the minimum speed for a blocking problem on the half plane we consider a blocking problem fire propagates on a half plane with unit speed in all directions to block it a barrier can be constructed in real time at speed we prove that the fire can be entirely blocked by the wall in finite time if and only if 1 the proof relies on a geometric lemma of independent interest namely let k r2 be a compact simply connected set with smooth boundary we define dk x y as the minimum length among all paths connecting x with y and remaining inside k then dk attains its maximum at a pair of points x y both on the boundary of k

optimal quantizer design for noisy channels an approach to combined source channel coding we present an analysis of the zero memory quantization of memoryless sources when the quantizer output is to be encoded and transmitted across a noisy channel necessary conditions for the joint optimization of the quantizer and the encoder decoder pair are presented and an iterative algorithm for obtaining a locally optimum system is developed the performance of this locally optimal system obtained for the class of generalized gaussian distributions and the binary symmetric channel is compared against the optimum performance theoretically attainable using rate distortion theoretic arguments as well as against the performance of lloyd max quantizers encoded using the natural binary code and the folded binary code it is shown that this optimal design could result in substantial performance improvements the performance improvements are more noticeable at high bit rates and for broad tailed densities

solitons in a nonlinear model medium the periodic stationary solutions of a model nonlinear evolution equation simulating the propagation of short wave perturbations in a relaxing medium are studied solutions expressed by a multiple valued function are shown to exist a method for determining the nonlinear interaction between solitary waves is suggested an example of a collision of solitons is given

index assignment for multichannel communication under failure we consider the problem of constructing multiple description scalar quantizers and describing the achievable rate distortion tuples in that setting we model this as a combinatorial optimization problem of number arrangements in a matrix this approach gives a general technique for deriving lower bounds on the distortion at given channel rates this technique is constructive thus allowing an algorithm that gives an upper bound for the case of two communication channels with equal rates the bounds coincide thus giving the precise lowest achievable distortion at fixed rates the bounds are within a small constant for a higher number of channels to the best of our knowledge this is the first result involving systems with more than two communication channels

existence of optimal strategies for a fire confinement problem we consider a class of variational problems for differential inclusions related to the control of forest fires the area burned by the fire at time t 0is modeled as the reachable set for a differential inclusion p x 2 f x starting from an initial setr0 to block the fire a barrier can be constructed progressively in time at a given speed in this paper we prove the existence of an optimal strategy which minimizes the value of the area destroyed by the fire plus the cost of constructing the barrier 2008 wiley periodicals inc

degenerate form of a three component degasperis procesi equation abstract we discuss the short wave limit of a new three component degasperis procesi equation and construct infinitely many conserved quantities for the degenerate system we study the reductions for a spectral problem of the degenerate system which are shown to contain spectral problems of some famous or new systems found recently

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

on the bandwidth of a hamming graph the bandwidth of the hamming graph the product kn d of complete graphs has been an open question for many years recently berger wolf and rheingold 1 pointed out that the bandwidth of a numbering of the hamming graph may be interpreted as a measure of the effects of noise in the multi channel transmission of data with that numbering they also gave lower and upper bounds for it in this paper we present better lower and upper bounds showing that the bandwidth of kn d is asymptotic to 2 d nd as d

local well posedness and blow up criteria for a three component camassa holm type equation considered herein is a three component camassa holm type equation that admits a bi hamiltonian structure and infinitely many conserved quantities in this paper we establish the local well posedness of this system in bp rs we also deduce two blow up criteria for this system

the rate distortion region for multiple descriptions without excess rate during recent years there has been strong interest in a certain source coding problem which some authors call the problem of multiple descriptions old and new wringing techniques enable us to establish a single letter characterization of the rate distortion region in the case of no excess rate for the joint description

alphabet constrained data compression the optimal data compression problem is posed in terms of an alphabet constraint rather than an entropy constraint solving the optimal alphabet constrained data compression problem yields explicit source encoder decoder designs which is in sharp contrast to other approaches the alphabet constrained approach is shown to have the additional advantages that 1 classical waveform encoding schemes such as pulse code modulation pcm differential pulse code modulation dpcm and delta modulation dm as well as rate distortion theory motivated tree trellis coders fit within this theory 2 the concept of preposterior analysis in data compression is introduced yielding a rich new class of coders and 3 it provides a conceptual framework for the design of joint source channel coders for noisy channel applications examples are presented of single path differential encoding delayed or tree encoding preposterior analysis and source coding over noisy channels

optimum quantizer performance for a class of non gaussian memoryless sources the performance of optimum quantizers subject to an entropy constraint is studied for a wide class of memoryless sources for a general distortion criterion necessary conditions are developed for optimality and a recursive algorithm is described for obtaining the optimum quantizer under a mean square error criterion the performance of entropy encoded uniform quantization of memoryless gaussian sources is well known to be within 0 255 bits sample of the rate distortion bound at relatively high rates despite claims to the contrary it is demonstrated that similar performance can be expected for a wide range of memoryless sources indeed for the cases considered the worst case performance is observed to be less than 0 3 bits sample from the rate distortion bound and in most cases this disparity is less at iow rates

minimum breakdown degradation in binary source encoding a memoryless binary equiprobable source produces one letter per second two people each are provided separately with private information about the source data at a rate of 1 2 bit per second suppose that by pooling their information they can produce a long mn reconstruction of the source output that has arbitrarily small error frequency we prove that then the least common asymptotic error frequency d that each can achieve without the other s help is sqrt 2 1 2 0 207 since it had been shown previously that 0 200 leq d leq 0 207 our result closes the so called 007 gap new analytical techniques introduced to effect the proof are of broader significance in multiuser information theory

