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

3
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2answers
86 views

Geometric description of a certain sphere bundle

It appears to be a standard fact in topology that $\mathbb{C}\mathbb{P}^2\#-\mathbb{C}\mathbb{P}^2$ has a structure of a $\mathbb{S}^2$ bundle over $\mathbb{S}^2$. Is there a nice geometric ...
1
vote
0answers
62 views

Different types of a test functions in weak solutions of a PDEs

In the literature about PDEs it is easy to find books that talk about weak solutions of a partial differential equations. A short reminder of the most usual definition is given bellow. Let's ...
-1
votes
1answer
52 views

Elementary quantum scattering problem on the line.

Let us consider the quantum scattering problem on the line with the Hamiltonian $$H=-\frac{d^2}{dx^2}+ V(x),$$ where $V(x)=1$ when $x\in (0,a)$, and $V(x)=0$ otherwise. It is easy to see that $H$ ...
11
votes
1answer
192 views

Roots of lacunary polynomials over a finite field

If $P$ is a polynomial over the field $\mathbb F_q$ of degree at most $q-2$ with $k$ nonzero coefficients, then $P$ has at most $(1-1/k)(q-1)$ distinct nonzero roots. Does this fact have any standard ...
4
votes
0answers
32 views

Is there an adaptation of the theory of standard forms and Tomita-Takesaki theory to the $\mathbb{Z}_{2}$-graded case?

Let $A$ be a von Neumann algebra acting on a Hilbert space $H$, and suppose that $\Omega \in H$ is a cyclic and separating vector for $A$. Then in Tomita-Takesaki theory one defines an unbounded ...
13
votes
1answer
181 views

Number guessing game with lying oracle

You are probably already familiar with the usual number guessing game. But for concreteness I restate it. The usual game The Oracle chooses a positive integer $n$ between 1 and 1024 (or any power of ...
4
votes
0answers
159 views

Hyperbolic subgroups of general linear groups

Is there a classification of hyperbolic subgroups of $GL_n(A)$ for $A$ some ring of characteristic $p$? $A$ for me is a finitely presented algebra over a finite field. More precisely I'm looking for ...
3
votes
1answer
121 views

How do modular functions of level $N>1$ transform under the full modular group?

Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$ First question What can we say in general about the factor $j(\...
2
votes
0answers
94 views

Categorical representations of absolute Galois groups

I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.
1
vote
0answers
75 views

Matrix Bernstein for spherical random variables

Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...
2
votes
1answer
124 views

What is the shortest length of an Egyptian fraction expansion for a given $p/q$?

An Egyptian fraction expansion is a sum of reciprocals of integers, for example: $$\frac{4}{17} = \frac{1}{5} + \frac{1}{29} + \frac{1}{1233} + \frac{1}{3039345}$$ Every positive rational number $p/...
4
votes
0answers
56 views

The ¨irreducible¨ representation variety of surface group

Let S be a closed surface of genus larger than 1, G be a compact, simply connected simple Lie group with finite center. Consider the representation variety M(S,G)=Rep($\pi_1$(S), G). Witten´s Formula ...
-2
votes
0answers
137 views

$A[x]$ points of an algebraic group

Let $K$ be a field and $G$ be an algebraic group. Specifically $O(n)$ or $Sp_{2n}$. Is it true that for any ring $A$ over $K$ , $G(A)\cong G(A[x])$. Is there any reference for such kind of results?
7
votes
0answers
589 views

Examples of set theory problems which are solved using methods outside of logic

The question is essentially the one in the title. Question. What are some examples of (major) problems in set theory which are solved using techniques outside of mathematical logic?
1
vote
0answers
48 views

Can sufficiently symmetric polytopes be uniquely reconstructed from their 1-skeleton?

