Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric) wavelets generated by interpolatory refinable functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

向量值双正交小波的存在性及滤波器的构造

引进了向量值多分辨分析与向量值双正交小波的概念.讨论了向量值双正交小波的存在性.运用多分辨分析和矩阵理论,给出一类紧支撑向量值双正交小波滤波器的构造算法.最后,给出4-系数向量值双正交小波滤波器的的构造算例.     

On the computation of all supported efficient solutions in multi-objective integer network flow problems

This paper presents a new algorithm for identifying all supported non-dominated vectors (or outcomes) in the objective space, as well as the corresponding efficient solutions in the decision space, for multi-objective integer network flow problems. Identifying the set of supported non-dominated vectors is of the utmost importance for obtaining a first approximation of the whole set of non-dominated vectors. This approximation is crucial, for example, in two-phase methods that first compute the supported non-dominated vectors and then the unsupported non-dominated ones. Our approach is based on a negative-cycle algorithm used in single objective minimum cost flow problems, applied to a sequence of parametric problems. The proposed approach uses the connectedness property of the set of supported non-dominated vectors/efficient solutions to find all integer solutions in maximal non-dominated/efficient facets.     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

紧支撑样条小波插值及其应用

基于紧支撑样条小波函数插值与定积分的思想,给出了由紧支撑样条小波插值函数构造数值积分公式的方法．并将该方法应用于二次、三次、四次和五次紧支撑样条小波函数,得到了相应的数值积分公式．最后,通过数值例子验证,发现该方法得到的数值积分公式是准确的,且具有较高精度．     

Solving combinatorial optimization problems using Karmarkar's algorithm

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.Computational facilities provided by the Cornell Computational Optimization Project supported by NSF Grant DMS-8706133 and by the Cornell National Supercomputer Facility. The Cornell National Supercomputer Facility is a resource of the Center for Theory and Simulation in Science and Engineering at Cornell Unversity, which is funded in part by the National Science Foundation, New York State, and the IBM Corporation. The research of both authors was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by ONR Grant N00014-90-J-1714.Research partially supported by NSF Grant ECS-8602534 and by ONR Contract N00014-87-K-0212.     

Buckling mode localization in rib-stiffened plates with misplaced stiffeners – Kantorovich approach

Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners is studied in this paper. The method of Kantorovich on reducing a partial differential equation to a system of ordinary differential equations is employed to obtain the deflection surface of the rib-stiffened plates under axial compressive load. The edges of the plates normal to the stiffeners can be either simply supported or clamped. The solutions of the deflection surface are then expressed in the form of transfer matrices. The expressions of the solutions obtained for the case of one edge simply supported and one edge clamped and the case of two edges clamped are similar to those for the case of two edges simply supported. When the two edges are simply supported, the method of Kantorovich yields the exact results. Localization factors, which characterize the average exponential rates of growth or decay of amplitudes of deflection, are determined using the method of transfer matrix. The method of Kantorovich is a general approximate method, which is applicable for various support conditions.     

Existence and stability of large scale nonlinear oscillations in suspension bridges

A nonlinear model of a suspension bridge is considered in which large-scale, stable oscillatory motions can be produced by constant loading and a small-scale, external oscillatory force. Loud's implicit-function theoretic method for determining existence and stability of periodic solutions or nonlinear differential equations is extended to a case of a non-differentiable nonlinearity.Author partially supported by NSF under Grant DMS 8318204 and AFOSR Grant 85-0330.Author partially supported by NSF under Grant DMS 9519882.Author partially supported by NSF under Grant DMS 8519776.     

An inequality of Clifford indices for a finite covering of curves

We prove that for a finite covering of curves the Clifford index of the source is at least that of the target.Partially supported by MURST (Italy).Partially supported by JSPS.     

On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.     

A MUSIC-type algorithm for detecting internal corrosion from electrostatic boundary measurements

We establish an asymptotic representation formula for the steady state current perturbations caused by internal corrosive boundary parts of small surface measure. Based on this formula we design a non-iterative method of MUSIC (multiple signal classification) type for localizing the corrosive parts from voltage-to-current observations. We perform numerical experiments to test the viability of the algorithm and the results clearly demonstrate that the algorithm works well even in the presence of relatively high noise ratios. H. Ammari is partially supported by the Brain Pool Korea Program at Seoul National University, H. Kang is partially supported by KOSEF grant R01-2006-000-10002-0, E. Kim is supported by BK21 Math. Division at Seoul National University, and M.S. Vogelius is partially supported by NSF grant DMS-0604999.     

