All Questions
153,421
questions
7
votes
2
answers
272
views
Motivations for the study of dual connections
I am intrigued by the notion of dual connections: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfy $$X(g(Y,Z))=g(\nabla_XY,Z)+g(Y,\nabla^*_XZ)$$
for a given (pseudo)-...
- 432
12
votes
2
answers
581
views
Steenrod powers of Pontryagin classes
It is well known that the Stiefel–Whitney classes $w_i$ of a smooth manifold are generated, over the Steenrod algebra, by those of the form $w_{2^{i}}$. I wonder if it the same statement is known/true ...
- 1,420
11
votes
1
answer
516
views
Oesterlé's unpublished bound on Uniform Boundedness
The bound in Merel's solution to the Uniform Boundedness conjecture is not explicit, as it relies on Falting's work on the Mordell conjecture. I think this still is the case.
But there are known ...
- 17.5k
15
votes
3
answers
371
views
Is primality essential in Varshamov's bound?
Let $v_q(n,r)=\sum_{i=0}^r \binom{n}i (q-1)^i$ denote a number of points in a ball of radius $r$ in the Hamming metric on the cube $\Sigma^n$, where $|\Sigma|=q$. What is the maximal number of points ...
- 103k
8
votes
1
answer
482
views
Concavity of the trace of a matrix power
Let $B$ be an $n\times n$ matrix, and define $f$ to be the function that maps positive semidefinite (PSD) $n\times n$ matrices $A$ to real numbers by
$$
f(A) = \mathrm{trace}( (B^*A^2B)^{1/3}).
$$
...
- 757
10
votes
0
answers
657
views
Fractional Matching version of Hall's Marriage theorem
Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent:
1) there exist a perfect matching in $G$;
2) there exist non-negative weights on edges such that the sum of ...
- 103k
5
votes
1
answer
509
views
Closed form for $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$
EDIT: Some additional details and corrections, I would appreciate any information about the highlighted expression.
I try to solve $\int_0^T e^{-x}\frac{I_n(\alpha x)}{x}dx$ where $I_n(x)$ is the ...
- 368
7
votes
1
answer
280
views
Trying to prove one of C.Taubes' theorems gauge-theory-freely
One of C.Taubes' theorems says that for a symplectic 4-manifold $X$ with $b^2_+>1$ (where $b^2_+$ denotes the dimension of a maximal positive-definite subspace of $H^2(X;\mathbb R)$ under the ...
- 81
8
votes
2
answers
787
views
Formula for the Frobenius-Schur indicator of a finite group?
Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $p \neq 2$.
Let $V$ be a finite-dimensional irreducible $kG$-module. If $V \cong V^*$, then $V$ admits a ...
- 2,791
1
vote
1
answer
283
views
How to prove that doubly regular tournaments are regular?
A doubly regular tournament is a tournament such that every two vertices have $j$ common out-neighbours. How can we prove such a tournament is $2j+1$-regular?
- 351
3
votes
1
answer
216
views
$L^2$-Euler number
Suppose $M$ is a closed manifold, and $\tilde M$ is the universal covering.
Q: Can we say that $\chi(M)=L^2\chi(\tilde M)$, where $L^2\chi(\tilde M)$ denotes the alternative sum of the dimension of $...
- 1,905
1
vote
0
answers
544
views
Orbifold line bundles
I was going through the paper - http://repository.ias.ac.in/3652/1/427.pdf, and I got this question. Let $Y$ be a connected smooth projective variety of dimension $n$ over $\mathbb{C}$. Let $G$ be a ...
- 643
3
votes
0
answers
150
views
Wolff's article: Note on counterexamples in strong unique continuation problems
I am reading Wolff's Note on counterexamples in strong unique continuation problems:
http://www.ams.org/journals/proc/1992-114-02/S0002-9939-1992-1014648-2/S0002-9939-1992-1014648-2.pdf
On Page 3, ...
