the broadwell model for initial values in l 1 _ bf r the cauchy problem for the broadwell model is shown to have a global mild solution for initial data inl 1 with smalll1 norm and a local solution for arbitrary initial data inl 1 for data which are small inl1 the asymptotic behaviour of the solutions ast is determined moreover it is shown that a global solution exists for all initial values inl 1 with finite entropy if theh theorem holds

emergence of fuzzy preferences for risk in a birkhoff von neumann logics environment we show that if a portfolio of a financial derivative asset and a stock is put in an environment where the value of an asset besides its price is formalized as a superposition of price states such portfolio may not be risk free and fuzzy preferences for risk premia may exist we argue for a modification of the classical brownian motion process as used in option pricing this modification on the classical brownian motion we call the email protected brownian motion and one specific format of this email protected brownian motion can be shown to have a connection with the quantum physical schrodinger equation

the milne and kramers problems for the boltzmann equation of a hard sphere gas existence uniqueness and properties of asymptotic behavior are proved for solutions of the milne and kramers problems for the linearized boltzmann equation for a gas of hard spheres the proof uses energy like estimates and follows the analysis of bardos santos and sentis for the equations of neutron transport

singular perturbations of first order hyperbolic systems with stiff source terms abstract this work develops a singular perturbation theory for initial value problems of nonlinear first order hyperbolic systems with stiff source terms in several space variables it is observed that under reasonable assumptions many equations of classical physics of that type admit a structural stability condition this condition is equivalent to the well known subcharacteristic condition for one dimensional 2 2 systems and the well known time like condition for one dimensional scalar second order hyperbolic equations with a small positive parameter multiplying the highest derivatives under this stability condition we construct formal asymptotic approximations of the initial layer solution to the nonlinear problem furthermore assuming some regularity of the solutions to the limiting inner problem and the reduced problem we prove the existence of classical solutions in the uniform time interval where the reduced problem has a smooth solution and justify the validity of the formal approximations in any fixed compact subset of the uniform time interval the stability condition seems to be a key to problems of this kind and can be easily verified moreover this presentation unifies and improves earlier works for some specific equations

boundary layer behavior in the fluid dynamic limit for a nonlinear model boltzmann equation in this paper we study the fluid dynamic limit for the one dimensional broadwell model of the nonlinear boltzmann equation in the presence of boundaries we consider an analogue of maxwell s diffusive and reflective boundary conditions the boundary layers can be classified as either compressive or expansive in terms of the associated characteristic fields we show that both expansive and compressive boundary layers before detachment are nonlinearly stable and that the layer effects are localized so that the fluid dynamic approximation is valid away from the boundary we also show that the same conclusion holds for short time without the structural conditions on the boundary layers a rigorous estimate for the distance between the kinetic solution and the fluid dynamic solution in terms of the mean free path in thel norm is obtained provided that the interior fluid flow is smooth the rate of convergence is optimal

quantum bohmian model for financial market we apply methods of quantum mechanics for mathematical modeling of price dynamics at the financial market the hamiltonian formalism on the price price change phase space describes the classical like evolution of prices this classical dynamics of prices is determined by hard conditions natural resources industrial production services and so on these conditions are mathematically described by the classical financial potential v q where q q1 qn is the vector of prices of various shares but the information exchange and market psychology play important and sometimes determining role in price dynamics we propose to describe such behavioral financial factors by using the pilot wave bohmian model of quantum mechanics the theory of financial behavioral waves takes into account the market psychology the real trajectories of prices are determined through the financial analogue of the second newton law by two financial potentials classical like v q hard market conditions and quantum like u q behavioral market conditions

emergent quantum mechanics of finances this paper is an attempt at understanding the quantum like dynamics of financial markets in terms of non differentiable price time continuum having fractal properties the main steps of this development are the statistical scaling the non differentiability hypothesis and the equations of motion entailed by this hypothesis from perspective of the proposed theory the dynamics of s p500 index are analyzed

the euler maclaurin summation formula the sampling theorem and approximate integration over the real axis abstract the euler maclaurin summation formula is used to deduce the whittaker shannon sampling theorem for not necessarily band limited functions as well as to study numerical integration over the real axis concerning the latter error estimates are determined in case the function to be integrated is smooth but not necessarily analytic two characteristic examples are given

on a class of exact two dimensional stationary solutions for the broadwell model of the boltzmann equation potential flows in steady state conditions are studied in detail in the framework of the plane broadwell discrete velocity model 1964 of the boltzmann equation it is shown that explicit exact solutions with physical meaning can be obtained in terms of elementary or special functions previously discovered analytical incompressible rows are recovered as particular cases new solutions for compressible flows are classified and discussed

