0
votes
0answers
3 views

Can every $p$-group with duality pairing be written as cokernel of a symmetric matrix over the $p$-adic integers?

Let $G$ be a $p$-group (where $p$ is a prime). Suppose there esists a symmetric bilinear map $\delta\colon G\times G\to \mathbb{Q}/\mathbb{Z}$ such that the induced map $g\to\langle g,\;\rangle$, is ...
0
votes
0answers
32 views

A morphism of elliptic schemes that preserves the identity is a homomorphism

I am trying to understand the proof of the fact that any morphism $f \colon E_1 \rightarrow E_2$ of elliptic curves over an arbitrary base scheme $S$ satisfying $f(0) = 0$ must respect the group ...
0
votes
0answers
33 views

Inversion, Koszul duality, combinatorics and geometry

According to this MO answer Koszul duality is related to operations on generating series; 1) multiplicative inversion for quadratic algebras, 2) compositional inversion for quadratic operads, 3) ...
0
votes
0answers
11 views

Largest eigenvalue & corresponding eigenvector of adjacency matrix of non-regular graph

Consider a graph $G$ that: is irregular, is connected, is not bipartite, has many cycles of various odd and even lengths. Its adjacency matrix $A$ has a degenerate dominant eigenvalue. The entire ...
4
votes
1answer
70 views

Can any object in a presentable category be written as a colimit of generators?

Let $\mathcal{C}$ be a presentable category, and let $S$ be a set of objects such that $S$ generates $\mathcal{C}$ under colimits, i.e., such that the smallest cocomplete subcategory of $\mathcal{C}$ ...
3
votes
1answer
52 views

Can all unit-distance graphs have their vertices at algebraic integers?

A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$. Obviously, we can ...
1
vote
0answers
29 views

References for elliptic integral

I'm trying to learn more about the most general elliptic integral, that is, an integral of the form $$\int\frac{A(x)}{B(x)\sqrt{S(x)}}$$ where $A(x), B(x)$ are arbitrary polynomials and $S(x)$ is ...
0
votes
0answers
9 views

Convergence in distribution of stochastic equation solutions

I post this post en MSE but I think that is more suitable for this site. I'm studying from Kurtz's book "Markov Processes Characterization and convergence" and I have a question about the ...
2
votes
0answers
48 views

Does a semistable curve descend to a regular base?

Let $f\colon X \rightarrow S$ be a semistable curve of genus $g \ge 0$. Being a semistable curve means that $f$ is a morphism of schemes such that $f$ is proper, flat, and of finite presentation; ...
-3
votes
0answers
15 views
-6
votes
1answer
74 views

The P VS NP problem as relates to factoring [on hold]

Good Day all, If someone could prove there is no simple way to factor an integer, and show searching is absolutely required, would that also prove P is not equal to NP? Thanks in advance for your ...
5
votes
0answers
45 views

When is a smooth function locally equivalent to a truncation of its Taylor series?

Let $U \subset \mathbb{R}^n$ be an open set and let $f:U \rightarrow \mathbb{R}$ be a smooth proper function. For $p \in \mathbb{R}^n$ let $$T_p(x_1,\ldots,x_n) = \sum_{i_1,\ldots,i_n=0}^{\infty} ...
2
votes
0answers
34 views

On subset of Deterministic games

Denote strings $u,v$ from $\{0,1\}^n$. Denote concatenated pair $[uv]$. Denote $$[uv]_{1}=\{[uv]\oplus e_i\}_{i=1}^{2n}$$ collection of pairs with Hamming distance $1$ from $[uv]$ string ...
0
votes
0answers
57 views

When does a hyperelliptic Riemann surface admit a map of degree 3

Let $X$ be a hyperelliptic curve of genus $g>1$. For which $g$ does $X$ admit a map $X\to \mathbb P^1$ of degree $3$? I think a genus two curve $X$ admits a map of degree $3$. Proof: Pick $P$ ...
2
votes
1answer
38 views

Legendre transform and Lipschitz aproximation

Let $(X,d)$ be a compact metric space and $f:X \to \mathbb{R}$ a real valued continuos function. Let us agree that a modulus of continuity means concave, nondecreasing, uniformly continuos function ...
0
votes
0answers
35 views

Existence of a complementary closed subspace extending a given subspace

Let $H$ be an infinite dimensional (separable) Hilbert space (or any infinite dimensional Banach space in which every closed subspace has a complementary subspace). Suppose that $X$ and $Y$ are closed ...
5
votes
0answers
44 views

Can a symplectic manifold be recovered from its Lagrangians?

