reachability and minimal times for state constrained nonlinear problems without any controllability assumption we consider a target problem for a nonlinear system under state constraints we give a new continuous level set approach for characterizing the optimal times and the backward reachability sets this approach leads to a characterization via a hamilton jacobi equation without assuming any controllability assumption we also treat the case of time dependent state constraints as well as a target problem for a two player game with state constraints our method gives a good framework for numerical approximations and some numerical illustrations are included in the paper

young invariant decomposition of spin interacting operators the young invariant decomposition of totally symmetric spin interacting operators is examined from the viewpoint of outer product isoscalar factors of the symmetric group the coefficients required for the decomposition of oneand two body operators are constructed and given explicitly the formalism enables a separate treatment of the spatial and spin parts of the matrix elements of the spin interacting operators

the riemann problem for the two dimensional zero pressure euler equations abstract the riemann problem for the two dimensional zero pressure euler equations is considered the initial data are constant values in each quadrant which satisfy an assumption that each initial discontinuity projects only one two dimensional wave the phenomenon of two dimensional delta shock wave with a dirac delta function in both density and internal energy is identified both generalized rankine hugoniot relation and entropy condition for this type of two dimensional delta shock wave are proposed the qualitative behavior of entropy solutions to this relation with certain special initial data is established based on these preparations we obtain twenty three explicit solutions and their corresponding criteria in particular the mach reflection like patterns arise in the exact solutions

a class of nonlinear nonhyperbolic systems of conservation laws with well posed initial value problems abstract with mild restrictions on the initial data we show well posedness of the initial value problem for systems of conservation laws in one space variable with real and equal characteristic speeds and a deficiency of one corresponding eigenvector the obtained weak solutions assume values as borel measures at each time after a smooth solution ceases to exist the concept of entropy density flux functions is extended to such systems finally we show that systems with well posed initial value problems admitting weak solutions of this type necessarily satisfy these structural assumptions

a numerical scheme for singular shock solutions and a study of its consistence in the sense of distributions in this paper we present a numerical scheme for the approximation of singular shocks as long as some properties such as boundedness of the velocity are verified when the space step h tends to 0 we prove that the scheme provides approximate solutions that tend to satisfy the equations more precisely when the approximate solutions are plugged into the equations the result tends to 0 when h 0 in the familiar definition of weak solutions with the requirement of a smooth test function these properties can be fully proved for general classes of systems extending the korchinski system that do not have distribution solutions in the usual sense in other cases such as the keyfitz kranzer singular shocks these properties have been checked up to very small values of h these results explain numerical observations on very different systems

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

coupling coefficient for the symmetric and unitary groups two recursion formulae are deduced for the construction of inner and outer product isoscalar factors respectively for the symmetric group by which the corresponding coupling coefficients can be easily obtained a formula for the evaluation of the isoscalar factor for the subgroup chain u n contains implies u n 1 is proposed based on the relation between the outer product coupling coefficient for the standard basis of the symmetric group and clebsch gordan coefficient for the canonical basis of the unitary group the calculation procedures are presented and the properties of those coefficients are investigated

shape analysis via oriented distance functions the object of this paper is twofold we first present constructions which induce topologies on subsets of a fixed domain or hold all d in rn by using set parameterized functions in an appropriate function space second we study the role of the family of oriented distance functions also known as algebraic or signed distance functions in the analysis of shape optimization problems they play an important role in the introduction of topologies which retain the classical geometric properties associated with sets convexity exterior normals mean curvature ck boundaries etc

the stochastic reach avoid problem and set characterization for diffusions in this article we approach a class of stochastic reachability problems with state constraints from an optimal control perspective preceding approaches to solving these reachability problems are either confined to the deterministic setting or address almost sure stochastic requirements in contrast we propose a methodology to tackle problems with less stringent requirements than almost sure to this end we first establish a connection between two distinct stochastic reach avoid problems and three classes of stochastic optimal control problems involving discontinuous payoff functions subsequently we focus on solutions of one of the classes of stochastic optimal control problems the exit time problem which solves both the two reach avoid problems mentioned above we then derive a weak version of a dynamic programming principle dpp for the corresponding value function in this direction our contribution compared to the existing literature is to develop techniques that admit discontinuous payoff functions moreover based on our dpp we provide an alternative characterization of the value function as a solution of a partial differential equation pde in the sense of discontinuous viscosity solutions along with boundary conditions both in dirichlet and viscosity senses theoretical justifications are also discussed to pave the way for deployment of off the shelf pde solvers for numerical computations finally we validate the performance of the proposed framework on the stochastic zermelo navigation problem

