cryptanalysis of a diophantine equation oriented public key cryptosystem c h lin et al 1995 proposed a new public key cipher system whose security is based upon the diophantine equations in this note we show that their scheme is insecure under the ciphertext only attack

diffraction of flexural waves by cracks in orthotropic thin elastic plates i formal solution the problem of flexural wave diffraction by a semi infinite crack in an infinite orthotropic thin plate is considered such models have application to the ultrasonic non destructive inspection of thin components such as aeroplane wings for simplicity the plate is modelled using kirchhoff theory and the crack is chosen to be aligned along one of the principal directions of material orthotropy for incident plane waves an exact analytical expression for the scattered field is derived by means of the wiener hopf technique in this model problem the wiener hopf kernel is scalar and its factorization is expressed in terms of simple definite non singular contour integrals a detailed numerical evaluation of the solution will be provided in the second part of this work

diffraction from the juncture of pressure release and locally reacting half planes solutions to two coupled half plane problems are derived in the first an exact solution to the canonical problem of acoustic plane wave diffraction from a pressure release zero impedance surface coupled to a perfectly rigid infinite impedance surface is given using the wiener hopf method the solution is expressed in terms of an angular spectrum of plane waves in the second the rigid surface is replaced by one characterized by a locally reacting finite impedance here an approximate kernel to the wiener hopf functional equation is used which leads to a complete and readily interpretable solution this solution is expressed in terms of the canonical angular spectrum for the first problem multiplied by a function that depends on material parameters of the locally reacting surface polar plots of the far field amplitude level for the diffracted field are presented

distribution of lattice points visible from the origin let be a region in the plane which contains the origin is star shaped with respect to the origin and has a piecewise c 1 boundary for each integer q 1 we consider the integer lattice points from which are visible from the origin and prove that the 1 st consecutive spacing distribution of the angles formed with the origin exists this is a probability measure supported on an interval m with m 0 its repartition function is explicitly expressed as the convolution between the square of the distance from origin function and a certain kernel

the pair correlation function of fractional parts of polynomials we investigate the pair correlation function of the sequence of fractional parts of n d n 1 2 n where d 2 is an integer and an irrational we conjecture that for badly approximable the normalized spacings between elements of this sequence have poisson statistics as n

on the solutions of a class of dual integral equations occurring in diffraction problems it is shown that there exists a category of two dimensional diffraction problems which can be put into a standard form of dual integral equations these diffraction problems include diffraction of electromagnetic waves by a finite strip a finite slit the diffraction of scalar or vector elastic waves by a rigid strip or crack etc a general method for solving such dual integral equations is given by the artifice of constructing a set of functions of compact support biorthogonal to another given set of functions the sufficient conditions for a given dual integral equations to be solvable in this manner are also determined hence the method forms a complement to the weiner hopf method to illustrate the method solutions are obtained for a bench mark problem the diffraction of light by a finite perfectly conducting strip or equivalently the diffraction of sh waves by a crack comparison with results obtained by others for low intermediate and high frequencies show the utility and accuracy of the method for the entire range of frequencies both the near field and the far field are obtained the latter is shown to correspond to the fraunhoffer diffraction pattern for high frequency it is also shown that for the equivalent crack problem the stress intensity factor sif fluctuates rapidly with changes in the angle of incidence for high frequencies thus making the sif especially sensitive to angle of incidence at high frequencies

from the von neumann equation to the quantum boltzmann equation ii identifying the born series in a previous paper ca1 the author studied a low density limit in the periodic von neumann equation with potential modified by a damping term the model studied in ca1 considered in dimensions d 3 is deterministic it describes the quantum dynamics of an electron in a periodic box actually on a torus containing one obstacle when the electron additionally interacts with say an external bath of photons the periodicity condition may be replaced by a dirichlet boundary condition as well in the appropriate low density asymptotics followed by the limit where the damping vanishes the author proved in ca1 that the above system is described in the limit by a linear space homogeneous boltzmann equation with a cross section given as an explicit power series expansion in the potential the present paper continues the above study in that it identifies the cross section previously obtained in ca1 as the usual born series of quantum scattering theory which is the physically expected result hence we establish that a von neumann equation converges in the appropriate low density scaling towards a linear boltzmann equation with cross section given by the full born series expansion we do not restrict ourselves to a weak coupling limit where only the first term of the born series would be obtained fermi s golden rule

