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

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3
votes
0answers
43 views

Bundles over Grassmanian with given top Chern class

So, I have been working on Chern classes for my master thesis and apparently (My proofs could be wrong and a few things are still vague) I was able to give a construction method and exhibit, via ...
4
votes
0answers
46 views

Reference for the Banach Manifold structure of $C^k(M,N)$

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following: Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set ...
0
votes
0answers
76 views

Universal property of complete linear systems

Let $X$ be a projective scheme over a field $k$ and $S$ a $k$-scheme. Fix a closed immersion $i:X \to \mathbb{P}^n$ for some $n$ and denote by $\mathcal{O}_X(1):=i^*\mathcal{O}_{\mathbb{P}^n}(1)$. Let ...
6
votes
1answer
87 views

Zeta-Determinant for shifted Laplacians on the circle

Consider on the circle $S^1$ the operator $$L := - \frac{\partial^2}{\partial \theta^2} + c$$ for some constant $c \in \mathbb{R}$. What is its $\zeta$-regularized determinant? This should be ...
1
vote
1answer
39 views

Distribution of Poles of solutions to the first Painleve equation

In the introduction of this paper the distribution of the poles of solutions to the first Painlevé equation is discussed. In particular it is said that the poles form a deformed lattice that ...
-1
votes
1answer
186 views

Class forcings and elementary embeddings

In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem: "Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial ...
1
vote
1answer
64 views

Essential surfaces in the Exterior of Montesinos knots

Hatcher and Oertel computed the boundary slopes of essential surfaces of Montesinos knots in this paper. But they do not consider surfaces that do not intersect the boundary of the exterior. An ...
3
votes
0answers
75 views

What does “control of a deformation problem” mean?

Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...
0
votes
0answers
112 views

Is the complex structure on a del-Pezzo surface a regular complex structure?

Let $(X, \omega, J)$ be a compact symplectic manifold with an almost complex structure. Fix some homology class $\beta \in H_2(X, \mathbb{Z})$. An almost complex structure $J$ is said to be ...
3
votes
0answers
18 views

Characterization of complete lattices with join-incomplete lattice endomorphisms

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) > \bigvee_L f(S).$ How can ...
6
votes
0answers
141 views

$\alpha$-minimal degrees for singular $\alpha$

An important question in $\alpha$-recursion theory is whether there is a minimal $\alpha$-degree at $\alpha=\aleph_\omega.$ Question 1. Who first introduced the above question, and where can I find ...
3
votes
1answer
293 views

Survey papers on the role played by PDE in mathematics

There are already several questions on Mathoverflow about the application of PDE to several other topics (e.g., algebraic and differential geometry and topology, number theory, harmonic analysis, ...
1
vote
0answers
95 views

Reference request: The compactness and compact embedding in Besov Space?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...
1
vote
0answers
56 views

What can we say about the boundary of the level set of a Sobolev function?

I'm a beginner of the area of free boundary problem. Let me first give some background: $\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph. ...
0
votes
0answers
69 views

Find motivation for calculating $\int_{2}^{X} A^2(t) A(\alpha t)dt$

I read a thesis of Kong Kar Lun (student of Tsang K.M) about the some mean value theorems for certain errors terms in analytic number theory and in which he gave the asymptotic formulas of the ...
5
votes
0answers
156 views

Examples of Rankin-Selberg L-functions from Eisenstein series

I've been digging for awhile to not much success, so I figure I would try here: I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...
-2
votes
0answers
107 views

question on collatz trajectories/pattern in form $2^n - 1$ [on hold]

I recently noticed this remarkable general "wedge" pattern in base 2 for Collatz iterates starting with values $2^n - 1$, displayed here for $n=20$. Has this been noticed or analyzed before? ...
4
votes
1answer
143 views

An example for a construction on monads/operads?

Suppose that $C$ is a bicategory. (I only need a monoidal category, i.e. one object bicategory, but I will stick with bicategories, since theory of monads is more commonly stated in that setting). A ...
2
votes
1answer
99 views

Where can I find a copy of Moussatat's 1976 thesis “On the Asymptotic Theory of Statistical Experiments and Some of Its Applications”?

