-4
votes
0answers
22 views

Loss of Kinetic Energy [on hold]

can somebody help me with how we walked from ((m1u12)/2)+(m2u22)/2) to ((m12u12 + m22u22 + m1m2(u12+u22))/2(m1+m2) . probably a walk through of how the loss of kinetic energy was gotten to be ...
-1
votes
0answers
49 views

Clear estimate is not so clear [on hold]

In a paper I found the estimate (there it is said that the estimate is clear) for $U \subset \mathbb{R}^n$ and $u \in W_0^{2,2}(U)$ saying that for all $\varepsilon >0 $ we have $$\int_{U} ...
0
votes
0answers
100 views

Morse theory in zero dimensions? [on hold]

Are there any known results for Morse theory of a compact 0-dimenionsal manifold (i.e. set of points)? In particular, can one define the analogue of a gradient flow for a finite set of points and ...
-1
votes
0answers
19 views

Find the expectation of function of binomial random variable [on hold]

$\mathbb{E}\left[x^{\frac{1}{n}}\right]=?$ where $n\sim Bi(N,p)$ Thanks in advance
2
votes
0answers
77 views

Linear sections of $\mathbb{G}(1,4)$

Let $G = \mathbb{G}(1,4)\subset\mathbb{P}^9$ be the Grassmannian of lines in $\mathbb{P}^4$. Let us take two general hyperplanes $H_1,H_2$ in $\mathbb{P}^9$, and let $X = H_1\cap H_2\cap G$. Now, let ...
-2
votes
0answers
57 views

On cyclic decomposition of element in $S_n$? [on hold]

Let $S_n$ be symmetric group and $x\in S_n$ be a permutation of $n$ numbers. Let $|x|=p$, where $n/2<p<n$ is prime. Consider $1^{t_1}2^{t_2}\ldots l^{t_l}$ to be the cyclic decomposition of $x$. ...
4
votes
0answers
62 views

Time averages and differentiability

Let $\varphi_t : M \rightarrow M$ be a smooth flow on a smooth manifold $M$. We may assume (although I'm not sure if this is important) that the flow preserves a smooth volume form on $M$. Given a ...
10
votes
3answers
496 views

How to write an abstract for a math paper? [on hold]

How would you go about writing an abstract for a Math paper? I know that an abstract is supposed to "advertise" the paper. However, I do not really know how to get started. Could someone tell me how ...
1
vote
0answers
137 views

Can triangulated categories be “approximated by countable subcategories” (that are triangulated but not full!)?

For a given (finite) set of (objects and) morphisms $f_i$ in a triangulated category $C$ I am interested in a (non-full!) triangulated subcategory $C'\subset C$ of "small size" that would contain ...
32
votes
2answers
1k views

Arctangents of odd powers of the golden ratio

While trying to answer this MSE question, I found that arctangents of many odd powers of the golden ratio $\varphi=\frac{1+\sqrt5}2$ are expressible as rational linear combinations of arctangents of ...
3
votes
1answer
81 views

A differential equation with continuous coefficient and no solution in a reflexive Banach space?

Is there a reflexive Banach space $B$ and a continuous map $f:B\to B$ such that the differential equation $$ \frac{d x (t)}{dt} = f(x(t)) $$ with some initial condition $x(0)=x_0$ has no solution?
50
votes
16answers
5k views

Solving algebraic problems with topology

Often, topologists reduce a problem which is - in some sense - of geometric nature, into an algebraic question that is then (partiallly) solved to give back some understanding of the original problem. ...
1
vote
1answer
348 views

How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$ if it's not open problem?

I proposed this question on MSE but some comments affirmed that is unsolved problem and no answer. I would like to see what MO say about it. How do I evaluate this sum ...
2
votes
0answers
257 views

Differential and pre-differential of Jacobi identity

Let M be a manifold. To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied? That is a Lie algebra structure for which $[X,fY]=f[X,Y]$. (For ...
2
votes
0answers
89 views

Swan conductor, representation Weil group

Let $F$ be a non-archimedean local field and $\mathcal W_F$ its Weil group. We consider a linear representation $\sigma$ of $\mathcal W_F$. Could someone explain to me the definition of the Swan ...
5
votes
1answer
211 views

Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
41
votes
5answers
4k views

Why higher category theory?

