0
votes
0answers
8 views

Books on First Hitting times

I am looking for books or expositional papers on First hitting times of sets by Brownian motion (any dimension). Here are some that are devoted to this subject or contain sections on it: 1)Sidney ...
0
votes
0answers
22 views

Generators of the algebra of invariant polynomials on a Lie algebra and the root-space decomposition

Let $G$ be a connected, simply-connected complex semisimple group with Lie algebra $\mathfrak{g}$. Fix a pair $T\subseteq B\subseteq G$ of a maximal torus and Borel subgroup, and let $\mathfrak{t}$ ...
3
votes
1answer
71 views

How many isolated roots can a polynomial in $z$ and $\overline{z}$ have?

This is probably well-known by I can't find it now online. My guess is that if the degree is $n$ then it's $2n$ but it's just a hunch. EDIT: This is an edited version. Before I asked about roots ...
-1
votes
0answers
23 views

question related to kernel methods [on hold]

i have some questions related to kernel methods in Machine Learning,hope someone can give proves. K(x,y) is kernel iff 1.K is symmetric 2.K is positive definite. ...
2
votes
0answers
30 views

Meaning of eigenvalue 1 and symmetry in Laplacian spectra of graphs

We often see normalized Laplacian spectra of graphs where density on eigenvalue 1 serves as an axis of symmetry, with particularly high (blue spectra in the figure) or low densities (red spectrum) ...
1
vote
0answers
41 views

state-of-the-art graph theory libraries like LEDA

I came across LEDA (library of efficient data types and algorithms) which implements graph theory algorithms: http://www3.cs.stonybrook.edu/~algorith/implement/LEDA/implement.shtml I want to ask: ...
3
votes
0answers
98 views

The zeta function and classical mechanics

In this paper, Guilherme França and André LeClair show that $$\gamma\sim 2 \pi \left(y-11/8\right)/W\left((y-11/8)e^{-1}\right)$$ where $W$ is the Lambert W function, and $\gamma$ are imaginary parts ...
-1
votes
0answers
20 views

Using Excel's NormInv and Proportion Estimates for Monte Carlo [on hold]

I have some data. Here's an example (ogive) 517 625 813 855 966 1143 1227 1248 1343 1367 1461 1465 1572 1574 1738 I can derive a % based on frequency of the value 0 0.071428571 0.142857143 ...
2
votes
0answers
38 views

Effective version of the Bombieri-Vinogradov theorem

Is there an effective version of the Bombieri-Vinogradov Theorem, in that have bounds on the implied constant been found?
2
votes
1answer
104 views

Is this inequality true?

Let $\Omega$ be a bounded domain with smooth boundary. Let $$ S=\{u\in C^2(\overline \Omega): \frac{\partial u}{\partial n}=0 \text{ on } \partial\Omega \}.$$ Fix $\Phi\in S$ with $\Phi(x)>0$ for ...
1
vote
0answers
32 views

A (different) foliation arising from Hopf fibration

In this question, first we fix an isomorphism between $TS^{3}$ and $S^{3}\times \mathbb{R}^{3}$.(To be more precise we consider the global trivialization of $TS^{3}$ with help of $3$ global ...
4
votes
0answers
90 views

Is there a simple proof that Milnor $K_2$ of a number field is torsion?

This is a theorem of Garland. I had a look at the original paper which looks pretty complicated. I was wondering if the proof has been simplified over the years or if a different approach is nowadays ...
2
votes
1answer
32 views

Minimum degree of rational form

Given a positive integer $g$, what is the minimum degree of the rational function that represents the real function $P(x)$ where $P(x)=\frac{x- x\mod g}{g}+1$ at $x \in ...
1
vote
0answers
51 views

Certain algebraic variete defining by multiplicative functional equation

In this question $B(M_{n}(\mathbb{C}))$ and $B^{2}(M_{n}(\mathbb{C}))$ are the linear space of all linear and 2-linear $M_{n}(\mathbb{C})$-valued maps on $M_{n}(\mathbb{C})$, respectively. ...
0
votes
0answers
17 views

Minimum degree rational function interpolation

Find a rational function $R(x)$ such that: $1)$ For $i\in\{1,\dots,g\}$, $x_{i}=x_{i-1}+g$ with $x_0=0$. $2)$ For $i\in\{0,\dots,g-1\}$ $R(x_i)=R(x_i+1)=\dots=R(x_i+g-1)=i+1$. $3)$ $R(x_g)=g+1$. ...
0
votes
2answers
25 views

Generalised “projection” of a metric space

Assume we have $n$ points $p_0\ldots p_{n-1}$ which form a discrete metric space $V$ with metric $d$. Can we define a function $f:V\rightarrow \mathbb{R}$ with $f(p_0) = 0$, $f(p_1) = d(p_0,p_1)$ and ...
1
vote
0answers
23 views

Visibility kernels of embedded graphs

Let $G$ be a connected graph embedded in the plane with all edges straight segments. For $\alpha \in (0,\pi)$, define an $\alpha$-path as a path in $G$ with all turns at vertices within ...
6
votes
0answers
52 views

Is such a map null-homotopic?

