provably good global routing of integrated circuits this paper investigates the global routing problem for integrated circuits we introduce a formulation on the basis of integer programming which minimizes the routing area among a limited set of steiner trees for each net indeed the involved cost function depends on the channel density of the routing which has a direct influence on the routing area our methods for solving the global routing problem employ local search heuristics sequential routing genetic routing and randomized procedures our methods for computing lower bounds are based on linear and lagrange relaxation an analysis on the tightness of the bounds indicates that the difference between the cost of the optimal integer solutions and the cost of the optimal fractional solutions is only a small number of tracks in practice moreover the analysis leads to the concept of linear preprocessing by which we exclude a large number of high cost solutions we introduce several versions of preprocessing one of which preserves the opportunity of obtaining a globally optimal solution in general all of them do so in practice linear preprocessing enables us to solve problem instances with several thousand nets provably optimal or at least provably close to optimal all methods have been implemented in the software package eridanus we present computational results the global routing problem assumes the placement of the chip components to be fixed an extension of the problem which we call global layout of integrated circuits allows the placement to be variable and searches for a placement that minimizes the routing area among a limited set of alternatives we show that the results concerning the global routing problem can be easily extended to global layout

the lax pairs for the holt system by using the known relation between the holt system and the henon heiles system the lax pairs for all the integrable cases of the holt system are constructed from the known lax representations for the henon heiles system

separation of variables for the generalized henon heiles system and system with quartic potential we consider two well known integrable systems on the plane using the concept of natural poisson bivectors on riemannian manifolds a geometric approach to construction of variables of separation and separated relations for the generalized henon heiles system and the generalized system with quartic potential is discussed in detail

non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

cubic theta functions abstract the theory of the cubic theta function q m 2 m n n 2 z 1 m z 2 n is developed analogously to the theory of the classical jacobian theta functions many inversion formulas addition formulas and modular equations are derived and these results extend previous work on cubic theta functions

performance bounds for erasure list and decision feedback schemes with linear block codes a message independence property and some new performance upper bounds are derived in this work for erasure list and decision feedback schemes with linear block codes transmitted over memoryless symmetric channels similar to the classical work of forney this work is focused on the derivation of some gallager type bounds on the achievable tradeoffs for these coding schemes where the main novelty is the suitability of the bounds for both random and structured linear block codes or code ensembles the bounds are applicable to finite length codes and to the asymptotic case of infinite block length and they are applied to low density parity check code ensembles

canonical transformation between integrable henon heiles systems a canonical transformation between two known integrable cases of the h enon heiles systems is given and the separation of variables for the corresponding hamilton jacobi equations is discussed

on the structure of picard fuchs type equations for liouville arnold integrable hamiltonian system on cotangent phase spaces there are studied in detail the structure properties of integral submanifold imbedding mapping for a class of algebraically liouville integrable hamiltonian systems on cotangent phase spaces and related with it so called picard fuchs type equations it is shown that these equations can be in general regularly constructed making use of a given a priori system of involutive invariants and proved that their solutions in the hamolton jacobi separable variable case give rise exactly to the integral submanifold imbedding mapping being as known a main ingredient for liouville arnold integrability by quadratures of the hamiltonian system under regard

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

the maximum concurrent flow problem the maximum concurrent flow problem mcfp is a multicommodity flow problem in which every pair of entities can send and receive flow concurrently the ratio of the flow supplied between a pair of entities to the predefined demand for that pair is called throughput and must be the same for all pairs of entities for a concurrent flow the mcfp objective is to maximize the throughput subject to the capacity constraints we develop a fully polynomial time approximation scheme for the mcfp for the case of arbitrary demands and uniform capacity computational results are presented it is shown that the problem of associating costs distances to the edges so as to maximize the minimum cost of routing the concurrent flow is the dual of the mcfp a path cut type duality theorem to expose the combinatorial structure of the mcfp is also derived our duality theorems are proved constructively for arbitrary demands and uniform capacity using the algorithm applications include packet switched networks 1 4 8 and cluster analysis 16

ratio list decoding we extend the notion of list decoding to ratio list decoding which involves a list decoder whose list size is specified as a function of the number of messages m n and the block length n we present the necessary and sufficient conditions on m n for the existence of code sequences which enable reliable list decoding with respect to the desired list size l m n n it is shown that the ratio capacity defined as the supremum of achievable normalized logarithms of the ratio r m n n m n l m n n is equal to the shannon channel capacity c for both stochastic and deterministic encoding allowing for random list size we are able to deduce some properties of identification codes where the decoder s output can be viewed as a list of messages corresponding to decision regions that include the channel output we further address the regime of mismatched list decoding in which the list constitutes of the codewords that accumulate the highest score values jointly with the channel output according to some given function we study the case of deterministic encoding and mismatched ratio list decoding we establish similar necessary and sufficient conditions for the existence of code sequences which enable reliable mismatched list decoding with respect to the desired list size l m n n and we show that the ratio capacity with mismatched decoding is equal to the mismatch capacity focusing on the case of an exponential list size l n e n theta its comparison with ordinary mismatched decoding shows that the increase in capacity is by theta bits per channel used for all channels and decoding metrics several properties of the average error probability in the setup of mismatched list decoding with deterministic list size are provided

