optimal binary index assignments for a class of equiprobable scalar and vector quantizers the problem of scalar and vector quantization in conjunction with a noisy binary symmetric channel is considered the issue is the assignment of the shortest possible distinct binary sequences to quantization levels or vectors so as to minimize the mean squared error caused by channel errors by formulating the assignment as a matrix or vector in the scalar case and showing that the mean squared error due to channel errors is determined by the projections of its columns onto the eigenspaces of the multidimensional channel transition matrix a class of source quantizer pairs is identified for which the optimal index assignment has a simple and natural form among other things this provides a simpler and more accessible proof of the result of crimmins et al 1969 that the natural binary code is an optimal index assignment for the uniform scalar quantizer and uniform source it also provides a potentially useful approach to further developments in source channel coding

design and analysis of optimal noisy channel quantization with random index assignment this paper studies the design of vector quantization on noisy channels and its high rate asymptotic performance given a tandem source channel coding system with vector quantization block channel coding and random index assignment a closed form formula is first derived for computing the average end to end distortion eed of the system which reveals a structural factor called the scatter factor of a noisy channel quantizer based on this formula we propose a noisy channel quantization design method by minimizing the eed experiments and simulations show that quantizers jointly designed with channel conditions significantly reduce the eed when compared with quantizers designed separately without reference to channel conditions which reveals a practical and effective design for noisy channel quantization as to simplify the channel model by considering a random index assignment furthermore we have presented the high rate asymptotic analysis of the eed for the tandem system while convergence analysis of the iterative algorithm is included in the appendix

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

boundary behaviour of the unique solution to a singular dirichlet problem with a convection term by karamata regular variation theory and constructing comparison functions we derive that the boundary behaviour of the unique solution to a singular dirichlet problem u b x g u u q u 0 x u 0 which is independent of u q w here is a bounded domain with smooth boundary in r n r q 0 2 lims 0 g s a ndb is nonnegative on which may be vanishing on the boundary

multi parameter bifurcation and asymptotics for the singular lane emden fowler equation with a convection term to the second variable and g is unbounded around the origin the asymptotic behaviour of the solution around the bifurcation point is also established provided g u behaves like u around the origin for some 0 1 our approach relies on finding explicit sub and supersolutions combined with various techniques related to the maximum principle for elliptic equations the analysis we develop in this paper shows the key role played by the convection term u p

singular nonlinear boundary value problems for second order ordinary differential equations abstract the boundary value problem y f x y 0 y 0 y 0 0 y 1 y 1 0 is studied under conditions which are modeled on the problem y a x y p 0y 0 y 1 0 or y 0 y 1 0 which occurs in a number of applications the technical arguments involve properties of solutions i e concavity and the use of iterative techniques a new fixed point theorem for cones for decreasing mappings is proved and used it is of interest in its own right since most such theorems are for monotone increasing mappings

the exact boundary behavior of the unique solution to a singular dirichlet problem with a nonlinear convection term abstract in this paper we analyze the exact boundary behavior of the unique solution to the singular nonlinear dirichlet problem u b x g u u q u 0 x u 0 where is a bounded domain with smooth boundary in r n q 0 2 0 g c 1 0 0 lim s 0 g s g is decreasing on 0 and b c loc is positive in may be vanishing or singular on the boundary we reveal that the nonlinear convection term u q does not affect the first expansion of the unique solution near the boundary to the problem

on channel constrained vector quantization and index assignment for discrete memoryless channels this correspondence introduces an approach to the design of channel robust vector quantizers design criteria are derived based on a new expression for the channel distortion as a function of the code vectors of the quantizer the introduced framework for design and analysis holds for arbitrary discrete memoryless channels simulations demonstrate good performance the correspondence also presents new results regarding the channel distortion as a function of the index assignment given to the code vectors of a vector quantizer

singular elliptic problems existence non existence and boundary behavior abstract we deal with the existence uniqueness and boundary behavior of positive solutions of the semilinear elliptic equation u a x g u b x f u in under dirichlet boundary conditions where r n is a bounded domain with smooth boundary a b g f are continuous non negative real valued functions the main feature here is that either g or f or both of them are singular at 0 in the sense that g t f t t 0 and 0 are parameters our results require no symmetry from either a or b and no monotonicity on f or g penalty arguments as well as variational principles are exploited

bit error resilient index assignment for multiple description scalar quantizers this paper addresses the problem of increasing the robustness to bit errors for two description scalar quantizers our approach is to start with an m diagonal index assignment and further apply a permutation to the indexes of each description to increase the minimum hamming distance d rm min of the set of valid index pairs in particular we show how to construct linear permutation pairs achieving d rm min geq 3 and establish a lower bound in terms of the description rate r for the highest value of m for which such permutations exist

