construction for binary asymmetric error correcting codes corresp a construction method of asymmetric error correcting codes is proposed in this method a t fold asymmetric error correcting code of length n 1 is constructed by expurgating and puncturing any t fold symmetric error correcting code of length n these codes are designed for use on binary asymmetric channels such as large scale integration lsi memory protection on which transition is one way

generalized sphere packing bound kulkarni and kiyavash recently introduced a new method to establish upper bounds on the size of deletion correcting codes this method is based upon tools from hypergraph theory the deletion channel is represented by a hypergraph whose edges are the deletion balls or spheres so that a deletion correcting code becomes a matching in this hypergraph consequently a bound on the size of such a code can be obtained from bounds on the matching number of a hypergraph classical results in hypergraph theory are then invoked to compute an upper bound on the matching number as a solution to a linear programming problem the problem of finding fractional transversals the method by kulkarni and kiyavash can be applied not only for the deletion channel but also for other error channels this paper studies this method in its most general setup first it is shown that if the error channel is regular and symmetric then the upper bound by this method coincides with the well known sphere packing bound and thus is called here the generalized sphere packing bound even though this bound is explicitly given by a linear programming problem finding its exact value may still be a challenging task the art of finding the exact upper bound or slightly weaker ones is the assignment of weights to the hypergraph s vertices in a way that they satisfy the constraints in the linear programming problem in order to simplify the complexity of the linear programming we present a technique based upon graph automorphisms that in many cases significantly reduces the number of variables and constraints in the problem we then apply this method on specific examples of error channels we start with the z channel and show how to exactly find the generalized sphere packing bound for this setup next studied is the nonbinary limited magnitude channel both for symmetric and asymmetric errors where we focus on the single error case we follow up on the deletion channel which was the original motivation of the work by kulkarni and kiyavash and show how to improve upon their upper bounds for single deletion correcting codes since the deletion and grain error channels have a similar structure for a single error we also improve upon the existing upper bounds on single grain error correcting codes finally we apply this method for projective spaces and find its generalized sphere packing bound for the single error case

adaptation of the mactaggart and jack complex multiplication algorithm for floating point operators with a suitable treatment of the exponents in the input operands a hardware implementation of the mactaggart and jack fixed point complex multiplication algorithm can also calculate a floating point product with no loss in accuracy from the greater dynamic range of the floating point inputs this floating point technique can be extended to any sum to two products operation such as encountered in matrix multiplication and vector cross products

sturm liouville eigenproblems with an interior pole the eigenvalues and eigenfunctions of self adjoint sturm liouville problems with a simple pole on the interior of the interval a b are investigated three general theorems are proved and it is shown that as n the eigenfunctions more and more closely resemble those of an ordinary sturm liouville problem and n m2 2 b a 2 just as if there were no singularity the low order modes however differ drastically from those of a nonsingular eigenproblem in that i both eigenvalues and eigenfunctions are complex despite the fact the problem is self adjoint ii the real and imaginary parts of the nth eigenfunction may both have ever increasing numbers of interior zeros as b instead of just n 1 zeros and iii as b the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem these results are general but in order to present quantitative solutions for the low order modes too special attention is gi

on group theoretic codes for asymmetric channels we study single error correcting codes for the asymmetric channel with input and output alphabets being 0 1 a 1 from an abelian group g of order n with elements g0 0 g1 gn 1 constantin and rao 1979 inform contr 40 20 36 define vg b1 b2 bn 1 o 1 a 1 n 1 n 1i 1 bigi 0 g and show that vg correct single errors we give explicit expressions for the size and weight distribution of these codes we further give a short discussion of some constant weight codes obtained by a similar construction

wave propagation in anisotropic layered media the determination of the natural modes of wave propagation in an anisotropic layered medium requires the solution of a transcendental eigenvalue problem that is usually approached numerically with the aid of search techniques such computations require great effort the method presented in this paper provides an alternate solution to this problem in terms of a quadratic eigenvalue problem involving tridiagonal matrices for which the eigenvalues can be found with great speed and accuracy the technique is then illustrated by means of an example involving a cross anisotropic gibson solid

distribution theory for discontinuous test functions and differential operators with generalized coefficients investigation of the differential operators with the generalized coefficients having singular support on a disjoint set of points requires the consideration of the distribution theory with the set of discontinuous test functions such a distribution theory for test functions having discontinuity at one point is developed a four parameter family of schrodinger operators formed by the operators with singular potential singular metrics and singular gauge field is considered it is proved that this family of singular interactions describes all possible selfadjoint extensions of the second derivative operator defined on the functions vanishing in a neighbourhood of the point approximation by operators with smooth coefficients is discussed c 1996 academic press inc