Suppose I have (semi-infinite) chain complexes $$ \cdots \rightarrow A_i \rightarrow A_{i+1}\rightarrow \cdots$$ $$ \cdots \rightarrow B_i \rightarrow B_{i+1}\rightarrow \cdots$$ over an additive ...
2
votes
0answers
42 views

Better Sobolev inequality holds in this case when assuming doubling and Poincare inequality?

Let $X$ be a Polish space and let $m$ be a locally finite Borel measure on $X$. Let $\epsilon$ be a strongly local, regular Dirichlet form on $L^2(X,m)$ with Domain $V :=\{f\in ...
1
vote
0answers
65 views

Can the Gaussian integers be covered by restricted recurrences?

Relaxation of the second question here. Let $a(n)$ be recurrence of the form $a(n)=f(n,a(n-1)\ldots(a(n-k))$ with fixed initial terms. (Observe that it might depend on $n$). $f$ may contain ...
3
votes
1answer
111 views

Fano varieties of cubic threefolds

Let $X$ be a smooth cubic threefold over $\mathbb{C}$. Let $F(X)$ denote the Fano variety of lines in $X$, which is a smooth surface of general type. Is this class of surfaces distingushed ...
-8
votes
0answers
65 views

The Riemann Zeta Function Works [on hold]

1 - Any counterexamples known for the Riemann Zeta Function? 2 - How to generalize the following? Here we have the visualization of the Riemann Zeta Function 3D Plot and the plane. We can observe ...
-2
votes
0answers
24 views

Can this specific Mixed-Integer Linear Program constraint be expressed? [on hold]

Thanks for your time. I have a linear program and no idea how I could express a form of constraint and even if it's possible. Maybe someone here know a solution. A company assembly and sells a ...
1
vote
0answers
137 views

What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?

The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
4
votes
0answers
174 views

Does Nelson try to prove PA inconsistent directly?

Edward Nelson is known for his serious attempts to show that Peano axioms, and sometimes even weaker theories, are inconsistent. I wasn't able to find Nelson's papers anywhere, so I wanted to ask a ...
3
votes
1answer
170 views

Question on effective Mordell conjecture

Suppose $F(x,y,z)$ is a homogeneous polynomial over $\mathbb{Q}$, where $C:F(x,y,z)=0$ is a curve of genus $g\geq 2$. Question: Faltings proved that $C$ has finite many rational points. Suppose that ...
4
votes
1answer
72 views

almost huge embeddings and stationary correctness

Suppose $\kappa$ is an almost huge cardinal. Using the characterization of Theorem 24.11 in Kanamori's book, one can show that if $j : V \to M$ is an embedding derived from an almost-huge tower of ...
-2
votes
0answers
61 views

Centralizer of element in group PSL(2,F_p) [migrated]

Is it true, that $\forall g\in PSL(2,F_p)\setminus\{e\}$, $Z(g)$ is Abelian? I think that this is true, but i can't find a simple proof.
-2
votes
1answer
22 views

Convert constraint to do convex optimization or use Lagrange multiplier method

$w_1, w_2, w_3 ... w_n$ are the weights I need to find I have the following constraint: $|w_1| + |w_2| + .. |w_n| <= 5$ That is the sum of the absolute values of the weights has to be less than ...
4
votes
1answer
65 views

Estimate of the sum Taylor's coefficients

Let $f(x) = \begin{cases}\ln\frac{x}{e^x-1}, \quad x > 0; \\ 0, \quad\qquad x=0; \\ \ln\frac{-x}{e^x-1}, \quad x < 0. \end{cases}$ Power series in 0: $f(x) = \sum_{n=1}^{\infty} a_n x^n = ...
-1
votes
0answers
42 views

Lower bound for sum of binomial coefficients without summation [on hold]

I am new here. So, I will be happy if can somebody help me to find an answer for this proof. I want to prove this lower bound: $$ \log {{n \choose n_1}} \leq nlog {n} - n_1log(n_1) - ...
2
votes
1answer
80 views

Curvatures preserved under the Kahler-Ricci flow

Maybe it is a trivial question. Is there any obvious reason that non-negative holomorphic bisectional curvature is preserved by (normalized) Kahler-Ricci flow, but non-negative Ricci curvature is not ...
3
votes
0answers
55 views

Some questions on analytic vectors and the integrability of Lie-algebra representations

I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this ...
9
votes
1answer
150 views

Is there an $L$ like inner model for $\sf Z$?

