3
votes
1answer
155 views

Ergodic theory reference for converging sequences of matrices

I have been told that the following is a well known theorem in ergodic theory & have been given the book by Furstenberg as a reference. However, I cannot find such a statement in it. Would anyone ...
5
votes
1answer
188 views

Is the Tate conjecture known for etale covers of products of curves

Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map ...
-2
votes
0answers
68 views

Property of $\limsup$ in $\ell_1$ [closed]

Let $w$ be any element in $\ell_1$, and $(w_n)$ be a bounded sequence in $\ell_1$ that converge to 0 pointwise. I want to prove ...
3
votes
1answer
247 views

Lower bound for a prime gap occurring infinitely often

In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound ...
5
votes
1answer
65 views

In the definition of the Heegard Floer surgery exact triangle, what exactly is the correspondence between Whitney triangles and periodic domains?

I'm reading Osváth-Szabó's notes on Heegard Floer homology, in particular about the surgery exact triangle. On page 14 (numbered 42 on the document), they describe an isomorphism between the space of ...
9
votes
1answer
485 views

Tight prime bounds

This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the ...
2
votes
0answers
46 views

Tensor product of commutators vs. commutator in a tensor product

Let $R$ be a (noetherian) commutative ring, and let $V$ and $W$ be finitely generated free $R$-modules. Let $X \subseteq \mathrm{End}_R(V)$ and $Y \subseteq \mathrm{End}_R(W)$ be finite subsets, and ...
1
vote
1answer
185 views

Integer points on $y^2=x^2-x^3+x^4$

Does the Diophantine equation $y^2=x^2-x^3+x^4$ have solutions other than $x=1,y=1$? Interestingly, the Diophantine equation $y^2=x^2-x^3+x^5$ has such solutions: $x=3,y=15$, $x=5,y=55$, ...
13
votes
2answers
472 views

History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...
2
votes
0answers
56 views

Cusp points in Alexandrov spaces

Given a space of bounded integral curvature (by which I mean a topological surface with an intrinsic metric, such that the sum of excesses of any finite collection of non-overlapping simple triangles ...
2
votes
0answers
34 views

Looking for N-dimensional spheres in the configuration space of the colorful Tverberg problem

Here we use standard notation for Tverberg's theorem: Dimension $d$, number of partition blocks $r$, and $N=(r-1)(d+1)$. The configuration space of Tverberg's theorem is the simplicial complex ...
3
votes
1answer
47 views

Are lightface \Delta-1-1 classes of reals describable with hyperarthmetic formulae?

I'd appreciate if someone could check my reasoning. Suppose $S$ is a lightface $\Delta^1_1$ class of reals. I want to argue that there is a computable $\Delta^0_\alpha$ formula $\phi(Y)$, for ...
-1
votes
0answers
56 views

Basis for the rational functions [closed]

The rational functions $f(x)$ are given by $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials. What bases for the rational functions are generally used in numerical analysis?
0
votes
1answer
54 views

Topology : Study on Separation Properties [closed]

I want an example of a completely regular space which is not a normal space. I have tried a lot but am unable to construct any example
10
votes
1answer
230 views

Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$

Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...
1
vote
0answers
94 views

Modular form, number of divisors [duplicate]

The Fourier expansion of Eisenstein series $E_k$ $(k \ge 4)$, which are modular forms, as well as the quasimodular $E_2$, involves powers-of-divisors $\sigma_{k-1}(n) = \sum_{d|n} d^{k-1}$. Is there ...
2
votes
3answers
239 views

Cardinality of $C^*([0,1])$ [closed]

What is the cardinality of the continuous dual of $C([0,1])$ (the set of continuous functions from $[0,1]\to \mathbb{R}$)?
-2
votes
0answers
78 views

k to 12 philippines [closed]

i just want to ask this question, hope you will answer..what will happen to students who didn't undergo the senior high school in k to 12, students who graduated in the old curriculum in ched the not ...
-1
votes
1answer
41 views

NonLinear Maps and homogeneity [closed]

