3
votes
1answer
88 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 ...
7
votes
0answers
111 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 ...
4
votes
0answers
132 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
2answers
252 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 ...
13
votes
1answer
269 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
0answers
26 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
1answer
52 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
1answer
375 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
1answer
146 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 ...
8
votes
1answer
295 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
1answer
184 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 ...
0
votes
0answers
62 views

Connected components $0-1$ matrices [on hold]

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
0answers
56 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 ...
3
votes
1answer
142 views

A question on models of set theory and Lebesgue measure

In a question and an answer at MO, Joel David Hamkins showed that (if ZFC is consistent) there are models of ZFC in which $V\neq HOD$ and every $\Sigma_2$-definable set has a definable member. Let ...
0
votes
1answer
93 views

equivalence in simplicial category

Let $(\mathcal{C},W)$ be a category with weak equivalences. One can build from $(\mathcal{C},W)$ its hammock localization $L^{H}(\mathcal{C},W)$ which is a simplicial category $\textit{ie}$ a category ...
3
votes
1answer
95 views

automorphism of prime order for group of Lie type in

Thanks for any help. Suppose $S$ is a simple group of Lie type of prime characteristic $p$. we know that every automorphism of $S$ is composite of inner, diagonal, field and graph automorphism of ...
0
votes
0answers
28 views

How do these two extensions of Sobolev spaces relate to each other?

In nonparametric statistics, the following space is often used $$H_{per}^\beta := \left\{f:[0,1]\to\mathbb{R}:\,D^{\beta-1}f\,\text{absolutely continuous and } D^\beta f\in L^2[0,1], \\D^{k}f(0) = ...
2
votes
0answers
67 views

Algorithm to minimally connect line segments in Euclidean plane

Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?" More formally and generally, what I'm looking for ...
3
votes
1answer
171 views

cycle class as Chern class

Let $X$ be a smooth projective complex variety and $Z \subset X$ a codimension $p$ closed algebraic subvariety. Then one can define the class of $Z$ in singular cohomology $[Z] \in H^{2p}(X, ...
-9
votes
0answers
57 views

Using the intuition from the arithmetic of cardinal numbers to ascribe values to functions on the extended real line? [on hold]

Please note, this question is about ascribing exact values to indefinite forms, not about fining corresponding limits, so please do not answer "there is no such limit, so it is indeterminate". First ...
17
votes
2answers
371 views

Unstable homotopy groups of spheres beyond Toda's range

In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are ...
1
vote
0answers
43 views

“Generators” for fusion rings

It's a rather obvious idea in the area of fusion rings, but I haven't found a reference yet. Start with the usual rules for a rank n fusion ring $X_i\bigotimes{X_j}=\Sigma_k{T_{ij}}^kX_k$ and ...
0
votes
0answers
36 views

Baire sets in locally compact Hausdorff spaces

I posed this on 14 Dec. at http://math.stackexchange.com/questions/1067751/baire-sets-in-locally-compact-hausdorff-spaces, but there has been no response: (This is a follow-up to ...
1
vote
0answers
34 views

finding dominating cycles in $2K_2$-free graphs

A cycle $C$ in a connected graph $G$ is called dominating if its complement $V(G)-V(C)$ is an independent set. H.J. Veldman proved in 1983 (Disc. Math. v.43, 281-96) a general result that in ...
2
votes
0answers
53 views

Varieties with few monomials and the n-conjecture

The n-conjecture is a generalization of abc and basically says that the if $a_1 + \ldots + a_n=0$, no proper subsum vanishes and $a_i$ are coprime, then the radical of $a_1\cdots a_n$ can't be too ...
1
vote
1answer
42 views

Is the orthogonal complement of an admissible subcategory admissible itself?

I´m studying Huybrechts book "Fourier-Mukai transforms in algebraic geometry" and I came up with the following: as an example of semi orthogonal decomposition of a triangulated category $\mathcal{D}$ ...
5
votes
0answers
60 views

Unbalanced equipartitions

Let $K$ be a compact convex set in the plane. Say that a perimeter-halving partition of $K$ is a partition of $K$ into two pieces by a chord (a segment with endpoints on the boundary $\partial K$) ...
4
votes
1answer
264 views

Cap product à la Poincaré

Recently, it became apparent to me that I was not the only one who always first thought in terms of cap product before actually computing a cup product. There is no denying this is evil, but I found ...
-8
votes
1answer
137 views

Do the mathematicians really know the exact values of what usually called “indeterminate forms”? [on hold]

First of all I would point out that exact value of a function and the limit of the function in that point do not necessarily coincide. For instance, it is often assumed that $0^0=1$ even though the ...
0
votes
0answers
51 views

Reference : Special case of Banach-valued function integration by parts

Let $u \in L^p (0,T; L^1(\Omega))$ with $\partial_t u \in L^p(0,T; L^1(\Omega))$ and $v \in L^q (0,T; L^\infty(\Omega))$ with $\partial_t u \in L^q(0,T; L^\infty(\Omega))$ (with $1/p+1/q=1$ and $p \in ...
3
votes
0answers
53 views

Is there an efficient way to visualize the bifurcation locus of this family of functions?

I have been trying to help out with this question from math.stackexchange. It concerns the family of functions: $$f_\alpha(z,w) = \frac{\alpha + z}{1 + w}.$$ and an iteration scheme: $$z_{n+1} = ...
9
votes
0answers
163 views

Is every projective space curve a set-theoretic intersection of two surfaces? What is known about this question?

