# All Questions

**2**

votes

**1**answer

88 views

### How to construct a graph with arbitrarily large girth and large chromatic number? [on hold]

Erdos theorem says it is possible and it is not so easy. What is the general procedure to construct graphs like Grötzsch graph?

**0**

votes

**0**answers

13 views

### Looking to derive bound for modulus of harmonic eigenfunction on weighted graph

We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus:
$\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$
and wish to lower ...

**-4**

votes

**0**answers

45 views

### Height of a tripod [on hold]

I am trying to determine the height of a tripod when the length on the tripod's legs (81") and the distance between the ends that touch the ground are 57" apart.
My thought process so far: When the ...

**4**

votes

**1**answer

401 views

### Beginners Guide to Cartan for Beginners [on hold]

I am working through parts of Cartan For Beginners by Ivey and Landsberg. Thankfully some exercises have solutions, but, we would benefit from some additional guidance.
Question: I am seeking ...

**-3**

votes

**0**answers

119 views

### Cylinder in a topological space? [on hold]

There is a notion of path in a topological space $X$, namely a continuous function f with domain X and codomain the interval $[0, 1]$.
Given that, in a Quillen model category, the dual of a path ...

**0**

votes

**0**answers

38 views

### Is the blowing up the rectifiable set cone?

Let $M$ be a rectifiable set in $\mathbb{R}^N$. For any point $p\in M$, is the following true?: $\lambda_i M$ subconverges to a cone in $p$ for $\lambda_i\to\infty$, i.e. $(\lambda_i M)\cap B(p,R)$ ...

**-2**

votes

**0**answers

48 views

### Proof of Kolomogrov-Sinai Theorem [on hold]

I've seen reference to the result, but have not been able to actually locate a proof of said theorem. If anybody here could point me that way, then I'd greatly appreciate it.

**-3**

votes

**0**answers

66 views

### Isoceles Triangles on a Grid Proof [on hold]

Given: A Finite Set of Unit Squares on a Large Grid. If we were to choose one of those sets of unit squares, we see that the squares of the set are tiled with isosceles right triangles, each with a ...

**-2**

votes

**0**answers

25 views

### Transformation Matrix Problem [on hold]

Can anyone break down this Transformation Matrix process for me after the characteristic polynomial?
http://i.stack.imgur.com/xdvyp.png

**3**

votes

**1**answer

134 views

### Pathological behavior of Lie algebra under a map of abelian schemes

I am trying to understand Example 7.5/9 from the book "Neron models". There one has a discrete valuation ring $R'$ that is the localization of $\mathbb{Z}[\zeta_p]$ at $p$, so that the absolute ...

**7**

votes

**0**answers

123 views

### Preserving Jonsson cardinals

I am (still) interested in trying to characterize and describe forcings that preserve Jonsson cardinals. A cardinal $\kappa$ is a Jonsson cardinal if there is no Jonsson algebra on $\kappa$, i.e. ...

**6**

votes

**1**answer

170 views

### Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$

Consider the following infinite matrix: $A_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for ...

**4**

votes

**1**answer

154 views

### Is there a “natural” bijection between models of $ZFC$ and $ZF\neg C$?

I'm quite confident that there is the same number of models of $ZFC$ and of $ZF\neg C$, which means that there would exist a bijection between the former and the latter, by definition. However, I was ...

**1**

vote

**1**answer

53 views

### Image of (right) inverse trace map $\xi\colon H^{\frac 12}(\partial\Omega) \to H^1(\Omega)$ dense in $H^1(\Omega)$?

Let $\gamma\colon H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ be the linear trace map which has a right continuous inverse $\xi\colon H^{\frac 12}(\partial\Omega) \to H^1(\Omega)$.
Is the image of ...

**2**

votes

**1**answer

91 views

### Hopf structures on “pictorial” descriptions of permutations

There is a well-known Hopf structure on permutations, due to Malvenuto and Reutenauer. Here, the F basis elements are permutations in one-line notation, the product of two permutations is their ...

**1**

vote

**0**answers

39 views

### Reducing enumeration of reduced words in general Coxeter groups to another #P-complete problem

There is a polynomial time algorithm that takes as inputs a Coxeter system $(W,S)$ with $S$ finite (but with $W$ not necessarily finite), say encoded as a Coxeter matrix, as well as two reduced words ...

**3**

votes

**1**answer

63 views

### Application of Egorov's Theorem for Pseudodifferential Operators

Let $\;P_{0} \in OPS^{m}_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic ...