new lower bounds for asymmetric and unidirectional codes new lower bounds on the sizes of asymmetric codes and unidirectional codes are presented various methods are used three of them of special interest the first is a partitioning method that is a modification of a method used to construct constant weight codes the second is a combining codes method that is used to obtain a new code from a few others the third method is shortening by weights that is applied on symmetric codes or on codes generated by the combining codes method tables for the sizes of codes of length n

response to comment on on the coulomb potential in one dimension by fischer leschke and muller response to comment on on the coulomb potential in one dimension by w fisher h leschke and p muller

the generalized gilbert varshamov bound is implied by turan s theorem code construction the generalization of the gilbert varshamov bound due to gu and fuja 1993 is a direct consequence of turan s theorem on the existence of a clique in a graph with many edges turan s theorem allows a slight improvement of gu and fuja s result this improved generalized gilbert varshamov bound is in fact equivalent to turan s theorem

the functional analytic versus the functional integral approach to quantum hamiltonians the one dimensional hydrogen atoma the capabilities of the functional analytic and of the functional integral approach for the construction of the hamiltonian as a self adjoint operator on hilbert space are compared in the context of non relativistic quantum mechanics differences are worked out by taking the one dimensional hydrogen atom as an example that is a point mass on the euclidean line subjected to the inverse distance potential this particular choice is made with the intent to clarify a long lasting discussion about its spectral properties in fact for the four parameter family of possible hamiltonians the corresponding energy dependent green functions are derived in closed form the multiplicity of hamiltonians should be kept in mind when modeling certain experimental situations as for instance in quantum wires

upper bounds on codes correcting asymmetric errors corresp a survey is given of known upper bounds on codes correcting asymmetric errors the bounds are improved by introducing new ideas by solving a linear programming problem an upper bound is given that is easy to compute for all codelengths and all minimum asymmetric distances

greedy and heuristic algorithms for codes and colorings many of the fundamental coding problems can be represented as graph problems these problems are often intrinsically difficult and unsolved even if the code length is relatively small with the motivation to improve lower bounds on the sizes of constant weight codes and asymmetric codes we suggest a few greedy algorithms and other heuristic methods which result in new record breaking codes some of the heuristics used are based on tabu search and evolutionary algorithms tables of new codes are presented

new bounds and constructions for granular media coding improved lower and upper bounds on the size and the rate of grain correcting codes are presented the lower bound is gilbert varshamov like combined with a construction by gabrys et al and it improves on the previously best known lower bounds on the asymptotic rate of lceil tau n rceil grain correcting codes of length n on the interval 0 0 0668 one of the two newly presented upper bounds improves on the best known upper bounds on the asymptotic rate of lceil tau n rceil grain correcting codes of length n on the interval tau in 0 1 8 and meets the lower bound of 1 2 for tau geq 1 8 moreover in a nonasymptotic regime both upper bounds improve on the previously best known results on the largest size of t grain correcting codes of length n for certain values of n and t constructions of 1 grain correcting codes based on a partitioning technique are presented for lengths up to 18 finally a lower bound of 1 2 log 2 n on the minimum redundancy of infty grain detecting codes of length n is presented

bounds and constructions for binary asymmetric error correcting codes corresp by use of known bounds on constant weight binary codes new uppper bounds are obtained on the cardinality of binary codes correcting asymmetric errors some constructions are exhibited that come close to these bounds for single error correcting codes some constructions are derived from the steiner system s 5 6 12 and for double error correcting codes some constructions are derived from the nordstrom robinson code

lee metric bch codes and their application to constrained and partial response channels shows that each code in a certain class of bch codes over gf p specified by a code length n spl les p sup m 1 and a runlength r spl les p 1 2 of consecutive roots in gf p sup m has minimum lee distance spl ges 2r for the very high rate range these codes approach the sphere packing bound on the minimum lee distance furthermore for a given r the length range of these codes is twice as large as that attainable by berlekamp s 1984 extended negacyclic codes the authors present an efficient decoding procedure based on euclid s algorithm for correcting up to r 1 errors and detecting r errors that is up to the number of lee errors guaranteed by the designed minimum lee distance 2r bounds on the minimum lee distance for r spl ges p 1 2 are provided for the reed solomon case i e when the bch code roots are in gf p the authors present two applications first lee metric bch codes can be used for protecting against bitshift errors and synchronization errors caused by insertion and or deletion of zeros in d k constrained channels second the code construction with its decoding algorithm can be formulated over the integer ring providing an algebraic approach to correcting errors in partial response channels where matched spectral null codes are used