This is a soft question. I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...
96
votes
27answers
13k views

Extremely messy proofs

Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what ...
54
votes
2answers
3k views

Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?

I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here: Is it possible to express ...
51
votes
36answers
11k views

What are some correct results discovered with incorrect (or no) proofs?

Many famous results were discovered through non-rigorous proofs, with correct proofs being found only later and with greater difficulty. One that is well known is Euler's 1737 proof that ...
0
votes
1answer
108 views

Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following: Suppose we have a uniformly parabolic equation with holder ...
15
votes
4answers
2k views

Condition for two matrices to share at least one eigenvector?

Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so ...
18
votes
6answers
1k views

Measures of non-abelian-ness

Let $G$ be a finite non-abelian group of $n$ elements. I would like a measure that intuitively captures the extent to which $G$ is non-commutative. One easy measure is a count of the non-commutative ...
8
votes
1answer
475 views

Does $(\mathbb{Z}/n\mathbb{Z})^2$ ever admit a difference set when $n$ is odd?

A difference set of a group $G$ is a subset $D\subseteq G$ with the property that there exists an integer $\lambda>0$ such that for every non-identity member $g$ of $G$, there exist exactly ...
14
votes
9answers
4k views

Commutator subgroup does not consist only of commutators?

Let $G$ be a group, $G'=[G, G]$. "Note that it is not necessarily true that the commutator subgroup $G'$ of $G$ consists entirely of commutators $[x, y], x, y \in G$ (see [107] for ...
31
votes
2answers
3k views

Open problems/questions in representation theory and around?

What are open problems in representation theory? What are the sources (books/papers/sites) discussing this? Any kinds of problems/questions are welcome - big/small, vague/concrete. Some estimation ...
13
votes
2answers
387 views

Model for the (infinity,1)-category of (homotopy-)limit preserving functors

I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial ...
13
votes
1answer
487 views

Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...
13
votes
3answers
2k views

Any more generalization of Fermat's Little Theorem? [closed]

Fermat's Little Theorem: If $p$ is a prime and $\gcd(a,p)=1$ then $a^{p-1} \equiv1\pmod p$. Over the years, Fermat's Little Theorem have been generalized in several ways. I am aware of four different ...
21
votes
2answers
3k views

A precise statement of the categorical version of geometric Langlands conjecture

The statement of the ordinary non-categorical version of geometric Langlands conjecture, which was proven for GL(n) in around 2002 by Frenkel, Gaitsgory and Vilonen, is quite well-known and is easy to ...
7
votes
3answers
944 views

Quadratic reciprocity and Weil reciprocity theorem

I was told that Weil reciprocity theorem (one has two meromorphic function $f,g$ on a complex curve $C$, so $\prod\limits_{x\in C} g(x)^{ord_xf}=\prod\limits_{x\in C}f(x)^{ord_xg} \ $ where $ord_xf$ ...
12
votes
3answers
3k views

Number of invertible {0,1} real matrices?

This question is inspired from here, where it was asked what possible determinants an $n \times n$ matrix with entries in {0,1} can have over $\mathbb{R}$. My question is: how many such matrices ...
12
votes
6answers
3k views

Reference for Mathematical Economics

I'm looking for a good introduction to basic economics from a mathematically solid(or, even better, rigorous) perspective. I know just about nothing about economics, but I've picked up bits and pieces ...
6
votes
2answers
593 views

simplicial spaces without degeneracies

Suppose I have a simplicial space $X_{\bullet}$ without degeneracies (sometimes called semi-simplicial space or incomplete simplicial space). There still is a geometric realization $\lVert X \rVert$ ...
4
votes
3answers
807 views

Canonical form of symmetric integer matrix M

Let $M$, $N$ be a symmetric matrix over a ring $R$. $M$ and $N$ are said to be equivalent if there exist an invertible matrix $U$ (over the same ring $R$) such that $N=U M U^T$ ($U^T$ is the transpose ...
14
votes
1answer
542 views

Can you name these orthogonal polynomials?

