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

closed solution for the spatially homogeneous kac s model of the nonlinear boltzmann equation finds a closed spatially homogeneous solution of the nonlinear kac s model 1956 in 1 1 dimensions velocity v and time t choosing for the even velocity part of the distribution function the bobylev krook wu mode 1975 6 the author adds an odd velocity part the author finds the possibility of the existence of the tjon relaxation effect when the time t is increasing this depends on both the initial condition and the cross section

bi solitons for a class of non linear equations with quadratic non linearity ii abstract we consider the class of non integrable non linear equations l q k k 2 l q 1 i j q aij i x i j t j 0 in 1 1 dimensions we seek rational solutions k 1 2 which we call bi solitons with exponential type variables i exp ix it in this paper we restrict to q 2 and 3 and investigate the general q case in the following paper we find that these bi solitons exist when the operator lq with can be factorized as the product of smaller order differential operators besides the trivial factorized bi solitons we show that there exist non trivial ones whenever k may be written as lmaxx l2fl z 1 2 in order to understand the origin of the factorization property to any polynomial k l2fl z we associate a linear transformation such that lqk has only the power l2 of k2 for q 2 and 3 we find that there exist particular polynomials of this type restraining lq to be a product of smallr order operators for the full non linear equations we verify that all the bi solitons can be obtained from these particular polynomials

hasse principle for hermitian spaces over semi global fields abstract in a recent paper colliot thelene parimala and suresh conjectured that a local global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p adic curves in this paper we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew hermitian spaces over central simple algebras with involutions the proof implements patching techniques of harbater hartmann and krashen as an application we obtain a springer type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p adic curves

learning coverage control of mobile sensing agents in one dimensional stochastic environments this technical note presents learning coverage control of mobile sensing agents without a priori statistical information regarding random signal locations in a one dimensional space in particular the proposed algorithm controls the usage probability of each agent in a network while simultaneously satisfying an overall network formation topology the proposed control algorithm is rather direct not involving any identification of an unknown probability density function associated to random signal locations our approach builds on diffeomorphic function learning with kernels the almost sure convergence properties of the proposed control algorithm are analyzed using the ode approach numerical simulations for different scenarios demonstrate the effectiveness of the proposed approach

convergence analysis of recursive identification algorithms based on the nonlinear wiener model recursive identification algorithms based on the nonlinear wiener model are presented a recursive identification algorithm is first derived from a general parameterization of the wiener model using a stochastic approximation framework local and global convergence of this algorithm can be tied to the stability properties of an associated differential equation since inversion is not utilized noninvertible static nonlinearities can be handled which allows a treatment of for example saturating sensors and blind adaptation problems gauss newton and stochastic gradient algorithms for the situation where the static nonlinearity is known are then suggested in the single input single output case the proposed methods can outperform conventional linearizing inversion of the nonlinearity when measurement disturbances affect the output signal for fir finite impulse response models it is also proved that global convergence of the schemes is tied to sector conditions on the static nonlinearity in particular global convergence of the stochastic gradient method is obtained provided that the nonlinearity is strictly monotone the local analysis performed for iir infinite impulse response models illustrates the importance of the amplitude contents of the exciting signals

linearity inside nonlinearity exact solutions to the complex ginzburg landau equation abstract systems of two linear partial differential equations pdes with constant coefficients define a natural basis of elementary functions to build some exact solutions to nonlinear pdes for the one dimensional complex ginzburg landau equation the usual representations of the complex field re a or a are multivalued which makes difficult the search for exact solutions in this manner fortunately the painleve analysis naturally introduces an elementary single valued like representation of a by two fields z grad respectively complex abd real uniquely defined by an explicit expression for and a ze i another important feature is the invariance by parity on a which increases the class of expected solutions this allows to retrieve quite easily the four famous solutions of nozaki and bekki represented by the constant coefficients of two linear partial differential equations and a finite set of constants

exact solution for the spatially homogeneous nonlinear kac model of the boltzmann equation with an external force term the author studies the spatially homogeneous kac model of the nonlinear boltzmann equation in 1 1 dimensions velocity v and time t when an external force term is present the author finds a closed solution the force term decreases exponentially with time like constant1 exp constant2 t where both constants depend explicitly on the moments of the cross section the author finds for the relaxation to equilibrium that the tjon overpopulation effect depends on both the initial condition and the microscopic model of cross section for the existence of this effect the author establishes a criterion which is a well defined linear combination of the moments of the author cross section

similarity solutions of 2 velocity models of the extended boltzmann equation abstract we analyse in the frame of the extended kinetic theory travelling wave solutions for the 2 velocity discrete models of the boltzmann equation the extended formalism allows for absorption creation due to scattering with a background as well as for the existence of external sources of test particles as a preliminary step existence uniqueness and stability of equilibrium states are investigated we find new soliton like solutions and analyse their dependence of the physical parameters of the model