a black scholes schrodinger option price bit versus qubit abstract the celebrated black scholes differential equation provides for the price of a financial derivative the uncertainty environment of such option price can be described by the classical bit a system with two possible states this paper argues for the introduction of a different uncertainty environment characterized by the so called qubit we obtain an information based option price and discuss the differences between this option price and the classical option price

an ℏ brownian motion and the existence of stochastic option prices abstract we introduce a particular type of brownian motion i e a brownian motion with a diffusion coefficient containing we show that under classical black scholes methodology we can obtain a pde with a stochastic rate of return in this environment of non classical uncertainty valued preferences for risk may exist

asymptotic behavior of one dimensional discrete velocity models in a slab we prove results on the asymptotic behavior of solutions to discrete velocity models of the boltzmann equation in the one dimensional slab 0 x 1 with general stochastic boundary conditions at x 0 and x 1 assuming that there is a constant wall maxwellian m mi compatible with the boundary conditions and under a technical assumption meaning strong thermalization at the boundaries we prove three types of results i if no velocity has x component 0 there are real valued functions 1 t and 2 t such that in a measure theoretic sense fi 0 t 1 t m i fi 1 t 2 t mi as t 1 and 2 are closely related and satisfy functional equations which suggest that 1 t 1 and 2 t 1 as t ii under the additional assumption that there is at least one non trivial collision term containing a product f k f l with k l where k denotes the x component of the velocity associated with f k we show that in a measure theoretic sense 1 t and 2 t converge to 1 as t this entails l1 convergence of the solution to the unique wall maxwellian for this result k l 0 is admissible iii in the absence of any collision terms but under the assumption that there is an irrational quotient i j l k here i l 0 and j k 0 renewal theory entails that the solution converges to the unique wall maxwellian in l

discretizations for the incompressible navier stokes equations based on the lattice boltzmann method a discrete velocity model with spatial and velocity discretization based on a lattice boltzmann method is considered in the low mach number limit a uniform numerical scheme for this model is investigated in the limit the scheme reduces to a finite difference scheme for the incompressible navier stokes equation which is a projection method with a second order spatial discretization on a regular grid the discretization is analyzed and the method is compared to chorin s original spatial discretization numerical results supporting the analytical statements are presented

broadwell model and conservative supersonic boundary in this paper we consider an initial boundary value problem for the broadwell model with a supersonic physical boundary here we pose a conservative boundary condition which preserves the total number of particles we show that the solution converges pointwise to a boundary layer exponentially when the perturbations of the initial data to the equilibrium state are sufficiently small by using green s function as established in lan et al netw heterog media 1 167 183 2006 weighted energy estimates and by constructing a pair of anti derivatives to convert the conservative boundary condition into a dissipative boundary condition with conservation laws together with an a priori chosen boundary layer

a discussion on embedding the black scholes option pricing model in a quantum physics setting abstract in this paper we consider the implications of embedding the black scholes option pricing model within a quantum physical setting the option price is considered to be a state function and a potential function is found which allows the option price to satisfy the schrodinger differential equation once this arbitrage free potential function is obtained we argue for the construction of a so called arbitrage potential function this functional is instrumental in determining the existence of a financial state function we show the existence of an arbitrage free price when the potential function converges to one the existence of arbitrage hinges on the non zero value of the planck constant this constant is then linked to a parameter which regulates the probability of occurence of strategy paths we call this parameter the belief parameter we argue that it is the belief parameter which may indeed proxy arbitrage the outcome of this paper shows that the black scholes model can be captured within a quantum physical setting and that the advantages of doing so may indeed provide for a first step to include arbitrage in a natural way in an otherwise arbitrage free model

positive 2 1 dimensional exact shock waves solutions to the broadwell model the exact solutions for the discrete boltzmann models are sums of similarity shock waves at least three for the 2 1 dimensional solutions two spatial coordinates here for the three dimensional six velocities broadwell model shock waves solutions are also constructed the difficult problem is the determination of solutions with positive density it is proven that in the arbitrary parameters space from which the solutions are constructed there exists a domain leading to positivity as an illustration a numerical example is discussed in the spatial coordinate plane both shock waves along one axis and bumps along the other are observed the movement of these structures with time is discussed

a new contribution to nonlinear stability of a discrete velocity model for a class of discrete velocity models of kinetic theory we prove exponential nonlinear conditional stability of the constant basic state in the slab 0 1

the discrete boltzmann equation with multiple collisions global existence and stability for the initial value problem this paper deals with the analysis of the initial value problem in all space rd d 2 for the discrete boltzmann equation with multiple collisions after a preliminary analysis of the mathematical modeling of the evolution equations global existence and asymptotic behavior are proven using suitable stability criteria some applications verify the analysis for specific models

