a note on the distribution of integer points on spheres in this note by applying the works of bocherer and schulze pillot we evaluate asymptotically the variance of the linnik distribution and show as y label eq1 sum n leq y n not equiv 0 rm mod 4 frac sum x 2 n p left frac x x right n 1 4 overline frac sum z 2 n q left frac z z right n 1 4 c lambda 1 2 p y o y for any spherical harmonics p q on s2 which are hecke eigenforms where is the usual euclidean norm on r3 and c is an explicit constant

pairwise comparison matrix with fuzzy elements on alo group this paper deals with pairwise comparison matrices with fuzzy elements fuzzy elements of the pairwise comparison matrix are applied whenever the decision maker is not sure about the value of his her evaluation of the relative importance of elements in question in comparison with pairwise comparison matrices with crisp elements investigated in the literature here we investigate pairwise comparison matrices with elements from abelian linearly ordered group alo group over a real interval we generalize the concept of reciprocity and consistency of crisp pairwise comparison matrices to matrices with triangular fuzzy numbers pcfn matrices we also define the concept of priority vector which is a generalization of the crisp concept such an approach allows for a generalization dealing both with the pcfn matrices on the additive multiplicative and also fuzzy alo groups it unifies several approaches known from the literature moreover we also deal with the problem of measuring the inconsistency of pcfn matrices by defining corresponding indexes the first index called the consistency grade g is the maximal alpha of alpha cut such that the corresponding pcfn matrix is still alpha consistent on the other hand the consistency index i of the pcfn matrix measures the distance of the pcfn matrix to the closest ratio matrix if the pcfn matrix is crisp and consistent then g is equal to 1 and the consistency index i is equal to the identity element e of the alo group otherwise g is less than 1 or i is greater than e four numerical examples are presented to illustrate the concepts and derived properties finally we show that the properties of reciprocity and consistency of pcfn matrices are saved by special transformations based on fuzzy extensions of isomorphisms between two alo groups

on consistency and ranking of alternatives in fuzzy ahp abstract in fuzzy ahp where pairwise comparisons are described by membership values the treatment of consistency has largely been sketchy this paper proposes a fuzzy consistency definition with consideration of a tolerance deviation essentially the fuzzy ratios of relative importance allowing certain tolerance deviation are formulated as constraints on the membership values of the local priorities the fuzzy local and global weights are determined via the extension principle the alternatives are ranked on the basis of the global weights by application of a maximum minimum set ranking method our analysis uses only the data which are considered as consistent the proposed approach is computationally efficient as it consists of solving only linear programs an example is included for illustrative purpose

ramanujan s ternary quadratic form do not seem to obey any simple law following i kaplansky we call a non negative integer n eligible for a ternary form f x y z if there are no congruence conditions prohibiting f from representing n by the classical theory of quadratic forms it is well known that any given genus of positive definite ternary quadratic forms represents every eligible integer consequently if a genus consists of a single class with representative f x y z then f represents every eligible integer in the case of ramanujan s form this only implies that an eligible integer one not of the form 4 16 6 is represented by 1 or 2

on fuzzy ratio comparisons in hierarchical decision models abstract this article develops a method for the interactive analysis of fuzzy pairwise comparisons in hierarchical weighting models in the proposed method fuzzy ratios of relative importance are employed as constraints on the membership values of local priorities so that fuzzy weights for the alternatives can be computed through linear programming moreover graph displays of fuzzy relations are used as a means of illustrating the dominance implications of the specified statements

towards an ultra efficient kinetic scheme part i basics on the bgk equation in this paper we present a new ultra efficient numerical method for solving kinetic equations in this preliminary work we present the scheme in the case of the bgk relaxation operator the scheme being based on a splitting technique between transport and collision can be easily extended to other collisional operators as the boltzmann collision integral or to other kinetic equations such as the vlasov equation the key idea on which the method relies is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time the main difference between the method proposed and semi lagrangian methods is that here we do not need to reconstruct the distribution function at each time step this allows to tremendously reduce the computational cost of the method and it permits for the first time to the author s knowledge to compute solutions of full six dimensional kinetic equations on a single processor laptop machine numerical examples up to the full three dimensional case are presented which validate the method and assess its efficiency in 1d 2d and 3d

