0
votes
0answers
2 views

Counting elements with certain word length in abelian groups

Given a (finite) abelian group $G = \langle S \mid R \rangle$, has the problem of counting the number of elements which can be expressed as a word (in $S$) of length $\leq k$ been studied? If so, ...
0
votes
0answers
21 views

Adjointable Abelian Monoidal Functor

Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?
1
vote
0answers
7 views

concentration inequalities for quadratic forms of correlated random vectors

Let $\mathbf{n}$ is a Gaussian random vector with mean $\mathbf{0}$ and co-variance matrix $\mathbf{H}$. Let $\mathbf{r} = Sign(\mathbf{n})$, where $Sign(n_i) = 1$ if $n_i>0$ and $Sign(n_i) = -1$ ...
0
votes
0answers
22 views

Purely inseparable field extensions of degree p

Take a field $k$. If $k'/k$ is a field extension of degree $p$, it is known that there are many possibilities for the isomorphism class of $k'$. See ...
1
vote
0answers
29 views

Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?

Let's say I have a linear differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator ...
1
vote
0answers
20 views

Does the following characterize local presentability?

Let $\mathcal C$ be a cocomplete category. Consider the following two conditions: $\mathcal C$ is locally presentable. The Yoneda embedding $$\mathcal C \hookrightarrow \{\text{continuous functors ...
-8
votes
0answers
25 views

Anyone need expert help/consulting with matlab or mathematica? [on hold]

Code Genius was started at MIT to connect programmers in the scientific community to experts of advanced languages for efficient and cost effective support. Our platform helps programmers find the ...
0
votes
0answers
39 views

Hilbert scheme of relative subschemes of lenght 2

Let $\mathfrak X \rightarrow S$ a smooth projective family over the spectrum of a dvr. We know that $(\mathfrak X _{\eta_R})^{[2]}$ and $(\mathfrak X _{p})^{[2]}$ are smooth, where $p$ is the closed ...
-2
votes
1answer
58 views

Recursion, Common Term, Combinatorics [on hold]

May we find the common term for recursive sequence? if yes that how to find the common term of recursive sequence such: 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 5 1 2 1 3 1 2 1 4 1 2 1 3 1 2 1 6 ... in a ...
4
votes
1answer
104 views

Frobenius $A_{\infty}$-bialgebras?

Recall that a finite dimensional associative algebra $A$ over a field $k$ is called a symmetric Frobenius algebra (sometimes called "closed" Frobenius algebra) if its equipped with a symmetric non ...
1
vote
0answers
28 views

A specific class of $(0,1)$-matrices

Let $T_n$ be the class of $(0,1)$-matrices of order n with no row sum equal to a column sum. Is there any name or research about this kind of matrix? For any $A\in T_n$, let $S_A$ be the sum of all ...
1
vote
0answers
20 views

Non-degenerate periodic orbits in the boundary of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk, and let $\Omega$ be a closed 2-form on $E$ such that it is non-degenerate fiberwise. For any $x \in E$, there is a ...
2
votes
0answers
37 views

Is there a quotient of $c_0$ without the approximation property?

The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property? This would follow if ...
0
votes
0answers
13 views

Markov chain matching local time

Let $\left(X_{t}\right)_{t\geq0}$ be a Markov process taking values in a finite state space $E$. Its local time at $y\in E$ started at $x\in E$ is defined as $$ ...
2
votes
2answers
84 views

Divisibility among discriminants

Let $f(x)$ be an algebraic function over the field $\mathcal F$ of algebraic numbers over $\mathbb{Q}$. Suppose that $r \in \mathcal F$. Does the discriminant of $f(r)$ divide the discriminant of ...
0
votes
0answers
24 views

Stationary distribution of two-dimensional Markov Process?

