a class of linearizable models and generation of material response functions to nonlinear hyperbolic heat conduction an approach is proposed for reducing to linear form 2 2 quasilinear nonhomogeneous systems of partial differential equations of first order in the process classes of material response functions to governing models of physical interest can be determined an application to nonlinear hyperbolic heat conduction is given

numerical simulation of air pollution due to traffic flow in urban networks abstract as it is well known traffic flow is the main pollution source in many urban areas where the number of vehicles ranges from many thousands to millions thus estimating the pollution emission rate due to traffic flow in big cities is a very hard task to approach this environmental issue in this paper we propose a methodology that consists of combining the 1d lighthill whitham richards traffic model for road networks with a classical 2d advection diffusion reaction pollution model for the atmosphere here the pollution model uses a source term that takes into account the traffic flow contamination by means of a radon measure supported on a road network within an urban domain furthermore we establish the existence of solution of the coupled model and detail a complete numerical algorithm to compute it mainly interfacing a finite volume scheme based on the supply demand method for the traffic model with a characteristics lagrange finite element method for the pollution model finally several numerical experiences for a real urban domain the guadalajara metropolitan area in mexico are presented

a method for determining exact solutions to a class of nonlinear models based on introduction of differential constraints a nonhomogeneous system of partial differential equations of first order involving two dependent and two independent variables is considered the introduction of suitable differential constraint equations permits to develop an approach to determining exact solutions to the model in point based on the use of the hodograph transformation within the present theoretical framework the governing systems of rate type materials and flows in fluid filled elastic tubes are investigated

essential dimension spinor groups and quadratic forms we prove that the essential dimension of the spinor group spinn grows exponentially with n and use this result to show that quadratic forms with trivial discriminant and hasse witt invariant are more complex in high dimensions than previously expected

conditionally invariant solutions of the rotating shallow water wave equations this paper is devoted to the extension of the recently proposed conditional symmetry method to first order nonhomogeneous quasilinear systems which are equivalent to homogeneous systems through a locally invertible point transformation we perform a systematic analysis of the rank 1 and rank 2 solutions admitted by the shallow water wave equations in 2 1 dimensions and construct the corresponding solutions of the rotating shallow water wave equations these solutions involve in general arbitrary functions depending on riemann invariants which allow us to construct new interesting classes of solutions

optimal management of an urban road network with an environmental perspective abstract within the framework of numerical simulation and optimal control of partial differential equations in this work we deal with the mathematical modelling and optimal management of urban road networks in particular we are interested in finding the optimal management of the network intersections in order to reduce traffic congestion and atmospheric pollution so we consider two different multi objective control problems the former from a cooperative viewpoint the latter within a hierarchical paradigm propose a complete numerical algorithm to solve them and finally present several numerical tests for a realistic case posed in the guadalajara metropolitan area mexico where the possibilities of our methodology are shown

when nonlinear differential equations are equivalent to linear differential equations a necessary and sufficient condition is established for the existence of a 1 1 transformation of a system of nonlinear differential equations to a system of linear equations the obtained theorems enable one to construct such transformations from the invariance groups of differential equations the hodograph transformation the legendre transformation and lie s transformation of the monge ampsre equation are shown to be special cases noninvertible transformations are also considered examples include burgers equation a nonlinear diffusion equation and the liouville equation introduction in this work we study transformations mapping nonlinear differen tial equations to linear differential equations in a 1 1 manner based upon the group analysis of differential equations we obtain necessary and sufficient conditions for the existence of such transformations the established theorems not only allow us to determine the existence of the transformations but also enable us to actually construct these transformations from invariance groups of the nonlinear equations in the following analysis two types of transformations are considered 1 the invariance groups of differential equations and 2 the mappings which transform nonlinear differential equations to linear differential equations theorems will be proved based upon the following observations clearly if there exists a 1 1 mapping between any two differential equations it must inject properties of one equation into the other including their invariance properties for this reason the often ignored fact that any linear differential equation admits an invariance group related to the superpo sition principle becomes very significant the generator of this particular group depends upon an arbitrary solution of the linear equation it follows then that any nonlinear equation transformable to a linear equation by a 1 1 mappingmust admit an invariance group whose generator depends upon an arbitrary solution of some linear differential equation the idea of comparing invariance groups of differential equations in the search of mappings connecting the equations was first used by bluman in his study of burgers equation 1 and it was applied to the study of the mappings of one dimensional linear parabolic equations to the heat equation 2 in the first section of this paper some basic properties of various transformations are summarized the theorems are presented in the second section the examples in the third section include the hodograph transformation the legendre transformation lie s transformation of themonge ampre equation and the equation ux uxx uyy 0 in the last section we examine properties of noninvertible mappings of burgers equation the diffusion equation u 2ux uy 0 and the liouville equation u y e u respectively into linear equations

