boundary singularity of poisson and harmonic bergman kernels abstract we give a complete description of the boundary behaviour of the poisson kernel and the harmonic bergman kernel of a bounded domain with smooth boundary which in some sense is an analogue of the similar description for the usual bergman kernel on a strictly pseudoconvex domain due to fefferman our main tool is the boutet de monvel calculus of pseudodifferential boundary operators and in fact we describe the boundary singularity of a general potential trace or singular green operator from that calculus

the statistical mechanics of the ising perceptron the authors investigate the problem of loading a single layered perceptron of ising or 1 weights focusing on the cases of linear and binary output neurons previous studies of this problem have been made using the canonical ensemble which in many cases require the breaking of replica symmetry they consider here an alternative approach based on the microcanonical ensemble and show that many results that were obtained previously using replica symmetry breaking within the canonical ensemble can be easily obtained using the replica symmetric assumption within the microcanonical ensemble since the nature of the replica symmetry breaking in many of the models with discrete weights is similar they believe that the results can immediately be extended to other cases such as learning a rule from examples and utilizing discrete weights of larger synaptic depth

statistical mechanics of the knapsack problem the knapsack problem is an np complete combinatorial optimization problem with inequality constraints using the replica method of statistical physics we study the space of its solutions for a large problem size it turns out that this problem is closely related to the theory of the binary perceptron

tight frames for cyclotomic fields and other rational vector spaces abstract here we consider the construction of tight frames for rational vector spaces this is a subtle question because the inner products on q d are not all isomorphic we show that a tight frame for c d can be arbitrarily approximated by a tight frame with vectors in q i q d and hence there are many tight frames for rational inner product spaces we investigate the minimal field for which there is a tight frame with a given gramian we then consider the rational vector space given the cyclotomic field q with a primitive n th root of unity we give a simple formula for the unique inner product which makes the n th roots 1 2 n 1 into a tight frame for q from this we conclude that the associated canonical coordinates have many nice properties e g multiplication in q corresponds to convolution which makes them well suited to computation along the way we give a detailed description of the space of q linear dependencies between the n th roots which includes a cyclically invariant tight frame

neural networks for optimization problems with inequality constraints the knapsack problem a strategy for finding approximate solutions to discrete optimization problems with inequality constraints using mean field neural networks is presented the constraints x d 0 are encoded by x x terms in the energy function a careful treatment of the mean field approximation for the self coupling parts of the energy is crucial and results in an essentially parameter free algorithm this methodology is extensively tested on the knapsack problem of size up to 103 items the algorithm scales like nm for problems with n items and m constraints comparisons are made with an exact branch and bound algorithm when this is computationally possible n d 30 the quality of the neural network solutions consistently lies above 95 of the optimal ones at a significantly lower cpu expense for the larger problem sizes the algorithm is compared with simulated annealing and a modified linear programming approach for nonhomogeneous problems these produce good solutions whereas for the more difficult homogeneous problems the neural approach is a winner with respect to solution quality and or cpu time consumption the approach is of course also applicable to other problems of similar structure like set covering

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

harmonic bergman functions on half spaces we study harmonic bergman functions on the upper half space of rn among our main results are the bergman projection is bounded for the range 1 p oo certain nonorthogonal projections are bounded for the range 1 p oo the dual space of the bergman ll space is the harmonic bloch space modulo constants harmonic conjugation is bounded on the bergman spaces for the range 1 p oo the bergman norm is equivalent to a normal derivative norm as well as to a tangential derivative norm

binary and quadriphase sequences with optimal autocorrelation properties a survey time discrete signals with good autocorrelation properties are used in various applications in communications engineering from an implementation point of view usually sequences with a small phase alphabet and a maximal energy efficiency i e a uniform envelope are most favorable for this reason known methods as well as several new construction techniques of binary and quadriphase sequences with optimal or best known autocorrelation properties are discussed in this correspondence in many cases the achievable correlation properties can be improved significantly if a single zero element per sequence is accepted these almost binary and almost quadriphase sequences are considered as well the optimality criteria used include the maximum absolute sidelobe and the merit factor of the periodic and the odd periodic autocorrelation function respectively for sequence lengths of up to 44 in the binary case and up to 32 in the quadriphase case the best known parameters obtained by computer search are compared with the constructed results

