reconstruction techniques for classical inverse sturm liouville problems this paper gives constructive algorithms for the classical inverse sturm liouville problem it is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator two methods of solving this latter problem are then provided and numerical examples are presented

surfaces of general type with p_g q 1 k 2 8 and bicanonical map of degree 2 we classify the minimal algebraic surfaces of general type with pg q 1 k 2 8 and bicanonical map of degree 2 it will turn out that they are isogenous to a product of curves i e if s is such a surface then there exist two smooth curves c f and a finite group g acting freely on c x f such that s c x f g we describe the c f and g that occur in particular the curve c is a hyperelliptic bielliptic curve of genus 3 and the bicanonical map of s is composed with the involution induced on s by r x id c x f c x f where is the hyperelliptic involution of c in this way we obtain three families of surfaces with pg q 1 k 2 8 which yield the first known examples of surfaces with these invariants we compute their dimension and we show that they are three generically smooth irreducible components of the moduli space m of surfaces with p g q 1 k 2 8 moreover we give an alternative description of these surfaces as double covers of the plane recovering a construction proposed by du val

l szpiro s conjecture on gorenstein algebras in codimension 2 abstract a gorenstein a algebra r of codimension 2 is a perfect finite a algebra such that r ext a 2 r a holds as r modules a being a cohen macaulay local ring with dim a dim a r 2 the aim of this article is to prove a structure theorem for these algebras improving on an old theorem of m grassi koszul modules and gorenstein algebras j algebra 180 1996 918 953 special attention is paid to the question how the ring structure of r is encoded in its hilbert resolution it is shown that r is automatically a ring once one imposes a very weak depth condition on a determinantal ideal derived from a presentation matrix of r over a graded analogues of the aforementioned results are also included questions of applicability to the theory of surfaces of general type namely canonical surfaces in p 4 have served as a guideline in these commutative algebra investigations

a new family of surfaces of general type with k 2 7 and p_g 0 we construct a new family of smooth minimal surfaces of general type with k 2 7 and p g 0 we show that a surface in this family has ample canonical divisor and birational bicanonical morphism we also prove that these surfaces satisfy bloch s conjecture

on the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues abstract consider the inverse acoustic scattering problem for spherically symmetric inhomogeneity of compact support define the corresponding homogeneous and inhomogeneous interior transmission problems we study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem these transmission eigenvalues are shown to be zeros of the scattering amplitude and also the set of eigenvalues of a special sturm liouville problem a uniqueness theorem for the potential of the derived sturm liouville problem is proved when the data are the given spectra and partial knowledge of the potential a corollary of this theorem is a uniqueness theorem for the original inverse acoustic scattering problem

an asymptotical inversion of the eigenfrequencies for a three dimensional problem with spherical symmetry anasymptotical method for the inversion of the eigenfrequencies of a three dimensional nonhomogenous body with spherical symmetry is presented the result is based on the uniform asymptotical approximation for the solution of linear differential equations of second order with turning points this approximation is of a higher order than the well known wentzel kramers brilbuin wkb approximation and provides the possibility of finding both the density and the elastic parameter of the media the wkb approximation provides the determination of sound speed only which is a combination of the mentioned functions the geophysical and helioseismological applications of the method are discussed

on unordered codes by extending the results obtained by d e knuth 1986 a parallel unordered coding scheme with 2 sup r information bits is described balanced codes in which each codeword contains equal amounts of zeros and ones with r check bits and up to 2 sup r 1 r 2 information bits are constructed unordered codes with r check bits and up to 2 sup r 2 sup r 1 1 information bits are designed codes capable of detecting 2 sup r 1 2 sup r 2 1 unidirectional errors using r check bits are also described a review of previous work is presented

coding for delay insensitive communication with partial synchronization assume that information is transmitted in parallel among many lines in such a way that an electrical transition represents a 1 and an absence of a transition represents a 0 the propagation delay in the wires varies and results in asynchronous reception the challenge is to find an efficient communication scheme that will be delay insensitive one of the common solutions to this problem is to use a handshake mechanism namely the transmitter sends the next vector only after getting an acknowledgment that the current vector was received a natural question is how does the receiver know that reception of the current vector is complete this problem was solved by verhoeff 1988 by using the so called unordered codes however in practice it is common that the communication lines are arranged in pairs double rail such that the propagation delay on the lines within a pair is identical in general the lines can be arranged in groups of size larger than 1 where transmission within a group is synchronized the authors have created a few delay insensitive schemes that take advantage of partial synchronization within groups to achieve that they have generalized to arbitrary alphabets the following known results sperner s theorem on unordered sets henry knuth s henry 1982 knuth 1986 construction of balanced codes and berger s 1961 construction of unordered codes finally they have focused on practice and constructed a code that uses double rail channels but has the advantage that it is a rate 3 4 code as opposed to the rate 1 2 double rail code that is the common code being used in real systems

