lattice covering time in d dimensions theory and mean field approximation abstract the covering time is the mean time for a lattice random walk rw to visit all n n sites at least once in d 1 the problem reduces to a first visit problem and it has been exactly solved in contrast for d 2 this novel problem is not reducible to any of the well known lattice rw problems the theory of the d dimensional lattice covering time is presented and contrasted against those of other rw problems such as first visit trapping and the number of distinct sites visited by the walk also a mean field approximation of the covering problem that considers rws of infinite range is introduced

travelling wave solutions for a generalized double dispersion equation abstract we analyze the generalization of a double dispersion equation by means of the theory of symmetry reductions of partial differential equations we consider travelling wave reductions depending on the form of an arbitrary function we present some explicit solutions soliton solutions kinks and anti kinks

covering of a finite lattice by a random walk abstract we study the covering process by a simple random walk of a d dimensional periodic hypercubic lattice of n sites in d 1 the probability l n x for site x to be the last site visited in this covering process does not depend on x as long as x is not the starting point of the walk we argue that in dimensions d 2 the probability l n x approaches a constant value according to a coulomb law l n x 1 2 1 const x d 2 valid for x small on the scale n 1 d whereas it behaves logarithmically in d 2 also there is a dimension dependent characteristic time scale on which the last site is visited the structure of the set of sites not yet visited on this characteristic time scale is fractal like in d 2 in d 3 on the other hand this set is essentially distributed randomly through the lattice

the design of weighted minimax quadrature mirror filters a technique for converting the constrained nonlinear optimization problem encountered in the design of weighted minimax quadrature mirror filters into an iterative unconstrained nonlinear optimization problem is presented this renders the design of weighted minimax quadrature mirror filters possible the technique is very efficient typically taking about seven iterations to converge a rapidly converging iterative procedure for solving the above nonlinear unconstrained optimization problem is also presented this procedure typically requires less than five iterations to converge

symmetry reductions exact solutions and painleve analysis for a generalized boussinesq equation abstract in this paper new nonclassical symmetry reductions and exact solutions are presented fora generalised boussinesq equation uxxxx putuxx quxuxt ru2xuxx utt o which has the modified boussinesq equation q o r 1 2 p2 and dispersive water wave equation or classical boussinesq equations q 2p r 3 2 p2 as special cases these symmetry reductions are obtained using the direct method originally developed by clarkson and kruskal to study symmetry reductions of the boussinesq equation which involves no group theoretic techniques and using these reductions we obtain exact solutions expressible in terms of solutions of the second and fourth painleve equations jacobi and weierstrass elliptic functions and elementary functions for certain values of the parameters p q and r furthermore in the case when q p and r 1 2 p2 symmetry reductions are obtained which are reminiscent of reductions of the 2 1 dimensional cubic nonlinear schrodinger equation arising from the talanov lens transformation

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

how uniformly a random walker covers a finite lattice we study the distribution of the number of visits a random walker makes at a given site on a finite lattice with n sites during a very long walk which visits each site a large number of times for regular hypercubic lattices in all dimensions we find normal central limit behavior but with anomalously large variance in 2 dimensions in 2 dimensions the ratio of the variance over the average number of visits increases logarithmically with n while it increases n in one dimension we confront this with the case of self repelling also called true self avoiding walks there the variance remains bounded for all times and increases logarithmically with n in 2 dimensions random walks are so ubiquitous in physics and in science in general that one might think that the question how a random walk covers a finite lattice should be well studied it seems that this is not the case in a series of recent papers brummelhuis and hilhorst 1 and others 2 4 have studied a related problem but while they were interested in the distribution of times when the last still unvisited site of the lattice disappears we address in the present paper the behavior at much later times 5 and we ask a different question how large will be the asymptotic fluctuations in the numbers of visits to any given site the only papers where this was studied 6 as far as we are aware of seem obsolete since the authors assumed that the distribution of visits is multifractal it is clear that finally on a uniform lattice with periodic boundary conditions each site will be visited equally often more precisely let us denote by n x the number of times a site x was visited during t steps on a lattice of n sites and let us average over all possible start positions then obviously