on the symmetries of the 6j symbol the 6j tensor for compact groups is shown to transform as a basis vector for the identity representation of the permutation group s4 this allows character theory to be used to determine the minimum number of independent components and a projection operator to determine the relations between components the symmetry properties

the perturbed riemann problem and delta contact discontinuity in chromatography equations abstract in this paper we discuss the perturbed riemann problem for the nonlinear chromatography equations the solutions to the problem are constructed with the method of the splitting delta function in solutions we find another kind of delta contact discontinuity on which both state variables simultaneously contain the dirac delta functions it is quite different from the previous ones on which only one state variable contains the dirac delta function this phenomenon is also predicted theoretically by mazzotti 2009 and mazzotti et al 2010 moreover by studying the limits of the solutions as the perturbed parameter e 0 it can be found that the riemann solutions are stable for such perturbations with the initial data

extended divergence measure fields and the euler equations for gas dynamics a class of extended vector fields called extended divergence measure fields is analyzed these fields include vector fields in l p and vector valued radon measures whose divergences are radon measures such extended vector fields naturally arise in the study of the behavior of entropy solutions of the euler equations for gas dynamics and other nonlinear systems of conservation laws a new notion of normal traces over lipschitz deformable surfaces is developed under which a generalized gauss green theorem is established even for these extended fields an explicit formula is obtained to calculate the normal traces over any lipschitz deformable surface suitable for applications by using the neighborhood information of the fields near the surface and the level set function of the lipschitz deformation surfaces as an application we prove the uniqueness and stability of riemann solutions that may contain vacuum in the class of entropy solutions of the euler equations for gas dynamics

state constrained stochastic optimal control problems via reachability approach this paper deals with a class of stochastic optimal control problems socps in the presence of state constraints it is well known that for such problems the value function is in general discontinuous and its characterization by a hamilton jacobi equation requires additional assumptions involving an interplay between the boundary of the set of constraints and the dynamics of the controlled system here we give a characterization of the epigraph of the value function without assuming the usual controllability assumptions to this end the socp is first translated into a state constrained stochastic target problem then a level set approach is used to describe the backward reachable sets of the new target problem it turns out that these backward reachable sets describe the value function the main advantage of our approach is that it allows us to easily handle the state constraints by an exact penalization however the target problem involves a new state variable and a new control variable that is u

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

non selfsimilar solutions for a hyperbolic system of conservation laws in two space dimensions the non selfsimilar riemann problem for a two dimensional nonstrictly hyperbolic system of conservation laws is considered where the initial data are two constant states separated by a smooth curve without dimension reduction or coordinate transformation the two dimensional global solutions are constructed for six cases according to the normal velocities on both sides of the initial discontinuity moreover the interactions of non selfsimilar elementary waves are discussed by respectively taking the initial discontinuity as a parabola and a circle which enable us to provide the global structures of the entropy solutions explicitly c 2012 elsevier inc all rights reserved

on the 3j symmetries the symmetry properties of the 3jm tensor for any finite or compact linear group are discussed using a wreath product construction this is shown to provide a complete group theoretic explanation for all symmetry properties whether essential or arbitrary the link with the similar but distinct method of inner plethysms is considered

a time dependent hamilton jacobi formulation of reachable sets for continuous dynamic games we describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game the algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time dependent hamilton jacobi isaacs partial differential equation while alternative techniques for computing the reachable set have been proposed the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters because the time dependent equation s solution is continuous and defined throughout the state space methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions a numerical implementation of our formulation is described and has been released on the web its correctness is verified through a two vehicle three dimensional collision avoidance example for which an analytic solution is available

coefficients of fractional parentage for u m p n q u m n u p q and u m n u m u n the outer product reduction coefficients orc which reduce the representation rep induced from the irreps of the permutation groups s f1 and s f2 into the irreps of s f1 f2 are shown to be the indirect coupling coefficients for the u m p n q u m n u p q irreducible basis the non standard orc for reducing the rep induced from the non standard irreps of s f13 s f1 s f3 and s f24 s f2 s f4 into that of s f s f12 s f34 with fij fi fj f f12 f34 are identified with the u f u f12 u f34 clebsch gordan coefficient for the special gel fand bases of u f12 and u f34 the u m p n q u m n u p q cfp as well as its special case the u m p u m u p cfp are identified with the s f s f12 s f34 outer product isoscalar factor the u m n u m u n cfp are obtained from the u m n u m u n cfp by simply changing all the partition labels for u n into their conjugates interchanging rows with columns and taking into account a phase change the cfp can be calculated from the orc numerical values of the one body cfp for systems with up to six particles are tabulated

