This tag is used if a reference is needed in a paper or textbook on a specific result.

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42 views

Proof that Markov shift is pointwise dual ergodic

I am looking for a reference of the proof that a Markov shift is pointwise dual ergodic, I tried google it but with no success.
2
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0answers
35 views

Local time for drifted Brownian motion and comparison results for reflected diffusion

Suppose $X(t) = x+ \mu t + \sigma W(t)$ where $x\ge 0$, $\mu, \sigma>0$ are real constants, and $W$ is a standard Brownian motion. The Skorohod decomposition of $X(t)$ can be written as $Z(t) = ...
5
votes
2answers
154 views

Formal languages with non-unique interpretations of terms

In mathematical logic and model theory, one considers interpretations of syntactic expressions: terms without free variables are interpreted as elements of some structure, formulas without free ...
1
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0answers
31 views

Introduction to free boundary problems (that are not Stefan problems)

Could someone recommend some notes/papers that deal with existence/regularity of free boundary problems arising from parabolic equations (excluding Stefan type equations)? I am thinking of eg. ...
6
votes
1answer
234 views

Euler's Triangular Number closure properties

Burton, in "Elementary Number Theory", states that the following problems are due to Euler 1775: If $n$ is a triangular number, then so are $9n+1$, $25n+3$ and $49n + 6$. R. F. Jordan in the J. ...
1
vote
1answer
91 views

Characterization of $d$-gonal curves on a K3 surface

Let $X$ be a K3 surface and $C$ a curve on $X$. We say that $C$ is $d$-gonal if it admits a pencil of degree $d$ (and none of smaller degree). I am wondering if there exist characterizations of ...
0
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0answers
74 views

Evolution of local oscillation of scalar curvature under Ricci flow

I apologize in advance if the question will turn out to have an obvious answer but my knowledge of Ricci flow is quite limited. Let $(M,g)$ be a smooth compact Riemannian manifold. I denote by $d_{g}$ ...
2
votes
2answers
160 views

Connected unipotent algebraic groups

Let $G$ be a connected unipotent algebraic (affine) group defined over a perfect field $k$. Then $G$ is $k$-isomorphic as an algebraic $k$-variety to the affine space $\mathbb A^n_k$. Is there any ...
3
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0answers
53 views

Lorenz attractor power spectrum

If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...
2
votes
0answers
38 views

Conjugacy classes of a triple

In the paper " THE COMPONENT GROUPS OF NILPOTENTS IN EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS" by D. King I am unable to proof the lemma 3.7 which is omitted there. The lemma is following: Let $\mathfrak ...
4
votes
1answer
105 views

PDEs on torus $\mathbb T$

(Hope this question is o.k. for MO) I have been learning PDE(non linear dispersive equations) techniques, mainly using harmonic analysis(kind of Strichartz estimates, estimates for unimodular ...
1
vote
1answer
106 views

Request a paper by Fred Cohen

I am looking for the following paper by Cohen, F. R.: On combinatorial group theory in homotopy. Homotopy theory and its applications (Cocoyoc, 1993), 57–63, Contemp. Math., 188, Amer. Math. ...
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0answers
68 views

Lucasian Primality Criterion for Specific Class of $k \cdot 2^n-1$

Definition Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ , where $m$ and $x$ are nonnegative integers . Conjecture Let $N=k\cdot 2^n-1$ such ...
7
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0answers
132 views

Singular cohomology of $BG$ and Borel cohomology of $G$

Stasheff, in "Continuous Cohomology of Groups and Classifying Spaces", attributes the following result to Wigner. For $A$ a discrete abelian group and $G$ a finite dimensional locally compact, ...
0
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0answers
68 views

Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example). Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...
2
votes
2answers
266 views

Geometrical interpretation of a Schrödinger operator

Consider a $2 \times 2$ Hermitian (or symmetric) matrix-valued function $$g(x) = \{ g_{jk}(x)\}_{j,k=1,2}, \quad x \in \mathbb{R}^{2},$$ such that $0 < m_{-}I \leq g(x) \leq m_{+}I$, for some ...
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2answers
611 views