a multiscale fast semi lagrangian method for rarefied gas dynamics in this paper we consider the extension of the method developed in dimarco and loubere 2013 22 23 with the aim of facing the numerical resolution of multi scale problems arising in rarefied gas dynamics the scope of this work is to consider situations in which the whole domain does not demand the use of a kinetic model everywhere this is the case of many realistic applications some regions of the computational domain require a microscopic description given by a kinetic model while the rest of the domain can be described by a coarser model of fluid type our aim is to show how the kinetic scheme developed in the pre cited articles is perfectly suited for building domain decomposition strategies which make the method more attractive with respect to classical numerical techniques for kinetic equations and multi scale realistic problems several numerical evidences are provided in this work in the two dimensional and three dimensional settings to assess the efficiency of the domain decomposition scheme

fuzzy hierarchical analysis the lambda max method abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the max method used by saaty in the analytical hierarchical process an example is presented where there are five criteria and three alternatives

averages of fourier coefficients of siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables let d be a a negative discriminant and let t vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant d we prove an asymptotic formula for d rightarrow infty for the average over t of the number of representations of t by an integral positive definite quaternary quadratic form and obtain bounds for averages of fourier coefficients of linear combinations of siegel theta series we also find an asymptotic estimate from below on the number of binary forms of fixed discriminant d which are represented by a given quaternary form in particular we can show that for growing d a positive proportion of the binary quadratic forms of discriminant d is represented by the given quaternary quadratic form

an acceleration procedure for discrete velocity approximation of the boltzmann collision operator abstract in this paper we investigate a method of realization of the discrete velocity approximation of the boltzmann collision operator studied by palczewski schneider and bobylev for this realization we propose an acceleration procedure which reduces the computational complexity of the method the efficiency of the acceleration procedure is demonstrated through a set of numerical tests which include space homogeneous relaxation problems and space nonhomogeneous problem of shock wave formation

analysis of spectral methods for the homogeneous boltzmann equation the development of accurate and fast algorithms for the boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis recently several works were devoted to the derivation of spectrally accurate schemes for the boltzmann equation but very few of them were concerned with the stability analysis of the method in particular there was no result of stability except when the method is modified in order to enforce the posivity preservation which destroys the spectral accuracy in this paper we propose a new method to study the stability of homogeneous boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution this method takes advantage of the spreading property of the collision together with estimates on regularity and entropy production as an application we prove stability and convergence of spectral methods for the boltzmann equation when the discretization parameter is large enough with explicit bound

saddle point principles and numerical integration methods for second order hyperbolic equations abstract this work describes a family of functionals whose stationarity often saddle point condition leads to well known so called variational formulations for structural dynamics such as the weak hamilton ritz formulation and the continuous discontinuous galerkin formulation and in turn to methods for the numerical integration of the equations of motion it is shown that all the time integration methods based on variational formulations do descend from such functionals moreover starting from the described family of functionals it is possible to construct new families of time integration methods which might exhibit computational advantages over the corresponding ones derived from variational formulations only

a convergence proof for nanbu s simulation method for the full boltzmann equation it is shown that the discrete measures given by the nanbu simulation method converge with respect to the weak topology of measures to solutions of the boltzmann equation the main conditions for this result are that the cauchy problem for the boltzmann equation has a sufficiently smooth solution and that the discretization parameters cell size timestep and test particle number satisfy suitable constraints

fast numerical method for the boltzmann equation on non uniform grids we introduce a new fast numerical method for computing discontinuous solutions to the boltzmann equation and illustrate it by numerical examples a combination of adaptive grids for approximation of the distribution function and an approximate fast fourier transform on non uniform grids for computing smooth terms in the boltzmann collision integral is used

