quasilinear parabolic systems of several components in this paper we investigate the blowup criteria of the quasilinear parabolic system u imath t c imath u imath alpha imath delta u imath prod jmath 1 n u jmath p imath jmath imath 1 2 cdots n with homogeneous dirichlet boundary conditions on a bounded domain r n where c i 0 i 0 p i 0 1 i n are constants denote by i the identity matrix and p p i which is assumed to be irreducible that i p is a singular m matrix is shown to be the critical case in which 1 plays a fundamental role where 1 is the first dirichlet eigenvalue of the laplacian on as a result we give a general answer to the question of galaktionov and levine on the porous medium systems

multilevel codes based on partitioning following v v ginzburg 1984 a hierarchy of codes is proposed to match the geometric partitioning of a signal set the authors show that coset codes including ungerboeck lattice and binary codes and indeed any codes which rely on partitioning of the signal set are all subclasses of the proposed coding scheme the combination of such codes in a multilevel scheme often leads to reduced complexity in comparison with previously published schemes a variety of decoder structures is discussed

structurally stable riemann solutions abstract we study the structure of solutions of riemann problems for systems of two conservation laws such a solution comprises a sequence of elementary waves viz rarefaction and shock waves of various types shock waves are required to have viscous profiles we construct a riemann solution by solving a system of equations characterizing its component waves a riemann solution is structurally stable if the number and types of its component waves are preserved when the initial data and the flux function are perturbed under the assumption that rarefaction waves and shock states lie in the stricly hyperbolic region we characterize riemann solutions for which the definition equations have maximal rank and we prove that such solutions are structurally stable structurally stable riemann solutions cannot contain overcompressive shock waves but they can contain transitional shock waves including doubly sonic transitional shock waves including doubly sonic transitional shock waves that have not been observed before

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

global and blow up solutions for nonlinear degenerate parabolic systems with crosswise diffusion abstract in this paper the nonlinear degenerate parabolic system u t v 1 u xx au v t u 2 v xx bv with dirichlet boundary condition is studied the regularization method and upper lower solutions technique are employed to show the local existence of a solution for the nonlinear degenerate parabolic system the global existence of a solution is discussed the finite time blow up result together with an estimate of the blow up time are found the blow up set with positive measure is analyzed in some detail

a class of optimum block codes in signal space by using convolutional coding coupled with expanded sets of channel signals ungerboeck showed that it is possible to obtain coding gains of several decibels at a given data rate without increasing the transmission bandwidth in this paper we construct a class of optimum signal space codes using short binary block as opposed to convolutional codes our results are compared to those obtained by ungerboeck a simplified decoding procedure that is asymptotically optimum is also presented

on nonlinear stability of general undercompressive viscous shock waves we study the nonlinear stability of general undercompressive viscous shock waves previously the authors showed stability in a special case when the shock phase shift can be determined a priori from the total mass of the perturbation using new pointwise methods by examining time invariants associated with the linearized equations we can now overcome a new difficulty in the general case namely nonlinear movement of the shock we introduce a coordinate transformation suitable to treat this new aspect and demonstrate our method by analyzing a model system of generic type we obtain sharp pointwise bounds andlp behavior of the solution for allp 1 p

analysis of blowup for a nonlinear degenerate parabolic equation in this paper the nonlinear degenerate parabolic equation ut u u u subject to dirichlet boundary conditions is studied the finite time blowup result together with an estimate of the blowup time is found the blowup set with positive measure and the asymptotic behavior of solution near the blowup time are analyzed in some detail via the similarity solutions

global existence and nonexistence for a class of degenerate parabolic systems this paper investigates the finite time blow up and global existence of the positive solution of the nonlinear degenerate parabolic system ut f1 u u a1v vt f2 v v a2u x t 0 with homogeneous dirichlet boundary data a necessary and sufficient condition for global existence of the solution is derived

multilevel trellis coded modulations for the rayleigh fading channel the authors present coded 8 phase shift keyed 8 psk modulations for the rayleigh fading channel the schemes are based on multilevel trellis coded modulation constructions and utilize maximum free hamming distance binary convolutional codes as building blocks a suboptimal multistage decoder that utilizes interstage interleaving and iterative decoding is proposed and evaluated examples are constructed to show that the proposed schemes outperform the best modified codes of the ungerboeck type due to significantly higher implicit time diversity yielding seven branches of built in time diversity whereas the ungerboeck code yields four branches of time diversity for a 64 state system the transmission delay is higher however the new schemes can provide three levels of unequal error protection when 8 psk or 8 differential phase shift keying 8 dpsk modulations are used they provide 10 14 db channel signal to noise ratio gain over uncoded 4 dpsk at a bit error rate of 10 sup 3 for a modest decoding complexity

