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

**9**

votes

**1**answer

332 views

### Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$?

Definition. $$\mathbb{J} = \{1,2,3,\ldots\}.$$
We can refer to the elements of $\mathbb{J}$ as "joiners."
The product of joiners is inherited from $\mathbb{Z}$.
The sum of joiners ...

**5**

votes

**0**answers

61 views

### Cap product on Leray-Serre spectral sequences

Let $\ p:X\to B\ $ be a fibration of spaces with fiber $F$. Then there are homological and cohomological Leray-Serre spectral sequences $E^{r}_{pq}$ and $E_r^{pq}$ that converge to $H_*(X)$ and ...

**0**

votes

**0**answers

24 views

### Generalized Plateau problem with non-Jordan boundary

Let $C_\pm$ be the two circles obtained by intersecting the cylinder $x^2+y^2=R^2$ with the planes $z=\pm 1$, on which we mark four points $A_\pm:(R,0,\pm 1)$ and $B_\pm:(-R,0,\pm 1)$. Assume that ...

**18**

votes

**2**answers

604 views

### References for $K_{4k}(\mathbb{Z})$

Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...

**0**

votes

**0**answers

13 views

### Measures for the Eccentricity of General Strictly Convex Smooth Closed Manifolds of Genus 0

Question:
Are there any measures for how much the shape of a strictly convex smooth closed manifold of genus 0 deviates from that of a hyper-sphere of equal dimension?
In euclidean 2-space ...

**2**

votes

**0**answers

54 views

### PDE Parameter-Dependent Center Manifolds

In the book Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems by Mariana Haragus, parameter-dependent center manifolds are discussed. Here it is assumed ...

**3**

votes

**1**answer

178 views

### Who said “the naive counting numbers don't exhaust $\Bbb N$”?

In the context of Robinson's framework, or more precisely its reformulation by Ed Nelson, one of the practitioners in the field expressed the sentiment something like "the naive counting numbers don't ...

**6**

votes

**3**answers

159 views

### Basic question about polytope duals

The following must be well known. Is there a beginning or midlevel
text where the answer is discussed? Thanks.
Along with a polytope one has the notion of its dual which is officially
defined via ...

**9**

votes

**0**answers

135 views

### Reference Request for Hilbert Schemes

I'm a physicist working on Fractional Quantum Hall effect. The mathematical subjects of study are symmetric, translational invariant, homogeneous polynomials on $\mathbb{C}$. Very early in my study I ...

**1**

vote

**0**answers

16 views

### How can I filter the effects of a variable from a correlation matrix?

I have a correlation matrix (it contains 500 columns and 500 rows) and I would like to make an other correlation matrix in which one variable (and its influences) is filtered from the initial matrix. ...

**2**

votes

**1**answer

67 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
$$
...

**10**

votes

**0**answers

213 views

### Multiplicative Structure of the Atiyah-Hirzebruch/Leray-Serre spectral sequence

This is related to this question (edit: now answered, see below). Is there a nice explanation of the multiplicative structure on the higher pages of that spectral sequence? I want to assume that $h$ ...

**1**

vote

**0**answers

79 views

### Intuition behind the Duistermaat-Guillemin version of Weyl's law

The theorem in question (see this paper), after a modification by Ivrii (see this paper) states the following:
Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 2$. Assume that the ...

**21**

votes

**2**answers

540 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
$$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$
...

**5**

votes

**2**answers

145 views

### Rational homology sphere that is not Seifert manifold

I wonder if there is an example of rational homology sphere that is not a Seifert manifold. If there is, how can one construct such a rational homology sphere from a surgery of a knot in $S^3$?

**5**

votes

**1**answer

553 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 ...

**1**

vote

**0**answers

40 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 ...

**6**

votes

**1**answer

267 views

### Non-negativity of invariant polynomials

Certifying the non-negativity of a symmetric polynomial is much easier than
certifying the non-negativity of an arbitrary polynomial function:
for instance, in (1) is proved that the complexity of ...

**2**

votes

**1**answer

125 views

### Existence of parametrizations of rational orthogonal matrices

I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this?
...

**5**

votes

**3**answers

250 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**

votes

**1**answer

149 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}\},
...

**3**

votes

**1**answer

102 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 ...

**-2**

votes

**0**answers

49 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 ...

**4**

votes

**0**answers

102 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

**2**answers

208 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 ...

**5**

votes

**2**answers

300 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 ...

**1**

vote

**0**answers

61 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 = ...

**14**

votes

**1**answer

365 views

### On convergence of convex bodies

Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$.
Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any ...

**4**

votes

**1**answer

90 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) ...

**6**

votes

**0**answers

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 ...

**1**

vote

**1**answer

236 views

### Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...

**2**

votes

**1**answer

871 views

### For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...

**3**

votes

**1**answer

167 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 ...

**1**

vote

**2**answers

256 views

### Looking for (information about) long diamonds

I was given an open problem as a birthday present recently. While I can probably handle spoilers at this point, what I really want are literature and other references. Also acceptable would be ...

**2**

votes

**1**answer

195 views

### Path integral methods

Are there detailed expositions of the path integral methods in (mathematical) physics other than Feynman-Hibbs and Glimm-Jaffe?

**12**

votes

**2**answers

375 views

### S-matrix for the HOMFLY/Hecke category

This question concerns the HOMFLY-PT category, closely related to Hecke algebras. (See here for example.)
The minimal idempotents of this category are indexed by pairs $(\lambda_+, \lambda_-)$ of ...

**1**

vote

**2**answers

348 views

### Was “arithmetical translation” (coding in the Goedel sense) ever a part of Hilbert's Program?

Was "arithmetical translation" (that is, coding in the Goedel sense) ever a part of Hilbert's Program? I ask this question for several reasons:
i) it gives the numerals |, ||, |||,.... an ersatz ...

**2**

votes

**1**answer

94 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 ...

**9**

votes

**1**answer

426 views

### Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?

It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: ...

**6**

votes

**0**answers

102 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 ...

**1**

vote

**1**answer

237 views

### Sheets in Lie algebras are classified by the pair $(\mathfrak{l},O)$. How about sheets in Lie groups?

Borho in this paper showed that a sheet (i.e. a suitable union of adjoint orbits of the same dimension) in a complex semisimple Lie algebra $\mathfrak{g}$ is classified by a pair $(\mathfrak{l},O)$ ...

**5**

votes

**3**answers

508 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 ...

**0**

votes

**0**answers

43 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?

**2**

votes

**1**answer

77 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 ...

**-1**

votes

**0**answers

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 ...

**0**

votes

**0**answers

30 views

### to find topological properties under a metric on a set [closed]

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 ...

**4**

votes

**0**answers

72 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 $S$ ...

**2**

votes

**0**answers

156 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**

votes

**2**answers

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 ...

**10**

votes

**3**answers

236 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 ...