construction of discrete kinetic models with given invariants we consider the general problem of the construction of discrete kinetic models dkms with given conservation laws this problem was first stated by gatignol in connection with discrete models of the boltzmann equation be and it has been addressed in the last decade by several authors even though a practical criterion for the non existence of spurious conservation laws has been devised and a method for enlarging existing physical models by new velocity points without adding non physical invariants has been proposed a general algorithm for the construction of all normal physical discrete models with assigned conservation laws in any dimension and for any number of points is still lacking in the literature we introduce the most general class of discrete kinetic models and obtain a general method for the construction and classification of normal dkms in particular it is proved that for any given dimension d 2 and for any sufficiently large number n of velocities for example n 6 for the planar case d 2 there exists just a finite number of distinct classes of dkms we apply the general method in the particular cases of discrete velocity models dvms of the inelastic be and elastic be using our general approach to dkms and our results on normal dvms for a single gas we develop a method for the construction of the most natural from physical point of view subclass of normal dvms for binary gas mixtures we call such models supernormal models snms they have the property that by isolating the velocities of single gases involved in the mixture we also obtain normal dvms

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

a random discrete velocity model and approximation of the boltzmann equation an approximation procedure for the boltzmann equation based on random choices of collision pairs from a fixed velocity set and on discrete velocity models is designed in a suitable limit the procedure is shown to converge to the time discretized and spatially homogeneous boltzmann equation

the linking number in systems with periodic boundary conditions periodic boundary conditions pbc are often used for the simulation of complex physical systems using the gauss linking number we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing pbc in the case of closed chains in pbc the periodic linking number is an integer topological invariant that depends on a finite number of components in the periodic system for open chains the periodic linking number is an infinite series that accounts for all the topological interactions in the periodic system in this paper we give a rigorous proof that the periodic linking number is defined for the infinite system i e that it converges for one two and three pbc models it gives a real number that varies continuously with the configuration and gives a global measure of the geometric complexity of the system of chains similarly for a single oriented chain we define the periodic self linking number and prove that it also is defined for open chains in addition we define the cell periodic linking and self linking numbers giving localizations of the periodic linking numbers these can be used to give good estimates of the periodic linking numbers in infinite systems we also define the local periodic linking number associated to chains in the immediate cell neighborhood of a chain in order to study local linking measures in contrast to the global linking measured by the periodic linking numbers finally we study and compare these measures when applied to a pbc model of polyethylene melts

local global principles for representations of quadratic forms we prove a local global principle for the problem of representations of quadratic forms by quadratic forms over in codimension 5 the proof uses the ergodic theory of p adic groups together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety

a consistency result for a discrete velocity model of the boltzmann equation in this paper we study the link between a certain class of discrete velocity models dvms and the boltzmann equation those models possess an infinite number of velocities laying on a regular grid of step h in r3 our aim is to prove that it is possible to construct models consistent with the boltzmann equation i e such that the discrete collision term can be seen as an approximation of the collision integral of the boltzmann equation

a discrete boltzmann equation based on a cub octahedron in mathbb r 3 in this article we introduce a discrete boltzmann equation in mathbb r 3 velocity space based on a cub octahedron we present a layer wise construction of a regular collision model in a bounded grid of the sphere packing problem the paper contains the analytical results about the basic kinetic features of the regular collision model we analyze the computational complexity report the efficiency of the model and perform numerical experiments based on the discrete model boltzmann equation for some standard test problem

fuzzy multi attribute selection among transportation companies using axiomatic design and analytic hierarchy process the basic concept in axiomatic design ad is the existence of design axioms first of these axioms is the independence axiom and the second one is the information axiom information axiom proposes the selection of the best alternative that has minimum information analytic hierarchy process ahp is another multi attribute method which is a decision making method for selecting the best among a set of alternatives given some criteria the method has been extensively applied especially in large scale problems where many criteria must be considered and where the evaluation of alternatives is mostly subjective multi attribute transportation company selection is a very important activity for effective supply chain selection of the best company under determined criteria such as cost time damage loss flexibility and documentation ability using both multi attribute ad and ahp will be realized in this study the fuzzy multiattribute ad approach is also developed and it is compared by one of fuzzy ahp methods in the literature the selection process has been accomplished by aiding a software that includes crisp ad and fuzzy ad