A two-dimensinal Markov process $\{\theta_{t},S_{t}\}_{t=1}^{\infty}$ where $\theta_{t} \in \Theta$ and $S_{t} \in S$.$\Theta$ is a continuous state space and $S$ is a discrete state space. Suppose I ...
0
votes
0answers
29 views

problem in meaning of symbol in commutator subgroup

i was reading paper "OUTER AUTOMORPHISMS IN NILPOTENT p-GROUPS OF CLASS 2, H. LlEBECK" in page 2 there is a symbol i dont get. if G is generated by a basis $a_\lambda$, $\lambda \in \Lambda$ and $z ...
0
votes
0answers
46 views

Graphs from which two vertices can be exchanged

A graph is vertex transitive if $x \mapsto y$ by an automorphism. Let $P$ denote the stronger property that $x \mapsto y \mapsto x$ by an automorphism. Simple facts: $P \rightarrow$ unimodular. ...
-3
votes
0answers
43 views

Which branch should I choose for my master degree? PDE or Dynamical Systems [on hold]

I'm finishing my undergraduate in Industrial Engineering and I've applied for a mathematical master in which I can choose all the subjects I want. https://mamme.masters.upc.edu/en/study-program ...
1
vote
2answers
130 views

Genus of a plane curve of the form $\prod_{i=1}^n (a_iX+b_iY+Z) = Z^n$

Does anybody know the genus of the following (projective) plane curve?: $$\prod_{i=1}^n (a_iX+b_iY+Z) = Z^n$$ where the $a_i$'s and the $b_i$'s are complex numbers with $a_j \ne a_i\ne b_i \ne b_j$ ...
2
votes
0answers
79 views

A true statement about sets; expressible? [on hold]

For any pair $(x,y)$ of sets, we obtain a unique set $z=F(x,y)$ by letting $F(\emptyset,y)=y$ and $F(\{x_i\}_{i\in I},y)=\{F(x_i,y)\}_{i\in I}$ if $I\ne\emptyset$ (the recursion eventually reaches the ...
1
vote
0answers
23 views

Reference request - Compact embedding of intermediate space

Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$. ...
2
votes
1answer
80 views

degeneration of reductive group

If $A$ is a mixed characteristic complete DVR (I'm only actually interested in $\mathbf{Z}_p$) and $G/A$ is a closed subgroup scheme of $GL(n)$ whose generic fibre is connected reductive and split, is ...
3
votes
0answers
39 views

Surjectivity of self-isometries as property of metric spaces

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$. A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
7
votes
2answers
391 views

Distributing points evenly on a sphere

I am looking for an algorithm to put $n$-points on a sphere, so that the minimum distance between any two points is as large as possible. I have found some related questions on stackoverflow but ...
0
votes
1answer
153 views

Supplementary notes to Mumford's The Red Book of Varieties and Schemes

I am a graduate student with good mathematical maturity (I took advanced courses like category theory, commutative algebra...). I want to study algebraic geometry from Mumford's red book. I find it ...
0
votes
1answer
20 views

Existenc conditions of single crossing [on hold]

There are two density function $f(x)$ and $g(x)$.They have the commnon support set $[\underline{x},\bar{x}]$. Condition (1)$f(\underline{x})<g(\underline{x})$ and $f(\bar{x})>g(\bar{x})$. ...
0
votes
0answers
11 views

c-superdifferential is unique +cost function is differentiable, then the potential function is differentiable?

Let $M$ be a compact Riemannian manifold, $\mu$ and $\nu$ are two Borel probability measures, the cost function $c(x,y)=\frac{d^2(x,y)}{2}$. It's well known that the infimum of the Kontorovich's ...
5
votes
0answers
65 views

A tensor product for triangulated categories?

Many triangulated categories which show up in mathematics, such as derived categories of various sorts, arise as the homotopy category of a stable $\infty$-category. Stable $\infty$-categories give ...
2
votes
0answers
23 views

integrability of Brownian motion stopped at some stopping time

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion starting at zero and denote by $S=(S_t)_{t\ge 0}$ its running maximum, i.e. $S_t=\sup_{0\le s\le t}B_s$. Given a fixed number $p>1$, define the ...
0
votes
0answers
72 views

Irreducible component of a scheme over a dvr

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
0
votes
0answers
17 views

Tensor product of algebra group and banach space

Let G be a locally compact group and A be a banach space. It is known that the tensor product L^1(G)⊙A is isometrically isomorphic to L^1(G,A). I need proof of it.
3
votes
0answers
30 views

Catenarity of monoid algebras

Let $R$ be a commutative ring, let $M$ be a commutative monoid, and let $R[M]$ denote the corresponding monoid algebra. Suppose further that $R$ is universally catenary. One may ask for conditions on ...
1
vote
1answer
13 views

Reducing join-incomplete lattice homomorphisms to homomorphisms with co-domain ${\bf 2}$