on the canonical rings of covers of surfaces of minimal degree let s be a regular surface of general type with at worst canonical singularities and with basepoint free canonical system let x be its canonical image it is well known that x must be a canonical surface or a minimal degree surface the main result of the authors completely describes the number and degree of the generators of the canonical ring of s in the second case more concretely if r deg x and n is the degree of the canonical map then 1 if n 2 and r 1 the canonical ring is generated in degree 1 plus one generator in degree 4 2 in the other cases the canonical ring is generated in degree 1 plus r n 2 generators in degree 2 and r 1 generators in degree 3 this result together with previous results of ciliberto and green describes when the canonical ring of s is generated in degree less than or equal to 2 x is not a surface of minimal degree other than the plane and in this last case n 6 2 the authors also construct a series of non trivial examples of the theorem and prove that some expected ones do not exist finally the authors apply their results to calabi yau threefolds obtaining analogous results the key point here is that for a calabi yau threefold the general member of a big and base point free linear system is a surface of general type

the uniqueness for inverse discrete transmission eigenvalue problems abstract the inverse problem for the discrete analog of the transmission eigenvalue problem with a spherically symmetric index of refraction is considered some uniqueness results are provided which imply that can be completely determined if only partial entries are given on together with partial transmission eigenvalues furthermore some similar results are presented for a related schrodinger equation

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

a comparison of two methods for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium abstract a comparison is made of two methods for solving the inverse scattering problem for acoustic waves in an inhomogeneous absorbing medium these methods were recently introduced by colton kirsch and monk and are based on considering a weighted average of the far field data instead of the far field data itself the comparison is based on numerical examples for the special case of spherically stratified media

canonical double covers of minimal rational surfaces and the non existence of carpets his article delves into the relation between the deformation theory of finite morphisms to projective space and the existence of ropes embedded in projective space with certain invariants we focus on the case of canonical double covers x of a minimal rational surface y embedded in p n by a complete linear series and carpets on y canonically embedded in p n we prove that these canonical double covers always deform to double covers and that canonically embedded carpets on y do not exist this fact parallels the results known for hyperelliptic canonical morphisms of curves and canonical ribbons and the results for k3 double covers of surfaces of minimal degree and enriques surfaces and k3 carpets that canonical double covers of minimal rational surfaces should deform to double covers is not a priori obvious for the invariants of most of these surfaces lie on or above the castelnuovo line thus in principle deformations of such covers could have birational canonical maps in fact many canonical double covers of non minimal rational surfaces do deform to birational canonical morphisms we also map the region of the geography of surfaces of general type corresponding to the surfaces x and we compute the dimension of the irreducible moduli component containing x in certain cases we exhibit some interesting moduli components parameterizing surfaces s with the same invariants as x but with birational canonical map unlike x

on the base space of a semi universal deformation of rational quadruple points this paper is devoted to the study of a particular class of normal surface singularities the rational quadruple points we will determine the base space of a semi universal deformation of such a singularity the answer turns out to be unexpectedly simple the isomorphism type of the base space of a rational quadruple point is completely determined by two numbers s and n the base space then is isomorphic to s x b n where s is a smooth germ of dimension s and where b n is a certain universal space defined in 3 5 a rational quadruple point with the star shaped resolution graph shown in figure 1 has the factor b n in its space we call such a singularity an n star

