow}{\mathbb{C}}^*]$. We set $$ \mu=\varinjlim_n \mu_n\subset ...
0
votes
0answers
64 views

Lefschetz hyperplane theorem for Neron-Severi

Suppose that $X$ is a smooth projective variety of dimension at least $3$, and that $D$ is a smooth ample divisor. I am wondering to about the status of the Lefschetz hyperplane theorem for the map ...
0
votes
0answers
13 views

Probability distribution of the distances between N mobile nodes in a square plain of length l [on hold]

For N randomly moving nodes enclosed in a square plain of length l. The (Nchoose2)W samples of the distances between the nodes are collected over a window of length W. We can assume W is large, what ...
2
votes
1answer
26 views

Analytical value for the first eigenvalue of a certain spherical triangle

I am testing some numerical algorithms for computing the Laplace-Beltrami eigenvalues on the sphere. One thing that came up was computing the first eigenvalue of the "equilateral" spherical triangle ...
-1
votes
0answers
54 views

Is countably complete lattice bounded? [on hold]

I wonder if countably complete lattice is bounded and, if it is why ?
3
votes
1answer
90 views

Proof for additivity of cumulants

If one does not define cumulants via the cumulant generating function (cgf), e.g. because the cgf does not exist, then an alternative way is to use the recusion \begin{align*} ...
-6
votes
0answers
38 views

Help with derivative rules [on hold]

Hi I need som help with some rules Y to Y'. 1: Y = A(^2)/B 2: Y = E(^2)/1 3: Y = E^(X+X) 4: Y = sin(X(^3))/cos(X(^2)) 5: Y = -X(^B)cos(B(^X)) What is the Y' of all functions?
1
vote
1answer
143 views

On Q-Cartier Divisors

I have my question on Q-Cartier Weil divisor. People say $D$ is Q-Cartier divisor if $nD$ is Cartier for some $n \geq 1$. Especially for $n > 1$, I have never seen the `rigorous' definition of ...
1
vote
1answer
282 views

Numbers $n$ such that the sum of the divisors of $n$ is a nontrivial power

Let $\sigma (n)$ be the sum-of-divisors function. For example, $\sigma(7)=1+7=2^3$. I know some results about triplets of positive integers $(n,a,b)$ where $a,b\ge 2$ such that $\sigma (n)=a^b$, but ...
0
votes
0answers
46 views

What can we say about variational energies here?

Let $V_{ij}^{lk}$ be any $nm \times nm$ real symmetric matrix, $\forall i,j,k,l$ \begin{equation} V_{ij}^{kl}=V_{ji}^{lk} \end{equation} (So for the indices we have $1 \leq k,l \leq m$ and $1 \leq i,j ...
-1
votes
0answers
42 views

The product of the power and the natural number in the short interval [on hold]

It is obvious that if $a,b,x\in\mathbb{N}$ and $a^n\leq 2x+1$ then there exists $b\in\mathbb{N}$ such that $a^nb\in\left[x^2,(x+1)^2\right]$. For example for $n=3$, $a=2$ and $x=4$ we have $b=2$ and ...
4
votes
3answers
296 views

Automatically generate BibTeX item from arxiv [on hold]

I'm looking for a tool which generates a BibTeX item for a given arxiv id. I only found http://www.crcg.de/arXivToBibTeX/ using Google but this tool always tells me that the arxiv ids I enter don't ...
-2
votes
0answers
38 views

periodic function satisfy the condition f’’(x)f(x)>0 at -inf < x < +inf? [on hold]

Can a periodic function f(x) satisfy the condition f’’(x)f(x)>0 at -inf < x < +inf?
11
votes
2answers
598 views

Which way for reading the proofs?

I am a master student in mathematics. For me a large part of doing mathematics is thinking about, reading and verifying the proof of theorems that I find them in my field of study. I can do this ...
4
votes
0answers
71 views

Interesting triple integral

Some time ago I stumbled on an alleged identity $$\int\limits_0^\infty \frac{dx}{x} \int\limits_0^x \frac{dy}{y} \int\limits_0^y \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]= ...
5
votes
1answer
180 views

Strange problem about triplets of differential forms

Suppose we have the following map: $$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$ ...
22
votes
4answers
743 views

