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

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0
votes
0answers
18 views

Pseudodifferential operators on spaces with boundary

Consider the upper half space $\mathbb{R}^n_{+} = \{x = (x_1,..,x_n) \in \mathbb{R}^n : x_n \geq 0\}$. Consider the Laplacian on this space with either the Dirichlet boundary condition or the Neumann ...
3
votes
1answer
86 views

Enriques classification of algebraic surfaces in characteristic zero

I am searching for a reference about the classification of algebraic surfaces over an arbitrary algebraically closed field of characteristic zero. In the 1949 book "le superficie algebriche" by ...
0
votes
0answers
17 views

Automorphisms of a differential field and transcendence degree

Let $(\mathcal{F},+,\times,\partial)$ be a differential field, and let's define its automorphism group $Aut(\mathcal{F})$ as the group, under composition, consisting of all bijective maps ...
2
votes
0answers
77 views

Earliest source for a Lie algebra construction

I am looking for the earliest reference to the fact that any associative algebra becomes a Lie algebra with bracket $AXB-BXA$, where $X$ is a fixed element of the algebra. This is observed in the ...
1
vote
0answers
77 views

Level-Lowering in Weight 2

Let $N$ and $p$ be relatively prime integers with $p$ a prime. Suppose $f$ is a weight $k=2$ (normalized, cuspidal, etc) newform of level $\Gamma_1(N) \cap \Gamma_0(p)$. I seem to recall the existence ...
-5
votes
0answers
158 views

Are there “adelic” L-functions? [on hold]

Following Tom163's answer to this question, I would like to know whether L-functions defined through adelic representations (as defined in https://projecteuclid.org/euclid.em/1317758108) have been ...
3
votes
2answers
144 views

$\mathcal S'(\mathbb R^d)$ is separable [on hold]

I Think the statement is true, but I struggle to find a reference for the fact that the space of tempered distributions equipped with the weak-* topology is separable. Thank you for your help!
-2
votes
1answer
60 views

How does deletion-contraction affect chromatic number? Can it increase chromatic number? [on hold]

Question: In graph theory, contracting an edge or deleting an edge are basic operations in many topics such as graph minors or Wagner's theorem on planar graphs. And I'm interested in how these ...
0
votes
0answers
30 views

Reference request for some “irregularities of distribution” papers

I would like to ask if anyone has access to any of the following papers: 1. J. G. van der Corput, Proc. Kon. Ned. Alcad. v. Wetensch., Amsterdam, 38, 813-821 (1935). 2. J. G. van der Corput, ibid. 38, ...
3
votes
2answers
148 views

Examples of toric threefolds

I am looking for examples of smooth projective toric threefolds $\mathbb P_\Delta$ such that the rational polytope $\Delta$ has only pentagonal faces and hexagonal faces. I quickly searched for ...
5
votes
1answer
161 views

Higher-dimensional category theory on objects

I would like to know if there exists a satisfying generalization of higher-dimensional category theory on objects, that doesn't forget the inner structure of objects. Usually, what people do is to ...
3
votes
0answers
129 views

Correspondence between real forms and real structures on complex Lie groups

I asked this in MSE, but without success, so I hope, it will be suitable here. E.B.Vinberg and A.L.Onishchik in their book give the following two definitions. For a complex Lie group $G$ its real ...
0
votes
0answers
32 views

interpolation between Bochner spaces

Is there a reference for the interpolation result stating the existence of an embedding \begin{equation} L^2(I;W^{2,p}(\Omega)) \cap H^1(I;L^p(\Omega)) \hookrightarrow ...
6
votes
2answers
185 views

What methods do we have to understand the spectrum of matrices with restricted entries?

Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"), What is the largest possible spectral radius of a $n \times n$ matrix ...
4
votes
0answers
77 views

On various “extension closures” and “orthogonals” in triangulated categories

A vague form of my question is the following one: for a class of objects $D$ of a triangulated category $C$ we consider the class $E$ of objects that satisfy $Mor_{C}(d,e)=\{0\}\ \forall d\in D$; ...
2
votes
0answers
42 views

Multiplication operators are sectorial [migrated]

Consider the multiplication operator $M_a$ on $L^p(M)$, where $M$ is a Riemannian manifold, and $a$ is a non-negative function. An operator $A$ is said to be sectorial if there exists $\theta \in (0, ...
3
votes
1answer
130 views

Preprint by Wall on Sjogren's theorem

In their account http://dx.doi.org/10.1016/0022-4049(87)90048-X of Sjogren's theorem, Cliff and Hartley refer to two articles: [9] B. Hartley, A note on a lemma of Sjogren relating to. dimension ...
3
votes
0answers
87 views

n-homology of a Harish-Chandra module

Let $G$ be a connected real reductive Lie group and let $K$ be its maximal compact subgroup. Let $P=MAN$ a parabolic subgroup. Let $K_M^0=M^0\cap K$ be connected component of the maximal compact ...
2
votes
0answers
62 views

Resolvent estimate of hyperbolic Laplacian [on hold]

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form $$\Vert (-\Delta - \lambda ...
3
votes
3answers
181 views

Enumerating cosets of the modular group

Suppose we chose the generators $f(z) = z+1, g(z) = z-1, h(z) = -1/z$ for the modular group $\Gamma$ (i.e. the group of fractional linear tranformations $z \mapsto (az +b)/(cz+d)$ with $a,b,c,d \in ...
2
votes
0answers
159 views

References for 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'

Presently I am reading the 'Theory of $p$-adic Galois Representations by Fontaine & Ouyang'. I am finding it difficult for eg. the initial sections on $l$-adic geometric representation of finite ...
3
votes
1answer
167 views

Model structure on non-negative differential graded algebras with homological grading

I was wondering if there exists a model structure on the category of non-negative differential graded algebras with homological grading. To be more precise: Let $Ch_{k}$ the model category of ...
3
votes
0answers
71 views

Localized at $p$ integral representations of finite elementary $p$-groups

Let $C_p$ be a cyclic group of prime order $p$. Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times). I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$. However, ...
0
votes
1answer
63 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C. I'd like a function μ:Cn×n→[0,∞) ...
7
votes
0answers
113 views

Complete list of exceptions and efficient algorithm for Waring's problem

2 weeks ago, Samir Siksek http://arxiv.org/abs/1505.00647 proved more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, 303, ...
0
votes
0answers
48 views

Global existence solutions NLS

Let's consider the following NLS in $\mathbb{R}^3$ $$i\partial_t\psi=-\Delta\psi+\vert\psi\vert^2\psi$$ How to prove that $H^2$-solutions are globally in time? Can someone suggest references to me?
3
votes
0answers
59 views

Integrability of Continuous Tangent Subbundles

Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space? More specifically given a smooth manifold of $M$ and a ...
4
votes
2answers
172 views

hyperbolic structure on Figure–8 knot complement

I was trying to understand the proof of the fact that there is a hyperbolic structure on Figure–8 knot complement initially from Thurston's notes and then from some online notes; but unfortunately I ...
0
votes
0answers
55 views

Sum of weighted permutations over natural numbers [closed]

If we consider the sum $\sum\limits_{i=1}^n i\sigma(i)$ where $\sigma$ is a permutation of the first $n$ natural numbers, then this sum has a few interesting properties for various values of $n$. My ...
6
votes
2answers
207 views

Where is the exponential map a diffeomorphism?

Let $M$ be a closed compact Riemannian manifold. The exponential map $\mathrm{exp}:TM\to M\times M$ takes $(p,v)$ to $(p,\gamma_v(1))$, where $\gamma_v$ is the geodesic flow at $p$ in the direction ...
1
vote
0answers
77 views

On moduli space of torsion free semi-stable sheaves on nodal curves

Let $X$ be a projective irreducible nodal curve of genus at least $2$. Denote by $U(r,d)$ the moduli space of semi-stable torsion free sheaves on $X$ of rank $r$ and degree $d$. There are several ...
4
votes
1answer
136 views

Upper bound of the waiting time of a sum process

Let $n \in \mathbb{N}$, $x_1, \ldots, x_n \in (0,1)$ fix but arbitrary, s.t. $\sum_{i=1}^n x_i = 1$. Let $X_i \sim \operatorname{Unif}(\{x_1, \ldots, x_n\})$ i.i.d., and $T_n = \min\{t \in \mathbb{N} ...
3
votes
1answer
136 views

