low firing rates in a compartmental model neuron a nonlinear relationship is obtained between the membrane potential of a compartmental model neuron as a function of input stimulation this is used to show how shunting effects can lead to low output firing rates in associative memory networks

on the existence of supervisory policies that enforce liveness in discrete event dynamic systems modeled by controlled petri nets we consider discrete state plants represented by controlled petri nets ctlpns where a subset of transitions can be prevented from firing by a supervisor a transition in a ctlpn can fire at a marking if there are sufficient tokens in its input places and it is permitted to fire by the supervisor a ctlpn is live if it is possible to fire any transition from every marking that is reachable under supervision in this paper we derive a necessary and sufficient condition for the existence of a supervisory policy that enforces liveness in ctlpns we show this condition cannot be tested for an arbitrary ctlpn however for bounded ctlpns or ctlpns where each transition is individually controllable we show the existence of a supervisory policy which enforces that liveness is decidable we also show the existence of a supervisory policy that enforces liveness is necessary and sufficient for the existence of a minimally restrictive supervisory policy

o n3 noniterative heuristic algorithm for linear programs with error free implementation we present an o n 3 noniterative heuristic algorithm and describe an implementation of p adic arithmetic to solve the linear programming problems error free an optimality test as well as numerical examples depict the scope of such an algorithm in the few cases where it did not lead to the optimal solution it did provide a nonnegative solution of the system ax b which could be used as the basic feasible solution for any one of the widely used simplex methods say the revised simplex algorithm without the need to include artificial variables in ax b for consistency check thus the algorithm could be at least a pre processor for any simplex algorithm and avoids the enhancement of dimensions of the constraint matrix a and consequent additional space computational complexities and errors

moments of inertia and the shapes of brownian paths we compute the joint probability law of the principal moments of inertia of brownian paths open or closed using constrained path integrals and random matrix theory the case of two dimensional paths is discussed in detail in particular we show that the ratio of the average values of the largest and smallest moments is equal to 4 99 open paths and 3 07 closed paths we also present results of numerical simulations which include investigation of the relationships between the moments of inertia and the arithmetic area enclosed by a path

shapes and sizes of arbitrary random walks at o 1 d3 ii asphericity and prolateness parameters abstract for arbitrary random walks in any d dimensional space expansions in powers of 1 d of asphericity and prolateness parameters and moments of the inverse size ratio have been developed which at o 1 d 3 yield very good approximations to exact values of the parameters for chains rings dumbbells and 3 and 5 arm regular stars the 1 d expansions have also been used to obtain an estimate of these shape asymmetry parameters for 3d edwards chains rings dumbbells and 3 arm stars and to give a mathematical proof that infinitely large random nets such as bethe lattice or starburst and mckay s net exhibit spherical symmetry for arbitrary random walks at d it is proved that these parameters coincide with their corresponding factors while for an end looped self avoiding walk it is found that its shape asymmetry is even larger than that of an open saw an 1 f expansion of the parameters for f arm regular stars has also been obtained and a comparison of the dimensionality dependence of the parameters with that of the corresponding factors has been made for the four types of random walks

weighted complementarity problems a new paradigm for computing equilibria this paper introduces the notion of a weighted complementarity problem wcp which consists of finding a pair of vectors x s belonging to the intersection of a manifold with a cone such that their product in a certain algebra x circ s equals a given weight vector w when w is the zero vector then wcp reduces to a complementarity problem the motivation for introducing the more general notion of a wcp lies in the fact that several equilibrium problems in economics can be formulated in a natural way as wcps moreover those formulations lend themselves to the development of highly efficient algorithms for solving the corresponding equilibrium problems for example fisher s competitive market equilibrium model can be formulated as a wcp that can be efficiently solved by interior point methods moreover it is shown that the quadratic programming and weighted centering problem which generalizes the notion of a linear programming and weighted centering problem recently proposed by anstreicher