Something I have wondered idly about from time to time is: If $(M,\omega), (M',\omega')$ are symplectic manifolds, and you "know what the Lagrangians $L \subset M$ resp. $L' \subset M'$ are," can ...
-2
votes
0answers
34 views

How do you describe this ordinal filter? [on hold]

I am creating a filter that takes the 4th highest out of each 13 values and does this to filter an array (so it is almost like a median filter). I want to describe it in my research paper. Could I go ...
7
votes
0answers
119 views

Do canonical stacks exist over Spec(Z)?

Suppose a scheme $X$ has tame quotient singularities. Does there exist a smooth DM stack $\mathcal X$ with coarse space $X$ so that the coarse space morphism $\mathcal X\to X$ is an isomorphism ...
5
votes
0answers
56 views

What Is The Minimal Monomial of the Symmetric Group?

In the symmetric group $S_n$ what is the shortest sequence $c_1,\ldots,c_k\in S_n$ such that, for all $x\in S_n$ the following product of conjugates of $x$: $$x^{c_1}x^{c_2}\ldots x^{c_k}$$ equals the ...
8
votes
2answers
161 views

Certain signed sum over $S_n$

The following question appeared in my research: Let $G_1,G_2,G_3$ all be subgroups of $S_n$, and consider the sum $$ \sum_{g_i \in G_i, g_1g_2g_3 = id} \epsilon(g_1) $$ that is, we only consider ...
0
votes
0answers
15 views

Interesting properties of complex Gateaux derivatives

The complex Gateaux derivatives are for $\psi=\psi_{1}+i\psi_{2}$ and functional F $dF(\psi,\xi)=d_{\psi_{1}}F(\psi_{1},\xi_{1}-id_{\psi_{2}}F(\psi_{2},\xi_{2}$ and ...
0
votes
0answers
17 views

Dependence of weak solution of an equation on a parameter

For each $p \in [a,b]$, let $X_p$ be a Hilbert space with $Y_p \subset X_p$ a subspace and we are given a bilinear form $a_p(\cdot,\cdot):X_p \times X_p \to \mathbb{R}$. Given $u_p$ with $p \mapsto ...
1
vote
1answer
45 views

Invariant generalized sections of dual vector bundles

Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My ...
1
vote
0answers
12 views

Phase of the inner product between the elements of an ETF

I am doing research in compressive sampling for Cognitive Radio applications. While working on a project I came across with the following question: Is there any research about the phase of inner ...
5
votes
0answers
51 views

A positivity problem involving the number of ways of expressing $n$ as a product of $k$ factors

Let $d_k(n)$ denote the number of ways of expressing $n$ as a product of $k$ factors, and let $$D_k(x)=\sum_{n\leq x}d_k(n)$$ be the summatory function. During a study of Mertens' function I was lead ...
0
votes
0answers
71 views

The amenable aspect of $F_{2}$

We fix an embedding of $F_{2}$, the free group on two generators, in $SO3$ and we put $G=\bar{F_{2}}$. Then we construct the following $C^{*}$ algebra: $$A=C^{*} G \sim C^{*}_{r} G $$ What ...
0
votes
0answers
21 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, ...
2
votes
0answers
21 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 ...
-3
votes
0answers
49 views

What is the full subcategory of $\it{Cat}$ consisting of categories $\mathbb{C}$ where $\mathsf{ob}(\mathbb{C}) = \mathsf{arr}(\mathbb{C})$ hold? [on 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
54 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
63 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 ...
1
vote
0answers
32 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
21 views

Ky Fan norms and nuclear norm [on hold]

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 ...
7
votes
0answers
145 views

Open problems in Berkovich geometry

I would like to know if there is a state of the art recent reference on non-archimedean analytic spaces mentioning/listing open problems, conjectures, unresolved questions in the theory (*). I have ...
2
votes
0answers
19 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
1answer
64 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
35 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 ...
6
votes
0answers
95 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 ...
5
votes
2answers
157 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
93 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 ...
2
votes
0answers
34 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 Galois extension of $K$, and $B$ be the integral closure of $A$ in $L$. Let $M_A$ be a maximal ideal of $A$, and ...
5
votes
2answers
125 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 these references expose still up-to-date and relevant to the ...
0
votes
0answers
27 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 ...
0
votes
2answers
131 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
71 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
60 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
31 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
76 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 ...

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