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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18 views

Feasibility Criteria in Integer Linear Programming

Consider an integer linear programming problem: For $A\in M(m, n, \mathbb{Z})$ and $b\in \mathbb{Z}^m$ find $x=(x_1,\ldots,x_n)^T\in \mathbb{Z}^n_{\geqslant0}$ such that $Ax=b$. Sometimes one ...
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47 views

Provenance of a result on regular simplices with integer vertices

There are several MO questions related to the question of characterizing those integers $n$ for which there exists a regular $n$-simplex in $\mathbb{R}^n$ with integer vertices, e.g., coordinates of ...
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1answer
33 views

Reference request for (weak*) metrizability of a bounded space of signed Radon measures on a compact set

I know the following is true and I know how to prove it (cf. exercise 50 on page 171 in Folland, Theorem 7.18 in Folland), but per my adviser's instructions, it would be better to find a source to ...
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Quantitive and computational improvement of the Oseledets multiplicative ergodic theorem for irrational rotation

Consider irrational rotation $T:S^1\to S^1, T(x) = x + \alpha$ where $\alpha\notin \mathbb{Q}$ (you may assume additional number theoretic properties of $\alpha$, say $\alpha = \sqrt{2}$ is already ...
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227 views

A (surprising?) expression for $e$

I apologise if this is off topic. Consider the quantity $$ F(m,n,k)=\frac{(m)_k}{k!n^{k-1} } $$ where $m,n \in \mathbb{N}.$ For moderately large $n$, it seems that the approximation $$ \sum_{k=1}^{K} ...
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1answer
77 views

Uniqueness of presentation for semi-abelian varieties

Let $k$ be any field and $G$ a semi-abelian variety over $k$, i.e., an algebraic group that fits into an exact sequence $$ 1 \to T \to G \to A \to 1$$ of algebraic groups, where $T$ is an algebraic ...
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1answer
381 views

Reference request: The unit of an adjunction of $\infty$-categories in the sense of Riehl-Verity is a unit in the sense of Lurie

I'm looking for a reference (or proof) for the statement given in the title: that when we have an adjunction between quasicategories in the sense of Riehl and Verity (defined e.g. in Section 4 of ...
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72 views

Is a presentation of the hyperbolic orthogonal group of rank 2 over the integers known?

The hyperbolic orthogonal group $O_{g,g}(\mathbb{Z})$ often appears in the study of high-dimensional manifolds, see e.g. work of Kreck or Galatius and Randal-Williams. Let $H$ denote the lattice $\...
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31 views

Reference request for solving pde numerically

What is the reference should i read to solve this pde numerical ? $$\frac{\partial u}{\partial t} - r (\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2})+\sin(y)\frac{\partial u}{\...
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76 views

Independent conditions imposed by a collection of double points

Let's consider the following statement : There exist a collection of $d$ points $\gamma \subset \mathbb{P}^{n}$, so that $h_{\Bbb P^n}(\gamma^{2} ,m) = \min\{(n+1)d, \binom {n+m}{n}\}$ implies for any ...
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882 views

What fraction of a sphere's volume lies within a cone?

Let $B \subset \mathbb{R}^n$ be a collection of $n$ (not necessarily independent) unit vectors which we will label $v_1,\ldots, v_n$ for convenience. The cone $K_B \subset \mathbb{R}^n$ associated to $...
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23 views

Beck-Fiala Discrepency Type Results for Arbitrary Graph Labelings

Suppose we have a graph $G$ on $n$ vertices $x_1 , \dots, x_n$ attached with weights of values from $1$ to $n$. We will write $\text{weight}(x_i)$ as simply $x_i$ and let $\text{diff}(G) = \min _{(x_i,...
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Errata for Atiyah-Macdonald

Is there a good errata for Atiyah-Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists inaccuracies ...
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1answer
128 views

Mapping Problems to Boolean Formulas for SAT Solvers

I came across Marijn Heule and Oliver Kullmann's paper on recent techniques in highly efficient SAT solvers. In particular they describe the Pythagorian Triple Problem, which they solved using that ...
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2answers
160 views

