0
votes
0answers
2 views

Showing existence of positive weak solution of a PDE by CoV

Given the following PDE $$ \begin{cases} -\Delta u+\alpha=u^q &x\in\Omega\\ u=0 &x\in\partial\Omega \end{cases} $$ where $\Omega\subset\mathbb R^3$ is open bounded with smooth boundary, ...
0
votes
0answers
5 views

Is $LIS(\pi)+LIS(\sigma)+LIS(\sigma\pi^{-1})$ lower bounded?

In the title, $LIS$ stands for the length of longest increasing subsequence and Greek letters stand for permutations from symmetric group $S_n$. Considering some cases such as ...
0
votes
0answers
12 views

What is the full subcategory of $\it{Cat}$ consisting of categories $\mathbb{C}$ where $\mathsf{ob}(\mathbb{C}) = \mathsf{arr}(\mathbb{C})$ hold?

Let $\it{Cat}'$ be the full subcategory of $\it{Cat}$ such that $ \mathsf{ob}(\it{Cat}') = \{ \mathbb{C} \in \it{Cat} \mid \mathsf{ob}(\mathbb{C}) = \mathsf{arr}(\mathbb{C}) \}$. The questions are as ...
0
votes
0answers
20 views

Some calculus in $O(n)$

How can one compute each of the following matrices, explicitly: $$\int_{O(n)} e^{g}dg$$ or $$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$ What is the explicite entries of the resulting ...
0
votes
0answers
17 views

Coefficients of Hilbert polynomials

Recall that we can define the Hilbert series of a graded commutative algebra $$\displaystyle S = \bigoplus_{n \geq 0} S_n$$ over a field $K$ by $$\displaystyle \mathcal{H}_S(t) = \sum_{n=0}^\infty ...
2
votes
0answers
16 views

Reference for exercises with solutions for affine Lie algebras

I am doing self-study of affine Lie algebras, and while I have all recommended reference material, I find it hard to prepare for the exam. It was quite easy to study finite-dimensional simple Lie ...
-1
votes
0answers
15 views

Ky Fan norms and nuclear norm

Ky Fan norms and the nuclear norm seem to be very relevant to my research so I would like to be familiar with what is already known. Can anybody recommend a reference discussing any aspects of these ...
3
votes
0answers
32 views

Open problems in Berkovich geometry

I would like to know if there is a state of the art recent reference on non-archimedean analytics spaces mentioning/listing open problems, conjectures, unresolved questions in the theory (*). I have ...
1
vote
0answers
14 views

Probability of a giant component existing in a $G(n,p)$ random graph with $p=\omega(\frac 1n)$

Consider a random $G(n,p)$ graph where $p=\omega(\frac 1n)$, and let $x$ denote the probability that the graph has a connected component of size linear in $n$. It is well known that $x$ tends to $1$ ...
2
votes
0answers
24 views

Is a matrix element of a norm continuous representation always a trigonometric polynomial?

I asked a similar question for the case of compact groups not long ago in math.stackexchange. Now I understand that the answer was "yes", and I want to modify that question. This is also related to my ...
0
votes
0answers
24 views

Extending a model to a given compactification of its generic fiber

Let $R$ be a discrete valuation ring and $K$ its field of fraction. Let $X$ be a proper $K$-variety, $U$ a dense open and consider an $R$-model $\mathcal{U}$ of $U$. Can we embed $\mathcal{U}$ in a ...
4
votes
0answers
45 views

Exactness of pure functors

I can't prove this lemma in "Notes on motivic cohomology, Beilinson, Macpherson, Schechtman": Lemma. A pure functor is exact. Definitions: A mixed category $\mathcal{M}$ is a ...
3
votes
2answers
80 views

Geodesics and Riemannian submersions

Let $X,Y$ be Riemannian manifolds, and $f\colon X\to Y$ be a Riemannian submersion. Let $\gamma$ be a geodesic on $X$ starting at a point $x\in X$ and which is orthogonal to the fiber $f^{-1}(f(x))$. ...
5
votes
1answer
50 views

Trigonometric polynomials on non-compact and non-abelian groups

I asked this initially in math.stackexchange, but it disappeared almost immediately, so I hope it will be proper to aks this here. Hewitt and Ross define trigonometric polynomial on a locally compact ...
1
vote
0answers
88 views

general formula for volume of a simplex? [migrated]

I am looking for a general formula to calculate the volume of a euclidean simplex in any number of dimensions. On Wikipedia I found that a formula similar to Heron's formula can be applied to ...
1
vote
0answers
21 views