a survey of search directions in interior point methods for linear programming a basic characteristic of an interior point algorithm for linear programming is the search direction many papers on interior point algorithms only give an implicit description of the search direction in this report we derive explicit expressions for the search directions used in many well known algorithms comparing these explicit expressions gives a good insight into the similarities and differences between the various algorithms moreover we give a survey of projected gradient and newton directions for all potential and barrier functions this is done both for the affine and projective variants

a cutting plane algorithm for convex programming that uses analytic centers anoracle for a convex sets n accepts as input any pointz in n and ifz s then it returns yes while ifz s then it returns no along with a separating hyperplane we give a new algorithm that finds a feasible point ins in cases where an oracle is available our algorithm uses the analytic center of a polytope as test point and successively modifies the polytope with the separating hyperplanes returned by the oracle the key to establishing convergence is that hyperplanes judged to be unimportant are pruned from the polytope if a ball of radius 2 l is contained ins ands is contained in a cube of side 2l 1 then we can show our algorithm converges after o nl2 iterations and performs a total of o n4l3 tnl2 arithmetic operations wheret is the number of arithmetic operations required for a call to the oracle the bound is independent of the number of hyperplanes generated in the algorithm an important application in which an oracle is available is minimizing a convex function overs

interior point methods an old and new approach to nonlinear programming in this paper we discuss the main concepts of structural optimization a field of nonlinear programming which was formed by the intensive development of modern interior point schemes

ellipsoids that contain all the solutions of a positive semi definite linear complementarity problem this paper deals with the lcp linear complementarity problem with a positive semi definite matrix assuming that a strictly positive feasible solution of the lcp is available we propose ellipsoids each of which contains all the solutions of the lcp we use such an ellipsoid for computing a lower bound and an upper bound for each coordinate of the solutions of the lcp we can apply the lower bound to test whether a given variable is positive over the solution set of the lcp that is if the lower bound is positive we know that the variable is positive over the solution set of the lcp hence by the complementarity condition its complement is zero in this case we can eliminate the variable and its complement from the lcp we also show how we efficiently combine the ellipsoid method for computing bounds for the solution set with the path following algorithm proposed by the authors for the lcp if the lcp has a unique non degenerate solution the lower bound and the upper bound for the solution computed at each iteration of the path following algorithm both converge to the solution of the lcp

an o n 3 l adaptive path following algorithm for a linear complementarity problem in this paper we propose an o n3l algorithm which is a modification of the path following algorithm 8 for a linear complementarity problem the path following algorithm has to take a short step size in each iteration in order to bound the number of overall arithmetic operations by o n3l in practical computation we can determine the step size adaptively mizuno yoshise and kikuchi 11 reported that such an adaptive algorithm required about o l iterations for some test problems here we show that we can use a rank one update technique in the adaptive algorithm so that the number of overall arithmetic operations is theoretically bounded by o n3l

the aspherity of random walks the asphericity a of a random walk is defined and calculated for non self avoiding walks the definition of a generalises to self avoiding walks percolating clusters and other fractal objects and thus provides a generalised quantitative measure of the departure from spherical symmetry of the gross shape of these objects

karmarkar s linear programming algorithm and newton s method this paper describes a full dimensional version of karmarkar s linear programming algorithm theprojective scaling algorithm which is defined for any linear program in n having a bounded full dimensional polytope of feasible solutions if such a linear program hasm inequality constraints then it is equivalent under an injective affine mappingj n m to karmarkar s original algorithm for a linear program in m havingm n equality constraints andm inequality constraints karmarkar s original algorithm minimizes a potential functiong x and the projective scaling algorithm is equivalent to that version of karmarkar s algorithm whose step size minimizes the potential function in the step direction