a direct approach to the design of qmf banks via frequency domain optimization this paper studies the design of quadrature mirror filter qmf banks via frequency domain optimization a direct approach is adopted that gives the necessary and sufficient condition for perfect reconstruction pr while analysis filter banks are designed to achieve frequency domain specifications required for subband coding synthesis filter banks are designed to minimize the reconstruction error in frequency domain the criterion used to measure the reconstruction error is h sub spl infin or chebyshev norm sup norm state space solutions are derived for the h sub spl infin optimization and numerical algorithms are developed to obtain the optimal synthesis filter bank moreover the asymptotic pr property is established for optimal h sub spl infin solution of the synthesis filter bank

multicriterion optimized qmf bank design the quadrature mirror filter qmf bank with multicriterion constraints such as minimal aliasing and or minimal error coding is among the most important problems in filterbank design for solving which linear algebra based methods are still heuristic and do not always work especially for large filter length it is shown in this paper that this problem can be reduced either to convex linear matrix inequality lmi optimization when filters are of nonlinear phase or to semi infinite linear sip programming when filters are of linear phase which can be very efficiently solved either by the standard lmi solvers or our previously developed sip solver the proposed computationally tractable optimization formulations are confirmed by several simulations

multi solitonic solutions for the variable coefficient variant boussinesq model of the nonlinear water waves abstract for the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth of the water we consider a variable coefficient variant boussinesq vcvb model with symbolic computation we construct the connection between the vcvb model and a variable coefficient ablowitz kaup newell segur vcakns system under certain constraints using the n fold darboux transformation of the vcakns system we present two sets of multi solitonic solutions for the vcvb model which are expressed in terms of the vandermonde like and double wronskian determinants respectively dynamics of those solutions are analyzed and graphically discussed such as the parallel solitonic waves shape changing collision head on collision fusion fission behavior and elastic fusion coupled interaction

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

a mixed norm performance measure for the design of multirate filterbanks a mixed norm performance measure is presented to design the synthesis filters of a multirate filterbank the mixed norm performance measure is based on the energy as well as the peak value of the error signal mathematically the performance measure minimizes the l sub 2 norm of the error signal subject to the l sub spl infin norm of the error being bounded by some positive value v this imposes a bound on the peak value of the error signal the design problem is shown to be that of a mixed spl hscr sub 2 spl hscr sub spl infin optimization problem the theory of linear matrix inequalities lmis offers a tractable solution to such multiobjective synthesis problems the synthesis filters designed with the new performance measure are compared with those obtained by similar induced norm minimization techniques in terms of degree of reconstruction order of the synthesis filters snr and aliasing distortion

design of multirate filter banks by spl hscr sub spl infin optimization a procedure is developed to design iir synthesis filters in a multirate filter bank the filters minimize the l sub 2 induced norm of the error system between the multirate filter bank and a desired pure time delay system this criterion is reduced to one of spl hscr sub spl infin optimization for which there is ready made software

linear periodic systems and multirate filter design we present a systematic procedure for the design of filters intended for multirate systems this procedure is motivated by viewing the equiripple design of filters in linear time invariant systems as a process of obtaining optimum minimax filters for a class of bounded energy input signals the philosophy of designing optimum minimax filters for classes of input signals is extended to multirate systems which are not time invariant we develop a generalized fourier analysis appropriate for linear periodic systems and use it to derive new error criteria for multirate filter design using such criteria yields optimum minimax multirate filters for the input signal class the utility of our method is demonstrated by using it to analyze several multirate systems we give numerical results on the design of a multirate implementation of a narrowband filter and compare our work to previous work on multirate filter design our numerical analysis is based upon a new formulation of the design as a semi infinite linear programming problem

n fold darboux transformation and solitonic interactions of a variable coefficient generalized boussinesq system in shallow water abstract under consideration in this paper is a variable coefficient generalized boussinesq system for the long weakly nonlinear and weakly dispersive surface waves in shallow water with the aid of symbolic computation n fold darboux transformation n dt is constructed for that system analytic solutions of the system are obtained via the n dt elastic interactions of three bell shaped and periodic bell shaped solitons are obtained fusion interactions and periodic fusion fission interactions of the solitary waves are graphically analyzed which are inelastic