- 751
5
votes
2
answers
367
views
Manifold of mappings between $M$ and $N$, with non-compact source $M$
EDIT: Let $M$ and $N$ are two smooth manifold and suppose $N$ is compact but $M$ is not necessarily compact. For my purpose, I just need to consider the case $M=\mathbb R \times S^1$ or $\mathbb R \...
- 2,719
2
votes
1
answer
131
views
Polyhedral structure of functions writable as a finite signed sum of max of linear functions
For any two positive integers $k,n$ consider the space of functions writable as,
$\sum_i \sigma_i \max \{ L_{i1},L_{i2},..,L_{ik} \}$ (a finite sum) where each $L_{*} : \mathbb{R}^n \rightarrow \...
- 555
4
votes
0
answers
174
views
Explicit examples of finite unramified group schemes
What are some explicit examples (e.g., by explicitly describing its Hopf algebra) of finite unramified group schemes? (Ie, the sort of group schemes which appear as automorphism groups of objects ...
- 10k
3
votes
0
answers
55
views
Equivalence Classes of a Subgroup of Similarity Transformations
Let $X$ be a real, finite-dimensional vector space and $A, B, C,$ and $D$ be matrices on $X$. I'm interested in the similarity classes of the block matrices
$$
\begin{bmatrix}
A & B\\
C & D\\
...
- 263
6
votes
0
answers
257
views
Analytical point of view of Kawamata's Unipotent reduction condition for Calabi-Yau family
Motivation: Unipotent reduction condition is very important for study of family of algebraic varieties. For example for algebraic fiber space if we have such condition then the direct image of ...
user21574
9
votes
0
answers
257
views
Reference request: $H^* X$-module structure on the Mayer–Vietoris coboundary
Is the following presumed folklore fact written anywhere?
Let $E^*$ a multiplicative cohomology theory. Then the coboundary map in the Mayer–Vietoris sequence of an excisive triad $(X;U,V)$ preserves ...
- 2,984
4
votes
2
answers
327
views
estimate for a sum of products of Weil's sum
Let $p$ be a prime and consider the field $\mathbb{F}_p$. Fix $f\in\mathbb{F}_p[X]$ a polynomial of degree $d\ge 2$. Define
$$
K(x,y)=\frac{1}{\sqrt{p}}\sum_{z\in\mathbb{F}_p}e_p(xz+yf(z)),
$$
where $...
- 443
6
votes
1
answer
504
views
Rationality of the Tate module of an abelian variety relative to the algebra of its endomorphisms
Suppose that $K/\mathbb{Q}_p$ is a finite extension and $k_K$ the residue field of $K$. Let $A/K$ be an abelian variety with good reduction. Suppose that $E\to\mathrm{End}^0_K(A)$ is an inclusion of a ...
- 323
11
votes
2
answers
361
views
Harmonic congruence
There are a number of interesting congruences for harmonic sums, not the least of which is Wolstenholme's theorem: $H_{p-1}:=\sum_{j=1}^{p-1}\frac1j\equiv 0\mod p^2$.
It appears that $\sum_{j=1}^{p-1}...
- 309
9
votes
1
answer
527
views
Fefferman's article: Pointwise convergence of Fourier series
I have some problems reading Pointwise convergence of Fourier series by Fefferman https://www.jstor.org/stable/1970917
When I proceed to Lemma 2, Chapter 6, I could not verify either of the following:...
- 751
3
votes
0
answers
126
views
quantitative winding number?
In light of Tao's discussion of Quantitative Continuity I've decided to post some thoughts on Winding number.
The Jordan curve theorem is non-trivial because closed curves can be very complicated. (...
- 22.6k
5
votes
1
answer
317
views
Distance function on a curve on a manifold
Suppose that we are given a non-negative even function $b\in C^\infty[-1,1]$ satisfying $b(0)=0$, $\sqrt{b(x+y)}\le \sqrt{b(x)}+\sqrt{b(y)}$ for any $x,y\in[-\frac12,\frac12]$. Can we always find a 3-...