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

semi inhomogeneous solutions of the kac model of boltzmann equations constructs semi inhomogeneous solutions of the kac model for which the spatial dependence is only present in the odd velocity part of the distribution at the macroscopic level the local density the local energy and the associated current components are determined and discussed while at the microscopic level there exist two different classes of distributions as an illustration the author gives two solutions explicitly for one class the distributions relax towards a maxwellian equilibrium solution and they correspond either to a contraction or an expansion while for the other they go to zero when the time increases to infinity the space variable appears linearly and in order to maintain the positivity stays inside a finite interval assuming that the distributions are zero outside such intervals a physical interpretation can be obtained for the whole space axis provided appropriate elastic walls sinks and sources are introduced these boundary conditions seem more natural for the class of distributions with maxwellian relaxation

iterative control for networked heterogeneous multi agent systems with uncertainties this note addresses the convergence of iterative control for networked heterogeneous multi agent systems where each agent has potentially different dynamics and dissimilar uncertainties the major contribution of this work is to quantify the acceptable modeling uncertainty for ensuring convergence of the proposed iterative approach for collaborative tracking convergence conditions are established for the case when inversion based iterative control of each individual agent designed separately independent of the iterative controllers of the other agents are conjoined using a network graph structure

spatial prediction with mobile sensor networks using gaussian processes with built in gaussian markov random fields in this paper a new class of gaussian processes is proposed for resource constrained mobile sensor networks such a gaussian process builds on a gmrf with respect to a proximity graph over a surveillance region the main advantages of using this class of gaussian processes over standard gaussian processes defined by mean and covariance functions are its numerical efficiency and scalability due to its built in gmrf and its capability of representing a wide range of non stationary physical processes the formulas for predictive statistics such as predictive mean and variance are derived and a sequential field prediction algorithm is provided for sequentially sampled observations for a special case using compactly supported weighting functions we propose a distributed algorithm to implement field prediction by correctly fusing all observations simulation and experimental results illustrate the effectiveness of our approach

distributed learning and cooperative control for multi agent systems this paper presents an algorithm and analysis of distributed learning and cooperative control for a multi agent system so that a global goal of the overall system can be achieved by locally acting agents we consider a resource constrained multi agent system in which each agent has limited capabilities in terms of sensing computation and communication the proposed algorithm is executed by each agent independently to estimate an unknown field of interest from noisy measurements and to coordinate multiple agents in a distributed manner to discover peaks of the unknown field each mobile agent maintains its own local estimate of the field and updates the estimate using collective measurements from itself and nearby agents each agent then moves towards peaks of the field using the gradient of its estimated field while avoiding collision and maintaining communication connectivity the proposed algorithm is based on a recursive spatial estimation of an unknown field we show that the closed loop dynamics of the proposed multi agent system can be transformed into a form of a stochastic approximation algorithm and prove its convergence using ljung s ordinary differential equation ode approach we also present extensive simulation results supporting our theoretical results

distributed parametric and nonparametric regression with on line performance bounds computation in this paper we focus on collaborative multi agent systems where agents are distributed over a region of interest and collaborate to achieve a common estimation goal in particular we introduce two consensus based distributed linear estimators the first one is designed for a bayesian scenario where an unknown common finite dimensional parameter vector has to be reconstructed while the second one regards the nonparametric reconstruction of an unknown function sampled at different locations by the sensors both of the algorithms are characterized in terms of the trade off between estimation performance communication computation and memory complexity in the finite dimensional setting we derive mild sufficient conditions which ensure that a distributed estimator performs better than the local optimal ones in terms of estimation error variance in the nonparametric setting we introduce an on line algorithm that allows the agents to simultaneously compute the function estimate with small computational communication and data storage efforts as well as to quantify its distance from the centralized estimate given by a regularization network one of the most powerful regularized kernel methods these results are obtained by deriving bounds on the estimation error that provide insights on how the uncertainty inherent in a sensor network such as imperfect knowledge on the number of agents and the measurement models used by the sensors can degrade the performance of the estimation process numerical experiments are included to support the theoretical findings

the kernel of the rost invariant serre s conjecture ii and the hasse principle for quasi split groups 3 6d4 e6 e7 we prove that for a simple simply connected quasi split group of type 3 6d4 e6 e7 defined over a perfect field f of characteristic 2 3 the rost invariant has trivial kernel in certain cases we give a formula for the rost invariant it follows immediately from the result above that if cd f 2 resp vcd f 2 then serre s conjecture ii resp the hasse principle holds for such a group for a c2 field in particular x y we prove the stronger result that serre s conjecture ii holds for all not necessary quasi split exceptional groups of type 3 6d4 e6 e7

