# All Questions

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### 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 ...
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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 ...
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### $\Gamma$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Strange problem about triplets of differential forms

Suppose we have the following map: $$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb{R}^n$, ...
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### 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 ...
Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that $$\int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} ... 0answers 46 views ### Subclass of semimartingales for which all characteristics can be estimated? I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ... 0answers 170 views ### Does this inequality always hold? Denote the adjacency matrix of a given undirected graph by g. It is an n-by-n symmetric Boolean matrix with elements on the diagonal to be zero (n\geq 3). Let g_{12}=g_{21}=g_{13}=g_{31}=1 ... 0answers 13 views ### Is an open map with open relative diagonal necessarily a local homeomorphism? Let f : X \to Y be an open (and continuous) map of locales. Suppose the relative diagonal \Delta_f : X \to X \times_Y X is an open embedding of locales. Does it follow that f : X \to Y is a ... 0answers 35 views ### Solving a tough a PDE shifting data [closed] How would I solve this one: u_t-\nabla^2u = f(r,\theta, t) \quad r<a, t>0 u(r,\theta, 0)=\phi(r,\theta) \quad r<a u=h(\theta) \quad r=a So I guess I need to make the BC's ... 1answer 164 views ### Evaluation maps for moduli of stable maps Let \overline{M}_{0,n}(\mathbb{P}^N,d) be the moduli space of stable maps of degree d from curves of genus zero with n-marked points to \mathbb{P}^N. Consider the product of the evaluation ... 1answer 92 views ### Is there any simpler form of this function Assume that n is a positive integer. Is there any simple form of this hypergeometric value$$_2\mathrm{F}_1\left[\frac{1}{2},1,\frac{3+n}{2},-1\right]?$$0answers 45 views ### On ranks of matrix products [on hold] Tensor product of two matrices increases simultaneously sizes of product matrix, size of rank multiplicatively. Is there a function on two matrices which increases size multiplicatively while rank ... 0answers 54 views ### completion of non-finitely generated ideal Let consider A=k[x_{1},x_{2}...], the polynomial ring with countably many indeterminates. Then we can consider the completion ... 0answers 99 views ### Solving the transcendental equation Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3 I need to solve the following equation: Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3 for x\in\mathbb{R}^{\ast} and where k\in\mathbb{R}^{+}. Here Li_{3} and Li_{2} are the third and ... 0answers 51 views ### Clarification on notation of “left invariant fields” (Lie groups) [migrated] In these notes in Definition 1.4 we learn that A vector field X on a Lie group G is called left invariant if d(L_g)_h(X(h))=X(g(h)) for all g,h \in G, or for short (L_g)_*(X)=X. where ... 1answer 151 views ### Isometries of some simple Cayley graphs Consider a Cayley graph of a group G with respect to a symmetric finite generating set S. There are some obvious candidates to isometries of this graph - for example, translation by elements of ... 0answers 11 views ### Boundary Condition for LevelSet Reinitialization Recently, I'm curious about the boundary condition of Levelset Reinitialization. Generally, when we try to express the interface, we use the levelset advection$$\phi_t + (V\cdot\nabla)\phi = 0 But ...
How compute $\sum_{j=1}^k \binom{x}{j}\binom{k-1}{j-1}\alpha^j, \quad x, \alpha\in\mathbb{R}$