approximation of reachable sets using optimal control and support vector machines we propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems it is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel hilbert space in some sense the method can be considered as an extension to the optimal control algorithm approach recently developed by baier gerdts and xausa the convergence of the method is illustrated numerically for selected examples

sticky particles and scalar conservation laws one dimensional scalar conservation laws with nondecreasing initial conditions and general fluxes are shown to be the appropriate equations to describe large systems of free particles on the real line which stick under collision with conservation of mass and momentum

a convergent scheme for a non local hamilton jacobi equation modelling dislocation dynamics we study dislocation dynamics with a level set point of view the model we present here looks at the zero level set of the solution of a non local hamilton jacobi equation as a dislocation in a plane of a crystal the front has a normal speed depending on the solution itself we prove existence and uniqueness for short time in the set of continuous viscosity solutions we also present a first order finite difference scheme for the corresponding level set formulation of the model the scheme is based on monotone numerical hamiltonian proposed by osher and sethian the non local character of the problem makes it not monotone we obtain an explicit convergence rate of the approximate solution to the viscosity solution we finally provide numerical simulations

uniqueness and stability of riemann solutions with large oscillation in gas dynamics we prove the uniqueness of riemann solutions in the class of entropy solutions in with arbitrarily large oscillation for the 3 3 system of euler equations in gas dynamics the proof for solutions with large oscillation is based on a detailed analysis of the global behavior of shock curves in the phase space and the singularity of centered rarefaction waves near the center in the physical plane the uniqueness of riemann solutions yields their inviscid large time stability under arbitrarily large perturbation of the riemann initial data as long as the corresponding solutions are in l and have local bounded total variation satisfying a natural condition on its growth with time no specific reference to any particular method for constructing the entropy solutions is needed the uniqueness result for riemann solutions can easily be extended to entropy solutions u x t piecewise lipschitz in x for any t 0 with arbitrarily large oscillation

on reachability and minimum cost optimal control questions of reachability for continuous and hybrid systems can be formulated as optimal control or game theory problems whose solution can be characterized using variants of the hamilton jacobi bellman or isaacs partial differential equations the formal link between the solution to the partial differential equation and the reachability problem is usually established in the framework of viscosity solutions this paper establishes such a link between reachability viability and invariance problems and viscosity solutions of a special form of the hamilton jacobi equation this equation is developed to address optimal control problems where the cost function is the minimum of a function of the state over a specified horizon the main advantage of the proposed approach is that the properties of the value function uniform continuity and the form of the partial differential equation standard hamilton jacobi form continuity of the hamiltonian and simple boundary conditions make the numerical solution of the problem much simpler than other approaches proposed in the literature this fact is demonstrated by applying our approach to a reachability problem that arises in flight control and using numerical tools to compute the solution

two dimensional riemann problem for a hyperbolic system of nonlinear conservation laws i four j cases abstract the riemann problem for a two dimensional 2 2 hyperbolic system of nonlinear conservation laws has been solved analytically the solutions reveal a variety of geometric structures 57 exact solutions and the corresponding criteria have been obtained a kind of new nonlinear hyperbolic wave which we call delta shock appears in some solutions the delta shock is a discontinuity surface which is the support of a dirac function

hamilton jacobi equations with state constraints in the present paper we consider hamilton jacobi equations of the form h x u vu 0 x e q where q is a bounded open subset of rn h is a given continuous real valued function of x s p e q x r x rn and vu is the gradient of the unknown function u we are interested in particular solutions of the above equation which are required to be supersolutions in a suitable weak sense of the same equation up to the boundary of q this requirement plays the role of a boundary condition the main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system as well from related questions in constrained geodesics

