Questions tagged [reference-request]

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

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Matroids with no relaxations (~ weak maps)

There's an operation in matroid theory which is called "relaxation". To keep things simple, let's consider a matroid $M$ with set of bases $\mathcal{B}$. If $M$ has a subset $H$ of $M$ that is both ...
8 votes
3 answers
636 views

Representation theorem for matrices (reference request)

Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that $$ A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k, $$ where $\lambda_1,\dots,\lambda_n$...
4 votes
2 answers
259 views

Finite models for torsion-free lattices

Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$? I know this to be true in many instances (e.g. ...
6 votes
1 answer
285 views

A conjecture (or theorem?) on unit vectors in a Euclidean space

I have heard (if I am not mistaken) that there exists the following conjecture (or theorem?). Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists ...
4 votes
0 answers
243 views

Eigenvalues of structured matrices

Let $A=(a_{i,j})$ be an $n\times n$ matrix with $a_{j,j+1}>0,\; 1\leq j\leq n-1,$ and $a_{j,j-2}>0,\; 3\leq j\leq n$, the rest of the entries are zeros. Is the following fact known: All ...
7 votes
1 answer
365 views

Are $f\sqrt{1+g^2}$ and $fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?

Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb R)$. Does it then necessarily follow that the ...
1 vote
0 answers
79 views

Reference for the following flow equation

I'm looking for reference on the following partial differential equation. $\partial_tF(t,x) = G(t)((\partial_xF(t,x))^2 + \partial_x^2F(t,x))$, where G(t) is a fixed Schwartz function. If possible, I ...
3 votes
1 answer
114 views

Flatness directions of the operator norm

It is known that the standard operator norm $\|\cdot\|_2$ over ${\bf M}_n({\mathbb R})$ is very flat, as is any operator norm (= subordinated norm) actually. The set of extremal points of the unit ...
3 votes
0 answers
226 views

What is rigidity of Hirzebruch, and Witten genera?

I would like to find some good references (or any insight) that would help me to understand a few articles mentioning rigidity of Hirzeburch genus. One of the consequences of this phenomenon is that ...
23 votes
3 answers
1k views

Existence of subset with given Hausdorff dimension

Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension. For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff ...
4 votes
1 answer
212 views

reference request: unbounded operators on normed spaces

I'm looking for books where the theory (basic properties, adjoints etc.) of unbounded linear operators between locally convex spaces or at least Banach spaces is developed. In Brezis' functional ...
42 votes
2 answers
2k views

Fermat's Last Theorem for integer matrices

Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
2 votes
2 answers
406 views

For what automorphic representations is Ramanujan-Petersson known?

I had in mind that Ramanujan-Petersson conjecture was essentially unknown in the case of number fields. I however recently heard that If an automorphic representation on $GL(2)$ is ramified at a ...
1 vote
0 answers
250 views

General Hoeffding inequality with two uncorrelated random vectors

Let $X_1,\ldots, X_n$ be independent sub-Gaussian variables with sub-Gaussian norm $K$. Let $a_1,\ldots,a_n$ be a random vector such that $\mathbb{E}[a_j X_i]=0$ for all $i,j\in \{1,\ldots ,n\}$ and ...
12 votes
1 answer
449 views

Bi-orderability of Baumslag-Solitar group $\langle a,b \mid a^{-1} b^m a = b^n\rangle$ and of $\langle a,b \mid a^{-1} b a^m = b^n\rangle$

We say that a group $(A, \cdot)$ is bi-orderable if there exists a total order $\preceq$ on $A$ such that $xz \prec yz$ and $zx \prec zy$ for all $x,y,z \in A$ with $x \prec y$. Let $m,n$ be non-zero ...
1 vote
2 answers
272 views

Faithfully flat modules over a group algebra

Suppose we have the following data: 1) A group ring $\mathbb{Z}[G]$, where $G$ is a torsion free group. 2) $M_{\bullet}$ a bounded (above and below) chain complex of $\mathbb{Z}[G]$-modules such ...
4 votes
2 answers
1k views

Every abelian torsion-free group is strictly totally orderable (via the compactness theorem)

Let $\mathbb G = (G, +)$ be a group. We say that $\mathbb G$ is strictly totally orderable (others would say bi-orderable) if there exists a total order $\preceq$ on $G$ such that $x+z \prec y + z$ ...
18 votes
2 answers
825 views

Reference to a conjecture on unit vectors in Euclidean space

I have heard that there exists the following conjecture (if I am not mistaken). Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists another unit vector ...
2 votes
1 answer
168 views