self similar fluid dynamic limits for the broadwell system this report discusses a new approach for the resolution of the fluid dynamic limit for the broadwell system of the kinetic theory of gases appropriate in the case of riemann maxwellian data since the formal limiting system is expected to have self similar solutions we are motivated to replace the knudsen number in the broadwell model so that the resulting model admits self similar solutions x t and then let go to zero the limiting procedure is justified and the resulting limit is a solution of the riemann problem for the fluid dynamic limit equations a class of riemann data for which this program can be carried out is exhibited furthermore it is shown that for the carleman model the complete program can be done successfully for arbitrary riemann data

black scholes schrodinger zipf mandelbrot model framework for improving a study of the coauthor core score abstract the data and findings by miskiewicz 2013 on the relationship between the number j of publications ranked according to their decreasing importance for some scientist with her his coauthors ca i e j 1 r as found in ausloos 2013 when specific types of publications i e proceedings in a generalized sense and peer review journals are considered are reexamined along the zipf mandelbrot law mandelbrot 1977 i e j 1 r the statistics are much improved the exponent and are compared the m a core value i e the core number of cas 1 is unaffected of course a black scholes schrodinger model framework is proposed to describe the findings

optimal quadrature formulae on the real line in the present paper we treat quadrature formulae on the real line with equidistant nodes involving a sequence of not necessarily consecutive derivatives in theorem 3 1 we obtain the optimal quadrature formula for sobolev space of functions with an additional restriction on its fourier transforms the quadrature which has been proved to be optimal was introduced in 111 to the best of our knowledge the quadrature formulae considered in the present paper were studied in 7 with respect to the optimal degree of precision an explicit expression for the error of the optimal quadrature formula is given a comparison of the results with those for quadrature formulae without derivatives given in 2 under the same computational complexity is given other results concerning numerical integration on the real line can be found in s 6 131

the quantum dark side of the optimal control theory abstract in a recent article a generic optimal control problem was studied from a physicist s point of view contreras et al 2017 through this optic the pontryagin equations are equivalent to the hamilton equations of a classical constrained system by quantizing this constrained system using the right ordering of the operators the corresponding quantum dynamics given by the schrodinger equation is equivalent to that given by the hamilton jacobi bellman equation of bellman s theory the conclusion drawn there were based on certain analogies between the equations of motion of both theories in this paper a closer and more detailed examination of the quantization problem is carried out by considering three possible quantization procedures right quantization left quantization and feynman s path integral approach the bellman theory turns out to be the classical limit 0 of these three different quantum theories also the exact relation of the phase s x t of the wave function x t e i s x t of the quantum theory with bellman s cost function j x t is obtained in fact s x t satisfies a conjugate form of the hamilton jacobi bellman equation which implies that the cost functional j x t must necessarily satisfy the usual hamilton jacobi bellman equation thus the bellman theory effectively corresponds to a quantum view of the optimal control problem

convergence of a splitting scheme applied to the ruijgrok wu model of the boltzmann equation this paper deals with upwind splitting schemes for the ruijgrok wu model physica a 113 1982 401 416 of the kinetic theory of rare7ed gases in the 8uid dynamic scaling we prove the stability and the convergence for these schemes the relaxation limit is also investigated and the limit equation is proved to be a 7rst order quasi linear conservation law the loss of quasi monotonicity of the present model makes it necessary to give a more careful analysis of its structure we also obtain global error estimates in the spaces w s p for 16s61 p 16p6 and pointwise error estimates for the approximate solution the proof naturally uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 1505 1519 due to the convexity of the 8ux function c

exact solutions in 1 1 dimensions of the general two velocity discrete illner model the illner model is the most general two velocity discrete model of a boltzmann equation in one spatial dimension which satisfies an h theorem it includes as particular cases both the carleman and the mckean models solitons one dimensional solutions and bisolitons two dimensional space plus time solutions which are defined as rational fractions and solutions with one or two exponential variables are determined the model is treated as a nonintegrable nonlinear one and from the solitons the possible class of bisolitons is guessed two classes of physically acceptable bisolitons are found the first class is distributions positive only along one semiaxis and identically zero outside these are interpreted physically by introducing elastic walls plus source or sink terms which become negligible at infinite time the second class is periodic solutions which can be seen as damped sound waves essentially the same tools are used as in a companion paper for the six velocity broadwell model where the two bisoliton classes mentioned above also exist this suggests that general methods for obtaining nontrivial exact solutions could exist for the hyperbolic semilinear discrete boltzmann models