the distribution of the free path lengths in the periodic two dimensional lorentz gas in the small scatterer limit we study the free path length and the geometric free path length in the model of the periodic two dimensional lorentz gas sinai billiard we give a complete and rigorous proof for the existence of their distributions in the small scatterer limit and explicitly compute them as a corollary one gets a complete proof for the existence of the constant term c 2 3 ln 2 frac 27 zeta 3 2 pi 2 in the asymptotic formula h t 2 ln epsilon c o 1 of the ks entropy of the billiard map in this model as conjectured by p dahlqvist

free path lengths in quasicrystals previous studies of kinetic transport in the lorentz gas have been limited to cases where the scatterers are distributed at random e g at the points of a spatial poisson process or at the vertices of a euclidean lattice in the present paper we investigate quasicrystalline scatterer configurations which are non periodic yet strongly correlated a famous example is the vertex set of a penrose tiling our main result proves the existence of a limit distribution for the free path length which answers a question of wennberg the limit distribution is characterised by a certain random variable on the space of higher dimensional lattices and is distinctly different from the exponential distribution observed for random scatterer configurations the key ingredients in the proofs are equidistribution theorems on homogeneous spaces which follow from ratner s measure classification

power law distributions for the free path length in lorentz gases it is well known that in the boltzmann grad limit the distribution of the free path length in the lorentz gas with disordered scatterer configuration has an exponential density if on the other hand the scatterers are located at the vertices of a euclidean lattice the density has a power law tail proportional to xi 3 in the present paper we construct scatterer configurations whose free path lengths have a distribution with tail xi n 2 for any positive integer n we also discuss the properties of the random flight process that describes the lorentz gas in the boltzmann grad limit the convergence of the distribution of the free path length follows from equidistribution of large spheres in products of certain homogeneous spaces which in turn is a consequence of ratner s measure classification theorem

on the shielding of sound from a source near one or two coaxial cones abstract the shielding of the sound from a source near a cone or two coaxial circular cones is addressed by an exact analytical method considering free spherical waves satisfying a rigid or impedance boundary condition on the wall of the cone s the boundary condition is applied for impedance proportional to the distance from the vertex the exact solution involves the eigenvalues of the problem that are generally real or complex not integers and coincide with the degree of the associated legendre functions and order of the spherical bessel functions which specify respectively the latitudinal and radial dependencies of the wave field these eigenfunctions or conical wave harmonics can be chosen in more than one way e g a as standing wave modes which are finite at the vertex of the cone s but do not satisfy a radiation condition at infinity b as propagating waves satisfying a radiation condition at infinity and generally singular at the vertex of the cone s the method of calculation of eigenvalues and eigenfunctions is presented both for a single cone and two coaxial cones with the same vertex and arbitrary aperture s the method specifies the eigenvalues and the corresponding radial azimuthal and latitudinal eigenfunctions an asymptotic formula is obtained for the eigenvalues which gives reasonable good agreement with the exact results the eigenvalues and eigenfunctions appear with suitable amplitudes in the green function representing a monopole source near the vertex of the cone s the acoustic field is plotted also for a longitudinal and a transverse dipole and mixed quadrupole source near the vertex

general wiener hopf factorization of matrix kernels with exponential phase factors the factorization of wiener hopf matrices with exponentially growing elements has long remained an unsolved problem such matrices often occur in the calculation of scattering from complex canonical geometries abrahams and wickham proc roy soc london ser a pp 131 156 and other problems in mathematical physics here a general method is given for resolving the difficulty and obtaining a single scalar integral equation the solution of which generates the required factors it is proven that this solution has a simple infinite series representation the method is illustrated by reference to three particular examples the first is a model matrix arising from a system of simple delay differential equations and the second occurs in the diffraction of sound by two knife edges finally a heat conduction problem is solved for an infinite wedge with asymmetric mixed boundary conditions on its faces

