0
votes
0answers
3 views

preservation of localness among certain Krull domains

The following question essentially appeared (http://math.stackexchange.com/questions/931801/preservation-of-localness-among-certain-krull-domains) on math.SE a while ago, but nobody has done anything ...
2
votes
1answer
55 views

Asymptotic property of a quadratic form

suppose $x=\Delta$, $y=M \Phi \Delta$, where $\Delta\in N\times 1$, $M^T=M \in N \times N$ and $\Phi^T=\Phi \in N \times N$. Define $Z=xy^T+yx^T$. It is known from my previous question that $Z$ has ...
0
votes
0answers
5 views

On the global and local hermitian space

Let $E/F$ be a quadratic extension of number fields and $v$ a finite place of $F$. Then I am wondering if there is a global hermition vector space over $E$ such that for all finite places $v$, local ...
0
votes
0answers
39 views

numerical method (implicit) for nonlinear pde

If $\newcommand{\lbar}{\underline{\lambda}}$ $$ \lambda(t)= \lbar+(\lambda_0-\lbar)\exp \left( (\mu-\frac{1}{2}\sigma^2)t+\sigma W^\lambda_t\right) $$ and $\mu$ , $\sigma$ , $\lbar$ , $\lambda_0$ , ...
7
votes
0answers
70 views

Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...
2
votes
0answers
10 views

Periodic group with bound on order of finite subgroups

I have asked the same question previously on stackexchange without any answer (http://math.stackexchange.com/questions/923638/periodic-group-with-bounded-subgroups): I am looking for infinite ...
0
votes
0answers
18 views

“Bounded” measures" ? = Dual space of the continuous integrable functions

I am trying to find if the dual space of continuous integrable functions on $\mathbb{R}$ is a well-defined subject. Ideally, it denotes the set of "bounded" measures, for example, the Lebesgue measure ...
18
votes
1answer
126 views

Does iterating a certain function related to the sums of divisors eventually always result in a prime value?

Let define the following function for integers (from 2): $f(x)=\sigma(x)-1$, where $\sigma$ is the sum of the divisors of $x$. For example $f(6)=6+3+2=11$, $f(5)=5$. Note that $x$ is a fixed point for ...
0
votes
1answer
17 views

Strictly positive solutions of a random linear system

Suppose $B\in\mathbb{R}^{m\times n}$ is a random binary matrix with i.i.d entries and $c\in \mathbb{R}^m$ is a strictly positive vector, that is $c_i>0$ for $i=1,2,\cdots m$. Also assume $m<n$, ...
3
votes
1answer
62 views

Unusual augmentation of a filtration

consider a probablity space $(\Omega,\mathcal{F}, \mathcal{P})$ and a filtration $(\mathcal{F}^0_t)$. In general $(\mathcal{F}^0_t)$ doesn't satisfy the usual conditions (it is not both complete at ...
0
votes
2answers
305 views

Morse matching with 0-cells and (n-1)-cells

Suppose one has a Morse matching (acyclic matching) of a poset $P$ of rank $n$ in which the only unmatched cells are 0-cells and $(n-1)$-cells, the same number $k$ of each one. If $P$ is connected ...
10
votes
1answer
348 views

Discrete Morse theory and chess

There are many mathematical objects that are similar to groups and Cayley graphs of groups but lack homogeneity in some sense. Graphs of webpages with edges corresponding to links are one example. ...
-3
votes
0answers
26 views

Limit with two variables. Headscratch or not [on hold]

Hi I have a limit with two variables in front of me and the book says directly that it is equal with 1 but for the life of me I dont understand why/?? maybe the answer is stupid but I am excausted and ...
4
votes
2answers
148 views

Using Discrete Morse Theory to represent hom classes

In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical ...
0
votes
2answers
280 views

Eigenfunction of local fractional derivative

Let $E_{\alpha}(x^{\alpha})$ be a Mittag-Leffler function, $\alpha \in (0,1)$. It is an eigenfunction for nonlocal fractional derivative, defined as a convolution with $$ \Phi_{\lambda}(x) = ...
1
vote
1answer
24 views

Unbiased sample from a product

Let $X = (x_1,\ldots,x_n)$ be an i.i.d sample from distribution $F%$ and let $y = \prod_{i=1}^n x_i$ Can we derive a randomized, unbiased. estimator $\hat{y}$ of $y$ that on average considers only a ...
0
votes
0answers
27 views

Containment of two varieties with a lot of intersection

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...
-1
votes
0answers
8 views

What does it mean for a stochastic sequence to be “stochastically smaller” than some other stochastic sequence?

A relatively simple type of random graph is the Erdős-Rényi random graph. The graph created by means of the following process: Let $V$ consists of $n \in \mathbb{N}$ vertices, and let each edge of the ...
4
votes
1answer
185 views

Construction of generalized Eilenberg-MacLane spaces

The Eilenberg-MacLane spaces $K(G,q)$ are readily generalized to study cohomology with local coefficients.The generalized Eilenberg-MacLane space $K_{\pi}(G,q)$ are spaces with only two nnvanishing ...
0
votes
0answers
55 views

Squarefree Parts of Mersenne Numbers with prime exponent [on hold]

The $n$-th Mersenne number is $M_n=2^n−1$. Write $M_n=a_n b^2_n$ where $a_n$ is positive and squarefree. In the discussion Squarefree Parts of Mersenne Numbers , the lower bound of $a_n$ has been ...
8
votes
1answer
918 views

Go I Know Not Whither and Fetch I Know Not What

$f$ is a polynomial in four variables. Take matrices $$ 1 = \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & ...
17
votes
22answers
2k views

Results true in a dimension and false for higher dimensions

Some theorems are true in vector spaces for a given dimension $n$ but become false in higher dimensions. Here are two examples: A positive polynomial not reaching its minimum. Impossible in ...
-4
votes
0answers
28 views

Help? Algebra question about ratios and proportions [on hold]

A community center is going on a trip to Philadelphia via several buses. The ratio of men to women to children is 1:2:3. If there are 150 people going on the trip, how many men are going? How many ...
16
votes
2answers
378 views

Stability of real polynomials with positive coefficients

Say that a polynomial in an indeterminate $x$ with real coefficients of degree $d$ has positive coefficients if each of the coefficients of $x^d,\ldots,x^1,x^0$ is (strictly) positive. For $f$ a ...
27
votes
1answer
1k views

Zeros of polynomials with real positive coefficients

The following problem arose in collaborative work with Subhro Ghosh: Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros ...
9
votes
4answers
285 views

The cohomology group $H^{1}(GL_{2}(\mathbb{F}_{p}), M_{2}(\mathbb{F}_{p}))$

Let $M_{2}(\mathbb{F}_{p})$ be the vector space of 2$\times$2 matrices over the finite field $\mathbb{F}_{p}$ where $p$ is a prime number, and let $GL_{2}(\mathbb{F}_{p})$ be the group of invertible ...
12
votes
7answers
2k views

How $a+b$ can grow when $a!b! \mid n!$

Let $a,b,n$ be natural numbers such that $a!b! \mid n!$. I am looking for a (somehow best) upper bound of $a+b$ in terms of $n$ (for large values on $n$). For example it is clear to see that we must ...
4
votes
2answers
55 views

Measure on hyperspace of compact subsets

For a Polish space $X$, let $K(X)$ be the set of compact subsets of $X$. Given the topology with basis $\{K\in K(X):K\subset U_0, K\cap U_1\neq\emptyset,\ldots,K\cap U_n\neq\emptyset\}$ for open sets ...
0
votes
1answer
23 views

Monotonicity of the gap of permutated sequence

Let $a$ be an arbitrary sequence and denote by $\mbox{gap}_k(a) = a_{(k)} - a_{(k+1)}$, where $a_{(k)}$ is the $k$th largest component of $a$. Of course, $k+1$ should be no larger than the length of ...
0
votes
1answer
13 views

Zeroes of Sturm-Liouville solutions as a function of the (complex) eigenvalue

Given the Sturm-Liouville type (time independent Schroedinger) equation \begin{equation} \frac{d^2 y}{d x^2} - \left(\mu + V(x)\right) y = \lambda \, y,\quad x \in \mathbb{R} \end{equation} where ...
2
votes
1answer
153 views

Can the Grothendieck ring of varities over a field $k$ be defined for non separated schemes?

The Grothendieck ring of varieties over a field $k$ is the abelian group generated by isomorphim classes $[X]$ of separated, reduced $k$-schemes $X$ of finite type with the relation $[X]=[Y] + ...
12
votes
1answer
334 views

When are Jones-Wenzl projectors defined?