deformations of the normalization of hypersurfaces introduction a deformation theory has been introduced by siersma si 11 for the simplest class of non isolated singularities x hypersurfaces with a smooth one dimensional singular locus and transverse to a general point of e an ai singularity he considered deformations loosely speaking such that z stays inside the singular locus of the deformed x this notion has been extended by pellikaan pe 1 and has been studied further by the authors in j s 11 and j s 2 let us be more precise about the deformation functor we are interested in let c be a category of spaces e g those of germs of analytic spaces a diagram of spaces z s xs flat over some base space s is called admissible iffzs xsls where ccxs s is the relative critical space as defined by teissier l te p 5871 now let z x be an admissible diagram over the spectrum of the ground field then the functor of admissible deformations def z x c set is defined by s isomorphism classes of deformations of x over s which are admissible of course in general one expects this deformation functor to be obstructed that this is also the case in general when x is a hypersurface in c 3 with a one dimensional reduced 2 is shown by a beautiful example due to pellikaan he considered the singularity x defined by yz 2 xz 2 xy 2 0 with z defined by the ideal yz xz xy he gave an admissible deformation over the space s defined by the equations ea eb ec 0 in c 4 and showed that there are obstructions this example has been worked out further in j s 1 ij s 2 and j s 3 example 3 3 the first example of an obstructed rational singularity has been given by pinkham pil he showed that the base space of a semi universal deformation of the cone over the rational normal curve of degree 4 is isomorphic to the space s above the nice thing is that the examples of pellikaan and pinkham are closely related the normalization of pellikaan s example is pinkham s example or to put it in another way pellikaan s example is a projection into 3 of pinkham s example

liaison classes of modules abstract we propose a concept of module liaison that extends gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative gorenstein ring analyzing the resulting equivalence classes we show that several results known for gorenstein liaison are still true in the more general case of module liaison in particular we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules as a consequence we show that the intermediate cohomology modules and properties like being perfect cohen macaulay buchsbaum or surjective buchsbaum are preserved in even module liaison classes furthermore we prove that the module liaison class of a complete intersection of codimension one consists of precisely all perfect modules of codimension one

stability in the inverse nodal solution for the interior transmission problem abstract the inverse nodal problem of determining a spherically symmetric wave speed v is considered in a bounded spherical region of radius b from zeros for eigenfunctions corresponding to the transmission eigenvalues it is shown that the space of potential functions q which correspond to interior transmission problems characterized by q q l 2 0 1 under a certain metric is homeomorphic to the partition set of the space of quasinodal sequences as a consequence the inverse nodal problem defined on the partition set of admissible sequence is stable

determination of a finite range potential from discrete phase shift data by inverse scattering a practical method of inverse scattering is proposed to determine a finite range scattering potential from phase shift data evaluated at a set of discrete energies by using the asymptotic behaviour of the phase shifts and the connection between short and long wavelength scales a relatively few input data are found to be adequate the method works well for repulsive and moderately attractive potentials useful in many applications

inverse problem with transmission eigenvalues for the discrete schrodinger equation the discrete schrodinger equation with the dirichlet boundary condition is considered on a half line lattice when the potential is real valued and compactly supported the inverse problem of recovery of the potential from the corresponding transmission eigenvalues is analyzed the marchenko method and the gel fand levitan method are used to solve the inverse problem uniquely except in one unusual case where the sum of the transmission eigenvalues is equal to a certain integer related to the support of the potential it is shown that in the unusual case there may be a unique potential corresponding to a given set of transmission eigenvalues there may be a finite number of distinct potentials for a given set of transmission eigenvalues or there may be infinitely many potentials for a given set of transmission eigenvalues the theory presented is illustrated with several explicit examples

the one dimensional inverse problem of reflection seismology an impulsive normal traction is applied uniformly over the surface of a perfectly stratified plane layered earth the ensuing particle velocity at the surface is assumed to be measured the problem of calculating the subsurface characteristic impedance from knowledge of the input pulse and the measured data is the one dimensional inverse problem of reflection seismology the problem is set up mathematically as an inverse problem for a first order 2 times 2 hyperbolic system the role of propagation of singularities is explained and the relation between the solution of the inverse problem and the cholesky factorization of certain matrices constructed from the data is elaborated for both the continuum problem and the related discrete problem it is found that recursive numerical techniques such as the levinson algorithm and certain other fast methods for cholesky factorization are intimately related to explicit finite difference schemes for solving hyperbolic systems the order of approximation between the

on the branch curve of a general projection of a surface to a plane in this paper we prove that the branch curve of a general projection of a surface to the plane is irreducible with only nodes and cusps this is a basic result in surface theory extremely useful in various applications however its proof in this general setting was so far lacking our approach substantially uses a powerful tool from projective differential geometry i e the concept of focal schemes