a generalization of circulant hadamard and conference matrices abstract we study the existence and construction of circulant matrices c of order n 2 with diagonal entries d 0 off diagonal entries 1 and mutually orthogonal rows these matrices generalize circulant conference d 0 and circulant hadamard d 1 matrices we demonstrate that matrices c exist for every order n and for d chosen such that n 2 d 2 and we find all solutions c with this property furthermore we prove that if c is symmetric or n 1 is prime or d is not an odd integer then necessarily n 2 d 2 finally we conjecture that the relation n 2 d 2 holds for every matrix c which generalizes the circulant hadamard conjecture we support the proposed conjecture by computing all the existing solutions up to n 50

prescribed absolute values character sums and spectrum integrality abstract let g be a group and let reg be the complex left regular representation of g we consider the following problem for which inverse closed subsets s g the spectrum of the matrix g s reg g is integral that is all of the eigenvalues of the matrix are integers for abelian g a complete characterization due to bridges and mena is known here we are interested in the case where g is the dihedral group d 2 n which turns out to be a far reaching generalization of the abelian case we obtain a complete characterization when n is an odd prime and present partial results when n is a prime power

nonexistence of abelian difference sets lander s conjecture for prime power orders in 1963 ryser conjectured that there are no circulant hadamard matrices of order 4 and no cyclic difference sets whose order is not coprime to the group order these conjectures are special cases of lander s conjecture which asserts that there is no abelian group with a cyclic sylow p subgroup containing a difference set of order divisible by p we verify lander s conjecture for all difference sets whose order is a power of a prime greater than 3

planar functions relative difference sets and character theory planar functions from2nto3nare studied in this paper by investigating the character values of the corresponding relative difference sets we obtain some nonexistence results of planar functions in particular we show that there is no planar functions fromzpqto4pq wherepandqare any primes we also show that except for four undecided cases there is no planar function from5nto6nifnis not a prime andn 50 000

a statistical mechanics analysis of the set covering problem the dependence of the optimal solution average cost of the set covering problem on the density of 1 s of the incidence matrix and on the number of constraints p is investigated in the limit where the number of items n goes to infinity the annealed approximation is employed to study two stochastic models the constant density model where the elements of the incidence matrix are statistically independent random variables and the karp model where the rows of the incidence matrix possess the same number of 1 s lower bounds for are presented in the case that p scales with ln n and is of order 1 as well as in the case that p scales linearly with n and is of order 1 n it is shown that in the case that p scales with exp n and is of order 1 the annealed approximation yields exact results for both models

sharp moser trudinger inequalities for the laplacian without boundary conditions we derive a sharp moser trudinger inequality for the borderline sobolev imbedding of w2 n 2 bn into the exponential class where bn is the unit ball of rn the corresponding sharp results for the spaces w0d n d are well known for general domains and are due to moser and adams when the zero boundary condition is removed the only known results are for d 1 and are due to chang yang cianchi and leckband the proof of our result is based on a new integral representation formula for the canonical solution of the poisson equation on the ball that is the unique solution of the equation u f which is orthogonal to the harmonic functions on the ball the main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation expressed in terms of its distribution function

projections for harmonic bergman spaces and applications on the setting of general bounded smooth domains in rn we construct l1 bounded nonorthogonal projections and obtain related reproducing formulas for the harmonic bergman spaces in addition we show that those projections satisfy sobolev lp estimates of any order even for p 1 among applications are gleason s problems for the harmonic bergman sobolev and little bloch functions on star shaped domains with strong reference points

