1
vote
0answers
12 views

Order theory as a foundation of mathematics?

I know the followings kinds of formalization of mathematics: based on set theory (e.g. ZFC) based on type theory (e.g. the formalism of Coq proof assistant, as an advanced example) based on category ...
6
votes
1answer
59 views

A combinatorial question about orthonormal bases

Suppose that $F:S^{n-1}\to A$ is a map of sets from the unit sphere in $\mathbb R^n$ to an abelian group, and that the sum $F(v_1)+\dots +F(v_n)$ over an orthonormal basis is independent of the basis. ...
0
votes
0answers
23 views

Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory

I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces. I got quite stuck in Corollary 3.27 ...
-2
votes
0answers
10 views

Partition on a Closed Set A= [2,3] [migrated]

Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty?
1
vote
0answers
11 views

Computionally efficient vertex enumeration for (convex) polytopes

Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The vertex enumeration problem asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be ...
1
vote
0answers
12 views

Optimization with random matrix

Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where ...
4
votes
0answers
51 views

Coloring a Ferrers diagram

I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed. Say that a coloring of the dots of a Ferrers ...
0
votes
0answers
65 views

All non-split Cartan subroups of $GL_2(\mathbb{Z}/n\mathbb{Z})$ are conjugate

Let $n>1$ be a positive integer and let $R$ be an order in an imaginary quadratic field with discriminant prime to $n$. Let $A=R/nR$ and let $\lbrace 1, \alpha \rbrace$ be a ...
2
votes
0answers
60 views

SubGROUPs of Banach spaces, when are they dense in a vector subspace?

It’s relatively easy to show that if $J$ is a closed subgroup of a finite-dimensional real Banach space, $B$, then it is a vector subspace iff for all bounded linear functionals $\sigma$ of $B$, ...
9
votes
0answers
178 views

Decidability of $x^3+y^3+z^3 = c$

I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that ...
1
vote
1answer
27 views

Is an arbitrary Brownian-motion path a viscosity solution of every differential equation?

Is an arbitrary Brownian path a viscosity solution of every differential equation? My intuition is that a path of Brownian motion is so ill-behaved that it not only does not have derivatives ...
1
vote
0answers
27 views

For a non-convex function f, how to find a function g such that $g\circ f$ is strictly convex? [migrated]

The following function $f(x)={1\over (1+e^{-x})}$ is non-convex but $\ln(f(x))$ is convex. Given a non-convex function $f$, can we find a function $g$ such that $g\circ f$ is strictly convex? If yes, ...
5
votes
1answer
120 views

A survey for various $K$-homology theories and their relationship

The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology ...
7
votes
0answers
33 views

Are the failure of SCH and “$cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular” equiconsistent?

Is it true that the following two statements are equiconsistent? (1) $2^\mu>\mu^+$ for some strong limit singular cardinal $\mu$ (2) $cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular ...
-4
votes
0answers
40 views

how should one locate ambulance stations so as to best serve the needs of the community..tnx [on hold]

how should one locate ambulance stations so as to best serve the needs of the community i don't know what algorithm to use, any suggestion/s?
0
votes
0answers
17 views

Gaussian gabor frame

It is widely known that $\phi(x)=e^{-\frac{x^2}{2}}$ does not define a Gabor frame if we consider translations by units of $1$ and multiplication by $e^{2 \pi inx}$for $n \in \mathbb{N}.$ A way to ...
7
votes
0answers
129 views

End of the Ext functors

Let $R$ be a ring, and consider the hom functor $\hom\colon Mod(R)^\text{op}\times Mod(R)\to Mod(R)$; the end of $\hom$ is well-known to be the set of endomorphisms (endonatural transformations) of ...
3
votes
0answers
18 views

Connection between connectivity and cohesion of a graph

Tutte [1] proved that, for every $3$-connected graph $G$ and vertices $u$ and $v$, there exists a nonseparating $uv$-path. A graph $G$ is $t$-cohesive if $G$ is connected, has at least two vertices, ...
-1
votes
0answers
28 views

Contour Integration [on hold]

I am trying to integrate $\frac{\sin x dx}{x(x-1)}$ over the real line except at an arbitrarily small neighborhood around 1, where the function has a singularity. My idea is to do an contour ...
3
votes
0answers
173 views

Are those varieties associated to L-functions actually motives?

Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product $\otimes$ such that $\forall ...
1
vote
1answer
73 views

Frame-bundle reduction from spinor-bundle reduction

Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin manifold, and let us denote by $F(M)$ its frame bundle, by $SP(M)$ its spin bundle and by $S = P(M)\times_{\rho}\Delta$ its spinor bundle, ...
1
vote
1answer
72 views

Tangent cone of a complete intersection

Let $X$ be a quasi projective variety over $\mathbb{C}$. By the tangent cone of $X$ at a point $p \in X$, I mean the subvariety of the tangent space of $X$ at $p$ as it is defined in Harris' ...
1
vote
0answers
90 views

Unusual generalization of the law of large numbers

I have seen in physical literature an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...
2
votes
0answers
115 views

Is this diagram of sheaves actually Cartesian as claimed?

The question is about Corollary 1.6.2 (b) in the book by Laumon and Moret-Bailly on algebraic stacks. There we have a scheme $S$ and morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ of sheaves on a ...
0
votes
0answers
41 views

the inverse of Trace theorem when $p = 2$

I can see there is an answer the inverse for the trace theorem and Image of the trace operator My question is that given $f\in H^{1/2}(\partial\Omega)$, is it possible to extended it into $\Omega$ ...
10
votes
1answer
288 views

Can Shor's Algorithm be modified to run efficiently on a classical computer?

Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that ...
-3
votes
0answers
34 views

construction of nonocommutative division rings [on hold]

I am studing "A first course in noncommutative rings by T.Y.Lam " Please introduce books or articles to better understand the contents of section 14 (noncommutative division rings) of this book. thank ...
-1
votes
0answers
23 views

Solving el Gammal given D can solve DDH [on hold]

I have a crypto final in 2 days and I am reviewing past finals but the prof does not give solutions. There is 1 question I cannot solve it and I spend a good 2-3 hours on it. Here is the question: ...
0
votes
0answers
50 views

What is the relation between the $K_0$ of a singular curve and its normalization?

Let $X$ be a singular curve over a field $k$. We define $K_0(X)$ to be the Grothendieck group of the category of coherent sheaves on $X$. For $X$ we have its normalization $\widetilde{X}$ and hence ...
6
votes
0answers
259 views

Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system, $$x_1^2+x_2^2+\dots+x_n^2 = y^2$$ $$x_1^3+x_2^3+\dots+x_n^3 = z^3$$ and define the function, $$F(s_m) = x_1+x_2+\dots+x_n$$ For $n\geq3$, using an ...
-2
votes
0answers
22 views

Determining odds of a slot machine given a payout value of the icon [on hold]

So most slot machines base the payout on the probability of the combination coming up. What I would like to do is flip that and set a payout and then have the probability based off of that if ...
3
votes
2answers
93 views

Density of polynomials which are soluble with respect to a set of primes

Suppose that $p$ is a prime, and $f(x)$ is a polynomial with integer coefficients and positive degree. Then there exists an integer $n_p$ such that $p | f(n_p)$ if and only if $f(x)$ has a linear ...
4
votes
0answers
106 views

On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...
5
votes
0answers
144 views

Are curves over imperfect fields defined over a smaller field?

Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...
4
votes
1answer
246 views

Are reduced residue systems relative primorials an active area of research? If not, why not?

