a blow up criterion for two dimensional compressible viscous heat conductive flows abstract we establish a blow up criterion in terms of the upper bound of the density and temperature for the strong solution to 2d compressible viscous heat conductive flows the initial vacuum is allowed

finite dimensional schwinger basis deformed symmetries wigner function and an algebraic approach to quantum phase schwinger s finite d dimensional periodic hilbert space representations are studied on the toroidal lattice with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the wigner function these subalgebras are shown to be admissible endowed with the non negative norm of hilbert space vectors hence they provide the desired canonical basis for the algebraic formulation of the quantum phase problem certain equivalence classes in the space of labels are identified within each subalgebra and connections with area preserving canonical transformations are examined the generalized representations of the wigner function are examined in the finite dimensional cyclic schwinger basis these representations are shown to conform to all fundamental conditions of the generalized phase space wigner distribution as a specific application of the schwinger basis the number phase unitary operator pair in is studied and based on the admissibility of the underlying q oscillator subalgebra an algebraic approach to the unitary quantum phase operator is established this being the focus of this work connections with the susskind glogower carruthers nieto phase operator formalism as well as standard action angle wigner function formalisms are examined in the infinite period limit the concept of continuously shifted fock basis is introduced to facilitate the fock space representations of the wigner function

strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data we extend to general polytropic pressures p k 1 the existence theory of 8 for isothermal 1 flows of navier stokes fluids in two and three space dimensions with fairly general initial data specifically we require that the initial density be close to a constant in l2 and l and that the initial velocity be small in l2 and bounded in l2n in two dimensions the l2 norms must be weighted slightly solutions are obtained as limits of approximate solutions corresponding to mollified initial data the key point is that the approximate densities are shown to converge strongly so that nonlinear pressures can be accommodated even in the absence of any uniform regularity information for the approximate densities

on spherically symmetric solutions of the compressible isentropic navier stokes equations we prove the global existence of weak solutions to the cauchy problem for the compressible isentropic navier stokes equations in n n 2 3 when the cauchy data are spherically symmetric the proof is based on the exploitation of the one dimensional feature of symmetric solutions and use of a new multidimensional property induced by the viscous flux the present paper extends lions existence theorem 15 to the case 1 n for spherically symmetric initial data where is the specific heat ratio in the pressure n 3 2 for n 2 and n 9 5 for n 3 dedicated to professor rolf leis on the occasion of his 70th birthday

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

an extended weyl wigner transformation for special finite spaces abstract we extend the weyl wigner transformation to those particular degrees of freedom described by a finite number of states using a technique of constructing operator bases developed by schwinger discrete transformation kernels are presented instead of continuous coordinate momentum pair system and systems such as the one dimensional canonical continuous coordinate momentum pair system and the two dimensional rotation system are described by special limits expressions are explicitly given for the spin one half case

quasiprobability distribution functions for periodic phase spaces i theoretical aspects an approach featuring s parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed for such an approach a suitable set of angle angular momentum coherent states must be constructed in an appropriate fashion

wigner distributions for finite state systems without redundant phase point operators we set up wigner distributions for n state quantum systems following a dirac inspired approach in contrast to much of the work in this study requiring a 2n 2n phase space particularly when n is even our approach is uniformly based on an n n phase space grid and thereby avoids the necessity of having to invoke a quadrupled phase space and hence the attendant redundance both n odd and even cases are analysed in detail and it is found that there are striking differences between the two while the n odd case permits full implementation of the marginal property the even case does so only in a restricted sense this has the consequence that in the even case one is led to several equally good definitions of the wigner distributions as opposed to the odd case where the choice turns out to be unique

on a parabolic hyperbolic chemotaxis system with discontinuous data well posedness stability and regularity abstract the global dynamics and regularity of parabolic hyperbolic systems is an interesting topic in pdes due to the coupling of competing dissipation and hyperbolic effects this paper is concerned with the cauchy problem of a parabolic hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis it is shown that as time tends to infinity the cauchy problem with large amplitude discontinuous data admit global weak solutions which converge to a constant state resp a viscous shock wave if the asymptotic states of initial values at far field are equal resp unequal our results improve the previous results where initial value was required to be continuous and have small amplitude numerical simulations are performed to verify our analytical results illustrate the possible regularity of solutions and speculate the minimal regularity of initial data required to obtain the smooth classical solutions of the concerned parabolic hyperbolic system