Stable mixed moving averages

Summary The class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable mixed moving averages. Their distribution determines a certain combination of the filter function and the mixing measure, leading to a generalization of a theorem of Kanter (1973) for usual moving averages. Stable mixed moving averages contain sums of independent stable moving averages, are ergodic and are not harmonizable. Also a class of stable mixed moving averages is constructed with the reflection positivity property.Research supported by AFSOR Contract 91-0030Research also supported by ARO DAAL-91-G-0176Research also supported by AFOSR 90-0168Research also supported by ONR N00014-91-J-0277     

The exponential growth of codimensions for Capelli identities

By applying the theorem that every positive integer is a sum of four squares, we calculate the exponential growth of the codimensions for the relatively free algebra satisfying Capelli identities. Work partially supported by RFFI grants 96-01-00146 and 98-01-01020. Work partially supported by ISF grant 6629/1. Work partially supported by RFFI grants 96-01-00146 and 96-15-96050.     

Stable theories without dense forking chains

Summary We define a generalized notion of rank for stable theories without dense forking chains, and use it to derive that every type is domination-equivalent to a finite product of regular types. We apply this to show that in a small theory admitting finite coding, no realisation of a nonforking extension of some strong type can be algebraic over some realisation of a forking extension.Research supported by the Deutsche ForschungsgemeinschaftThanks as always to Alistair. Research supported by NSERC and FCARPartially supported by NSF grant DMS90 06628     

Polynomial systems with few real zeroes

We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is ℤⁿ, this bound is 2n+1, while the Khovanskii bound is exponential in n². The bound 2n+1 can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic. Part of work done at MSRI was supported by NSF grant DMS-9810361. Work of Sottile is supported by the Clay Mathematical Institute. Sottile and Bihan were supported in part by NSF CAREER grant DMS-0134860. Bertrand is supported by the European research network IHP-RAAG contract HPRN-CT-2001-00271.     

Malnormality is undecidable in hyperbolic groups

In answer to a question of Myasnikov, we show that there exist hyperbolic groups for which there is no algorithm to decide which finitely generated subgroups are malnormal or quasiconvex. Bridson’s research is supported by an EPSRC Advanced Fellowship and Wise is supported in part by a grant from the NSF.     

The subgaussian constant and concentration inequalities

We study concentration inequalities for Lipschitz functions on graphs by estimating the optimal constant in exponential moments of subgaussian type. This is illustrated on various graphs and related to various graph constants. We also settle, in the affirmative, a question of Talagrand on a deviation inequality for the discrete cube. Research supported in part by NSF Grant No. DMS-0405587 and by EPSRC Visiting Fellowship. Research supported in part by NSF Grant No. DMS-9803239, DMS-0100289. Research supported in part by NSF Grant No. DMS-0401239.     

Linear orders with distinguished function symbol

We consider certain linear orders with a function on them, and discuss for which types of functions the resulting structure is or is not computably categorical. Particularly, we consider computable copies of the rationals with a fixed-point free automorphism, and also ω with a non-decreasing function. D. Cenzer was partially supported by National Science Foundation grants DMS 0532644 and 0554841 and 652372. B. Csima was partially supported by Canadian NSERC Discovery Grant 312501. B. Khoussainov has partially been supported by Marsden Fund of Royal New Zealand Society.     

Nonlinear diffusion and viral spread through the leaf of a plant

The spread of a virus through the leaf of a plant is both spatially and temporally causal in that the present status depends on the past and the spatial spread is compactly supported and progresses outwards. Such spatial spread is known to occur for certain nonlinear diffusion processes. The first compactly supported solution for nonlinear diffusion equations appears to be that of Pattle published in 1959. In that paper, no explanation is given as to how the solution was derived. Here, we show how the solution can be derived using Lie symmetry analysis. This lays a foundation for exploring the behavior of other choices for nonlinear diffusion and exploring the addition of reaction terms which do not eliminate the compactly supported structure. The implications associated with using the reaction–diffusion equation to model the spatial–temporal spread of a virus through the leaf of a plant are discussed.     

General interpolation on the lattice h{\Bbb Z}^s: Compactly supported fundamental solutions

Summary. In this paper we combine an earlier method developed with K. Jetter on general cardinal interpolation with constructions of compactly supported solutions for cardinal interpolation to gain compactly supported fundamental solutions for the general interpolation problem. The general interpolation problem admits the interpolation of the functional and derivative values under very weak restrictions on the derivatives to be interpolated. In the univariate case, some known general constructions of compactly supported fundamental solutions for cardinal interpolation are discussed together with algorithms for their construction that make use of MAPLE. Another construction based on finite decomposition and reconstruction for spline spaces is also provided. Ideas used in the latter construction are lifted to provide a general construction of compactly supported fundamental solutions for cardinal interpolation in the multivariate case. Examples are provided, several in the context of some general interpolation problem to illustrate how easy is the transition from cardinal interpolation to general interpolation. Received May 11, 1993 / Revised version received August 16, 1994