**0**

votes

**1**answer

60 views

### contracting homotopy on simplicial sets

Let $X$ be a topological space and let $PX$ be its space of paths. Let $I=[0,1]$ with coordinate $s$. There is an homotopy
$$
F\: : \: I\times PM\to PM
$$
Defined by $F(s,y)(t):=y(st)$. This map is an ...

**-2**

votes

**0**answers

31 views

### Solving nonlinear system of ODEs

I have the following system of differential equations:
$$
\begin{cases}
\frac{dx}{dt} = (1 - y) x - 0.4 xu
\\
\frac{dy}{dt} = (x - 1)y - 0.2yu
\\
\psi_1' = - \frac{dH}{dx} = (-1 + 0.4u)\psi_1 + y ...

**4**

votes

**0**answers

69 views

### Is there any study about positive definiteness of some matrix space whose matrices don't have to be positive-definite?

--Updated description--
I'm trying to investigate the stability of tensegrity structures, and this question is related to the second order test.
Suppose there is a vector space ...

**4**

votes

**1**answer

150 views

### Which criteria for “uniformly splitting” polynomials?

Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...

**0**

votes

**1**answer

80 views

### References: Solutions of the Bethe Ansatz Equations [on hold]

Could someone show me good references to find solutions of the Bethe Ansatz Equations, for simple cases (using algebraic geometry or others interfaces with mathematics)?

**5**

votes

**1**answer

137 views

### When is $\vartheta(x)>x$? [Skewes number analog]

Let $\vartheta(x)=\sum_{p\le x}\log p$. What is known about the first time $\vartheta(x)>x?$
Bays & Hudson give good upper bounds (slightly improved by Chao & Plymen) on the first crossing ...

**0**

votes

**0**answers

110 views

### Understanding the homotopy category of chain complexes [on hold]

In the definition of the Homotopy category of chain complexes http://en.wikipedia.org/wiki/Homotopy_category_of_chain_complexes , One defines maps between chain complexes in a certain way that I ...

**0**

votes

**0**answers

105 views

### Shimura reciprocity law

I have spent quite some time understanding class fields generated by Kummer extensions and class fields generated by modular forms. Now, I am turning the notch of sophistication a bit to study class ...

**0**

votes

**1**answer

114 views

### On understanding Orlov's LG B model

I try to read Orlov's papers on Landau-Ginzburg model, but I am quite puzzled，there are several questions：
1 the method of truncation is used frequently，(that is: using a bounded above complex $Q$ ...

**-2**

votes

**0**answers

64 views

### Handwaving explanation of “unit root” sought [on hold]

I am struggling to get my head around the concept of "unit root" in relation to time series. And it would be a great help if someone could give me a two or three sentence handwaving explanation of the ...

**-4**

votes

**0**answers

41 views

### Weak convergence in Lp [on hold]

Got a little problem with this ex. I could use some help.
Let $U := \Pi_{i=1}^d(a_i, b_i) \subset \mathbb{R}$ ($a_i < b_i$ for each $i$) and let $f \in L^p(U)$ for some $1<p<+\infty$. Let us ...

**0**

votes

**0**answers

84 views

### A consequenc of a Lie group act on a Riemannian manifold by isometry

I am learning differential geometry for using this topic in my research. I am stuck to prove following Result which I got in a article.
Formulation:
Let $ f: [0, 1]\rightarrow \mathbb{R}^2$ be a ...

**1**

vote

**1**answer

47 views

### An optimization problem in complex space

Consider the following optimization problem
$$
\min \| \textbf{Ax-B}\|
$$
$$
s.t.|x_i|=1,i=1,...,n
$$
where $\textbf{x}\in \mathbb{C}^{n}$ is the optimization varaible, $x_i$ is the $i$-th ...

**1**

vote

**1**answer

340 views

### Numerical evidence and argument against Littlewood conjecture

This is joint work with someone. We got numerical evidence and argument
against Littlewood conjecture, though mistakes are certainly possible.
Littlewood conjecture states that for any two real ...

**2**

votes

**2**answers

97 views

### Uniformly bounded operator family and pointwise convergence

Let $1 \leq p < \infty$ be fixed and let $\Omega \subseteq \mathbb{R}^n$ be open. Let $(Q_n)_{n \in \mathbb{N}}$ be a uniformly bounded family of operators on $L^p(\Omega)$, i.e. there exists ...