new bounds and optimal binary signature sets part ii aperiodic total squared correlation we derive new bounds on the aperiodic total squared correlation atsc of binary antipodal signature sets for any number of signatures k and any signature length l we then present optimal designs that achieve the new bounds for several k l cases as interesting arguably side results we show that individual maximal merit factor sequences for example barker sequences are single user atsc optimal while neither the familiar gold nor the kasami set designs are atsc optimal in general the atsc optimal signature set designs provided in this work are in this sense better suited for asynchronous and or multipath code division multiplexing applications

cyclotomic integers and finite geometry the most powerful method for the study of finite geometries with regular or quasiregular automorphism groups g is to translate their definition into an equation over the integral group ring 2 g and to investigate this equation by applying complex representations of g for the definitions and the basic facts see section 2 if g is abelian this approach boils down to proving or disproving the existence of elements of 2 g with 0 1 coefficients whose character values are cyclotomic integers of certain prescribed absolute values up to now there have been two general methods to tackle this problem namely hall s multiplier concept and turyn s selfconjugacy approach however both methods need severe technical assumptions and thus are not applicable to many classes of problems despite many efforts over a period of more than 30 years no general method has been found to overcome these difficulties in this paper we present a new approach to the study of combinatorial structures via group ring equations which works without any restrictive assumptions in order to understand the method of the present paper it will be instructive to briefly discuss the self conjugacy concept first turyn 51 demonstrated that the character method for the study of group ring equations works very nicely under the so called self conjugacy condition an integer n is called self conjugate modulo m if all prime ideals above n in the mth cyclotomic field q m are invariant under complex conjugation under this condition it is possible to find all cyclotomic integers in q m of absolute value nt 2 for any positive integer t it is the complete knowledge of the cyclotomic integers of prescribed absolute value which makes the character method work so well under the self conjugacy condition since turyn s fundamental work 51 there have been dozens of papers extending and refining his approach however all these results are restricted to the case of self conjugacy and that is a very severe restriction indeed namely the probability that n is selfconjugate modulo m decreases exponentially fast in the number of distinct prime divisors of n and m see remark 2 2 this means that the self conjugacy method fails in almost all cases one may ask if it is possible to extend turyn s method in order to get rid of the self conjugacy assumption it turns out that in general this is impossible at least with present day methods the required complete knowledge of the cyclotomic integers of prescribed absolute value would yield an

estimates for the bergman and szego kernels in mathbf c 2 the purpose of this paper some of whose conclusions were announced in nrsw is to study the bergman and szegd projection operators on pseudoconvex domains q of finite type in c2 the results we obtain are of three kinds i the size estimates of the bergman and szeg6 kernels and their derivatives ii the cancellation properties of those kernels expressed in terms of the actions of these operators on suitable bump functions these properties have an interest in their own right but are also crucial in iii below iii the sharp mapping properties of these operators on functions spaces such as lp the nonisotropic sobolev spaces and the nonisotropic lipschitz spaces which are naturally attached to the geometry of the boundary this also leads ultimately to sharp results in the regularity properties of solutions of the equation du f in q and 9bu f on the boundary let us describe these matters in more detail our starting points are certain basic geometric constructs associated to q a metric p defined in terms of the vector fields x1 and x2 which are the real and imaginary parts of the tangential levi vector field on d 2 and a function a p 8 a polynomial in 8 for p e d 2 which represents the higher levi invariant attached to d 9 the significance of these to the geometry of d q can be understood from the following facts the ball b p 8 c d q centered at p of radius 8 in the metric p when viewed in the appropriate coordinates has width 8 in the complex tangential directions but its width is a p 8 in the complementary direction thus the volume of the ball is 82a p 8 among other crucial properties

