# All Questions

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### How to transform matrix to this form by unitary transformation?

Without loss of gernerality, we can only consider $n$-dimensional diagonal matrix $M$ whose elements are all nonnegative, i.e. $$M=\operatorname{diag}(m_1,m_2,\cdots,m_n)\ (m_i \geq 0).$$ Then is ...
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### What was the Question that led Euler to his Investigations on Polyhedra?

The question that led Euler to his investigations on graphs is the well-known question related to the seven bridges of Königsberg, and that story is a must in every introduction to graph theory. ...
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A Coalgebra $C$ is called metrizable if there is a base $B$ for $C$(as a vector space) and a metric $d:B \times B \to \mathbb{R}$ on $B$ such that the linear extension $\tilde{d}: C\otimes C ... 2answers 85 views ### l-functions of calabi-yau varieties This question might not be suitable for MO since i know nothing about Calabi-yau varieties aside the fact that they are used in string theory to compactify additional dimensions, but still, it makes ... 1answer 80 views ### Properties to have matrices that commute in$\mathrm{GL}_n(\mathbb C)$Let$G$be a finite subgroup of$\mathrm{GL}_n(\mathbb C)$,$A,B \in G$whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of$A$are included in a circle arc of ... 0answers 18 views ### Idempotent fractional ideals of a noetherian domain Let$R$be a commutative Noetherian domain,$K$its fraction field, and$J$a fractional ideal (i.e. a finitely generated sub-$R$-module of$K$) such that$J^2=J$. Is it true that$J=0$or$J=R$? If ... 0answers 40 views ### Parabolic bundles on elliptic curves as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ... 0answers 10 views ### Convex interaction energy Does anybody know examples of absolutely continuous probability measures$\mu_0,\mu_1$on$\mathbb{R}^nwith finite 2nd moments such that $$\frac{d^2}{dt^2}\left(\int_{\mathbb{R}^n\times ... 0answers 6 views ### TSP: Approximation Ratio of the Double Tree Heuristic after Diagonals have been Removed In their article "Double-Tree Approximations for the Metric TSP: Is the Best One Good Enough?", Vladimir Deineko and Alexander Tiskin derive a lower bound for the approximation ratio of the ... 1answer 119 views ### Generalization of notion of convexity I am searching for the correct term for the following, if it exists. A set X\subset \mathbb{R}^2 is called r-convex if for any two points x_1, x_2\in X such that there exists an arc of radius ... 1answer 84 views ### Theorem with an example i have this theorem in the paper they gives an example: but here H_1 is not satisfied ! How to correct it please? 0answers 190 views +100 ### Topological razors (ball-like spaces) Introduction Many admire the Euclidean space, and I am not an exception. I will try to catch the topological roundness of the n-ball in its greatest generality. I call the resulting axiomatized ... 0answers 49 views ### Prove that (AxB)∩(CxD)=(A∩C)x(B∩D) [on hold] Prove that (A\times B)\cap (C\times D) = (A\cap C)\times (B\cap D) where \times represents the Cartesian product. 2answers 335 views ### Trapping a convex body by a finite set of points In \mathbb{R}^n, let K be a convex body and T a finite set of points disjoint from the interior of K. Say that T traps K if there is no continuous motion of K carrying K arbitrarily ... 0answers 18 views ### What algebras does the hidden subgroup problem for finite abelian groups apply to? Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, of which factoring and the discrete logarithm problem for integers belong to. Apparently the HSP for finite ... 0answers 54 views ### How to show non-existence of elements in the intersection of two ideals? Given l, k any two natural numbers, define I_1 =\langle y^{l+2k}, (x+y)^{3l+2k} \rangle: x^{l+2k} + \langle y^{2l+3k}, (x+y)^{3l} \rangle: x^{l+2k}; I_2 = \langle x^{l+k} \rangle. I want to ... 1answer 106 views ### Entropy for Haar measure on O(n) Let G be a locally compact group. A measure \mu is the right-Haar measure on G if for every g\in G and E\subseteq G Borel set \mu(Eg)=\mu(E). It is known that every locally compact group ... 