on the commutative factorization of n n matrix wiener hopf kernels with distinct eigenvalues in this article we present a method for factorizing n x n matrix wiener hopf kernels where n 2 and the factors commute we are motivated by a method posed by jones jones 1984a proc r soc a 393 185 192 to tackle a narrower class of matrix kernels however no matrix of jones form has yet been found to arise in physical wiener hopf models in contrast the technique proposed herein should find broad application to illustrate the approach we consider a 3 x 3 matrix kernel arising in a problem from elastostatics while this kernel is not of jones form we shall show how it can be factorized commutatively we discuss the essential difference between our method and that of jones and explain why our method is a generalization the majority of wiener hopf kernels that occur in canonical diffraction problems are however strictly non commutative for 2x2 matrices abrahams has shown that one can overcome this difficulty using pade approximants to rearrange a non commutative kernel into a partial commutative form an approximate factorization can then be derived by considering the dynamic analogue of antipov s model we show for the first time that abrahams pade approximant method can also be employed within a 3x3 commutative matrix form

on the solution of two coupled wiener hopf equations vector wiener hopi equations previously solved in closed form by using different techniques involve matrices having a particular structure a general procedure is given to factorize these matrices it leads to an approach which even if sometimes cumbersome is uniform and allows the solution of new problems in addition a new method is proposed which transforms simultaneous wiener hopf equations in the form of fredholm integral equation of the second kind

on two way fa with monotonic counters and quadratic diophantine equations we show an interesting connection between two way deterministic finite automata with monotonic counters and quadratic diophantine equations the automaton m operates on inputs of the form a1i1 anin for some fixed n and distinct symbols a1 an where i1 in are nonnegative integers we consider the following reachability problem given a machine m a state q and a presburger relation e over counter values is there i1 in such that m when started in its initial state on the left end of the input a1i1 anin with all counters initially zero reaches some configuration where the state is q and the counter values satisfy e in particular we look at the case when the relation e is an equality relation i e a conjunction of relations of the form ci cj we show that this case and variations of it are equivalent to the solvability of some special classes of systems of quadratic diophantine equations we also study the nondeterministic version of two way finite automata augmented with monotonic counters with respect to the reachability problem finally we introduce a technique which uses decidability and undecidability results to show separation between language classes

a combinatorial particle swarm optimization for solving multi mode resource constrained project scheduling problems the particle swarm optimization pso has been widely used to solve continuous problems the discrete problems have just begun to be also solved by the discrete pso however the combinatorial problems remain a prohibitive area to the pso mainly in case of integer values in this paper we propose a combinatorial pso cpso algorithm that we take up challenge to use in order to solve a multi mode resource constrained project scheduling problem mrcpsp the results that have been obtained using a standard set of instances after extensive experiments prove to be very competitive in terms of number of problems solved to optimality by comparing average deviations and percentages of optima found our cpso algorithm outperforms the simulated annealing algorithm and it is close to the pso algorithm

on the boltzmann equation for the lorentz gas we consider the boltzmann grad limit for the lorentz or wind tree model we prove that if is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the poisson distribution with parameter 0 then the evolution of an initial a c particle density tends in the boltzmann grad limit to the solution of the boltzmann equation for the model as an intermediate step we prove that the process of the free path lengths and impact parameters induced by the lebesgue measure on a small region tends to a limiting independent process

the brownian motion as the limit of a deterministic system of hard spheres we provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard spheres as the number of particles n goes to infinity and their diameter varepsilon simultaneously goes to 0 in the fast relaxation limit alpha n varepsilon d 1 rightarrow infty with a suitable diffusive scaling of the observation time as suggested by hilbert in his sixth problem we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions we use indeed the linear boltzmann equation to describe one tagged particle in a gas close to global equilibrium our proof is based on the fundamental ideas of lanford the main novelty here is the detailed study of the branching process leading to explicit estimates on pathological collision trees

on the probability of a random lattice avoiding a large convex set given a set d let p be the probability that a random d dimensional unimodular lattice chosen according to haar measure on sl d sl d is disjoint from 0 for special convex sets we prove bounds on p that are sharp up to a scaling of by a constant we also prove bounds on a variant of p where the probability is conditioned on the random lattice containing a fixed given point p 0 our bounds have applications among other things to the asymptotic properties of the collision kernel of the periodic lorentz gas in the boltzmann grad limit in arbitrary dimension d

ant colony optimization theory a survey research on a new metaheuristic for optimization is often initially focused on proof of concept applications it is only after experimental work has shown the practical interest of the method that researchers try to deepen their understanding of the method s functioning not only through more and more sophisticated experiments but also by means of an effort to build a theory tackling questions such as how and why the method works is important because finding an answer may help in improving its applicability ant colony optimization which was introduced in the early 1990s as a novel technique for solving hard combinatorial optimization problems finds itself currently at this point of its life cycle with this article we provide a survey on theoretical results on ant colony optimization first we review some convergence results then we discuss relations between ant colony optimization algorithms and other approximate methods for optimization finally we focus on some research efforts directed at gaining a deeper understanding of the behavior of ant colony optimization algorithms throughout the paper we identify some open questions with a certain interest of being solved in the near future