solving the boltzmann equation in n log 2 n in c mouhot and l pareschi fast algorithms for computing the boltzmann collision operator math comp to appear c mouhot and l pareschi c r math acad sci paris 339 2004 pp 71 76 fast deterministic algorithms based on spectral methods were derived for the boltzmann collision operator for a class of interactions including the hard spheres model in three dimensions these algorithms are implemented for the solution of the boltzmann equation in two and three dimensions first for homogeneous solutions then for general nonhomogeneous solutions the results are compared to explicit solutions when available and to monte carlo methods in particular the computational cost and accuracy are compared to those of monte carlo methods as well as to those of previous spectral methods finally for inhomogeneous solutions we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium which was suggested in the work l desvillettes and c villani invent math 159 2005 pp 245 316

the role of the ramanujan conjecture in analytic number theory we discuss progress towards the ramanujan conjecture for the group gln and its relation to various other topics in analytic number theory

deriving priority vector from pairwise comparisons matrix with fuzzy elements abstract this paper deals with the pairwise comparisons matrix with fuzzy elements fpcm fuzzy elements of the fpcm are applied whenever the decision maker is not sure about the value of his her evaluation of the relative importance of elements in question we investigate the pairwise comparisons matrices with elements from abelian linearly ordered group alo group over a real interval introduce the concept of reciprocity and weak consistency of fpcms and derive their properties such an approach unifies several approaches known from the literature moreover we propose a new general procedure for deriving the priority vector from the fpcm enabling ranking or rating a given set of alternatives

fuzzy hierarchical analysis revisited abstract we present a new method of finding the fuzzy weights in fuzzy hierarchical analysis which is the direct fuzzification of the original method used by saaty in the analytic hierarchy process we test our new procedure in two cases where there are formulas for the crisp weights an example is presented where there are five criteria and three alternatives

acceptability measurement and priority weight elicitation of triangular fuzzy multiplicative preference relations based on geometric consistency and uncertainty indices this paper investigates consistency and uncertainty measurements as well as acceptability checking for triangular fuzzy multiplicative preference relations tfmprs a geometric consistency index is proposed to measure the consistency degree of a tfmpr and thresholds of acceptable consistency are discussed for tfmprs uncertainty indices are introduced to measure indeterminacy degrees of triangular fuzzy judgments and of tfmprs and a notion of acceptable tfmprs are put forward by considering both acceptable consistency and acceptable uncertainty in tfmprs we also explore the elicitation of triangular fuzzy weights from acceptable tfmprs and the aggregation of local triangular fuzzy weights for solving hierarchical multi criteria decision making problems a geometric mean based equation and a logarithmic least square model are developed to elicit normalized acceptable triangular fuzzy multiplicative weights from an acceptable tfmpr an equation and a linear program are established to transform triangular fuzzy multiplicative weights into triangular fuzzy additive weights a geometric weighting based equation and a pair of linear programs are devised to generate global priority fuzzy weights from local triangular fuzzy multiplicative weights three numerical examples including a hierarchical multi criteria decision making problem are used to illustrate how to apply the developed framework and comparative studies are provided to show its availability and advantages

discrete velocity models for mixtures without nonphysical collision invariants an important aspect of constructing discrete velocity models dvms for the boltzmann equation is to obtain the right number of collision invariants it is a well known fact that dvms can also have extra collision invariants so called spurious collision invariants in plus to the physical ones a dvm with only physical collision invariants and so without spurious ones is called normal for binary mixtures also the concept of supernormal dvms was introduced meaning that in addition to the dvm being normal the restriction of the dvm to any single species also is normal here we introduce generalizations of this concept to dvms for multicomponent mixtures we also present some general algorithms for constructing such models and give some concrete examples of such constructions one of our main results is that for any given number of species and any given rational mass ratios we can construct a supernormal dvm the dvms are constructed in such a way that for half space problems as the milne and kramers problems but also nonlinear ones we obtain similar structures as for the classical discrete boltzmann equation for one species and therefore we can apply obtained results for the classical boltzmann equation