I have a collection of orthogonal polynomials in infinitely commuting variables $x_1, x_2, x_3, \ldots$. I think they must be well known (perhaps Schur or Hermite polynomials or some variant ...
7
votes
3answers
830 views

What is the term analogous to “Wronskian” for difference equations?

I am currently following a course on differential equations and difference equations (recurrence relations). The teacher tries to make parallels between the two concepts, because the methods for ...
4
votes
5answers
495 views

How should I think about correspondences?

I know at least two definitions of correspondence, and my question might as well be about both of them. Let $X,Y$ be objects in your favorite category. A correspondence is a span, namely a diagram ...
13
votes
0answers
296 views

Is there a motivic Cauchy integral formula?

Let $R$ be a complete dvr with fraction field $K$ and residue field $k$, and let $X, Y$ be two smooth projective $R$-schemes with isomorphic generic fibers. Is it true that $[X_k]=[Y_k]$ in ...
2
votes
3answers
596 views

A Diophantine equation with prime powers

Let $p$ and $q$ be prime numbers such that $p^2+p+1=3q^a$: is it true that $a=1$? This specific equation appears when computing order components of finite groups.
3
votes
2answers
232 views

Acyclic complexes for extraordinary cohomology theories

Let $X$ be a CW complex such that for all extraordinary homology theories, if you plug $X$ into them you get the same value as plugging in a point. Must $X$ be contractible?
3
votes
0answers
151 views

Kähler differentials, intuition behind $\text{div}(\omega)$, canonical divisor on algebraic curves?

See my two previous questions here: Intuition for thinking about R-module of Kähler differentials, universal receptacles, derivations? and Kähler differentials, define valuation? for background. If ...
4
votes
3answers
325 views

$A \wedge A \wedge A$ in Chern-Simons

I am confused with the wedging operations of Lie algebra valued differential forms. Especially, for instance, I have some problems with the Chern-Simons 3-form $$A \wedge dA + \frac{2}{3}A \wedge A ...
1
vote
0answers
65 views

Differential form heat kernel on hyperbolic space

Is there an explicit formula in the literature for the heat kernel of the Hodge Laplacian on differential forms? I found some on functions, but not on forms of higher degree. What at least about ...
4
votes
2answers
629 views

Polish by compact is Polish?

Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish? I have a specific space in mind, so if the ...
8
votes
1answer
230 views

Pull-back of a fibration along a homotopy equivalence and homotopy classes of sections

I previously asked this on Math.SE but didn't receive a satisfactory answer. Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: ...
2
votes
0answers
98 views

Kähler differentials, define valuation? [migrated]

See my previous question for a definition of the $K$-module of Kähler differential $\Omega_{K/k}$. This question is sort of a follow up on it. Suppose $k$ is a field of characteristic $0$, $R$ is a ...
11
votes
3answers
382 views

Searching for $C^*$

I am trying to search on MathSciNet for articles which contains $C^*$ in their title (as in $C^*$-algebras) however I can't figure out how to get MathSciNet not to interpret the '*' as a stand in for ...
0
votes
0answers
40 views

Signs and value of higher order terms in the Taylor expansion of a strongly convex function

Say that I have a function $f: S \rightarrow \mathbb{R}^+$ such that: $S \subset \mathbb{R}^n$ is a closed convex set such as $S=[-10,10]^n$ $f$ is continuous and infinitely differentiable at all ...
0
votes
1answer
196 views

Provability of unprovability

I have three questions (without any real background, this is just something I've been wondering about recently) Can PA prove that PA can't prove nor disprove, say, Goodstein theorem (or any natural ...

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