optimal unidirectional error detecting correcting codes in this correspondence a class of t error correcting and multiple unidirectional error detecting systematic codes is presented these codes are significantly more efficient than the earlier codes the efficiency of these codes approaches the efficiency of the bch codes asymptotically furthermore it is shown that these codes can be easily decoded also in this correspondence we have presented a generalization of berger codes over zq these new codes are also shown to be optimal

classification of quadruple galois canonical covers i in this article we classify quadruple galois canonical covers of smooth surfaces of minimal degree the classification shows that they are either non simple cyclic covers or bi double covers if they are bi double then they are all fiber products of double covers we construct examples to show that all the possibilities in the classification do exist there are implications of this classification that include the existence of families with unbounded geometric genus in sharp contrast with triple canonical covers and families with unbounded irregularity in sharp contrast with canonical covers of all other degrees together with the earlier known results on double and triple covers a pattern emerges that motivates some general questions on the existence of higher degree canonical covers some of which are answered in this article

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

numerical technique for the inverse resonance problem motivated by the work of regge nuovo cimento 8 1958 671 9 1958 491 we are interested in the problem of recovering a radial potential in r3 from its resonance parameters which are zeros of the appropriately defined jost function for a potential of compact support these may in turn be identified as the complex eigenvalues of a nonselfadjoint sturm liouville problem with an eigenparameter dependent boundary condition in this paper we propose and study a particular computational technique for this problem based on a moment problem for a function g t which is related to the boundary values of the corresponding gelfand levitan kernel

koszul modules and gorenstein algebras abstract i first define koszul modules which are a generalization to arbitrary rank of complete intersections after a study of some of their properties it is proved that gorenstein algebras of codimension one or two over a local or graded cm ring are koszul modules thus generalizing a well known statement for rank one modules the general techniques used to describe koszul modules are then used to obtain a structure theorem for gorenstein algebras in codimension one and two over a local or graded cm ring

the inverse interior transmission eigenvalue problem with mixed spectral data abstract the inverse spectral problem for the interior transmission eigenvalue problem with the unit time is studied by given spectral data the authors show that the refractive index on the whole interval can be uniquely determined by parts of its transmission eigenvalues together with the partial information on the refractive index in particular we pose and solve a new type of inverse spectral problems involving the interior transmission eigenvalue problem with complex transmission eigenvalues except for at most finite real eigenvalues

maximal families of gorenstein algebras the purpose of this paper is to study maximal irreducible families of gorenstein quotients of a polynomial ring r let gradalg h r be the scheme parametrizing graded quotients of r with hilbert function h we prove there is a close relationship between the irreducible components of gradalg h r whose general member is a gorenstein codimension c 1 quotient and the irreducible components of gradalg h r whose general member b is a codimension c cohen macaulay algebra of hilbert function h related to h if the castelnuovo mumford regularity of the gorenstein quotient is large compared to the castelnuovo mumford regularity of b this relationship actually determines a well defined injective mapping from such cohen macaulay components of gradalg h r to gorenstein components of gradalg h r in which generically smooth components correspond moreover the dimension of the gorenstein components is computed in terms of the dimension of the corresponding cohen macaulay component and a sum of two invariants of b using linkage by a complete intersection we show how to compute these invariants linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem

some remarks on the universal cover of an open k3 surface we shall give in an optimal form a sufficient numerical condition for the finiteness of the fundamental group of the smooth locus of a normal k3 surface we shall moreover prove that if the normal k3 surface is elliptic and the above fundamental group is not finite then there is a finite covering which is a complex torus