positive versus compact support solutions to a singular elliptic problem abstract the semilinear elliptic problem u k x u q u p u 0 in u 0 on is considered in this paper where is a bounded domain of r n n 2 with c 2 boundary 0 q p 1 and k x as x we mainly study the effect of the blow up rate of k x near to the existence of positive and compact support solutions furthermore an optimal compact support principle is given for a class of elliptic differential inequalities

the hadamard transform a tool for index assignment we show that the channel distortion for maximum entropy encoders due to noise on a binary symmetric channel is minimized if the vector quantizer can be expressed as a linear transform of a hypercube the index assignment problem is regarded as a problem of linearizing the vector quantizer we define classes of index assignments with related properties within which the best index assignment is found by sorting not searching two powerful algorithms for assigning indices to the codevectors of nonredundant coding systems are presented one algorithm finds the optimal solution in terms of linearity whereas the other finds a very good but suboptimal solution in a very short time

high rate vector quantization for noisy channels with applications to wideband speech spectrum compression this paper considers the high rate performance of source coding for noisy discrete symmetric channels with random index assignment ia accurate analytical models are developed to characterize the expected distortion performance of vector quantization vq for a large class of distortion measures it is shown that when the point density is continuous the distortion can be approximated as the sum of the source quantization distortion and the channel error induced distortion expressions are also derived for the continuous point density that minimizes the expected distortion next for the case of mean squared error distortion a more accurate analytical model for the distortion is derived by allowing the point density to have a singular component the extent of the singularity is also characterized these results provide analytical models for the expected distortion performance of both conventional vq as well as for channel optimized vq as a practical example compression of the linear predictive coding parameters in the wideband speech spectrum is considered with the log spectral distortion as performance metric the theory is able to correctly predict the channel error rate that is permissible for operation at a particular level of distortion

on the performance and complexity of channel optimized vector quantizers the performance and complexity of channel optimized vector quantizers are studied for the gauss markov source observations on the geometric structure of these quantizers are made which have an important implication on the encoding complexity for the squared error distortion measure it is shown that an operation equivalent to a euclidean distance measurement with respect to an appropriately defined set of points used to identify the encoding regions can be used to perform the encoding this implies that the encoding complexity is proportional to the number of encoding regions it is then demonstrated that for very noisy channels and a heavily correlated source when the codebook size is large the number of encoding regions is considerably smaller than the codebook size implying a reduction in encoding complexity

a three solution theorem for singular nonlinear elliptic boundary value problems abstract we establish a three solution theorem for singular elliptic boundary value problems of the form u f u u in u 0 on where is a bounded domain in r n n 1 with a smooth boundary here f 0 0 is a c 1 function in 0 with f 0 0 and 0 1 in particular if there exist two pairs of sub supersolutions 1 1 and 2 2 where 1 2 1 1 2 1 with 2 2 and 2 2 are strict sub and supersolutions then we establish existence of three solutions u 1 u 2 and u 3 for the above boundary value problem such that u 1 1 2 u 2 2 1 and u 3 1 1 1 2 2 1 our results extend the work in 1 and also in 17 where such multiplicity results were discussed in the non singular case 0 further our results strengthen the multiplicity results in 12 and 13 for such singular problems 0 namely establish an additional third solution we also establish a sub super solution theorem for infinite semipositone problems i e the case when f 0 0

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

joint source and noisy channel trellis encoding corresp in a trellis encoding communication system the decoder is a time invariant nonlinear filter with finite memory sliding block code and the encoder is a trellis search algorithm matched to the decoder a coding theorem is established for a trellis encoding of a stationary and ergodic source over a discrete memoryless noisy channel which shows that such communication systems can perform arbitrarily close to the source distortion rate function evaluated at the channel capacity

regularity of an elliptic problem with a singular nonlinearity we study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular we prove that the positive solution is in general holder continuous up to the boundary and has even better regularity in some special cases

the existence and non existence of positive solutions to a singular quasilinear elliptic problem in rn abstract this paper deals with the existence and nonexistence of entire positive solutions of the quasilinear elliptic equation p u a x g u b x f u in r n 1 p n n 3 with u x 0 as x here either g or f or both of them are singular at 0 in the sense that g t f t as t 0 by using a perturbation method which eliminates the singularity on a ball with radius r and then let r tends to infinity with help of the bounded super solution of the original problem we obtain the existence of a weak solution of the problem the main results of this paper improve the corresponding results of dragos patru covei the existence and asymptotic behavior of a positive solution to a quasilinear elliptic problem in r n nonlinear anal 69 2008 2615 2622 j v goncalves c a santos singular elliptic problems existence non existence and boundary behavior nonlinear anal 66 2007 2078 2090