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 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

stability of planar shock fronts for multidimensional systems of relaxation equations abstract we investigate stability of multidimensional planar shock profiles of a general hyperbolic relaxation system whose equilibrium model is a system under the necessary assumption of spectral stability and a standard set of structural conditions that are known to hold for many physical systems our main result generalizing the work of kwon and zumbrun in the scalar relaxation case is to establish the bounds on the green s function for the linearized equation and obtain nonlinear l 2 asymptotic behavior sharp decay rate of perturbed weak shock profiles to establish green s function bounds we use the semigroup approach in the low frequency regime and use the energy method for the high frequency bounds separately for the system equilibrium case the analysis of the linearized equation is complicated due to glancing phenomena we treat this difficulty similarly as in the inviscid and viscous systems under the constant multiplicity condition

planar x z mixtures discrete boltzmann equations with species either in odd or even x z values we present large size physical discrete boltzmann equations or discrete velocity models results for binary mixtures of gas light mass 1 and heavy mass m in a half plane coordinates x z interface at z 0 without velocities except the rest particle where the sums x z are either odd or even for heavy or light species the models fill all integer coordinates z 0 of the plane with only a z spatial dependence densities with x z are equal here to previous results we generalize with new mathematical tools to any binary mixture with m ratio of any even to odd values m 2p 2q 1 and ratio of odd to odd values m 2p 1 2q 1 p q arbitrary integers with only binary collisions we construct large size physical discrete models only mass energy and momentum along the z axis invariants without other invariants which are called spurious we prove the existence of a physical square diagonals along the x and z axes grid where for the first component all x z a sufficient value functio

multi asset black scholes model as a variable second class constrained dynamical system abstract in this paper we study the multi asset black scholes model from a structural point of view for this we interpret the multi asset black scholes equation as a multidimensional schrodinger one particle equation the analysis of the classical hamiltonian and lagrangian mechanics associated with this quantum model implies that in this system the canonical momentums cannot always be written in terms of the velocities this feature is a typical characteristic of the constrained system that appears in the high energy physics to study this model in the proper form one must apply dirac s method for constrained systems the results of the dirac s analysis indicate that in the correlation parameters space of the multi assets model there exists a surface called the kummer surface k where the determinant of the correlation matrix is null on which the constraint number can vary we study in detail the cases with n 2 and n 3 assets for these cases we calculate the propagator of the multi asset black scholes equation and show that inside the kummer k surface the propagator is well defined but outside k the propagator diverges and the option price is not well defined on k the propagator is obtained as a constrained path integral and their form depends on which region of the kummer surface the correlation parameters lie thus the multi asset black scholes model is an example of a variable constrained dynamical system and it is a new and beautiful property that had not been previously observed

l 1 stability of multi dimensional discrete velocity boltzmann equations we study the l 1 stability of multi dimensional discrete velocity boltzmann equations under suitable smallness assumption on initial data we show that bounded mild solutions are l 1 stable for a stability estimate we employ bony s multi dimensional analysis for total interactions over characteristic planes

kinetic limits for a class of interacting particle systems we study one dimensional particle systems in which particles travel as independent random walks and collide stochastically the collision rates are chosen so that each particle experiences finitely many collisions per unit time we establish the kinetic limit and derive the discrete boltzmann equation for the macroscopic particle density

quantum like microeconomics statistical model of distribution of investments and production in this paper we demonstrate that the probabilistic quantum like ql behavior the born s rule interference of probabilities violation of bell s inequality representation of variables by in general noncommutative self adjoint operators schrodinger s dynamics can be exhibited not only by processes in the micro world but also in economics in our approach the ql behavior is induced not by properties of systems here systems commodities are macroscopic they could not be superpositions of two different states in our approach the ql behavior of economical statistics is a consequence of the organization of the process of production as well as investments in particular hamiltonian financial energy is determined by rate of return

approximation of the boltzmann equation by discrete velocity models two convergence results related to the approximation of the boltzmann equation by discrete velocity models are presented first we construct a sequence of deterministic discrete velocity models and prove convergence as the number of discrete velocities tends to infinity of their solutions to the solution of a spatially homogeneous boltzmann equation second we introduce a sequence of markov jump processes interpreted as random discrete velocity models and prove convergence as the intensity of jumps tends to infinity of these processes to the solution of a deterministic discrete velocity model

2d and 3d exact shock wave solutions with specular reflection to the discrete boltzmann models the authors construct exact solutions which are 2d two spatial coordinates for the four velocity planar model 3d for the cubic broadwell models and p dimensional for the p dimensional hypercubic discrete boltzmann model these solutions are shock waves which satisfy a specular reflection boundary condition at a wall