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

new results in binary multiple descriptions an encoder whose input is a binary equiprobable memoryless source produces one output of rate r 1 and another of rate r 2 let d 1 d 2 and d 0 respectively denote the average error frequencies with which the source data can be reproduced on the basis of the encoder output of rate r l only the encoder output of rate r 2 only and both encoder outputs the two descriptions problem is to determine the region r of all quintuples r 1 r 2 d 1 d 2 d 0 that are achievable in thc usual shannon sense let r d 1 d log 2 d 1 d log 2 1 d denote the error frequency rate distortion function of the source the no excess rate case prevails when r 1 r 2 r d 0 and the excess rate case when r 1 r 2 r d 0 denote the section of r at r 1 r 2 d 0 by d r 1 r 2 d 0 d 1 d 2 r 1 r 2 d 1 d 2 d 0 in r in the no excess rate case we show that a portion of the boundary of d r 1 r 2 d 0 coincides with the curve frac 1 2 d 1 2d 0 frac 1 2 d 2 2d 0 frac 1 2 1 2d 0 2 this curve is an extension of witsenhausen s hyperbola bound to the case d 0 0 it follows that the projection of r onto the d 1 d 2 plane at fixed d 0 consists of all d 1 geq d 0 and d 2 geq d 0 that lie on or above this hyperbola in the excess rate case we show by counterexample that the achievable region of el gamal and cover is not tight

index assignment for two channel quantization this paper concerns the design of a multiple description scalar quantization mdsq system for two identical channels for an unbounded discrete information source this translates to the combinatorial problem of finding an arrangement of the integers into the infinite plane square grid so that each row and each column contains exactly n numbers such that the difference between any two numbers in the same row or column is at most d with d to be minimized for a given n the best previous lower and upper bounds on the lowest d were n sup 2 3 o n and n sup 2 2 o n we give new lower and upper bounds both of the form 3n sup 2 8 o n we also consider minimizing the maximal variance in any row or column and show that it must be at least n sup 4 60 o n sup 3 and that it does not have to be more than 3n sup 4 160 o n sup 3

on the bandwidth of 3 dimensional hamming graphs this paper presents strategies for improving the known upper and lower bounds for the bandwidth of hamming graphs k n d and 0 1 d our labeling strategy lowers the upper bound on the bandwidth of the continuous hamming graph 0 1 3 from 5 to 4497 a lower bound of 4439 on bw 0 1 3 follows from known isoperimetric inequalities and a related dynamic program is conjectured to raise that lower bound to 4 9 4444 in particular showing the power of our method we prove that the bandwidth of k 6xk 6xk 6 is exactly 101

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

differential inclusions and the control of forest fires abstract this paper introduces a new class of variational problems for differential inclusions motivated by the control of forest fires the area burned by the fire at time t 0 is modelled as the reachable set for a differential inclusion x f x starting from an initial set r 0 to block the fire a wall can be constructed progressively in time at a given speed in this paper we study the possibility of constructing a wall which completely encircles the fire moreover we derive necessary conditions for an optimal strategy which minimizes the total area burned by the fire

achievable rates for multiple descriptions consider a sequence of independent identically distributed i i d random variables x l x 2 cdots x n and a distortion measure d x i x i on the estimates x i of x i two descriptions i x in 1 2 cdots 2 nr 1 and j x in 1 2 cdots 2 nr 2 are given of the sequence x x 1 x 2 cdots x n from these two descriptions three estimates i x x2 j x and hat x o i x j x are formed with resulting expected distortions e frac 1 n sum n k 1 d x k hat x mk d m m 0 1 2 we find that the distortion constraints d 0 d 1 d 2 are achievable if there exists a probability mass distribution p x p hat x 1 hat x 2 hat x 0 x with ed x hat x m leq d m such that r 1 i x hat x 1 r 2 i x hat x 2 where i cdot denotes shannon mutual information these rates are shown to be optimal for deterministic distortion measures

hamilton jacobi equations with obstacles we consider a problem in the theory of optimal control proposed for the first time by bressan we characterize the associated minimum time function using tools from geometric measure theory and we obtain as a corollary an existence theorem for a related variational problem

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem

perturbative symmetry approach the aim of our paper is to formulate a perturbative version of the symmetry approach in the symbolic representation and to generalize it in order to make it suitable for the study of nonlocal and non evolution equations our formalism is the development and incorporation of the perturbative approach of zakharov and schulman the symbolic method of sanders and wang and the standard symmetry approach of shabat et al we apply our theory to describe integrable generalizations of the benjamin ono type equations and to isolate integrable cases of the camassa holm type equations