the free path in a high velocity random flight process associated to a lorentz gas in an external field we investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity the random flight models we study arise naturally as the boltzmann grad limit of a random lorentz gas in the presence of an external field by analyzing the time duration of the free path we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers as a consequence we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight

homogenization of transport equations this article is devoted to the mathematical analysis of various formulas giving the equivalent absorption and scattering cross section for mixed materials in linear transport theory we begin with a general result on the treatment of high frequency oscillations in linear transport equations which is partly based upon the velocity averaging results and is the analogue for transport equations of the compensated compactness class of results the case of periodic inhomogeneities is then studied in detail in particular we show the essential difference with periodic homogenization of diffusion equations due to small divisor problems these results were announced in f golse c r acad sci paris ser i math 305 1987 pp 801 804 f golse mathematical aspects of fluid and plasma dynamics g toscani v boffi and s rionero eds lecture notes in math 1460 springer verlag berlin new york 1991 pp 152 169 finally we treat a case of stochastic inhomogeneities in linear transport theory insp

santalo s formula and stability of trapping sets of positive measure abstract billiard trajectories broken generalised geodesics are considered in the exterior of an obstacle k with smooth boundary on an arbitrary riemannian manifold we prove a generalisation of the well known santalo s formula as a consequence it is established that if the set of trapped points has positive measure then for all sufficiently small smooth perturbations of the boundary of k the set of trapped points for the new obstacle obtained in this way also has positive measure more generally the measure of the set of trapped points depends continuously on perturbations of the obstacle k some consequences are derived in the case of scattering by an obstacle k in i r n for example it is shown that for a large class of obstacles k the volume of k is uniquely determined by the average travelling times of scattering rays in the exterior of k

the boltzmann grad limit of the periodic lorentz gas we study the dynamics of a point particle in a periodic array of spherical scatterers and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density boltzmanngrad limit a generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a markov process with memory two

on the solution of wiener hopf problems involving noncommutative matrix kernel decompositions many problems in physics and engineering with semi infinite boundaries or interfaces are exactly solvable by the wiener hopf technique it has been used successfully in a multitude of different disciplines when the wiener hopf functional equation contains a single scalar kernel for complex boundary value problems however the procedure often leads to coupled equations which therefore have a kernel of matrix form the key step in the technique is to decompose the kernel into a product of two functions one analytic in an upper region of a complex transform plane and the other analytic in an overlapping lower half plane this is straightforward for scalar kernels but no method has yet been proposed for general matrices in this article a new procedure is introduced whereby pade approximants are employed to obtain an approximate but explicit noncommutative factorization of a matrix kernel as well as being simple to apply the use of approximants allows the accuracy of the factorization to be increased

on the non commutative factorization of wiener hopf kernels of khrapkov type there are a great many problems in mathematical physics and engineering which when modelled mathematically are reduced to wiener hopf functional equations defined in some region of a complex plane in simple models this equation is scalar but in more complicated situations inherent coupling etc can often give rise to coupled systems of equations and so the resulting wiener hopf kernels are of matrix form the crucial step in the solution of any wiener hopf equation is to decompose the kernel which can be a quite general function of the complex transform variable into a product of two factors which are regular in overlapping half planes this can be accomplished explicitly for scalar kernels and may be generalized to a particular class of matrix kernels namely those which permit a commutative decomposition although this class is wide and therefore contains some valuable physical problems it is found that many such kernels yield factorization elements with exponentially large behaviour at infinity in their half planes of analyticity this prevents a later step in the wiener hopf procedure being carried through and therefore introduces a serious limitation to the usefulness of khrapkov s approach previous researchers have offered schemes to eliminate the exponential behaviour of the khrapkov factors but for reasons discussed herein these have problems in their successful implementation this article offers a new procedure for quenching the exponential growth which is based on a method proposed recently by the author for fully coupled kernels which do not have commutative factors the approach is two step first a commutative factorization is obtained which is algebraic at infinity as required but which has spurious branch cuts in the regions of intended analyticity second an approximate procedure is used to eliminate the offending branch points by introduction of pade approximants for a certain scalar function and by inclusion of a regularizing meromorphic matrix for one particular example kernel in which an exact factorization happens to be known the explicit approximate kernel factors thus derived are shown to yield very close agreement to the exact factors the application of this work to problems in the area of fluid structural interaction is discussed