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

on trellis codes with a delay processor and a signal mapper it is already known that a trellis code t which is constructed by using the encoder of a convolutional code c with short constraint length followed by a delay processor and a signal mapper is equivalent to a trellis code with large constraint length in this paper we derive a new lower bound on the free distance of t which in some cases is better than the previously derived bound moreover instead of the decoding used in earlier publications we apply iterative decoding on both tailbiting and zero tail representations of t to take advantage of the new lower bound and in the meantime to decrease the associated error coefficient caused by the decoding used in earlier publications comparisons among various designs of such a trellis code and some well known coding methods are also provided

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

multiple symbol differentially detected multilevel codes for the rayleigh fading channel conventional differential detection is known to perform poorly on the fast fading rayleigh fading channel due to the rapid variation of the channel state the technique of multiple symbol differential detection msdd improves the error performance by detecting over a sequence of symbols and exploiting the inherent correlation of the fading process to further reduce the error rate we need to introduce some form of coding successful coding for the rayleigh fading channel requires interleaving to decorrelate the channel a conflicting requirement to msdd we propose a solution consisting of interleaving blocks of l m ary phase shift keying mpsk symbols that are multiple symbol differentially detected the blocks are treated as elements of an m sup l symbol alphabet over which a multilevel code based on geometrically uniform partitions is defined

multilevel trellis mpsk modulation codes for the rayleigh fading channel the multilevel coding technique is used for constructing multilevel trellis m ary phase shift keying mpsk modulation codes for the rayleigh fading channel in the construction of a code all the factors which affect the code performance and its decoding complexity are considered the error performance of some of these codes based on both one stage optimum decoding and multistage suboptimum decoding has been simulated the simulation results show that these codes achieve good error performance with small decoding complexity

some degenerate and quasilinear parabolic systems not in divergence form abstract this paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form u t u p u av v t v q v bu with null dirichlet boundary conditions and positive initial conditions where p q a and b are all positive constants the local existence and uniqueness of classical solution are proved moreover it will be proved that all solutions exist globally if and only if ab 1 2 where 1 is the first eigenvalue of in with homogeneous dirichlet boundary condition

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

new multilevel codes over gf q set partitioning is applied to multidimensional signal spaces over gf q i e gf sup n1 q n1 or d are presented these codes use reed solomon codes as component codes longer multilevel block codes are also constructed using q ary block codes with block length longer than q 1 as component codes some quaternary multilevel block codes are presented with the same length and number of information symbols as but larger distance than the best previously known quaternary one level block codes it is proved that if all the component block codes are linear the multilevel block code is also linear low rate q ary convolutional codes word error correcting convolutional codes and binary to q ary convolutional codes can also be used to construct multilevel trellis codes over gf q or binary to q ary trellis codes

some ternary and quaternary codes and associated sphere packings tables of good ternary and quaternary codes are presented and they are used in the construction of dense sphere packings results include 1 tables of the best ternary and quaternary constacyclic codes including cyclic codes up to block length 50 2 a class of optimal n 2 codes over gf q 3 the u v w mod 2u v mod u construction a new ternary code construction technique that can be used to construct the ternary reed muller codes and others and 4 tables of linear ternary and quaternary codes obtained by modifying and combining various codes here and in the literature packings are generated in even dimensions up to 100 using these codes and a sphere packing construction in dimensions 36 and 60 new record densities appear to have been achieved

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

the leech lattice the octacode and decoding algorithms new multilevel constructions of the golay code and the leech lattice are presented these are derived from the turyn construction and the holy construction with the octacode as the glue code further we show that the holy construction of the leech lattice with the octacode as the glue code is essentially different from the permuted turyn construction although both constructions rely on the octacode the turyn construction is based on an odd type of the octacode whereas any type of the octacode can be used in the holy construction moreover the multilevel representation of the holy construction leads to a novel lattice partition chain based on these structures we derive new bounded distance decoders for the golay code and the leech lattice whose effective error coefficient is smaller than that of any previously known bounded distance decoder we provide a general theorem for computing the effective error coefficient of coset decoding with bounded distance decoding for the subcode

improved geometric goppa codes i basic theory in this paper we present a construction of improved geometric goppa codes which for the case of r 2g are often more efficient than the current geometric goppa codes derived from some varieties which include algebraic curves hyperplanes surfaces and other varieties for the special case of a plane in a three dimensional projective space the improved geometric goppa codes are reduced to linear multilevel codes for these improved geometric goppa codes a designed minimum distance can be easily determined and a decoding procedure which corrects up to half the designed minimum distance is also given

