Finite Intervals in the Lattice of Topologies

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T ₀-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.     

Finite Intervals in the Lattice of Topologies

We prove a conjecture of Reinhold: that a finite lattice is isomorphic to an interval in the lattice of topologies on some set if and only if it is isomorphic to an interval in the lattice of topologies on a finite set.     

On the Lattice of Antichains of Finite Intervals

Motivated by applications to information retrieval, we study the lattice of antichains of finite intervals of a locally finite, totally ordered set. Intervals are ordered by reverse inclusion; the order between antichains is induced by the lower set they generate. We discuss in general properties of such antichain completions; in particular, their connection with Alexandrov completions. We prove the existence of a unique, irredundant ∧-representation by ∧-irreducible elements, which makes it possible to write the relative pseudo-complement in closed form. We also discuss in detail properties of additional interesting operators used in information retrieval. Finally, we give a formula for the rank of an element and for the height of the lattice.     

On the Green Function for a Complement of a Finite Number of Real Intervals

We give a new proof (and a generalization) of recent remarkable result by V. Totik concerning the smoothness properties of the equilibrium measure for compact subsets of the real line and the Green function for complement of these subsets taken with respect to the complex plane.     

Sojourn Time in an Union of Intervals for Diffusions

We give a method for computing the iterated Laplace transform of the sojourn time in an union of intervals for linear diffusion processes. This random variable comes from a model occurring in biology concerning the clustering of membrane receptors. The way used hinges on solving differential equations. We finally have a look on the particular case of Brownian motion and we provide a representation for the Laplace transform of its local time in a finite set.     

Diagonalization of the Finite Hilbert Transform on Two Adjacent Intervals

We continue the study of stability of solving the interior problem of tomography. The starting point is the Gelfand–Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function f along a collection of lines. Pick one such line, call it the x-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting f to the x-axis. Let \(I_1\) be the interval where f is supported, and \(I_2\) be the interval where the Hilbert transform of f can be computed using the Gelfand–Graev formula. The equation to be solved is \(\left. {\mathcal {H}}_1 f=g\right| _{I_2}\), where \({\mathcal {H}}_1\) is the FHT that integrates over \(I_1\) and gives the result on \(I_2\), i.e. \({\mathcal {H}}_1: L^2(I_1)\rightarrow L^2(I_2)\). In the case of complete data, \(I_1\subset I_2\), and the classical FHT inversion formula reconstructs f in a stable fashion. In the case of interior problem (i.e., when the tomographic data are truncated), \(I_1\) is no longer a subset of \(I_2\), and the inversion problems becomes severely unstable. By using a differential operator L that commutes with \({\mathcal {H}}_1\), one can obtain the singular value decomposition of \({\mathcal {H}}_1\). Then the rate of decay of singular values of \({\mathcal {H}}_1\) is the measure of instability of finding f. Depending on the available tomographic data, different relative positions of the intervals \(I_{1,2}\) are possible. The cases when \(I_1\) and \(I_2\) are at a positive distance from each other or when they overlap have been investigated already. It was shown that in both cases the spectrum of the operator \({\mathcal {H}}_1^*{\mathcal {H}}_1\) is discrete, and the asymptotics of its eigenvalues \(\sigma _n\) as \(n\rightarrow \infty \) has been obtained. In this paper we consider the case when the intervals \(I_1=(a_1,0)\) and \(I_2=(0,a_2)\) are adjacent. Here \(a_1 < 0 < a_2\). Using recent developments in the Titchmarsh–Weyl theory, we show that the operator L corresponding to two touching intervals has only continuous spectrum and obtain two isometric transformations \(U_1\), \(U_2\), such that \(U_2{\mathcal {H}}_1 U_1^*\) is the multiplication operator with the function \(\sigma (\lambda )\), \(\lambda \ge (a_1^2+a_2^2)/8\). Here \(\lambda \) is the spectral parameter. Then we show that \(\sigma (\lambda )\rightarrow 0\) as \(\lambda \rightarrow \infty \) exponentially fast. This implies that the problem of finding f is severely ill-posed. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators \(U_1\), \(U_2\) as \(\lambda \rightarrow \infty \). When the intervals are symmetric, i.e. \(-a_1=a_2\), the operators \(U_1\), \(U_2\) are obtained explicitly in terms of hypergeometric functions.     