solving wiener hopf problems without kernel factorization a new approach to solving problems of wiener hopf type is expounded by showing its implementation in two concrete and typical examples from fluid mechanics the new method adapts mathematical ideas underlying the so called unified transform method due to a s fokas and collaborators in recent years the method has the key advantage of avoiding what is usually the most challenging part of the usual wiener hopf approach the factorization of kernel functions into sectionally analytical functions two example boundary value problems involving both harmonic and biharmonic fields are solved in detail the approach leads to fast and accurate schemes for evaluation of the solutions

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

a new public key cipher system based upon the diophantine equations a new public key two key cipher scheme is proposed in this paper in our scheme keys can be easily generated in addition both encryption and decryption procedures are simple to encrypt a message the sender needs to conduct a vector product of the message being sent and the enciphering key on the other hand the receiver can easily decrypt it by conducting several multiplication operations and modulus operations for security analysis we also examine some possible attacks on the presented scheme

commutative wiener hopf factorization of a matrix a class of n x n matrices is given for which commutative wiener hopf factorization is possible and the explicit factors are obtained it is also shown that there are no other matrices with factors which commute if the factors possess distinct eigenvalues

on l1 exponential trend to equilibrium for conservative linear kinetic equations on the torus we compute the growth bound of weighted advection semigroups on the n dimensional flat torus with general not necessarily bounded measurable nonnegative collision frequencies and give a compactness theorem in l1 spaces of interest for the stability of essential type of kinetic semigroups under perturbation by scattering operators this provides us with a characterisation of the existence of a spectral gap for general conservative neutron transport like equations various related questions are also dealt with

on the scattering of sound by two semi infinite parallel staggered plates i explicit matrix wiener hopf factorization this paper discusses the two dimensional scattering of sound waves by two semi infinite rigid parallel plates the plates are staggered so that a line in the plane of the motion passing through both edges is not in general perpendicular to the plane of either plate the problem is formulated as a matrix wiener hopf functional equation which exhibits the difficulty of a kernel containing exponentially growing elements we show how this difficulty may be overcome by constructing an explicit product decomposition of the matrix kernel with both factors having algebraic behaviour at infinity this factorization is written in terms of a single entire auxiliary function that has a simple infinite series representation the wiener hopf equation is solved for arbitrary incident wave fields and we derive an asymptotic expression for the field scattered to infinity the latter includes the possibility of propagating modes in the region between the plates in part ii of this work we will evaluate our solution numerically and obtain some analytical estimates in a number of physically interesting limits

symbolic dynamics and arnold diffusion we consider hyperbolic tori of three degrees of freedom initially hyperbolic hamiltonian systems we prove that if the stable and unstable manifold of a hyperbolictorus intersec t transversaly then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a bernoulli shift the proof is based on a new geometricodynamical feature of partially hyperbolic systems the transversality torsion phenomenon which produces complete hyperbolicity from partial hyperbolicity we deduce the existence of infinitely many hyperbolicperiodicorbits near the given torus the relevanc e of these results for the instability of near integrable hamiltonian systems is then discussed for a given transition chain we construct chain of hyperbolic periodic orbits then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolicsets r 2002 elsevier science usa all rights reserved

diffraction by three semi infinite planes an explicit solution to the scattering of a plane wave by three parallel equidistant semi infinite planes is derived the solution involves a wiener hopf factorization of a 2 2 matrix various properties of the scattered field are obtained

numerical solution of scattering problems using a riemann hilbert formulation a fast and accurate numerical method for the solution of scalar and matrix wiener hopf wh problems is presented the wh problems are formulated as riemann hilbert problems on the real line and a numerical approach developed for these problems is used it is shown that the known far field behaviour of the solutions can be exploited to construct numerical schemes providing spectrally accurate results a number of scalar and matrix wh problems that generalize the classical sommerfeld problem of diffraction of plane waves by a semi infinite plane are solved using the approach

