# Tagged Questions

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

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**1**answer

59 views

### Character kernels in the lattice of subgroups of a finite abelian group

I am looking for any efforts that have been made to characterize the character kernels (equivalently, the subgroups yielding cyclic quotients) inside the lattice of subgroups of a finite abelian ...

**7**

votes

**1**answer

87 views

### Integer Recursion Reference Request

I've run across the following recursion which at times seems very steady and predictable and at other times seems very chaotic.
Let $c_1, \dots c_k, b_0, m \in \mathbb{Z}$ with $b_0>m\ge 3$ and ...

**3**

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**0**answers

80 views

### Eilenberg-Moore spectral sequence for path-loop fibration over Q\Sigma X (reference request)

Related to the question here, here is another question. Consider the kernel of the map $H_*(QY;Z/p)\rightarrow H_{*+1}(Q\Sigma Y;Z/p)$. restricted to
$PH_*(QY)$, and let's say $Y$ itself is a ...

**2**

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**0**answers

48 views

### Constant in a trace Sobolev theorem for concave domains

I wonder is the following inequality is true/known:
Let $\Omega\subset \mathbb{R}^n$ be a (locally) Lipschitz domain which is the complement of a convex set, then
$$
\int_{\partial\Omega} |u|^2 ds
...

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vote

**1**answer

60 views

### Positive solutions to simultaneous real quadratic equations

I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as
$diag(x)Ax=1$
$x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal ...

**0**

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**0**answers

56 views

### When does this system of equations has a non-trivial solution?

Let $A$ be a non-negative matrix whose rows and columns are indexed by the elements of $2^M$ - the subsets of some finite set $M$. The subsets of $2^M$ are ordered according to some pre-specified ...

**4**

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**1**answer

146 views

### Generalized ordering on simplicial complex

The vertices of simplicial complexes are usually totally ordered so that face maps of each simplex can be defined easily for the purposes of homology. That gives an "oriented" simplicial complex.
But ...

**2**

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**0**answers

91 views

### Hypergeometric function asymptotics

I came across the following hypergeometric function recently:
$$
_2F_1(1-n,p-2n+1;p-n+1;x)
$$
where $p > 0$ is a non-integer constant, $n$ some large positive integer, and $x > 0$ a small ...

**8**

votes

**2**answers

288 views

### Examples of calculating perverse sheaves on algebraic varieties with easy stratification

I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book
...

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**6**answers

2k views

### Looking for a source for Intended Interpretation

Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...

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**1**answer

304 views

### Dimension of a homotopy type

What is the state of knowledge about the dimension of homotopy types? By the latter I mean the minimal number which is the dimension of a topological space representing the homotopy type. The open ...

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**0**answers

43 views

### Conceptual question about partitions in a given rectangular grid

Suppose we have a Young diagram $\lambda$ inside an $r \times n$ rectangular grid, i.e. $\lambda \subset [r] \times [n]$. If I were to add just one more box to $\lambda$, obtaining a new partition ...

**3**

votes

**1**answer

165 views

### How many second-order PDEs can be obtained from a contact EDS?

Let $(M,[\theta])$ be a contact manifold, $\dim M=2n+1$, and denote by $\mathcal{I}^\theta$ the differential ideal generated by the contact form $\theta$.
An exterior differential system on $M$ of ...

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vote

**1**answer

182 views

### Is it proved that for every integer $p>0$ there exists an integer $k>0$ such that every integer $n>0$ can be expressed as $j_1^p+\dots+j_k^p$?

It has been shown, by elementary methods, that every positive integer can be expressed as the sum of $4$ squares. This type of result has been proven for many different powers $p$, for example, when ...

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**2**answers

344 views

### Reference for Nori motives

I would like to study Nori motives and I am a complete outsider of the subject. I do, however, have background on Chow motives, Voevodsky motives $\mathrm{DM}$ and his stable homotopy category ...

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**1**answer

331 views

### Reference request: seminar report of Serre from late 60s on possibility of Galois representations attached to modular forms?

See here for a comment of Matt Emerton.
There are also various seminar reports of Serre, e.g. his report on mod p modular forms, but also his report from the late 60s on the possibility of Galois ...

**12**

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**1**answer

347 views

### Counting lattice points inside a three-dimensional ellipsoid

I want to answer the following simple question:
Given a three-dimensional ellipsoid defined by $Q(x, y, z) \leq Z$ for a positive definite quadratic form $Q$, how many lattice points in ...

