# All Questions

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### totally uni modular matrix with binary variables

please can you help me in solving this problem I have totally uni-modular matrix A multiplied by set of Binary variables X. it means that A*X<=B how to solve this problem in polynomial time .. I ...
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### How complicated is the formula expressing that a set is non-measurable?

The question is exactly stated by the title, i.e. how complicated is the formula $\psi(x)$ (in the language of set theory) expressing that a given set of reals $x$ is non-measurable? A second ...
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### Does the coordinate ring of affine variety admit a structure of infinite dimensional variety?

We work in the category of algebraic varieties over some algebraically closed field $k$. By infinite dimensional variety I mean a filtration: $$V_0\subset V_1\subset V_2\subset\ldots$$ where each ...
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### Traveling Salesmen Variation [on hold]

Can anyone give me an elegant heuristic solution to a variation of the traveling salesman problem with a non-NP running time? For a set P of pairs of locations (X,Y), what is the shortest path that ...
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### Name or references for minimal $N$ such that $\left(\frac{a}{b}\right)_n = \left(\frac{a}{b'}\right)_n$ whenever $b \equiv b' \bmod (N)$

Let $\left( \dfrac{a}{b} \right)_n$ denote the nth power residue symbol, a generalization of the Legendre symbol. I have recently seen it quoted that there is a minimal ideal $N$ (minimal by ideal ...
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### A $2$-torsion version of the motivic stable homotopy category?

For a field $k$ there exists the motivic stable homotopy $SH(k)$; it is compactly generated. My question: does there exist a 'reasonable' functor $p$ from $SH(k)$ to a certain triangulated category ...
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### Relationship between curvature tensor, algebraic Bianchi identity and sectional curvature

I am currently trying to understand the algebraic Bianchi identity, and I am clearly missing some purely algebraic fact. Let $M$ be a Riemannian manifold, $R$ its curvature tensor (with index ...
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### Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$. It is clear that there is a constant ...
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### Which power of $2$ kills $W(k)$?

Is the following fact "well-known": if $-1$ is a sum of squares in a field $k$, then the Witt group $W(k)$ of quadratic forms is killed by multiplication by $2^N$ for some $N\ge 0$? What can one say ...
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### evaluation map $ev_t$ on loop space

Considering parameter of $S^1$ as $t$, we define. $$ev_t: C^\infty(S^1, \mathbb R^n)\to \mathbb R^n$$ $$ev_t(\gamma):=\gamma(t)$$ I am looking for a possible topology on $C^\infty(S^1,\mathbb R^n)$ ...
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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

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 $X$ over reals. 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 ...
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### 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 ...
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 ...