(I am hoping that someone well-versed in the literature of Temperley-Lieb algebras or of quantum groups at roots of unity can answer my question. Fingers crossed.) Consider the Temperley-Lieb algebra ...
-6
votes
0answers
100 views

Does $\pi$ encode the prime numbers? [on hold]

I have a question regarding whether or not $\pi$ encodes the sequence of primer numbers. It is common knowledge that $$ \zeta (2) = \sum_{i = 1}^{\infty} \frac{1}{n^2} = \prod_{p \in \mathbb{P}} ...
4
votes
1answer
308 views

Is there a generalization of Sobolev spaces for certain locally compact groups?

I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square ...
3
votes
1answer
205 views
+50

Helmholtz equation Poynting vector integral

The Maxwell's equation for harmonic time dependent field in vacuum is \begin{align} \nabla \times B + i\omega E &= 0\\ \nabla \times E - i\omega B &= 0 \\ \nabla \cdot B &= 0 \\ \nabla ...
3
votes
0answers
34 views

Geodesic flow on infinite surfaces

The geodesic flow on a compact hyperbolic surface (i.e. a surface with a riemannian metric of constant curvature $-1$) has been well-studied, in particular it has been known for a long time that it is ...
1
vote
2answers
65 views

Non-DS circulant graphs

Let $p$ be a prime number greater than or equal to 11. Are there any cospectral non-isomorphic circulant graphs on $p$ vertices.
6
votes
1answer
618 views

Is the Duflo polynomial conjecture open?

Let $G/K$ be a symmetric space. Let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan decomposition, with the odd part $\mathfrak{p}$. It is well known that the algebra of invariant ...
6
votes
3answers
893 views

Concentration results for inner products of two independent random gaussian vectors

Hi, I wanted to know if there are standard results on concentration of absolute value of inner products of two random vectors. Thus if $X, Y \in R^m$ are two independent random vectors with each ...
13
votes
3answers
525 views

Probing the generalization of the abc conjecture to more than 3 variables

Browkin and Brzezinski, in "Some remarks on the $abc$-conjecture", Math. Comp. 62 (1994), no. 206, 931–939, state the following generalization of the $abc$ conjecture to more than three variables: ...
1
vote
1answer
65 views

Matrix Generator for M/M/1 Queue Waiting Time Distribution

I "believe" that generator, $\bf W$, of the waiting time distribution for the M/M/1 queue is given by the following (I'm not sure if this is even correct): ${\bf W} =\left( \begin{array}{ccccc} 0 ...
0
votes
2answers
430 views

divisorial ideals

Let $I$ be an ideal of a domain. Then is there an ideal $J$ properly located between $I$ and $I^{\nu}$? Here $I^{\nu}$ is divisor of $I$.
1
vote
0answers
67 views

Hopf-Borel theorem over polynomial rings

Hello! There is a theorem of Borel saying that: For a connected cocommutative Hopf algebra $A$ over the perfect field $k$ such that underlying graded vector space is finite-dimensional over $k$ we ...
9
votes
1answer
171 views

Steady state expectation of dynamic system of urns & balls

We have a large number of urns $N+1$. (Large means that the relative difference between $N$ and $N+1$ is well within the error bounds that I care about. The reason for the $+1$ will be apparent ...
0
votes
1answer
134 views

Name for class of matrix determinants

Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions $\alpha_i(x)$. Now, ...
1
vote
0answers
107 views

Accessible subgroups of free groups

Let $F$ be a nonabelian free finitely generated group, and $F = G_0 \rhd G_1 \rhd G_2 \dots$ a strictly descending subnormal chain of subgroups ($G_n \lhd G_{n-1}$ for each $n \in \mathbb{N}$) each ...
2
votes
1answer
109 views

Vanishing homology of simplicial complexes with few facets

Let $K$ be a simplicial complex with $n$ vertices and $n-t$ facets, where $t \geq 3$. Is it true that the $(n-3-j)$th reduced homology (with coefficients in a field) of $K$ vanishes for $0 \leq j ...
-8
votes
0answers
41 views

New formulas for Generating sin(nx) & cos(nx) [on hold]

I have proved formulas for generating $\sin (nx)$ and $\cos(nx)$ in term of only $\sin x$ and integer $n$, or $\cos x$ and integer $n$, such as: $$\sin(9x) = 9\sin x -120\sin^3 x + 432\sin^5 x – ...
2
votes
0answers
30 views

Weighted graph similarity

I have the following problem. Consider an undirected biconnected graph with $n$ vertices and $m$ edges ($n \leq m \leq n(n-1)/2$). The $m$ edges of the graph are then "populated" by integer weights ...
0
votes
0answers
90 views

Decomposition of symmetric homogeneous polynomials

Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of ...

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