knot concordance whitney towers and l 2 signatures we construct many examples of nonslice knots in 3 space that cannot be distinguished from slice knots by previously known invariants using whitney towers in place of embedded disks we define a geometric filtration of the 3 dimensional topological knot concordance group the bottom part of the filtration exhibits all classical concordance invariants including the cassongordon invariants as a first step we construct an infinite sequence of new obstructions that vanish on slice knots these take values in the l theory of skew fields associated to certain universal groups finally we use the dimension theory of von neumann algebras to define an l 2 signature and use this to detect the first unknown step in our obstruction theory

structural stability of solutions to the riemann problem for a scalar conservation law the aim of this paper is to study the structural stability of solutions to the riemann problem for a scalar conservation law with a linear flux function involving discontinuous coefficients it is proved that the riemann solution is possibly instable when one of the riemann initial data is at the vacuum furthermore we point out that the riemann solution is also possibly instable even when the riemann initial data stay far away from vacuum in order to deal with it we perturb the riemann initial data by taking three piecewise constant states and then the global structures and large time asymptotic behaviors of the solutions are obtained constructively it is also proved that the riemann solutions are unstable in some certain situations under the local small perturbations of the riemann initial data by letting the perturbed parameter e tend to zero in addition the interaction of the delta standing wave and the contact vacuum state is considered which appear in the riemann solutions

one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

lagrangians of physics and the game of fisher information transfer the lagrangians of physics arise out of a mathematical game between a smart measurer and nature personified by a demon each contestant wants to maximize his level of fisher information i the game is zero sum by conservation of information in the closed system the payoff of the game introduces a variational principle extreme physical information epi which fixes both the lagrangian and the physical constant of each scenario the epi approach provides an understanding of the relationship between measurement and physical law epi also defines a prescription for constructing lagrangians the prior knowledge required for this purpose is a rule of symmetry or conservation that implies a unitary transformation for which i remains invariant as an example when applied to the smart measurement of the space time coordinate of a particle the symmetry used is that between position time space and momentum energy space then the unitary transformation is the fourier one and epi derives the following the equivalence of energy momentum and mass the constancy of planck s parameter h and the lagrangian that implies both the klein gordon equation and the dirac equation of quantum mechanics

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

on the relative slice problem and four dimensional topological surgery a central open problem in the classification theory of topological fourmanifolds is to determine the validity of four dimensional surgery and fivedimensional s cobordism theorems without fundamental group restrictions by work of m freedman 2 3 the class of groups for which these theorems hold good groups includes the groups of polynomial growth recently freedman and teichner 6 showed that more generally the groups of subexponential growth are good it is expected 3 that the theorems fail for free non abelian groups this conjecture is known as the a b slice problem a more precise conjecture states that the whitehead double of the borromean rings wh bor is not freely slice in this paper we study the relative slice reformulation of this problem introduced in 4 our main theorem may be viewed as a result in link theory while providing some evidence towards the conjecture recall the following definition from 4

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

remarks on the extended characteristic uncertainty relations three remarks concerning the form and the range of validity of the state extended characteristic uncertainty relations urs are presented a more general definition of the uncertainty matrix for pure and mixed states is suggested some new urs are provided

two dimensional riemann problem for a 2 2 system of hyperbolic conservation laws involving three constant states abstract zhang and zheng 1990 conjectured on the structure of a solution for a two dimensional riemann problem for euler equation to resolve this illuminating conjecture many researchers have studied the simplified 2 2 systems in this paper 3 pieces riemann problem for two dimensional 2 2 hyperbolic system is considered without the restriction that each jump of the initial data projects one planar elementary wave we classify twelve topologically distinct solutions and construct analytical and numerical solutions the computed numerical solutions clearly confirm the constructed analytic solutions