Divisibility of the degree of an extension by the degree its residual field

Let $A$ be and integrally closed domain whose quotient field is $K$, $L$ be a finite extension of $K$, and $B$ be the integral closure of $A$ in $L$. Let $M_A$ be a maximal ideal of $A$, and $M_B$ be ...
4
votes
0answers
44 views

Recent trends in effective analysis

The references listed at http://en.wikipedia.org/wiki/Computable_analysis have all been published 30-15 years ago. Are the approaches which they expose still relevant to the current paths of research ...
0
votes
0answers
20 views

About the selection of reals $u_0,u_1$ such that $u_{n}$ is a positive integer

Let $r\geq 4$ and $n≥1$ be two positive integers. Let us consider the sequence $(u_{n})$ defined by: $$u_{n}=r^{n^2}\sum_{m=1}^{n}\frac{u_1-ru_0+2(m-1)}{r^{m^2}}+u_0$$ where $u_0,u_1$ are real ...
-1
votes
2answers
82 views

On independent sets of graph

Given $G$ a regular graph on $n$ vertices, denote $\alpha(G)>1$ to be independence number. Denote $\Gamma(G)$ to be collection of possible subset of independent vertices in $G$ of cardinality ...
2
votes
0answers
58 views

References for general Hasse-Weil zeta function

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case. I am ...
0
votes
0answers
41 views

Tensor product of complexes

Let $A$ be a ring and let the modules that are involved be left and right A-modules (not necessarily bimodules over A). I'll denote as $\mathcal{E}^n_R(M, N)$ the category of n-fold extensions of M ...
-2
votes
0answers
29 views

calculating E(Xt^2,Xt-h^2) with Xt normal(0,sigma^2) [on hold]

the problèm is let {X_t} all normal N(0, sigma^2) défine rho_X(h)=cov(X_t,X_t-h)/var(X) and Y_t=(X_t)^2 proove that rho_Y(h)=[rho_X(h)]^2 i know that i have to use expected value E(E(Y/X)) but i don't ...
2
votes
0answers
44 views

Comparing the size of two sums

Let $q$ be a prime power and $\mathbb{F}_q$ be the field of $q$ elements. Let $\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. I am working on a research project, where I bounded a ...
1
vote
1answer
35 views

1-wasserstein distance v.s. total variation distance

Suppose that $\mu_1$ and $\mu_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma_x$ denote the ...
-1
votes
0answers
40 views

Complementary polynomials

Denote $S=\{0,1\}^n$. $\mathsf{MLP}_{d,n}=\{p\in\Bbb R[x_1,\dots,x_n]:p\mathsf{\mbox{ is mutilinear with total degree}}(p)=d\}$. Is there an $n\geq d^2+1$ such that there exists distinct polynomials ...
1
vote
0answers
40 views

Analogous to a PDE but where independent variable is a function

Consider, as an example for my question, a density function $u(\boldsymbol{x},t)$ on a vector field $\boldsymbol{x}$ at some time $t$. The flow velocity vector of the density is given by ...
0
votes
0answers
47 views

Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?

Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective? The same question for $\ell_\infty$-module $c_0$. Both these modules are not dual, so standard arguments ...
5
votes
0answers
41 views

Choosing a metric in which homeomorphism is Holder continuous

Let $X$ be a compact metrizable space, and let $f:X \to X$ be a homeomorphism. Is it always possible to choose a compatible metric on $X$ in which $f$ is Holder continuous? I've tried some simple ...
5
votes
1answer
141 views

Identity for Power Series and Binomial Coefficients

This question concerns a combinatorial identity obeyed by power series coefficients. Throughout we let $[x^{M}]\{\phi(x)\}$ denote the coefficient of $x^{M}$ in a power series $\phi(x)$. Let $k$ be ...
-3
votes
0answers
32 views

Matrix Algebra reduction [on hold]

I am trying to reduce the following: $x$ and $y$ column vectors $y^t$ is the transposed column vector $(I - \frac{1}{(1+ y^t x)} * x y^t) (I + x y^t) = I$ I am stuck at $x y^t * y^t X = x y^t (x ...
3
votes
0answers
78 views

On the boundary of the twindragon

Let $\mathcal T$ be the famous twindragon, i.e., $$ \mathcal T=\left\{\sum_{n=0}^\infty a_n\left(\frac{1+i}2\right)^n : a_n\in\{0,1\}\right\}. $$ Then, as is well known, $\mathcal T$ has a ...
8
votes
2answers
121 views

Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.