new developments of delta shock waves and its applications in systems of conservation laws a theory of delta shock waves with dirac delta functions developing in both state variables for a class of nonstrictly hyperbolic systems of conservation laws is established in this paper we solve constructively the riemann problems for the system under consideration in solutions we find another kind of delta shock waves on which both state variables simultaneously contain the dirac delta functions it is quite different from the previous ones on which only one state variable contains the dirac delta function we strictly prove it satisfies the system in the sense of distributions the generalized rankine hugoniot relation and entropy condition are proposed to solve the delta shock waves furthermore we show all of the existence and stability of solutions including the delta shock waves to reasonable viscous perturbations the generalized rankine hugoniot relation is also confirmed in particular our theory on the delta shock waves possesses the generality and practicability which can be conveniently and successfully applied to those systems investigated by korchinski 1977 tan zhang and zheng 1991 ercole 2000 cheng and yang 2011 etc and we also give a simplified approach to solve a 2 d riemann problem for the system studied by tan and zhang 1990 for the case 2j 2j and obtain the explicit formulae of the delta shock waves finally the numerical simulations completely coinciding with theoretical analysis are presented

high mach number limit of one dimensional piston problem for non isentropic compressible euler equations polytropic gas we study the high mach number limit of the one dimensional piston problem for the full compressible euler equations of polytropic gas for both cases that the piston rushes into or recedes from the uniform still gas at a constant speed there are two different situations and one needs to consider measure solutions of the euler equations to deal with the concentration of mass on the piston or formation of vacuum we formulate the piston problem in the framework of radon measure solutions and show its consistency by proving that the integral weak solutions of the piston problems converge weakly in the sense of measures to singular measure solutions of the limiting problems as the mach number of the piston increases to infinity we study the high mach number limit of the one dimensional piston problem for the full compressible euler equations of polytropic gas for both cases that the piston rushes into or recedes from the uniform still gas at a constant speed there are two different situations and one needs to consider measure solutions of the euler equations to deal with the concentration of mass on the piston or formation of vacuum we formulate the piston problem in the framework of radon measure solutions and show its consistency by proving that the integral weak solutions of the piston problems converge weakly in the sense of measures to singular measure solutions of the limiting problems as the mach number of the piston increases to infinity

shadow waves entropies and interactions for delta and singular shocks this paper deals with two key problems for delta or singular shock solutions to systems of conservation laws that of their entropy admissibility conditions which is connected to the notorious uniqueness problem and that of their interaction we choose to represent systems of conservation laws by nets of functions which are piecewise constant or constant with respect only to the space variable here called shadow waves all the calculations can then be done on each element of such nets using only the usual rankine hugoniot conditions a 3 3 pressureless gas dynamics model is the main example in the paper

on nonlinear optimal control problems with state constraints this paper is concerned with an optimal control problem where the state is constrained to stay either in a smooth open set omega or in its closure overline omega under a higher order sufficient condition for the viability of omega and overline omega we prove that the optimal cost function v omega is the unique continuous constrained solution of the hamilton jacobi bellman equation furthermore we show that v omega coincides with the optimal cost function v overline omega on omega

eulerian droplet model delta shock waves and solution of the riemann problem abstract we study an eulerian droplet model which can be seen as the pressureless gas system with a source term a subsystem of this model and the inviscid burgers equation with source term the condition for loss of regularity of a solution to burgers equation with source term is established the same condition applies to the eulerian droplet model and its subsystem the riemann problem for the eulerian droplet model is constructively solved by going through the solution of the riemann problems for the inviscid burgers equation with a source term and the subsystem respectively under suitable generalized rankine hugoniot relations and entropy condition the existence of delta shock solution is established the existence of a solution to the generalized rankine hugoniot conditions is proven some numerical illustrations are presented

two dimensional riemann problems for zero pressure gas dynamics with three constant states abstract the riemann problems for two dimensional zero pressure gas dynamics are solved completely when the initial data take three constant states having discontinuities on x y positive and x negative axes with the help of characteristic analysis by studying interactions among delta shocks vacuums and contact discontinuities the riemann solutions constructed exhibit nine different explicit configurations the mach reflection like configurations appear in some solutions

minimum time control problems for non autonomous differential equations in this paper we investigate a minimum time problem for controlled non autonomous differential systems with dynamics depending on the final time the minimal time function associated to this problem does not satisfy the dynamic programming principle however we will prove that it is related to a standard front propagation problem via the reachability function two simple numerical examples are given to illustrate our approach

two dimensional riemann problem for a hyperbolic system of conservation laws in three pieces abstract this paper studies the two dimensional riemann problem for a hyperbolic system of conservation laws with three constant initial data separated by x positive y positive and x negative axes using the generalized characteristic analysis method by considering the interactions of rarefaction waves shock waves and contact discontinuities 13 exact solutions and the corresponding criteria are obtained delta shock waves appear in some solutions moreover the numerical illustrations agree completely with the geometric structures of the exact solutions

