# All Questions

**0**

votes

**0**answers

86 views

### Polynomials interpolation without X's values [on hold]

Consider we evaluate a polynomial P of degree d on some points (say 2d+1 points or more) to obtain Y's. If we have the second distinct polynomial P2 with the same degree as before, and evaluate it on ...

**1**

vote

**0**answers

28 views

### “Bad” lower functions for a Bessel process?

Let $(X_t, t \ge 0)$ be a Bessel($\delta$) process, for some dimension $\delta > 2$, starting, say, from $1$.
Let $f: \mathbb{R}_+ \to \mathbb{R}_+$ be an upper semicontinuous function; assume ...

**1**

vote

**2**answers

156 views

### The cohomology groups of $\Omega U(n)$

Let $\Omega U(n)$ be the loop space of $U(n)$. Is it true that the cohomology groups $H^*(\Omega U(n); \mathbb{Z})$ are torsion-free? How can one calculate these groups?

**1**

vote

**0**answers

25 views

### How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

It is well known that several definite integrals $I_n$ containing a parameter $n\in\mathbb N$ can be expressed recursively (e.g. doing integration by parts) in terms of $I_{n-1} $ or $I_{n-2} $, and ...

**1**

vote

**0**answers

68 views

### Puzzling CAS-detected factorization by cyclotomic polynomials [on hold]

If one factorizes by CAS the expression $$x^{\frac{m(m+1)}{2}}\prod_{k=1}^m(x^k+(\frac{1}{x})^k)$$ a puzzling perfect factorization seems to be possible for all natural values of $m$.
E.g. for ...

**8**

votes

**2**answers

241 views

### cardinality of perfect sets in generalized Baire space

I've been unable to find an answer to the following question in the literature
on generalized descriptive set theory. Consider Baire space $\kappa^{\kappa}$
where $\kappa$ is inaccessible. The basic ...

**-1**

votes

**0**answers

39 views

### Question about total derivative matrix [on hold]

I'm new here and would like to ask a question concerning the total derivative of a function from $\mathbb{R}^n \rightarrow \mathbb{R}^p$.
I know the definition of the total derivative, but I don't ...

**2**

votes

**0**answers

37 views

### two questions about weak inclusion of unitary representations and Fell topology

I need several facts about weak containment of unitary representations of locally compact groups $G$, and I am reading the exposition in the book "Kazhdan's Property (T)" by B. Bekka, P. de la Harpe ...

**0**

votes

**0**answers

52 views

### A distributional normal derivative for functions in $H^1(\Omega)$

Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this.
For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the ...

**5**

votes

**0**answers

133 views

### Presentation of Homotopy Pure Braid Group?

Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to self-intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group.
...

**2**

votes

**1**answer

192 views

### On a property of the Grothendieck group of a smooth projective curve

Let $K$ be a complete DVR of characteristic $0$, $X$ a smooth projective curve over $K$. Denote by $K^0(X)$ the Grothendieck group of locally free sheaves on $X$ and by $\mbox{det}$ the natural group ...

**3**

votes

**0**answers

109 views

### polynomials in many variables and Hasse principle

I was wondering whether there exists any result of the form
"if $f \in \mathbb{Z}[x_1, ..., x_k]$ is a polynomial (not form! I don't require homogeneity) of total degree $n$, with $k \geq \delta ...

**0**

votes

**0**answers

26 views

### Corestriction map for the relative homology (cohomology) group

Let $G$ be a group and $N$ be its normal subgroup. Is there any concept of corestriction map for the relative homology (cohomology) group $H_n(G,N,-)$ ($H^n(G,N,-)$) such that when $N=G$ it is the ...

**2**

votes

**2**answers

295 views

### Conjectured relation between alternating Prime zeta series and Riemann zeta

Let $P(s)$ be the Prime zeta function.
Numerical evidence suggests these identities:
$$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad ...

**7**

votes

**0**answers

106 views

### Excluding exotic PL structures on S^4

Suppose you have a finite group $G<SO(5)$ such that $S^4/G$ is homeomorphic to $S^4$ and such that $S^4/G$ is a PL manifold with respect to a PL structure induced by a standard structure on $S^4$. ...

**7**

votes

**2**answers

270 views

### When do you go hunting for Lagrangian submanifolds?

Similar to this question, I'm trying to figure out why one would be interested in Lagrangian submanifolds. But from a more geometric point of view. My best find so far is Exercise 12.4 in McDuff, ...

**0**

votes

**0**answers

28 views

### basis span of space necessary to be orthogonal? [on hold]

If a vector space V that span of {v1,v2,....,vk},can the basis vector of V are not mutually orthogonal?
For vector space V, are there exist set of orthogonal basis {o1,o2,....,ok} that can span V? ...

