0
votes
0answers
11 views

Infinite sequence avoiding a countable set of words

As an application in group theory, I would need an infinite sequence over a finite alphabet, that avoids countably many words, each of length at least 10^8. I have found several results about avoiding ...
1
vote
1answer
36 views

Periodic points in C^2

I came up with a problem which is similar to the following quesitons: Consider a map: $f(x,y)=(y^2-2,xy-2)$. It is seems that the number of periodic points of given period is bounded. I want to ...
0
votes
0answers
9 views

$IM=mM$. can we say that $I$ is a reduction ideal of $m$?

Definition. Let $R$ be a Noetherian ring􀀀, $I$ a proper ideal,􀀀 and $M$ a finite $R$-module. An ideal $J\subset I$ is called a reduction ideal of $I$ with respect to $M$ if $JI^nM = I^{n+1}􀀀M$ for ...
0
votes
0answers
40 views

Can someone elaborate on a categorical/homotopical point made in the proof that $\Sigma 2 = S^1$, given in the HoTT book?

In chapter 6, specifically in the section about suspensions a proof is given that ∑2 = S^1. The book says that $\mathrm{transport}^{x \mapsto g(f(x)) = x}(\mathrm{refl}_N, \mathrm{merid}(y)) = ...
1
vote
1answer
96 views

Topological Grothendieck Construction

Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The Grothendieck construction is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and ...
0
votes
0answers
62 views

Computing coefficients to power sums

Is it possible to find the (distinct) coefficients of monomials such as $$(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3})^4\cdot(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})\cdot(x_{1}+x_{2}+x_{3}+x_{4})^{2}$$ ...
-2
votes
0answers
53 views

Why does the critical line for Riemann's zeta function lie at real part 1/2 rather than real part 0? [on hold]

Sorry for the un-mathematical way of formulating this question in the title, feel free to edit the title if that seems more appropriate. What I actually like to know is: Is this yet another instance ...
0
votes
1answer
40 views

Homotopy bounds in simply connected complete Riemannian manifolds

Let $M$ be a simply connected complete Riemannian manifold, and let $x\in M$. Does there exist a nondecreasing function $R:\mathbb R_+\to\mathbb R_+$ such that, for every $r>0$ and all paths ...
0
votes
2answers
52 views

Matrices congruent to each other via a permutation

Consider the collection of all integer matrices and partition them via an equivalence relation $A\sim B\Leftrightarrow \exists$ a permutation matrix $P$ such that $B=PAP^T$. Is some canonical form ...
2
votes
0answers
40 views

Explict form of $E_\infty$-morphisms between differential graded commutative algebras

This is a partial duplicate to this MO question, I apologize for that. I'm asking since the answers there still do not allow me to work out an answer to my question, which is a bit more specific. ...
0
votes
0answers
27 views

regular locus of an affine domain

Let $A$ be an affine domain over a field $k$ (need not be algebraically closed). Let $\mathfrak{p}$ be a prime ideal of $A$, such that $A_{\mathfrak{p}}$ is a regular local ring. Does there always ...
1
vote
0answers
35 views

Aperiodic graphs

The concepts of being non-periodic and aperiodic for tilings have obvious versions for connected graphs with a countable set of vertices and a finite number of edges meeting at each vertex. A graph ...
1
vote
1answer
19 views

Linear intersection number and maximum degree

This question is inspired by a Andrew D. King's comment in Linear intersection number and coloring (not chromatic) number A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set ...
-2
votes
0answers
23 views

Inverse Trigonometric Functions [on hold]

I have the following question: for which I need to prove the above to be x/2 I tried to first convert it to : then multiply and divide by to get this: But have no idea what to do next , ...
10
votes
3answers
529 views

Maryam Mirzakhani's works

Maryam Mirzakhani has made several contributions to the theory of moduli spaces of Riemann surfaces. Mirzakhani was awarded the Fields Medal in 2014 for "her outstanding contributions to the dynamics ...
1
vote
0answers
35 views

K-theory of coherent sheaves on complex manifolds: references and gamma-filtration?