positive steady state solutions of a competing reaction diffusion system with large cross diffusion coefficients abstract this paper deals with a coupled system of reaction diffusion equations modeling the competition of two species with cross diffusion we discuss the existence of positive steady state solutions in relation to the intrinsic birth rates a 1 a 2 of species and the cross diffusion coefficients 12 21 our results show that positive steady state solutions exist if a 1 and a 2 lie in certain range or if 12 and 21 are sufficiently large

on multilevel block modulation codes the multilevel technique for combining block coding and modulation is investigated a general formulation is presented for multilevel modulation codes in terms of component codes with appropriate distance measures a specific method for constructing multilevel block modulation codes with interdependency among component codes is proposed given a multilevel block modulation code c with no interdependency among the binary component codes the proposed method gives a multilevel block modulation code c that has the same rate as c a minimum squared euclidean distance not less than that of c a trellis diagram with the same number of states as that of c and a smaller number of nearest neighbor codewords than that of c finally a technique is presented for analyzing the error performance of block modulation codes for an additive white gaussian noise awgn channel based on soft decision maximum likelihood decoding error probabilities of some specific codes are evaluated by simulation and upper bounds based on their euclidean weight distributions

statistical analysis of the product high order ambiguity function the high order ambiguity function haf was introduced for the estimation of polynomial phase signals pps embedded in noise since the haf is a nonlinear operator it suffers from noise masking effects and from the appearance of undesired cross terms and possibly spurious harmonics in the presence of multicomponent mc signals the product haf phaf was then proposed as a way to improve the performance of the haf in the presence of noise and to solve the ambiguity problem in this correspondence we derive a statistical analysis of the phaf in the presence of additive white gaussian noise awgn valid for high signal to noise ratio snr and a finite number of data samples the analysis is carried out in detail for single component pps but the multicomponent case is also discussed error propagation phenomena implicit in the recursive structure of the phaf based estimator are explicitly taken into account the analysis is validated by simulation results for both single and multicomponent ppss

on the performance evaluation of trellis codes generating function techniques for analyzing the error event and the bit error probabilities for trellis codes are considered the conventional state diagram approach for linear codes where the number of states is equal to the number of trellis states cannot be applied directly to arbitrary trellis codes and instead a state diagram where the number of states is equal to the square of the number of trellis states must be used it is shown that for an interesting class of trellis codes a modified generating function can be defined for which the number of states em is equal to the number of trellis states the class of codes considered includes trellis codes of rate r n 1 n based upon set partitioning whenever the first partition breaks the signal constellation into two subsets which have the same configuration matrix i e the same ordered set of mutual distances the complexity of calculating this modified generating function is the same as for the ordinary generating function of a convolutional code with the same number of trellis states bounds on the performance of some interesting codes are given based upon this method

coded phase frequency modulation channel coding combined with expanded signal sets has been shown 1 to improve error performance over uncoded modulation without expanding the bandwidth of the transmitted signals in this paper new coded modulation formats defined over an expanded set of signals varying both in phase and frequency are presented the new schemes combine fsk and psk modulation and make use of trellis coding and viterbi decoding to improve error performance over uncoded modulation the free euclidean distance is calculated for several classes of codes and upper bounds and simulation results are also presented for some simple codes the spectral characteristics of the proposed coded modulation formats are evaluated and compared to conventional two dimensional modulation formats differential encoding and various extensions of the basic scheme are also discussed

two level coding based on trellis coded modulation and reed solomon codes a number of two level codes based on trellis coded modulation tcm schemes and their matched single or double error correcting reed solomon codes using qam constellations are presented and evaluated compared to the well known 4d 16 state tcm scheme with a comparable bandwidth efficiency these two level codes provide greater immunity from both gaussian and impulse noise at the cost of a reasonably higher complexity and a longer decoding delay on the order of a few hundred symbols the authors also show how to cope with phase ambiguities of the constellation in these two level codes both rotationally invariant coding and rotation detecting techniques are considered

finite state codes a class of codes called finite state fs codes is defined and investigated the codes which generalize both block and convolutional codes are defined by their encoders which are finite state machines with parallel inputs and outputs a family of upper bounds on the free distance of a given fs code is derived a general construction for fs codes is given and it is shown that in many cases the fs codes constructed in this way have a free distance that is the largest possible catastrophic error propagation cep for fs codes is also discussed it is found that to avoid cep one must solve the graph theoretic problem of finding a uniquely decodable edge labeling of the state diagram