General convex polytopes can not be uniquely reconstructed from their 1-skeleton1, as explained here. Not even the dimension is known from the skeleton, as e.g. the complete graph $K_n,n\ge 5$ is the ...
1
vote
1answer
126 views

Combinatorial games with infinite paths, and generalized Sprague-Grundy theory

Generalized Sprague-Grundy theory has been used to analyze finite impartial loopy games with normal play. There is a nice short account by Mark S. in this answer. It was introduced by Cedric Smith in ...
4
votes
0answers
52 views

Higher order variations of Riemannian geodesics

Consider a mapping $\Gamma$ from the Euclidean plane or an open subset to a Riemannian manifold $M$, so that each $\Gamma(s,\cdot)$ is a geodesic. There is a well established theory of the first order ...
7
votes
0answers
55 views

Is anything known about this class of series involving the divisor function?

I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it! Let $N\in\mathbb{N}$, let $q$ be a point in the open ...
2
votes
0answers
48 views

Convex hull of piece-wise linear functions

Let $K>1$ be a positive integer. Consider a function class $$\mathcal{F}_K:=\Big\{\max_{1\leq k\leq K} a_k^\top x + b_k:\ ||a_k||\leq 1, |b_k|\leq 1, \forall 1\leq k\leq K \Big\}$$ on some compact ...
-4
votes
0answers
33 views

Focal point (Definition ) [closed]

I Am a bigginer in the differential geometry And I need the definition of focal points, all the books i see is defined in riemannian submanifolds with jacobi fields ... I know just the notions of ...
0
votes
0answers
29 views

Moser inequality involving the trace operator

Can anyone give me a reference of whether there exists a Moser-Trudinger type inequality involving the trace operator. More precisely, $$ \int_{a}^{b} \exp\bigg[ \beta U(x, 0)^2\bigg]\leq C$$ for ...
1
vote
1answer
127 views

references on group representation over local fields / a question on an argument of a Ralph Greenberg's paper

I'm currently studying Iwasawa theory. 1) There are many $\mathbb{Z}_p$-modules on which some Galois groups act. So I often face some facts on the group representation over local fields or p-adic ...
0
votes
0answers
11 views

Converse results for vertex expansion implies spectral expansion?

For an $r$-regular expander digraph $G = (V, E)$ with adjacency matrix $A$, it is known that if for some $\delta, c > 0$, the second largest eigenvalue of the normalized adjacency matrix $M = r^{-...
3
votes
3answers
179 views

Extending a continuous map over projective space

Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a bijective, continuous map $\phi:K \to K$ which satisfies the ...
4
votes
0answers
82 views

$\ell_p$ geodesic distance on smooth Riemannian manifold and Logarithmic Sobolev Inequalities

Bear with me, I'm not a professional geometer. Recently, I've been studying Logarithmic Sobolev Inequalities (LSI) for probability distributions on manifolds (e.g as done in works of Bobkvo et al. ...
10
votes
2answers
285 views
+50

A conformal map whose Jacobian vanishes at a point is constant?

Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$. Assume $d \ge 3$ ...
34
votes
7answers
3k views

Paradoxical Mathematical Objects Pending for Construction [duplicate]

The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly paradoxical description ...
2
votes
0answers
39 views

Ranks of cycles with base field coefficients as a generalization of ranks of multivectors?

This must be probably only reference request since I am inclined to believe that I am asking about something well known but just cannot pin down appropriate keywords for searching. The starting point:...
1
vote
1answer
77 views

Zeros of Multivariate Complex Functions [need reference]

I am looking for a good accessible reference that would summarize properties of zeros of complex analytic functions. For my purpose, it would be interesting to see a discussion on the following ...
8
votes
2answers
608 views

The Stacks project

I have a question concerning the admirable Stacks Project. Which comparable projects are there: approach-wise: "an open source textbook on algebraic stacks and the algebraic geometry that ...
17
votes
6answers
963 views

Lebesgue measure theory applications

I'm looking for reasonably simple examples of applications of Lebesgue measure theory outside the measure theory setting. I give an example. Theorem: Let $X$ be a differentiable submanifold of $\...
3
votes
0answers
95 views