an asymptotic method of factorization of a class of matrix functions a novel method of asymptotic factorization of nn matrix functions is proposed the considered class of matrices is motivated by certain problems originated from the elasticity theory an example is

asymmetric channel divider in stokes flow this article examines the classic problem of stokes flow into a divided channel with in contrast to previous literature the divider barrier asymmetrically placed with respect to the moving parallel channel walls the boundary value problem is reduced to a wiener hopf equation that is of matrix form and of a class for which no exact solution is known an explicit approximate solution in general accurate to any specified degree is obtained by a recent method which employs pade approximants numerical results exhibit the flows due to moving walls or various combinations of downstream pressure gradients

on the correlations of directions in the euclidean plane let r v x y q denote the repartition of the v level correlation measure of the finite set of directions p x y p where p x y is the fixed point x y 0 1 2 and p is an integer lattice point in the square q q 2 we show that the average of the pair correlation repartition r 2 x y q over x y in a fixed disc d 0 converges as q oo more precisely we prove for every a r and 0 1 10 the estimate formula math we also prove that for each individual point x y 0 1 2 the 6 level correlation r 6 x y q diverges at any point a r 5 as q and we give an explicit lower bound for the rate of divergence

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

stability analysis of matrix wiener hopf factorization of daniele khrapkov class and reliable approximate factorization this paper presents new stability results for matrix wiener hopf factorization the first part of the paper examines conditions for stability of wiener hopf factorization in the daniele khrapkov class the second part of the paper concerns the class of matrix functions which can be exactly or approximately reduced to the factorization of the daniele khrapkov matrices the results of the paper are demonstrated by numerical examples with partial indices 1 1 0 0 and 1 1

mode conversion corner singularities and matrix wiener hopf factorization in diffraction theory the diffraction of acoustic electromagnetic and elastic waves by an edge is a subject of continuing mathematical and physical importance the physical motivation for research is the vast number of technological and geophysical applications of the fundamental models of linear wave propagation in continuous media while the mathematical drive is to understand and devise new analytical and numerical techniques for their quantification this paper is concerned with certain physical aspects of diffraction which have proven to give the most troublesome of mathematical difficulties these are mode conversion that is when the incident wave is scattered among different modes supported by the propagating medium and edge or corner singularities when the diffracting obstacle creates a stress singularity it is shown by way of a particular example that the exact solution of diffraction problems involving these phenomena will in general give rise to a vector wiener hopf equation with a matrix kernel whose factorization is non commutative the main result of this paper is a new procedure motivated by a precise treatment of both the corner singularity and mode conversion for determining the factors in this case

on the convergence to equilibrium for degenerate transport problems we give a counterexample which shows that the asymptotic rate of convergence to the equilibrium state for the transport equation with a degenerate cross section and in the periodic setting cannot be better than t 1 2 in the general case we suggest moreover that the geometrical properties of the cross section are the key feature of the problem and impose through the distribution of the forward exit time the speed of convergence to the stationary state

aubry mather theory and hamilton jacobi equations we review and extend some results of jauslin kreiss moser and weinan e on the smooth approximation of mather sets

the statistics of the trajectory of a certain billiard in a flat two torus we consider a billiard in the punctured torus obtained by removing a small disk of radius 0 from the flat torus 2 with trajectory starting from the center of the puncture in this case the phase space is given by the range of the velocity only let and respectively r denote the first exit time length of the trajectory and respectively the number of collisions with the side cushions when 2 is being identified with 0 1 2 we prove that the probability measures on 0 associated with the random variables and r are weakly convergent as varepsilon rightarrow 0 and explicitly compute the densities of the limits

gaps in n mod 1 and ergodic theory cut the unit circle s r z at the points 1 2 n where x x mod 1 and let j1 jn denote the complementary intervals or gaps that remain we show that in contrast to the case of random points whose gaps are exponentially distributed the lengths ji n are governed by an explicit piecewise real analytic distribution f t dt with phase transitions at t 1 2 and t 2 the gap distribution is related to the probability p t that a random unimodular lattice translate r meets a fixed triangle st of area t in fact p t f t the proof uses ergodic theory on the universal elliptic curve e sl2 r r 2 sl2 z z and ratner s theorem on unipotent invariant measures