boundedness and blow up for a semilinear reaction diffusion system abstract we consider the semilinear parabolic system s u t u p t u q where x rnn 1 t 0 and p q are positive real numbers at t 0 nonnegative continuous and bounded initial values u0 v0 x are prescribed the corresponding cauchy problem then has a nonnegative classical and bounded solution u t x v t x in some strip st 0 t r n 0 t sup t 0 u v remain bounded in s t we show in this paper that if 0 t so that solutions can be continued for all positive times when pq 1 and 1 pq 1 n 2 with max p q one has t for every nontrivial solution u v t is then called the blow up time of the solution under consideration finally if 1 pq 1 n 2 both situations coexist since some nontrivial solutions remain bounded in any strip s while others exhibit finite blow up times

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

stable viscosity matrices for systems of conservation laws abstract a natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension u 1 f u x 0 u r m is studied these matrices are admissible in the sense that small amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u t f u x v du x x as ifv 0 if d is in this class the class is determined by a linearized stability requirement the cauchy problem for the equation u 1 f u 0 u x vdu xx should be well posed in l 2 uniformly in v as v 0 previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion

two classes of convolutional codes over gf q for q ary orthogonal signaling two classes of nonbinary convolutional codes for error control in communication systems employing q ary orthogonal signaling are discussed these classes include as special cases the dual k codes and the codes treated by b d trumpis 1975 the types of code extensions of previous bounds on free distance the noncatastrophic condition for these codes and conditions for catastrophic error propagation are described codes obtained by a computer search optimal in that the truncated transfer function bound on the probability of symbol error is minimized are presented it is shown how good convolutional codes can be constructed from cyclic block codes the performance of some of these codes over additive white gaussian noise channels rayleigh channels and partial band interference channels is also discussed

multilevel codes and multistage decoding h imai and s hirakawa have proposed 1977 a multilevel coding method based on binary block codes that admits a staged decoding procedure the author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets the multilevel structure allows the redundancy in the coset selection procedure to be allocated efficiently among the different levels it also allows the use of suboptimal multistage decoding procedures that have performance complexity advantages over maximum likelihood decoding

boundedness of global solutions for a nonlinear degenerate parabolic porous medium system with localized sources abstract this paper deals with nonlinear degenerate parabolic porous medium system with localized sources it is shown that under certain conditions solutions of the equation blow up in finite time for large a and b or large initial data while there exist global positive solutions for small a and b or small initial data moreover it is also shown that all global positive solutions of the above equation are uniformly bounded these extend a recent work of chen et al in y chen q liu h gao boundedness of global solutions of a porous medium equation with a localized source nonlinear anal 64 2006 2168 2182 which considered the single equation with localized source

trellis coded multidimensional phase modulation a 2l dimensional multiple phase shift keyed l mpsk signal set is obtained by forming the cartesian product of l two dimensional mpsk signal sets a systematic approach to partitioning l mpsk signal sets that is based on block coding is used an encoder system approach is developed it incorporates the design of a differential precoder a systematic convolutional encoder and a signal set mapper trellis coded l 4psk l 8psk and l 16psk modulation schemes are found for 1

an approach to ungerboeck coding for rectangular signal sets corresp a method of finding good ungerboeck codes for large rectangular quadrature amplitude modulation qam signal sets is described using the concept of euclidean weights due to ungerboeck we prove that a 2 n point basic constellation may be employed to determine exactly the free distance for an ungerboeck coded rectangular 2 m point set when m n 1 bits are uncoded and the remaining bits pass through a rate n 1 n convolutionai encoder it is shown that rate 2 3 encoders may be used to achieve most of the theoretically possible coding gain in the proposed scheme where the effect of the error coefficient on the coding gain has been considered

reduced lists of error patterns for maximum likelihood soft decoding a method whereby a substantially reduced family of error patterns called survivors may be created for maximum likelihood soft decoding is introduced the family of survivor depends on the received word a decoder based on this approach first forms the survivors then scores them rather than obtaining the survivors by online elimination of error patterns the use of predetermined lists that represent all families of survivors is proposed

dual family viscous shock waves in n conservation laws with application to multi phase flow in porous media we consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws we study si j dual family shock waves which are associated with a pair of characteristic families i and j we explicitly introduce defining equations relating states and speeds of si j shocks which include the rankine hugoniot conditions and additional equations resulting from the viscous profile requirement we then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of si j shocks under change of problem parameters all possible structures of solutions to the riemann problems containing si j shocks and classical waves are described as a physical application all types of si j shocks with i j are detected and studied in a family of models for multi phase flow in porous media