reduction of nonhomogeneous quasilinear 2 2 systems to homogeneous and autonomous form by using the invariance with respect to suitable lie groups of point transformations necessary and sufficient conditions are determined allowing one to map through an invertible point transformation nonhomogeneous and nonautonomous quasilinear 2 2 systems to homogeneous and autonomous form the procedure is constructive and the new independent and dependent variables are obtained by determining the canonical variables associated with the lie groups of point symmetries admitted by the source system various examples of physical interest arising from different contexts are considered

hodograph transformation and differential constraints for wave solutions to 2 2 quasilinear hyperbolic nonhomogeneous systems the differential constraint method is used to work out a reduction approach to determine solutions in a closed form to the highly nonlinear hodograph system arising from 2 2 hyperbolic nonhomogeneous models these solutions inherit all of the features of the standard wave solutions obtainable via the classical hodograph transformation and in the meantime incorporate the dissipative effects induced on wave processes by the source like term involved in the governing equations within such a theoretical framework the problem of integrating the standard linear hodograph system associated with 2 2 homogeneous models is also revisited and a number of results obtained elsewhere of relevant interest in wave problems are recovered as a particular case along the lines of the proposed reduction approach different examples of 2 2 governing models are analysed thoroughly in order to highlight the flexibility of the provided solutions to describe hyperbolic dissipative wave processes

the torsion index of the spin groups the torsion index is a positive integer associated by grothendieck to any connected compact lie group g 10 as explained in section 1 knowing the torsion index of a group has direct consequences for the abelian subgroups of g the integral cohomology of the classifying space the complex cobordism of the classifying space the chow ring of the classifying space 26 and the classification of g torsors over fields 24 demazure 5 marlin 14 and tits 25 have given upper bounds for the torsion index in this paper we compute the torsion index exactly for all the spin groups spin n in another paper we will compute the torsion index of the exceptional group e8 as discussed below this completes the calculation of the torsion index for all the simply connected simple groups the topology of the spin groups becomes more and more complicated as the dimension increases and so it was not at all clear that it would be possible to do the calculation in all dimensions indeed the answer is rather intricate and the proof in high dimensions requires some deep information from analytic number theory bauer and bennett s theorem on the binary expansion of 2 1

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

on the decoupling problem of general quasilinear first order systems in two independent variables abstract the paper deals with the decoupling problem of general quasilinear first order systems in two independent variables we consider either the case of homogeneous and autonomous systems or the one of nonhomogeneous and or nonautonomous systems necessary and sufficient conditions for the partial or full decoupling of the systems at hand are provided the conditions involve the properties of eigenvalues and eigenvectors of the coefficient matrix and provide the differential constraints whose integration leads to the decoupling transformation some applications of physical interest are also given