ellipsoidal techniques for reachability under state constraints the paper presents a scheme to calculate approximations of reach sets and tubes for linear control systems with time varying coefficients bounds on the controls and constraints on the state the scheme provides tight external approximations by ellipsoid valued tubes the tubes touch the reach tubes from the outside at each point of their boundary so that the surface of the reach tube is totally covered by curves that belong to the approximating tubes the result is an exact parametric representation of reach tubes through families of external ellipsoidal tubes the parameters that characterize the approximating ellipsoids are solutions of ordinary differential equations with coefficients given partly in explicit analytical form and partly through the solution of a recursive optimization problem the scheme combines the calculation of external approximations of infinite sums and intersections of ellipsoids and suggests an approach to calculate reach sets of hybrid systems

inner approximating reachable sets for polynomial systems with time varying uncertainties in this paper we propose a convex programming based method to address a long standing problem of inner approximating backward reachable sets of state constrained polynomial systems subject to time varying uncertainties the backward reachable set is a set of states from which all trajectories starting will surely enter a target region at the end of a given time horizon without violating a set of state constraints in spite of the actions of uncertainties it is equal to the zero sublevel set of the unique lipschitz viscosity solution to a hamilton jacobi partial differential equation hje we show that inner approximations of the backward reachable set can be formed by zero sublevel sets of its viscosity supersolutions consequently we reduce the inner approximation problem to a problem of synthesizing polynomial viscosity supersolutions to this hje such a polynomial solution in our method is synthesized by solving a single semidefinite program we also prove that polynomial solutions to the formulated semidefinite program exist and can produce a convergent sequence of inner approximations to the interior of the backward reachable set in measure under appropriate assumptions this is the main contribution of this paper several illustrative examples demonstrate the merits of our approach

calculation of primitive 6 j symbols the 6 j symbols of a group are independent of the subgroup chain chosen to define the basis states the authors present an improved algorithm for calculating the primitive 6 j symbols for a compact group with a faithful irrep by recursive building up using only the kronecker product rules and the general properties of 6 j previously one has sometimes needed to search for the useful equations by systematically trying all equations which involve unknown 6 j the authors show that the primitive 6 j may all be easily solved in terms of a subclass the core 6 j they discuss how the core 6 j can usually be solved proving that the method is complete for so3 the authors conjecture that the algorithm is complete for all groups

classification selection rules and symmetry properties of the clebsch gordan coefficients of symmetric group new analytical expressions for clebsch gordan coefficients cgcs of symmetric group are formulated based on the racah factorization lemma as examples cgcs for the coupling f 1 1 f 1 1 and f 1 1 f 2 2 are tabulated we have introduced a new technique called the clebsch gordan sequences cgss cgss provide a partial understanding of vanishing of cgcs and an explanation of new symmetry properties of multiplicity free cgcs

3jm and 6j tables for some bases of su6 and su3 tables of 6j symbols for su6 and su3 and tables of 3jm factors for su6 contains implies su2 su3 su3 contains implies u1 su2 and su3 contains implies so3 are presented the tables are computer produced using a program that implements the building up principle in a general form these tables are useful for calculations in high energy nuclear and solid state physics some other tabulations contain errors and none uses all the symmetries available the n independence of sun results is discussed by using the various symmetric group unitary group duality relations

level set approach for reachability analysis of hybrid systems under lag constraints this study aims at characterizing a reachable set of a hybrid dynamical system with a lag constraint in the switch control the setting does not consider any controllability assumptions and uses a level set approach the approach consists of the introduction of an adequate hybrid optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set the value function is in turn characterized by a system of quasi variational inequalities sqvi we prove a comparison principle for the sqvi which shows uniqueness of its solution a class of numerical finite difference scheme for solving the system of inequalities is proposed and the convergence of the numerical solution toward the value function is studied using the comparison principle some numerical examples illustrating the method are presented our study is motivated by an industrial application we are interested in the maximum range of hybrid vehicles

optimal control with state space constraint i optimal control of piecewise deterministic processes with state space constraint is studied under appropriate assumptions it is shown that the optimal value function is the only viscosity solution on the open domain which is also a supersolution on the closed domain finally the uniform continuity of the value function is obtained under a condition on the deterministic drift