existence and asymptotic behavior of measure valued solutions for degenerate conservation laws abstract we consider two classes of typical degenerate hyperbolic systems of conservation laws to provide a general approach for solving the existence and large time asymptotic behavior of measure valued solutions for initial boundary value problems some existence theorems of the measure valued solutions are established the convergence of large time averages of the measure valued solutions to a dirac mass concentrated at the input state on the boundary is proved for almost each fixed space variable although the measure valued solutions of the initial boundary problems may not be unique in general our results indicate that the asymptotic equilibrium of these measure valued solutions is unique

global structure of riemann solutions to a system of two dimensional hyperbolic conservation laws abstract the riemann problem for a two dimensional nonstrictly hyperbolic system of conservation laws is considered without the restriction that each jump of the initial data projects one planar elementary wave ten topologically distinct solutions are obtained by applying the method of generalized characteristic analysis some of these solutions involve the nonclassical waves i e the delta shock wave and the delta contact discontinuity for which we explicitly give the expressions of their strengths locations and propagation speeds moreover we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one dimensional cauchy problem which provides a verification that our construction produces the correct unique global solutions

the riemann problem for a resonant nonlinear system of conservation laws with dirac measure solutions the riemann problem for a resonant nonlinear system of conservation laws is considered here the riemann solution was constructed by employing the viscosity approximation approach one kind of new discontinuity which is called the dirac contact wave appeared in the riemann solution because the strict hyperbolicity as well as the genuine nonlinearity of the system considered failed the solution we obtained in this paper is not unique for some initial data an additional condition was explored to guarantee the uniqueness of the riemann problem

riemann problem for the relativistic chaplygin euler equations abstract the relativistic euler equations for a chaplygin gas are studied the riemann problem is solved constructively there are five kinds of riemann solutions in which four only contain different contact discontinuities and the other involves delta shock waves under suitable generalized rankine hugoniot relation and entropy condition the existence and uniqueness of delta shock solutions are established

two dimensional riemann problem involving three j s for a hyperbolic system of nonlinear conservation laws in this paper we study the riemann problem for a two dimensional hyperbolic system of conservation laws with initial data projecting exactly three contact discontinuities we use the generalized characteristic analysis method to analyze interactions of contact discontinuities and obtain six analytical solutions and corresponding criteria delta shock waves and spiral structures appear in some solutions we also conduct numerical experiments and numerical results confirm the geometric structures of the constructed solutions

existence and uniqueness of discontinuous solutions for a hyperbolic system in this paper we prove the global existence and uniqueness of solutions to the cauchy problem of a hyperbolic system which probably contains so called waves

casson towers and slice links we prove that a casson tower of height 4 contains a flat embedded disc bounded by the attaching circle and we prove disc embedding results for height 2 and 3 casson towers which are embedded into a 4 manifold with some additional fundamental group assumptions in the proofs we create a capped grope from a casson tower and use a refined height raising argument to establish the existence of a symmetric grope which has two layers of caps data which is sufficient for a topological disc to exist with the desired boundary as applications we present new slice knots and links by giving direct applications of the disc embedding theorem to produce slice discs without first constructing a complementary 4 manifold in particular we construct a family of slice knots which are potential counterexamples to the homotopy ribbon slice conjecture

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

towards a compact high order method for non linear hyperbolic systems ii the hermite hllc scheme in a finite volume framework we develop an approximate hll riemann solver specific to weakly hyperbolic systems those systems are obtained by considering not only the variable but also its first spatial derivative as unknowns to this aim we rely upon the theory of d shock waves newly developed in the scalar case first we demonstrate that the extended version of the hlle scheme to weakly hyperbolic systems is compatible with the existence of dirac measures in the solution then we develop a specific hermite least square hlsm interpolation that enables to generate a high order and compact scheme without creating spurious oscillations in the reconstruction of the variable or its first derivative extensive numerical experiments make it possible to validate the method and to check convergence to entropy solutions relying upon those results we construct a new hll riemann solver suited for the extended one dimensional euler equations for this purpose we introduce the contribution of a contact discontinuity inside the definition of the solver by using a formal analogy with the scalar study we demonstrate that this solver tolerates the existence of d shock waves in the solution numerical experiments that follow help to validate some of the assumptions made to generate this scheme

riemann problem and wave interactions for the one dimensional relativistic string equation in minkowski space abstract in this paper we consider the riemann problem for the one dimensional relativistic string equation in minkowski space it is shown that the delta shock wave will be constructed in the riemann solutions in some certain situations furthermore the interactions of elementary waves including contact discontinuity and delta shock wave are obtained