some new binary codes correcting asymmetric unidirectional errors the authors give a tabulation of the numbers of codewords in new binary codes with the asymmetric unidirectional error correcting capabilities of 3 4 5 6 for lengths 14 15 23 the new codes have greater sizes than the known codes for 14

spectral enumerators for certain additive error correcting codes over integer alphabets the paper contains a study of certain block codes with integer coordinate symbols devised for the correction of some types of additive errors the codes are defined by a check equation over a finite abelian group thus they appear as an extension of the varshamov and constantin rao codes the main results of the paper are concerned with the error correcting capability including a decoding method for generalized varshamov codes and principally with enumeration problems a general formula for spectral enumerators is obtained and applied to the varshamov and constantin rao codes

new lower bounds for constant weight codes some new lower bounds are given for a n 4 w the maximum number of codewords in a binary code of length n minimum distance 4 and constant weight w in a number of cases the results significantly improve on the best bounds previously known

on the coulomb potential in one dimension a mathematically rigorous definition of the one dimensional schrodinger operator d 2 dx 2 gamma x is given it is proven that the domain of the operator is defined by the boundary conditions connecting the values of the function on the left and right half axes the investigated operator is compared with the schrodinger operator containing the coulomb potential gamma x

dynamic responses of a layered medium subjected to anti plane loadings abstract the dynamic response of a layered medium subjected to anti plane loadings is investigated in this study an effective analytical method is used to present transient full field solutions in the layered medium the boundary value problem is solved using laplace transform with respect to time and fourier transform with respect to space the transient response is then implemented analytically by cagniard s method that obviates the contour integration difficulties and whose terms represent successively reflected and transmitted waves the static solution is obtained by the extreme case of transient response while the time is sufficiently long the objective in this study is to investigate the transition phenomena from transient response to static value the numerical examples of a layered half space are illustrated and discussed in detail it is found that the characteristic time of the transition not only depends on the distance between source and receiver but also is strongly related to the wave velocities of layer and half space furthermore very interesting transition behavior is observed when the layer has faster wave velocity than the half space

outer bounds on the capacity of interference channels corresp new outer bounds are demonstrated for the capacity regions of discrete memoryless interference channels and gaussian interference channels the bound for discrete channels coincides with the capacity region in special cases the bound for gaussian channels improves previous knowledge when the interference is of medium strength

comment on on the coulomb potential in one dimension by p kurasov the mathematical analysis in 1996 j phys a math gen 29 1767 is not sufficient to decide whether in one dimension the singularities of the potentials and split the corresponding one particle quantum systems at the origin into two completely decoupled subsystems in fact it is argued that this question cannot be answered by mathematical considerations alone

a general error correcting code construction for run length limited binary channels a general method is presented for constructing a single error correcting run length limited code for the class of input restricted binary symmetric channels described by the parameters d k where d is the minimum and k the maximum number of consecutive zeros in any allowable channel input sequence upper and lower bounds on the rates of these codes are derived

new upper bounds on the size of codes correcting asymmetric errors corresp new upper bounds on the size of codes correcting asymmetric errors are derived by sharpening some of the constraints in the integer programming problem of delsarte and piret it is shown that their code for length 9 and asymmetric distance 2 is optimal

constructions and bounds for systematic tec aued codes several methods of constructing systematic t error correcting all unidirectional error detecting codes are described these codes can be constructed by adding a tail to a linear t error correcting code but other constructions presented are more of an ad hoc nature these codes will often be found as suitably chosen subsets of nonsystematic tec aued codes further bounds on the word length of systematic tec aued codes are derived and extensive tables are given

a computationally efficient representation for propagation of elastic waves in anisotropic solids a new closed form representation is developed for the exact solution of the christoffel equation for wave propagation in solids the new representation is numerically more efficient than the traditional representations based on the use of fourier and laplace transforms using the new representation the retarded green s functions are derived for an infinite anisotropic solid and an anisotropic half space the method is applied to calculate the elastic wave response of an anisotropic cubic solid to highly localized delta function and step function type impulses both surface and bulk wave responses have been calculated the effect of anisotropy is discussed by considering cubic solids with different anisotropy parameters interestingly it is found that for certain values of the anisotropy parameter two distinct longitudinally polarized components can be observed to propagate along an axis of cubic symmetry one of the signals is the normal longitudinal wave signal while the other results from the concave shape of the transverse slowness surface

