# All Questions

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### Given a set of Y Points (locations) with Longitude and Latitude, what is the shortest path that crosses a set of X points? [on hold]

Is there a good algorithm for this? I don't even know where to begin! I don't need an exact answer (as the running time on that would probably be polynomial, which is unacceptable), a guaranteed ...
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### proof of Goldbach's conjecture [on hold]

Let's say I have solved the Goldbach's conjecture, where should I submit it? is there a website? Or someone organization I can talk to?
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### Gysin sequence for $\mathbb S^3$ bundle

Let the group $G=\mathbb S^3$ act semi freely on a paracompact space $X$. Then exercise 12 of G.E. Breadon's book , Introduction to compact transformation groups pg 169 asks to derive the following ...
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### Maximizing an integral over a convex region

Let $C$ denote a compact, convex region in the plane containing the origin with unit area, and let $f$ be a probability distribution on $C$. Let $f^\ast$ denote the distribution that maximizes the ...
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### Proof Help: Getting Started [on hold]

I am having some trouble starting off this proof. I am not sure if I need to prove by the contrapositive or if it is a direct proof. Prove: If $x, y,$ and $z$ are natural numbers such that ...
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### Is a group scheme determined by its category of representations?

More precisely, let $G$ be an affine group scheme over a field $k$, $Rep_k(G)$ be the category of finite dimensional representations of G, and $\omega_0$ be the forgetful functor from $Rep_k(G)$ to ...
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### Polyhedra with minimal edge length

Given a fixed volume and fixed surface area I would like to construct polyhedra that minimize the total length of the edges. This seems like a straight-forward problem to solve by brute force for ...
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### On numerical approximation to stationary distribution of diffusion process

Suppose a vector-valued diffusion process X satisfies the stochastic differential equation $$dX_t = b(X_t)dt + \sigma(X_t) dW_t,$$ in which $W$ is a Brownian motion and $b,\sigma$ are such that strong ...
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Calculate the parametric equations of the xz plane parallel to the surface tangent. z = x ^ 2 + y ^ 2-4x-6y +13 in (3,3,1)
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### Brun's Theorem for twin primes and its generalization [on hold]

Brun's Theorem given in 1919 ensures that the sum of the reciprocals of the twin primes converges. Do you know a different proof of this same result? Moreover, you know if the "generalization" of it ...
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### Polyhedra from a tropical variety

it is known that tropicalization of a variety(irreducible and subvariety of some torus.) is a support of a polyhedral complex. I wonder which kinds of polyhedra can occur in this polyhedral complex. ...
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### Norm of a matrix exponential [on hold]

Prove the following inequality $$||e^{Pt}||\leq e^{t\alpha{(P)}}\sum_{k=0}^{r-1}\frac{(||P||\sqrt{r}\,t)^k}{k!}$$, where $r$ is the order of the matrix $P$ and $\alpha(P)$ be the maximum eigenvalue of ...
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### Equiareal shapes in $\mathbb{R}^d$

There was quite a bit of work on the so-called equichordal problem throughout the 20th century, to decide if some plane convex curve could have two equichordal points. A point is equichordal for a ...
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### Cotangent bundle lift theorem

Let $M$ be a smooth manifold and $T^\ast M$ be its cotangent bundle. Consider the tautological 1-form $\theta$ on $T^\ast M$ ($\theta=\sum y_i dx^i$ in local canonical coordinate systems). A ...
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### Equivariant Stratifications of a Variety

Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$. For fixed $\beta\in S$, we ...
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### Find all Linear spaces on an orbit (by a connected linear algebraic group)

Suppose $V$ is a vector space over $\mathbb{C}$ and $G\subset \textrm{GL}(V)$ is a connected linear algebraic group. Consider the orbit closure $Y = \overline{G.\mathbb{P} L}$, for a subspace ...
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### Is this a well-known matrix? [on hold]

All, Does the following matrix $P$ fall in a class of known marices $p_{i,j} = \frac{1}{\mu_i + \mu_j - \mu_i\mu_j}$ It looks almost like Cauchy Matrix but not quite.
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### Existence of $n$-connected topological groups with $m$-dimensional action extending that of $GL(m)$ on $\mathbb{R}^m$

I'll first state the question as concisely as I can and then provide some motivation. Consider two positive integers $m$ and $n$ such that $m < n+2$. Does there exist a topological group $G$ ...
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### Inner product on a complexification and complexification of an inner product [on hold]

Let's consider an inner product on the complexification of a vector space $X$. When there exists an inner product on $X$ whose complexification gives the original inner product on the ...
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### Is there a one-dimensional subshift of positive entropy s, all of whose sub-subshifts also have entropy s?

A subshift is a subset $X$ of $A^\mathbb{N}$ or $A^\mathbb{Z}$ (with $A$ finite), such that $X$ is topologically closed and closed under the shift operation. The shift operation is defined by ...
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### How to describe all normal subgroups of the dihedral group Dn? [migrated]

The dihedral group consists of rotations and symmetries. But the symmetry group is a group only if n is even, thus the group of rotations is a normal subgroup of the dihedral group. So how to ...
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### References for ordinary singular points

Let $X$ be a $n$-dimensional complex projective algebraic variety, let us suppose that $X$ has only isolated singularities. Edit: Let us say that an ordinary $m$-ple singular point is an isolated ...
203 views

### Best known bounds on tensor rank of matrix multiplication of 3×3 matrices

Years ago I attended a conference where they taught us that matrix multiplication can be represented by a tensor. The rank of the tensor is important, because putting it into minimal rank form ...
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### Interpret some coefficients in algebras

Let $A$ be a real vector space equipped with a scalar product $\langle \,,\,\rangle$, and assume moreover that a multiplication is define on it so that becomes an algebra (e.g. polynomials with the ...
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### On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
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### Jacobson ring = a ring whose nilradical and Jacobson radical coincide?

In Wikepedia (Nilradical), It claims that "A ring is called a Jacobson ring if the nilradical coincides with the Jacobson radical." Here the word "ring" means a commutative ring. However, I remember ...
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### Homology basis of minimal resolution

Let $S$ be a compact algebraic surface with du Val singularities and $T$ be the minimal resolution of it. Is it true that $$H_2(T,\mathbb{Z})/(\oplus_{i}\mathbb{Z}E_i)\cong H_2(S,\mathbb{Z}),$$ ...
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Consider a real-valued radom variable X with EX^4=1,we know that EX^3<=1. If also EX<=0,find an constant c<1 such that EX^3<=c and find the smallest constant c for which this inequality ...
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### Two questions about proof in Hilbert Space [on hold]

I am currently studying Hilbert Space in Real analysis, and I have a part not understandable. This is a theorem for Hilbert Space $H$. $Theorem$ : If $L$ is a bounded linear functional on $H$, then ...
Let $p$ be a positive real number. For any fixed $\epsilon>0$ does there exist a positive integer $n$ such that fractional part of $p^n$ is less than $\epsilon$? Add-on: $p$ is rational. ...
Can anybody tell me how to construct the character table of $PSL(2，8)$? I need a specific method.