As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. For example, it is fascinating to me that it is ...
0
votes
0answers
27 views

Euler transformation of pochhammer symbol

From the Euler transformation of Pochhammer symbol $$\sum_{n=0}^{\infty}\frac{(b)_n}{n!}a_nz^n=(1-z)^{-b}\sum_{n=0}^{\infty}\frac{(b)_n}{n!}\Delta^na_0(\frac{z}{1-z})^n$$ the following ...
1
vote
1answer
96 views

Counting primes powers $p^{\frac{k}{t}} \left( t \in \mathbb{R}{+}, k \in \mathbb{N} \right)$ by changing $\rho$'s in $\psi(x)$?

With $\rho=\beta+ \gamma \,i$ being a non-trivial zero of $\zeta(s)$, the logarithmic prime counting function is: $$\psi(x) = x - \log(2\pi) - \frac12 \log\left(1- \frac{1}{x^2}\right) - ...
1
vote
1answer
108 views

Eisenstein series of weight $2$ for $\Gamma_0(N)$ : where am I wrong?

Let $A_{N,2}$ be the set of triples $(\psi,\varphi,t)$ such that $\psi$ and $\varphi$ are primitive Dirichlet characters modulo $u$ and $v$ with $(\psi\varphi)(-1)=1$, and $t$ is an integer such ...
-2
votes
0answers
43 views

How to solve this triple integral? [on hold]

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
0
votes
0answers
43 views

Is the following set of infinite absolutely convex combinations closed? [on hold]

Let X be an infinite dimensional Banach space and let $(x_n)$ be a weakly-null sequence in X. Let A:=$\{\sum_{n=1}^{\infty} a_nx_n: (a_n) \in B_{l_1}\}$, where $B_{l_1}$ is the closed unit ball of ...
0
votes
1answer
70 views

Irreducible representation of $C^*(D_\infty)$, group $C^*$-algebra of an infinite dihedral group

I have a question about an irreducible representation of the (full) group $C^*$-algebra of an infinite dihedral group $D_\infty$, denoted by $C^*(D_\infty)$. Ultimately, I'm interested in finding a ...
0
votes
0answers
18 views

Simple proof for a simple second-order approximation of a convex function [on hold]

I am wondering if someone can prove (or outline the proof) for the following statement from p. 459 of Boyd and Vandenberge's Convex Optimization textbook. Consider a strongly convex function ...
0
votes
0answers
65 views

How generic are Cayley graphs of non-Abelian groups with logarithmic girth?

Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know, For which $G$ can the ...
-1
votes
0answers
19 views

How to describe behavior of population system, given by system of ODEs? [on hold]

So I have a system of equations:$$x'(t)=x(t)(4-2x(t)-y(t))\\y'(t)=y(t)(3-x(t)-y(t)) $$ What I understand so far is: if we have x being the population of prey and y is the population of predators. x ...
4
votes
1answer
138 views

Meager set of full measure

Let X be a compact Hausdorff topological group and let m be the Haar measure on X. Can we find a meager set in X whose complement is m-null? I can do it when X is separable but I don't know if there ...
1
vote
1answer
121 views

Explicit examples of Dehn presentations of hyperbolic groups

It is well known fact that a (f.g.) group is hyperbolic if and only if it admits a (finite) Dehn presentation. My question concerns something I'm struggling with since the first time I read the proof ...
2
votes
0answers
106 views

Do J-holomorphic curves “very nearly” fail to be an immersion near the bubbling points?

Let $u_{t}: \mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be a family of degree $2$ maps defined (for $t$ small and non zero) by $$u_t([X,Y]) := [X^2, t Y^2, XY].$$ Note that as $t$ goes to zero, ...
4
votes
0answers
64 views

Analysis of Nim-Like Game? [on hold]

There are a finite number of heaps, each with a finite number of counters. Two players take turns; on each move, they may remove exactly one counter from any heap, and also, if the heap is of size ...
0
votes
0answers
30 views

Contraction of simplicial presheaves

Let $X,Y$ be two simplicial presheaves on a small category $\mathcal{C}$, let $*$ be the final simplicial presheaf. Consider the category of simplicial presheaves equipped with its projective model ...
7
votes
0answers
107 views

What is the role of fiber functor in Deligne's theorem on Tannakian categories?

The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to ...

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