the riemann problem admitting δ δ shocks and vacuum states the vanishing viscosity approach in this paper using the vanishing viscosity method we construct a solution of the riemann problem for the system of conservation laws ut u2 x 0 vt 2 uv x 0 wt 2 v2 uw x 0 with the initial data u x 0 v x 0 w x 0 u v w x 0 this problem admits shock wave type solutions and vacuum states shock is a new type of singular solutions to systems of conservation laws first introduced in e yu panov v m shelkovich shock waves as a new type of solutions to systems of conservation laws j differential equations 228 2006 49 86 it is a distributional solution of the riemann problem such that for t 0 its second component v may contain dirac measures the third component w may contain a linear combination of dirac measures and their derivatives while the first component u has bounded variation using the above mentioned results we also solve the shock cauchy problem for the first two equations of the above system since shocks can be constructed by the vanishing viscosity method they are natural solutions to systems of conservation laws we describe the formation of the shocks and the vacuum states from smooth solutions of the parabolic problem the results of this paper as well as those of the above mentioned paper show that solutions of systems of conservation laws can develop not only dirac measures as in the case of shocks but their derivatives as well

hamilton jacobi formulation for reach avoid differential games a new framework for formulating reachability problems with competing inputs nonlinear dynamics and state constraints as optimal control problems is developed such reach avoid problems arise in among others the study of safety problems in hybrid systems earlier approaches to reach avoid computations are either restricted to linear systems or face numerical difficulties due to possible discontinuities in the hamiltonian of the optimal control problem the main advantage of the approach proposed in this paper is that it can be applied to a general class of target hitting continuous dynamic games with nonlinear dynamics and has very good properties in terms of its numerical solution since the value function and the hamiltonian of the system are both continuous the performance of the proposed method is demonstrated by applying it to a case study which involves the target hitting problem of an underactuated underwater vehicle in the presence of obstacles

generalized variational principles global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics we study systems of conservation laws arising in two models of adhesion particle dynamics the first is the system of free particles which stick under collision the second is a system of gravitationally interacting particles which also stick under collision in both cases mass and momentum are conserved at the collisions so the dynamics is described by 2 2 systems of conservations laws we show that for these systems global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the lax oleinik variational principle for scalar conservation laws however this weak solution is not unique among weak solutions satisfying the standard entropy condition we also study a modified gravitational model in which instead of momentum some other weighted velocity is conserved at collisions for this model we prove both existence and uniqueness of global weak solutions we then study the qualitative behavior of the solutions with random initial data we show that for continuous but nowhere differentiable random initial velocities all masses immediately concentrate on points even though they were continuously distributed initially and the set of shock locations is dense

spaces of weighted measures for conservation laws with singular shock solutions abstract we study a model system of two strictly hyperbolic conservation laws which is genuinely nonlinear but for which the riemann problem has no global solution singular solutions are defined by means of a generalized rankine hugoniot relation and an overcompressive condition on the discontinuity we show that approximate solutions which can be constructed by several standard methods converge in a weighted measure space and that the error in the approximation converges to zero viscous approximations satisfy approximate entropy inequalities which imply the overcompressive condition

level sets of viscosity solutions some applications to fronts and rendez vous problems the authors treat some applications of hamilton jacobi equations to the study of a flame front propagation model and the rendez vows problem the solution of both problems requires the determination of the level sets of the viscosity solution for the corresponding equation in the flame front propagation model described here it is assumed that the evolution is driven by a vector field satisfying a transversality condition at time t 0 the evolution in the normal direction with variable velocity c x geq 0 is considered as a special case this approach is constructive permitting the numerical solution of such problems

on the symmetry properties of outer product reduction the symmetry properties of outer product coupling coefficients for the symmetric group and of clebsch gordan coefficients for the unitary group and of the corresponding isoscalar factors are discussed a simple rule is given for determination of a phase factor of outer reduction symmetry of sn and u n the rule is provided with a relationship between the phase factor of a young diagram with its subdiagrams

numerical approximations of pressureless and isothermal gas dynamics we study several schemes of first or second order accuracy based on kinetic approximations to solve pressureless and isothermal gas dynamics equations the pressureless gas system is weakly hyperbolic giving rise to the formation of density concentrations known as delta shocks for the isothermal gas system the infinite speed of expansion into vacuum leads to zero timestep in the godunov method based on exact riemann solver the schemes we consider are able to bypass these difficulties they are proved to satisfy positiveness of density and discrete entropy inequalities to capture the delta shocks and to treat data with vacuum