a simple dynamical model of two dimensional vesicles two dimensional vesicles are modelled as pressurized self intersecting rings the fluctuating motion of the bounding walls of the vesicle when it is subjected to the random collisions of fluid molecules is treated along the lines suggested by rouse 1953 in his study of dilute polymer solutions within the model exact solutions of the dynamical equation governing the probability distribution of shapes when expressed in an appropriate set of normal coordinates is obtained numerical simulations of the dynamical evolution of the distribution of the principal components of the square of the radius of gyration are also presented

a nonlinear analytic center cutting plane method for a class of convex programming problems a cutting plane algorithm for minimizing a convex function subject to constraints defined by a separation oracle is presented the algorithm is based on approximate analytic centers the nonlinearity of the objective function is taken into account yet the feasible region is approximated by a containing polytope this containing polytope is regularly updated by adding a new cut through a test point each test point is an approximate analytic center of the intersection of a containing polytope and a level set of the nonlinear objective function we establish the complexity of the algorithm our complexity estimate is given in terms of the problem dimension the desired accuracy of an approximate solution and other parameters that depend on the geometry of a specific instance of the problem

study of the shape of random walks a comprehensive study of the shape of discrete random walks considering the general case of arbitrary length and any space dimension is presented the probability distributions for several magnitudes such as the principal inertia moments the asphericity and the angle between the principal axis of inertia and the end to end vector are evaluated numerically in most cases and especially in low dimensional spaces these probability distributions spread widely and possess a non zero skewness this implies that any description of the shapes of random walks which is based only on mean values of related magnitudes is incomplete this situation does nor vary when random walks of large lengths are considered since the relative fluctuations ratio between the standard deviation and the mean value do not go to zero in that limit on the other hand when the space dimension grows the distributions become more peaked reaching the expected asymptotic limit for infinite dimension the present study also indicates that the probability distributions for the principal inertia moments have approximately the form of chi squared distributions analysing the probability distribution for the asphericity it is shown that the distributions for the different moments are not completely independent

a method for the parametric center problem with a strictly monotone polynomial time algorithm for linear programming given a system of linear inequalities ax b dt and equalities mx g ht where the right hand sides are parametrically deformed over the scalar t the parametric center problem is to trace the parametric family of approximate solutions x t to the center problems p t where p t is the problem maximize i 1m ln bi dit aix subject to ax b dt and mx g ht we present an algorithm for tracing the parametric family of solutions x t over some given range at the kth iterate of the algorithm the point x tk is an approximate solution to the center problem p tk in an appropriate measure of approximation the value of tk is increased to tk 1 and a newton step is used to generate x tk 1 which is an approximate solution to the center problem p tk 1 the sequence of values of t exhibit at least one of the following geometric rates of change tmax tk 1 tmax tk 1 1 128m tk 1 tmin tk tmin 1 1 128m where tmin tmax is the maximum minimum number t such t

the shapes and sizes of two dimensional pressurized self intersecting rings as models for two dimensional vesicles two dimensional vesicles are modelled as pressurized unrestricted closed random walks the model though suffering from undesirable physical characteristics has the decided advantage of yielding exact analytical statistical analysis of the size and shapes of these objects closed form expressions for the average of the square of the radius of gyration and the asphericity are derived in addition exact expressions for the full probability distribution are also obtained numerical computations of the probability function for several pressures are presented the authors arrive at a new exact formula for the distribution function p r12 r22 of the principal components of the square of the radius of gyration for two dimensional rings

the shapes of open and closed random walks a 1 d expansion a new technique for calculating the shapes of random walks is presented the method is used to derive an exact analytical expression for the asphericity of an unrestricted closed or ring walk embedded in d spatial dimensions a graphical procedure is developed to systematise a 1 d series expansion for the individual principal radii of gyration and their respective probability distribution functions p r2i 1 or i or d the average principal radii of gyration are calculated to o 1 d2 for both open and closed walks and selected terms in the 1 d expansion are summed to all orders in 1 d in the determination of p r2i this leads to an explicit analytical form for p r2i for open walks the distribution of the largest eigenvalue is compared with a distribution obtained from numerical simulations of walks in three dimensions the agreement between the two is extremely good other predictions for various parameters that characterise the average shape of open and closed walks in three dimensions are also found to agree remarkably well with the results of simulations the error being of the order of 5