References for Neumann eigenfunctions

I am looking for references on eigenfunctions with Neumann boundary condition. In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it ...
4 votes
0 answers
410 views

Computing the volume of intersection between a ball and a box

$C$ is the set of vectors which are coordinate-wise less than $\overline{c}\in [-1,1]^d$ and greater than $\underline{c}\in [-1,1]^d.$ Is there a procedure not exponentially complex in $d$ that ...
1 vote
1 answer
278 views

Polylogarithm : reference request for proof of integral representation

On page 494 of the book Integrals and series, volume I : elementary functions, Gordon and Breach, 1986, by A. P. Prudnikov, Y. A. Brychkov, and O.I. Marichev, (perhaps translated from Russian), the ...
3 votes
0 answers
77 views

Logarithmic Sobolev growth of time-space-periodic Schrödinger solutions

Consider the following Schrödinger equation $$i\partial_t \psi (t,x) + \Delta \psi - V(t,x) \psi = 0 \, ,$$ where $x\in \mathbb{T}^d$ and $V(t,\cdot)$ is real, smooth, and periodic (with a diophantine ...
2 votes
0 answers
415 views

An equivariant Hahn embedding theorem?

The Hahn Embedding Theorem asserts that for any (linearly) ordered abelian group $\Lambda$, there exists a linearly ordered indexing set $\Omega$ such that $\Lambda$ admits an order-preserving group ...
5 votes
1 answer
169 views

Hopf C-star algebra/comodules using a Fubini tensor product rather than the minimal tensor product?

Throughout, $\otimes$ denotes the minimal tensor product of $\newcommand{\Cst}{{\rm C}^{\ast}}\Cst$-algebras. and $\odot$ denotes the tensor product of (underlying) vector spaces. Given two $\Cst$-...
9 votes
0 answers
459 views

Category of metric spaces

Is there a standard/good reference text that does category of metric spaces? Say, it seems that by looking at this category one can recover everything about particular metric space up to scaling --- ...
4 votes
0 answers
477 views

analytic approximations of the min and max operators

Question: What is the state of the art on analytic approximations of $\min$ and $\max$? My hunch is that numerical analysts probably have a better solution than the one I propose here. For any $\...
33 votes
4 answers
2k views

Hahn-Banach theorem with convex majorant

At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and $f:L\...
4 votes
0 answers
85 views

Is there a name for this kind of structure? (Not quite a lattice-ordered group)

I'm looking at a certain class of groups $G$ that come with a partial order $\le$ on the elements. So far it looks like $(G,\le)$ has the following properties: The partial order is invariant under ...
6 votes
1 answer
321 views

"Sub-logarithmic" zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem

For each $n\in\mathbb{N}$, let: $\chi_n\pmod{q_n}$ a real non-principal Dirichlet character ($q_1 < q_2 < \cdots$), $\beta_n$ the largest real zero of $L(s,\chi_n)$, $\delta_n := (1-\beta_n)\...
5 votes
2 answers
265 views

Is there a similar theory as for symmetric polynomials, that deals with polynomials on the entries of matrices that are symmetric in both dimensions?

I am looking at a polynomial of the entries of a matrix, and this polynomial is invariant under permutation of the rows or columns of the matrix. Is there a similar characterization as in the case of ...
0 votes
0 answers
131 views

“Chapman-Kolmogorov”-convolution vs. smoothness

Let $K:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a so-called "integral-kernel": we certainly require $K(x,.)$ and $K(.,y)$ to be Lebesgue measurable for almost all $x,y \in \mathbb{R}^n$. An ...
4 votes
0 answers
633 views

Scalar curvature in terms of second fundamental form, reference request

I would like to cite a reference for the following formula for scalar curvature: If $\Sigma$ is a hypersurface in Euclidean space, then $R=H^2-\lvert A\rvert^2$, where $R$ is the scalar curvature ...
2 votes
2 answers
366 views

Why we use Caputo fractional derivative in application?

