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3 votes
0 answers
549 views

Can additivity of the Euler characteristic be interpreted in terms of the Poincaré–Hopf theorem? [closed]

Whenever there is a long exact sequence in homology induced by a short exact sequence of chain complexes one finds that the corresponding Euler characteristics are additive. For example, if $Y \subset ...
13 votes
1 answer
865 views

Most discriminants are almost squarefree

Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$. Does anyone know of a reference that would allow me to show that the proportion of $f$ with ...
4 votes
0 answers
172 views

Are prime gaps of even index essentially larger than those of odd index?

Let $g_{n}:=p_{n+1}-p_{n}$ be the $n$- th prime gap, and let's introduce the following summatory functions: $$G_{1}(x):=\sum_{1\leq n\leq x}g_{2n-1}$$ $$G_{2}(x):=\sum_{1\leq n\leq x}g_{2n}$$. Let's ...
5 votes
0 answers
219 views

Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$: $e^*\Omega^1_{E/X}$ and $\...
11 votes
1 answer
1k views

How does $f_* O_X$ measure ramification and Grothendieck-Riemann-Roch

Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective varieties over a field $k$ of characteristic zero, where $\dim X=\dim Y$. Then $f$ is flat. Hence $f_\ast \mathcal{O}_X$ is a ...
5 votes
2 answers
281 views

How to simplify the proof of right-properness?

Question. Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback ...
12 votes
1 answer
661 views

A sum over characters of the symmetric group

Let $C_\mu$ be the size of the conjugacy class in $S_n$ of permutations whose cycletype is the partition $\mu\vdash n$. Let $\chi$ be the characters of the irreducible representations of $S_n$. Let $\...
36 votes
2 answers
4k views

How can we detect the existence of almost-complex structures?

Any smooth $k$-manifold $M$ comes with a well-defined map $f:M\rightarrow BGL_{k}(\mathbb{R})$ (up to homotopy) classifying its tangent bundle. Since $GL_{k}(\mathbb{R})$ deformation-retracts onto $...
3 votes
1 answer
156 views

Is this additive equivalence a triangulated equivalence?

Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, and suppose that there exists an additive equivalence $F: \mathcal{C} \to \mathcal{D}$. Suppose further that $\mathcal{C} = \text{add}(...
3 votes
2 answers
155 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. ...
3 votes
0 answers
139 views

Square integral of finite Euler product

Consider the finite Euler product $$ P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right). $$ (Here $p_1, p_2, \dots$ are of course the primes.) Question: What is a good asymptotic upper bound for $$ \...
9 votes
3 answers
371 views

Decay of real continuous algebraic functions at infinity

Let $f$ be a real valued continuous algebraic function on $\mathbb R^n$. Suppose the zero set of $f$ is bounded, i.e., if $|x|$ is large enough, $f(x)\neq 0$. Is there any estimate of the sort $|f(x)|\...
5 votes
0 answers
298 views

Non-universally trivial Chow group of zero-cycles on Fano hypersurfaces

Let $X$ be a smooth projective variety over a field $k$. By (one) definition, the Chow group of zero-cycles $CH_0(X)$ is universally trivial if, for every field extension $k \subset K$, the degree map ...
11 votes
1 answer
704 views

Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?

$\newcommand{\til}{\tilde}$ Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds. Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...
3 votes
1 answer
832 views

GIT quotient of variety with finite quotient singularities

Let $X$ be a variety over $\mathbb{C}$ with finite quotient singularities, i.e. every point has a Zariski-open neighbourhood isomorphic to $U/H$ where $U$ is a smooth variety and $H$ is a finite group ...
37 votes
1 answer
3k views

Community experiences writing Lamport's structured proofs

About two years ago, I came across this paper by Lamport http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf on writing proofs hierarchically. It changed how I wrote ...
0 votes
1 answer
73 views

Mapping an arrow from the direct limit of a diagram to the family of arrows from the diagram

Consider the direct limit of an indexed family $\{a_n\}_{n\in \omega}$: $\require{AMScd}$ \begin{CD} a_0 @>>> \ldots @>>> a_n @>>> a_{n+1} @>>> ...
2 votes
1 answer
206 views

Strange limit problem involving $\binom{z}{n} e^{-xn\log n}$ with $z \in \mathbb{C}$

Is the following limit result correct: $$\lim\limits_{x \to 0^{+}}\sum\limits_{n=0}^{\infty} \binom{z}{n} e^{-xn\log n} = 2^{z}$$ where, $z \in \mathbb{C}$, and the notation $\displaystyle \binom{z}{n}...
9 votes
1 answer
3k views

Hard maths on viXra? [closed]

A few years ago a nice paper surveyed the differences in quality between papers submitted to arXiv and those submitted to arXiv's rough cousin, viXra. However, that paper was about generic ...
1 vote
2 answers
110 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 ...
4 votes
2 answers
4k views

Pointwise convergence for continuous functions

Let $f_n:[0,1]\rightarrow \mathbb R$ be a sequence of continuous functions converging pointwise, i.e. such that $\forall x\in [0,1]$, the sequence $(f_n(x))_{n\in \mathbb N}$ converges. We set $f(x)=\...
1 vote
1 answer
637 views

The total variation of a complex measure

Let $\Omega$ be a locally compact and Hausdorff topological space. The Riesz representation theorem says that $C_0(\Omega)^*$ , dual of the commutative C*-algebra $C_0(\Omega)$, is just the space of ...
2 votes
1 answer
202 views

What is the formal name of this set-related concept?