gaussian process regression for sensor networks under localization uncertainty in this paper we formulate gaussian process regression with observations under the localization uncertainty due to the resource constrained sensor networks in our formulation effects of observations measurement noise localization uncertainty and prior distributions are all correctly incorporated in the posterior predictive statistics the analytically intractable posterior predictive statistics are proposed to be approximated by two techniques viz monte carlo sampling and laplace s method such approximation techniques have been carefully tailored to our problems and their approximation error and complexity are analyzed simulation study demonstrates that the proposed approaches perform much better than approaches without considering the localization uncertainty properly finally we have applied the proposed approaches on the experimentally collected real data from a dye concentration field over a section of a river and a temperature field of an outdoor swimming pool to provide proof of concept tests and evaluate the proposed schemes in real situations in both simulation and experimental results the proposed methods outperform the quick and dirty solutions often used in practice

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

brief paper leader localization in multi agent systems subject to failure a graph theoretic approach in this paper structural controllability of a leader follower multi agent system with multiple leaders is studied a graphical condition for structural controllability based on the information flow graph of the system is provided the notions of p link and q agent controllability in a multi leader setting are then introduced which provide quantitative measures for the controllability of a system subject to failure in the agents and communication links the problem of leader localization is introduced which is concerned with finding the minimum number of agents whose selection as leaders results in a p link or q agent controllable network polynomial time algorithms are subsequently presented to solve the problem for both cases of undirected and directed information flow graphs

hermitian analogue of a theorem of springer with involutions and in fact in a much more general setupse for algebraic groups the weak form of springer s theorem forhermitian forms was proved by bayer fluckiger and lenstra in bl andthe analogue of springer s theorem for hermitian forms was raised as anopen question in bst the aim of this paper is to settle this question inthe affirmative for hermitian forms over quaternion algebras with respectto involutions of the first kind we prove that if

orthogonal involutions on central simple algebras and function fields of severi brauer varieties abstract an orthogonal involution on a central simple algebra a after scalar extension to the function field f a of the severi brauer variety of a is adjoint to a quadratic form q over f a which is uniquely defined up to a scalar factor some properties of the involution such as hyperbolicity and isotropy up to an odd degree extension of the base field are encoded in this quadratic form meaning that they hold for the involution if and only if they hold for q as opposed to this we prove that there exists non totally decomposable orthogonal involutions that become totally decomposable over f a so that the associated form q is a pfister form we also provide examples of nonisomorphic involutions on an index 2 algebra that yield similar quadratic forms thus proving that the form q does not determine the isomorphism class of even when the underlying algebra has index 2 as a consequence we show that the e 3 invariant for orthogonal involutions is not classifying in degree 12 and does not detect totally decomposable involutions in degree 16 as opposed to what happens for quadratic forms

isotropy of orthogonal involutions an orthogonal involution on a central simple algebra becoming isotropic over any splitting field of the algebra becomes isotropic over a finite odd degree extension of the base field provided that the characteristic of the base field is not 2 the proof makes use of a structure theorem for chow motives with finite coefficients of projective homogeneous varieties of incompressibility of certain generalized severi brauer varieties and of steenrod operations

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

a proof of the pfister factor conjecture it is shown that any split product of quaternion algebras with orthogonal involution is adjoint to a pfister form this settles the pfister factor conjecture formulated by d b shapiro a more general problem on decomposability for algebras with involution is posed and solved in the case where the algebra is equivalent to a quaternion algebra

connection between the existence of bisolitons for quadratic nonlinearities and a factorization of the associate linear operator in a first part we investigate the exponential type bisolitons of the class of equations e lq k e i j 1i j q aij xitji j k x k2 e 0 0 or e 0 which are rational solutions and we assume that their denominators have no coupling between the solitons we have found that their existence requires a factorization property of either the operator e lq for e 0 0 q 3 or lq for e 0 q 3 with the exception of the burgers equation we find three kinds of nontrivial bisolitons either those associated with the mixed nonlinearity x k2 alone or those common to the mixed nonlinearity and kkx or those belonging to kkx alone in the second part we look at the two last types of bisolitons for q 3 and give a constructive method leading both to factorized linear operators and to the explicit determination of the bisolitons we consider mainly lq 2 t a t b x c xx g xg 2 k gx with the linear operator factorizing the linear part of the burgers equation lq 2 t is a differential operato