constants of cyclotomic derivations let k x k x0 xn 1 and k y k y0 yn 1 be the polynomial rings in n 3 variables over a field k of characteristic zero containing the n th roots of unity let d be the cyclotomic derivation of k x and let be the factorisable derivation of k y associated with d that is d xj xj 1 and yj yj yj 1 yj for all j zn we describe polynomial constants and rational constants of these derivations we prove among others that the field of constants of d is a field of rational functions over k in n n variables and that the ring of constants of d is a polynomial ring if and only if n is a power of a prime moreover we show that the ring of constants of is always equal to k v where v is the product y0 yn 1 and we describe the field of constants of in two cases when n is power of a prime and when n pq

comparing the bergman and szego projections on domains with subelliptic boundary laplacian we prove that the difference between the bergman and szego projections on a bounded pseudoconvex domain with c boundary is smoothing whenever the boundary laplacian is subelliptic an equivalent statement is that the bergman projection can be represented as a composition of the szego and harmonic bergman projections along with the restriction and poisson extension operators modulo an error that is smoothing we give several applications to the study of optimal mapping properties for these projections and their difference

statistical mechanics of the multi constraint continuous knapsack problem we apply the replica analysis established by gardner to the multi constraint continuous knapsack problem which is one of the linear programming problems and a most fundamental problem in the field of operations research or for a large problem size we analyse the space of solution and its volume and estimate the optimal number of items to go into the knapsack as a function of the number of constraints we study the stability of the replica symmetric rs solution and find that the rs calculation cannot estimate the optimal number of items in the knapsack correctly if many constraints are required in order to obtain a consistent solution in the rs region we try the zero entropy approximation for this continuous solution space and get a stable solution within the rs ansatz on the other hand in the replica symmetry breaking rsb region the one step rsb solution is found by parisi s scheme it turns out that this problem is closely related to the problem of optimal storage capacity and of generalization by maximum stability rule of a spherical perceptron

new binary sequences with optimal autocorrelation magnitude new binary sequences of period n 4 2m 1 for even m ges 4 are found where the sequences are described by a 4 x 2m 1 array structure the new sequences are almost balanced and have four valued autocorrelation i e n 0 plusmn4 which is optimal with respect to autocorrelation magnitude the complete autocorrelation distribution and the exact linear complexity of the sequences are mathematically derived finally it is shown that the sequences are implemented by a combination of linear feedback shift registers and a simple logic

solving optimization problems with mean field methods abstract a brief review is given for the use of feed back artificial neural networks ann to obtain good approximate solutions to combinatorial optimization problems the key element is the mean field approximation mft which differs from conventional methods and feels its ways towards good solutions rather than fully or partly exploring different possible solutions the methodology which is illustrated for the graphs bisection and knapsack problems is easily generalized to potts systems the latter is related to the deformable templates method which is illustrated with the track finding problem the mean field approximation is based on a variational principle which also turns out to be very profitable when computing correlations in polymers

a statistical analysis of the knapsack problem we investigate the dependence of the multi knapsack objective function on the knapsack capacities and on the number of capacity constraints p in the case when all n objects are assigned the same profit value and the weights are uniformly distributed over the unit interval a rigorous upper bound to the optimal profit is obtained employing the annealed approximation and then compared with the exact value obtained through the lagrangian relaxation method the analysis is restricted to the regime where n goes to infinity and p remains finite

berezin transforms on pluriharmonic bergman spaces we show that perhaps surprisingly in several aspects the behaviour of the reproducing kernels of toeplitz operators and of the berezin transform on some weighted pluriharmonic bergman spaces is the same as in the holomorphic case

frames for vector spaces and affine spaces a finite frame for a finite dimensional hilbert space is simply a spanning sequence we show that the linear functionals given by the dual frame vectors do not depend on the inner product and thus it is possible to extend the frame expansion and other elements of frame theory to any finite spanning sequence for a vector space the corresponding coordinate functionals generalise the dual basis the case when the vectors are linearly independent and are characterised by the fact that the associated gramian matrix is an orthogonal projection existing generalisations of the frame expansion to banach spaces involve an analogue of the frame bounds and frame operator the potential applications of our results are considerable whenever there is a natural spanning set for a vector space computations can be done directly with it in an efficient and stable way we illustrate this with a diverse range of examples including multivariate spline spaces generalised barycentric coordinates and vector spaces over the rationals such as the cyclotomic fields