**2**

votes

**1**answer

55 views

### Does directional limits along any given direction, always exist for a function of bounded variation?

If a function $f:\mathbb{R}^N\to\mathbb{R}$ is of bounded variation (in modern sense or in Tonelli sense or according to any of the existing definitions), then can we say that, given any point $x\in ...

**0**

votes

**0**answers

98 views

### Descent datum for a line bundle

Let $\pi:C \to \mathbb P^1$ be a double cover branched at $r$ points. To understand the theory of descent better, I would like, if possible, to construct by hands the descent datum of a line bundle ...

**1**

vote

**1**answer

81 views

### Can we count isospectral graphs?

On n-vertices, how many isospectral graphs exist?
[..I saw this previous "historic" discussion between two of the stalwarts in this field - Operation on Isospectral graphs ]
Given a graph are ...

**0**

votes

**2**answers

150 views

### Gradient Ricci soliton

I am reading Cao and Chen's paper "On Bach-flat gradient shrinking Ricci solitons".
A complete Riemannian manifold $(M^n,g_{ij})$ is called a gradient shrinking Ricci soliton if there exists a smooth ...

**0**

votes

**0**answers

22 views

### Proving that some property on a chain complex of groups implies isomorphism between direct sums of these groups [on hold]

Let $C_*$ be a chain complex such that every $C_i$ is a torsion-free finitely generated abelian group, with $C_i=0$ for every $i<0$ and every $i>N$ for some sufficiently large integer $N$. If ...

**11**

votes

**2**answers

750 views

### Classification of rings satisfying $a^4=a$

We have the famous classification of rings satisfying $a^2=a$ (for each element $a$) in terms of Stone spaces, via $X \mapsto C(X,\mathbb{F}_2)$. Similarly, rings satisfying $a^3=a$ are classified by ...

**-2**

votes

**0**answers

38 views

### Circular variation with repetition formula [on hold]

I am looking for formula for circular variation with repetition. This mean :
You have 2 letter and you need to place it in corner of a square.
all variation ...

**2**

votes

**0**answers

132 views

### l-adic cohomology and perverse sheaves

Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial ...

**9**

votes

**1**answer

311 views

### Holomorphic cusp forms and cohomology of GL(2,Z)

Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute ...

**2**

votes

**0**answers

130 views

### What is the inverse kernel of this integral transform?

I am looking for the associated inverse kernel to the integral transform $T$ defined by
$(Tf)(u) = \int_{-\infty}^{+\infty} K(u,t)f(t) \ dt,\ \ u \in \mathbb{R^+}$
whose kernel is $K(u,t) = ...

**0**

votes

**0**answers

53 views

### Elementary Linear Algebra Maximization Problem [on hold]

Can someone show me the proof for the following:
min $\frac{q^TLq}{q^TWq}$ where q is not 0 and subject to $qWe =0$ is solved when
q is the eigenvector corresponding to the 2nd smallest eigenvalue ...

**1**

vote

**1**answer

107 views

### Tangent space of the Fourier algebra $A(G)$

Let $G$ be a real Lie group and $A(G)$ be its Fourier algebra. Let us call a linear continuous functional $f:A(G)\to{\mathbb C}$ a tangent vector of $A(G)$ in the point $a\in G$, if it satisfies the ...

**0**

votes

**0**answers

16 views

### Wavelet transform stability to deformations

I've come across the following claim in a paper of Mallat:
"High frequency instabilities [of a signal representation] to deformations can be avoided by grouping frequencies into dyadic packets in ...

**1**

vote

**0**answers

76 views

### Intuition for the structure theorem for connected solvable algebraic groups over an algebraically closed field of positive characteristic

In the theory of algebraic groups one of the fundamental results is the structure theorem for connected solvable groups. I never understood the proof in any of the standard textbooks, so I just moved ...

**1**

vote

**1**answer

64 views

### Equivalence of definitions of the Milnor $K$-groups

In Kurihara's paper: "The exponential homomorphisms for the Milnor $K$-groups and an explicit reciprocity law" he difines, in the first page, the $q$-th Milnor K-group for the ring $R$ as
...

**1**

vote

**1**answer

139 views

### Relative Compactness vs Way Below in Locally Compact Hausdorff Spaces

Let $Y$ be a subset of a locally compact Hausdorff topological space $X$ and consider the following properties.
$\overline{Y}$ is compact.
Every open cover of $X$ has a finite subcover of $Y$.
...

**0**

votes

**0**answers

46 views

### Check equality of sigma-algebras [on hold]

let $X$ be a set, $\mathcal{C}, \mathcal{E} \subseteq \mathcal{P}(X)$ two families of subsets and $\mathcal{A}$ a $\sigma$-algebra on $X$. Assume that $\mathcal{E} \subseteq \mathcal{C} \cap ...