on symmetric error correcting and all unidirectional error detecting codes the design of binary block codes that are capable of correcting up to t symmetric errors and detecting all unidirectional errors is discussed a class of systematic t symmetric error correcting all unidirectional error detecting t syec aued codes are proposed when t 0 the proposed codes become berger codes for t 1 the proposed codes are shown to be of asymptotically optimal order methods to construct nonsystematic t syec aued codes for t 2 and 3 are also presented

a coding scheme for single peak shift correction in d k constrained channels a two step coding scheme for peak shift correction in d k constrained sequences is described the first step is based on q ary q k d 1 is a prime block codes that allow correction of specific types of double errors caused by single peak shifts the second step is a simple conversion of q ary symbols to binary strings of the type 00 01 the concatenation of these strings satisfies the d k constraint within the codeword and in concatenation with neighboring words the length of the codewords is controlled and if necessary can be fixed the rate r sub 1 of the overall encoding approaches 2 log sub 2 k d 1 k d 2 for large codeword lengths codes for correction of peak shift deletions and insertions of zeros are presented as well encoding and decoding are done by simple algorithms without using look up tables enumeration or denumeration procedures and therefore the codelength may be large

response to comment on penetrability of a one dimensional coulomb potential by roger g newton for original paper see ibid vol 26 p 2445 1993 by expressing in matrix form the one dimensional hamiltonians where the potentials are proportional to mod x mod 1 and x 1 i show that we are dealing with two very different problems thus in my viewpoint the criticism of r g newton of the paper mentioned in the title which is based on results relating to mod x mod 1 does not apply to a potential of the form x 1

lower bounds on achievable rates for limited bitshift correcting codes limited codes runlength limited charge constrained and so on capable of correcting shifts of their symbols are considered the error correction ability is characterized by the minimal bitshift distance d sub bs of a code for a given spl delta d sub bs n where n is the code length the achievable code rate r sub spl alpha is lower bounded we prove the existence of codes of rate r spl ges r sub spl alpha

bounds and constructions for binary codes of length less than 24 and asymmetric distance less than 6 upper bounds to the maximum number of codewords in a binary code of length n and asymmetric distance delta are derived for some values of n and delta a method is given in which a code of length n m and asymmetric distance at least t 1 is constructed by expurgating and puncturing a code of length n and hamming distance at least 2t 1 novel asymmetric error correcting codes are constructed by applying this method to some celebrated symmetric error correcting codes a table is presented on the size of optimal asymmetric error correcting codes of length less than 24 and asymmetric distance less than 6

unidirectional covering codes a code c spl sube z sup n sub 2 where z sub 2 0 1 has unidirectional covering radius r if r is the smallest integer so that any word in z sup n sub 2 can be obtained from at least one codeword c spl isin c by replacing either 1s by 0s in at most r coordinates or 0s by 1s in at most r coordinates the minimum cardinality of such a code is denoted by e n r upper bounds on this function are here obtained by constructing codes using tabu search lower bounds on the other hand are mainly obtained by integer programming and exhaustive search best known bounds on e n r for n spl les 13 and r spl les 6 are tabulated

conditional limit theorems under markov conditioning let x 1 x 2 cdots be independent identically distributed random variables taking values in a finite set x and consider the conditional joint distribution of the first m elements of the sample x 1 cdots x n on the condition that x 1 x 1 and the sliding block sample average of a function h cdot cdot defined on x 2 exceeds a threshold alpha eh x 1 x 2 for m fixed and n rightarrow infty this conditional joint distribution is shown to converge m the m step joint distribution of a markov chain started in x 1 which is closest to x l x 2 cdots in kullback leibler information divergence among all markov chains whose two dimensional stationary distribution p cdot cdot satisfies sum p x y h x y geq alpha provided some distribution p on x 2 having equal marginals does satisfy this constraint with strict inequality similar conditional limit theorems are obtained when x 1 x 2 cdots is an arbitrary finite order markov chain and more general conditioning is allowed

on constant composition codes over z sub q a constant composition code is a special constant weight code under the restriction that each symbol should appear a given number of times in each codeword in this correspondence we give a lower bound for the maximum size of the q ary constant composition codes with minimum distance at least 3 this bound is asymptotically optimal and generalizes the graham sloane bound for binary constant weight codes in addition three construction methods of constant composition codes are presented and a number of optimum constant composition codes are obtained by using these constructions