a hybrid rank based evolutionary algorithm applied to multi mode resource constrained project scheduling problem abstract we consider the multi mode resource constrained project scheduling problem mrcpsp where a task has different execution modes characterized by different resource requirements due to the nonrenewable resources and the multiple modes this problem is np hard therefore we implement an evolutionary algorithm looking for a feasible solution minimizing the makespan in this paper we propose and investigate two new ideas on the one hand we transform the problem of single objective mrcpsp to bi objective one to cope with the potential violation of nonrenewable resource constraints relaxing the latter constraints allows to visit a larger solution set and thus to simplify the evolutionary operators on the other hand we build the fitness function not on a priori grid of the bi objective space but on an adaptive one relying on clustering techniques this proposed idea aims at more relevant fitness values we show that a clustering based fitness function can be an appealing feature in multi objective evolutionary algorithms since it may promote diversity and avoid premature convergence of the algorithms clustering heuristics require certainly computation time but they are still competitive with respect to classical niche formation multi objective genetic algorithm

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

on the distribution of free path lengths for the periodic lorentz gas consider the domain and let the free path length be defined as the distribution of values of is studied in the limit as for all it is shown that the value is critical for this problem in other words the limiting behavior of depends only on whether is larger or smaller than

on the pair correlation density for hyperbolic angles we prove a conjecture of boca popa zaharescu on the density function for the pair correlation of hyperbolic angles in a growing orbit of a discrete lattice in psl2 r

a particle swarm optimization algorithm for makespan and total flowtime minimization in the permutation flowshop sequencing problem abstract in this paper a particle swarm optimization algorithm pso is presented to solve the permutation flowshop sequencing problem pfsp with the objectives of minimizing makespan and the total flowtime of jobs for this purpose a heuristic rule called the smallest position value spv borrowed from the random key representation of bean j c bean genetic algorithm and random keys for sequencing and optimization orsa journal of computing 6 2 1994 154 160 was developed to enable the continuous particle swarm optimization algorithm to be applied to all classes of sequencing problems in addition a very efficient local search called variable neighborhood search vns was embedded in the pso algorithm to solve the well known benchmark suites in the literature the pso algorithm was applied to both the 90 benchmark instances provided by taillard e taillard benchmarks for basic scheduling problems european journal of operational research 64 1993 278 285 and the 14 000 random narrow random and structured benchmark instances provided by watson et al j p watson l barbulescu l d whitley a e howe contrasting structured and random permutation flowshop scheduling problems search space topology and algorithm performance orsa journal of computing 14 2 2002 98 123 for makespan criterion the solution quality was evaluated according to the best known solutions provided either by taillard or watson et al the total flowtime criterion was evaluated with the best known solutions provided by liu and reeves j liu c r reeves constructive and composite heuristic solutions to the p ci scheduling problem european journal of operational research 132 2001 439 452 and rajendran and ziegler c rajendran h ziegler ant colony algorithms for permutation flowshop scheduling to minimize makespan total flowtime of jobs european journal of operational research 155 2 2004 426 438 for the total flowtime criterion 57 out of the 90 best known solutions reported by liu and reeves and rajendran and ziegler were improved whereas for the makespan criterion 195 out of the 800 best known solutions for the random and narrow random problems reported by watson et al were improved by the vns version of the pso algorithm

finding numerical solutions of diophantine equations using ant colony optimization the paper attempts to find numerical solutions of diophantine equations a challenging problem as there are no general methods to find solutions of such equations it uses the metaphor of foraging habits of real ants the ant colony optimization based procedure starts with randomly assigned locations to a fixed number of artificial ants depending upon the quality of these positions ants deposit pheromone at the nodes a successor node is selected from the topological neighbourhood of each of the nodes based on this stochastic pheromone deposit if an ant bumps into an already encountered node the pheromone is updated correspondingly a suitably defined pheromone evaporation strategy guarantees that premature convergence does not take place the experimental results which compares with those of other machine intelligence techniques validate the effectiveness of the proposed method

numerical solution of certain exponential and non linear diophantine systems of equations by using a discrete particle swarm optimization algorithm this article proposes the numerical solution of some non linear systems of diophantine equations including some exponential cases by its transformation into an optimization problem for the first example the implemented algorithm is able to find all the non negative integer roots in a given region of the domain of the objective function when compared to the analytic solution systems of different sizes are solved as an example for these their answers were found being them of excellent quality in terms of accuracy and precision even though there is no way of being certain that they are the only ones following matiyasevich theorem however and for systems of modest size the computation time increased considerably when the search space was increased

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results