For a complex manifold $X$ one has an exact category of locally free coherent sheaves; so it seems to be no problem to define certain $K$-theory (I do not know whether the $K$-groups given by the ...
0
votes
0answers
33 views

Norm on C$^*$-algebra [on hold]

Given two orthogonal elements $a,b$ in a C$^*$-algebra $A$ (i.e. $a b^* = b^* a=0$) we have $\| a + b\| = \max\{ \|a\|, \|b\|\}$. How do I show?
7
votes
1answer
135 views

Abstract connectedness

Is there an abstract structure that characterizes connectedness, analogously to how topological spaces characterize continuity? Here's one way to make this question more precise: if $(X,T_X)$ is a ...
2
votes
1answer
62 views

Exact reference for Liouville theorem

It seems hard for me to find that the solution of the following equation $$ \Delta u+e^u=0 $$ defined on a simply-connected domain $D\subset R^2$ must be of form $$ ...
8
votes
4answers
153 views

Application of Fraïssé construction in set theory

As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property. Now I would like to know ...
0
votes
0answers
53 views

Matrices over a finite field with given Jordan normal form over the algebraic closure [migrated]

Can one describe the (conjugacy classes of) square matrices over a finite field such that over the algebraic closure of this finite field their Jordan normal form consists of one Jordan block? (Such ...
0
votes
0answers
16 views

Alternative form for weighted least squares

Coefficients $\beta$ can be estimated from $y$ by weighted least squares with: $ \beta = (X^T\Sigma^{-1}X)^{-1} X^T \Sigma^{-1} y $ where $\Sigma$ is the covariance matrix of the noise. Let $N$ be ...
1
vote
0answers
76 views

Averages over integer points of the sphere

A paper of William Duke sketches a proof that integer points on the sphere are equidistributed. $$ V_N = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = N \} $$ Up to reflections across the $x$, $y$ ...
2
votes
0answers
23 views

Reference that contains examples of absolutely indecomposable representations of quivers over a finite field

Is there a reference that lists/discusses examples of absolutely indecomposable representations of quivers over a finite field (absolutely indecomposable = does not decompose into a direct sum over ...
1
vote
0answers
34 views

$\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?

I am trying to prove the following Lemma, which seems intuitive, but I still have doubts: Lemma Given a Brownian motion $\{W_t,\mathcal F_t:0\le t \le1\}$, two progressivley measurable processes, ...
1
vote
1answer
92 views

Dihedral extension of 2-adic number field

Sorry if the question is too long and maybe elementary. I am reading a paper by Hirotada Naito on "Dihedral extensions of degree 8 over the rational p-adic fields". To generate dihedral extension ...
8
votes
0answers
82 views

Variety acquiring rational point over any quadratic extension

Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$? If ...
2
votes
1answer
104 views

Converse to Weil Restriction of Scalars

Let $k$ be a field of characteristic zero (I'm only interested in number fields), and let $\mathbb{G}_{/k}$ be a linear algebraic group defined over $k$ which is almost $k$-simple (all normal ...
5
votes
0answers
68 views

Degenerate linear recurrence sequences

Let $(u_n)_{n \geq 0}$ be a linear recurrence given by $$u_n = a_1 u_{n-1} + \cdots + a_k u_{n-k} \quad \forall n \geq k ,$$ where $u_0, \ldots, u_{k-1}, a_1, \ldots, a_k \in \mathbb{Z}$. We recall ...
1
vote
0answers
36 views

Diffusion in a bounded domain

Let us consider an $\mathbb{R}^d$ diffusion $$dX_t = dW_t +\mu(X_t)dt.$$ Let further $D\subset \mathbb{R}^d$ be a bounded connected open domain. By $Y^D$ we denote the diffusion $X$ restricted to ...
0
votes
0answers
22 views

Integral Domains [migrated]

I have to proof that ($\ N_A,1_A,T)$ are a Peano system where $\ T:N_A\rightarrow N_a$ $\ x \mapsto x+1_A , x \in N_A$ and $\ N_A= \{n 1_A|n\epsilon N \} $ where N are the natural numbers and A is an ...
1
vote
0answers
33 views

$H^1$ convergence of eigenfunctions of Schrödinger operators [migrated]

Consider the Schrödinger-Operator with Potential $V\in L^\infty(\Omega)$ with Dirichlet boundary conditions $$ H^D=-\Delta + V $$ and let $u_{i,n}\in H_0^1(\Omega)$ be the first, nonnegative ...
0
votes
0answers
51 views

Asymptotic sequence and asymptotic expansion [on hold]

If $(f_k(x))$ is an asymptotic sequence as x to infinity and $\phi=a_0f_0+a_1f_1+a_2f_2+...$ (equality) where $a_i$ are constant. Is $a_0f_0+a_1f_1+a_2f_2+...$ an asymptotic expansion of $\phi$?
1
vote
0answers
81 views