Fixed point property for intersection of spaces which are homeomorphic to a disk

The following question is question 9.8 from Miller's paper ``Some interesting problems '': Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every $n\in \omega, ...
7
votes
0answers
252 views

Godel's second incompleteness theorem for non-r.e. theories

R. Jeroslow in this paper proves that a non-recursively enumerable theory whose set of theorems is $\Delta_2$-definable may prove consistency of itself but it can not prove 2-consistency of itself. ...
3
votes
0answers
62 views

Idempotent relations on the unit square with closed graphs

A colleague and I are interested in idempotent relations from $I=[0,1]$ to $I$ - relations such that $R\circ R(x)=R(x)$ for all $x\in I$. Specifically, the graphs of the relations we care about must ...
7
votes
0answers
167 views

May's infinite loop machine for Friedlander's result for Adams conjecture

Eric M. Friedlander in the paper The infinite loop Adams conjecture via Classification Theorem for $\Gamma$-spaces proved the infinite loop Adams conjecture using techniques involved $\Gamma$-space. ...
5
votes
1answer
381 views

Vectorisation of a category

I have no experience with category theory at all, but I recently stumbled upon the following construction. Since it is extremely elementary and seems rather natural, it should be known, but I have not ...
0
votes
0answers
40 views

Looking for Information on Local Degree of Maps on Homology Manifolds

By a homology $n$-manifold, we mean a topological space $X$ such that for all $x \in X$: 1: if $k \neq n$ then $H_k(X, X-x)=0$ 2: $H_n(X,X-x) \cong \mathbb{Z}$. Given homology $n$-manifolds $X$ and ...
6
votes
1answer
130 views

On the independence of lower and upper asymptotic and Banach densities

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := ...
1
vote
0answers
139 views

Reference request: Cohomology of Elliptic Curves

Is it true that the group $$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$ is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility? ...
9
votes
1answer
394 views

On a result attributed to W. Ljunggren and T. Nagell

I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation $$\frac{x^{n}-1}{x-1} = y^{2}$$ doesn't admit solutions in ...
9
votes
2answers
653 views

How to learn QFT from mathematical perspective?

I want to learn QFT, because I have heard of its applications in mathematics, I am not interested in scattering cross sections and such. Where can I start to learn? Only books I found are either way ...
9
votes
1answer
239 views

Sard's Theorem For Banach Spaces

Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...
1
vote
0answers
22 views

Estimates for derivatives of a positive discrete harmonic function

There is the following estimation (Duffin, Discrete potential theory, Theorem 5): Let $f$ be a discrete harmonic function in a sphere of radius $R$ with the center $p$, all in $\mathbb Z^3$. Then, if ...
1
vote
2answers
147 views

On the natural density of almost perfect numbers

This question is pretty basic, so I apologize in advance if it is unsuitable for MO. If so, please do let me know and I will migrate it over to MSE. Essentially, by work of Kanold, we know that the ...
3
votes
1answer
110 views

Classification of 3-forms in dimension 7

I'm looking for a classification of $3$-forms over a real vector space of dimension $7$ as for the $3$-forms in dimension $6$. References on the latter case are R. Bryant On the geometry of almost ...
4
votes
1answer
161 views

Learning Quantum (Co)Homology and Landau Ginzburg Superpotential

I am learning about Quantum Homology which I have to use in my research, and I see that in many papers (For example in FOOO, "Spectral invariants with bulk, Quasimorphisms and Lagrangian Floer ...
2
votes
2answers
239 views

Magnus' embedding theorem

I am looking for a (preferably modern) reference to the following old result of Magnus. Let $F$ be a free group of finite rank and $$ F_1 = [F,F], F_2 = [F_1,F_1], \dots , F_{n+1} = [F_n,F_n], \dots ...
6
votes
1answer
160 views

Henstock, Differentiation under the integral sign

Does anyone know, where I can find the proof of necessary and sufficient conditions for differentiating under the integral sign in case of Henstock integral? Here are the theorems but not all the ...
2
votes
0answers
77 views

Extension of the Hilbert-Mumford Criterion

Let $X$ be a smooth variety, $L$ a line bundle on $X$ and $G$ a reductive group actin on $X$ with a linearization of the action to $L$. Say we are over the complex numbers. Both the concept of GIT ...