# All Questions

**-4**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**16**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**5**answers

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

**27**answers

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

**2**answers

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

**36**answers

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

**1**answer

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

**4**answers

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

**6**answers

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

**1**answer

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

**9**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**3**answers

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

**2**answers

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

**3**answers

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

**3**answers

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

**6**answers

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

**2**answers

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

**3**answers

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

**1**answer

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

**3**answers

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

**5**answers

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

**0**answers

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

**3**answers

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

**2**answers

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

**0**answers

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

**3**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**0**answers

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

**1**answer

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 ...