a discontinuous galerkin scheme for front propagation with obstacles we are interested in front propagation problems in the presence of obstacles we extend a previous work bokanowski et al siam j sci comput 33 2 923 938 2011 to propose a simple and direct discontinuous galerkin dg method adapted to such front propagation problems we follow the formulation of bokanowski et al siam j control optim 48 7 4292 4316 2010 leading to a level set formulation driven by min u t h x nabla u u g x 0 min u t h x u u g x 0 where g x g x is an obstacle function the dg scheme is motivated by the variational formulation when the hamiltonian h h is a linear function of nabla u u corresponding to linear convection problems in the presence of obstacles the scheme is then generalized to nonlinear equations written in an explicit form stability analysis is performed for the linear case with euler forward a heun scheme and a runge kutta third order time discretization using the technique proposed in zhang and shu siam j numer anal 48 1038 1063 2010 several numerical examples are provided to demonstrate the robustness of the method finally a narrow band approach is considered in order to reduce the computational cost

delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws abstract for simple models of hyperbolic systems of conservation laws we study a new type of nonlinear hyperbolic wave a delta shock wave which is a dirac delta function supported on a shock we prove that delta shock waves are w limits in l1 of solutions to some reasonable viscous perturbations as the viscosity vanishes further we solve the multiplication problem of a delta function with a discontinuous function to show that delta shock waves satisfy the equations in the sense of distributions under suitable generalized rankine hugoniot and entropy conditions we establish the existence and uniqueness of solutions involving delta shock waves for the riemann problems the existence of solutions to the cauchy problem is also investigated

6 j for mixed symmetry triads in k20 the authors discuss the possible choices of conjugation and permutation matrices for mixed symmetry triads this requires a short review of the previous choices for all triads the primitive 6 j of the finite group k20 are then calculated as an example of a group that contains mixed symmetry triads

the clebsch gordan coefficients of permutation groups s 2 s 6 the clebsch gordan cg coefficients of the permutation groups s 2 s 6 are calculated by using the eigenfunction method phases have been chosen in a consistent way to exhibit as many symmetries for the coefficients as is possible including non multiplicity free cases the cg coefficients of s 2 s 5 and those for 321 321 to 321 5 of s 6 are tabulated in the square root form of rationals these tables together with the s 6 contains implies s 5 isoscalar factors published earlier provide a complete tabulation of the s 2 s 6 cg coefficients given in the young yamanouchi basis

representations and clebsch gordan coefficients of z metacyclic groups we investigate finite groups of the form zmszn and give all irreducible representations and clebsch gordan coefficients in analytic form two subclasses are considered which seem to be inportant for applications the m metacyclic groups which are important for spin systems and the k metacyclic groups which are the smallest finite groups that have an irreducible representation of dimension p 1 where p is prime

robustness of adaptive control under time delays for three dimensional curve tracking we analyze the robustness of a class of controllers that enable three dimensional curve tracking by a free moving particle the free particle tracks the closest point on the curve by building a strict lyapunov function and robustly forward invariant sets we show input to state stability under predictable tolerance and safety bounds that guarantee robustness under control uncertainty input delays and a class of polygonal state constraints including adaptive tracking and parameter identification under unknown control gains such an understanding may provide certified performance when the control laws are applied to real life systems

exit time problems in optimal control and vanishing viscosity method the authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first order hamilton jacobi hj equation up to the boundary this equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary when these solutions are equal they can be obtained through the vanishing viscosity method finally when the hj equation has a continuous solution it is proved to be the value function for the first exit time of the domain it is also the vanishing viscosity limit arising in particular in some large deviations problems

phase conventions consistent with duality between sun and sl the schur weyl construction of irreducible tensors by symmetrized lth rank exterior products of a defining n dimensional vector space establishes a duality between the coupling algebra of the unitary group un and the symmetric group sl the coupling coefficients shared by these two groups are real and possess symmetries due to complex conjugation in un and association in sl the unitary group is factored by its unitary unimodular subgroup sun un u1 so it is usual to identify the coupling algebras of un and sun by choosing the trivial phaseless coupling for u1 this establishes a set of equivalence relations since all pseudoscalar irreducible representations irreps of un subduce onto the scalar irrep of sun the question remains can this trivial identification with the coupling algebra of sun be made consistent with the symmetries which follow from duality specifically the symmetries which follow from duality must be consistent with the equivalence relations we argue such consistency may be obtaine