on the zero set of a holomorphic one form on a compact complex manifold on any compact complex surface m divisors of nonnegative selfintersection which are contained in the zero set or in the integral set of a holomorphic 1 form are shown to induce a fibration of m onto a riemann surface this result is extended to higher dimensions for m projective applications to zero sets of holomorphic 1 forms on surfaces are given introduction the zero set of a vector field on a compact manifold m has for some time been known to convey structural information about m in the compact holomorphic category zero sets of holomorphic vector fields provide extensions of the gauss bonnet theorem and for m kahler the dimension of the zero set of a holomorphic vector field relates to cohomological vanishing via the carrelllieberman theorem 2 in the dual case the zero set and more generally the integral set of a holomorphic 1 form can convey structural information for a compact complex manifold m in theorem 1 we show that structural information is obtained for any compact surface which carries a holomorphic 1 form pulling back to zero on a divisor d of nonnegative self intersection in theorem 2 we extend this result to the case m is higher dimensional and projective we state theorem 1 let s be any compact complex surface having a holomorphic 1form q if d is any divisor on s satisfying d d 0 and x pulls back to zero on d then there exists a holomorphic map f s r onto a compact riemann surface satisfying 1 x f r for some or e h rs 2 r 2 f has connected fibers 3 each component of d is setwise contained in a fiber of f 4 each connected component of d is a rational multiple of the natural divisor associated to the fiber of f containing it and so d d 0 notationally d d is the intersection pairing of d with itself qm denotes the sheaf of germs of holomorphic 1 forms on the manifold m a holomorphic 1 form x is said to pull back to zero on a divisor d d n3d1 if and only if iq 0 on d3 for each j where i3 dl s is inclusion we remark that if x vanishes on d then it necessarily pulls back to zero on d but the converse need not hold received by the editors july 8 1986 and in revised form may 20 1987 1980 mathematics subject classification 1985 revision primary 14j25 14j40 secondary 14c20 32j15 32l99 partially supported by nsf grant dms 8405207 1988 american mathematical society 0002 9947 88 1 00 25 per page

reconstruction of a spherically symmetric speed of sound consider the inverse acoustic scattering problem for a spherically symmetric inhomogeneity of compact support that arises among other places in nondestructive testing define the corresponding homogeneous and inhomogeneous interior transmission problems see e g d colton and p monk quart j mech math 41 1988 pp 97 125 here the authors study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem it is shown in mclaughlin and polyakov j differential equations to appear that these eigenvalues are the zeros of an average of the scattering amplitude and a uniqueness theorem for the inverse acoustic scattering problem is presented where these eigenvalues are the given data in the present paper an algorithm for finding the solution of the inverse acoustic scattering problem from this subset of transmission eigenvalues is developed and implemented the method given here completely determines the sound

multifrequency inverse problem for the reduced wave equation with sparse data the inverse problem for the reduced wave equation u k2n2 x u 0 is considered where the quantity to be determined is the value of n x in a compact domain d in r3 the data consists of a finite set of measurements of the scattered field produced by different incident fields the measurements are made at various points exterior to d and at possibly different frequencies the mathematical problem involves solving a system of nonlinear functional equations conditions are developed which indicate when the measured body is close to a particular comparison body a linearized perturbation is valid a new higher order nonlinear iterative procedure is developed for the full nonlinear problem which reduces to the usual solution in the linearized region the method is illustrated by computational results for the one dimensional case

optimal asymmetric error detecting codes some of the properties of codes capable of detecting errors when used on a binary asymmetric or z channel are examined and in fact the maximum cardinality codes are determined these results are extended to the q ary asymmetric channel introduced by varshamov 1973 ieee trans inform theory 19 92 95

a four dimensional deformation of a numerical godeaux surface a numerical godeaux surface is a surface of general type with invariants p9 q 0 and k2 1 in this paper the moduli space of a numerical godeaux surface with order two torsion is computed to be eightdimensional whether or not the moduli space of such a surface is irreducible is still unknown the surface in this paper is constructed as one member of a four parameter family of double planes there is a natural involution on the surface inherited from the double plane construction which acts on the moduli space we show that the invariant subspace is four dimensional and coincides with the family of double planes

transmission eigenvalues and a problem of hans lewy abstract subject to certain assumptions on the matrix n n x we show that except for a discrete set of values of k called transmission eigenvalues the only solution of n u k 2 u 0 v k 2 v 0 in d u v u v on d where d r n is a domain with smooth boundary d having unit outward normal is the trivial solution u v 0 this problem has important applications to the inverse scattering problem for anisotropic media and is in a class of problems first considered by lewy bull amer math soc 65 1959 37 58

a compactification of a family of determinantal godeaux surfaces in this paper we present a geometric description of the compactification of the family of determinantal godeaux surfaces via the study of the bicanonical pencil and using classical prym theory in particular we reduce the problem of compactifying the space of bicanonical pencils of determinantal godeaux surfaces to the compactification of the family of twisted cubic curves in p3 with certain given tangent conditions

multifrequency inverse problem for the reduced wave equation resolution cell and stability the multifrequency inverse problem associated with the reduced wave equation u k2n2 x u 0 x r3 is examined for the case where the data set is sparse the resolution cell or solution set is examined in detail and is shown to be an infinite dimensional manifold the concept of stability is introduced it is shown that the intrinsic condition of structural stability to the inverse process selects out a preferred set of solutions from the solution set the structural stability of various iterative schemes used in the inverse process are examined

local structure of abelian covers abstract we study normal finite abelian covers of smooth varieties in particular we establish combinatorial conditions so that a normal finite abelian cover of a smooth variety is gorenstein or locally complete intersection