on almost universal mixed sums of squares and triangular numbers in 1997 k ono and k soundararajan invent math 130 1997 proved that under the generalized riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous ramanujan form x 2 y 2 1oz 2 equivalently the form 2x 2 5y 2 4t z represents all integers greater than 1359 where t z denotes the triangular number z z 1 2 given positive integers a b c we employ modular forms and the theory of quadratic forms to determine completely when the general form ax 2 by 2 ct z represents sufficiently large integers and to establish similar results for the forms ax 2 bt y ct z and at x bt y ct z here are some consequences of our main theorems i all sufficiently large odd numbers have the form 2 x 2 y 2 z 2 if and only if all prime divisors of a are congruent to 1 modulo 4 ii the form x 2 y 2 t z is almost universal i e it represents sufficiently large integers if and only if each odd prime divisor of a is congruent to 1 or 3 modulo 8 iii x 2 t y t z is almost universal if and only if all odd prime divisors of a are congruent to 1 modulo 4 iv when 2 3 the form t x t y tz is almost universal if and only if all odd prime divisors of are congruent to 1 modulo 4 and 2 5 7 where 2 a is the 2 adic order of

normal high order discrete velocity models of the boltzmann equation abstract this paper aims at approximations of the collision operator in the boltzmann equation the developed framework guarantees the normality of the approximation which means correct collision invariants h theorem and equilibrium solutions it fits into the discrete velocity model framework is given in such a way that it is understandable with undergraduate level mathematics and can be used to construct approximations with arbitrary high convergence orders at last we give an example alongside a numerical verification here the convergence orders range up to 3 2 and the time complexity is given by 3 1 2 4 2 3 in 2 3 dimensions

equidistribution of heegner points and ternary quadratic forms we prove new equidistribution results for galois orbits of heegner points with respect to single reduction maps at inert primes the arguments are based on two different techniques primitive representations of integers by quadratic forms and distribution relations for heegner points our results generalize an equidistribution result with respect to a single reduction map established by cornut and vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow moreover for fixed fundamental discriminant and variable conductor we deduce an effective surjectivity theorem for the reduction map from heegner points to supersingular points at a fixed inert prime our results are applicable to the setting considered by kolyvagin in the construction of the heegner points euler system

convergence of a discrete velocity model for the boltzmann bgk equation abstract we prove the convergence of a conservative and entropic discrete velocity model for the bathnagar gross krook bgk equation in this model the approximation of the maxwellian is based on a discrete entropy minimization principle the main difficulty due to its implicit definition is to prove that this approximation is consistent we also demonstrate the existence and uniqueness of a solution to the discrete velocity model by using a fixed point theorem finally the model is written in a continuous equation form and we prove the convergence of its solution toward a solution of the bgk equation

fast algorithms for computing the boltzmann collision operator the development of accurate and fast numerical schemes for the five fold boltzmann collision integral represents a challenging problem in scientific computing for a particular class of interactions including the so called hard spheres model in dimension three we are able to derive spectral methods that can be evaluated through fast algorithms these algorithms are based on a suitable representation and approximation of the collision operator explicit expressions for the errors in the schemes are given and spectral accuracy is proved parallelization properties and adaptivity of the algorithms are also discussed

finiteness theorems for positive definite n regular quadratic forms an integral quadratic form f of m variables is said to be n regular if f globally represents all quadratic forms of n variables that are represented by the genus of f for any n 2 it is shown that up to equivalence there are only finitely many primitive positive definite integral quadratic forms of n 3 variables that are n regular we also investigate similar finiteness results for almost n regular and spinor n regular quadratic forms it is shown that for any n 2 there are only finitely many equivalence classes of primitive positive definite spinor or almost n regular quadratic forms of n 2 variables these generalize the finiteness result for 2 regular quaternary quadratic forms proved by earnest 1994