**3**

votes

**1**answer

99 views

### Classify set theories whose transitive models sharing the same sets of ordinals are equal

This question is a follow-up from my recent question, Classifying set theories whose standard models sharing the same ordinals are equal
Let's say that a (recursively axiomatizable) set theory $T$ ...

**5**

votes

**2**answers

159 views

### “All retracts are closed” as separation axiom

The starting point of this question is the fact that any retract of a $T_2$-space is closed.
Let's say a topological space $(X,\tau)$ is $T_{\textrm{rc}}$ if all retracts of $X$ are closed.
All ...

**1**

vote

**0**answers

72 views

### Lefschetz fixed notation

If $f\colon X\to X$ is a self-map of a nice space with isolated fixed points, then the Lefschetz fixed point theorem relates a global number to local numbers. Some write: $L(f)=\sum_{x\in ...

**3**

votes

**2**answers

184 views

### When does Skolemization require the axiom of choice?

Skolemization is often used for eliminating existential quantifiers, which is often useful for proving theorems, especially in automated resolution theorem proving. Skolemization in first order ...

**2**

votes

**1**answer

98 views

### Fourier transform of a matrix represented compact lie group

In physics, I come across this kind of integration (in the nonlinear sigma model):
\begin{equation}
S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}]
\end{equation}
...

**3**

votes

**1**answer

95 views

### Balanced binary code that “resists” local decoding?

I am looking for a construction that can be stated as the following coding problem: a binary code with good distance ($d = \Omega(n)$ where codeword length is $n$) that "resists local decoding" in the ...

**8**

votes

**1**answer

155 views

### Higher moments of information and Renyi entropy

For a given discrete probability distribution, Shannon entropy can be though as an expectation value $\langle - \log p \rangle$ (see also: What is entropy, really?, What is the role of the logarithm ...

**6**

votes

**1**answer

234 views

### abelian $\ell$-adic representations in $\widehat{SL(2,Z)}$

In Grothendieck's Esquisse he claims that the action of
$$\text{Gal}(\mathbb Q)\to\text{Out}(\pi_1(M_{1,1})=\text{Out}(\widehat{SL(2,Z)})$$
obtained from the homotopy exact sequence of the étale ...

**11**

votes

**2**answers

294 views

### A bit of history of Verdier duality

I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ?
I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...

**14**

votes

**1**answer

312 views

### Lurie's approach to the bar-cobar adjunction

I've been trying to read Jacob Lurie's approach to the bar-cobar constructions (Higher algebra §5.2 in the 2014-09 version) but I don't recognize what I know about these constructions. I wonder if ...

**-4**

votes

**0**answers

34 views

### Inverse Laplace Transformation [on hold]

I have a question about laplace transformation.
$\frac{8s+4}{s^2+23}$
I tried to split them. $\frac{8s}{s^2+23}$ is similar to cos and $\frac{4}{s^2+23}$ is similar to sin. Can anyone help?
...

**1**

vote

**1**answer

56 views

### Limit involving modified Bessel Function of the second kind

I'm looking for the following limit
$$\lim_{x\rightarrow 0} \frac{\sqrt{\frac{\text{BesselK}^{(2,0)}(0,x)}{\text{BesselK}(0,x)}}}{\log (x)}$$
I believe the limit is finite, and is near -0.578. ...

**4**

votes

**1**answer

410 views

### Modular forms and “too many symmetries”

How do we interpret Barry Mazur's quote of
Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence ...

**3**

votes

**1**answer

165 views

### Coherent cohomology of an abelian scheme and base change

Let $f\colon A \rightarrow S$ be an abelian scheme of dimension $d$. I would like a reference or an argument for the fact that $R^1f_* \mathcal{O}_A$ is locally free of dimension $d$ and that its ...

**9**

votes

**1**answer

367 views

### Is a free group a product of f.g subgroups of infinite index?

Let $F$ be a free group, and let $H,K \leq F$ be finitely generated subgroups of infinite index in $F$. Is it possible that for the set of products we have $HK = F$ ?

**3**

votes

**1**answer

200 views

### A More Advanced Version of Aluffi's Chapter 0

This is a crosspost of this question from MSE.
Paulo Aluffi's Book, Algebra, Chapter 0 aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce ...

**10**

votes

**1**answer

383 views

### Connected components $0-1$ matrices

Let $M$ be a $0-1$ matrix.
Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where ...

**0**

votes

**1**answer

87 views

### Complete solution set of a Convolutional Equation?

Here is a problem that am I stuck and I appreciate any help. In essence, I am trying to show that the only solutions for the described problem are the ones provided below. Best..
Setup: In what ...