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

uniform global existence and parabolic limit for partially dissipative hyperbolic systems abstract this work concerns smooth solutions to the cauchy problem for first order partially dissipative hyperbolic systems with a small parameter the systems are written in non conservative form in several space variables we introduce algebraic conditions on the structure of the systems under these conditions together with a partial dissipation condition and the shizuta kawashima stability condition we prove three main results around constant equilibrium states these results are uniform global existence with respect to the parameter global in time convergence of the systems to second order nonlinear parabolic systems in a slow time variable and global existence when the parameter is fixed we also give examples of physical models to which the above results can be applied

the fluid dynamic limit of the broadwell model of the nonlinear boltzmann equation in the presence of shocks this paper studies the asymptotic equivalence of the broadwell model of the nonlinear boltzmann equation to its corresponding euler equation of compressible gas dynamics in the limit of small mean free path e it is shown that the fluid dynamical approximation is valid even if there are shocks in the fluid flow although there are thin shock layers in which the convergence does not hold more precisely by assuming that the fluid solution is piecewise smooth with a finite number of noninteracting shocks and suitably small oscillations we can show that there exist solutions to the broadwell equations such that the broadwell solutions converge to the fluid dynamical solutions away from the shocks at a rate of order e as the mean free path e goes to zero for the proof we first construct a formal solution for the broadwell equation by matching the truncated hilbert expansion and shock layer expansion then the existence of broadwell solutions and its convergence to the fluid dynamic solution is reduced to the stability analysis for the approximate solution we use an energy method which makes full use of the inner structure of time dependent shock profiles for the broadwell equations

a discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws abstract we present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities under a suitable multidimensional generalization of the whitham relaxation subcharacteristic condition we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

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

a completely solvable model of the nonlinear boltzmann equation abstract in one space and one time dimension a model of the nonlinear boltzmann equation is presented that is exactly solvable for all initial conditions furthermore this model has the following desirable properties i conservation of the number of particles ii energy conservation iii nonlinearity iv positivity of distribution functions and v unique equilibrium state for any given density which is approached as t for most physically interesting initial conditions some of the simple properties of this model are studied

galerkin petrov approach for the boltzmann equation abstract in this work we propose a new galerkin petrov method for the numerical solution of the classical spatially homogeneous boltzmann equation this method is based on an approximation of the distribution function by associated laguerre polynomials and spherical harmonics and test in a variational manner with globally defined three dimensional polynomials a numerical realisation of the algorithm is presented the algorithmic developments are illustrated with the help of several numerical tests

construction of positive exact 2 1 dimensional shock wave solutions for two discrete boltzmann models it is proved that 2 1 dimensional spacex y timet positive exact shock wave solutions of two discrete boltzmann models exist for each densityni these solutions are linear combinations of three similarity shock waves n i n0i j n ji 1 d j exp j y y j x j t j 1 2 3 two models with four independent densities are investigated the square discrete velocity boltzmann model and the model with eight velocities oriented toward the eight corners of a cube the positivity problem for the densities is nontrivial two classes of solutions are considered for which the two first similarity shock wave components depend on only one spatial dimension j const j j 1 2 for the positivity if 1 2 0 it is sufficient to prove that the 16 asymptotic shock limitsn0i n0i n3i j 0 2 n ji j 0 3 n ji are positive the density solutions are built up with five arbitrary parameters and we prove that there exist subdomains of the arbitrary parameter space in which the 16 shock limits are positive we study numerically two explicit shock wave solutions we are interested in the movement of the shock front when the time is growing and in the possible appearance of bumps in the space at intermediate times these bumps represent populations of particles which are larger than at initial time or at equilibrium time

measure solutions of the steady boltzmann equation in a slab it is shown that the steady boltzmann equation in a slab 0 a has solutionsx x such that the ingoing boundary measures 0 0 and 0 can be prescribed a priori the collision kernel is truncated such that particles with smallx component of the velocity have a reduced collision rate

regular article a new discrete velocity method for navier stokes equations the relation between the lattice boltzmann method which has recently become popular and the kinetic schemes which are routinely used in computational fluid dynamics is explored a new discrete velocity method for the numerical solution of navier stokes equations for incompressible fluid flow is presented by combining both the approaches the new scheme can be interpreted as a pseudo compressibility method and for a particular choice of parameters this interpretation carries over to the lattice boltzmann method

kinetic schemes in the case of low mach numbers originally kinetic schemes have been used as numerical methods to solve the system of compressible euler equations in gas dynamics the main idea in the approach is to construct the numerical flux function based on a microscopical description of the gas in this article the schemes are investigated in the case of isentropic euler equations and low mach numbers expanding the microscopical velocity distribution naturally leads to new kinetic schemes with strong resemblance to lattice boltzmann methods by adjusting the parameters of the kinetic scheme the numerical viscosity can be used to reproduce a given physical viscosity in this way a finite difference solver for the incompressible navier stokes equation is obtained its close relation to the lattice boltzmann approach is highlighted

