0
votes
0answers
67 views

asymptotic behavior of minimum dilatations on punctured surfaces

Let $l_{g,n}$ be the logarithm of minimum dilatation for pseudo-Anosov homeomorphisms on surface of genus $g$ with $n$ punctures. Let $n$ be fixed and $g$ varies. Is the asymptotic behavior of ...
3
votes
3answers
211 views

reference for list of left-regular representations of real associative algebras

Suppose $\mathcal{A}$ is a unital associative algebra over $\mathbb{R}$. If we identify $\mathcal{A} = \mathbb{R}^n$ then the $\mathcal{A}$ multiplication corresponds to particular linear maps on ...
7
votes
3answers
1k views

Monad arising from operad

It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way ...
0
votes
0answers
11 views

Conditioned sum of n Poissons versus unconditioned Poissons

Let $\theta >1$ and take independent random variables $Z_k \sim \text{Poisson}(\theta/k)$ for $1 \leq k \leq n$ and let $Z_k^*$ have marginals like the $Z_k$ conditioned on $\sum_1^n k Z_k = n$: ...
0
votes
0answers
21 views

a solution of nonlinear wave equation

I am looking for a solution for the following wave equation: $u_{xx}+u_{yy}=F(x,y)u_{tt}$ where $F(x,y)=\sqrt{E(x,y)/P(x,y)}$ and $E(x,y)$, $P(x,y)$ are linear functions. Is there any analytic ...
-5
votes
0answers
33 views

Mathematics example that didn't quite undestand [on hold]

Find all value k such that the function x^3-3x+k has one real solution ? Thanks
1
vote
0answers
56 views

Applications of the Weak and Weak$^*$ topologies to PDEs?

Chapter $3$ of Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis constructs and explains the Weak and Weak$^*$ topologies over a Banach Space $E$. The most ...
2
votes
2answers
615 views

Delauney triangulation in high (>20) dimensions

Hi all, I know that its very hard to find the Delauney triangulation of high dimensional spaces, especially if there are several thousand points that need to be triangulated. So I was wondering . . ...
13
votes
5answers
2k views

Matrix representation for $F_4$

Has anyone ever bothered to write down the 26-dimensional fundamental representation of $F_4$? I wouldn't mind looking at it. Is it in $\mathfrak{so}(26)$? I'm familiar with the construction of the ...
75
votes
16answers
19k views

What if Current Foundations of Mathematics are Inconsistent? [closed]

The title of the question is also the title of a talk by Vladimir Voevodsky, available here. Had this kind of opinion been expressed before? EDIT. Thanks to all answerers, commentators, voters, ...
14
votes
1answer
498 views

Operator norms of circulant matrices

The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix. For complex numbers $a_1,\ldots,a_n$, I will use the notation $$ ...
9
votes
2answers
2k views

roots of sum of two polynomials

I believe that there is no common theory for finding roots of polynomial sum. In my case I have $$P_{n}(x)+AQ_{n}(x)$$. I am wondering how roots of this sum depend on $A$?
9
votes
1answer
766 views

Wrong-way Frobenius reciprocity for finite groups representations

This is a typical lazy mathematician question, so do not hesitate to close it and recommend me to do my homeworks... Let $H$ be a subgroup of a finite group $G$, and let $Res_H^G$ and $Ind_H^G$ the ...
9
votes
3answers
697 views

Can a metric conformal to a Kahler metric be Kahler?

Let $X$ be a non-compact complex manifold of dimension at least 2 equipped with a Kahler metric $\omega$. Take a smooth positive function $f : X \to \mathbb R$, and define a new hermitian metric on ...
155
votes
27answers
19k views

What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC." For example, suppose A is an abelian group such that every ...
3
votes
0answers
231 views

Restriction of a global moduli functor that admits a coarse moduli space

Let $F:(Sch/k)^{o}\to Sets$ be a functor, where $Sch/k$ is the category of schemes over a field $k$. Suppose that $F$ admits a coarse moduli space, let it be $M$. Consider a $k$-point $x\in M$ (which ...
6
votes
1answer
114 views

Algebraic cobordism (of a point) outside the geometric diagonal

This question is about the state of current knowledge regarding Voevodsky's algebraic cobordism of a point $\mathrm{MGL}^{*,*}(\mathrm{Spec}\,k)$. That the geometric diagonal ...
12
votes
5answers
491 views

Applications of space filling curves

I am seeking articles where a space filling curve has been used as a theoretical application, such as in the study of general orthogonal polynomials.
6
votes
1answer
320 views

Is it true that irreducible generic representations of $G_2(F)$ are self-dual?