generating functions and lower bounds on rates for limited error correcting codes runlength limited rll and digital sum limited dsl codes are considered for these codes the finite and asymptotic lower bounds on achievable rates for the given minimum hamming distance are derived using generating functions and trellis diagram techniques the authors prove the existence of rll and dsl codes of rate r and minimum distance d delta n such that r or 2 log lambda sub 1 0 min log lambda sub 1 z delta log z 0

new constructions of codes for asymmetric channels via concatenation we present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets based on methods of generalized code concatenation for the binary asymmetric channel our methods construct nonlinear single error correcting codes from ternary outer codes we show that some of the varshamov tenengol ts constantin rao codes a class of binary nonlinear codes for this channel have a nice structure when viewed as ternary codes in many cases our ternary construction yields even better codes for the nonbinary asymmetric channel our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors

comment on penetrability of a one dimensional coulomb potential by m moshinsky see ibid vol 26 p 2445 1993 the analysis of the one dimensional schrodinger equation with the potential lambda x given is found to be incorrect contrary to the author s claim the origin is indeed impenetrable

bounds on the capacity of the bit shift magnetic recording channel a simple statistical channel model is suggested to account for single position bit shifts that have been identified as one of the major impairments in magnetic recording the authors investigate bounds on the capacity of this channel where the channel inputs are the d k codes commonly used in magnetic recordings for d or 2 this channel is conveniently described in terms of phrase lengths where a bit shift causes a phrase either to shrink or to expand the inherent correlation present in consecutive shift affected phrases introduces memory into the channel model sequences of nondecreasing lower bounds and nonincreasing upper bounds on the capacity are evaluated and investigated for a variety of parameters lower bounds on the zero error capacity are also studied the channel model is extended to a concatenated scheme of a bit shift channel connected in tandem with the binary symmetric channel thus capturing both major error generating mechanisms the bit shifts as well as randomly generated errors lower and upper bounds on the capacity of this concatenated channel model with d k input sequences are derived and discussed

a single asymmetric error correcting code with 2 sup 13 codewords of dimension 17 a new single asymmetric error correcting code is proposed this code is constructed using a product of two codes of smaller dimensions the proposed code is of dimension 17 and of size 8192 i e with 2 sup 13 codewords the best known code of dimension 17 has size 7968 and capable of handling 12 information bits only the only other three known cases of single asymmetric codes accommodating one more extra information bit than the symmetric case are for code dimensions 2 4 and 16

nonasymptotic upper bounds for deletion correcting codes explicit nonasymptotic upper bounds on the sizes of multiple deletion correcting codes are presented in particular the largest single deletion correcting code for q ary alphabet and string length is shown to be of size at most qn q q 1 n 1 an improved bound on the asymptotic rate function is obtained as a corollary upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length and this idea is demonstrated by application to sets of run length limited strings the problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph this problem is formulated as an integer linear program the upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program the nonasymptotic bounds derived imply the known asymptotic bounds of levenshtein and tenengolts and improve on known nonasymptotic bounds numerical results support the conjecture that in the binary case the varshamov tenengolts codes are the largest single deletion correcting codes

new lower bounds for binary codes of asymmetric distance two lower bounds for asymmetric single error correcting codes are derived the codes are constructed by puncturing constant weight codes and by using a random coding argument

perfect d k codes capable of correcting single peak shifts codes for the multibit peak shift recording channel called d k codes of reduced length n are considered arbitrary d k and perfect d k codes capable of correcting single peak shifts of given size t are defined for the construction of perfect codes a general combinatorial method connected with finding good weight sequences in abelian groups is used and the concept of perfect t shift n designs is introduced explicit constructions of such designs for t 1 t 2 and t p 1 2 are given where p is a prime this construction is universal in that it does not depend on the d k constraints it also allows automatic correction of those peak shifts that violate d k constraints the construction is extended to d k codes of fixed binary length and allows the beginning of the next codeword to be determined the question whether the designed codes can be represented as systematic codes with minimal redundancy is considered as well

constructions of error correcting dc free block codes two dc free codes are presented with distance 2d b or 1 length 2n 2r d 1 for d 3 where r is the least integer or log sub 2 2n 1 for the first code l 4 c 2 and the asymptotic rate of this code is 0 7925 for the second code l 6 c 3 and the asymptotic rate of this code is 0 8858 asymptotically these rates achieve the channel capacity for small values of n these codes do not achieve the best rate as an example of codes of short length with good rate the author presents a 30 10 6 4 dc free block code with 2 sup 21 codewords a construction is presented for which from a given code c sub 1 of length n even weight and distance 4 the author obtains a 4n l c 4 dc free block code c sub 2 where l is 4 5 or 6 and c is not greater than n 1 but usually significantly smaller the codes obtained by this method have good rates for small lengths the encoding and decoding procedures for all the codes are discussed