- 187
20
votes
3
answers
2k
views
Integral cohomology of $SU(n)$ - looking for constants
I am interested in explicit generators of the cohomology $H^\bullet(SU(n),\mathbb{Z})$. Let $\omega = g^{-1} dg$ be the Maurer-Cartan form on $SU(n)$. The forms $\alpha_3,\alpha_5,\dots,\alpha_{2n-1}$,...
- 633
1
vote
1
answer
64
views
Inequality about the minimum vertex degree in $k$-uniform hypergraphs
Let $H=(V,E)$ be a $k$-uniform hypergraph with $n$ vertices, that is, $V:=V(H)$ is a $n$-element finite set of vertices and $E:=E(H)\subset\binom{V}{k}$ is a family of $k$-element subsets of $V$.
...
- 139
5
votes
0
answers
513
views
Reduction of torsion points on Neron Model
Let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $R$ and residue field $k$. Let $A/K$ be an abelian variety with Neron model $\mathcal{A}/R$. We denote by $\tilde{\mathcal{A}}/k$ the ...
- 411
1
vote
0
answers
366
views
Kawamata covering lemma - question on the branch divisor
Let $X$ be a non-singular projective variety over the field of complex numbers, of dimension $\geq 2$. Suppose $D$ is a non-singular and irreducible divisor of $X$.
The Kawamata covering lemma (...
- 169
1
vote
0
answers
201
views
Galois Cohomology and $\sqrt{k} \notin \mathbb{Q}$
I know it seems excessive, but I have been trying to understand the relationship between two concepts:
Galois cohomology
Fermat Descent
The first one is very abstract and I know very little about it....
- 22.6k
3
votes
1
answer
350
views
Is Leray's theorem on commutative Hopf algebras proven in Milnor-Moore?
Question 1. Is a correct proof of Leray's theorem (the one that says that
a connected graded Hopf algebra $H$ over a field of characteristic $0$ is
isomorphic as an algebra to the symmetric ...
- 33.2k
6
votes
2
answers
415
views
Classification of weak 3-groups
Weak 2-groups can be classified by the data $(\pi_1,\pi_2, t, \omega)$, where $\pi_1$ is a group, $\pi_2$ an Abelian group, $t: \pi_1 \to Aut(\pi_2)$, and $\omega \in H^3(B\pi_1,\pi_2)$.
I wonder do ...
- 4,736
5
votes
2
answers
1k
views
Triviality of the adjoint and endomorphism bundles
Let $P$ be be a principal bundle over a manifold $M$ with structure group $G$, where $G$ is a Lie group. Let $E = P\times_{\rho} \mathbb{R}^{k}$ be a vector bundle associated to $P$ through a faithful ...
- 3,064
9
votes
1
answer
473
views
Using Jordan's theorem to find Galois group for a polynomial
I'm trying to apply the result of Jordan's theorem (cited below) to find the Galois group for a given polynomial. My goal is to provide an example where Jordan's theorem is useful, so the polynomial I'...
- 191
3
votes
1
answer
1k
views
Relation between real part of eigenvalues of $A$ and $(A+A^{T})/2$
I saw the following theorem in a very old paper of Bendixson. Does anybody know a shorter and beautiful proof of that?
Theorem. If $A$ is a real matrix, then for each of its eigenvalues $(\lambda)$, ...
- 351
4
votes
1
answer
160
views
What is the probability that you roll a dice with s sides for n times and t sides appeared once?
You roll a dice with $s$ sides for $n$ times. What is the probability that $t$ sides appeared once?
Example with $s=2$, $n=3$: you roll a dice with 2 sides for 3 times.
possible outcomes: '1' and '2'....
- 43
3
votes
2
answers
537
views
Does there exist a rational point on the elliptic curve: $y^2=x^3+6x^2+x$ ? If yes, how to find one? (relations to the 'rational distance problem') [closed]
The elliptic curve $y^2=x^3+6x^2+x$ is associated with the Rational Distance problem, which asks whether there exists a point in the plane, that is at rational distances from the four vertices of the ...