inconsistency of pair wise comparison matrix with fuzzy elements based on geometric mean in this paper inconsistency of pair wise comparison matrix with fuzzy elements is investigated inconsistency is based on the distance of the matrix to a special matrix measured by a particular metric new inconsistency index of reciprocal matrix with fuzzy elements is introduced and newly designed method of logarithmic least squares for eliciting associated weights is applied some basic properties of the index are derived simple illustrating examples and results of numerical simulation for comparing the new index with saaty s consistency ratio are supplied

regularized boltzmann operators abstract in this paper we propose two regularization approaches for the boltzmann collision operator the constructed operators preserve the mass momentum and energy their equilibrium states are maxwellians and they satisfy the h theorem in the first approach the regularization consists in allowing microscopic collisions which do not exactly preserve energy and momentum however the limit of the mollified operator when the cut off parameter tends to 0 is not the usual boltzmann operator unless a certain condition on the distribution function is satisfied in the second approach the regularization relies on a smoothing of the masses of the particles and leads to a regularized operator which formally tends to the boltzmann operator for any arbitrary distribution function when the cut off parameter tends to zero

lattice points in rational ellipsoids we combine exponential sums character sums and fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids

a self dual variational approach to stochastic partial differential equations abstract unlike many of their deterministic counterparts stochastic partial differential equations are not amenable to the methods of calculus of variations a la euler lagrange in this paper we show how self dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields we construct solutions as minima of suitable non negative and self dual energy functionals on ito spaces of stochastic processes we show how a stochastic version of bolza s duality leads to solutions for equations with additive noise we then use a hamiltonian formulation to construct solutions for non linear equations with non additive noise such as the stochastic navier stokes equations in dimension two

towards an ultra efficient kinetic scheme part ii in a recent paper we presented a new ultra efficient numerical method for solving kinetic equations of the boltzmann type dimarco and loubere 2013 17 the key idea on which the method relies is to solve the collision part on a grid and then to solve exactly the transport part by following the characteristics backward in time on the contrary to classical semi lagrangian methods one does not need to reconstruct the distribution function at each time step this allows to tremendously reduce the computational cost and to perform efficient numerical simulations of kinetic equations up to the six dimensional case without parallelization however the main drawback of the method developed was the loss of spatial accuracy close to the fluid limit in the present work we modify the scheme in such a way that it is able to preserve the high order spatial accuracy for extremely rarefied and fluid regimes in particular in the fluid limit the method automatically degenerates into a high order method for the compressible euler equations numerical examples are presented which validate the method show the higher accuracy with respect to the previous approach and measure its efficiency with respect to well known schemes direct simulation monte carlo finite volume muscl weno

discrete kinetic models in the fluid dynamic limit we investigate discrete kinetic models in the fluid dynamic limit described by the euler system and the navier stokes correction obtained by the chapman enskog procedure we show why reliable small systems can be expected only for small mach numbers and derive a calculus for designing models for given prandtl numbers

antisymmetric hamiltonians variational resolutions for navier stokes and other nonlinear evolutions the theory of anti self dual asd lagrangians introduced in 6 is developed further to allow for a variational resolution of nonlinear pdes of the form u au u f 0 where is a convex lower semicontinuous function on a reflexive banach space x f x a d a x x is a positive linear operator and where d x x is a nonlinear operator that satisfies suitable continuity and antisymmetry properties asd lagrangians on path spaces also yield variational resolutions for nonlinear evolution equations of the form u t u t au t u t f 0 starting at u 0 u0 in both stationary and dynamic cases the equations associated to the proposed variational principles are not derived from the fact that they are critical points of the action functional but because they are also zeroes of the lagrangian itself for that we establish a general nonlinear variational principle that has many applications in particular to navier stokes type equations to generalized choquard pekar schrodinger equations with nonlocal terms as well as to complex ginsburg landau type initial value problems the case of navier stokes evolutions is more involved and will be dealt with in 9 the general theory of antisymmetric hamiltonians and its applications is developed in detail in an upcoming monograph 7 2007 wiley periodicals inc