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

the wave equivalent of the black scholes option price an interpretation abstract we propose an interpretation of the wave equivalent of the black scholes option price we consider nelson s version of the brownian motion dynamical theories of brownian motion princeton university press princeton nj 1967 and we use this specific motion as an input to produce a black scholes pde with a risk premium

exact solutions of the broadwell model in 1 1 dimensions the author studies the one spatial dimensional 6 velocity broadwell model with four identical densities and three independent ones the author determines solitons one dimensional shock wave solutions and bisolitons two dimensional space plus time solutions which are rational fractions with one or two exponential variables the author obtains three classes of positive exact solutions in 1 1 dimensions space x time t the first one is periodic in the space variable and for large time the solutions correspond to propagating damped linear waves the second is positive only along one semi x axis while the third positive along the whole x axis represents non planar damped shock waves using the same tools in a companion paper for the discrete 2 velocity models the author obtains in a two dimensional space the first two classes of solutions mentioned above this suggests that for the discrete boltzmann models general methods exist for the determination of non trivial exact solutions

decay theorems for the broadwell equations in this note we establish boundedness and decay theorems for finitely supported solutions to the broadwell equations with four velocity components there is a vast literature on these and similar equations for details see the survey of platkowski illner and the references contained therein the particular variant of them we examine seems new though nonlinear they are homogeneous of degree one

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

a discrete bgk approximation for strongly degenerate parabolic problems with boundary conditions abstract we consider a class of bgk systems with a finite number of velocities depending on a positive relaxation parameter that approximate strongly degenerate hyperbolic parabolic equations with initial boundary conditions we prove a priori estimates for the solutions of the systems showing that these functions converge towards the entropy solutions of strongly degenerate problems when the relaxation parameter goes to zero

space localization and well balanced schemes for discrete kinetic models in diffusive regimes we derive and study well balanced schemes for quasi monotone discrete kinetic models by means of a rigorous localization procedure we reformulate the collision terms as nonconservative products and solve the resulting riemann problem whose solution is self similar the construction of an asymptotic preserving ap godunov scheme is straightforward and various compactness properties are established within different scalings finally some computational results are supplied to show that this approach is realizable and efficient on concrete 2 2 models

a conjecture on the possible exact 1 1 dimensional solutions to the discrete boltzmann models i the exact multidimensional exponential type solutions to the discrete boltzmann models which have been found are sums of unidimensional similarity waves our goal is to obtain at a linear level a physical signature of the class of possible exact rational solutions in the discrete kinetic theory contrary to the continuous theory the macroscopic conservation laws are included into the set of nonlinear equations for the densities these conservation laws being linear differential relations underline the above recalled result for a class of l l dimensional rational solutions with two exponential variables satisfying at least two conservation laws we prove that the possible solutions are sums of similarity waves we first consider the class of models with three independent densities and two conservation laws second we investigate what happens when the two conservation laws of mass and momentum exist third we sketch briefly results for five densities and three conservation laws finally we conjecture that all exact bounded l l dimensional solutions are sums of similarity waves

the boltzmann equation global existence for a rare gas in an infinite vacuum solutions of the boltzmann equation are proved to exist globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough it is also proved as might be expected in this case that the density of the gas at each point in space goes to zero as time goes to infinity

exact 2 1 dimensional solutions for two discrete velocity boltzmann models with four independent densities for the square discrete velocity boltzmann model the authors obtain exact 2 1 dimensional space x y time t solutions these are rational solutions with three exponential variables relaxing towards non uniform maxwellian equilibrium states with two exponential variables these are also solutions of another model with four independent densities and eight velocities oriented towards the eight corners of a cube the positivity problem for the densities is non trivial

a boundary value problem for the two dimensional broadwell model it is shown that a certain boundary value problem for the steady two dimensional broadwell model on a rectangle has a solution the boundary conditions specify the ingoing particle densities on each side of the rectangle

dynamic optimization and its relation to classical and quantum constrained systems abstract we study the structure of a simple dynamic optimization problem consisting of one state and one control variable from a physicist s point of view by using an analogy to a physical model we study this system in the classical and quantum frameworks classically the dynamic optimization problem is equivalent to a classical mechanics constrained system so we must use the dirac method to analyze it in a correct way we find that there are two second class constraints in the model one fix the momenta associated with the control variables and the other is a reminder of the optimal control law the dynamic evolution of this constrained system is given by the dirac s bracket of the canonical variables with the hamiltonian this dynamic results to be identical to the unconstrained one given by the pontryagin equations which are the correct classical equations of motion for our physical optimization problem in the same pontryagin scheme by imposing a closed loop strategy the optimality condition for the action gives a consistency relation which is associated to the hamilton jacobi bellman equation of the dynamic programming method a similar result is achieved by quantizing the classical model by setting the wave function x t e i s x t in the quantum schrodinger equation a non linear partial equation is obtained for the s function for the right hand side quantization this is the hamilton jacobi bellman equation when s x t is identified with the optimal value function thus the hamilton jacobi bellman equation in bellman s maximum principle can be interpreted as the quantum approach of the optimization problem

