Strong bounds with cut and column generation for?class-teacher timetabling

This work presents an integer programming formulation for a variant of the Class-Teacher Timetabling problem, which considers the satisfaction of teacher preferences and also the proper distribution of lessons throughout the week. The formulation contains a very large number of variables and is enhanced by cuts. Therefore, a cut and column generation algorithm to solve its linear relaxation is provided. The lower bounds obtained are very good, allowing us to prove the optimality of previously known solutions in three formerly open instances.     

Almost tight bounds forɛ-Nets

Given any natural numberd, 0<<1, letf _d () denote the smallest integerf such that every range space of Vapnik-Chervonenkis dimensiond has an-net of size at mostf. We solve a problem of Haussler and Welzl by showing that ifd2, then

The Darling-Erdős theorem for sums of I.I.D. random variables

Summary We obtain a general Darling-Erds type theorem for the maximum of appropriately normalized sums of i.i.d. mean zero r.v.'s with finite variances. We infer that the Darling-Erds theorem holds in its classical formulation if and only ifE[X ² 1 {|X|t}]=o((loglogt)^-1) ast. Our method is based on an extension of the truncation techniques of Feller (1946) to non-symmetric r.v.'s. As a by-product we are able to reprove fundamental results of Feller (1946) dealing with lower and upper classes in the Hartman-Wintner LIL.     

Turán-type bounds for distance graphs

A lower bound is obtained for the number of edges in a distance graph G in an infinitesimal plane layer ?² × [0, ε] ^d , which relates the number of edges e(G), the number of vertices ν(G), and the independence number α(G). It is proved that \(e\left( G \right) \geqslant \frac{{19\nu \left( G \right) - 50\alpha \left( G \right)}}{3}\). This result generalizes a previous bound for distance graphs in the plane. It substantially improves Turán’s bound in the case where \(\frac{1}{5} \leqslant \frac{{\alpha \left( G \right)}}{{\nu \left( G \right)}} \leqslant \frac{2}{7}\).     

Combined perturbation bounds: Ⅱ. Polar decompositions

In this paper, we study the perturbation bounds for the polar decomposition A= QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ2r||△Q||2F ≤ ||△A||2F,1/2||△H||2F ≤ ||△A||2F and ||△∑||2F ≤ ||△A||2F, respectively, where ∑ = diag(σ1, σ2,..., σr, 0,..., 0) is the singular value matrix of A and σr denotes the smallest nonzero singular value. Here we present some new combined (asymptotic)perturbation bounds σ2r ||△Q||2F 1/2||△H||2F≤ ||△A||2F and σ2r||△Q||2F ||△∑ ||2F ≤||△A||2F which are optimal for each factor. Some corresponding absolute perturbation bounds are also given.     

Approximation and contamination bounds for probabilistic programs

In many applications of manufacturing and service industries, the quality of a process is characterized by the functional relationship between a response variable and one or more explanatory variables. Profile monitoring is for checking the stability of this relationship over time. In some situations, multiple profiles are required in order to model the quality of a product or process effectively. General multivariate linear profile monitoring is particularly useful in practice due to its simplicity and flexibility. However, in such situations, the existing parametric profile monitoring methods suffer from a drawback in that when the profile parameter dimensionality is large, the detection ability of the procedures commonly used T ²-type charting statistics is likely to decline substantially. Moreover, it is also challenging to isolate the type of profile parameter change in such high-dimensional circumstances. These issues actually inherit from those of the conventional multivariate control charts. To resolve these issues, this paper develops a new methodology for monitoring general multivariate linear profiles, including the regression coefficients and profile variation. After examining the connection between the parametric profile monitoring and multivariate statistical process control, we propose to apply a variable-selection-based multivariate control scheme to the transformations of estimated profile parameters. Our proposed control chart is capable of determining the shift direction automatically based on observed profile data. Thus, it offers a balanced protection against various profile shifts. Moreover, the proposed control chart provides an easy but quite effective diagnostic aid. A real-data example from the logistics service shows that it performs quite well in the application.     

Logarithmic lower bounds for Néel walls

Most mathematical models for interfaces and transition layers in materials science exhibit sharply localized and rapidly decaying transition profiles. We show that this behavior can largely change when non-local interactions dominate and internal length scales fail to be determined by dimensional analysis: we consider a reduced model for Néel walls, micromagnetic transition layers which are observed in a suitable thin-film regime. The typical phenomenon associated with this wall type is the very long logarithmic tail of transition profiles. Recently, we derived logarithmic upper bounds. Here, we prove that the latter result is indeed optimal. In particular, we show that Néel wall profiles are supported by explicitly known comparison profiles that minimize relaxed variational principles and exhibit logarithmic decay behavior. This lower bound is established by a comparison argument based on a global maximum principle for the non-local field operator and the qualitative decay behavior of comparison profiles.Received: 17 June 2003, Accepted: 18 November 2003, Published online: 25 February 2004Mathematics Subject Classification (2000): 78A30, 49S05, 45G15, 35B25     