quasi periodic and periodic solutions for coupled nonlinear schrodinger equations of manakov type we consider travelling periodic and quasi periodic wave solutions in coupled nonlinear schrodinger equations in fibre optics these equations can be used to model single mode fibres with strong birefringence and two mode optical fibres recently these equations appear as a model describing pulse pulse interactions in wavelength division multiplexed channels of optical fibre transmission systems in some cases this model reduces to the integrable manakov system ims two phase quasi periodic solutions for the ims are given in terms of two dimensional kleinian functions the reduction of quasi periodic solutions to elliptic functions is discussed new solutions are found in terms of generalized hermite polynomials which are associated with two gap treibich verdier potentials

packing steiner trees a cutting plane algorithm and computational results in this paper we describe a cutting plane algorithm for the steiner tree packing problem we use our algorithm to solve some switchbox routing problems of vlsi design and report on our computational experience this includes a brief discussion of separation algorithms a new lp based primal heuristic and implementation details the paper is based on the polyhedral theory for the steiner tree packing polyhedron developed in our companion paper this issue and meant to turn this theory into an algorithmic tool for the solution of practical problems

lax representation for two particle dynamics splitting on two tori the lax representation in terms of matrices is constructed for separable multiply periodic systems splitting on two tori hyperelliptic kleinian functions and their reduction to elliptic functions are used

exact solutions to a nonlinear reaction diffusion equation and hyperelliptic integrals inversion an approach is proposed to obtain some exact explicit stationary solutions in terms of elliptic functions to a nonlinear reaction diffusion equation the method is based on the reduction of the hyperelliptic integral to the elliptic one and its inversion via the weierstrass and jacobi elliptic functions the solutions for both polynomial reaction and diffusion functions include bounded periodic and localized in space functions such solutions seem to be the best candidates to describe periodic nanostructures observed in experiments on formation of thin films by means of molecular epitaxy the so called quantum wires generalization of the approach is discussed for reaction and diffusion functions distinctive from polynomials in particular explicit stationary solutions are found in terms of elliptic functions for arbitrary diffusion and relevant reaction terms

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

improved analysis of list decoding and its application to convolutional codes and turbo codes a list decoder generates a list of more than one codeword candidates and decoding is erroneous if the transmitted codeword is not included in the list this decoding strategy can be implemented in a system that employs an inner error correcting code and an outer error detecting code that is used to choose the correct codeword from the list probability of codeword error analysis for a linear block code with list decoding is typically based on the worst case lower bound on the effective weights of codewords for list decoding evaluated from the weight enumerating function of the code in this paper the concepts of generalized pairwise error event and effective weight enumerating function are proposed for evaluation of the probability of codeword error of linear block codes with list decoding geometrical analysis shows that the effective euclidean distances are not necessarily as low as those predicted by the lower bound an approach to evaluate the effective weight enumerating function of a particular code with list decoding is proposed the effective euclidean distances for decisions in each pairwise error event are evaluated taking into consideration the actual hamming distance relationships between codewords which relaxes the pessimistic assumptions upon which the traditional lower bound analysis is based using the effective weight enumerating function a more accurate approximation is achieved for the probability of codeword error of the code with list decoding the proposed approach is applied to codes of practical interest including terminated convolutional codes and turbo codes with the parallel concatenation structure

list decoding random coding exponents and expurgated exponents some new results are derived concerning random coding error exponents and expurgated exponents for list decoding with a deterministic list size l two asymptotic regimes are considered the fixed list size regime where l is fixed independently of the block length n and exponential list size where l grows exponentially with n we first derive a general upper bound on the list decoding average error probability which is suitable for both regimes this bound leads to more specific bounds in the two regimes in the fixed list size regime the bound is related to known bounds and we establish its exponential tightness in the exponential list size regime we establish the achievability of the well known sphere packing lower bound an immediate byproduct of our analysis in both regimes is the universality of the maximum mutual information list decoder in the error exponent sense finally we consider expurgated bounds at low rates both using gallager s approach and csiszar korner marton approach which is equivalent for the optimal input assignment the latter expurgated bound which involves the notion of multi information is also modified to apply to continuous alphabet channels and in particular to the gaussian memoryless channel where the expression of the expurgated bound becomes quite explicit

list decoding of turbo codes list decoding of turbo codes is analyzed under the assumption of a maximum likelihood ml list decoder it is shown that large asymptotic gains can be achieved on both the additive white gaussian noise awgn and fully interleaved flat rayleigh fading channels it is also shown that the relative asymptotic gains for turbo codes are larger than those for convolutional codes finally a practical list decoding algorithm based on the list output viterbi algorithm lova is proposed as an approximation to the ml list decoder simulation results show that the proposed algorithm provides significant gains corroborating the analytical results the asymptotic gain manifests itself as a reduction in the bit error rate ber and frame error rate fer floor of turbo codes