Connectedness in the language of path-connectedness

Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds? A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a ...
3
votes
3answers
164 views

how to find explicitly given component in a regular representation

Given a finite group $G$ and its irreducible representation $\pi$ I want to find explicit elements of the group algebra $\mathbb{C}[G]$ lying in components of the left regular representation ...
1
vote
0answers
199 views

An example of threefold

Its description is a little bit complicated but it would be great if anyone can give an example. I tried to construct it as a toric variety (See the previous question) but did not succeed. I am ...
4
votes
3answers
320 views

reflexive banach space

I want to ask this non-expert question: What does it mean geometrically for a Banach space to be reflexive? Well, we could say a Banach space is reflexive iff unit ball is weakly compact. Or some ...
-1
votes
0answers
29 views

Global existence of power-type nonlinear Schrodinger equations on compact manifolds [on hold]

Consider the nonlinear Schrodinger equation $$i\partial_t u + \Delta u = K|u|^ru$$ on a compact manifold, may be with boundary (with Dirichlet boundary conditions). It is known that on $\mathbb{R}^n$, ...
27
votes
8answers
4k views

Uninteresting questions with interesting answers [on hold]

What are best examples of questions in mathematics that are not interesting until one knows the answers, whose answers themselves are what is interesting? The thing that prompts me to post this is ...
3
votes
2answers
118 views

distributional Hessian for semiconvex functions on non-smooth manifolds

Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that $$ \int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ...
1
vote
0answers
46 views

Subclass of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ...
4
votes
0answers
170 views

Does this inequality always hold?

Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...
1
vote
0answers
13 views

Is an open map with open relative diagonal necessarily a local homeomorphism?

Let $f : X \to Y$ be an open (and continuous) map of locales. Suppose the relative diagonal $\Delta_f : X \to X \times_Y X$ is an open embedding of locales. Does it follow that $f : X \to Y$ is a ...
-4
votes
0answers
35 views

Solving a tough a PDE shifting data [closed]

How would I solve this one: $u_t-\nabla^2u = f(r,\theta, t) \quad r<a, t>0$ $u(r,\theta, 0)=\phi(r,\theta) \quad r<a$ $u=h(\theta) \quad r=a$ So I guess I need to make the BC's ...
5
votes
1answer
164 views

Evaluation maps for moduli of stable maps

Let $\overline{M}_{0,n}(\mathbb{P}^N,d)$ be the moduli space of stable maps of degree $d$ from curves of genus zero with $n$-marked points to $\mathbb{P}^N$. Consider the product of the evaluation ...
1
vote
1answer
92 views

Is there any simpler form of this function

Assume that $n$ is a positive integer. Is there any simple form of this hypergeometric value $$_2\mathrm{F}_1\left[\frac{1}{2},1,\frac{3+n}{2},-1\right]?$$
-3
votes
0answers
45 views

On ranks of matrix products [on hold]

Tensor product of two matrices increases simultaneously sizes of product matrix, size of rank multiplicatively. Is there a function on two matrices which increases size multiplicatively while rank ...
2
votes
0answers
54 views

completion of non-finitely generated ideal

Let consider $A=k[x_{1},x_{2}...]$, the polynomial ring with countably many indeterminates. Then we can consider the completion ...
0
votes
0answers
99 views

Solving the transcendental equation $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$

I need to solve the following equation: $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$ for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $Li_{3}$ and $Li_{2}$ are the third and ...
-1
votes
0answers
51 views

Clarification on notation of “left invariant fields” (Lie groups) [migrated]

In these notes in Definition 1.4 we learn that A vector field $X$ on a Lie group $G$ is called left invariant if $d(L_g)_h(X(h))=X(g(h))$ for all $g,h \in G$, or for short $(L_g)_*(X)=X$. where ...
6
votes
1answer
151 views

Isometries of some simple Cayley graphs

Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$. There are some obvious candidates to isometries of this graph - for example, translation by elements of ...
0
votes
0answers
11 views

Boundary Condition for LevelSet Reinitialization

Recently, I'm curious about the boundary condition of Levelset Reinitialization. Generally, when we try to express the interface, we use the levelset advection $$\phi_t + (V\cdot\nabla)\phi = 0$$ But ...
-1
votes
1answer
106 views

How compute combinatorial expression [closed]

How compute $\sum_{j=1}^k \binom{x}{j}\binom{k-1}{j-1}\alpha^j, \quad x, \alpha\in\mathbb{R}$

15 30 50 per page