# All Questions

0answers
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 ...
0answers
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 ...
2answers
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?
0answers
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 ...
0answers
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 ...
2answers
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 ...
0answers
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 ...
0answers
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 ...
0answers
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 ...
0answers
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. ...
1answer
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 ...
0answers
109 views

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 $k \geq \delta ... 0answers 26 views ### Corestriction map for the relative homology (cohomology) group Let$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$it is the ... 2answers 295 views ### Conjectured relation between alternating Prime zeta series and Riemann zeta Let$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 ... 0answers 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. ... 2answers 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, ... 0answers 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? ... 1answer 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 ... 0answers 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 ... 1answer 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 ... 2answers 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 ... 0answers 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 ... 2answers 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 ... 0answers 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 ... 0answers 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 ... 1answer 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 ... 0answers 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) = ... 0answers 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 ... 1answer 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 ... 0answers 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 ... 0answers 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 ... 1answer 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 ... 1answer 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. ... 0answers 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 ... 0answers 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 ... 1answer 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 ... 0answers 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 ... 1answer 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 ... 1answer 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} ... 1answer 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 ... 0answers 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 ... 1answer 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 ... 1answer 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 ... 0answers 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 ... 0answers 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) ... 0answers 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 ... 0answers 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} ... 0answers 62 views ### Modular forms related to G(q) and H(q) If G(q),H(q) are the functions appearing in Rogers-Ramanujan identitiesG(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})}$and ... 0answers 47 views ### RKHS norm and posterior of Gaussian process In Srinivas et al (2010) [appendix B], the authors claim the following "easy to see" property relating the norm of a function in a RKHS induced by a kernel$k(\cdot,\cdot)$, and its norm in the RKHS ... 2answers 340 views ### Special values of$\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 ...

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