4answers 829 views ### When is the time evolution of a Hamiltonian system described by the geodesic flow on a Riemannian manifold? Here is my precise question. Let M, \omega be a symplectic manifold and let H: M \to \mathbb{R} be any smooth function. The symplectic form gives rise to an isomorphism between the tangent ... 1answer 181 views ### Why is proving C^{\infty} regularity of sub Riemannian geodesics so hard? In Montgomery's A Tour of Subriemannian Geometries, Their Geodesics and Applications, problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?". Can someone explain what are the major ... 0answers 17 views ### Quick question about conjugate equivalence bimodules and inner products let A and B be W^{*}-algebras, let X be an A-B-equivalence bimodule (according to the definition given in "Morita equivalence for C^*-algebras and W^*-algebras" by Rieffel, ... 1answer 62 views ### Clique factorization I'm reading about Clique factorization in wikipedia: http://en.wikipedia.org/wiki/Gibbs_random_field#Clique_factorization but I'm unable to understand this: What is X_C here? Ok I understood ... 0answers 48 views ### Diameter of n-unit-vector closed scribble Suppose one creates a random, closed, likely self-crossing polygon from n unit-length vectors arranged head-to-tail, randomly oriented except for the requirement that their sum is zero (so the ... 1answer 68 views ### Bound for the Frattini subgroup of a p-group Assume that G is a finite p-group, p odd, with a non-trivial elementary abelian Frattini subgroup. Then both \Phi(G) and G/ \Phi(G) are vector spaces over \mathbb{F}_p. Is it possible to ... 1answer 77 views ### Geometric Intuition of P^+ in Modular Tensor Categories I'm currently reading through Bakalov and Kirillov's "Lectures on Tensor Categories and Modular Functors," and I am having some difficulty understanding the definition of p^\pm given on page 49. ... 1answer 107 views ### Characterization(?) of coersive(?) elements in the special linear group Take your favorite matrix norm \|\bullet\| (my favorite is the Frobenius norm \|A\| = \sqrt{\operatorname{tr} A A^t}). Now consider the set S_x of matrices A, such that \|A\| < x and ... 1answer 118 views ### What is the Essential Reason that allows a PTAS for the EUCLIDEAN TSP? Questions: Is there some understanding of the reason, why the euclidean TSP allows a PTAS, whereas the metric TSP in general does not and, is the PTAS stable under sufficiently small perturbation ... 1answer 277 views ### Does there exist a countable partition of [0,1] by disjoint closed subsets? [duplicate] Possible Duplicate: Why are the integers with the cofinite topology not path-connected? As in the title, is it possible to find closed, disjoint subsets C_n of [0,1] such that [0,1] = ... 0answers 41 views ### Reference request: has this semilinear version of Navier Stokes been studied? I have noticed that the Navier Stokes equations can be written as a semilinear symmetric first order system$$ u_t+A_1u_{x_1}+A_2u_{x_2}+A_3u_{x_3} = f(u) for a 9 by 1 vector u containing the ... 1answer 278 views ### Topological Problems Solved by Lattice Duality It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ... 1answer 54 views ### submatrix of a given size with maximum frobenius norm Let I\subset \{1,2,\ldots,n\}, and let |I| denote its cardinality. Now given a Hermitian matrix \mathbf{A}\in\mathbf{C}^{n\times n}. I am interested in finding the subset I that maximizes the ... 0answers 90 views ### Where are there defined objects between gerbes and bundle gerbes? Consider a special kind of/something like a gerbe where there is first given local trivialization data with equivalence over 2-fold overlaps but not isomorphism. Does this exist in the literature? 1answer 108 views ### Subgradient of Minimum Eigenvalue Consider three N \times N Hermitian matrices A_0, A_1, A_2. Consider the function \begin{align} f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2) \end{align} where \lambda_{\text{min}} ... 