Banach代数中有限序列的多项式谱(英文)

In this paper,we discuss that the polynomial spectrum,relative spectrum and Spectrum of a Pair of Elements are all compact,so their resolvent sets are all open.     

A Periodicity Theorem in K-Theory for Finite Covariant Systems

Let A be a complex Banach algebra. In this paper, we prove a periodicity theorem that K(S ² A, G, )K(A, G,), where (A,G,) is a finite covariant system and K(A,G,) is an Abelian group associated with (A,G,).     

A Note on Periodicity in Representation Theory of Finite Dimensional Algebras

Let Λ be an algebra over an algebraically closed field K of infinite transcendence degree over its prime subfield. We prove in particular that if every s-dimensional indecomposable nonprojective Λ-module is Ω-periodic (resp. DTr-periodic) then there exists a common bound for the Ω-periods (resp. DTr-periods) of s-dimensional indecomposable nonprojective Λ-modules.     

Quasiconvex Minimization on a Locally Finite Union of Convex Sets

Extending the approach initiated in Aussel and Hadjisavvas (SIAM J. Optim. 16:358–367, 2005) and Aussel and Ye (Optimization 55:433–457, 2006), we obtain the existence of a local minimizer of a quasiconvex function on the locally finite union of closed convex subsets of a Banach space. We apply the existence result to some difficult nonconvex optimization problems such as the disjunctive programming problem and the bilevel programming problem. Dedicated to Jean-Pierre Crouzeix on the occasion of his 65th birthday. The authors thank two anonymous referees for careful reading of the paper and helpful suggestions. The research of the second author was partially supported by NSERC/Canada.     

All Finite Distributive Lattices Occur as Intervals Between Hausdorff Topologies

It is shown that a finite lattice L is isomorphic to the interval betweentwo Hausdorff topologies on some set if and only if L is distributive. Thecorresponding results had previously been shown in ZFC for intervals between T₁ topologies and, assuming the existence of infinitely manymeasurable cardinals, for intervals between T₃ topologies.     

Bounds and Inequalities for General Orthogonal Polynomials on Finite Intervals

In this paper some characterizations of the regularity of (0, 2) interpolation are given.     

有限个真子空间的并的进一步探讨

无限域上的有限维线性空间不能由有限个真子空间的并来覆盖,这是高等代数中一个重要的结论.本文在规定"真子空间的个数是极小的"前提下给出此结论的两种简洁的证明方法;并给出线性空间的不属于任意给定的有限个真子空间的一组基;最后利用此结论给出两道研究生入学考试试题的证明.     

Laplacian Spectrum of Non-Commuting Graphs of Finite Groups

In this paper, we compute the Laplacian spectrum of non-commuting graphs of some classes of finite non-abelian groups. Our computations reveal that the non-commuting graphs of all the groups considered in this paper are L-integral. We also obtain some conditions on a group so that its non-commuting graph is L-integral.     

Lincoln-Petersen模型中群体大小的置信限和置信区间

对于Lincoln-Petersen模型,基于捕获-再捕获方法得到的数据,我们给出了群体大小的置信限和置信区间的有效计算方法.     

Normally Admissible Stratifications and Calculation of Normal Cones to a Finite Union of Polyhedral Sets

This paper considers computation of Fréchet and limiting normal cones to a finite union of polyhedra. To this aim, we introduce a new concept of normally admissible stratification which is convenient for calculations of such cones and provide its basic properties. We further derive formulas for the above mentioned cones and compare our approach to those already known in the literature. Finally, we apply this approach to a class of time dependent problems and provide an illustration on a special structure arising in delamination modeling.