positive solutions of the p laplace equation with singular nonlinearity abstract for a given bounded domain in r n with c 1 boundary for some 0 1 and a possibly singular nonlinearity f on 0 we give sufficient conditions on f so that the p laplace equation p u f x u admits a solution u w 0 1 p on the basis of a comparison principle we will give a sufficient condition under which such a problem admits a unique solution

the existence and multiplicity of classical positive solutions for a singular nonlinear elliptic problem with any growth exponents a b s t r a c t in this paper we consider a class of singular elliptic boundary problems u u p x u u 0 in u 0 on with 1 0 1 and p c 0 0 1 we emphasize that there is no restriction on the boundedness of from above by means of the approximating variational arguments and truncation two classical positive solutions are obtained 2014 the authors published by elsevier ltd this is an open access article under the cc by nc nd license http creativecommons org licenses by nc nd 3 0

binary lattice vector quantization with linear block codes and affine index assignments we determine analytic expressions for the performance of some low complexity combined source channel coding systems the main tool used is the hadamard transform in particular we obtain formulas for the average distortion of binary lattice vector quantization with affine index assignments linear block channel coding and a binary symmetric channel the distortion formulas are specialized to nonredundant channel codes for a binary symmetric channel and then extended to affine index assignments on a binary asymmetric channel various structured index assignments are compared our analytic formulas provide a computationally efficient method for determining the performance of various coding schemes one interesting result shown is that for a uniform source and uniform quantizer the natural binary code is never optimal for a nonsymmetric channel even though it is known to be optimal for a symmetric channel

robust vector quantization by a linear mapping of a block code in this paper we propose a novel technique for vector quantizer design where the reconstruction vectors are given by a linear mapping of a binary block code lmbc the lmbc framework provides a relation between the index bits and the reconstruction vectors through mapping properties we define a framework show its flexibility and give optimality conditions we consider source optimized vector quantization vq where the objective is to directly obtain a vq with inherent good channel robustness properties several instructive theoretical results and properties of the distortion experienced due to channel noise are demonstrated these results are used to guide the design process both optimization algorithms and a block code selection procedure are devised experimental results for gauss markov sources show that quantization performance close to an unconstrained vq is obtained with a short block code which implies a constrained vq the resulting vqs have better channel noise robustness than conventional vqs designed with the generalized lloyd algorithm gla and splitting initialization even when a post processing index assignment algorithm is applied to the gla based vq we have thus demonstrated a unique method for direct design resulting in an inherent good index assignment combined with small losses in quantization performance

existence and regularity results for fully nonlinear equations with singularities we consider the dirichlet boundary value problem for a singular elliptic pde like f u p x u where p 0 in a bounded smooth domain of mathbb r n the nondivergence form operator f is assumed to be of hamilton jacobi bellman or isaacs type we establish existence and regularity results for such equations

quantizers with uniform decoders and channel optimized encoders scalar quantizers with uniform decoders and channel optimized encoders are studied for a uniform source on 0 1 and binary symmetric channels two families of affine index assignments are considered the complemented natural code cnc introduced here and the natural binary code nbc it is shown that the nbc never induces empty cells in the quantizer encoder whereas the cnc can nevertheless we show that the asymptotic distributions of quantizer encoder cells for the nbc and the cnc are equal and are uniform over a proper subset of the source s support region empty cells act as a form of implicit channel coding an effective channel code rate associated with a quantizer designed for a noisy channel is defined and computed for the codes studied by explicitly showing that the mean squared error mse of the cnc can be strictly smaller than that of the nbc we also demonstrate that the nbc is suboptimal for a large range of transmission rates and bit error probabilities this contrasts with the known optimality of the nbc when either both the encoder and decoder are not channel optimized or when only the decoder is channel optimized

new periodic and soliton solutions for the generalized bbm and burgers bbm equations abstract in this paper we consider the benjamin bona mahony equation bbm and we obtain new exact solutions for it by using a generalization of the well known tanh coth method new periodic and soliton solutions for the generalized bbm and burgers bbm equations are formally derived

quantizers with uniform encoders and channel optimized decoders scalar quantizers with uniform encoders and channel optimized decoders are studied for uniform sources and binary symmetric channels it is shown that the natural binary code nbc and folded binary code fbc induce point density functions that are uniform on proper subintervals of the source support whereas the gray code gc does not induce a point density function the mean squared errors mse for the nbc fbc gc and for randomly chosen index assignments are calculated and the nbc is shown to be mean squared optimal among all possible index assignments for all bit error rates and all quantizer transmission rates in contrast it is shown that almost all index assignments perform poorly and have degenerate codebooks