Let $G_2$ be the split exceptional group of type $G_2$ and $F$ be a p-adic field. Is it true that every irreducible smooth representation of $G_2(F)$ is self-contragredient? If the answer is Yes, can ...
9
votes
0answers
129 views

Simplest example of failure of finite Galois descent in algebraic $K$-theory?

Let $E \to F$ be a $G$-Galois extension of fields. What is the simplest example where the natural map $K(E) \to K(F)^{hG}$ is not an equivalence on connective covers (i.e., where finite Galois ...
7
votes
0answers
70 views

Are the unwound thin realization and fat realization homotopy equivalent?

This is a question about a theorem (proposition 2) in the article--On the homotopy type of classifying spaces Recall some definitions first: Given a category $\mathcal{C}$ internal in ...
6
votes
1answer
235 views

Is polynomial convexity a topological invariant?

Is the property of being polynomially convex a topological invariant? In other words, assume that $M$ and $N$ are two compact subset of $n$-dimensional complex Euclidean space which are homeomorphic. ...
11
votes
1answer
406 views

Elements of arbitrary large order in the first Galois cohomology of an elliptic curve

Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$. In this answer Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I ...
274
votes
15answers
38k views

Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
16
votes
2answers
827 views

Probability of two vectors lying in the same orthant

Let $S^{d-1} = \{x \in \mathbb R^d: \|x\| = 1\}$ denote the unit sphere in $\mathbb R^d$. Let $v$, $w$ be drawn uniformly at random from $S^{d-1}$, conditioned on their inner product being equal to ...
6
votes
0answers
216 views

A non-hyperfinite type III factor from an action of the free group on the circle

We define below a von Neumann algebra $\mathcal{M}$ from an action of the free group on the circle, and we prove that $\mathcal{M}$ is a non-hyperfinite type III factor. Question : Is ...
3
votes
1answer
2k views

The Paley-Wiener theorem and exponential decay.

Consider a function whose Fourier transform is supported on a half-ray: $$ A(t)=\int_0^\infty \omega(E) e^{-iEt}d E, $$ where I can suppose $\omega(E)\geq 0$ and any suitable regularity conditions on ...
5
votes
1answer
510 views

Books about Capacity theory

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...
3
votes
1answer
428 views

About Theorem 3.2 in 'introduction to spectral theory of automorphic forms' by Iwaniec

In Theorem 3.2 of 'Introduction to spectral theory of automorphic forms' by Iwaniec,the first bound is about the coefficients of automorphic forms $$\sum_{|n|\le ...
2
votes
1answer
148 views

Fundamental Units in Totally Real Cubic Fields

How much is known about the fundamental units in totally real cubic fields? For example, Daniel Shanks has a family of totally real cubic fields for which the fundamental units are known; those with ...
19
votes
3answers
1k views

A generalization of the triangle counting problem for simple weighted graphs

One nice identity is that $$\operatorname{tr}(A^3)/6$$ counts the number of triangles of a graph with adjacency matrix $A$. It also implies that triangle counting in a graph can be performed in ...
14
votes
4answers
2k views

What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
13
votes
2answers
3k views

Separable and algebraic closures?

I have no intuition for field theory, so here goes. I know what the algebraic and separable closures of a field are, but I have no feeling of how different (or same!) they could be. So, what are the ...
1
vote
1answer
209 views

derivative of Riemann zeta function

One can of course give a power series for the derivative of RiemannZeta. But is there a "closed form", e.g. expressible in the field of functions generated by Zeta and Gamma and some constants?
7
votes
1answer
650 views

Iterated forcing and CH

I need some help with this theorem: if $P_\beta=\langle P_\alpha,\dot{Q}_\alpha:\alpha\leq\beta\rangle$, $\beta<\omega_2$, is a CSI of proper forcings, $P_\alpha\Vdash \lvert ...
1
vote
2answers
353 views

Models of the natural numbers in ultrapowers in the universe.