**1**

vote

**0**answers

38 views

### Invariants of Permutations with Predicate and Equivalency Relation

Has the following kind of problem been investigated previously and, where can I find information about it:
Given
the set $\mathbb{P}_{n_0}$ of all permutations of $n_0$ elements
a ...

**2**

votes

**1**answer

110 views

### Regularity - mean curvature equation

In my research I arrived at the following equation:
$$ \int_B \frac{\nabla u \cdot \nabla \varphi}{ \sqrt{1+|\nabla u|^2}}=\int_B f \varphi, (*)$$
for every $\varphi \in C^1(B)$,
which is a weak form ...

**4**

votes

**1**answer

232 views

### Complexifying a real-analytic singularity

This is probably a well-known issue, but I could not find a clear discussion in the literature, and I think others could find it useful.
Consider a real-analytic function germ $f:(\mathbb R^2,0) ...

**3**

votes

**1**answer

207 views

### bounded analytic function as a power series

Suppose $$f(x)=\sum_{k=0}^\infty a_k\frac{(i x)^k}{k!}$$ where $$a_k=k!\int_0^1 p_k(y_{k-1})\int_0^{y_{k-1}}p_{k-1}(y_{k-2})\cdots \int_0^{y_1}p_1(y_0) dy_0\cdots dy_{k-2}\;dy_{k-1}$$ for functions ...

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567 views

### Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...

**3**

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**2**answers

195 views

### open subgroup scheme closed

Let $G/S$ be a group scheme and $H \leq G$ an open subgroup scheme. Is $H \subseteq G$ closed? I want to apply this to $G^0 \leq G$ (see SGA 3, VI_B, Théorème 3.10) for $G$ commutative.
(*) If $S = ...

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**2**answers

304 views

### Principal bundle approach to general relativity

I am curious if there is any literature (texbooks, mainly, but articles will do too, though I don't have easy access to any paid journal) that deals with general relativity by using Ehresmann ...

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**1**answer

140 views

### Is there an algebraic way to characterise the ordinary integral flags?

Fix a vector space $V$ and an integer $1\leq n<\dim V$.
If $\mathcal{I}\subseteq\Lambda^\bullet V^*$ is an ideal, I denote by $\mathcal{I}^i:=\mathcal{I}\cap\Lambda^iV^*$ its $i^\textrm{th}$ ...

**16**

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**1**answer

527 views

### Finiteness Conjecture (New Doomsday conjecture)

This is completely out of curiosity.
I wonder if there has been any recent progress reported on the Finiteness or New Doomsday conjecture, in the form of a talk, preprint or possibly a paper?
Just ...

**2**

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**0**answers

203 views

### Identity with Ramanujan's generalized continued fraction

Let $F(x,q)=\sum_{n\geq 0}x^n\dfrac{q^{n^2}}{(q)_n}$, where $(q)_n=(1-q)(1-q^2)\dots(1-q^n)$. Then:
$$H(x,q)=\frac{F(-xq,q)}{F(-x,q)}=\dfrac{1}{1-\dfrac{qx}{1-\dfrac{q^2x}{1-\dots}}}$$ is the ...

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votes

**2**answers

316 views

### Determinacy of (infinite, possibly loopy) combinatorial games

I am looking for references and hopefully enlightening proofs of the following statement(s) concerning the determinacy of not-necessarily-well-founded (i.e., possibly infinite, possibly loopy) ...

**25**

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**0**answers

441 views

### “Three great cocycles” in Complex Analysis as cohomology generators

In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$
and the Schwarzian ...

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**0**answers

185 views

### Reference request: Whitehead product and the Borel construction

This is a question about signs.
Fix
a based space $(X,x_0)$,
a topological group $G$
acting on $X$ from the left, so that the basepoint $x_0$ is fixed,
a based map $\alpha\colon S^p\to G$ ...

**4**

votes

**1**answer

234 views

### Kalman filters and stock price prediction

Could someone be so kind as to direct me to a good source that would explain time series (more specifically) stock price prediction using Kalman filters, Extended kalman filters or particle filters. ...

**3**

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**1**answer

199 views

### Generators K-theory of Cuntz algebras

The Cuntz algebra $O_n$ is the C*-algebra generated by n isometries $S_1$, ..., $S_n$ such that $S_i^* S_j=\delta_{i,j}$ and $\sum_{i=1}^nS_iS_i^*=1$. Cuntz proved that this algebra has the following ...