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

interactions of delta shock waves for the transport equations with split delta functions this paper is concerned with the interactions of shock waves and the vacuum states between the two contact discontinuities for the transport equations the solutions are obtained constructively when the initial data are three piecewise constant states the global structure and large time asymptotic behaviors of the solutions are analyzed case by case moreover it can be found that the riemann solutions are stable for such small perturbations with initial data by studying the limits of the solutions when the perturbed parameter e tends to zero

two multi dimensional uncertainty relations two multi dimensional uncertainty relations one related to the probability density and the other one related to the probability density current are derived and discussed both relations are stronger than the usual uncertainty relations for the coordinates and momentum

numerical method for the bone regeneration model defined within the evolving 2d axisymmetric physical domain abstract a numerical algorithm used to obtain a solution for a peri implant osseointegration model is constructed the model is formulated in terms of a system of three nonlinear coupled time dependent advection diffusion reaction equations which are defined within the irregular two dimensional physical domain which evolves in time the embedded boundary method and the level set function which is approximated on the fixed regular rectangular grid are used to track the changes of the irregular geometry of the physical domain the method of lines is applied to separate the discretizations in time and in space the advection diffusion and reaction terms are discretized separately by means of the cell centered finite volume method the exact solution of the riemann problem for the nonstrictly hyperbolic system without genuine nonlinearity is obtained an approach for the determination of the gradients of the unknown variables on the edges of the irregular control volumes is proposed the explicit second order trapezoidal rule is used for the time integration since it allows to maintain positivity of the solution which is critical for the considered problem some results of the numerical simulations are presented contact and distance osteogenesis are predicted for micro rough and smooth implants

generalized characteristic analysis and guckenheimer structure abstract we develop a generalized characteristic analysis method using this method we study interactions of elementary waves and construct the riemann solutions for two dimensional scalar conservation laws especially we obtain the necessary and sufficient condition of appearance of guckenheimer structure which is something like mach reflection in gas dynamics

4 manifold topology i subexponential groups the technical lemma underlying the 5 dimensional topologicals cobordism conjecture and the 4 dimensional topological surgery conjecture is a purely smooth category statement about locating 1 null immersions of disks these conjectures are theorems precisely for those fundamental groups good groups where the 1 null disk lemma ndl holds we expand the class of known good groups to all groups of subexponential growth and those that can be formed from these by a finite number of application of two operations 1 extension and 2 direct limit the finitely generated groups in this class are amenable and no amenable group is known to lie outside this class

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

delta shock wave formation in the case of triangular hyperbolic system of conservation laws abstract we describe shock wave generation from continuous initial data in the case of triangular conservation law system arising from generalized pressureless gas dynamics model we use smooth approximations in the weak sense that are more general than small viscosity approximations

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

δ shock waves as a new type of solutions to systems of conservation laws abstract a concept of a new type of singular solutions to systems of conservation laws is introduced it is so called n shock wave where n is n th derivative of the dirac delta function n 1 2 in this paper the case n 1 is studied in details we introduce a definition of shock wave type solution for the system u t f u x 0 v t f u v x 0 w t f u v 2 f u w x 0 within the framework of this definition the rankine hugoniot conditions for shock are derived and analyzed from geometrical point of view we prove shock balance relations connected with area transportation finally a solitary shock wave type solution to the cauchy problem of the system of conservation laws u t u 2 x 0 v t 2 u v x 0 w t 2 v 2 u w x 0 with piecewise continuous initial data is constructed these results first 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