shapes and sizes of arbitrary random walks at o 1 d3 i most probable size ratio abstract for arbitrary random walks in any d dimensional space a 1 d expansion of the most probable size ratio i e squared radius of gyration s 2 divided by s 2 of open random walks has been developed which at o 1 d 3 yields a very good approximation to the exact value for chains d 2 and rings d 1 and for the first time gives an estimate of the most probable size ratio for end looped random walks asymptotic distribution functions for large and small size ratio have also been investigated analytically for open and closed random walks with explicit results given up to the fourth order for any values of d for random walks at d it has been proved that the most probable size coincides with mean size and the th shape factor is inversely proportional to the th eigenvalue of the architecture matrix for the walks

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

an ol n 3 primal interior point algorithm for convex quadratic programming we present a primal interior point method for convex quadratic programming which is based upon a logarithmic barrier function approach this approach generates a sequence of problems each of which is approximately solved by taking a single newton step it is shown that the method requires o sqrt n l iterations and o n3 5l arithmetic operations by using modified newton steps the number of arithmetic operations required by the algorithm can be reduced to o n3l

a scaling technique for finding the weighted analytical center of a polytope let a bounded full dimensional polytope be defined by the systemax b wherea is anm n matrix letai denote theith row of the matrixa and define theweighted analytic center of the polytope to be the point that minimizes the strictly convex barrier function i 1mwi ln aitx bi the proper selection of weightswi can make any desired point in the interior of the polytope become the weighted analytic center as a result the weighted analytic center has applications in both linear and general convex programming for simplicity we assume that the weights are positive integers if some of thewi s are much larger than others then newton s method for minimizing the resulting barrier function is very unstable and can be very slow previous methods for finding the weighted analytic center relied upon a rather direct application of newton s method potentially resulting in very slow global convergence we present a method for finding the weighted analytic center that is based on the scaling technique of edmonds and karp and is an enhancement of newton s method the scaling algorithm runs in o sqrt m log w iterations wherem is the number of constraints defining the polytope andw is the largest weight given on any constraint each iteration involves taking a step in the newton direction and its complexity is dominated by the time needed to solve a system of linear equations

a cutting plane method from analytic centers for stochastic programming the stochastic linear programming problem with recourse has a dual block angular structure it can thus be handled by benders decomposition or by kelley s method of cutting planes equivalently the dual problem has a primal block angular structure and can be handled by dantzig wolfe decomposition the two approaches are in fact identical by duality here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations the only significant difference form the aforementioned methods is that new cutting planes or columns by duality will be generated not from the optimum of the linear programming relaxation but from the analytic center of the set of localization

on vaidya s volumetric cutting plane method for convex programming we describe a simplified and strengthened version of vaidya s volumetric cutting plane method for finding a point in a convex set x1d49e rn at each step the algorithm has a system of linear inequality constraints which defines a polyhedron x1d4ab x1d49e and an interior point x x1d4ab the algorithm then either drops one constraint or calls an oracle to check if x x1d49e and if not obtain a new constraint that separates x from x1d49e following the addition or deletion of a constraint the algorithm takes a small number of newton steps for the volumetric barrier v progress of the algorithm is measured in terms of changes in v the algorithm is terminated when either it is discovered that x x1d49e or v becomes large enough to demonstrate that the volume of x1d49e must be below some prescribed amount the complexity of the algorithm compares favorably with that of the ellipsoid method especially in terms of the number of calls to the separation oracle compared to vaidya s original analysis we decrease the total number

a new algorithm for minimizing convex functions over convex sets let s subseteq mathbb r n be a convex set for which there is an oracle with the following property given any pointz n the oracle returns a yes ifz s whereas ifz s then the oracle returns a no together with a hyperplane that separatesz froms the feasibility problem is the problem of finding a point ins the convex optimization problem is the problem of minimizing a convex function overs we present a new algorithm for the feasibility problem the notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm the algorithm for the feasibility problem easily adapts to the convex optimization problem