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

exact solution for the spatially inhomogeneous nonlinear kac model of the boltzmann equation we study the spatially inhomogeneous kac model of the nonlinear boltzmann equation in 1 1 1 dimensions velocity v time t position x we obtain an exact solution which is the product of a maxwellian with a time dependent width by a second order polynomial in the velocity variable the solution satisfies a specular reflection condition at the boundary x x0 the position x0 x appears linearly and only in the odd part of the velocity distribution the range of x0 x being arbitrarily large but finite in order to maintain the positivity of the distribution the local density is spatially homogeneous further a particular linear relation between the moments of the cross section must be satisfied the most general maxwellian width has two relaxation times and their ratio is a function of the moments of the cross sections depending on whether this ratio is larger than or smaller than 1 we find contraction or expansion the solution relaxes towards a maxwellian equilibrium solution studying the tjon overpopu

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

new classes of similarity solutions for the nonlinear carleman boltzmann equation abstract we apply the lie group based similarity method to the carleman model and calculate the generators of the group depending on four group constants heretofore undiscovered classes of similarity solutions will be presented and already discovered solutions by ernst and wick will be rediscovered and classified by specifying the group constants

on a class of nonintegrable equations in 1 1 dimensions with factorized associated linear operators we consider a class of nonintegrable nonlinear equations with powerlike nonlinearities kn xkn k being the solutions and n 2 n integer building explicitly their exponential type bisolitons the denominators of the bisolitons have no soliton couplings and the linear differential operators of the linear part of the equations are factorized operators we extend our study to a larger class of nonlinearities polynomial nonlinearities which are linear combinations of powerlike nonlinearities we study the two extreme possibilities either the bisoliton is specific of a mixed nonlinearity not being a solution of any component nonlinearity or the bisoliton is common to all components different properties occur depending whether the components are kn or xkn for kn nonlinearities in order to understand the origin of the factorization of the linear operators we give a criterion which is easily checked at an almost entirely linear level of constraints we conjecture that all possible bisolitons are of th

powerlike decreasing solutions of the boltzmann equation for a maxwell gas we study the homogeneous isotropic nonlinear boltzmann equation for a maxwellian interaction we show that solutions decreasing like inverse powers of the energy are physically acceptable both in the linearized and the quadratic problem because all moments may not exist we introduce a generalized generating function and a finite differential system for generalized sonine moments is derived these new solutions may lead to small relaxation rates and justify in most cases the linear approximation

asymptotic optimality of multicenter voronoi configurations for random field estimation this technical note deals with multi agent networks performing estimation tasks consider a network of mobile agents with sensors that can take measurements of a spatial stochastic process using the kriging statistical technique a field estimate may be calculated over the environment with an associated error variance at each point we study a single snapshot scenario in which the spatial process mean is known and each agent can only take one measurement we consider two optimization problems with respect to the measurement locations using as objective functions the maximum error variance and the extended prediction variance as the correlation between distinct locations vanishes we show that circumcenter and incenter voronoi configurations become network configurations that optimize the maximum error variance and the extended prediction variance respectively we also present distributed coordination algorithms that steer the network towards these configurations

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

distributed kriged kalman filter for spatial estimation this paper considers robotic sensor networks performing spatially distributed estimation tasks a robotic sensor network is deployed in an environment of interest and takes successive point measurements of a dynamic physical process modeled as a spatio temporal random field taking a bayesian perspective on the kriging interpolation technique from geostatistics we design the distributed kriged kalman filter for predictive inference of the random field and of its gradient the proposed algorithm makes use of a novel distributed strategy to compute weighted least squares estimates when measurements are spatially correlated this strategy results from the combination of the jacobi overrelaxation method with dynamic average consensus algorithms as an application of the proposed algorithm we design a gradient ascent cooperative strategy and analyze its convergence properties in the absence of measurement errors via stochastic lyapunov functions we illustrate our results in simulation

isotropy of unitary involutions we prove the so called unitary isotropy theorem a result on isotropy of a unitary involution the analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal symplectic and unitary involutions are formal consequences of this theorem a component of the proof is a detailed study of the quasi split unitary grassmannians

recursive estimators with markovian jumps recursive stochastic algorithms have various applications in the literature it is assumed that the true value lies in a connected domain but in many cases it is known that the true value is contained in the union of a finite number of pairwise disjoint sets instead of a connected domain in these situations the existing algorithms may be not applicable to cope with this problem this paper proposes recursive stochastic algorithms with event triggered markovian jumps and presents sufficient conditions for almost sure convergence of the proposed algorithms as an example of applications this paper significantly improves an existing adaptive algorithm with the proposed method for consistent estimation of non minimum phase zeros

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

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

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

semiorthogonal decompositions and birational geometry of del pezzo surfaces over arbitrary fields we study the birational properties of geometrically rational surfaces from a derived categorical perspective in particular we give a criterion for the rationality of a del pezzo surface s over an arbitrary field namely that its derived category decomposes into zero dimensional components when s has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of s from brauer classes and chern classes associated to these vector bundles

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