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

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0
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0answers
4 views

Suggestion for books in Pertubation theory with an emphasis on the theory

As the title suggest I am looking for another good coverage of the theory of Pertubation theory. Currently I am working through Murodock's book: Pertubations: Theory and Methods. But I am rest assure ...
12
votes
2answers
176 views

Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element $$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$ ...
0
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0answers
55 views

Handbook Homogeneous Space Examples?

Homogeneous spaces are super common in differential geometry, with many classy tricks for linking them together. Is there anywhere available a list covering the well-known examples of the homogeneous ...
2
votes
1answer
81 views

Seeking more information regarding the “rigoidal category” of $\mathbb{N}$-graded sets

Definitions. By an $\mathbb{N}$-graded set, I mean a set $X$ together with a function $|\Box|_X:\mathbb{N} \leftarrow X,$ called the grading. These will simply be called graded sets hereafter. If ...
4
votes
3answers
209 views

Asymptotics for the number of abelian groups of order at most $x.$

The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for ...
4
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0answers
89 views

Another interpretation of the $16$ dimensional Severi Vairety

I asked about an interpretation of this variety here. There is another one that could be easier. Let $K$ be an algebraically closed field of characteristic $0$. We denote the set of terns of $3\times ...
8
votes
2answers
193 views

Topological Derivation of Leray Spectral Sequence

I'm interested in computing - to the extent possible - the Leray spectral sequence for a particular map which is almost, but not quite, a fiber bundle (e.g. a Seifert fiber space). The hardest step ...
1
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0answers
60 views

Intuitive understanding of the mean curvature flow [on hold]

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $S = ...
-2
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0answers
38 views

Finding topological properties under a metric on set of composition operators of L2 [on hold]

We define a new metric on all composition operators in $L^2$: $$ d‎‎_R (‎A‎,B)=‎ \sqrt{‎\Vert ‎P‎_{R(A)}‎‎- ‎‎‎‎‎‎P‎‎_{R(B)} \Vert^2+‎\Vert A‎‎-‎B\Vert^2 }‎. $$ Now we would li‎‎‎‎‎ke to find ...
3
votes
1answer
132 views

Looking for Severi varieties

Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let $$ \mathfrak{J}_{3}=\{A\in\mathcal{M}_{3}(\mathbb{O}):A\text{ is Hermitian}\}, ...
4
votes
1answer
82 views

Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...
2
votes
1answer
93 views

First proof of the integral representation of the hypergeometric function $F(a,b,c;\cdot)$

Assuming $\lvert x\lvert<1$ and $0<a<c$, the following formula holds true $$F(a,b,c;x)=\sum_{n=0}^{+\infty} \frac{(a)_n(b)_n}{(c)_n(1)_n} x^n=\frac{\Gamma ( c ...
5
votes
0answers
97 views

Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1. Has anyone seen these trees? The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
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0answers
35 views

Stochastic Pontryagin Principle with a final state constraint

I am searching for information about the Stochastic Pontryagin Principle with a final state constraint. Someone knows a paper or a book where this case is treated in depth?
-1
votes
0answers
75 views

Twisted Hodge numbers in a family

It is well known (e.g. Voisin's book) that for a smooth family $\pi: \mathcal{X} \to B$ of smooth projective varieties (and projectively normal) over $\mathbb{C}$, the Hodge numbers $h^{p,q}(X_b)$ are ...
2
votes
1answer
74 views

Minimal Support Solutions of a Linear System (Dissertation)

For a given $n \times m$ matrix A with $m>>n$ and a given vector $\vec b \in \mathbb{F}^{n \times 1}$, and given that $A\vec{x}=\vec{b}$ for at least one $\vec{x} \in \mathbb{F}^{m \times ...
3
votes
1answer
159 views

Local product structure of determinantal variety

The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending ...
0
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0answers
30 views

to find topological properties under a metric on a set [on hold]

we define a metric d on a set of composition operators on L2. I would like to find connected component and path connected component and other topological properties by d . Is there any book or paper ...
2
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0answers
151 views

About the limitation by size

This could be a big post, so I'll try to summarize my thoughts and divide them into several questions. When working in category theory, I used to choose the following definition. A category $C$ is ...
2
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0answers
54 views

Weighted global Holder property for Brownian motion paths

It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1 $$ ...
2
votes
2answers
112 views

A reference about Grassmannian over finite fields

Suppose $Gr_k(k,n)$ the Grassmannian which classifies all the dimension $k+1$ sub-spaces of a dimension $n+1$ linear space over the field $k$. For the case over a finite field $\mathbb F_{q}$, we can ...
9
votes
3answers
228 views

Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture: A non-trivial connected sum $M_1\# M_2$ admits a ...
0
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0answers
38 views

Open Hamiltonian Gromov-Witten Invariants

Both open Gromov-Witten invariants and Hamiltonian Gromov-Witten invariants have been studied. I am interested in knowing whether anyone has considered open Hamiltonian Gromov-Witten invariants ...
2
votes
1answer
100 views

Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
5
votes
0answers
139 views

Flat + locally of finite presentation + monomorphism = open immersion

It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$: Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then ...
1
vote
1answer
253 views

research articles in topology/geometry [closed]

There is a saying "Do you read the masters?" I want to read some basic papers in Topology/geometry... I can not clearly state what is basic as of now... My back ground includes course in ...
1
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0answers
58 views