schwinger and pegg barnett approaches and a relationship between angular and cartesian quantum descriptions ii phase spaces following the discussion in state space language presented in a preceding paper we work on the passage from the phase space description of a degree of freedom described by a finite number of states without classical counterpart to one described by an infinite and continuously labelled number of states with this it is possible to relate an original schwinger idea to the pegg barnett approach to the phase problem in phase space language this discussion shows that one can obtain the weyl wigner formalism for both cartesian and angular coordinates as limiting elements of the discrete phase space formalism

jacobi ϑ functions and discrete fourier transforms properties of the jacobi 3 function and its derivatives under discrete fourier transforms are investigated and several interesting results are obtained the role of modulo n equivalence classes in the theory of functions is stressed an important conjecture is studied

existence and uniqueness of low energy weak solutions to the compressible 3d magnetohydrodynamics equations abstract we prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics mhd equations we first obtain the existence of weak solutions with small l 2 norm which may display codimension one discontinuities in density pressure magnetic field and velocity gradient the weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible mhd equations our results generalise and extend those for the intermediate weak solutions of compressible navier stokes equations

hyperbolic compressible navier stokes equations abstract we consider the non isentropic compressible navier stokes equations with hyperbolic heat conduction and a law for the stress tensor which is modified correspondingly by maxwell s law these two relaxations turning the whole system into a hyperbolic one are not only treated simultaneously but are also considered in a version having galilean invariance for this more complicated relaxed system the global well posedness is proved for small data moreover for vanishing relaxation parameters the solutions are shown to converge to solutions of the classical system

extended cahill glauber formalism for finite dimensional spaces i fundamentals the cahill glauber approach for quantum mechanics on phase space is extended to the finite dimensional case through the use of discrete coherent states all properties and features of the continuous formalism are appropriately generalized the continuum results are promptly recovered as a limiting case the jacobi theta functions are shown to have a prominent role in the context

continuum quantum systems as limits of discrete quantum systems i state vectors abstract dynamical systems on continuum hilbert spaces may be realized as limits of dynamical systems on discrete possibly finite dimensional hilbert spaces in this first of four papers on the topic the continuum and discrete spaces are interfaced to one another algebraically convergence of vectors is defined in such a way as to preserve inner products and a necessary and sufficient coordinate wise criterion for convergence is proved

discrete quantum phase spaces and the mod n invariance instituto de fisica teorica universidade estadual paulista rua pamplona 145 cep 01405 sao paulo sp

discrete wigner function for finite dimensional systems a phase space approach to finite dimensional systems is developed from basic principles for a system describable by a hilbert space of dimension d we define a one to one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d the properties fulfilled by this correspondence and its uniqueness are examined this formalism is applied to the number difference and phase difference of a two mode field this case is compared with the marginal distribution for these variables arising from a two mode wigner function for number and phase

global solutions of the navier stokes equations for multidimensional compressible flow with discontinuous initial data abstract we prove the global existence of weak solutions of the navier stokes equations for compressible isothermal flow in two and three space dimensions when the initial density is close to a constant in l 2 and l and the initial velocity is small in l 2 and bounded in l 2 n in two dimensions the l 2 norms must be weighted slightly a great deal of qualitative information about the solution is obtained for example we show that the velocity and vorticity are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a crucial role in the entire analysis particularly in closing the required energy estimates understanding rates of regularization near the initial layer and most important obtaining time independent pointwise bounds for the density

from the discrete weyl wigner formalism for symmetric ordering to a number phase wigner function the general weyl wigner formalism in finite dimensional phase spaces is investigated then this formalism is specified to the case of symmetric ordering of operators in an odd dimensional hilbert space a respective wigner function on the discrete phase space is found and the limit when the dimension of hilbert space tends to infinity is considered it is shown that this limit gives the number phase wigner function in quantum optics analogous results for the almost symmetric ordering in an even dimensional hilbert space are obtained relations between the discrete wigner functions introduced in our paper and some other discrete wigner functions appearing in literature are studied

hudson s theorem for finite dimensional quantum systems we show that on a hilbert space of odd dimension the only pure states to possess a non negative wigner function are stabilizer states the clifford group is identified as the set of unitary operations which preserve positivity the result can be seen as a discrete version of hudson s theorem hudson established that for continuous variable systems the wigner function of a pure state has no negative values if and only if the state is gaussian turning to mixed states it might be surmised that only convex combinations of stabilizer states give rise to non negative wigner distributions we refute this conjecture by means of a counterexample further we give an axiomatic characterization which completely fixes the definition of the wigner function and compare two approaches to stabilizer states for hilbert spaces of prime power dimensions in the course of the discussion we derive explicit formulas for the number of stabilizer codes defined on such systems