a generalized cubic moment and the petersson formula for newforms using a cubic moment we prove a weyl type subconvexity bound for the quadratic twists of a holomorphic newform of square free level trivial nebentypus and arbitrary even weight this generalizes work of conrey and iwaniec in that the newform that is being twisted may have arbitrary square free level and also that the quadratic character may have even conductor one of the new tools developed in this paper is a more general petersson formula for newforms of square free level

almost strong approximations for definite quadratic spaces a fundamental result in the arithmetic of integral quadratic forms is the strong approximation theorem for the spin group spin of a non degenerate inde nite quadratic form in three or more variables in practice one invokes the corresponding version for the kernel o0 of the spinor norm function this result is due to the works of eichler and kneser ei kn in the 1950s and has ever since played a central role in the proofs of a number of important theorems for both inde nite as well as de nite quadratic forms on the other hand it is well known that such a strong approximation cannot possibly hold for i the group so of a inde nite quadratic form as it is not simply connected and ii for the group spin of a de nite quadratic form as spin1 is compact the purpose of this paper is to show that there exists a version of an almost strong approximation theorem for de nite quadratic forms one forom and one foro m m 3 and show that they too have serious applications in the arithmetic theory e g in the asymptotic distributions of classes in a genus and in the representation theory of de nite forms it is anticipated that such almost strong approximations would also extend to other classical groups and will be treated in a future article we give a brief outline of the contents of this paper terminology and notations are generally those from om for simplicity as well as convenience we shall consider here only positive de nite integral z lattices a key step in the proof of the strong approximation theorem for o0 is the following result on inde nite representations with approximation property invent math 129 471 487 1997

half space problems for a linearized discrete quantum kinetic equation we study typical half space problems of rarefied gas dynamics including the problems of milne and kramer for a general discrete model of a quantum kinetic equation for excitations in a bose gas in the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations these systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete boltzmann equation then a classification of well posed half space problems for the homogeneous as well as the inhomogeneous linearized discrete kinetic equation can be made the number of additional conditions that need to be imposed for well posedness is given by some characteristic numbers these characteristic numbers are calculated for discrete models axially symmetric with respect to the x axis when the characteristic numbers change is found in the discrete as well as the continuous case as an illustration explicit solutions are found for a small sized model

a fuzzy approach to deriving priorities from interval pairwise comparison judgements abstract in this paper we study the problem of priority elicitation in the analytic hierarchy process and propose a new approach to deriving crisp priorities from interval pairwise comparison judgements by introducing linear or non linear membership functions representing the decision maker s degree of satisfaction with various crisp priority vectors the interval judgements are transformed into fuzzy inequality constraints the interval prioritisation problem is then formulated as a fuzzy mathematical programming problem for obtaining an optimal crisp priority vector that maximises the overall degree of satisfaction the proposed approach yields linear or non linear mathematical programs capable of deriving priorities from consistent and inconsistent interval judgements the presence of a consistency index that measures the level of inconsistency of interval judgements is an attractive feature of our approach another feature which does not exist in the known prioritisation methods is the opportunity for additional prioritisation of the initial judgements numerical examples are given and comparisons with other interval prioritisation methods are carried out

deterministic solution of the spatially homogeneous boltzmann equation using discontinuous galerkin discretizations in the velocity space abstract we present a new deterministic approach for the solution of the spatially homogeneous boltzmann kinetic equation based on nodal discontinuous galerkin dg discretizations in the velocity space in the new approach the collision operator has the form of a bilinear operator with a pre computed kernel its evaluation requires o n 5 operations at every point of the phase space where n is the number of degrees of freedom in one velocity dimension the method is generalized to any molecular potential results of numerical simulations are presented for the problem of spatially homogeneous relaxation for the hard spheres potential comparison with the method of direct simulation monte carlo showed excellent agreement