Hausdorff dimension of the graph of the sum of two continuous functions

How can one prove the following result on the Hausdorff dimension of the graph of the sum of two continuous functions: Let $f,g:[0,1] \to \mathbb R$ be two continuous functions. Suppose that $$\...
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27 views

Neumann equation on manifold with edge or corner

Let $(M,g)$ be a compact Riemannian manifold with boudnary and corner, i.e. locally mdoelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$. ...
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58 views

Defining pull-back of Chow groups under a morphism of special type

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. Let $\pi : Y\rightarrow X$...
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1answer
177 views
+100

Symmetric monoidal category with trivial switch morphisms

Is there a specific terminology for a symmetric monoidal category in which for any object $x$ the switch map $x\otimes x\to x\otimes x$ is the identity ? (Or alternatively the action of the symmetric ...
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1answer
446 views

Simplicial set of permutations

Let $S_k$ be the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone ...
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56 views

Spectra of one dimensional Schrodinger operators [on hold]

I am trying to understand how to compute the spectra of one-dimensional Schrödinger operators $$ \mathcal{L}:=-\partial_x^2+V, $$ where $V$ is a bounded function in the whole line. I am particularly ...
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Textbook recommendations for undergraduate proof-writing class

I am teaching the proof-writing class (for the 3rd time) in the Fall and plan to buck the party line and use a different text than the default Bond and Keane. My parameters are as follows: Logic, ...
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86 views

Hermitian sublattices of a given type

Consider an unramified quadratic extension $E/F$ of non-archimedean local fields, and suppose that $\langle\cdot,\cdot\rangle$ is a fixed Hermitian form on $E^d$ such that $\mathcal{O}_E^d$ is self-...
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1answer
145 views

For $G$ an adjoint Chevalley group, are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?

Let $G$ be an adjoint Chevalley group. Are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups? I read a theorem that states: When $G$ is the universal Chevalley group and it's not of ...
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139 views

Reference request: A knot is tame if and only if it has a tubular neighbourhood

Definitions: A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal). Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...
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1answer
142 views

Is there a name for a “stable” physical measure?

Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a continuous map, and let $\mu$ be a probability measure on $M$ with compact support. Definition. The ...
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1answer
93 views

Flat scalar curvature on 4 manifold

Let $(M,g)$ be a closed(oriented) Riemannian 4 manifold. It is well-known that, if $scal^g\geq0$ and not identically zero, then $M$ admits a PSC metric by conformal transformation. Q Is $T^4$ the ...
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1answer
86 views

Representation-finite quivers over dual numbers

Given a Dynkin quiver $Q$ and a field $K$. Question 1: For which such $Q$ are there only finitely many indecomposable representations over the dual numbers $K[x]/(x^2)$? Note that those ...
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155 views

Diffeomorphism classification of Grassmannian manifolds

Is anything known about the diffeomorphism classification of Grassmannian manifolds? I know that there are some results on projective spaces (for example in Lopez de Medrano's "Involutions on ...
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0answers
72 views

On semicontinuity of Hilbert function for a zero dimensional scheme

Let, $X$ be a zero dimensional subscheme of $\Bbb P^n$ and let us define a function as follows: $h_{\Bbb P^n}(X ,d) = \binom {n+d}{n}$ - dim $I_{X}(d)$ , where dim $I_{X}(d)$ = the dimension of ...
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81 views

The space of $k$ differential forms as a Fréchet space

Given a smooth manifold $M$, can define define seminorms on $\Gamma(U,\bigwedge^kT^{\ast}M)$ for $U$ a coordinate open set by the following: $p^{s}_L(u = \sum_{I}u_I dx_I) = \sup_{x \in M}\max_{|I|=p, ...
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1answer
173 views

Tannaka duality for closed monoidal categories

I asked this some time ago at mathstackexchange, and people there explained to me the mathematical part of what I was asking, but the question about references remains open. In my impression, people ...
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231 views

Reference request for some result of de Bruijn on zeros of some holomorphic function

In a video lecture on Youtube, ''Vaporizing and freezing the Riemann zeta function'', Terry Tao states that ''de Bruijn proved that if for some $t_0$ the zeros of $H_{t_0}$ are contained in the strip $...
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139 views

Kuga-Satake in characteristic $p$ [on hold]

Have Kuga-Satake correspondences been investigated in characteristic $p$? (I'm being intentionally vague about what this would mean.)
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90 views