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

viscous singular shock profiles for a system of conservation laws modeling two phase flow abstract this paper is concerned with singular shocks for a system of conservation laws via the dafermos regularization u t f u x t u x x for a system modeling incompressible two phase fluid flow the existence of viscous profiles is proved using geometric singular perturbation theory the weak convergence and the growth rate of the viscous solution are also derived the weak limit is the sum of a piecewise constant function and a measure supported on a shock line and the maximum value of the viscous solution is of order exp 1

concentrations in the one dimensional vlasov poisson equations i temporal development and non unique weak solutions in the single component case abstract weak solutions of the one component vlasov poisson equation in a single space dimension are proposed and studied here as a simpler analogue problem for the behavior of weak solutions of the two dimensional incompressible euler equations with non negative vorticity the physical structural and functional analytic analogies between these two problems are developed in detail here with this background explicit solutions for electron sheet initial data the analogue of vortex sheet initial data are presented which display the phenomena of singularity formation in finite time as well as the explicit temporal development of charge concentrations other rigorous explicit examples with charge concentration are developed where there are non unique weak solutions with the same initial data in one of these non unique weak solutions an electron sheet completely collapses to a point charge in finite time the detailed limiting behavior of regularizations such as the diffusive fokker planck equation are developed through a very efficient numerical method which yields extremely high resolution for these simpler analogue problems a striking consequences of the numerical results reported here is the fact that there is not a selection principle for a unique weak solution in some situations where there are several weak solutions with charge concentration for the same initial data in particular two explicit weak solutions with the same initial data are constructed here where it is demonstrated that the zero smoothing limit of time reversible particle methods converges to one of these solutions while the zero diffusion limit of the fokker planck equation converges to the other weak solution

higher order intersections in low dimensional topology we show how to measure the failure of the whitney move in dimension 4 by constructing higher order intersection invariants of whitney towers built from iterated whitney disks on immersed surfaces in 4 manifolds for whitney towers on immersed disks in the 4 ball we identify some of these new invariants with previously known link invariants such as milnor sato levine and arf invariants we also define higher order sato levine and arf invariants and show that these invariants detect the obstructions to framing a twisted whitney tower together with milnor invariants these higher order invariants are shown to classify the existence of twisted whitney towers of increasing order in the 4 ball a conjecture regarding the nontriviality of the higher order arf invariants is formulated and related implications for filtrations of string links and 3 dimensional homology cylinders are described

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

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

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

state extended uncertainty relations a scheme for construction of uncertainty relations for n observables and m states is presented several lowest order inequalities are displayed and briefly discussed for two states and and canonical observables the entangled extension of the heisenberg relation reads p 2 q 2 p 2 q 2

global structure of admissible solutions of multi dimensional non homogeneous scalar conservation law with riemann type data abstract we investigate the global expression and structure of admissible weak solutions of an n dimensional non homogeneous scalar conservation law with the initial data that has two constant states separated by an n 1 dimensional smooth manifold we obtain the unique global existence of non self similar solutions it is the first result about the global structure of non self similar shock waves and rarefaction waves of n dimensional non homogeneous scalar conservation law the shock wave and the rarefaction wave can be directly expressed and studied by a global implicit function finally we give some applications to discover some interesting phenomena

the interaction of nonlinear hyperbolic waves nonlinearities in wave equations lead to focusing and defocusing of solutions focusing causes sharply defined wave fronts the interaction of such sharply defined wave fronts and more generally of nonlinear hyperbolic waves is of fundamental importance and includes such phenomena as mach triple point formation shock wave diffraction patterns and the study of riemann problems in one and higher dimensions recent progress in the study of nonlinear hyperbolic wave interactions has revealed a surprising range of new mathematical phenomena and structures this mathematical theory should be useful in the design of improved computational algorithms and in part was motivated by such considerations it is also of considerable interest for its own sake as new mathematical phenomena as well as in terms of the direct insight it provides into physical phenomena within the subject matter and point of view adopted here we have attempted to present a broad and we hope a representative account of recent progress

