5
votes
1answer
376 views

Is there a sideways-walking rolling convex body?

Let $K$ be a solid, homogenous convex body in $\mathbb{R}^3$. Place $K$ on an inclined plane, and let it roll down the plane, under some reasonable assumptions of friction between $K$ and the plane, ...
0
votes
0answers
34 views

additive support functions of a convex set

Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by $$h_K(u) = \sup \{ \sum_{i=1}^d x_i ...
-2
votes
0answers
40 views

n-th prime in first order arithmetics [on hold]

Recently I have thought about formalizing Turing machine in first order arithmetics, step by step, starting from the most basic things. But I quickly struck a problem - to continue, I need to find a ...
1
vote
0answers
158 views

A generalized urn-ball matching problem; Complicated combinatoric/probabilistic limit

I'm looking for a generalization to the urn-ball matching problem. As a reminder of what I've got in mind, here's the simple version: Randomly assign (with replacement) $N$ balls to $M$ urns. ...
-1
votes
0answers
82 views

Why integer should have finite many digits? [on hold]

for example, if we take the real part of pi 3.1415926... and write it from right to left like ...6295141, we can get a number, but this number is not a integer, why ? why it is not a integer? Can we ...
1
vote
1answer
24 views

a class of directed hypergraphs

I am interested in a certain class of directed hypergraphs, more precisely in the class of those hypergraphs each of whose hyperedges contain an even number of nodes (not necessarily the same even ...
6
votes
0answers
60 views

When were bordered Heegaard Floer homology's DA bimodules invented?

This is less of a strictly mathematical question and more of a reference request. In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston ...
9
votes
2answers
294 views

What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$

The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...
3
votes
1answer
217 views

What year was Hechler forcing created?

Hechler forcing is described on page 278, Jech. Does anyone know when Hechler forcing was first used in a publication?
-2
votes
0answers
34 views

Simplifying Trig Equation with Identifities [on hold]

I have an equation I have been given to solve, I know how to start but I do not know what to do after I use the Trig Identities. Any help? Here is what I was given (cos(A + B) + cos(A - B)) / ...
4
votes
2answers
145 views

Can finitely generated subgroups of limit groups be detected in free group quotients?

In Henry Wilton's excellent paper "Hall's Theorem for limit groups" (Geom. Funct. Anal. 18, pp. 271–303, 2008 ) he proves the following result (see his paper or here and here for the relevant ...
4
votes
0answers
121 views

Universal property of module categories over monads

Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of $T$-modules $\mathsf{Mod}(T)$ is ...
3
votes
1answer
71 views

Extremal graph theory for directed graphs

In extremal graph theory, there are results such as $$t(C_4,G)\geq t(K_2,G)^4,$$ where $G$ is an undirected graph, $C_4$ is a cycle graph on 4 nodes, $K_2$ is a complete graph of $2$ nodes, and ...
4
votes
1answer
106 views

What is the early history of the concepts of probabilistic independence and conditional probability/expectation?

In the 1738 second edition of The Doctrine of Chances, de Moivre writes, Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards ...
3
votes
1answer
87 views

Zero currents localized along a submanifold

Let $\mathcal{D}(\mathbb{R})$ be the continuous dual of $C^\infty_c(\mathbb{R})$, the space of compactly-supported smooth functions. There is a nice characterization of distributions ...
0
votes
0answers
24 views

Grassmann algebra morphism with universal property

I'm pretty sure that the following doesn't work, but nevertheless i wanted to ask, maybe this is a kind of well-known construction i've never heard of: Let $\Lambda(\mathbb{R}^n)$ be a finite ...
1
vote
0answers
71 views

Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the Fourier transform of a random variable $X$ $$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$ ...
1
vote
0answers
49 views

A comparison principle for degenerate parabolic equation

Let $\Omega$ be a bounded smooth domain and let $p < 0$ be real. Suppose that $u, v \in L^2(0,T;H^1(\Omega) \cap L^2(0,T;L^2(\partial\Omega))$ with $|u|^{p+1} \in L^2(0,T;L^2(\partial\Omega))$ and ...
3
votes
1answer
83 views

Simple example of isolated critical point with non-semisimple monodromy

Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some ...
-1
votes
0answers
11 views

regu tools l_curve regularization stanford ee 263 [migrated]

I am trying to solve one of the famous stanford EE263 problems, which gives me matrix A representing blurring of an image and y, representing the blurred image. For that I have been trying to use ...
4
votes
4answers
738 views

Advice for number theory library

Hi I just got a faculty position and it comes with a generous start up funds for "office supplies", which I must use or lose. What does a pure mathematician need? I have good computers already. I ...
3
votes
0answers
75 views

Infinite series of determinants

I am interested in what is known about the following class of sums. For a sequence of matrices $A_i$ (which possibly have different size), I am wondering about examples and methods for evaluating sums ...
2
votes
0answers
60 views

$p$-groups with $\Omega_1(G)\leq\Phi(G)$

Let $G$ be a finite $p$-group with $\Omega_1(G)\leq\Phi(G)$. What do we have information about this group?
2
votes
0answers
76 views

Are the only discrete groups with nontrivial p-adic Haar measure finite?

Let K be a complete non-archimedean field (say $\mathbb{C_p}$) and let G be a discrete group. Since {e} is an open p-compatible compact subgroup of G, G admits a (left) K-valued Haar measure $\mu$. ...
3
votes
1answer
65 views

Reference for Frobenius’s proof of Schur’s finite version of the Rogers - Ramanujan identities

In his paper “Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche” I. Schur has stated that Frobenius has communicated to him a simple direct proof of his finite version of the ...
0
votes
0answers
39 views

First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric

This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions. We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
1
vote
1answer
123 views

What are finite groups $H$ such that $H^n(H,\mathbb{Q/Z}) \cong H_n(G,\mathbb{Z})$?

Let $G$ be a finite group and $G^{\prime}$ be its commutator subgroup. Let $\mathbb{Z}$ and $\mathbb{Q}$ denote the integers and rationals. $\mathbb{Z}$ and $\mathbb{Q/Z}$ treated as trivial ...
-5
votes
1answer
59 views

An elementary question in abstract algebra [on hold]

Let K be a field and A be a K-algebra, then how can we show that $\lambda1_A=\lambda$ for $\lambda\in$K and $1_A$ is the identity of A?
-2
votes
1answer
40 views

How to find a matrix by its characteristic value and characteristic vectors? [on hold]

Now I am studying linear algebra course, In that for a given matrix we are finding the characteristic values (eigen vlaues) and characteristic vectors (eigen vectors). But my qustion is why cant we ...
1
vote
0answers
36 views

A counterpart of Karhunen theorem

According to the Karhunen theorem, if the correlation function of a process $X(t)$ can be represented as $$ R(t,s)= \int_{\Lambda} f(t, \lambda) \overline{f(s, \lambda)}d\nu(\lambda) $$ then the ...
-1
votes
0answers
24 views

departure time/overlap algorithm [on hold]

i'm looking for "departure time/overlap algorithm" or any other idea. Suppose you have n trains and each one has a performance profile(how much electricity they need at the current time while driving ...
3
votes
2answers
161 views

Interpolation of periods for a Hida family of modular forms

Let $\mathbf{f}$ be a Hida family of ordinary $p$-adic modular forms, and let $V(\mathbf{f})$ be the corresponding $\Lambda$-adic Galois representation (a quotient of the inverse limit $$ ...
5
votes
0answers
68 views

Comparison of K-groups of (affine) singular schemes with K'=G-groups

It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In ...
4
votes
1answer
188 views

Normal Subgroup Growth

Let $F$ be a free group on $d$ generators. Denote by $F_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F_k$. What is the normal subgroup growth of $G$? Explicitly, for each natural number ...
1
vote
0answers
57 views

Cancellations in products of two elements of a hyperbolic group

Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...
1
vote
0answers
36 views

Lanczos algorithm with thick restart on a dynamic matrix

currently, I'm working on a way to compute the 2 biggest eigenvalues of a real, symmetric, huge and sparse matrix that changes a few entries from time to time. The problem should be solved using an ...
3
votes
2answers
88 views

Tomography problem involving a set of point masses

Consider a set of $N$ points in $n$-dimensional space, i.e. \begin{align*} \{x_1, \dots, x_N\} \subset \mathbb R^n. \end{align*} Let us be given a finite family of non-injective matrices ...
-2
votes
0answers
37 views

Information geometry divergence [on hold]

on http://en.wikipedia.org/wiki/Information_geometry How to derive this equation. I tried but always got 0 for each item. $$ D[\partial_i\partial_j||\cdot]= ...
9
votes
4answers
247 views

Is any quadric birational to a product of Brauer-Severi varieties?

Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let $$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n_k,$$ be a non-singular ...
8
votes
1answer
157 views

Immersions of $n$-manifolds in $\mathbb{R}^n$ versus embeddings in $\mathbb{R}^{n+1}$

Let $M$ be a connected noncompact parallelisable smooth $n$-manifold. (By Hirsch-Smale theory, $M$ can be immersed into $\mathbb{R}^n$.) I am interested in the following two properties: P1: $M$ can ...
2
votes
1answer
50 views

Eigenvalues of a graph and its one-edge-delation graph

Let $G$ be any graph with at least one edge and let $e$ be any edge of $G$. Let $G-e$ denote the subgraph of $G$ obtained by deletion of the edge $e$. Assume that $G$ has $n$ vertices. Suppose ...
0
votes
0answers
6 views

Contraction between basis vectors and basis one-forms [migrated]

Discretion: The title may be misleading, because I am not certain whether the one-forms are actually basis one-forms. I always thought by definition, $dx^i (e_j) =\delta^i_j $. But, I am confused ...
4
votes
0answers
118 views

$\infty$-Borel Determinacy?

An $\infty$-Borel set is a set $X\subseteq\mathbb{R}$ which has an $\infty$-Borel code - a set $r$ coding the construction of $X$ via open sets, complementation, and well-ordered unions (see ...
7
votes
1answer
142 views

Semiring of vector bundles on $\mathbb{C}\mathbb{P}^1$

Consider the semiring $$\mathbb{N}[H,H^{-1}]/(H^p+H^q = H^{p+q}+1)_{p,q \in \mathbb{Z}}.$$ Is it finitely presentable? Is there any simplification of the relations (except for $p \geq q \geq 0$)? ...
-4
votes
0answers
70 views

Generating function which has no singularity [on hold]

We can know the growth rate of coefficients from singularities of generating functions, but if a generating function which has no singularity at all, for example, the exponential function. What ...
1
vote
2answers
49 views

Approximating Probability Distribution by Sampling

Consider a discrete probability distribution over $n$ events. Assume that the probabilistic kernel is a black box, that is, we can only sample from it without knowing anything about the type or ...
13
votes
1answer
596 views

How to prove that every polynomial in an infinite family is irreducible over Q?

Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...
1
vote
0answers
49 views

1d random walk probability of previous n positions

I have the following question. (May be it is very simple, but I cannot find the answer). Suppose I have a 1d random walk on integer numbers with equal pobabilities of unit step in either direction ...
1
vote
1answer
73 views

Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?
-2
votes
0answers
86 views

Formal definition of function: equality [on hold]

Consider this formal definition of function, which does not require $A$ to be domain. \vspace{0.5cm} \textit{Let $A$, $B$ be sets, then $F\subseteq A \times B$ is said to be a function iff ...

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