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

reciprocal maximum length sequence pairs for acoustical dual source measurements in this paper we propose and demonstrate a method to obtain simultaneous dual source receiver impulse responses in acoustical systems using binary maximum length sequences mls a binary mls and its reversed order sequence form a reciprocal mls pair their correlation property includes a two valued pulse like autocorrelation function and a relatively smaller valued cross correlation function this unique property along with other number theory properties makes the reciprocal mls pair suitable for simultaneous dual source cross correlation measurements in the measurement of a dual source system each of the reciprocal mls pairs simultaneously excite one of two separate sources one or several receiver signals cross correlate in turn with each of the mls pairs resulting in impulse responses associated with two separate sources the proposed method is particularly valuable for system identification tasks with multiple sound vibration sources and receivers that have to be accomplished in a limited time period a fast algorithm called a fast mls transform is exploited for the cross correlation in this paper we propose a fast mls transform pair for the reciprocal mls pairs its efficiency lies in the requirement of one single permutation matrix for a pair of two fast mls transforms its feasibility and usefulness in the acoustical measurements are demonstrated using experimental results

approximation algorithms for inventory problems with submodular or routing costs we consider the following two deterministic inventory optimization problems with non stationary demands submodular joint replenishment problem this involves multiple item types and a single retailer who faces demands over a finite planning horizon of t periods in each time period any subset of item types can be ordered incurring a joint ordering cost which is submodular moreover items can be held in inventory while incurring a holding cost the objective is to find a sequence of orders that satisfies all demands and minimizes the total ordering and holding costs inventory routing problem this involves a single depot that stocks items and multiple retailer locations facing demands over a finite planning horizon of t periods in each time period any subset of locations can be visited using a vehicle originating from the depot there is also cost incurred for holding items at any retailer the objective here is to satisfy all demands while minimizing the sum of routing and holding costs we present a unified approach that yields mathcal o left frac log t log log t right ologtloglogt factor approximation algorithms for both problems when the holding costs are polynomial functions a special case is the classic linear holding cost model wherein this is the first sub logarithmic approximation ratio for either problem

an algorithm for linear programming which requires o m n n 2 m n 1 5 n l arithmetic operations we present an algorithm for linear programming which requires o m n n 2 m n 1 5 n l arithmetic operations wherem is the number of constraints andn is the number of variables each operation is performed to a precision of o l bits l is bounded by the number of bits in the input the worst case running time of the algorithm is better than that of karmarkar s algorithm by a factor of sqrt m n

the nonlinear geometry of linear programming ii legendre transform coordinates and central trajectories karmarkar s projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem this paper studies the set of trajectories obtained by integrating this vector field called p trajectories as well as a related set of trajectories called a trajectories the a trajectories arise from another linear programming algorithm the affine scaling algorithm the affine and projective scaling vector fields are each defined for linear programs of a special form called standard form and canonical form respectively these trajectories are studied using a nonlinear change of variables called legendre transform coordinates which is a projection of the gradient of a logarithmic barrier function the legendre transform coordinate mapping is given by rational functions and its inverse mapping is algebraic it depends only on the constraints of the linear program and is a one to one mapping for canonical form linear programs when the polytope of feasible solutions is bounded there is a unique point mapping to zero called the center the a trajectories of standard form linear programs are linearized by the legendre transform coordinate mapping when the polytope of feasible solutions is bounded they are the complete set of geodesics of a riemannian geometry isometric to euclidean geometry each a trajectory is part of a real algebraic curve each p trajectory for a canonical form linear program lies in a plane in legendre transform coordinates the p trajectory through 0 in legendre transform coordinates called the central p trajectory is part of a straight line and is contained in the a trajectory through 0 called the central a trajectory each p trajectory is part of a real algebraic curve the central a trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form c x u it is also the set of minima of a parametrized family of logarithmic barrier functions power series expansions are derived for the central a trajectory which is also the central p trajectory these power series have a simple recursive form and are useful in developing higher order analogues of karmarkar s algorithm a trajectories are defined for a general linear program using this definition it is shown that the limit point x 0 of a central a trajectory on the boundary of the feasible solution polytope p is the center of the unique face of p containing x 0 in its relative interior received by the editors october 8 1986 and in revised form june 9 1987 and march 25 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 the first author was partially supported by onr contract n00014 87 k0214 1989 american mathematical society 0002 9947 89 1 00 25 per page