comment on improved analysis of list decoding and its application to convolutional codes and turbo codes in a recent paper bai ieee trans inf theory 2007 an improved analysis concerning the analysis of list decoding was presented the event that the correct codeword is excluded from the list is central for the additive white gaussian noise awgn channel an important quantity is the in bai ieee trans inf theory 2007 called effective euclidean distance this was earlier considered in aulin ieee trans commun 1999 under the name vector euclidean distance ved where also a simple mathematical expression for this quantity was easily derived for any list size in bai ieee trans inf theory 2007 a geometrical analysis gives this when the list size is 1 2 or 3

packing steiner trees polyhedral investigations letg v e be a graph andt v be a node set we call an edge sets a steiner tree fort ifs connects all pairs of nodes int in this paper we address the following problem which we call the weighted steiner tree packing problem given a graphg v e with edge weightswe edge capacitiesce e e and node sett1 tn find edge setss1 sn such that eachsk is a steiner tree fortk at mostce of these edge sets use edgee for eache e and the sum of the weights of the edge sets is minimal our motivation for studying this problem arises from a routing problem in vlsi design where given sets of points have to be connected by wires we consider the steiner tree packing problem from a polyhedral point of view and define an associated polyhedron called the steiner tree packing polyhedron the goal of this paper is to partially describe this polyhedron by means of inequalities it turns out that under mild assumptions each inequality that defines a facet for the single steiner tree polyhedron can be lifted to a facet defining inequality for the steiner tree packing polyhedron the main emphasis of this paper lies on the presentation of so called joint inequalities that are valid and facet defining for this polyhedron inequalities of this kind involve at least two steiner trees the classes of inequalities we have found form the basis of a branch cut algorithm this algorithm is described in our companion paper in this issue

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

separability and lax pairs for henon heiles system the hamiltonian system corresponding to the generalized henon heiles hamiltonian h 1 2 px2 py2 1 2ax2 1 2by2 x2y ey3 is known to be integrable in the following three cases a b e 1 3 e 2 b 16a e 16 3 in the first two the system has been integrated by making use of genus one and genus two theta functions we show that in the third case the system can also be integrated by making use of elliptic functions finally using the fairbanks theorem we find lax pairs for each of the three integrable systems under consideration

unveiling turbo codes some results on parallel concatenated coding schemes a parallel concatenated coding scheme consists of two simple constituent systematic encoders linked by an interleaver the input bits to the first encoder are scrambled by the interleaver before entering the second encoder the codeword of the parallel concatenated code consists of the input bits to the first encoder followed by the parity check bits of both encoders this construction can be generalized to any number of constituent codes parallel concatenated schemes employing two convolutional codes as constituent codes in connection with an iterative decoding algorithm of complexity comparable to that of the constituent codes have been previously shown to yield remarkable coding gains close to theoretical limits they have been named and are known as turbo codes we propose a method to evaluate an upper bound to the bit error probability of a parallel concatenated coding scheme averaged over all interleavers of a given length the analytical bounding technique is then used to shed some light on some crucial questions which have been floating around in the communications community since the proposal of turbo codes

the stackel systems and algebraic curves we show how the abel jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves we prove that derivative of the abel jacobi map is just the stackel matrix which determines n orthogonal curvilinear coordinate systems in a flat space the lax pairs r matrix algebras and explicit form of the flat coordinates are constructed an application of the weierstrass reduction theory allows us to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space

new deterministic interleaver designs for turbo codes it is well known that an interleaver with random properties quite often generated by pseudo random algorithms is one of the essential building blocks of turbo codes however randomly generated interleavers have two major drawbacks lack of an adequate analysis that guarantees their performance and lack of a compact representation that leads to a simple implementation we present several new classes of deterministic interleavers of length n with construction complexity o n that permute a sequence of bits with nearly the same statistical distribution as a random interleaver and perform as well as or better than the average of a set of random interleavers the new classes of deterministic interleavers have a very simple representation based on quadratic congruences and hence have a structure that allows the possibility of analysis as well as a straightforward implementation using the new interleavers a turbo code of length 16384 that is only 0 7 db away from capacy at a bit error rate ber of 10 sup 5 is constructed we also generalize the theory of previously known deterministic interleavers that are based on block interleavers and we apply this theory to the construction of a nonrandom turbo code of length 16384 with a very regular structure whose performance is only 1 1 db away from capacity at a ber of 10 sup 5

treibich verdier potentials and the stationary m kdv hierarchy here x x col co3 denotes the elliptic weierstrass function with fundamental periods f p 2a l 2co3 im co3 wl 5 0 see 1 ch 18 in the special case where col is real and co3 is purely imaginary the potential q is real valued and ince s striking result 31 in modern spectral theoretic terminology yields the fact that the self adjoint operator l associated with the differential expression d 2 q in l2 ir has a finite gap spectrum of the type