All Questions
152,887
questions
1
vote
1
answer
137
views
Wave equation with linear coefficients
The following pde came up in a physics problem:
$$
(Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y),
$$
A,B,C,D are fixed ...
3
votes
1
answer
180
views
When are all sums of the elements of a set different?
Consider a set $S = \{x_1, \dots, x_n\} \subset \mathbb{Q}\setminus\{0\}$ and assume that for any $I, J \subset [n]$ with $I \neq J$ we have that
\begin{equation}
\sum_{i \in I} x_i \neq \sum_{j \in ...
1
vote
0
answers
80
views
Recognizing bridgeless cubic graph with special 2-factor
A 2-factor of graph $G(V, E)$ is a set of vertex-disjoint cycles that cover $V$. It is known that every connected bridgeless cubic graph contains a 2-factor (and a perfect matching).
I conjecture ...
15
votes
4
answers
1k
views
Painting $n$ balls from $2n$ balls red, and guessing which ball is red, game
The game
Lucy has $2n$ distinct white colored balls numbered $1$ through $2n$. Lucy picks $n$ different balls in any way Lucy likes, and paint them red. Lucy then giftwrap all the balls so that it is ...
2
votes
2
answers
942
views
What is the difference between the moduli space of curves and the moduli space of orbi-curves?
Edit: In my original framing of this question it was not so clear what I was looking for, so this is basically a re-write.
I feel that I should already know the answer to this, but it never sits ...
2
votes
1
answer
733
views
Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$
Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory.
I consider the regular representation of $\mathbb S_n$ over reals $\mathbb R$ ($\mathbb R \mathbb ...
2
votes
0
answers
339
views
Looking for the manuscript "Uniform polytopes" by N. Johnson
The manuscript Uniform Polytopes (1991) by Norman Johnson is cited in the wikipedia page on uniform polytopes (http://en.wikipedia.org/wiki/Uniform_polytope).
Is there an electronic copy of this ...
2
votes
1
answer
716
views
Functional representation of adapted jointly measurable stochastic processes
It seems like the question stated here in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO.
Let $X_t : \Omega \to E, \ t \geq 0$ be ...
5
votes
1
answer
307
views
Meaning of $g_d^r$ in algebraic geometry
As an editor I often encounter the symbol $g_d^r$ as a noun. I tried googling but I only get papers where the symbol is used without a definition. Can someone supply a reference to a definition? ...
2
votes
1
answer
203
views
Projective dimension of a sub-ideal
Let $\mathbf{k}$ be a field, and let $S=\mathbf{k}[x_1,x_2,\ldots,x_n]$. Let $I\subset J$ be finitely generated monomial ideals in $S$. Is it true that the projective dimension of $I$ is either ...
9
votes
2
answers
374
views
Which finite p-groups occur as commutators of finite p-groups?
Let $p$ be a prime number. For which finite $p$-groups $H$ is there a finite $p$-group $G$ such that $[G,G] \cong H$?
4
votes
1
answer
177
views
"Inverse problem" for the zeta function [duplicate]
Let $C$ be a smooth, projective, geometrically irreducible curve, of genus $g$, over a finite field $\mathbb{F}_q$. By the Weil conjectures, the zeta function has the shape
$$
Z_C(t)=\frac{P(t)}{(1-t)(...
11
votes
1
answer
1k
views
How can I have a copy of this old paper by Frobenius?
How can I have a copy of this old paper and a translation of it?
Frobenius, G. (1902). Uber primitive Gruppen des Grades n und der Klasse n - 1. S. B. Akad. Berlin 1902, 455-459.
8
votes
1
answer
661
views
What does the sum of the reciprocals of all the highly composite numbers converge to?
I've calculated the sum of the reciprocals of all the $156$ first highly composite numbers up to $10^{18}$:
$\sum \dfrac{1}{HCC(n)} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{12} + \...
12
votes
12
answers
4k
views
Obscure Names in Mathematics [closed]
I recently stumbled over the "Happy Ending Problem" (cf e.g. http://en.wikipedia.org/wiki/Happy_Ending_problem), which made we wonder, if there are other conjectures, theorems or problems, whose names ...