Godel proved the consistency of the axiom of choice with the axioms of $\sf ZF$ by showing that given any model of $\sf ZF$, there is a definable class which satisfies $\sf ZFC$. The proof uses a lot ...
11
votes
1answer
204 views

Who first noticed that Stirling numbers of the second kind count partitions?

When the Stirling numbers of the second kind were introduced by James Stirling in 1730, it was not combinatorially; rather, the numbers ${n \brace k}$ were defined via the polynomial identity $$ x^n = ...
1
vote
0answers
47 views

The number of blocks in Szemerédi Regularity Lemma

In mathematics, the Szemerédi regularity lemma states that every large enough graph can be divided into subsets of about the same size so that the edges between different subsets behave almost ...
2
votes
1answer
78 views

Resolving nodes of a quintic CY 3-fold

Let's consider the following quintic 3-fold $X$: \begin{equation} \{(x_i) \in \mathbb{P}^4 \ | \ x_1f(x)-x_2g(x)=0\} \end{equation} for generic homogeneous polynomials $f(x),g(x)$ of degree four. It ...
0
votes
0answers
68 views

Understanding Prof. Keevash's proof on the “Existence of Designs” [on hold]

I have tried to read Prof. Kevvash's paper on the "Existence of designs". I am finding it very tough to read it linearly. I am comfortable with the nibble ideas and probabilistic methods in general. ...
3
votes
1answer
72 views

the eigenvalues of a generalized circulant matrix

A $2k\times 2k$ circulant matrix $\ C$ takes the form \begin{align} C= \begin{bmatrix} c_0 & c_{2k-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{2k-1} & & c_{2} ...
2
votes
0answers
34 views

Behavior of elementary symmetric polynomials near zero sets

It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with ...
-4
votes
0answers
32 views

Compare time it takes to travel a curve and a line [on hold]

First posted in Math Stack Exchange: http://math.stackexchange.com/questions/989690/compare-time-it-takes-to-travel-a-curve-and-a-line?noredirect=1#comment2025799_989690 Suppose you have a right ...
5
votes
1answer
199 views

Polynomial recurrence relation covering the integers (and then Gaussian integers)

Say that a polynomial recurrence relation (my terminology) for $f_i$ is: $k$ initial conditions setting $f_1,\ldots,f_k$ to integers ($\in \mathbb{Z})$. A recurrence equation of the form $f_i =$ a ...
3
votes
0answers
57 views

Number of orbits of $\mathrm{SL}_2(\mathcal O_A)$ on $\mathbf P^1(A)$ when $A$ is a quaternion algebra

This is a reference request. Let $A$ be an anisotropic quaternion algebra over $\mathbf Q$. Let $\mathcal O_A$ be a maximal order in $A$. Then $\mathrm{SL}_2(A)$ acts transitively on the right on ...
0
votes
0answers
37 views

What is the state of art MIQP solver

I used Gurobi with a MIQP with 26 binary variables and 26*4 interaction term without any other constraint. The speed is very slow already.... I want to ask what is the state of art of MIQP solvers. ...
0
votes
0answers
38 views

Intersection points of closed curves inscribed in a convex polygon

Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...
-2
votes
0answers
83 views

Arranging books into bags [on hold]

I'm trying to find an algorithm to answer the following question (informal): given a (finite) set of distinct books of different (positive integer) sizes and a (finite) set of bags of different ...
-4
votes
0answers
50 views

In Dedekind' construction of real numbers, what's wrong with this understanding [on hold]

in this prove, every cut corresponds to a real number. and a cut is a subset of Q.and cut have these three properties.1.is not empty 2.if p belong to this cut,any qp so in my understanding, every cut ...
-5
votes
0answers
44 views

transitive actions of automorphism group [on hold]

Can a finite group be extended to a group whose group of automorphisms acts transitively on the first group. Or more generally, given any finite module, can we extend this module so that the group ...
-2
votes
0answers
48 views

The space $W = \{u \in L^2(0,T;V) : u_t \in L^2(0,T:V^*)\}$ without having identified $H$ and $H^*$

Let $V \subset H \subset V^*$ be a Gelfand triple with the Hilbert space $H$ identified with its dual space and $V$ a reflexive separable Banach space. Define $W = \{u \in L^2(0,T;V) : u_t \in ...
4
votes
1answer
98 views

A lower bound on the $L^2$ norm of a Dirichlet polynomial

The Question. Suppose $0 < \alpha < \beta$ are fixed, and $a_n$ is an arbitrary sequence of real numbers. Is it known how to bound from below \begin{equation*} \int_0^{T} \Big| \sum_{\alpha T ...

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