minimal vanishing sums of roots of unity with large coefficients a vanishing sum where n is a primitive nth root of unity and the a i s are non negative integers is called minimal if the coefficient vector a 0 a n 1 does not properly dominate the coefficient vector of any other such non zero sum we show that for every c there is a minimal vanishing sum of nth roots of unity with its greatest coefficient equal to c where n is of the form 3pq for odd primes p q this solves an open problem posed by lenstra jr

on aperiodic and periodic complementary binary sequences we give direct and recursive constructions for aperiodic and periodic complementary sequences using these constructions many missing entries in the table of bomer and antweiler 1990 can be filled

binary sequences with merit factor greater than 6 34 the maximum known asymptotic merit factor for binary sequences has been stuck at a value of 6 since the 1980s several authors have suggested that this value cannot be improved in this paper we construct an infinite family of binary sequences whose asymptotic merit factor we conjecture to be greater than 6 34 we present what we believe to be compelling evidence in support of this conjecture the numerical experimentation that led to this construction is a significant part of the story

on vanishing sums of roots of unity an unsolved problem in number theory asked the following for a given natural number m what are the possible integers n for which there exist mth roots of unity 1 n c such that 1 n 0 we show in this paper that the set of all possible n s is exactly the collection of n combinations of the prime divisors of m where n denotes the set of all non negative integers the proof is long and involves a subtle analysis of minimal vanishing sums of mth roots of unity couched in the setting of integral group rings of finite cyclic groups our techniques also recovered with ease some of the classical results on vanishing sums of roots of unity such as those of redei de bruijn and schoenberg

birational mappings and matrix subalgebra from the chiral potts model we study birational transformations of the projective space originating from lattice statistical mechanics specifically from various chiral potts models associating these models with stable patterns and signed patterns we give general results which allow us to find all chiral q state spin edge potts models when the number of states q is a prime or the square of a prime as well as several q dependent family of models we also prove the absence of monocolor stable signed pattern with more than four states this demonstrates a conjecture about cyclic hadamard matrices in a particular case the birational transformations associated with these lattice spin edge models show complexity reduction in particular we recover a one parameter family of integrable transformations for which we give a matrix representation when the parameter has a suitable value

the multidimensional 0 1 knapsack problem an overview abstract the multidimensional 0 1 knapsack problem is one of the most well known integer programming problems and has received wide attention from the operational research community during the last four decades although recent advances have made possible the solution of medium size instances solving this np hard problem remains a very interesting challenge especially when the number of constraints increases this paper surveys the main results published in the literature the focus is on the theoretical properties as well as approximate or exact solutions of this special 0 1 program

sufficient conditions for a conjecture of ryser about hadamard circulant matrices let h be a hadamard circulant matrix of order n 4h2 where h 1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial n t are bounded by a constant independent of n then for all the n n th primitive roots w of 1 p w n is not an algebraic integer in the cyclotomic field k q w where p t is the representer polynomial of h and is the euler function this implies that p w is not a real number

estimates of the harmonic bergman kernel on smooth domains abstract we obtain optimal size estimates of the harmonic bergman kernel and its derivatives on smooth domains based on these estimates we derive mapping properties of the harmonic bergman projection on lebesgue spaces and lipschitz spaces

probabilistic analysis of the number partitioning problem given a sequence of n positive real numbers the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of over the two sets is minimized in the case in which the s are statistically independent random variables uniformly distributed in the unit interval this np complete problem is equivalent to the problem of finding the ground state of an infinite range random antiferromagnetic ising model we employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large n limit furthermore we calculate analytically the fraction of metastable states i e states that are stable against all single spin flips and found that it vanishes like