Combined perturbation bounds:Ⅱ.Polar decompositions

In this paper,we study the perturbation bounds for the polar decomposition A=QH where Q is unitary and H is Hermitian.The optimal (asymptotic) bounds obtained in previous works for the unitary factor,the Hermitian factor and singular values of A areσ_r~2||ΔQ||_F~2≤||ΔA||_F~2, 1/2||ΔH||_F~2≤||ΔA||_F~2 and ||Δ∑||_F~2≤||ΔA||_F~2,respectively,where∑=diag(σ_1,σ_2,...,σ_r,0,...,0) is the singular value matrix of A andσ_r denotes the smallest nonzero singular value.Here we present some new combined (asymptotic) perturbation boundsσ_r~2||ΔQ||_F~2 1/2||ΔH||_F~2≤||ΔA||_F~2 andσ_r~2||ΔQ||_F~2 ||Δ∑||_F~2≤||ΔA||_F~2 which are optimal for each factor.Some corresponding absolute perturbation bounds are also given.     

Are there elimination algorithms for the permanent?

We define the class of elimination algorithms. There are algebraic algorithms for evaluating multivariate polynomials, and include as a special case Gaussian elimination for evaluating the determinant. We show how to find the linear symmetries of a polynomial, defined appropriately, and use these methods to find the linear symmetries of the permanent and determinant. We show that in contrast to the Gaussian elimination algorithm for the determinant, there is no elimination algorithm for the permanent.     

Berry-Esseen bounds and Cramér type large deviations for eigenvalues of random matrices.

We establish Berry-Esseen bounds and Cramér type large deviations for the eigenvalues of Wigner Hermitian matrices in the bulk and at the edge cases. Similar results are also obtained for covariance matrices.     

Sharp bounds for multilinear Hausdorff operators北大核心CSCD

We give the necessary conditions of boundedness of multilinear Hausdorff operators on Lebesgue spaces and λ-central Morrey spaces, respectively, when the kernel functions are nonnegative. Meanwhile, the corresponding operator norms are worked out. ©, 2015, Chinese Academy of Sciences. All right reserved.     

A-posteriori residual bounds for Arnoldi’s methods for nonsymmetric eigenvalue problems

Convergence of the implicitly restarted Arnoldi (IRA) method for nonsymmetric eigenvalue problems has often been studied by deriving bounds for the angle between a desired eigenvector and the Krylov projection subspace. Bounds for residual norms of approximate eigenvectors have been less studied and this paper derives a new a-posteriori residual bound for nonsymmetric matrices with simple eigenvalues. The residual vector is shown to be a linear combination of exact eigenvectors and a residual bound is obtained as the sum of the magnitudes of the coefficients of the eigenvectors. We numerically illustrate that the convergence of the residual norm to zero is governed by a scalar term, namely the last element of the wanted eigenvector of the projected matrix. Both cases of convergence and non-convergence are illustrated and this validates our theoretical results. We derive an analogous result for implicitly restarted refined Arnoldi (IRRA) and for this algorithm, we numerically illustrate that convergence is governed by two scalar terms appearing in the linear combination which drives the residual norm to zero. We provide a set of numerical results that validate the residual bounds for both variants of Arnoldi methods.     

Improved bounds on weak ε-nets for convex sets

LetS be a set ofn points in ℝ ^d . A setW is aweak ε-net for (convex ranges of)S if, for anyT⊆S containing εn points, the convex hull ofT intersectsW. We show the existence of weak ε-nets of size , whereβ ₂=0,β ₃=1, andβ _d ≈0.149·2 ^d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/ε) log^1.6(1/ε)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/ε), which improves a previous bound of Capoyleas. Work by Bernard Chazelle has been supported by NSF Grant CCR-90-02352 and the Geometry Center. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-89-21421. Work by Michelangelo Grigni has been supported by NSERC Operating Grants and NSF Grant DMS-9206251. Work by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Emo Welzl and Micha Sharir has been supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Work by Micha Sharir has also been supported by NSF Grant CCR-91-22103, and by a grant from the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Berry–Esseen bounds for von Mises and U-statistics

Let X ₁,... ,X _n be i.i.d.\ random variables. An optimal Berry–Esseen bound is derived for U-statistics of order 2, that is, statistics of the form _{j> k} H (X _j , X _k ), where H is a measurable, symmetric function such that E | H (X₁, X₂)| > , assuming that the statistic is non-degenerate. The same is done for von Mises statistics, that is, statistics of the form _j,k H (X _j , X _k ). As a corollary, the central limit theorem is derived under optimal moment conditions.