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

uniqueness of the interior transmission problem with partial information on the potential and eigenvalues abstract the inverse spectral problem of determining a spherically symmetric wave speed v is considered in a bounded spherical region of radius b a uniqueness theorem for the potential q of the derived sturm liouville problem b q is presented from the data involving fractions of the eigenvalues of the problem b q on a finite interval and knowledge of q over a corresponding fraction of the interval the methods employed rest on weyl function techniques and properties of zeros of a class of entire functions

on balanced codes in a balanced code each codeword contains equally many 1 s and 0 s parallel decoding balanced codes with 2 sup r or 2 sup r 1 information bits are presented where r is the number of check bits the 2 sup 2 r 1 construction given by d e knuth ibid vol 32 no 1 p 51 3 1986 is improved the new codes are shown to be optimal when knuth s complementation method is used

on efficient balanced codes over the mth roots of unity let spl phi sub m spl sube spl copf be the set of all mth roots of unity m spl isin in a balanced code over spl phi sub m is a block code over the alphabet spl phi sub m such that each code word is balanced that is the complex sum of its components or weight is equal to 0 let b sub m n be the set of all balanced words of length n over spl phi sub m in this correspondence it is shown that when m is a prime number the set b sub m n is not empty if and only if m divides n in this case the minimum redundancy for a balanced code over spl phi sub m of length n is on the other hand it is shown that when m 4 the set b sub 4 n is not empty if and only if n is even and in this case the minimum redundancy for a balanced code over spl phi sub 4 of length n is further this correspondence completely solves the problem of designing efficient coding methods for balanced codes over spl phi sub m when m 4 in fact it reduces the problem of designing efficient coding schemes for balanced codes over spl phi sub 4 to the design of efficient balanced codes over the usual bipolar alphabet spl phi sub 2 1 1

variable length unordered codes in an unordered code no code word is contained in any other code word unordered codes are all unidirectional error detecting aued codes in the binary case it is well known that among all systematic codes with k information bits berger codes are optimal unordered codes with r log2 k 1 log2k check bits this paper gives some new theory on variable length unordered codes and introduces a new class of systematic instantaneous unordered codes with variable length check symbols the average redundancy of the new codes presented here is r 1 2 log2k c where c 1 0470 1 1332 ir and k in is the number of information bits when k is large it is shown that such redundancy is at most 0 6069 bits off the redundancy of an optimal systematic unordered code design with fixed length information symbols and variable length check symbols and at most 2 8075 bits off the redundancy of an optimal variable length unordered code design the generalization is also given for the nonbinary case and it is shown that similar results hold true

on the determination of radially dependent lame coefficients in this paper we prove that one can determine radially dependent lame coefficients lambda and mu of an isotropic elastic ball by choosing only two correspondences in the dirichlet to neumann map i e twice making measurements of displacement and stress on the surface of the ball this is a one dimensional multiunknown undetermined coefficient problem for a system of partial differential equations we use a matrix form of the transmutation mapping for solving this problem the procedure for proving the uniqueness based on transmutation can be viewed as a computational algorithm also the method we develop has the potential of solving other one dimensional multiunknown inverse problems

design of some new efficient balanced codes a balanced code with r check bits and k information bits is a binary code of length k r and cardinality 2 sup k such that each codeword is balanced that is it has k r 2 1 s and k r 2 0 s this paper contains new methods to construct efficient balanced codes to design a balanced code an information word with a low number of 1 s or 0 s is compressed and then balanced using the saved space on the other hand an information word having almost the same number of 1 s and 0 s is encoded using the single maps defined by knuth s 1986 complementation method three different constructions are presented balanced codes with r check bits and k information bits with k spl les 2 sup r 1 2 k spl les 3 spl times 2 sup r 8 and k spl les 5 spl times 2 sup r 10r c r c r spl isin 15 10 5 0 5 are given improving the constructions found in the literature in some cases the first two constructions have a parallel coding scheme