on self repelling walks the authors investigate the properties of self repelling walks otherwise known as true self avoiding walks in both one and two dimensions for a range of values of the repulsion parameter g 0 2 or g or 10 0 in one dimension they have obtained 24 terms of the generating function of the mean square end to end distance rn2 while on the two dimensional square lattice they have obtained 12 15 terms in one dimension they find the data to be well fitted by rn2 n43 a b n13 c n o 1 n and in two dimensions by rn2 dn mod lnn mod alpha 1 0 ln mod lnn mod lnn with alpha approximately 0 5 estimates of the amplitudes a and d are also obtained

optimal approximate inverse of linear periodic filters we propose a method for constructing optimal causal approximate inverse for discrete time single input single output siso causal periodic filters in the presence of measurement noise the analysis is based on block signals and multi input multi output mimo time invariant models for periodic filters the objective function to be minimized is the asymptotic block mean square error the optimization problem is formulated in terms of transfer matrices as an optimal model matching problem with nonsquare model and plant based on an inner outer factorization on the transpose of the plant rational matrix it is shown that the problem can be further reduced to one with a lower dimensional square model and plant which is then solved in the time domain and a closed form solution is obtained a lower bound on the objective function is given it is shown that the lower bound can be asymptotically achieved as the order of the optimal transfer matrix increases the proposed method is extended to mimo periodic systems numerical examples are used to illustrate the performance of the proposed approximate inverse

new solutions for solving boussinesq equation via potential symmetries method this work deals with the boussinesq equation that describes the propagation of the solitary waves with small amplitude on the surface of shallow water firstly the equation is written in a conserved form a potential function is then assumed reducing it to a system of partial differential equations the lie group method has been applied for determining symmetry reductions of the system of partial differential equations the solution of the problem by means of lie group method reduces the number of independent variables in the given partial differential equation by one leading to nonlinear ordinary differential equations the resulting non linear ordinary differential equations are then solved numerically using matlap package

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

how a random walk covers a finite lattice abstract a random walker is confined to a finite periodic d dimensional lattice of n initially white sites when visited by the walk a site is colored black after t steps of the walk for t scaled appropriately with n we determined the structure of the set of white sites the variance of their number has a line of critical points in the td plane which separates a mean field region from a region with enhanced fluctuations at d 2 the critical point becomes a critical interval moreover for d 2 the set of white sites is fractal with a fractal dimensionality whose t dependence we determine

on the time taken by random walks on finite groups to visit every state consider a random walk on a finite group g suppose that the time taken for the transition probabilities to approach the uniform distribution is small compared with g then the time taken for the random walk to visit every state is approximately r g log g where r is the mean number of returns to the initial state in the short term

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

discrete wavelet transform based on cyclic convolutions a filterbank in which cyclic convolutions are used in place of linear convolutions will be referred to as a cyclic convolution filterbank ccfb a dyadic tree structured ccfb can be used to perform a discrete wavelet transform suitable for coding based on symmetric extension methods this paper derives two types of efficient implementation techniques for the tree structured ccfb one using complex arithmetic and one using only real arithmetic in addition the present paper analyzes in detail the perfect reconstruction pr condition for the two channel ccfb and shows that this condition is much less restrictive than that of usual two channel filterbanks when each of the two channel ccfbs constituting a tree structured ccfb is designed to be pr the whole system is pr a quadrature mirror filter qmf ccfb having the pr property is demonstrated to be easily designed using a standard filter design subroutine in contrast designing a linear phase pr fir qmf bank that has good frequency response is not possible when filters are realized by linear convolutions

multiple soliton solutions for the ninth order kdv equation and sixth order boussinesq equation abstract two nonlinear dispersive equations namely the ninth order kdv equation and the sixth order boussinesq equation are formally derived by generalizing the bilinear forms of the kdv and boussinesq equations respectively the two equations are approached by using the tanh coth method to obtain single soliton solutions and by the hirota bilinear method to determine the multiple soliton solutions the study highlights the fact that both equations are completely integrable and admits n soliton solutions