riemann problems for nonstrictly hyperbolic 2 2 systems of conservation laws on resout le probleme de riemann pour des systemes 2 2 de lois de conservation hyperboliques ayant des fonctions flux quadratiques

two dimensional riemann problem involving three contact discontinuities for 2 2 hyperbolic conservation laws in anisotropic media abstract in this study we consider the two dimensional riemann problem for a system of conservation laws which models polymer flooding in an anisotropic oil reservoir the initial data are constants in three sectors centered at the origin which only involve three contact discontinuities by the generalized characteristic analysis method and the phase plane analysis method we find that the solutions are not unique for certain values of the initial data we propose an additional stability condition for the interaction of the contact discontinuities by this stability condition all of the exact solutions and their corresponding criteria can be obtained the solutions exhibit various geometrical structures in particular envelope rarefaction waves develop in some solutions

the riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws abstract in this paper we solve the riemann problem with the initial data containing dirac delta functions for a class of coupled hyperbolic systems of conservation laws under suitably generalized rankine hugoniot relation and entropy condition the existence and uniqueness of solutions involving delta shock waves are proved further four kinds of different structure for solutions are established uniquely

knot concordance and von neumann rho invariants we present new results announced in t on the classical knot concordance group c we establish the nontriviality at all levels of the n solvable filtration fn f1 f0 c introduced in cot1 recall that this filtration is significant due to its intimate connection to tower constructions arising in work of a casson and m freedman on the topological classification problem for 4 manifolds and due to the fact that all previously known concordance invariants are reflected in the first few terms in the filtration in cot1 nontriviality at the first new level n 3 was established here we prove the nontriviality of the filtration for all n hence giving the ultimate justification to the theory a broad range of techniques is employed in our proof including cut and paste topology and analytical estimates we use the cheeger gromov estimate for von neumann invariants a deep analytic result we also introduce a number of new algebraic arguments involving noncommutative localization and blanchfield forms we have attempted to make this article accessible to readers with only passing knowledge of cot1

stochastic variational approach to minimum uncertainty states we introduce a new variational characterization of gaussian diffusion processes as minimum uncertainty states we then define a variational method constrained by kinematics of diffusions and schrodinger dynamics to seek states of local minimum uncertainty for general non harmonic potentials

renyi formulation of the entropic uncertainty principle for povms uncertainty relations for a pair of arbitrary measurements are posed in the form of inequalities using the renyi entropies the formulation deals with povm measurements on finite dimensional hilbert space both the entropic uncertainty relations with state dependent and state independent bounds are presented the results are illustrated on the example of measurements for state discrimination

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

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

from probabilities to hamilton s principle it is shown that hamilton s principle of classical mechanics can be derived from the relativistic invariance of the generalized spacetime fisher information corresponding to the statistical description of results of measurement of the coordinates and the motion in space by means of the probability density and probability density current

vanishing viscosity limit for a system of balance laws with general type initial data arising from 1d saint venant model we study the vanishing viscosity limit of a non strictly hyperbolic system of balance laws known as the 1d saint venant model whose solution admits waves the balance term depends on both the variables x and t and is unbounded in the space variable x unlike the work of olenik usp math nauk 12 3 3 73 1957 in russian we also studied the large time behavior when the viscosity parameter is positive when the balance term depends only on the time variable we obtained an explicit formula for the solution

interactions of delta shock waves in a strictly hyperbolic system of conservation laws we study interaction of generalized solutions to a strictly hyperbolic system of two conservation laws from magnetohydrodynamics the riemann problem of this system is known to be solvable by a combination of shock waves contact discontinuities rarefaction waves and delta shock waves we derive how the solutions continue beyond points of interaction of these types of solutions we introduce a type of generalized solution a delta contact discontinuity a complete description of all global solutions with initial data given by two jump points and three constant states is obtained