new single asymmetric error correcting codes new single asymmetric error correcting codes are proposed these codes are better than existing codes when the code length n is greater than 10 except for n 12 and n 15 in many cases one can construct a code c containing at least 2 sup n n codewords it is known that a code with c spl ges 2 sup n n 1 can be easily obtained it should be noted that the proposed codes for n 12 and n 15 are also the best known codes that can be explicitly constructed since the best of the existing codes for these values of n are based on combinatorial arguments useful partitions of binary vectors are also presented

relationship of anisotropic and isotropic materials for antiplane problems the material symmetry is assumed to be such that the inplane and antiplane deformations are uncoupled a correspondence is obtained between the anisotropic and the isotropic problem through such a correspondence the relationship of the stresses and displacement for the anisotropic and the corresponding isotropic problem is established for both polar and cartesian coordinate systems

improved gilbert varshamov bound for constrained systems nonconstructive existence results are obtained for block error correcting codes whose codewords lie in a given constrained system each such system is defined as a set of words obtained by reading the labels of a finite directed labeled graph for a prescribed constrained system and relative minimum distance delta the new lower bounds on the rate of such codes improve on those derived recently by v d kolesnik and v y krachkovsky 1991 the better bounds are achieved by considering a special subclass of sequences in the constrained system namely those having certain empirical statistics determined by delta

exact transient solutions of buried dynamic point forces and displacement jumps for an elastic half space abstract the exact transient closed form solutions for applying time dependent point forces and displacement jumps at a depth below the surface of an elastic half plane are obtained in this study a new methodology for constructing the reflected field is proposed and is shown to be both powerful and efficient the solution of an exponentially distributed loading in the laplace transform domain at the surface of a half plane is considered as the fundamental solution the waves reflected from the free surface caused by the incident waves can be constructed by superimposing the fundamental solution the numerical results of stress and displacement fields during the transient process are obtained and compared with the corresponding static value it is shown in this study that the dynamic transient solution will approach the static value after the last reflected wave has passed

a new table of constant weight codes a table of binary constant weight codes of length nl28 is presented explicit constructions are given for most of the 600 codes in the table the majority of these codes are new the known techniques for constructing constant weight codes are surveyed and a table of unrestricted binary codes of length nl28 is given

on the capacity of generalized ising channels nearly tight lower and upper bounds on the capacity of generalized ising channels are presented for the case where feedback is allowed a closed form expression for the capacity is found for channel error probability p in 0 p 0 where p 0 approx 0 398324 a near capacity achieving family of encoders is presented for the values p in 0 p 0 two lower bounds on that capacity for larger values of p are presented one of which is tight on the interval p 0 0 5

perturbation theory for a sturm liouville problem with an interior singularity the perturbation theory of operators and forms is used to construct sturm liouville differential operators for potentials with 1 x pf 1 x and for epsilon rightarrow 0 1 x i epsilon interior singularities the norm resolvent convergence of approximating sequences of operators with smooth potentials is established and various qualitative and quantitative properties of their spectra are obtained for an infinite interval a comparison is made with the one dimensional coulomb problem possessing a potential 1 x

transient analysis of anisotropic multilayered media subjected to dynamic antiplane loadings the transient responses of anisotropic multilayered media subjected to arbitrarily distributed antiplane loadings are investigated to solve the complicated problem a linear coordinate transformation is introduced and successfully used to transform the anisotropic layered medium problem to an isotropic problem the relationship between field quantities of the anisotropic problem and the corresponding isotropic problem are established for a cartesian coordinate system the boundary value problem is solved by using the integral transform method the solutions in the laplace transform domain are constructed in the form of a power series of the phase related reflection and transmission matrices each term in the series represents a reflected or transmitted wave the transient solution is then obtained by means of cagniard s method the corresponding static solution is also derived by application of the final value theorem for numerical calculations the transient responses of an anisotropic thin layer overlying a half space are considered the transition behavior from transient response to static value is presented and discussed in detail

a lower bound for a n 4 w corresp an explicit lower bound is given for a n 4 w the maximum number of codewords in a binary code of length n minimum distance four and constant weight w