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

on the passage from local to global in number theory would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase in number theory the distinction local versus global with various connotations has found a home in almost every part of mathematics local problems being often a stepping stone to the more difficult global problems to illustrate what the earthy geometric terms local and global signify in number theory let us consider any diophantine equation say

consistency analysis and priority derivation of triangular fuzzy preference relations based on modal value and geometric mean existing consistency notions of triangular fuzzy preference relations tfprs are illustrated to be non robust new consistency and acceptable consistency definitions are put forward for tfprs we propose a notion of normalized triangular fuzzy multiplicative weights ntfmws transformation formulae are provided to convert ntfmws into consistent tfprs a logarithmic least square model is developed to derive a ntfmw vector from a tfpr triangular fuzzy preference relation tfpr is an effective framework to model pairwise estimations with imprecision and vagueness in order to obtain a reliable and rational decision result it is important to investigate consistency and priority derivation of tfprs the paper analyzes existing definitions and properties of consistent tfprs and illustrates that they have no invariance with respect to permutations of decision alternatives a new triangular fuzzy arithmetic based transitivity equation is introduced to define consistent tfprs the new transitivity equation reflects multiplicative consistency of modal values and multiplicative consistency of geometric means of triangular fuzzy estimations some properties are presented for consistent tfprs and a notion of acceptable consistency is put forward for tfprs geometric mean and uncertainty ratio based transformation formulae are devised to convert normalized triangular fuzzy multiplicative weights into consistent tfprs a logarithmic least square model is further established for deriving a normalized triangular fuzzy multiplicative weight vector from a tfpr with acceptable consistency a geometric mean based method is developed to compare and rank triangular fuzzy multiplicative weights three numerical examples including a group decision making problem are examined to demonstrate validity and advantages of the proposed models

duality for non convex variational principles abstract this paper treats the construction of dual variational principles for non convex problems using lagrangian methods besides the well known saddle point theory there are dualities involving minimizing the lagrangian in both variables and dualities where one first maximizes and then minimizes the lagrangian in each variable when the primal problem has certain structure an associated canonical lagrangian is defined which leads to the appropriate dual variational principle and to a hamiltonian description the properties of and correspondence between the critical points are analyzed the last five sections are devoted to examples of these dual principles

existence stability and convergence of solutions of discrete velocity models to the boltzmann equation we prove the convergence of finite difference approximations to solutions of the boltzmann equation an essential step is the proof of convergence of discrete approximations to the collision integral this proof relies on our previous results on the consistency of this approximation for the space homogeneous problem we prove strong convergence of our discrete approximation to the strong solution of the boltzmann equation in the space dependent case we prove weak convergence to diperna lions solutions

deriving priorities from fuzzy pairwise comparison judgements a new approach for deriving priorities from fuzzy pairwise comparison judgements is proposed based on cuts decomposition of the fuzzy judgements into a series of interval comparisons the assessment of the priorities from the pairwise comparison intervals is formulated as an optimisation problem maximising the decision maker s satisfaction with a specific crisp priority vector a fuzzy preference programming method which transforms the interval prioritisation task into a fuzzy linear programming problem is applied to derive optimal crisp priorities aggregating the optimal priorities which correspond to different cut levels enables overall crisp scores of the prioritisation elements to be obtained a modification of the linear fuzzy preference programming method is also proposed to derive priorities directly from fuzzy judgements without applying cut transformations the formulation of the prioritisation problem as an optimisation task is similar to the previous approach but it requires the solution of a non linear optimisation program the second approach also derives crisp priorities and has the advantage that it does not need additional aggregation and ranking procedures both proposed methods are illustrated by numerical examples and compared to some of the existing fuzzy prioritisation methods

variational principles for operator equations and initial value problems on decrit et on etudie des principes variationnels pour certaines classes d equations operateur et de problemes aux valeurs initiales