3
votes
1
answer
169
views
Uniqueness of the maximum derivative of a rational function
This may seem like an elementary question, but bear with me; you'll find that it is actually quite hard. Consider the function
$$f(x)=\frac{a_nx^n}{\sum_{i=0}^n a_ix^i}$$
with all $a_i\geq 0$ and $a_0=...
2
votes
2
answers
319
views
Deformation quantization of a closed Riemann surface with genus >1
Quantization of of an elliptic curve can be done in different ways.
In C^*-algebraic version,
one can start with the C^*-algebra ...
29
votes
2
answers
2k
views
Making $\mathbb{Q}$-cohomology integral
Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$
with $i>0$.
Does there always exist a variety $Y$ and a ...
3
votes
2
answers
618
views
Intuition behind $\zeta(2) = \frac{\pi^2}{6}$ [closed]
In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$
However, as far as I know, they do not offer an intuitive explanation of why this result should be true. ...
2
votes
3
answers
550
views
How did the summation operation come into use? [closed]
So we've been using summations at least since the dawn of calculus. I'm wondering how the process of summing a function came to be known? Are there events that led to the invention of the summation ...
1
vote
1
answer
627
views
Quotients and radicals
Let $I, J$ be ideals in a commutative ring with identity $R$. Define the quotient ideal $(I : J)$ by $$(I : J)=\{x\in R : xJ\subseteq I\}.$$
Define the radical $r(A)$, of an ideal $A$ of $R$ by
$$...
5
votes
0
answers
100
views
The name for the quotient property
I asked this question on math@stackoverflow and was suggested to ask it here as well.
We call a surjective $f:X\rightarrow Y$ a quotient mapping if it satisfies, for every $U\subset Y$
(continuity,...
26
votes
2
answers
3k
views
Teaching the fundamental group via everyday examples
This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...
2
votes
1
answer
794
views
Various definitions of the Bruhat decomposition of the affine Grassmannian
Let $G$ be a connected, simply connected, complex, semi-simple group with affine Grassmannian $\mathcal Gr \cong G(\mathbb C[t,t^{-1}])/G(\mathbb C[t])$. Fix a choice of maximal torus $T \subset G$ ...
3
votes
2
answers
273
views
A problem on chains of squares — can one find an easy combinatorial proof?
Consider the unit square $ S = [0,1] \times [0,1] $. For each $ n \in \mathbb{N} $, we can tessellate $ S $ by the collection
$$
A
= \left\{
\left[ \frac{i}{n},\frac{i + 1}{n} \right] \times
...
3
votes
3
answers
572
views
What should be considered a finite size of an infinite dimensional space? [closed]
I've got a map between two infinite dimensional spaces, $f: A\to B$, where $A$ seems "larger" than $B$. For the sake of conversation let's assume that $A$ is the set of smooth maps $\mathbb R^3\to \...
3
votes
1
answer
163
views
Ising model: probability of a long path of minus under plus boundary conditions
Consider for example the Ising model on a square lattice. Fix zero magnetic field and plus boundary conditions.
Low temperature, one minus spin. With a Peierls argument one can prove that, given a ...
9
votes
2
answers
550
views
Splitting integers 1, 2, 3, … n to avoid least possible sum
For each positive integer n, partition the integers 1, 2, 3, … 2n into two sets of n integers each. Let g(n) be the least integer such that there is such a partition in neither of whose parts there is ...
4
votes
0
answers
193
views
What is the significance of the median eigenvalue of a graph Laplacian?
Crossposted on Mathematics SE
When I look at the spectral density plots of my (usual) Laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which ...
14
votes
2
answers
293
views
Which real Pin groups agree?
In the Lie theory notes on my website it is claimed (Example 7.3.3.5) that $\mathrm{Pin}(4,0)$ and $\mathrm{Pin}(0,4)$ are not isomorphic. As Nigel Ray pointed out to me, this claim is not quite ...
0
votes
0
answers
159
views
Generalized weight space
In their paper Lepowsky and Mcmollum sketch theory of weights in a more general setting. Here is their definition of a weight space:
If $A$ is a subset of $\mathfrak g$ and $\lambda$ is a function ...
