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

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6
votes
1answer
105 views

Continuity of conjugation actions of Polish groups

Let $G$ and $H$ be Polish groups and let $\psi: G \rightarrow H$ be a continuous injective homomorphism such that $\psi(G)$ is normal in $H$. Then $H$ acts on $G$ by conjugation via $\psi$, in other ...
17
votes
1answer
327 views

$\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?

QUESTION Numerical calculation with gp (first to the default 38-digit precision, then tripled) supports the conjecture that $$ \int_0^\infty x \, [J_0(x)]^5 \, dx = \frac{\Gamma(1/15) \, \Gamma(2/15) ...
6
votes
1answer
270 views

An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...
5
votes
1answer
97 views

In the category of uniform spaces, is the completion of a quotient map also a quotient map?

I asked this question about 2 months ago on math.stackexchange, but so far I received neither comments nor answers. Let $X$ and $Y$ be two Hausdorff uniform spaces. A surjective uniformly continuous ...
22
votes
1answer
1k views

Algebraic proof of Five-Color Theorem using chromatic polynomials by Birkhoff and Lewis in 1946

I'm guessing everyone is familiar with Four Color Theorem which was proved by Appel and Haken using computers. A weaker version of this theorem is Five Color Theorem which states that a planar graph ...
0
votes
0answers
50 views

Criterion for transverse boundary intersection of one-parameter family in $\overline M_{g,n}(X,\beta)$

Suppose we have a projective, nonsingular, convex variety $X$, $\beta \in H_2(X,\mathbb{Z})$ and a family $$ \begin{array}{ccc} \mathcal{C}& \to & X \cr \downarrow& & & \cr B ...
3
votes
1answer
113 views

Mixed (anisotropic) Sobolev spaces

Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, ...
2
votes
1answer
332 views

A question on the bounds of the $n$-th composite $c_n$

While trying to prove the inequality $$c_{p_n-m}+c_{m-n}>p_n+2$$ I tried the bounds of $c_n$ (denotes the $n$-th composite number) given in this paper to prove that the sum $c_{p_n-m}+c_{m-n}$ ...
0
votes
1answer
100 views

Tradeoff among symmetries [closed]

I posted this is mathematics SE and it didn't get much attention, so I'm trying here. I apologize if it is not relevant. Take a set $X \in \mathbb{R}^2$ of nonzero measure $\mu(X) \neq 0$. I am ...
1
vote
1answer
1k views

Does this linear algebra construction based on a graph have a name, and where has it been studied?

In the paper Kochen-Specker set with seven contexts by Lisonek, Badziag, Portillo and Cabello, the following construction is used : Question : Have such constructions been used elsewhere, and if so ...
5
votes
1answer
181 views

$\text{cov}(\mathcal{M})$ vs. $\mathfrak{b}$ vs. $\mathfrak{s}$

Let me first recall some pretty standard notations: $\text{cov}(\mathcal{M})$ is the covering number of the ideal $\mathcal{M}$ of all meager subsets of $\mathbb{R}$; $\mathfrak{b}$ is the bounding ...
8
votes
1answer
216 views

Primer on Eisenstein series

My apologies if this question is a duplicate. I seached, and the closest I could locate is this question, which has very intriguing and intractable (for me) responses. In my continuing journey of ...
5
votes
0answers
104 views

Local version of the Hardy-Littlewood-Sobolev theorem for Riesz potentials: $\|I_\alpha(f)\|_{L^q} \le C \|f\|_{L^p}$?

Recently, I have been studying the properties of the Riesz potential $$ I_\alpha(f)(x) = c_{d,\alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy. $$ The classical Hardy-Littlewood-Sobolev ...
3
votes
1answer
80 views

Horizontal Sobolev space on Carnot group

This question is connected with my previous: Heisenberg group: function without vertical derivative. Here I am trying to look from another side: what is a difference between Sobolev space and ...
8
votes
5answers
499 views

Representation Theory of Lie Groups: Reference Request

I am looking for a reference that describes the correspondence between the (finite-dimensional) representations of (real) Lie groups and the representations of their Lie algebras. More precisely, ...
5
votes
1answer
158 views

What is the monoidal equivalent of a locally cartesian closed category?