**0**

votes

**0**answers

28 views

### Differential equation with fourier transform and convolution [on hold]

We have differential equation $3s(t)-2s''(t)=r(t)\,$ and $s(t)$ is convolution $s=g*r\,$ where $g(t)=ae^{-b\left | t \right |}\,$ $\\a,b\in\mathbb R+$ Solve constans a and b.
I tried to solve ...

**3**

votes

**1**answer

159 views

### Spec of an injective ring map contains minimal primes in its image?

Let $f\colon A \rightarrow B$ be an injective ring homomorphism. One knows (from EGA I, 1.2.7 or elsewhere) that the image of $\mathrm{Spec}(f)$ is dense. Does that image necessarily contain all the ...

**0**

votes

**0**answers

36 views

### Optimization related question [on hold]

I would like to find a complex symmetric Toeplitz matrix $C$ from (huge) set of the data { $y$ }, where every complex-scalar $y$ can be described by a following model which includes the matrix $C$ ...

**0**

votes

**1**answer

92 views

### Units of an extension of $\mathbb{Z}$ [on hold]

Let $P(x)\in\mathbb{Z}[x]$ be monic and irreducible over $\mathbb{Q}[x]$, and let $\theta$ be a root of $P(x)$. Let $K = \{a + b\theta\} \subseteq \mathbb{Z}[\theta]$. When is it the case that there ...

**1**

vote

**1**answer

48 views

### The uniform integrability of exponential of Poisson process

Let $\left\{N_t,\mathcal{F}_t\right\}_{t\ge0}$ be a Poisson process with intensity $\lambda>0$. Define
$$X_t=\exp{\left[N_t-\lambda t(e-1)\right]}$$
I can show that $\{X_t,\mathcal{F}_t\}_{t\ge0}$ ...

**2**

votes

**1**answer

168 views

### Is the centralizer $Z_G(A)=\{g\in G| a g= g a\}$ of a finite $A\subset G$ connected for a connected compact Lie group?

Let $G$ be a connected compact Lie group, consider the left/right action on itself.
For any finite $A\subset G$, consider the centralizer
$Z_G(A):=\{g\in G| a g= g a\}$.
Q: is $Z_G(A)$ a connected ...

**-1**

votes

**0**answers

60 views

### Why solving a system of linear equation produces the intersection of the equation [on hold]

Let us consider two equations
1)x+y=1
2)-x+y=1
Consider the solution of the equations using Gaussian Elimination,
Geometrically I am able to understand the ...

**1**

vote

**1**answer

149 views

### Basics on lattice in classical groups

as a beginner,I am not sure whether this question is too basic to post here./-\。
Many textbook will talk about the prototypical example SL(n,Z)\SL(n,R), which can be identified with the space of ...

**3**

votes

**1**answer

185 views

### Dirichlet polyhedra for hyperbolic manifolds

Let $H$ be a simply-connected, complete
space of constant negative curvature, that
is, a hyperbolic space, $\Gamma$ a discrete group of
isometries, and and $M=H/\Gamma$ its quotient space;
we assume ...

**1**

vote

**0**answers

70 views

### Morita Equivalence of Full Corners in $C^*$-algebras

Suppose $\mathcal{A}$ is a $C^*$-algebra with a unique normalized trace and $p \in \mathcal{A}$ is a projection so that $\mathcal{B} = p\mathcal{A}p$ is a full corner.
Does $\mathcal{B}$ have a ...

**0**

votes

**0**answers

104 views

### Non-linear diophantine equation [on hold]

I'm preparing a PHD research paper and I encounter this interesting problem.
Let $k$ and $n$ be positive integers.
Under which conditions is there a solution $(x,y)$ to the equation $y(n−x)=(k+nx)$ ...

**6**

votes

**0**answers

68 views

### CAT(0) groups that does not act on CAT(0) cubical complex

CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...

**-2**

votes

**0**answers

30 views

### How to solve this optimization problem with abs object function? [on hold]

Helo, every one. May I ask for help about how to solve this problem.
$\begin{align}
& \text{max}_{x_i} \quad |\sum_{i=1}^{4} a_i x_i | \\
& s.t. \quad \sum_{i=1}^4 x_i^2=1
\end{align}
$
...

**2**

votes

**0**answers

62 views

### Modular forms related to $G(q)$ and $H(q)$

If $G(q),H(q)$ are the functions appearing in Rogers-Ramanujan identities $$G(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(q;q)_{n}}=\prod_{n=1}^{\infty}\frac{1}{(1-q^{5n-1})(1-q^{5n-4})}$$ and ...

**1**

vote

**0**answers

47 views

### RKHS norm and posterior of Gaussian process

In Srinivas et al (2010) [appendix B], the authors claim the following "easy to see" property relating the norm of a function in a RKHS induced by a kernel $k(\cdot,\cdot)$, and its norm in the RKHS ...

**10**

votes

**2**answers

340 views

### Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line ...