quantum discrete phase space dynamics and its continuous limit the main aspects of a discrete phase space formalism are presented and the discrete dynamical bracket suitable for the description of time evolution in finite dimensional spaces is discussed a set of operator bases is defined in such a way that the weyl wigner formalism is shown to be obtained as a limiting case in the same form the moyal bracket is shown to be the limiting case of the discrete dynamical bracket the dynamics in quantum discrete phase spaces is shown not to be attained from discretization of the continuous case

lagrangian structure for two dimensional non barotropic compressible fluids abstract we study the lagrangian structure for weak solutions of two dimensional navier stokes equations for a non barotropic compressible fluid i e we show the uniqueness of particle trajectories for two dimensional compressible fluids including the energy equation our result extends partially the result obtained for barotropic fluids by hoff and santos 10

d dimensional levy flights exact and asymptotic the analytic and asymptotic properties of the spherically symmetric d dimensional levy stable probability density function p d r are discussed in detail these isotropic stable probability density functions pdfs are analogous to the one dimensional symmetric levy stable pdfs previously studied by the present authors j math phys 43 2670 2002 we construct a hypergeometric representation of p d r when is rational and find a number of new representations of p d r in terms of special functions for various values of d and a recursion relation is found between p d r and p d 2 r which in particular implies there exists a simple map between p 1 r and p 3 r as in our previous paper we discuss the properties of p d r for both the cases 2 and 2 we demonstrate the existence of intricate exponentially small series in the large r asymptotics of p d r when is an integer which are dominant when is even we explicitly construct this beyond all orders expansion of p d r for arbitr

analytic representation of finite quantum systems a transform between functions in r and functions in zd is used to define the analogue of number and coherent states in the context of finite d dimensional quantum systems the coherent states are used to define an analytic representation in terms of theta functions all states are represented by entire functions with growth of order 2 which have exactly d zeros in each cell the analytic function of a state is constructed from its zeros results about the completeness of finite sets of coherent states within a cell are derived

eigenvalues and eigenvectors of the finite fourier transform the eigenvalues and eigenvectors of the n n unitary matrix of finite fourier transform whose j k element is 1 n 1 2 exp 2 i n jk i 1 1 2 is determined in doing so a multitude of identities some of which may be new are encountered a conjecture is advanced

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

continuum quantum systems as limits of discrete quantum systems ii state functions in this second of four papers on the eponymous topic pointwise convergence of a discrete state function to a continuum state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel as examples and as a prerequisite for the sequels the normal approximation theorem and the convergence of the kravchuk functions to the hermite gaussians are expressed in terms of the algebraic notion of convergence

lagrangean structure and propagation of singularities in multidimensional compressible flow we study the propagation of singularities in solutions of the navier stokes equations of compressible barotropic fluid flow in two and three space dimensions the solutions considered are in a fairly broad regularity class for which initial densities are nonnegative and essentially bounded initial energies are small and initial velocities are in certain fractional sobolev spaces we show that if the initial density is bounded below away from zero in an open set v then each point of v determines a unique integral curve of the velocity field and that this system of integral curves defines a locally bi holder homeomorphism of v onto its image at each positive time this lagrangean structure is then applied to show that if the initial density has a limit at a point of such a set v from a given side of a continuous hypersurface in v then at each later time both the density and the divergence of the velocity have limits at the transported point from the corresponding side of the transported hypersurface which is also a continuous manifold if the limits from both sides exist then the rankine hugoniot conditions hold in a strict pointwise sense showing that the jump in the divergence of the velocity is proportional to the jump in the pressure this leads to a derivation of an explicit representation for the strength of the jump in the logarithm of the density from which it follows that discontinuities persist for all time convecting along fluid particle paths and in the case that the pressure is strictly increasing in density having strengths which decay exponentially in time

continuum quantum systems as limits of discrete quantum systems iii operators convergence of a discrete operator to a continuum operator is defined as examples the circular rotor the one dimensional box the harmonic oscillator and the fractional fourier transform are realized as limits of finite dimensional quantum systems limits thus defined preserve algebraic structure the results prepare for a sequel in which some affine canonical transforms will be discretized

global existence of weak solutions to two dimensional compressible viscoelastic flows abstract the global existence of weak solutions of the compressible viscoelastic flows in two spatial dimensions is studied in this paper we show the global existence if the initial velocity u 0 is small in h with an arbitrary 0 1 2 and the perturbation of 0 f 0 around the constant state 1 i are small in l 2 b p 1 2 p with p 1 1 16 2 4 one of the main ingredients is that the velocity and the effective viscous flux g i are sufficiently regular for positive time the regularity of g i helps to obtain the l estimate of density and deformation gradient and hence eliminates the possible concentration and oscillation issues