Reference Request: Fourier Mukai on non Weierstrass Elliptic Fibration

I'm aware of the standard results about Fourier Mukai on Weierstrass elliptic fibration. What I need is the references about Fourier Mukai on elliptic fibration which can have reducible fibers and/...
0
votes
1answer
182 views

Book on algebraic structures

What is the most complete book on algebraic structures that deals with the complete taxonomy from magmas to Lie algebras and inner product spaces?
8
votes
5answers
322 views

Reference request: Recovering a Riemannian metric from the distance function

Let $M = (M, g)$ be a Riemannian manifold, and let $p \in M$. Writing $d$ for the geodesic distance in $M$, there is a function $$ d(-, p)^2 : M \to \mathbb{R}. $$ This function is smooth near $p$. ...
3
votes
1answer
115 views

Sums of squares in global fields (Reference Request)

There is a result due to Siegel that, for a number field $K$, any totally positive element of $K$ is the sum of four squares of $K$. This is discussed in another question (sum of squares in ring of ...
4
votes
0answers
87 views

Simplicial homotopy groups - reference request

I am looking for a reference for the definition 2.6 in https://ncatlab.org/nlab/show/simplicial+homotopy+group, which states "The simplicial homotopy groups of any simplicial set, not necessarily Kan,...
1
vote
0answers
41 views

Matchings in infinite, not necessarily bipartite, graphs

Aharoni, Nash-Williams, and Shelah have extended the famous marriage theorem for finite bipartite graphs due to Hall to arbitrary graphs. Is there a similar generalization of Tutte's theorem on ...
6
votes
3answers
206 views

$S$-dual of filtered spectra

I hope this is research level. Suppose $E$ is the direct limit of finite spectra, say $E=\mathrm{colim }\ E_i$, which itself is not finite. I wonder how much and under which conditions the inverse ...
5
votes
0answers
128 views

Reference request: A commutative variant of the Exterior Algebra

Consider $A = \mathbb{R}[t_1,\ldots,t_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t_1,\ldots,t_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through $$ p(y) = ...
2
votes
4answers
189 views

Complex differential equations

I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs. Mostly, I'm just ...
1
vote
1answer
145 views

Subschemes of projective varieties

I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective variety any zero-locus $X$ of homogeneous polynomials in ...
1
vote
1answer
93 views

Giving Uniform Bound on Differences of Sums of Converging Polynomials

The title does not quite capture the essence of the difficulty, please allow me to be more explicit here. I thought of this question when I was trying out an open problem by Ovidiu Furdui(See problem ...
2
votes
1answer
253 views

Shafarevich's theorem about solvable groups as Galois groups

I am seeking references to any proofs of Shafarevich's theorem about solvable groups being Galois groups.
2
votes
2answers
150 views

Random walk and isoperimetric constant

I assume that a result of the following kind is known, and I would really appreciate a reference for it... Or at least, some hints as to where to start looking. Theorem(?): Let $\varepsilon>0$ ...
7
votes
1answer
180 views

Proof of Green's formula for rectifiable Jordan curves

$\newcommand{\Ga}{\Gamma}$ I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
2
votes
0answers
102 views

theories where angles exist without a metric

The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible exotic/nonintuitive models of Euclid's axioms/postulates, outside the one where "lines" are interpreted ...
3
votes
0answers
98 views

Diffeomorphism group action on the space of embeddings

Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is ...
2
votes
1answer
160 views

Naive question on the Jacobian of a curve

Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the ...
5
votes
0answers
82 views

Convolution theorem on a non-abelian Lie group

Let $\mathrm{G}$ be a compact (simple, if it helps) non-abelian Lie group and let $\hat{\mathrm{G}}$ be its unitary dual of (equivalence classes) of irreducible unitary representations. Defining the ...
4
votes
0answers
78 views

Ref. request: Enumerating elements of Bruhat cells

Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then $$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$ where we embed the ...