Perturbating the boundary of a helicoid

I prepare a long helix with many periods, so that I can obtain a helicoid with soap film, i.e. a minimal surface whose boundary is the helix. The helix is not perfect, it is unavoidable that some ...
0
votes
0answers
19 views

Markov Modulated Markov Chain

Consider a discrete time Markov chain $X_t$ on some finite state space $\mathcal{S}$ with transition matrix $P$. Now consider a process $Y_t$ also on $\mathcal{S}$, which conditioned on $X_{t}=s$ ...
3
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1answer
97 views

Least ordinal not embedded in a total order

If $(E,<)$ is a linear order, let $s(E,<)$ denote the least ordinal which doesn't embed in $(E,<)$. I am trying to prove the following: If $(M,+,.,0,1)$ is a model of open induction, (or ...
4
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0answers
66 views

Some questions on the nodal geometry of Dirac operators

Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows: Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...
5
votes
3answers
504 views

If the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equal

On p. 76 of the 1996 edition of Serre's A Course in Arithmetic, one reads the following (inline) remark: One can prove that, if $A$ has natural density $k$, the analytic density of $A$ exists and ...
2
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0answers
71 views

A Künneth-Theorem version for relative singular cohomology

I'm not an expert in algebraic topology, but sometimes I need some results from this area, for example tools to determine singular cohomology groups of product spaces. The Künneth-Theorem which I ...
3
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0answers
144 views

A generalization of Rogers-Ramanujan identity

The generalized Rogers-Ramanujan identity has the following form $$\sum_{k_1\geq\cdots\geq k_r\geq 0}\frac{x^{k_1^2+\cdots +k_r^2+k_i+\cdots +k_r}}{(x)_{k_1-k_2}\cdots ...
1
vote
1answer
239 views

Reference request for an introduction to deformation theory in algebraic geometry

I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through ...
5
votes
2answers
523 views

mod 5 partition identity proof

I am looking for a proof that: $$\prod_\limits{m=0}^\infty \dfrac{1}{(1-x^{5m+1})}=\sum_\limits{i=0}^\infty \dfrac{x^i}{\prod_\limits{j=1}^i (1-x^{5j})}$$ The left hand side expands into: ...
1
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1answer
94 views

A question about Borel sets on the unit interval

It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. ...
4
votes
1answer
487 views

Algebraic Geometry needed for Kähler-Einstein metric

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic ...
5
votes
2answers
293 views

Symplectic orthogonality and projective duality: how do they work together?

Let $(V,\omega)$ be a $2n$-dimensional linear symplectic space, and $(\mathbb{P}V,\theta_\omega)$ the corresponding $(2n-1)$-dimensional contact manifold. Given a smooth $(n-1)$-dimensional smooth ...
3
votes
0answers
187 views

Roadmap for the ideas expressed in Grothendieck's Esquisse d'un Programme

I would like to understand Grothendieck's Esquisse d'un Programme more. Are there any references that would help me, and are there modern works pursuing the same themes? At this point I am still ...
0
votes
0answers
64 views

Reference for generalized model categories [closed]

I would like to know some references for "Stanculescu's notion" of Generalized model category. Motivation The idea is find some works that use the notion and increase my knowledge about it. ...
-1
votes
1answer
156 views

Distinguished triangle and short exact sequence [closed]

Forgive me for asking an elementary question. Given coherent sheaves $A$, $B$, $C$ and morphisms $B\xrightarrow{f} C\xrightarrow{g} A$ which give rise to the distinguished triangle $A[-1] \rightarrow ...
10
votes
1answer
204 views

Elementary prime-generating sequences

A student of mine keeps coming again and again and telling "I've found a formula $n\mapsto f(n)$ giving all primes" or sometimes "infinitely many primes", where $f$ is a classical function (I mean ...
6
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0answers
110 views

Understanding homotopy t-structure

The following question came up while reading Hoyois' From algebraic cobordism to motivic cohomology. Let $S$ be a Noetherian scheme of finite Krull dimension and let $SH(S)$ denote the homotopy ...
0
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1answer
81 views

In the proof of the existence of weak solutions to the NSE

In the proof of the existence of weak solutions to the NSE (Navier-Stokes Equations by Constantin and Foias, Chapter 8), the following argument is made: Let $u_m$ converges weakly to $u$ in ...
1
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1answer
97 views

The space of loops as a Banach space [closed]

Let $S$ be the space of all loops in $\mathbb{R}^2$, i.e. all continuous mappings $\gamma:[0,1]\to \mathbb{R}^2$ such that $\gamma(0)=\gamma(1)$. Is there any canonical interpretation of $S$ as a ...
1
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2answers
51 views

Maximal Minimum Weight DAGs

In the case of undirected, connected graphs the name for the maximal cycle-free subgraph of minimal weight is called Minimum Spanning Tree, and the efficient algorithms for their calculation are well ...
2
votes
1answer
76 views

Almgren's mimeographed lectures notes on varifolds

I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to ...
3
votes
2answers
90 views

References about the matrix generators of the finite subgroups of the orthogonal group O(4)

"On Quaternions and Octonions" by Conway and Smith gives the classification of the finite subgroups of the orthogonal group O(4). I want to get the explicit matrix generators of the finite subgroups. ...
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0answers
74 views

Fiber of the specialization map of Picard groups

Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...
4
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0answers
88 views

Ref request: modelling regular theories as an injectivity condition

The background to the following observation is standard material in categorical logic, and I thought this was was too — I don’t remember learning it, but I don’t think it is original — but I can’t now ...