1answer 95 views ### Transition probabilities in coupled CTMCs I know that for a CTMC, the probability of transition from time 0 to t is given by P(t)=e^{Q(t)t}. I have a system of N interdependent CTMCs evolving simultaneously. Each of the N CTMCs can ... 1answer 379 views ### On quantities with no very small odd prime factors; a response to Wlodzimierz Holsztynski In response to a comment posted under Powers of 2 and the products of initial odd primes , I shall raise some questions about quantities near O_n= P_{n+1}/2, the product of the first n odd ... 1answer 584 views ### Is compass and straight edge geometry complete? Euclid's first three postulates are the basis of compass and straight edge constructions which are as complex as arithmetic. The constructions themselves may be expressed as a formula with each of ... 5answers 267 views ### procedure-based (as opposed to definition-based) concepts Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated ... 1answer 91 views ### Sufficient conditions for equality of measures related to harmonic functions In Axler's book "Harmonic function theory" on this link http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Livres/Harmonic%20Function%20theory.pdf on page 112 ... 1answer 54 views ### Integral representation of the resolvent of a semigroup Let T(t) be a C_{0}-semigroup with the generator A. Now, does the so called integral representation of the resolvent (\lambda - A)^{-1} = \int_{0}^{\infty} e^{-t\lambda}T(t) dt $$hold for ... 1answer 66 views ### A question about uniformly bounded semigroups Let A be an unbounded linear operator of domain D(A) defined on a Banach space X. Suppose that A generates a C_0-semigroup T(t) which is uniformly bounded. I would like to know if there ... 1answer 148 views ### Initial ideal of k-th power of an ideal Hi, Let I be an ideal in a polynomial ring S = k[x_1, \ldots, x_n], where k is an algebraically closed field of characteristic zero. Fix a term order on S (e.g. a lexicographic order) and ... 2answers 124 views ### When distance nonincreasing map is an isometry Let f: M \to M be a distance nonincreasing map between a closed Riemannian manifold M and f is homotopic to the idendity map. Is it then f an isometry? 0answers 46 views ### Characteristic subgroups of the limit group Let \{ G_i \}_{i=1}^\infty be a direct spectrum of groups with respect to embeddings \varphi_i:G_i \mapsto G_{i+1}, i \in \mathbb{N}, and let G be the limit group of this spectrum. Suppose ... 1answer 764 views ### Invariance of dynamical system under a transformation I have come across an interesting property of a dynamical system, being transformed by a map, but i haven't been able to figure out why this is happening (for quite some time now actually). Any help ... 0answers 210 views ### What is the best lower bound for 3-sunflowers? A collection of t sets A_i is called a t-sunflower if A_i \cap A_j = Z for all i \neq j for some fixed Z. A well-known conjecture of Erdos and Rado says that in any k-uniform family of ... 2answers 255 views ### Subgroups of SL_3(\mathbb{Z}) that are finitely generated, Zariski-dense, infinite index, and torsion-free My question stems from Misha's answer of a MathOverflow question. Misha supplied the following question in his answer: Open question: Does there exist a finitely generated Zariski-dense ... 0answers 54 views ### Example of flasque but non-soft sheaves? Does anyone have an interesting examples of a flasque but not soft \mathscr{O}_X-module over a ringed space? Of course with X being non-paracompact. 4answers 365 views +100 ### Is the diameter of a centrally symmetric convex body realized by a pair of antipodal points? Let S \subset \mathbb{R}^n be the boundary of a centrally symmetric convex body and provide S with the geodesic metric given by its embedding in Euclidean space (i.e., the distance between two ... 0answers 56 views ### Fundamental theorem of calculus for iterated stochastic integrals I'm trying to find the rate (or a bound for it) with which an iterated integral of the type$$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$converges to zero (in probability/distribution) for$h ...
My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...
Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of ...