Fix $q$ to be a positive integer. Let $$f : \mathbb{N} \to \{-1 ,0, 1\}$$ be a $q$-periodic arithmetic function such that $$\sum_{n = 1}^q f(n) = 0.$$ If $f$ is not identically zero, is it true that ...
2
votes
0answers
55 views

Asymptotic expansion for the Bell numbers

The Bell numbers $B(n)$ (that is, the numbers that count the set partitions of a set, and have exponential generating function $\exp(e^x -1)$ ) admit the asymptotic expansion $$\frac{\log B(n)}{n} = ...
1
vote
0answers
27 views

Measurability of eigenelements $s \mapsto (\varphi_k(s), \lambda_k(s))$ of Laplace-Beltrami on $M_s$

For each $s \in [a,b]$, let $M_s$ be a compact Riemannian manifold with no boundary. Under what conditions on $s \mapsto M_s$ do the eigenvalues and eigenfunctions $(\varphi_k(s), \lambda_k(s))_{k ...
-3
votes
0answers
34 views

Maximum - Minum area [on hold]

The problem is listed by this link because i couln't post images here (something about reputation). http://math.stackexchange.com/questions/1117894/maximum-minumum-area
0
votes
1answer
85 views

Schematic image of a relative Cartier divisor of a fiberwise dense open

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise ...
0
votes
1answer
43 views

Help with notation for the state of a dynamical system defined by a PDE

Before my question let me briefly describe a simplified version of the dynamical system I'm working with. Suppose that I have a density function $m(\boldsymbol{x},t)$, that describes the abundance of ...
5
votes
0answers
78 views

Groupoid cardinality of DM stack and point counting on coarse moduli spaces

Let $X$ be a finite type DM stack over a finite field $k$ with a coarse moduli space $X_c$. (We only assume $X_c$ is an algebraic space and $X$ might have infinite inertia stack.) Under which ...
0
votes
0answers
57 views

Moduli space of points - Gorenstein ideal

I've been working on algebraic covers, $\varphi\colon X\rightarrow Y$, ($\varphi_*\mathcal{O}_X$ is a locally free $\mathcal{O}_Y$-algebra of rank d). I'm more interested in the algebraic point of ...
2
votes
1answer
61 views

Statistical distance between discrete and continuous distributions

Are there any statistical distance functions that are capable of comparing a continuous and a discrete distribution? From reading this list http://en.wikipedia.org/wiki/Statistical_distance the only ...
2
votes
0answers
64 views

Reference for cdh topology

Let $f:X\rightarrow Y$ be a proper surjective morphism over some base scheme $S$ of finite type, suppose $f$ restricts to an isomorphism over some open $U$ of $X$, we also suppose both $X$ and $Y$ are ...
2
votes
1answer
118 views

Rankin-Selberg convolution and product of degrees

As I'm kinda obsessed with the Selberg class and because of the general converse conjecture, I'm still trying to get a rough idea of what automorphic representations and their L-functions as well as ...
1
vote
0answers
82 views

Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$. If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to ...
1
vote
0answers
87 views

What is the technical difference between a deformation and a perturbation?

What is the technical difference between a deformation and a perturbation? Do they exist in somewhat different categories?
1
vote
1answer
104 views

Sequence of smooth maps converging to the identity

Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...
-6
votes
0answers
106 views

New algorithm discovered to find prime numbers [on hold]

I think that I have discovered a new algorithm to find prime numbers. It uses all prime numbers less than a particular number to find prime numbers within a range. I am not sure whether it generates ...
0
votes
0answers
170 views

Can mathematics get from other sciences what it got from physics? [on hold]

Throughout history, physics has been an unparalleled source of '' inspiration'' for discovering/inventing mathematical ideas, which is due to its ability to describe the physical world. But can this ...
3
votes
3answers
210 views

Survey papers of results (relevant to mathematics) produced during research on the relationship between mathematics and music theory

It is well-known that there is a relationship between music and mathematics, and there are many references that explore this topic (for example, Benson's book). However, I would like to ask if there ...
0
votes
0answers
95 views

Lower bound for constructing $n!$ with additions, subtractions, and multiplications starting from $1$

Found this on Complexity Zoo warning expired certificate check NP Over The Complex Numbers. [BCS+97] show the following striking result. For a positive integer $n$, let $t(n)$ denote the minimum ...
1
vote
0answers
50 views

Relations in a space generated by indicator functions

Simple Question I ran into the following seemingly simple question. For an arbitrary set $M$ consider the real vector space generated by indicator functions $\chi_A$ of all subsets $A\subset M$. ...

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