Good covering of a sphere

Consider a sphere $S_r(0)$ with center at zero and radius $r$ in the Hamming space $\{0,1\}^n$. We will be interested in covering this sphere with balls of radius $\rho < r$. We know that there ...
0
votes
0answers
45 views

The right expansion of a square root matrix

I have some issues in finding an asymptotic expansion for a square root matrix and I have already posted a question (Asymptotic expansion square root matrix). Somebody redirected me to a post where ...
5
votes
0answers
61 views

Spectral Sequences of Parametrized Spectra

I apologize if this question is of the form "what are some interesting problems in bla" but I was wondering if anybody have studied the following set-up: Suppose that I have a parametrized spectra ...
0
votes
0answers
34 views

Intersection and union of torsion classes

One of the main result in Cassidy, C., M. Hébert, and G. M. Kelly. "Reflective subcategories, localizations and factorization systems." Journal of the Australian Mathematical Society (Series A) ...
1
vote
0answers
132 views

“For sufficiently large” vs. “For all sufficiently large” [on hold]

A purely grammatical question: Do people generally prefer: "For sufficiently large x,..." or "For all sufficiently large x,..." or not care? Or might you use either according to context? The meaning ...
3
votes
2answers
346 views

Category of Gödel Codings? [Reference Request]

Consider computation with the integers $\mathbb{Q}$. The traditional theory of recursive functions on $\mathbb{N}$ applies to $\mathbb{Q}$ by the identification of $\frac{a}{b} \in \mathbb{Q}$ with ...
2
votes
0answers
33 views

Global existence for infinite dimensional ODE

Let us consider the ODE $\hskip3pt \dot x=F(t,x)\hskip3pt $ in an infinite-dimensional Banach space $E$, where the flux $F$ is defined and continous from the whole $\mathbb R\times E$ into $E$. ...
2
votes
1answer
51 views

Bounds on Hilbert-Schmidt norm of difference of products of matrices

I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in). I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and ...
1
vote
0answers
63 views

dimension of a scheme and degree of an L-function [on hold]

I try to understand correctly the notion of scheme, as Serre in the third volume of his Oeuvres defines zeta and L-functions in this context. What seems interesting to me is that he states a theorem ...
3
votes
0answers
77 views

Deligne-Lusztig and Character sheaves

Consider: $G$ - a nice group ($GL_n$) over a finite field $F$. $X$ - the flag variety. Consider a nice $G$-equivariant $l$-adic sheaf $\mathcal{M}$ on $X \times X$, equipped with Weil structure. Fix ...
1
vote
1answer
51 views

Stalks of higher direct image under open embedding

Let $U$ be an open subset of $\mathbb P^1$ without two points (say $t=0$ and $t=\infty$) and $j: U\to \mathbb P^1$ be an open immersion. Ground field $k$ is algeraically closed. Let $G$ be the group ...
2
votes
0answers
59 views

Errata for Getzler-Kapranov “Cyclic operads and Cyclic homology”

Do you know if anyone made an errata for the Getzler-Kapranov paper "Cyclic operads and Cyclic homology"? I was trying to read it, and I have found (at least I think I did) quite a few typos not all ...
0
votes
2answers
82 views

Inverse of a matrix expression

Let $$X_i = \left(I - P\left(I - t_it_i^T\right)\right)^{-1}$$ where $P$ is an $N\times N$ matrix and $t_i$ is a vector of $N$ elements. Is there a way to simplify this expression in order to ...
0
votes
0answers
7 views

Encoding and Transforming Data for a Logistic Regression

When running a logistic regression, the result of the regression is a value that could fall in $(-\infty, \infty)$. You run it through the logistic function and get a value in $(0, 1)$. So far, so ...
5
votes
0answers
90 views

Is the moduli space of curves arising from wild ramification smooth?

Fix a natural number $g$, a prime $p$, and a $p$-group $P$. Let $C$ be a smooth projective curve of genus $g$ with a faithful action of $P$ and an isomorphism $C / P \cong \mathbb P^1$ such that $P$ ...
1
vote
0answers
48 views

Yang-Mills Functional and Energy

I have a question about the meaning of Yang-Mills Functional. It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is: ...
3
votes
1answer
160 views

Motivic integration in positive characteristic: how much is known?

It seems that in papers on motivic integration people usually assume the base field to have characteristic $0$ (and algebraically closed?). My question is: how much can one prove over a positive ...

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