numerical approximations of one dimensional linear conservation equations with discontinuous coefficients conservative linear equations arise in many areas of application including continuum mechanics or high frequency geometrical optics approximations this kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions in this paper we study the convergence of a class of finite differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem some basic examples including computational results are also supplied

on the riemann problem for 2d compressible euler equations in three pieces abstract the riemann problem for two dimensional isentropic euler equations is considered the initial data are three constants in three fan domains forming different angles under the assumption that only a rarefaction wave shock wave or contact discontinuity connects two neighboring constant initial states it is proved that the cases involving three shock or rarefaction waves are impossible for the cases involving one rarefaction shock wave and two shock rarefaction waves only the combinations when the three elementary waves have the same sign are possible impossible

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

two dimensional riemann problem for a single conservation law on construit les solutions d entropie pour l equation aux derivees partielles t u t x y x f u t x y y g u t x y 0 avec donnees initiales constantes dans chaque quadrant du plan x y ces solutions sont lisses par morceaux sous la condition f u 0 g u 0 f u g u 0

a cartoon for the climbing ramp problem of a shock and von neumann paradox the climbing ramp problem of a shock for two dimensional scalar conservation laws is considered by use of the generalized characteristic analysis method we prove constructively that there are only three configurations of solutions regular reflection like mach reflection like and von neumann reflection like for a kind of flux function of the conservation laws and some initial data the criteria for these three configurations are obtained furthermore a necessary condition for the appearance of regular reflection in gas dynamics is presented

the riemann problem for pressureless fluid dynamics with distribution solutions in colombeau s sense the riemann problem for the equations of pressureless fluid dynamics was considered solutions of this problem were constructed by employing the viscosity vanishing approach for some initial data solutions showed high singularity around the shock waves a new mathematical theory of generalized functions initiated by j f colombeau was applied to dealing with the multiplication of singular distributions as a byproduct the entropy condition was obtained for singular distribution solutions

stability of the riemann solutions for a nonstrictly hyperbolic system of conservation laws abstract we prove that the riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the riemann initial data the proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves during the interaction process of the delta shock wave with the rarefaction wave a new kind of nonclassical wave namely a delta contact discontinuity is discovered here which is a dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations m nedeljkov and m oberguggenberger interactions of delta shock waves in a strictly hyperbolic system of conservation laws j math anal appl 344 2008 1143 1157 moreover the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case

a counterexample to the strong version of freedman s conjecture a long standing conjecture due to michael freedman asserts that the 4 dimensional topological surgery conjecture fails for non abelian free groups or equivalently that a family of canonical examples of links the generalized borromean rings are not a b slice a stronger version of the conjecture that the borromean rings are not even weakly a b slice where one drops the equivariant aspect of the problem has been the main focus in the search for an obstruction to surgery we show that the borromean rings and more generally all links with trivial linking numbers are in fact weakly a b slice this result shows the lack of a non abelian extension of alexander duality in dimension 4 and of an analogue of milnor s theory of link homotopy for general decompositions of the 4 ball

on a regularization of a scalar conservation law with discontinuous coefficients this paper is devoted to a scalar conservation law with a linear flux function involving discontinuous coefficients it is clear that the delta standing wave should be introduced into the riemann solution in some nonclassical situation in order to study the formation of delta standing wave we consider a regularization of the discontinuous coefficient with the helmholtz mollifier and then obtain a regularized system which depends on a regularization parameter 0 the regularization mechanism is a nonlinear bending of characteristic curves that prevents their finite time intersection it is proved rigorously that the solutions of regularized system converge to the delta standing wave solution in the 0 limit compared with the classical method of vanishing viscosity here it is clear to see how the delta standing wave forms naturally along the characteristics