If a closed monoidal category is the monoidal equivalent of a Cartesian closed category, is there an analogous equivalent for locally cartesian closed categories? Is there a standard terminology or ...
6
votes
1answer
150 views

Reference for Hopf algebra applications to Feynman diagrams

I need to give a talk about Hopf algebras and I would like to give a (at least) 5 minutes introduction using Feynman diagrams as a motivation. I'm looking for a not-so-heavy reference explaining how ...
1
vote
0answers
57 views

Largest dimensional Lie subgroup of $SU(N)$ [duplicate]

What is the largest (Lie) subgroup of $SU(n)$ in the sense of its dimension. I am aware of this potential duplicate subgroup of SU(N) with maximal manifold dimension , however, the title of this ...
3
votes
1answer
175 views

Quasicoherent sheaves on superschemes

I am interested in learning about super algebraic geometry (some objects I am studying seem to be naturally superstacks, at least in some sense). What would be the best reference for the subject? I am ...
3
votes
1answer
176 views

Kan extensions in the $2$-category of monoidal categories

Kan extensions make sense in any $2$-category. But so far I have only really seen them in the case of the $2$-category of categories, functors, natural transformations and the $2$-category of ...
2
votes
1answer
108 views

Paper of Denis Simon on quadratic equations in dimensions 4, 5?

In several places I have come across references to a 2005-6 preprint of Denis Simon entitled Quadratic equations in dimensions 4, 5, and more This paper gives fast algorithms to find isotropic ...
8
votes
0answers
94 views

Reference for class of involutions containing longest element of finite Coxeter group?

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...
1
vote
1answer
88 views

Combinatorial polynomials from general diagram fillings?

There is a plethora of polynomials defined on partition shaped Young diagrams, (Schur, Jack, Grothendieck,...), and skew Young diagrams. There are also composition shaped diagrams that are responsible ...
15
votes
1answer
424 views

Whatever happened to $L(j)$?

So this question probably shows my inner model theoretic ignorance, but: In "Two remarks on elementary embeddings of the universe" (http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969567), ...
1
vote
0answers
37 views

Second derivative estimates for gradient dipendent elliptic equations

Caffarelli in the article " Interior a priori estimates for solutions of Fully non linear Equations" Ann. Math 130,No.1, 1989 proved that a continuous viscosity solution $ u $ of a uniformly elliptic ...
3
votes
0answers
134 views

Reference request: Flipping the factors in the Künneth formula

I would like to know if there is a reference for the fact that the following diagram commutes: $$ \begin{array}{ccccccccc} 0 & \to & H_*(X) \otimes H_*(Y) & \to & H_*(X\times Y) & ...
2
votes
0answers
65 views

When do Borel $\sigma$-algebras generated by the total variation norm and the weak* topology coincide?

I am almost certain that I read somewhere that the following is true, but I cannot seem to locate the reference. I would be most appreciative if someone could point me to a reference. The result was ...
1
vote
1answer
149 views

Non-Pythagorean proof for the square root of 2 and solution to YBC7289 [closed]

My name is J. Frederic Teubner I am an independent researcher. I wish to publish a proof for the non-Pythagorean solution to the Babylonian tablet YBC7289 and am currently inquiring as to whether or ...
13
votes
3answers
615 views

Inverse problem of Chern Classes

For my graduate (master) thesis I am studying the theory of Chern Classes. As a possible personal development the only sensible idea I have so far, and which I frankly think is impossible, is to work ...
2
votes
1answer
169 views

Open problems books [closed]

As the title might indicate , I would like to look for recommendations for mathematical book that present open problems in depth with commentary. The only book of this type that I've come across is ...
5
votes
1answer
228 views

Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface

It is well known that Grayson's dumbbell neck-pinch1,2 separates into disconnected pieces under mean curvature flow:                     Image ...
9
votes
1answer
253 views

Assuming AD, is every infinite cardinal closed under power set in a choice model?