I'm working on some papers which use Caputo fractional evolution equation (see on Wikipedia) as application for thier main result: For example: $$\left\{\begin{matrix} ^CD^{\sigma}_tx(t)+Ax(t)=&...
0 votes
1 answer
53 views

Monotonicity of $\mathbf{P} ( \bar{X}_N > 0 )$ in $N$

Let $X$ be a real-valued random variable with positive expectation (wlog, $\mathbf{E}[X] = 1$, say). For $N \in \mathbf{N}$, let $X_1, \cdots, X_N$ be independent, identically-distributed copies of $...
3 votes
1 answer
151 views

Reference for Dedekind's problem

Dedekind's problem is about enumerating antichains in the Boolean lattice. Is there an explicit reference where Dedekind stated this problem? Is there a good motivation to study this problem except ...
5 votes
1 answer
182 views

References for systems of elliptic PDEs

I was wondering if there were any recent references dealing with the theory of systems of elliptic PDEs: in particular, someone was telling me about something which sounded like 'Schur complementarity'...
12 votes
4 answers
2k views

Book on manifolds from a sheaf-theoretic/locally ringed space PoV

I'm looking for an introductory (or rather, non-advanced) book on manifolds as locally ringed spaces, i.e., from the algebraic geometric viewpoint. Most introductory texts only introduce manifolds ...
6 votes
1 answer
327 views

Naive question on local cohomology

Let $X$ be a smooth, projective variety and $Z_1, Z_2$ two smooth, projective subvarieties in $X$ of the same dimension. Let $E$ be a locally free sheaf on $X$. Recall, there are natural morphims: $$...
2 votes
1 answer
463 views

Difference quotient for functions of bounded variation

Let $u:\mathbb{R}^N \to \mathbb{R}^N$, $u \in BV(\mathbb{R}^N)$, be a function of bounded variation. We have that the following holds $$(\ast) \qquad \frac{1}{|B_r(0)|}\int_{B_r(0)} \frac{|u(x+z)-...
5 votes
1 answer
205 views

Classification of $\operatorname{Rep}D(H)$

Question Let $H$ be a finite dimensional complex Hopf algebra and $D(H)$ its quantum double. Can we classify the simple objects in $\operatorname{Rep}D(H)$ if the representations of $H$ are well-...
31 votes
5 answers
8k views

How many binary operations are associative?

Let $X$ be a finite set of $n$ elements, and consider a binary operation $\odot: X \times X \rightarrow X$. There are $n^{n^2}$ such binary operations, as the $n \times n$ table entries can each be ...
1 vote
0 answers
81 views

Embedding random variables in infinite-dimensional spaces

Let $H$ be a reproducing kernel Hilbert space of functions $f:E\to F$ with kernel $k$. A point in $E$ may be embedded into $H$ via the canonical embedding $x\mapsto k(x,\cdot)$. Similarly, a random ...
6 votes
1 answer
284 views

Stability of fractional Sobolev spaces under diffeomorphisms

Let $H^s_p(\mathbb{R}^n)$ be the fractional Sobolev space of fractional order $s\in \mathbb{R}$, for $1<p<\infty$, and let $\phi:\mathbb{R}^n\to\mathbb{R}^n$ be a diffeomorphism. Assume that the ...
3 votes
0 answers
449 views

Accuracy of Richardson's error estimate in the presence of rounding errors

Richardson extrapolation is a well known technique in scientific computing. It is used to compute error estimates when our approximations can be viewed as a function of a parameter $h$. Common ...
3 votes
1 answer
2k views

Singular value decomposition of random rectangular matrices

Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance). What is the ...
7 votes
0 answers
343 views

What is known about "almost orthogonal vectors"?

Motivation: Suppose we have a kernel $k(a,b)$ defined over the natural numbers. Then by the Moore–Aronszajn theorem, we can embedd the natural number $a$ in some Hilbert space $\mathbb{H}$, which we ...
4 votes
0 answers
81 views

A Krull-Schmidt theorem for partially ordered groups

If $G$ is a po-group (ie. partially ordered group), we say that $G$ is po-indecomposable if it's not the direct product of two non trivial subgroups (such subgroups are necessary convex and normal). ...
2 votes
2 answers
569 views

Reference for weak*-semigroup

Let $X$ a dual Banach space (there exists a Banach space $Y$ such that $X=Y'$). A weak* semigroup on $X$ is a semigroup $(T_t)_{\geq 0}$ on $X$ such that, for all $x\in X$, we have $T_tx\to x$ in the ...
1 vote
1 answer
101 views

Name of the class of linearly ordered groups with no minimal positive element

Is there a special name for a linearly ordered group $G$ such that for every positive element $g\in G$ there exists an element $h\in G$ such that $e<h<g$?
4 votes
1 answer
480 views

Strictly totally ordered semigroups - Looking for references

Let $\mathfrak A = (A, \cdot)$ be a semigroup (written multiplicatively). We say that $\mathfrak A$ is linearly orderable if there exists a total order $\le$ on $A$ such that $ac < bc$ and $ca < ...

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