an efficient filtered scheme for some first order time dependent hamilton jacobi equations we introduce a new class of filtered schemes for some first order nonlinear hamilton jacobi equations the work follows recent ideas of froese and oberman siam j numer anal 51 2013 pp 423 444 and oberman and salvador j comput phys 284 2015 pp 367 388 for steady equations here we mainly study the time dependent setting and focus on fully explicit schemes furthermore specific corrections to the filtering idea are also needed in order to obtain high order accuracy the proposed schemes are not monotone but still satisfy some epsilon monotone property a general convergence result together with a precise error estimate of order sqrt delta x are given delta x is the mesh size the framework allows us to construct finite difference discretizations that are easy to implement and high order in the domain where the solution is smooth a novel error estimate is also given in the case of the approximation of steady equations numerical tests including evolutive convex and non

semicontinuous solutions of hamilton jacobi bellman equations with degenerate state constraints abstract in this paper the value function of a bolza optimal control problem with state constraints is characterized as the unique lower semicontinuous solution of a hamilton jacobi equation the state constraints are given by an arbitrary closed set with possibly empty interior in particular soner s inward pointing condition is extended here to the case of degenerate state constraints

pursuit evasion problems and viscosity solutions of isaacs equations the dirichlet problem for first order hamilton jacobi equations arising in differential games of pursuit and evasion is studied local and global sub and superoptimality principles are stated for respectively viscosity sub and supersolutions these results are applied to obtain a general existence theorem and to prove the existence of the value of the game the main application concerns the problem of stability terminability of a dynamical system with two competitive controls and the opposite one of evadability from a general closed set the approach used in this paper allows lyapunov functions satisfying the usual condition in the weak sense of viscosity solutions

relationship between sn and u n isoscalar factors and higher order u n invariants general relationships expressing u n and sn coupling and transformation isoscalar factors in terms of u n racah and 9 coefficients are derived the absolute values of u n racah coefficients involving at most k column irreducible representations are shown to be identical with su k racah coefficients in particular an explicit relationship is established between the u n and su 2 approaches to the many electron correlation problem

well posedness for pressureless flow we study the uniqueness problem for pressureless gases previous results on this topic are only known for the case when the initial data is assumed to be a bounded measurable function this assumption is unnatural because the solution is in general a radon measure in this paper the uniqueness of the weak solution is proved for the case when the initial data is a radon measure we show that besides the oleinik entropy condition it is also important to require the energy to be weakly continuous initially our uniqueness result is obtained in the same functional space as the existence theorem

two dimensional riemann problem for a hyperbolic system of nonlinear conservation laws ii initial data involving some rarefaction waves abstract this paper is a continuation of our first paper j differential equations in press in this paper we solve the 2 d riemann problem with the initial data projecting some contact discontinuities and rarefaction waves the solutions reveal a variety of geometric structures for the interaction of elementary waves such as contact discontinuities rarefaction waves shocks and or shocks defined in our first paper

delta shock waves in chromatography equations abstract the previous investigations on delta shock waves were mostly focused on those with dirac delta function in only one state variable in this paper we obtain another kind from the nonlinear chromatography equations in which the dirac delta functions develop simultaneously in both state variables it is strictly proved to satisfy the system in the sense of distributions the generalized rankine hugoniot relation and entropy condition are clarified the numerical results completely coinciding with the theoretical analysis are presented

concentration and cavitation in the vanishing pressure limit of solutions to the euler equations for nonisentropic fluids numerical simulations 2 d riemann problem in gas dynamics and formation of spiral in nonlinear problems in engineering and science numerical and analytical approach beijing 1991 science press beijing 1992 pp 167 179 discrete contin dyn syst 1 1995 555 584 6 2000 419 430 for the euler equations for gas dynamics in the regime of small pressure showed that for one case the particles seem to be more sticky and tend to concentrate near some shock locations and for the other case in the region of rarefaction waves the particles seem to be far apart and tend to form cavitation in the region in this paper we identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions of the full euler equations for nonisentropic compressible fluids with a scaled pressure it is rigorously shown that any riemann solution containing two shocks and possibly one contact discontinuity to the euler equations for nonisentropic fluids tends to a shock solution to the corresponding transport equations and the intermediate densities between the two shocks tend to a weighted measure that along with the two shocks and possibly contact discontinuity forms the shock as the pressure vanishes by contrast it is also shown that any riemann solution containing two rarefaction waves and possibly one contact discontinuity to the euler equations for nonisentropic fluids tends to a two contact discontinuity solution to the transport equations and the nonvacuum intermediate states between the two rarefaction waves tend to a vacuum state as the pressure vanishes some numerical results exhibiting the processes of concentration and cavitation are presented as the pressure decreases 2003 elsevier b v all rights reserved msc 35l65 35b30 76e19 35q35 35l67 35b25 65m06