simultaneous acoustic channel measurement via maximal length related sequences a wide variety of acoustic systems has multiple sources and receivers this paper proposes a technique for making acoustic measurements simultaneously for multiple sources the proposed technique features a collection of excitation signals of the maximum length sequence mls and mls related classes each signal in the set has a pulse like autocorrelation function and the cross correlation functions between arbitrary pairs of signals drawn from the set have peak values that are significantly lower than the peak magnitude of the autocorrelation functions the proposed method is particularly valuable when characterization of multisource multireceiver system must be accomplished in a limited time period both simulation and experimental results are presented that demonstrate the feasibility and fidelity of the proposed techniques in characterizing acoustic systems

on the complexity of quantified linear systems in this paper we explore the computational complexity of the conjunctive fragment of the first order theory of linear arithmetic quantified propositional formulas of linear inequalities with k 1 quantifier alternations are log space complete in s k p or p k p depending on the initial quantifier we show that when we restrict ourselves to quantified conjunctions of linear inequalities i e quantified linear systems the complexity classes collapse to polynomial time in other words the presence of universal quantifiers does not alter the complexity of the linear programming problem which is known to be in p our result reinforces the importance of sentence formats from the perspective of computational complexity

long step strategies in interior point primal dual methods in this paper we analyze from a unique point of view the behavior of path following and primal dual potential reduction methods on nonlinear conic problems we demonstrate that most interior point methods with o root n ln 1 epsilon efficiency estimate can be considered as different strategies of minimizing a convex primal dual potential function in an extended primal dual space their efficiency estimate is a direct consequence of large local norm of the gradient of the potential function along a central path it is shown that the neighborhood of this path is a region of the fastest decrease of the potential therefore the long step path following methods are in a sense the best potential reduction strategies we present three examples of such long step strategies we prove also an efficiency estimate for a pure primal dual potential reduction method which can be considered as an implementation of a penalty strategy based on a functional proximity measure using the convex primal dual potential we prove efficiency estimates for karmarkar type and dikin type methods as applied to a homogeneous reformulation of the initial primal dual problem

random walks on polytopes and an affine interior point method for linear programming let k be a polytope in rn defined by m linear inequalities we give a new markov chain algorithm to draw a nearly uniform sample from k the underlying markov chain is the first to have a mixing time that is strongly polynomial when started from a central point we use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately

the shape of self avoiding walks an exhaustive analysis of the shape properties of discrete self avoiding random walks is presented the dependence of the main parameters of the probability distributions from the length of the walk and the differences between these distributions for conventional and self avoiding walks are studied by means of high precision monte carlo simulations it is shown that the characteristics of the shape of self avoiding random walks when regarded as a function of the walk length present the expected asymptotic behaviour the differences with conventional random walks depend upon the observable considered the probability distributions of the principal inertia eigenvalues of self avoiding walks spread around the most probable value more widely than the corresponding distributions for unrestricted walks while in the cases of the asphericity or ratios of inertia eigenvalues the distributions for self avoiding walks are somewhat more peaked than their counterparts for conventional random walks analytical expressions for the probability distributions of inertia moment ratios and the two dimensional asphericity are given for common random walks there is an excellent agreement between these analytical distributions and the monte carlo data in this work it is established that this concordance is maintained if the same analytical distributions are applied to the self avoiding case this means that within the precision of the simulations the functional form of the mentioned distributions does not vary when passing from conventional to self avoiding walks