11
votes
2
answers
2k
views
Green's function of the Ornstein-Uhlenbeck operator
The Ornstein-Uhlenbeck operator $L$ is given by
$$
Lu = \Delta u- \frac{1}{2}x\cdot \nabla u.
$$
Is there a known closed form expression of the Green's function of $L$ on $\mathbb R^d$ (for $d\geq 2$ ...
2
votes
0
answers
103
views
Ozsvath-Szabo orientation convention for Seifert fibred spaces
I am confused by the orientation convention that Ozsvath and Szabo use in On Heegaard Floer homology and Seifert Fibered Surgeries and would appreciate if someone clarifies this for me. On page 15 ...
3
votes
0
answers
435
views
Hitting time of two dimensional continuous martingale
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
1
vote
1
answer
391
views
Limit-circle and limit-point at endpoints
I was wondering if the following holds:
If you have an ODE $$-y''(x) + q(x) y(x) = \lambda y(x)$$ on a finite interval $(a,b)$ and you know that this equation is limit-circle or limit-point at the ...
4
votes
1
answer
697
views
A question about running MMP with scaling
Let $\pi:X \to U$ be a projective morphism, and $(X, \Delta = A + B)$ be a KLT pair, where $A$ is a general ample divisor and $B$ is effective.
Suppose $K_X + \Delta$ is not nef (over $U$) and there ...
3
votes
0
answers
277
views
Is there any progress on Problem 13 (from Schoen and Yau)?
This is closed related to the question asked here. I wonder if there is any progress on Problem 13 from the "Problem Section" in Schoen and Yau, page 281, problem 13, which asks:
Let $M_1$ and $M_2$ ...
4
votes
1
answer
271
views
Real points of zero-dimensional real algebraic varieties
There have been a number of discussions of zeros of random polynomials here (the most recent being: Why do roots of polynomials tend to have absolute value close to 1?).
Here is a closely related ...
1
vote
0
answers
185
views
Classification properties of fusion rings
Fusion rings have so many classification properties (I checked the literature a bit) that my head hurts. For practical reasons I define the following three new properties (which might coincide with ...
15
votes
2
answers
938
views
Mixed Hodge structure on configuration spaces
Let $X$ be a smooth complex projective variety. Let $F(X,n)$ be the configuration space parametrizing $n$ distinct ordered points in $X$. The cohomology groups $H^k(F(X,n),\mathbf Q)$ carry a mixed ...
4
votes
1
answer
211
views
Chances for a cosine polynomial to be positive at a point
Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a ...
3
votes
0
answers
436
views
How to define Product of Conditional Measures?
I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below.
If $(X,\Sigma)$ is a measurable space, then the function $\mu : \Sigma\...
3
votes
0
answers
166
views
Number of maximal chains in Bruhat order
Is there a formula for the number of maximal chains between two permutation in the (strong) Bruhat order?
7
votes
0
answers
281
views
Picard scheme of varieties over imperfect fields
Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...
3
votes
0
answers
160
views
Reference for existence results for 2D forced viscous Burgers equation
I am looking for results concering the following parabolic PDE
$$u\cdot\nabla u + \Delta u = F(x),$$
where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...
1
vote
2
answers
188
views
Calculating Exterior Distance from Measurements of Inner Geometry
Gauss has proven in his famous Theorema Egregium, that it is possible, to calculate the gaussian curvature from measuring angles and distances on the surface, irrespective of how the surface is ...
5
votes
2
answers
539
views
Integrals involving trigonometric functions and polynomials
Can one describe all the real polynomials $P(x)$ such that the following integrals converge:
$$
\int_0^{\infty} \sin(P(x))dx, \int_0^{\infty} \cos(P(x))dx ?
$$
Among special cases are such ...
1
vote
1
answer
224
views
On the Saito Kurokawa representation
I know Saito-Kurokawa(SK) representation is the famous non-tempered representation of $SO(5)$. But since the tempered or non-tempered terms are concerned with local phenomenon, I am wondering that ...
3
votes
1
answer
440
views
A number array related to colored necklaces and the primes
I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...
4
votes
1
answer
334
views
Reference request: Invariant sets of dynamical systems
(I should preface this with the disclaimer that this is a slightly elaborated version of a question that I posted onto math se recently to which I received no responses, and have since deleted the ...