I "invented" a concept and it feels like it has already been invented before. I would like to know whether such a concept exists and if so, what is its name? Let $S$ be a family of finite sets. Say ...
2 votes
1 answer
575 views

Inequality for square of the subgaussian distributions

Hi all, For my research I am trying to bound some exponential moments of subgaussian r.v.'s. And I am stuck with proving one of such inequalities. More specifically: Let $a$ be unit vector in $\...
3 votes
0 answers
237 views

Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
2 votes
0 answers
106 views

Does anyone know if there is a generalization of symplectic Kodaira dimension beyond 4-manifolds?

I'm aware that in algebraic geometry, one has the Kodaira-Iitaka dimension, which generalizes the Kodaira dimension, but does anyone know if a correspondent generalization in the symplectic category ...
4 votes
1 answer
289 views

Completeness of Localizations of Completions of Commutative Rings

Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r_n+Rx^n)_{n\ge0}$ where for $n\ge0$, $r_n\in R$ ...
0 votes
1 answer
263 views

p-summable sequence

Let Y be a closed linear subspace of X and suppose that Y does not have copy of l1 .Does each weakly p-summable sequence in X/Y has a subsequence that's the image of a weakly p-summable sequence in X ...
11 votes
0 answers
408 views

Sums of squares via semidefinite programming for the complex free group algebra

In the algebra of real noncommutative polynomials (the “free monoid algebra” over the real field) it is possible to reduce the question of whether an element is a sum of hermitian squares and ...
4 votes
1 answer
417 views

Birch's conjecture from Representation Theory

Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic ...
18 votes
1 answer
869 views

Two conjectures about zero inner products and dissociated sets

The following problems come from something I worked on (with my coauthors) related to proving a new time lower bound for streaming problems. Having worked on these problems for some time with little ...
21 votes
4 answers
2k views

Why are quantum groups so called?

I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of ...
6 votes
1 answer
838 views

Is there an alternate name for the symplectic convolution?

Looking into the Wigner-Weyl transformation mapping Hilbert space operators to functions on phase-space, I've run up against the need for a symplectic convolution $$[F\star G](x,p) = \int \!dy\,dk\, ...
10 votes
1 answer
467 views

Optimal exponent in the Lojasiewicz-Simon gradient inequality

Lojasiewicz's theorem asserts that if $F: \mathbb{R}^n\to \mathbb{R}$ is a real-analytic function in a neighborhood of its critical point $0$, then there exist constants $\theta\in (0,1/2]$, $\gamma\...
1 vote
1 answer
89 views

Vectors which average to zero over any graph neighborhood

Given an undirected connected graph on $n$ nodes, let $S$ be the subspace of vectors $x \in \mathbb{R}^n$ which satisfy $$\sum_{j \in N(i)} x_j = 0,$$ for all $i=1, \ldots, n$. Here $N(i)$ is the set ...
1 vote
1 answer
121 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 ...
7 votes
1 answer
383 views

When is a general sheaf (on the projective plane) globally generated?

Let $v$ be a chern character on $\mathbb P^2$ so that the moduli of sheaves of chern character $v$ is non-empty of the expected dimension. When is it true that the general sheaf in moduli is globally ...
1 vote
0 answers
165 views

Conjugacy scheme, fppf versus GIT

I would be glad to have some guidance in the following. Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the ...
0 votes
1 answer
395 views

Asymptotics of "ugly" function elucidate Goldbach's conjecture?

Question We now define the following "ugly" function: $$ A_c(s,r,n,m) = \begin{cases} 1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise} \end{cases} $$ How does the "ugly"...
4 votes
0 answers
134 views

henselizations along closed subscheme

Where can I find some references about henselizations ablong a closed subscheme? For example if I take a map $Y\times\mathbb{A}^{1}\rightarrow Y$ and $Z$ a closed subscheme. Let $Y_{Z}^{h}$ the ...
8 votes
0 answers
194 views

Non-Standard Derived Equivalences of Non-Flat Algebras

I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an ...
1 vote
0 answers
106 views

parametrizing a conic in $F_p$ [closed]

Let $F_p$ be a finite field and $p\equiv 3 \pmod 4$, and $a,c$ are non-square elements in $F_p$. I want to parametrize the conic: $$cy^2=-3x^2-2ax-16a$$ ($-1$ and $3$ are non-squares in this field ...
1 vote
1 answer
112 views

A relation among projections of a von Neumann algebra

This is a follow-up question on this. Let $A$ be a von Neumann algebra and $P$ be its projection lattice. For $p,s,q \in P$, let us define $ p \perp q \mid s \iff ps^\perp q = 0$ where $s^\perp = 1-...
2 votes
0 answers
470 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 ...
9 votes
2 answers
901 views

differential geometry using Robinson's infinitesimals?

Is there a detailed treatment of differential geometry using Robinson's infinitesimals?
2 votes
1 answer
499 views

Representation of rationals by quadratic form

In one paper about number theory author stated 2 lemmas Lemma 1. If $p$ is a prime $\equiv3(mod $ $4)$ then $x^2+y^2-pz^2$ represents a non-zero rational number $m$ if and only if $m$ is not of the ...
7 votes
2 answers
11k views

Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$? I ask this because I am looking into the relation between $A$ and $A+cI$, ...
5 votes
1 answer
204 views

How often can subsets of a universe intersect exactly once?

My question is inspired by the following observation: Claim: It is not possible to choose $n$ subsets of the universe $[n]$, each of size $\Omega(n)$, such that for each subset $S$ and each element $...
8 votes
1 answer
225 views

Classifying two-faces of four-polytopes

Motivation: This question is related to my study of hyperbolic Coxeter polytopes. In general, if one put some restrictions on the type of their dihedral angles (say, all dihedral angles are equal to $\...
8 votes
2 answers
2k views

Applications of Gauss-Bonnet theorem

In wikipedia,I was pretty amazed to find a proof of fundamental theorem of algebra using Gauss Bonnet theorem. I think given how central it is to mathematics with its far reaching generalizations ...

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