An example of a function $\phi : R^2 \to R$ such that $\phi(av) = a \phi(v)$ but $\phi$ is not linear. So I know that I need to find a function that has linear homogeneity but doesn't have the ...
2
votes
0answers
73 views

$G$-invariant part of products of determinants of minors

Let $G = SL_n$; then for any tuple $\lambda$ such that $\sum \lambda_i = n$, define $f_\lambda(g)$ as the product of the determinants of successive minors of lengths $\lambda_i$ of $g$ (e.g. for ...
5
votes
0answers
173 views

Factors of the polynomial $X^n-a$

I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a ...
-3
votes
0answers
71 views

Is there a group-theoretic proof of the Riemann rearrangement theorem? [closed]

The analytic proofs are easy to understand but they don't explain why commutativity breaks down when you go from finite sums to infinite sums of real numbers. I suspect that this involves the action ...
2
votes
0answers
36 views

Generators for lattice polytopes

Let $V$ be a finite set of integral vectors in $\mathbb{R}_n$. Consider the real (positive) convex cone $\mathbb{R}^+V$ spanned by $V$. We can assume that all vectors in $V$ are extremal in ...
12
votes
3answers
850 views

Not-lonely runners

The lonely runner conjecture has several formulations. They all involve a number $n$ runners running on a circular track, each with a different speeds, and the conjecture is that each runner is ...
2
votes
1answer
36 views

Compact imbedding - reference request

I am looking for reference to the following imbedding theorem Theorem For any $s>1/2$ fractional Sobolev space $W^{s}_2(0,1)$ imbeds compactly into $C([0,1])$. I know how to prove it but I need ...
-3
votes
0answers
76 views

How subset is a set is proved in ZF system? [closed]

I guess that all subset is a set is guaranteed by the axiom of separation in ZF system. Otherwise the notion of power-set will not make sense. But I wander how it's proved. I guess that the prove ...
1
vote
0answers
116 views

Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1]. Is there a known tight upper bound in the number of polytopes in ...
8
votes
1answer
657 views

Removing an article from arxiv [closed]

I put up my paper on arxiv before sending it for submission. But now the journal I wish to submit it to is not accepting it since its already been published (or because its publicly available). Is ...
4
votes
1answer
201 views

The embeddings $S^{6} \to S^{7}$ and $S^{7}\to S^{8}$

Is the standard smooth structure of $S^{7}$ the only structure for which each of the following canonical embedding is an smooth embedding?: 1)$S^{6}\to S^{7}$ 2)$S^{7}\to S^{8}$
2
votes
0answers
125 views

Universal cover of elliptic curve without a point

This is something that I perhaps would be expected to know, but don't. Let $E_\tau$ be the elliptic curve ${\mathbb C}/({\mathbb Z +\mathbb Z} \tau)$. Consider the complement of a point in $E_\tau$, ...
2
votes
0answers
67 views

Degree of join of two varieties‏

Let $X\subset\mathbb{P}^n$ be an irreducible variety of degree $d$ and dimension $n-4$. Let $L\subset X$ be a line of mulitplicity $m$ in $X$. Assume that $m+1\leq d$ so that the general line spanned ...
5
votes
1answer
249 views

Serious introduction to the Langlands program for nonspecialist

I recently became interested in the Langlands program and hope to learn more. For context, I am an analytic number theorist but have some light background in algebraic number theory and modular ...
0
votes
0answers
30 views

Any software that can symmetrize input sets?