It was apparently written at Berkeley under the direction of Le Cam, and it is cited in a number of contributions to mathematical statistics, for example in Strasser's (1985) book "Mathematical Theory ...
5
votes
1answer
62 views

Continuous section inside a family of rank-varying operators

Good morning everybody, my question is as follows: let $K$ be a compact set and assume $F:K\to L(\mathcal H,\mathbb R^{m+1})$ be a continuous map from the compact set $K$ to the space of linear ...
2
votes
0answers
64 views

$Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]

This is some sort of "follow-up" to the (unanswered) question posted here. Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$ Then $\varphi $ is an automorphism of $O(2n)$, and ...
0
votes
1answer
34 views

References for non-zero boundary value problem

I studied linear elliptic, parabolic and hyperbolic PDEs (boundary/initial value problem) in terms of existence, uniqueness and regularity. I studied always, following Evans book "PDE", the case with ...
1
vote
0answers
83 views

One-sided local $L^p$ spaces

Consider the vector space $L^p_{\text{left-loc}}$ of measurable functions $f:[0,1]\to\mathbb R$ so that for all $x\in(0,1]$ there exists $\delta>0$ so that $f|_{[x-\delta,x]}\in L^p$. Does this ...
0
votes
2answers
212 views

Linear Algebra classic books [closed]

I'm learning linear algebra at the moment, so I'm looking for some great old classic books. Something like Fermat's or Gauss books of some great mathematians. I don't really like the nowadays books ...
7
votes
0answers
230 views

Residue class sufficiency sets for the Collatz conjecture

I have recently managed to show a sequence of sufficiency sets for the Collatz conjecture whose natural density approaches 0 (the set theoretic limit approaches the set $\{1\}$). It is an extension of ...
6
votes
0answers
311 views

“Forthcoming paper” of Goldston-Graham-Pintz-Yıldırım

The above-named authors of [1] and its (significantly different) published version [2] write: In a forthcoming paper, we will show how the methods here can be extended to prove corresponding ...
2
votes
0answers
58 views

Ultraproducts and subobjects of projectives

Usually the question whether the class of projective algebras in a given variety is closed under taking subalgebras seems to be quite hard. In varieties with well understood dual geometry (e. g. ...
2
votes
0answers
46 views

Terminology for torsion semigroups where the order of elements is uniformly finite

A (multiplicatively written) semigroup $\mathbb A = (A, \cdot)$ with the property that ${\rm ord}_\mathbb{A}(a) := |\{a^n: n \in \mathbf N^+\}| < \infty$ for every $a \in A$ is called a periodic ...
0
votes
0answers
16 views

Is there any relation between number or nature of cycles and sub matrix of adjacency or incidence matrix of a graph? [closed]

Is there any relation between number or nature of cycles and sub-matrix of adjacency or incidence matrix of a graph?
0
votes
0answers
109 views

Minimum regular open set containing a given set in a T0 Alexandrov topological space

What is known about the minimum regular open set containing a given set in a T$_0$ Alexandrov topological space? I'm particularly interested in the condition for the minimum set happening to be ...
7
votes
1answer
515 views
+50

Counting subspaces

We are given the finite vector space $V = V(n,p) = \mathbb{F}_p^n$ and two fixed subspaces $W_1, W_2 \subseteq V$ of dimensions $m_1$, $m_2$ respectively. Suppose that the dimension of $W_1 \cap W_2$ ...
2
votes
0answers
83 views

Distribution of residue classes of totients of (univariate) polynomials

Let $\phi$ denote Euler's totient function and $f$ a non-constant univariate polynomial with integer coefficients such that $f(u) \in \mathbf N^+$ for all $u \in \mathbf N^+$ (assume $f$ is ...
3
votes
2answers
185 views

Products of elliptic isometries

A well-known property on groups acting on trees is: Theorem: Let $T$ be a tree and $g,h \in \mathrm{Isom}(T)$ two elliptic isometries. If $\mathrm{Fix}(g) \cap \mathrm{Fix}(h) = \emptyset$ then ...
15
votes
0answers
350 views

What sort of cardinal number is the Löwenheim-Skolem number for second-order logic?