History of integral notation for coends

I'm searching the wheres and whys about the integral notation for co/ends. Who was the first to adopt it? Can you give me a precise pointer or tell me the whole story about it? Was s/he motivated by ...
0
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0answers
105 views

Solution to system of polynomial equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
1
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0answers
129 views

Sets of coprime numbers

Consider the set $\{0, 3, 7, 15\}$ of four integers. If you add each of these numbers to a fixed power of 2, then the resulting four numbers are pairwise coprime. For example, $\{4, 7, 11, 19\}$ are ...
0
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1answer
117 views

Equivalence of two definitions of Sobolev spaces

Good morning, I am looking for a reference about the following fact that seems to be folklore. Define the Sobolev (Beppo Levi?) space $$ D^{1,p}(\mathbb{R}^N) = \left\{ u \in L^{p^*}(\mathbb{R}^N) ...
1
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0answers
100 views

Automorphism group of a modular curve and its action on the set of cusps

Let $X$ be a modular curve, that is the compact Riemann surface obtained by adding cusps to a quotient $Y=Y_\Gamma=\mathbb H/\Gamma$ of Poincaré upper half-plane $\mathbb H$ by a congruence subgroup ...
0
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1answer
92 views

Action of the pure braid group on the commutator subgroup of a free group

Let $P=P_n$ be the pure braid group on $n$ strands and $F=F_n$ the free group on $n$ generators. I'm interested in a nice description of the action of $P$ on the derived subgroup $F'$ which somehow ...
4
votes
1answer
97 views

Maps between spaces of non-empty compact subsets with the Hausdorff distance (reference request)

Let $X, Y$ be metric spaces, and let $PX$ (resp. $PY$) be the set of all non-empty compact subsets of $X$ (resp. $Y$) with the Hausdorff metric. A continuous map $f\colon X\to Y$ induces a continuous ...
0
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0answers
78 views

Defining density of a random function using Radon-Nikodym Theorem

Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$. Let $X$ be random function defined ...
4
votes
1answer
169 views

References to proofs of upper and lower bounds on the number of coprimes in an interval?

On the first page of the article "When the sieve works", the authors present upper and lower bounds for $S(T,T+x;\mathcal{E})$; the number of integers in the interval $(T,T+x]$ that are coprime to all ...
3
votes
1answer
253 views

Does this function have any exponential growth?

Has anyone seen any function of the following type? $$ g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0. $$ The question is whether for some constant ...
2
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1answer
147 views

Axiomatization of Degree Theory

I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies Normality ...
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0answers
108 views

Proper monomorphisms in complex analytic spaces

In the algebraic geometry of schemes, we know that monomorphisms which are universally closed (= every pullback is a closed map) and of finite type are closed immersions. See Gortz, Wedhorn "Algebraic ...
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0answers
43 views

Reference for a special case of the Hanson-Wright inequality

I would like find tail bounds for the expression $$ \begin{align*} \left|\left\langle a,\phi\right\rangle \left\langle \phi,b\right\rangle -\left\langle a,b\right\rangle\right|, \end{align*} $$ where ...
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1answer
121 views

Decidable theorem or result that is not weaker than Tarski's theorem

I am wondering what other decidable theorem or results that is not weaker or stronger than Tarski's theorem. Could any one give reference or a simple introduction about such result known in their ...
4
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0answers
186 views

Etale Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
1
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0answers
61 views

Kontsevich integral for 2-bridge knots

Are there any articles that explain a formula for Kontsevich integral of 2-bridge knots?
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3answers
285 views

Reference for a strong intermediate value theorem for measures

Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
4
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1answer
157 views

Nuclearity noncommutative torus

I read that the Noncommutative torus (rotation algebra) is nuclear when $\theta\in\mathbb{R}\setminus\mathbb{Q}$. Unfortunately, I haven't found a proof. Could someone give me a reference and/or an ...
0
votes
1answer
106 views

Stability conditions of coherent sheaves on abelian 3-folds

My work for now consists on understanding stability conditions of coherent sheaves on abelian 3-folds. I have the book by D. Huybrechts (the geometry of moduli spaces of sheaves), But I would like to ...
3
votes
1answer
160 views

Determinant of the oriented adjacency matrix of a tree

Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries $$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\ -1 ...
2
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3answers
567 views

Koebe–Andreev–Thurston theorem - where can I find a proof?

Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two ...
2
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0answers
63 views

What is the computational complexity to compute the integral numerically?

Given $$\int_{\Delta}\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}$$ where $P_i$ is polynomial(that is $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomial) whose coefficients are ...
2
votes
1answer
128 views

A perfect $(n,k)$ shuffle function

Suppose you have a deck of $n$ cards; e.g., $n{=}12$: $$ (1,2,3,4,5,6,7,8,9,10,11,12) \;. $$ Cut the deck into $k$ equal-sized pieces, where $k|n$; e.g., for $k{=}4$, the $12$ cards are partitioned ...
4
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0answers
84 views

$A_\infty$ structure on sum of twists of structure sheaf

Fix $n$ and let $P^n$ be projective $n$-space. Let $S = k[x_0, \dots, x_n]$. Set $A^0 = \bigoplus_{d \ge 0} H^0(P^n, \mathcal{O}(d))$ and $A^n = \bigoplus_{d < -n} H^n(P^n, \mathcal{O}(d))$. I ...
4
votes
2answers
251 views

Powers of finite simple groups

I have heard about the following result: for each finite simple non-abelian group $S$ and each natural number $r\ge 2$ there exists a number $n=n(r,S)$ such that the power $S^n$ is $r$-generator but ...
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2answers
260 views

Notion of manifold curvature?

Consider a particular embedding of a $C^2$ manifold $\mathcal{M}\subseteq\mathbb{R}^m$. Given $p\in\mathcal{M}$, suppose $\epsilon>0$ is small enough that the portion $U$ of $\mathcal{M}$ which is ...
2
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1answer
60 views

Calculating the “Belvedere Hull” of a Simple Planar Polygon

As an informal motivation the problem, imagine a tower with polygonal footprint, that is located in a beautiful landscape, the "Belvedere Hull" is then related to the directions, in which one would ...
3
votes
1answer
116 views

Systems of equations in Boolean Algebra

I have to study systems of equations in a Boolean algebra, the matrix is $m\times n$ with $m\neq n$. The Boolean algebra is actually the simplest one, it contains only $0$ and $1$, let us denote it by ...
11
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1answer
706 views

Von Neumann's consistency proof

In the paper Zur Hilbertschen Beweistheorie, John Von Neumann has proposed a consistency proof for a fragment of first-order arithmetic (the fragment without induction and with the successor axioms ...
1
vote
1answer
84 views

Piercing of subspaces in a projective space?

The "piercing subspace" problem may be stated as follows: There are given several subspaces in a projective space, rather non-intersecting. Find an additional subspace of a prescribed dimension that ...
4
votes
1answer
225 views

Picard of the product of two curves

Can anyone point to me where I can find the proof that the Picard group of the product of two curves is isomorphic to the product of the Picard groups times the hom among the Jacobians? Does the ...
8
votes
1answer
193 views

amenable + without $BS(m,n)$+finite $K(G,1)$implies virtually cyclic?

I heard from someone that the following problem is an open question. (Open Problem 1)For a countable discrete group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle ...
2
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1answer
292 views

fixed point and homotopy fixed points

Let $G$ be a group and $X$ be a $G$-space (finite G-CW-complexe when needed). Let $p$ a prime number and $G= \mathbf{Z}/p\mathbf{Z}$, If I'm not wrong Miller-Lannes,... theory provides tools and ...
5
votes
1answer
174 views

What does an endomorphism in a triangulated category give rise to?

Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the ...