two dimensional riemann problem for scalar conservation laws abstract using the generalized characteristic analysis method we study the two dimensional riemann problem for scalar conservation laws which is nonconvex along the y direction and interactions of its elementary waves give the classification of initial discontinuities and construct all riemann solutions which riemann data are two or three pieces of constants all kinds of guckenheimer structure appear in the solutions and the necessary and sufficient condition of appearance of it is given

dynamics of propagation and interaction of δ shock waves in conservation law systems abstract we introduce a new definition of a shock wave type solution for a class of systems of conservation laws in the one dimensional case the weak asymptotics method developed by the authors is used to construct formulas describing the propagation and interaction of shock waves the dynamics of merging two shocks is described by explicit formulas continuously in time

generalized plane delta shock waves for n dimensional zero pressure gas dynamics abstract with the help of a generalized plane wave solution we study a type of generalized plane delta shock wave for the n dimensional zero pressure gas dynamics and refine its generalized rankine hugoniot relation which is a system of ordinary equations this relation describes accurately the character of the generalized plane delta shock location propagation speed and weight under a suitable entropy condition four different explicit constructions of solutions for a kind of riemann problem with radon measure as initial data are established uniquely the overtaking of two plane delta shocks forming a new generalized plane delta shock is also investigated finally the 2 d riemann problem with four pieces of initial data is solved in a simplified situation

characteristic galerkin schemes for scalar conservation laws in two and three space dimensions in this paper we are concerned with a finite element method for multidimensional scalar conservation laws we describe a general formulation of the euler characteristic galerkin ecg scheme motivated by key features of the one dimensional ecg scheme the method is defined by projecting the transport collapse operator onto a finite element space spanned by piecewise constant basis functions the ecg scheme is em tvd monotone maximum norm nonincreasing and unconditionally stable in the l 1 norm to the best of our knowledge there is no other method which has all these properties however with piecewise constant basis functions the ecg scheme is at most first order accurate greater accuracy can be obtained through a recovery procedure unlike conventional dimension by dimension methods multidimensional ecg schemes contain important terms which describe corner effects nonuniform meshes and large time steps can be used with this class of schemes

riemann problems for a class of coupled hyperbolic systems of conservation laws abstract we study a class of coupled hyperbolic systems of conservation laws which contain the one dimensional zero pressure gas dynamics as a prototypical example the riemann problems are solved constructively the riemann solutions exactly include two kinds delta shock wave solutions and vacuum solutions under the generalized rankine hugoniot relation and entropy condition all of the existence uniqueness and stability of solutions to viscous perturbations are proved two typical examples are presented finally

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

dual decompositions of 4 manifolds iii s cobordisms the main result is that an s cobordism topological or smooth of 4 manifolds has a product structure outside a core sub s cobordism these cores are arranged to have quite a bit of structure for example they are smooth and abstractly forgetting boundary structure diffeomorphic to a standard neighborhood of a 1 complex the decomposition is highly nonunique so cannot be used to define an invariant but it shows that the topological s cobordism question reduces to the core case the simply connected version of the decomposition with 1 complex a point is due to curtis freedman hsiang and stong controlled surgery is used to reduce topological triviality of core s cobordisms to a question about controlled homotopy equivalence of 4 manifolds there are speculations about further reductions the decompositions on the ends of the s cobordism are dual decompositions with homotopically controlled handle structures and the main result is an application of earlier papers in the series

global solutions to the two dimensional riemann problem for a system of conservation laws we study the global solutions to the two dimensional riemann problem for a system of conservation laws the initial data are three constant states separated by three rays emanating from the origin under the assumption that each ray in the initial data outside of the origin projects exactly one planar contact discontinuity this problem is classified into five cases by the self similar transformation the reduced system changes type from being elliptic near the origin to being hyperbolic far away in self similar plane then in hyperbolic region applying the generalized characteristic analysis method a goursat problem is solved to describe the interactions of planar contact discontinuities while in elliptic region a boundary value problem arises it is proved that this boundary value problem admits a unique solution based on these preparations five explicit solutions and their corresponding criteria can be obtained in self similar plane

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2