Assume AD+DC. Assume $\kappa$ is an infinite cardinal and $N$ is a (set or class) transitive model of ZFC containing $\kappa$. Is it true that for all $\alpha<\kappa$, $N$ thinks that the power ...
6
votes
0answers
225 views

Heisenberg group: function without vertical derivative

Let $\mathbb H$ be Heisenberg group with vector fields $$ X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t $$ and $U\subset\mathbb H$ is an open set. I am ...
0
votes
1answer
142 views

Redundancy of the Cantor Enumeration of the Rationals

What is the cardinality of the set of values corresponding to the first $n$ rationals generated in Cantor's enumeration scheme for proofing of their countability? Edit: following the suggestion of ...
1
vote
1answer
233 views

When can we write fundamental units explicitly

Given a number field $K$, the Dirichlet Unit Theorem tells us about the structure of the unit group $O_K^\times$. However, the proofs do not seems to give any way to explicitly write out a set of ...
2
votes
3answers
337 views

A table for irreducible integral representation of finite cyclic groups

Is there such a table where the irreducible integral representations of finite cyclic groups are listed? Edited: Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...
0
votes
0answers
118 views

Heat asymptotics

Consider a compact manifold $M$ with smooth boundary, with either the Dirichlet or the Neumann boundary conditions. Consider a (time-dependent) open ball $B_t \subset M$. Given a fixed $u \in L^1(M)$, ...
1
vote
1answer
43 views

Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains

I am not really familiar with the topic, thus I am looking for some references about the following problem. Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...
2
votes
0answers
78 views

Quantitative estimate of heat dispersion - off diagonal estimates

Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = ...
1
vote
0answers
79 views

Fixed point theorem in ordered spaces

Can someone provide a proof or a source containing a proof of the following theorem Theorem: Let $D$ be a subset of the cone $K$ of partially ordered space $E,$ $F:D\rightarrow E$ be nondecreasing. ...
2
votes
2answers
212 views

Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?

Let $X$ be a singular curve over an algebraic closed field $k$ with characteristic zero. Let $Z$ be the closed subset of singular points on $X$ and $U=X-Z$ be the smooth part, which is an open subset ...
2
votes
2answers
145 views

constant rank theorem for banach spaces

Is there a similar statement to the constant rank theorem for finite dim real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dim Banach space ...
1
vote
1answer
95 views

Collections in direct products and freeness

I am looking for references about the following type of questions: Let $G$ and $H$ be two groups, let $(g_i:i\in I)\subset G$ and $(h_i:i\in I)\subset H$ be collections of group elements, and ...
2
votes
0answers
221 views

Programming workbooks in C++ and Research Math [closed]

I know the basics of C++ by taking a few courses and going through "C++ Primer" by Lippman. As a math graduate student, I would love to get my hands on some programming-math exercises geared towards ...
5
votes
2answers
283 views

References for the moduli space of complex structures

I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...
1
vote
1answer
115 views

Interchanging limits for doubly indexed random sequences

I've encountered the following problem which seems to be quite standard but for which I can't find any proper references (asking on mathematics SE didn't bring up any answers so I'm reposting my ...
1
vote
1answer
72 views

Derivative of a time evolution operator w.r.t. a parameter

Let $N\geq1$ be an integer and let $H:[0,1]^2\to\mathbb C^{N\times N}$ be a pointwise hermitean matrix valued function. For $y\in[0,1]$ and $0\leq a\leq b\leq 1$, let $U_y(b,a)$ be the time evolution ...
6
votes
1answer
236 views

3-manifolds with isomorphic fundamental groups

There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces ...
0
votes
0answers
32 views

A source for integral operators in the context of Arthur-Selberg trace formula

Could you suggest a textbook for integral operators (in the context of Arthur-Selberg trace formula)?
0
votes
0answers
78 views

Boundedness of heat semigroup on $L^1(\Omega)$

On a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary, consider the Laplacian $-\Delta$ with either the Dirichlet or Neumann boundary conditions. More generally, one can also consider ...