Is there any software that contains symmetrization techniques ex. polarization, Steiner Symmetrization etc. I suppose not. Which software would you suggest for rigid transformations? Thank you
0
votes
0answers
39 views

Calculating the the ratio of two Dirac delta functions as the limit of the ratio of nasent delta functions? [closed]

I am in a situation where I find myself with the ratio of Dirac delta functions. Specifically, I find myself with the ratio of the nascent deltas: $\frac{\lim_{\varepsilon \rightarrow ...
8
votes
0answers
159 views
+50

Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corvallis article

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}_f$ the finite adèles of $F$. Fix a maximal compact ...
-2
votes
1answer
64 views

What does “group size” mean in the -G option of directg in nauty? [closed]

To be sure I understand the definitions used in the nauty user manual: An automorphism group size (for a digraph) is the number of re-labelings (including the trivial original digraph) of the ...
16
votes
0answers
316 views

function field analogy and global/absolute geometry

The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...
4
votes
0answers
69 views

Symplectic sum and Symplectic cut

The symplectic sum of Gompf and the symplectic cut of Lerman are known to be inverse of each other, in the sense that if you apply one of these first and the other one afterward, you obtain the ...
1
vote
1answer
45 views

mixed semi definite and second order programming complexity order

Consider the following mixed semi definite and second order programming: $\begin{array}{l} \mathop {{\rm{min}}}\limits_{\bf{X}} \,{\rm{Tr}}\left( {{\bf{XA}}} \right)\\ {\rm{s}}{\rm{.t:}}\, & ...
0
votes
1answer
210 views

Computing the Chern class of $S^6$ [on hold]

I am trying to calculate the Chern class of the tangent bundle of the sphere $S^6$. I am told that this is an interesting case, since $S^6$ is not a complex manifold, but it has an almost complex ...
-4
votes
0answers
35 views

Integer solutions for multiple variable equations [closed]

Obviously it will take some brute-force. But how do I minimize the brute-force needed (optimize)? I know one can solve Diophantine equations and quadratic Diophantine equations. But what if I have ...
0
votes
0answers
68 views

Examples of manifolds whose second Stiefel-Whitney satisfies a nontriviality condition

I'm looking for examples of pairs $(M,L)$ where $M$ is a symplectic manifold, $L$ a (closed, connected) Lagrangian submanifold, such that the second Stiefel-Whitney of $L$, $w_2(TL)$, evaluates ...
5
votes
1answer
254 views

Detection of stable homotopy by K-theory spectra

This is primarily a reference request. Does anyone know of any writing about algebraic K-theory spectra picking up elements in the stable homotopy groups of spheres in their Hurewicz image coming from ...
-1
votes
0answers
37 views

Closed sets on the product space and operators [closed]

$H$ is an hilbert space and $C$ is a closed subset of $H\times H$ with the product topology. If $P$ is the projection $P: (x,y) \in F\times F \to y \in F$ do we have that the set $$P(C)= \{ P((x,y)), ...
1
vote
2answers
125 views

Identity involving shifted Legendre coefficients

For small values of $n$ ($2\leqslant n\leqslant 5$), the coefficients $a_k = (-1)^k{n\choose k}{n+k\choose k}$ of the shifted Legendre polynomial $\tilde{P}_n(x)$ satisfy the identity ...
-2
votes
1answer
102 views

AI / Machine Learning related to high/modern/front mathematics [closed]

I major math and cs. and i'm interested in ai/machine learning/data mining. so i want to know what math subjects are used in frontier of these technology. especially, high mathematical tool, like ...
3
votes
1answer
78 views

Boolean completion (of a forcing notion) isomorphic to each of its cones

Suppose $ \mathbb{P} := (P, {\leq_P}, 1_P) $ is a separative partial order. Let $ \mathbb{B} := \operatorname{RO}(\mathbb{P}) $ denote the Boolean completion. Fix some dense embedding $ i \colon P ...
0
votes
1answer
57 views

Invariance of mutual information

Let $I(X,Y):=H(X)+H(Y)-H(X,Y)$ be the mutual information of the joint probability distribution $p_{XY}$ (here $H(\cdot)$ is the Shannon entropy of its argument). I know that the mutual information is ...
1
vote
1answer
44 views

Uniqueness of solution of a nonconvex optimization problem

What conditions need to be hold for a nonconvex optimization problem to have a unique solution? Specifically, I have the following minimization problem that I'd like to know whether it has a unique ...
0
votes
0answers
60 views

Can we deduce that all the real zeros of those $k^{th}$ derivatives are also simple?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the ...

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