universal shape ratios for open and closed random walks exact results for all d the mean asphericities ad of open and closed gaussian polymer chains are computed exactly and analytically up to simple quadratures for arbitrary space dimensions d excellent agreement is found with existing simulation data for d 2 3 and 4 the authors technique can also be used to compute other averages of ratios of fluctuating variables as well as extended to include the effects of self avoiding walk interactions

a polynomial method of approximate centers for linear programming we present a path following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour this is done both for the problem in standard form and for its dual problem we also discuss some implementation strategies

asymptotic behavior of inflated lattice polygons we study the inflated phase of two dimensional lattice polygons with fixed perimeter n and variable area associating a weight exp pa jb to a polygon with area a and b bends for convex and column convex polygons we calculate the average area for positive values of the pressure for large pressures the area has the asymptotic behaviour langle a rangle a max 1 k j tilde p 2 mathcal o rho tilde p where tilde p pn gg 1 and 1 the constant k j is found to be the same for both types of polygons we argue that self avoiding polygons should exhibit the same asymptotic behavior for self avoiding polygons our predictions are in good agreement with exact enumeration data for j 0 and monte carlo simulations for j 0 we also study polygons where self intersections are allowed verifying numerically that the asymptotic behavior described above continues to hold

network coding a computational perspective in this work we study the computational perspective of network coding focusing on two issues first we address the computational complexity of finding a network code for acyclic multicast networks second we address the issue of reducing the amount of computation performed by network nodes in particular we consider the problem of finding a network code with the minimum possible number of encoding nodes i e nodes that generate new packets by performing algebraic operations on packets received over incoming links we present a deterministic algorithm that finds a feasible network code for a multicast network over an underlying graph g v e in time 0 e kh v k2h2 h4k3 k h where k is the number of destinations and h is the number of packets our algorithm improves the best known running time for network code construction in addition our algorithm guarantees that the number of encoding nodes in the obtained network code is upper bounded by 0 h3k2 next we address the problem of finding integral and fractional network codes with the minimum number of encoding nodes we prove that in the majority of settings this problem is np hard however we show that if h o 1 k o 1 and the underlying communication graph is acyclic then there exists an algorithm that solves this problem in polynomial time

an ol n 3 potential reduction algorithm for linear programming we describe a primal dual potential function for linear programming phi x s rho ln x t s sum limits j 1 n ln x j s j where n x is the primal variable ands is the dual slack variable as a result we develop an interior point algorithm seeking reductions in the potential function with rho n sqrt n neither tracing the central path nor using the projective transformation the algorithm converges to the optimal solution set in o sqrt n l iterations and uses o n3l total arithmetic operations we also suggest a practical approach to implementing the algorithm

a potential reduction variant of renegar s short step path following method for linear programming abstract we propose a new polynomial potential reduction method for linear programming which can also be seen as a large step path following method we do an approximate linesearch along the newton direction with respect to renegar s strictly convex potential function if the iterate is far away from the central trajectory if the iterate lies close to the trajectory we update the lower bound for the optimal value dependent on this updating scheme the iteration bound can be proved to be o n l or o nl our method differs from the recently published potential reduction methods in the choice of the potential function and the search direction