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**1**answer

92 views

### Does positive relative density imply asymptotic additive basis behaviour?

First definitions: let $A, B \ \subset \mathbb{Z_{>0}}$ and $1\in A, 1\in B$. We define the relative density of $A$ with respect to $B$ to be $$rel(A, B) = \inf_n \frac{|A \cap [1,n]|}{| B \cap ...

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**2**answers

778 views

### Does the functor Sch to Top have a right adjoint?

Let $S$ be a scheme, let $T$ be an $S$-scheme, and let $M$ be a set. Let $M_{S}$ be the disjoint union of $M$ copies of $S$, considered as an $S$-scheme. (Notation from [SGA 3, Exp. I, 1.8].) Then ...

**6**

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**1**answer

95 views

### Coefficient problem for univalent harmonic functions on unit disk

The Clunie Sheil Small conjecture for the second coefficient of a univalent harmonic function on the unit disk is as follows:
Suppose, $h(z)+\overline{g(z)}$ is a one-to-one harmonic function on the ...

**7**

votes

**3**answers

433 views

### Is there research on Machine Learning techniques to discover conjectures (theorems) in a wide range of mathematics beyond mathematical logic?

Although there already exists active research area, so-called, automated theorem proving, mostly work on logic and elementary geometry.
Rather than only logic and elementary geometry, are there ...

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votes

**1**answer

116 views

### Definition of the Teichmuller space via topological marking of the $\pi_1$

Let $S_0$ be a compact orientable and oriented surface, of genus $g$, fixed once for all. For a fixed point $s_0\in S_0$, one considers a symplectic basis $a_1,\ldots,a_g,b_1,\ldots,b_g$ of ...

**1**

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**0**answers

75 views

### What does the regular representation of the coinvariant ring of a unitary reflection group look like?

Let $V$ be a complex vector space of finite dimension $n$ and let $W$ be a finite unitary reflection group. This is, $W$ is a subgroup of $GL(V)$ generated by reflections, i.e., elements $r \in GL(V)$ ...

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**0**answers

57 views

### Linking circles inside an immersed surface

(Migrated from Math Stack Exchange)
A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the ...

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**3**answers

384 views

### $A_{\infty}$-structure on closed manifold

Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ?
Edit: First, ...

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**1**answer

168 views

### Heyting algebras originating from directed graphs

The category RefGph of reflexive directed graphs is the functor
category $\hat{∆}_1=\mbox{Fun}(∆^◦_1,$Set), where $∆_1$ is
the simplex category truncated at level 1.
Hence the poset Sub(X) of ...

**5**

votes

**1**answer

172 views

### Is there a polynomial-time algorithm to check if a signed graph contains an odd-K5 minor?

I suspect this exists, if anyone has a reference please that would be very helpful.
By signed graph, I mean each edge is designated either odd or even (e.g. as in Guenin's result for weakly bipartite ...

**6**

votes

**1**answer

121 views

### Class number of Burnside groups

Let $B(m,n)$ be the Burnside group on $m$ generators of exponent $n$. Suppose the class number - the number of conjugacy classes - of $B(m,n)$ is finite. Does it imply that $B(m,n)$ is finite?

**0**

votes

**1**answer

93 views

### Whitehead's lemma (Lie algebras) for reductive Lie algebras [closed]

Whitehead's lemma (Lie algebras) is: Let $\mathfrak{g}$ be a semisimple Lie algebra over a field of characteristic zero, $V$ a finite-dimensional module over it and $f$: $\mathfrak{g} \to V$ a linear ...

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**0**answers

22 views

### Homology of the subcomplexes of the “diamond shaped” sphere under 1-norm in $R^n$ as a simplicial complex

The 1-norm on $\mathbb{R}^n$ is defined by $\|v\| = |v_1| + |v_2| + \cdots + |v_n|$ for a vector $v = (v_1, \ldots, v_n) \in \mathbb R^n$.
The unit sphere $S^{n-1}_1$ under the 1-norm is a simplicial ...

**2**

votes

**1**answer

309 views

### A morphism-revealing category? [closed]

The constructs can be considered as subcategories of Set but when considered as subcategories of the category SubSet, of pairs of sets with pairs $(X,S)$, $S\subseteq X$, as objects and functions ...

**7**

votes

**1**answer

234 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 ...

**3**

votes

**1**answer

138 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 ...

**1**

vote

**0**answers

29 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 ...

**15**

votes

**1**answer

787 views

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

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 ...