2
votes
0answers
53 views

Lemma 2.1.1.4 in Lurie's HTT

I have encountered a problem in understanding Lurie's proof of the following fact: "Given a left fibration between simplicial sets $q:X \to S$, there exists a functor $$ho(S) \to Ho(sSet)$$ which is ...
1
vote
0answers
26 views

When is a Hankel matrix invertible?

I would like to have some conditions that render a general Hankel matrix of the form \begin{pmatrix}a_1 & a_2 & a_3 &\cdots & a_n \\ a_2 & a_3 & a_4 & \cdots & a_{n+1} ...
5
votes
0answers
34 views

Relations between functors in a recollement

Consider a recollement situation like the following by the very definition of the various functors it follows that $i^* j_*=0$, and $j^! i_* = 0 = j^* i_!$. Also, $j^! i_! = 0 = j^* i_*$ by ...
-3
votes
0answers
25 views

Discrete Mathematics [on hold]

Consider the following sets. A = {a ∈ Z | a = 4k + 2 for some integer k} B = {b ∈ Z | b = 8n + 2 for some integer n} C = {c ∈ Z | c = 4m for some integer m} a) Prove: B ⊆ A. b) Disprove: A ⊆ B. c) ...
1
vote
1answer
62 views

Brauer-Manin obstruction to surfaces of Kodaira dimension 1

Roughly speaking, the Kodaira dimension is an invariant of a variety that corresponds to curvature. One can show that curves of genus $\geq 2$ have Kodaira dimension 1 using Riemann-Roch. In Corollary ...
-1
votes
0answers
29 views

problem in one of H. LlEBECK paper [on hold]

in OUTER AUTOMORPHISMS IN NILPOTENT p-GROUPS OF CLASS 2, H. LlEBECK page 1 let G be p-group of class 2 and let G be such that G/G' is direct product \begin{equation} \prod gp( a_jG') \end{equation} ...
0
votes
0answers
35 views

Some inequality

Assume that $f:[0,2\pi]\to [0,2\pi]$ is a diffeomorphism, and let $\underline f = \min f'$ and $\overline f = \max f'$ and define $$g(s) = \int_0^s e^{if(t)} dt.$$ Assume that $g(0)=g(2\pi)$. It seems ...
1
vote
0answers
32 views

Hypercube edge-coloring problem

Question: Is there a pairing (a fixed point free involution) of the vertices of the $n$-dimensional cube graph, and a $2$-coloring of its edges such that the number of color changes needed to get from ...
0
votes
0answers
22 views

Cayley graphs for finitely presented Lie algebras

I have seen that an important tool of finitely presented groups consists in writing down its Cayley graph with respect to a given set of generators, and then try to extract data like the coarse ...
1
vote
0answers
22 views

About the small set expansion conjecture.

Given a graph $G=(V,E)$ and a $\delta > 0$ one wants to calculate $h(G,\delta)=min_{\vert S\vert \leq \delta \vert V \vert } \phi(S)$. ($\phi(S) = \frac{ E(S,\bar{S}) }{d min \{\vert S \vert , n - ...
-1
votes
0answers
75 views

Writing mathematics for your website? [on hold]

I am an undergraduate student in Computer Science, me and some of my friends are interested in creating a website to teach programming languages and some mathematics topic which are required to be a ...
-3
votes
0answers
35 views

does exist any group that it's derivation be S3? [on hold]

my question is that does exist any group that it's derivation be S3? at all, when we have a group, how we can find another group that it's derivation be the order one?
-2
votes
0answers
47 views

compute singular (co)homology and topological fundamental group of algebraic variety over finite field(or other arithmetic base fields) [on hold]

I was told that singular (co)homology and topological fundamental group do not behave well in varieties and that's why we should have the etale substitution. However I am curious of how these things ...
0
votes
0answers
22 views

Self-diffeomorphisms of fibered knot complements

A knot $K \subset S^3$ is fibered if the complement $S^3 \setminus K$ of (a small open neighborhood of) $K$ is a fiber bundle over $S^1$. (The fiber will be a surface with one boundary component.) ...
-4
votes
0answers
43 views

how can I make a grid [on hold]

I have 16 numbers, 1 to 16. 16 rows. 4 columns. I want each number to appear in each column but to only appear on a row with each of the other 15 numbers once.
0
votes
0answers
11 views

Uniform Boundedness of the Szasz-Durrmeyer Operators on Variable $L^{p(\cdot)}$ Spaces

I am stuck with the uniform boundedness problem associated with the Szasz-Durrmeyer Operators on $L^{p(\cdot)}$ spaces. The Szasz-Durrmeyer operator can be defined as $$M_{n}(f;x)=\sum ...
2
votes
1answer
123 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 a sequence of words $w_i$, where the length of $w_i$ is such that $l(w_i) > 10^8 ...
2
votes
1answer
97 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 ...
1
vote
0answers
26 views

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

Question. Let $(R,m)$ be a Noetherian local ring and $M$ be a finite faithful $R$-module. Let $I$ be an ideal of $R$ such that $IM=mM$. Can we say that $I$ is a reduction ideal of $m$? Recall that $I$ ...
2
votes
0answers
103 views

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

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)) = ...
2
votes
1answer
254 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
94 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
55 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 ...
1
vote
2answers
86 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
79 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
38 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
52 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
28 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
36 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 , ...
15
votes
4answers
1k 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 ...
2
votes
0answers
62 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
44 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?
8
votes
3answers
283 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
123 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
5answers
311 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 ...
4
votes
1answer
144 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
27 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
2answers
92 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, ...
3
votes
1answer
140 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 ...
11
votes
1answer
133 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 ...
3
votes
1answer
125 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
83 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
38 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
52 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
85 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
46 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 ...

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