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) > \bigvee_L f(S).$ Is it true ...
0
votes
0answers
44 views

The spring Markov chain on $\mathbb{N}$

I'm trying to understand and learn more about "almost surely bounded" Markov chains on countable state spaces. I'm looking for references where I can learn how to work with more complicated examples ...
-1
votes
1answer
56 views

Bounded metric spaces with non-surjective self-isometry

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$. A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
0
votes
1answer
128 views

Minimum number of people such that 2 can be expected to sit next to each other [on hold]

We are given a large, round table with $n$ seats. It is easy to see that whenever $p\geq \text{int}(\frac{n}{2}) + 1$ people are seated, at least $2$ people will sit next to each other (here ...
0
votes
0answers
30 views

How to find PDF of ordered random variables? [on hold]

Assumpion: Let $X_1, X_2, \ldots, X_L$ be $L$ independent and identical random variables (RVs). Let $F_{X_i}(x_i)$ and $f_{X_i}(x_i)$ be CDF and PDF of $X_i$. Suppose that $F_{X_i}(x_i) = F_X(x_i)$ ...
0
votes
0answers
22 views

probability of reaching a point in a 2d grid in a certain number of steps [on hold]

I have a random walk process in a 2d grid with N steps where N is small. How can I calculate the probability that any given cell was reached in N, N-1, N-2 ... 1 steps. That is, I would like to be ...
0
votes
0answers
18 views

Can I use proximal algorithms on complex real-valued functions?

There is a plethora of literature in proximal operators and proximal optimization algorithms specially for Compressive sensing. A proximal operator is defined as \begin{equation} ...
0
votes
0answers
22 views

Rearrangement of summation expression

Referring to the spherical harmonics expansion in this article: Méléard, P., Pott, T., Bouvrais, H., & Ipsen, J. H. (2011). Advantages of statistical analysis of giant vesicle flickering for ...
1
vote
0answers
56 views

Partitioning graph for Graph Isomorphism

Motivation: I am studying graph isomorphism problem. I am trying to construct a partitioning method to reduce search cases required to get solution of graph isomorphism. Construction: $G$ is a $r$ ...
3
votes
0answers
35 views

Asymptotics of a Bivariate Generating Function

I have the following generating function, $$G(x,y)=\sum_{n,k \geq 0}a(n,k)x^ny^k = \frac{(y^2-y)x+1}{(y-y^3)x^2-(y+1)x+1}$$ and I am interested in obtaining an asymptotic for the sequence $a(n,k)$ ...
1
vote
1answer
132 views

Are there infinitely many $k$ for which $\sigma(k)/k $ is a square? [on hold]

I would like to know if there are infinitely many $k$ for which $$\sigma(k)/k=n^2$$ such that: $m=kn$ with $m,n>0 $ ? Note: $\sigma(\frac{m}{ n})$ is the sum of divisors function of ...
3
votes
1answer
100 views

Polynomials with Unique Critical Value

My question is extremely simple to state: I am looking for a characterization of multivariate complex polynomials $f$ such that $f(Sing(f))=\{0\}$. My motivation is that I recently read somewhere that ...
1
vote
0answers
62 views

binomial coefficients and irrationals

The following, probably either currently impossible to deal with, or having a negative solution, arose from an ergodic theory question, presumably itself currently intractible. I am not a number ...
2
votes
0answers
70 views

What am I missing in this highly oscillatory integral? [on hold]

I want to numerically integrate this equation (in python): $\int_{0}^{\infty}{\rm d}k f(k) J_v(r k)J_v(s k) $, where f(k) is a non-smooth function, and $J_v$ are the Bessel function of the fist ...
2
votes
2answers
147 views

Fibrations on blow-ups of $\mathbb{P}^2$

Let $X_n = Bl_{p_1,...,p_n}\mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ in $n$ general points $p_1,...,p_n\in\mathbb{P}^2$. Let $f_i:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^1$ be the linear ...
2
votes
0answers
69 views

Set of smooth curves on the Hilbert scheme is open. H

Let $H = Hilb_{d,g,r}$ be the Hilbert scheme of genus $g$ curves of degree $d$ in proyective space $\mathbb{P}^r$, over an algebraically closed field $k$. Is it true that the set of points of $H$ ...
1
vote
0answers
27 views

Uniqueness of Riemann Constant Vector Solution

Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map $$A(P) = ...

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