Empirically random, quickly multiplicable matrices

I have encountered a need for fast computation of a transformation $Ax$ where $A\in \mathbb{C}^{K\times N},\ K\sim 10^7,\ N\sim 10^3$ is designed, and $x\in \mathbb{C}^N$ has iid $\mathcal{CN}(0,1)$ ...
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552 views

A model structure on marked simplicial sets

Do you have a reference for the following fact? And before that, is it true? The Joyal model structure on simplicial sets "lifts" to a model structure on the category of marked simplicial sets, ...
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152 views

Remaining models conjectured to converge to SLE(6) or CLE(6)

I am wondering which models are conjectured (eg. numerically) to converge to SLE(6) (Schramm-Loewner evolution with $\kappa=6$) or CLE(6) (conformal loop ensemble). I am searching for a research topic ...
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158 views

Are random convex polygons on a sphere themselves sphere-like?

Say $\mathbb{R}^n$ is divided by $k>n$ randomly chosen hyperplanes. Each connected region away from the hyperplanes is the intersection of $k$ half-spaces, so it is a convex cone. It is known that ...
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1answer
117 views

The radius of an interval's image through a space-filling curve

Take $f:[0,1]\to [0,1]^n$ a continuous tour around $[0,1]^n,$ say, some iteration of a Hilbert curve. For $\varepsilon \in (0,1)$ what is the following thing called and are there any nontrivial upper ...
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1answer
381 views

Canonical divisors and vector bundles

Let $E$, $X$ be irreducible smooth algebraic varieties over the complex numbers and let $p \colon E \to X$ be a morphism which is locally trivial with respect to the Zariski topology. Since $p$ is a ...
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163 views

Complex factorization of the angular part of the Laplacian

Some time ago some research led me to the following equality: \begin{equation} \frac{1}{\sin^2 \phi }\frac{\partial^2 }{\partial \theta^2} +\frac{\partial^2 }{\partial \phi^2} +\cot \phi \frac{\...
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81 views

On earlier references for $P=BPP$ and Kolmogorov's possible view on modern breakthroughs involving randomness?

Kolmogorov and Uspenskii in this paper 'http://epubs.siam.org/doi/pdf/10.1137/1132060' speculate $P=BPP$ in $1986$. They do this without getting into circuit lower bounds and from a different view ...
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When is an integral transform trace class?

Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator $$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$ the operator $K$ is Hilbert ...
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2answers
114 views

Proof of Isoperimetric Inequality using Curve Shortening Flow

I am aware that there is a nice result where one uses curve shortening flow to prove the isoperimetric inequality on the Euclidean plane, but could not seem to find the original article where this was ...
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1answer
69 views

Literature about solitons and Hirota derivatives

This summer I'm going to learn a mini-course about soliton theory ("Soliton equations and symmetric functions" in LHSM (Russian summer school in mathematics). The web-page of this course is https://...
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1answer
101 views

Spectrum of the Magnetic Stark Hamiltonians $H(\mu,\epsilon)$

I am looking for a document where I can find a proof the spectrum of the of the Magnetic Stark Hamiltonians $H(\mu,\epsilon)=\big(D_x-\mu y)^2+D^2_y+\epsilon x+V(x,y)$ cited on the article below for $\...
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0answers
47 views

Random graphs - multiple giant components

For a given random graph, a connected component that contains a finite fraction of the entire graph’s vertices is called giant. A well known result in random graphs is the existence and uniqueness of ...
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0answers
94 views

Computing Chow group of a variety which is almost a blow-up of another variety

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. I have a morphism which is ...
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46 views

Reference request ( Conductor of Galois representation associated to Dirichlet character)

(Sorry for my poor english...) Let $\chi$ be a Dirichlet character modulo $N$ and $\Psi_{\chi}$ be an one dimensional Galois representation such that \begin{equation} \Psi_{\chi}: \text{Gal}(\...
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94 views

Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?

Note: Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a (very ...
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3answers
137 views

Reference request: book on stochastic calculus (not finance)

I am looking at fractional Gaussian/Brownian noise from a signal theoretic and engineering point of view. In particular, I am looking at the math behind what defines these noise processes and what ...