4
votes
0
answers
299
views
Best Approximation in Operator/non-Frobenius Norm
Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$
$$\...
- 143
16
votes
2
answers
790
views
Klee's trick --- more applications
In his "Some topological properties..." (1955), Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ ...
- 43.7k
42
votes
9
answers
6k
views
What problem in pure mathematics required solution techniques from the widest range of math sub-disciplines?
(This is a restatement of a question asked on the Mathematics.SE, where the solutions were a bit disappointing. I'm hoping that professional mathematicians here might have a better solution.)
What ...
5
votes
1
answer
208
views
When do Gorenstein Stanley-Reisner rings have Du Bois singularities?
The question is pretty much as in the title. Given a simplicial complex $\Delta$, I can associate a Stanley-Reisner ring. I assume this ring is Gorenstein, when does it have Du Bois singularities?
...
- 51
4
votes
1
answer
99
views
Sign-expansion definition of Surreal arithmetical operations
Is there a way to define the addition and multiplication operations in Surreals numbers, defined directly on the sign-expansion notation {-,+}, i.e. without firstly convert them to the Conway notation ...
- 41
2
votes
1
answer
112
views
Smoothness of space of morphisms from a curve to a locally complete intersection
Let $C$ be an irreducible smooth project curve over $\mathbb C$ and $Y$ a variety over $\mathbb C$ locally of complete intersection. Write $Y^{\text sm}$ for the smooth locus of $Y$. Consider the '...
- 59
1
vote
0
answers
126
views
Volume growth of balls implies volume growth of spheres?
Suppose I have a complete, non-compact Riemannian manifold $M$ such that the volume of balls around a fixed point $p \in M$ satisfies
$$\mathrm{vol}(B_R(p)) \leq v(R)$$
for some function $v$. Can we ...
- 9,591
4
votes
2
answers
143
views
Covering all except one of the purple intersection points of $n$ red and $m$ blue lines efficiently
Consider a set of $n$ red lines and $m$ blue lines, suppose there are $nm$ distinct red-blue intersections.
What is the minimum number of lines $L_1,L_2,\dots, L_n$ such that the union contains all $...
- 1,825
3
votes
1
answer
115
views
Does a reductive group over $K$ always have a torus that becomes maximal split over $L$?
Let $K$ be a field, let $L$ be a field containing $K$, and let $G$ be a reductive group over $K$. Does there always exist a torus $T$ of $G$ so that $T_{/L}$ is a maximal split torus of $G_{/L}$? If ...
- 841
5
votes
1
answer
195
views
Multiplier norm vs cb norm
Let $f:G\to \mathbb{C}$ be a finitely supported functions and let $m_f$ denote the associated multiplier on $C^*_r(G)$, the reduced group $C^*$-algebra:
$$m_f(\alpha)(g)=f(g)\alpha(g)$$
for every $\...
- 165
3
votes
1
answer
190
views
"Künneth bigrading" for subsets of $X \times Y$?
Given two algebraic varieties $X$ and $Y$, the Künneth theorem implies that there is a relation between $H^*(X) \otimes H^*(Y)$ and $H^*(X \times Y)$, and in fact in many cases they are equal.
Given ...
1
vote
0
answers
120
views
Can the Inclusion-Exclusion Principle be used to establish a lower bound for the number of $i$ where $an < i \le an+n$ and $\gcd(i,w)=1$
Let $\Phi(x,w) = $ the number integers $i$ where $i \le x$ and $\gcd(i,w)=1$
Let $a, n > 1, w$ be integers.
Does it follow that $\Phi(an+n,w) - \Phi(an,w) \ge \sum\limits_{d|w}\lfloor\frac{\mu(d)n}{...
- 1,039
21
votes
2
answers
2k
views
Implications of the disproof of the "climb-to-a-prime" conjecture
Now that James Davis has found a counter example, 13532385396179, to John Conway's climb-to-a-prime conjecture, I would be interested to learn whether this has any implications of interest in number ...
- 178k