I am sorry if this question has been already asked, couldn't find anything similar myself. I have recently recalled this long standing open problem of whether every curve in $\mathbb{P}^3(\mathbb{C})$ ...
0
votes
0answers
32 views

Extension of homomorphism to place in quotient field, or of local ring to valuation ring in quotient field

Let be given a domain $R$ (that can be supposed to be integrally closed if this can help), and $\varphi$ an homomorphism of $R$ into a field $F$. $\varphi$ extends uniquely to a homomorphism ...
1
vote
1answer
113 views

intuitive interpretation of analytic spread

I am studying analytic spreads from Bruns-Herzog's book. The definition is clear but calculation of the analytic spread of an ideal is hard for me in practice. I wonder if it is hard for you too. ...
1
vote
1answer
37 views

Retractions and left-factoring morphisms

Let $\mathcal{C}$ be any category and let $A, B$ be objects. A retraction is a morphism $r: A\to B$ such that there is $s:B\to A$ such that $r\circ s:B\to B$ is the identity. A morphism $l: A\to B$ ...
2
votes
0answers
61 views

What are interesting open problems in pseudo-differential operators?

May I ask what are some interesting open problems in the field of micro-local analysis (or classical analysis, semi-classical analysis, etc) using pseudo-differential operators? To my knowledge the ...
1
vote
0answers
50 views

(Affine) Schemes and the point of view of morphisms with values in a field

Let $A$ be a commutative ring with unity. If $\varphi_1 : A \rightarrow K_1$ and $\varphi_2 : A \rightarrow K_2$ are ring morphisms from $A$ to fields, $\varphi_1$ and $\varphi_2$ are said to ...
0
votes
0answers
60 views

sufficient condition of complete intersection [on hold]

According to Corollary 2.8 and the front part of Section 3 of this paper, if $X:= Q_1\bigcap Q_2\bigcap Q_3\subset \mathbb{P} _{\mathbb{C}}^{4}$ be a connected and purely $1$-dimensional intersection ...
-4
votes
0answers
58 views

Flow in graph. Proof [on hold]

Let $(S_1, \overline{S_1} ) , (S_2, \overline{S_2} )$ be minimum cuts in some network. Is it true that $(S_1 \cap S_2, \overline{S_1 \cap S_2)}$ is a minimum cut in this network?
-4
votes
0answers
101 views

show that L(X,Y)banach then Y banach

Let {Xα}α∈A be a collection of Banach spaces. It is easy to show that P={(xα):supα∥xα∥<∞} with ∥(xα)∥=supα∥xα∥ is a banach space. If the indexing set A is finite, then it is easy to show that P ...
1
vote
1answer
65 views

Any two bivariate algebraically dependent polynomials are always in the same ring generated by some bivariate polynomial?

If $f(x,y)$ and $g(x,y)$ are two algebraically dependent polynomials over some field $k$, is it true that there exists a bivariate polynomial $p(x,y)$ such that both $f(x,y)$ and $g(x,y)$ are in the ...
4
votes
0answers
70 views

An extension of group schemes admitting Neron models

Let $R$ be a discrete valuation ring, $K$ its field of fractions, and $$ 0 \rightarrow G_K' \rightarrow G_K \rightarrow G_K'' \rightarrow 0$$ a short exact sequence of smooth $K$-group schemes of ...
2
votes
1answer
71 views

On transforming pair of bivariate polynomials to pair of univariate polynomials by applying polynomial map

We know that a polynomial map $f(x,y), g(x,y)$ is polynomial automorphism if there exists polynomials $p(x,y)$ and $q(x,y)$ such that $f(p,q)$=x and $g(p,q)=y$. Jacobian conjecture tries to ...
-6
votes
0answers
27 views

Writing an integral for a bounded volume [on hold]

Good day folks! I'm trying to write an integral which could represent the volume of the shaded area below. However my integral calculus is quite poor, so I need some help on that equation. ...
22
votes
3answers
698 views

Graduate program applications that require questionnaires and other non-letter material

In the December 2014 AMS Notices, a letter to the editor (http://www.ams.org/notices/201411/rnoti-p1311.pdf) by Deconinck and Medlock addresses the problem of (math) graduate programs requiring letter ...
4
votes
1answer
275 views

Is the space of real conics with a singular point an orientable manifold?

Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set of such a polynomial gives a real curve ...
0
votes
1answer
137 views

A question in Sasakian geometry

Let $(S,\eta, \xi)$ be a Sasakian manifold with killing vector field $\xi$, then we have the following exact sequence $$0\to <\xi>\to TS\to \frac{TS}{<\xi>}\to 0$$. Can ...
13
votes
1answer
249 views

ULU Decomposition of a matrix

Let $g \in GL_n(\mathbb{F}_q)$. Is it true that we can always write $g = u_1lu_2$, where $u_1$ and $u_2$ are upper-triangular and $l$ is lower-triangular? Note that I'm not requiring that the matrices ...
-1
votes
0answers
38 views

Gaussian Elimination for Orthogonal groups [on hold]

We known the classical Gaussian elimination algorithm where we can reduce any invertible matrix to a diagonal matrix by row-column operations. Is there row-column operations and Gaussian elimination ...
0
votes
0answers
87 views

Roots in the solution

It is known that for a one-dimensional self-adjoint operator with periodic boundary conditions, the number of roots is directly related to the eigenstate this eigenfunction belongs to. Now it is ...

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