In their paper "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic", Magidor and Väänänen make the following statement: "For second order logic, $LS(L^{2})$ ...
1
vote
0answers
69 views

Existence of the solution of a Dirichlet type differential equation

I'm reading the first chapter of the book A geometric approach to free boundary problems by Caffarelli and Salsa, see the PDF here. The question came from Page 14—15. Let me state my question: ...
1
vote
1answer
117 views

Noncommutative analogs of classical Banach geometric properties

The scale of Schatten-von Neumann classes is noncommutatitve analog of classical $\ell_p$-spaces. A lot of researchers devoted their lives to study Banach geometric structure of these spaces. ...
2
votes
2answers
89 views

Reference request: using integral equations to study asymptotics of ODEs

I was told by my supervisor that one way to study the asymptotic behaviour of solutions to ODEs is "to reformulate them as integral equations, and use fixed-point kind theorems on the resulting ...
0
votes
0answers
397 views

Commutative algebra books representing the edge of research

Recently I have come across the books Combinatorial Commutative Algebra by Miller and Sturmfels along with Combinatorics and Commutative Algebra by Stanley. I will soon own a copy of each. I also ...
3
votes
1answer
195 views

reference request: direct product of WOT-continuous unitary representations

In an article I'm revising, I spend some time giving a self-contained proof of the following result Let $G$ be a (Hausdorff) topological group and let $(\pi_i)$ be a family of unitary ...
2
votes
1answer
123 views

Gradient of Probability Distribution

Given a random walk on a lattice $L$ (not necessarily centered - we allow $E[X_i] \neq 0$ for the i.i.d. increments $X_i$), let $p_t(x)$ denote the probability measure of state $x \in L$ after $t$ ...
2
votes
1answer
154 views

The first Betti number of a finite covering space of a closed 3-manifold

Let $\tilde{M}\to M$ be a finite covering of closed 3-manifolds. Is it possible that $\beta_1(\tilde{M})-1> [\pi_1(M):\pi_1(\tilde{M})](\beta_1(M)-1)>0$? Here, ...
7
votes
1answer
262 views

A question related to Woodin's $HOD$ conjecture

Recall that an $\kappa$ is $\omega$-strongly measurable in $\text{HOD}$ if there exists $\lambda < \kappa$ such that $(2^{\lambda})^{\text{HOD}} < \kappa$ and such that there is no partition of ...
6
votes
0answers
88 views

Linear vs smooth actions of finite groups on spheres, euclidean spaces and closed disks

I would like to know examples (with references, if possible) of the following: (1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear ...
1
vote
1answer
115 views

Maximal $\pi/2$-separated subset of the sphere

A subset $A$ of a metric space is called $\varepsilon$-separated if $$dist(x,y)> \varepsilon \mbox{ for all } x\ne y\in A.$$ (Notice that the inequality in my definition is strict.) What is the ...
2
votes
1answer
80 views

Enumerator Polynomials for Linear Anytime Codes

Let $C = \{c \in \mathbb{F}^n_2 : Hc=0\}$ be a binary linear code where $H \in \mathbb{F}^{k \times n}_2$ is a block lower-triangular matrix of full rank called the parity-check matrix of $C$. Clearly ...
2
votes
1answer
120 views

How to construct the symmetric power function from a modular form?

I want to understand how we construct from a modular form $f$ its symmetric power function $Sym^rf.$ I read that there is a particular representation that does this but I am not familiar with this ...
8
votes
2answers
258 views

References on quaternionic geometry

Is there any analog, in the quaternionic setting, of Kahler potentials? In particular I'm interested in a similar construction of the Fubini-Study 2-form on $\mathbb{P}^1(\mathbb{C})$ over the ...
1
vote
1answer
76 views

A proof of the Ibragimov et al. commutation relation

Let $u_a(x),\,a=1,2,\ldots n$ be a $n$-component field in Minkowski spacetime $x^\mu,\,\mu=0,1,2,3$ and let $u_{a,\,\mu}=\frac{du_a}{dx^\mu}$. Let us introduce two operators (we use Einstein ...
8
votes
0answers
80 views

types of generating functions (ordinary, exponential, ???) closed under substitution

A nice feature of ordinary and exponential generating functions is that they are closed under substitution: if $F(z)$ and $G(z)$ both have integer coefficients, then $F(G(z))$ also has integer ...
5
votes
0answers
42 views

Possible values of eigenvalues of Hadamard product of Hermitian matrices

One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem: ...