on nonlinear stationary half space problems in discrete kinetic theory we show that every steady discrete velocity model of the boltzmann equation on the real line i d dx fi ci f which satisfies anh theorem and for which all i 0 has solutions on the half line 0 which take prescribed non negativefi o if i 0 and approach a certain manifold of maxwellians asx such solutions give the density distribution in a knudsen boundary layer in the discrete velocity case

large time behavior of discrete velocity boltzmann equations we study the asymptotic behavior of equations representing one dimensional motions in a fictitious gas with a discrete set of velocities the solutions considered have finite mass but arbitrary amplitude with certain assumptions every solution approaches a state in which each component is a traveling wave without interaction the techniques are similar to those in an earlier treatment of the broadwell model 1

quantum probability and financial market can the mathematical formalism of quantum mechanics and in particular quantum probability be applied outside of physics the answer is positive in this paper we apply methods of quantum mechanics for mathematical modelling of price dynamics of the financial market we propose to describe behavioral financial factors e g expectations of traders by using the pilot wave bohmian model of quantum mechanics on the one hand our bohmian model is a quantum like model for the financial market cf with works of w segal i e segal e haven e w piotrowski j sladkowski on the other hand since bohmian mechanics provides for the possibility to describe individual price trajectories it belongs to the domain of extended research on deterministic dynamics for financial assets our model emphasizes the complexity of the financial market the traditional description of price dynamics is completed by schrodinger s dynamics for the pilot wave of expectations of traders this is a kind of socio economic model for the financial market

1 1 dimensional hierarchies of multispeed discrete boltzmann model equations multispeed discrete boltzmann models in rd satisfying all conservation laws and leading to well defined temperatures are studied the 1 1 dimensional restriction of the system of partial differential equations pde satisfied by the densities of these models can be classified one class is characterized along a chosen coordinate by the number and location of the independent densities and by the number and physical meaning of the independent collision terms each class defines a hierarchy of pde that depends on the dimension d of a particular model in rd different classes associated either with two velocity speeds 1 and 1 d 1 2 and 2 and 2 d 1 2 or three velocity speeds 0 1 and 0 2 are obtained each class represents a d hierarchy of a well known discrete model the simplest exact solutions which are the similarity shock wave solutions are discussed and compared with the rankine hugoniot solutions

quantum model for the price dynamics the problem of smoothness of trajectories we apply methods of quantum mechanics to mathematical modelling of price dynamics in a financial market we propose to describe behavioral financial factors e g expectations of traders by using the pilot wave bohmian model of quantum mechanics our model is a quantum like model of the financial market cf with works of w segal i e segal e haven in this paper we study the problem of smoothness of price trajectories in the bohmian financial model we show that even the smooth evolution of the financial pilot wave t x representing expectations of traders can induce jumps of prices of shares

the analytical solutions of 2d stationary broadwell kinetic model the present communication is devoted to the calculation of some explicit stationary solutions of the four velocity broadwell kinetic model in two spatial dimensions the method of the truncated painleve expansion is employed and two classes of solutions are constructed it is shown that these solutions satisfy the special boundary value problem in a rectangle

wave structure induced by fluid dynamic limits in the broadwell model consider the fluid dynamic limit problem for the broadwell system of the kinetic theory of gases for maxwellian riemann initial data the formal limit is the riemann problem for a pair of conservation laws and is invariant under dilations of coordinates the approach of self similar fluid dynamic limits consists in replacing the mean free path in the broadwell model so that the resulting problem preserves the invariance under dilations the limiting procedure was justified in st here we study the structure of the emerging solutions we show that they consist of two wave fans separated by a constant state each wave fan is associated with one of the characteristic fields and is either a rarefaction wave or a shock wave the shocks satisfy the lax shock conditions and have the internal structure of a broadwell shock profile

on shock wave solutions in extended discrete kinetic theory exact traveling wave solutions joining asymptotic maxwellians in the frame of discrete velocity models of the boltzmann equation are investigated and discussed in an extended formulation which also allows for removal creation and scattering collisions with the particles of a given background it is shown that the classical broadwell shock wave structure does not carry over to any extended problem on the other hand families of exact shock wave solutions do exist for extended two velocity models for which such waves are lacking in the classical i e unextended case

l1nonlinear stability of traveling waves for a hyperbolic system with relaxation where x t r r the unknowns u v belong to r the function f f u is in c r and 0 is a fixed constant relaxation systems often arise in many physical situations let us recall for example gases not in thermodynamic equilibrium vk kinetic theory ce cip pi chromatography raa river flows traffic flows and more general waves wh more recently the study of a special class of hyperbolic systems with relaxation was developed in view of the numerical approximation of discontinuous solutions of conservation laws jx an the 2 2 relaxation hyperbolic systems of conservation laws were first analyzed by liu li2 who justified some nonlinear stability criteria for diffusion waves expansion waves and traveling waves article no 0180