Our question arises from wondering about the systems of natural numbers in models ZFC + Con(ZFC) and ZFC + $\neg$Con(ZFC). In thinking of the systems of natural numbers of these models, we came to ...
7
votes
4answers
439 views

Picard groups of quartic K3 surfaces

Does anyone know where I can find examples of quartic K3 surfaces for which the Picard group is known? I'm really interested in examples where there are explicit constructions of the divisors ...
10
votes
2answers
1k views

Hilbert transforms of measures

Given a finite measure $\mu$ on the real line $\mathbb R$, one definition of its Hilbert transform is $(H\mu)(y) =\frac{1}{\pi}(PV)\int \frac{d\mu(x)}{x-y}$ which is known to exist almost everywhere ...
10
votes
4answers
2k views

Which topics/problems could you show to a bright first year mathematics student?

I am teaching a one semester course (January to June) to first year students pursuing various different degrees. Because there are students studying actuarial science, physics, other sciences, other ...
0
votes
0answers
113 views

asymptotics of primes in arithmetic progressions

If $a$ and $q$ are given coprime positive integers, what is the best known error term for $$ \sum_{p<x,\,p\,\text{is prime},\,p\equiv a \pmod q} \frac{\log p}p-\frac{\log x}{\varphi(q)}? $$ Is it, ...
5
votes
1answer
676 views

Wanted: differential coming from higher genus surface in Heegaard Floer Homology

I am interested in studying moduli of complex surfaces which arise in computing the differential on the Heegaard Floer Homology chain complex. In particular, I am interested in the generic case, when ...
15
votes
17answers
15k views

Undergraduate Differential Geometry Texts

Can anyone suggest any basic undergraduate differential geometry texts on the same level as Manfredo do Carmo's Differential Geometry of Curves and Surfaces other than that particular one? (I know a ...
36
votes
25answers
7k views

Where can square roots come from when they are not distances?

In a recent survey "Supergeometry in Mathematics and Physics", Kapranov points out cases in which observable quantities of immediate interest are represented as bilinear combinations of more ...
6
votes
3answers
538 views

functions with same area

I have two real valued functions $f_1$ and $f_2$ such that $\int_0^Tf_1=\int_0^Tf_2=a_1$ $\int_0^Tf_1^2=\int_0^Tf_2^2=a_2$ $\forall \\ t, f_1(t),f_2(t)\in[0,1]$ Now,I want to construct a family ...
0
votes
1answer
113 views

spectrum of an induced algebra

Let $G$ be a reductive group defined over an algebraically closed field $k$ and $B$ be a fixed Borel subgroup of $G$. Suppose $X=Spec(R)$ is an affine scheme with $B$ rationally acting on it; hence ...
48
votes
17answers
11k views

Parodies of abstruse mathematical writing

Perhaps under the influence of a recent question on perverse sheaves, in conjunction with the impending $\pi$-day (3/14/15 at 9:26:53), I recalled a long-ago parody of abstruse mathematical language ...
8
votes
2answers
721 views

Does the category of topological symmetric spectra satisfy the monoid axiom ?

In the paper "Symmetric spectra"by Hovey, Smith and Shipley, they say that they don't know if the monoid axiom holds for topological symmetric spectra. This paper was written in 1998 so I am wondering ...
8
votes
4answers
905 views

Notation for eventually less than

Is there some existing notation for \[f(n)\leq g(n)\] for sufficiently large n Apart from just writing that itself? I'm thinking of something compact like the ...
1
vote
1answer
532 views

the existence of compact Kahler manifolds satisfying some Hodge numbers' restrictions

Given any $n\geq 2$, is there an example of $n$-dimensional compact Kahler manifold such that its Hodge numbers satisfy $h^{1,1} = h^{2,2} < h^{3,3} = h^{4,4} < h^{5,5} = h^{6,6} < \cdots ...
0
votes
0answers
48 views

a two dimensional integer bijection

I want to try to find a specific two-dimensional linear integer bijection. This is to be used in a double sum rearrangement. It's kind of complicated, but I would really appreciate if anybody has any ...

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