the mathematics of atmospheric dispersion modeling the gaussian plume model is a standard approach for studying the transport of airborne contaminants due to turbulent diffusion and advection by the wind this paper reviews the assumptions underlying the model its derivation from the advection diffusion equation and the key properties of the plume solution the results are then applied to solving an inverse problem in which emission source rates are determined from a given set of ground level contaminant measurements this source identification problem can be formulated as an overdetermined linear system of equations that is most easily solved using the method of least squares various generalizations of this problem are discussed and we illustrate our results with an application to the study of zinc emissions from a smelting operation

riemann problems and exact solutions to a traffic flow model within the theoretical framework of differential constraints method a nonhomogeneous model describing traffic flows is considered classes of exact solutions to the governing equations under interest are determined furthermore riemann problems and generalized riemann problems which model situations of interest for traffic flows are solved

reduction of balance laws to conservation laws by means of equivalence transformations a class of partial differential equations a conservation law and two balance laws with three independent variables and involving six arbitrary continuously differentiable functions is considered in the framework of equivalence transformations these are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent dependent and additional variables representing values taken by the arbitrary elements projecting the admitted symmetries into the space of independent and dependent variables we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements in particular we are interested in finding an equivalent autonomous system of conservation laws it is shown how the results apply to some physical problems

some new examples in the theory of quadratic forms we construct a 6 dimensional anisotropic quadratic form phi and a 4 dimensional quadratic form psi over some fieldf such that phi becomes isotropic over the function field f psi but every proper subform of phi is still anisotropic over f psi it is an example of non standard isotropy with respect to some standard conditions of isotropy for 6 dimensional forms over function fields of quadrics known previously besides of that we produce an 8 dimensional quadratic form phi with trivial determinant such that the index of the clifford invariant of phi is 4 but phi can not be represented as a sum of two 4 dimensional forms with trivial determinants using this we find a 14 dimensional quadratic form with trivial discriminant and clifford invariant which is not similar to a difference of two 3 fold pfister forms the proofs are based on computations of the topological filtration on the grothendieck group of certain projective homogeneous varieties to do these computations we develop several methods covering a wide class of varieties and being to our mind of independent interest

coupling traffic models on networks and urban dispersion models for simulating sustainable mobility strategies the aim of the present paper is to investigate the viability of macroscopic traffic models for modeling and testing different traffic scenarios in order to define the impact on air quality of different strategies for the reduction of traffic emissions to this aim we complement a well assessed traffic model on networks garavello and piccoli 2006 1 with a strategy for estimating data needed from the model and we couple it with the urban dispersion model sirane soulhac 2000 2

the constant astigmatism equation new exact solution in this paper we present a new solution for the constant astigmatism equation this solution is parameterized by an arbitrary function of a single variable

linearization procedure of nonlinear first order system of partial differential equations by means of canonical variables related to lie groups of point transformations an algorithm is given to linearize nonlinear first order systems of partial differential equations admitting an infinite parameter lie group of point trasformations the associated infinitesimal operator must be a linear combination of commuting operators which individually are not necessarily admitted by the basic system whereas the cofficents of the combination are given by arbitrary solutions of suitable linear system the procedure is based on the introduction of the canonical variables corresponding to the commuting operators within such a framework we reformulate a theorem already proved by kumei and bluman siam j appl math 42 1982 the paper concludes with some illustrative examples of the proposed algorithm

around 16 dimensional quadratic forms in i 3_q we determine the indexes of all orthogonal grassmannians of a generic 16 dimensional quadratic form in i 3 q this is applied to show that the 3 pfister number of the form is ge 4 other consequences are a new and characteristic free proof of a recent result by chernousov merkurjev on proper subforms in i 2 q originally available in characteristic 0 as well as a new and characteristic free proof of an old result by hoffmann tignol and izhboldin karpenko on 14 dimensional quadratic forms in i 3 q originally available in characteristic ne 2 we also suggest an extension of the method based on investigation of the topological filtration on the grothendieck ring of a maximal orthogonal grassmanian which applies to quadratic forms of dimension higher than 16

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant