# All Questions

**-2**

votes

**0**answers

23 views

### Prove determinant of nxn matrix is (a+(n-1)b)(a-b)^(n-1)? [on hold]

Prove det(mat) is (a+(n-1)b)(a-b)^n-1 where matrix is nxn matrix with a's on diagonal and all other elements b, off diagonal?
For example, suppose matrix with diagonal composed solely of a's. All ...

**2**

votes

**0**answers

27 views

### $SL(n) \times SL(n)$-invariants of $m$-tuples of matrices

I work over field of complex numbers. Let $G=SL(n) \times SL(n)$, and $(A,B) \in G$ acts on $m$-tuples of matrices $M_{n \times n}(\mathbb{C})^{\oplus m}$ as follows
$$
(A,B) \cdot (M_1, \ldots, M_m) ...

**1**

vote

**0**answers

48 views

### Circle method on things other than the integers

The circle method is often used to estimate the number of solutions to the equation $$x_1 + x_2 + ... x_k = N$$ if for all $i$ $x_i\in A\subseteq\mathbb{N}_0$ and some subset of the nonnegative ...

**0**

votes

**0**answers

55 views

### Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?

The usual cohomology theory on schemes uses injective or flasque resolutions of quasi-coherent sheaves. Hence it uses Axiom of Choice.
However, if the base scheme is a noetherian separated scheme, the ...

**-1**

votes

**0**answers

49 views

### The problem of Riemann zeta function [on hold]

$\zeta(2)=\sum_0^\infty 1/n^{2}<1.8$
From the popular knowledge
$\zeta(2)\Gamma(2)=\int_0^\infty x/(e^x-1)dx$
but
$\int_0^\infty x/(e^x-1)dx=\int_0^\infty xe^{-x}/(1-e^{-x})dx$
$=\int_0^\infty ...

**2**

votes

**1**answer

49 views

### Suslin lines hereditarily Lindelof

I need to prove that every suslin line is hereditarily Lindelof. Any idea will be helpfull.

**-1**

votes

**0**answers

86 views

### Recreating the wheel

I recently finished my Phd in pure maths and I am looking for open problems in my research area, functional analysis. Without going into the details, I stumbled onto an interesting problem and I ...

**0**

votes

**0**answers

17 views

### Cover one finite subset of integers by another one

Let $A$, $B$ be two finite subsets of integers. We denote by $C(A, B)$ the minimum number of shifts of $A$ to cover $B$. More formally, it can be written as
$$
C(A, B)=\min\{|S|: S\subseteq ...

**4**

votes

**1**answer

76 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

**0**answers

20 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

**0**answers

31 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

**0**answers

74 views

### A complicated combinatoric/probabilistic limit

I'm hoping to find a tractable expression for a limit of the following expression:
$P_i (\textbf{n},\textbf{U};M,N) =\frac{M U_i}{M^{\sum \limits_{j=1}^{N} n_j}} \sum \limits_{k_1 = \delta_{i,1}} ...

**3**

votes

**0**answers

40 views

### Positivity of Intersections in higher dimensions

"Positivity of Intersections" is a phenomenon in 4-dimensions, due to Gromov: Given two embedded $J$-holomorphic curves in an almost-complex 4-manifold $(X^4,J)$, all intersection points are isolated ...

**-1**

votes

**0**answers

76 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

**1**answer

19 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 ...

**5**

votes

**0**answers

42 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

**2**answers

188 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

**1**answer

182 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

**0**answers

32 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

**2**answers

101 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 ...

**3**

votes

**0**answers

48 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 $\mathsf{Mod}(T)$ is cocomplete, and I think that we have ...

**2**

votes

**1**answer

59 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 ...

**3**

votes

**0**answers

56 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

**1**answer

67 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

**0**answers

16 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 ...

**0**

votes

**0**answers

32 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

**0**answers

29 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

**1**answer

69 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

**0**answers

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

**0**answers

215 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 ...

**2**

votes

**0**answers

67 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

**0**answers

46 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?

**1**

vote

**0**answers

40 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 $\mu$. ...

**2**

votes

**0**answers

36 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

**0**answers

33 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

**1**answer

109 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

**1**answer

55 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

**1**answer

36 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

**0**answers

31 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

**0**answers

22 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 ...

**2**

votes

**2**answers

136 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

**0**answers

57 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 ...

**3**

votes

**1**answer

114 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 ...

**0**

votes

**0**answers

42 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

**0**answers

31 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

**2**answers

73 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

**0**answers

34 views

### Information geometry divergence

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]= ...

**5**

votes

**2**answers

140 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 ...

**7**

votes

**1**answer

112 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

**1**answer

49 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 ...