an improved image denoising model based on the directed diffusion equation by substituting an anisotropic diffusion operator for the isotropic laplace operator in the directed diffusion equation adding two different coefficients in the two diffusion terms and choosing the image denoised by soft wavelet shrinkage as the initial approximate image an improved directed diffusion equation model is proposed experiments show that the new model can solve the problem that edges and details will be rapidly blurred during the diffusion process in the directed diffusion equation

a quantum model of option pricing when black scholes meets schrodinger and its semi classical limit abstract the black scholes equation can be interpreted from the point of view of quantum mechanics as the imaginary time schrodinger equation of a free particle when deviations of this state of equilibrium are considered as a product of some market imperfection such as transaction cost asymmetric information issues short term volatility extreme discontinuities or serial correlations the classical non arbitrage assumption of the black scholes model is violated implying a non risk free portfolio from haven 2002 1 we know that an arbitrage environment is a necessary condition to embedding the black scholes option pricing model in a more general quantum physics setting the aim of this paper is to propose a new black scholes schrodinger model based on the endogenous arbitrage option pricing formulation introduced by contreras et al 2010 2 hence we derive a more general quantum model of option pricing that incorporates arbitrage as an external time dependent force which has an associated potential related to the random dynamic of the underlying asset price this new resultant model can be interpreted as a schrodinger equation in imaginary time for a particle of mass 1 2 with a wave function in an external field force generated by the arbitrage potential as pointed out above this new model can be seen as a more general formulation where the perfect market equilibrium state postulated by the black scholes model represent a particular case finally since the schrodinger equation is in place we can apply semiclassical methods of common use in theoretical physics to find an approximate analytical solution of the black scholes equation in the presence of market imperfections as it is the case of an arbitrage bubble here as a numerical illustration of the potential of this schrodinger equation analogy the semiclassical approximation is performed for different arbitrage bubble forms step linear and parabolic and compare with the exact solution of our general quantum model of option pricing

large time behavior of solutions for hyperbolic balance laws we study the existence and the large time behavior of global solutions to the initial value problem for hyperbolic balance laws in n space dimensions with n 3 admitting an entropy and satisfying the stable condition we first construct global existence of the solutions to such a system around a steady state if the initial energy is small enough then we show that k order derivatives of these solutions approach a constant state in the lp norm at a rate o t 12 k n2 np with p 2 and 0 n2 provided that initially z0 b 2 where b 2 is a homogeneous besov space these decay results do not impose an additional smallness assumption on lp norm of the initial data and we thus improve the results in 3 19 we also show faster decay results in the sense that if pz0 b 2 i p z0 b 2 1 with n2 n 22 k order derivatives of the solutions approach a constant state in the lp norm at a rate o t 12 k 1 n2 np

a constructive approach to steady nonlinear kinetic equations abstract we study steady boundary value problems of nonlinear kinetic theory using a continuation argument based on the variation of the knudsen number we derive a method for the construction of steady solutions of discrete velocity models in a slab this method is readily transformed into a numerical code in a preliminary numerical test case the numerical scheme turns out to yield solutions even for knudsen numbers small enough to calculate with high precision the asymptotic flow field adjacent to a kinetic boundary layer thus we are able to numerically simulate in a simplified situation the transition from a mesoscopic kinetic boundary layer to the macroscopic far field

pseudo hermitian formulation of the quantum black scholes hamiltonian abstract we show that the non hermitian black scholes hamiltonian and its various generalizations are pseudo hermitian the metric operator is explicitly constructed for this class of hamiltonians it is also shown that the effective black scholes hamiltonian and its partner form a pseudo supersymmetric system

a boundary value problem for discrete velocity models on considere deux problemes aux valeurs limites non lineaires concernant un ecoulement stationnaire a une dimension entre des plaques paralleles

exact solutions of a hierarchy of mixing speeds models this paper presents several new aspects of discrete kinetic theory dkt first a hierarchy of d dimensional d 1 2 3 models is proposed with 2d 3 velocities and three moduli speeds 0 2 and a third one that can be arbitrary it is assumed that the particles at rest have an internal energy which for microscopic collisions supplies for the loss of the kinetic energy in a more general way than usual collisions are allowed that mix particles with different speeds second for the 1 1 dimensional restriction of the systems of pde for these models which have two independent quadratic collision terms we construct different exact solutions the usual types of exact solutions are studied periodic solutions and shock wave solutions obtained from the standard linearization of the scalar riccati equations called riccatian shock waves then other types of solutions of the coupled riccati equations are found called non riccatian shock waves and they are compared with the previous ones the main new result