phase transitions in pressurized semiflexible polymer rings we propose and study a model for the equilibrium statistical mechanics of a pressurized semiflexible polymer ring in two dimensions the hamiltonian has a term which couples to the algebraic or signed area of the ring and a term which accounts for bending semiflexibility the model allows for self intersections using a combination of monte carlo simulations flory type scaling theory mean field approximations and lattice enumeration techniques we obtain a phase diagram in which collapsed and inflated phases are separated by a continuous transition the scaling properties of the averaged area as a function of the number of units of the ring are derived for large pressures the asymptotic behavior of the area is calculated for both continuum and lattice versions of the model for small pressures the area is obtained through a known mapping onto the quantum mechanical problem of an electron moving in a magnetic field the simulation data agree well with the analytic and mean field results

a specialized fast cross correlation for acoustical measurements using coded sequences in acoustics applications binary maximal length sequences and related sequences are increasingly used for acoustics system identification tasks a number of coded sequences such as binary maximal length related sequences and ternary sequences possess two valued or pulselike autocorrelation functions it is this correlation property that is exploited in most of acoustical applications however the length of some of these sequences is not directly suitable for fft based cross correlation algorithms this paper explores using standard ffts to calculate the cross correlation between two periodic finite length sequences of equal length where the lengths of the sequences are not a power of 2 we apply our specialized correlation algorithm to analyze data collected in a room acoustic environment to simultaneously obtain impulse responses between multiple sources and multiple receivers

polynomial time algorithms for linear programming based only on primal scaling and projected gradients of a potential function this paper presents extensions and further analytical properties of algorithms for linear programming based only on primal scaling and projected gradients of a potential function the paper contains extensions and analysis of two polynomial time algorithms for linear programming we first present an extension of gonzaga s o nl iteration algorithm that computes dual variables and does not assume a known optimal objective function value this algorithm uses only affine scaling and is based on computing the projected gradient of the potential function q ln x t s sum limits j 1 n ln x j wherex is the vector of primal variables ands is the vector of dual slack variables and q n sqrt n the algorithm takes either a primal step or recomputes dual variables at each iteration we next present an alternate form of ye s o sqrt n l iteration algorithm that is an extension of the first algorithm of the paper but uses the potential function q ln x t s sum limits j 1 n ln x j sum limits j 1 n ln s j where q n sqrt n we use this alternate form of ye s algorithm to show that ye s algorithm is optimal with respect to the choice of the parameterq in the following sense suppose thatq n n t wheret 0 then the algorithm will solve the linear program in o n r l iterations wherer max t 1 t thus the value oft that minimizes the complexity bound ist 1 2 yielding ye s o sqrt n l iteration bound

polynomial time computation of strong and n present value optimal policies in markov decision chains this paper studies the problem of finding a stationary strong present value optimal and more generally an n present value optimal policy for an infinite horizon stationary finite state action substochastic markov decision chain a policy is strong present value optimal if it has maximum present value for all small positive interest rates a policy is n present value optimal if its present value falls below the maximum present value for all small positive by o n 1 the importance of stationary n present value optimal policies arises from the following known facts the set of such policies diminishes with n and for n s is precisely the set of stationary strong present value optimal policies for 0 n s stationary n present value optimal policies are nearly strong present value optimal and are of independent interest the best algorithms for finding stationary strong present value optimal policies find stationary n present value policies for n 1 s in that order this paper decomposes th

supersymmetry approach in the field theory of ergodicity breaking transitions abstract the supersymmetry susy self consistent approximation for the model of non equilibrium thermodynamic system with quenched disorder is derived from the dynamical action calculated by means of generalized second legendre transformation technique the equations for adiabatic and isothermal susceptibilities memory and field induced parameters are obtained on the basis of asymptotic analysis of dynamic dyson equations it is shown that the marginal stability condition that defines the critical point is governed by fluctuations violating fluctuation dissipation theorem fdt the temperature of ergodicity breaking transition is calculated as a function of quenched disorder intensities transformation of superfields related to the mapping between an instanton process and the corresponding causal solution is discussed

solving linear programming problems exactly a linear programming problem in an inequality form having a bounded solution is solved error free using an algorithm that sorts the inequalities removes the redundant ones and uses the p adic arithmetic c elsevier science inc 1997