# All Questions

**0**

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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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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**

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**0**answers

15 views

**-6**

votes

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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 ...