domain of definition of levermore s five moment system the simplest system in levermore s moment hierarchy involving moments higher than second order is the five moment closure it is obtained by taking velocity moments of the one dimensional boltzmann equation under the assumption that the velocity distribution is a maximum entropy function the moment vectors for which a maximum entropy function exists consequently make up the domain of definition of the system the aim of this article is a complete characterization of the structure of the domain of definition and the connected maximum entropy problem the space homogeneous case of the equation and numerical aspects are also addressed

entropy production in second order three point schemes we discuss semi discrete three point finite difference methods for the numerical solution of system of conservation laws which are second order accurate in space in the sense of truncation error particular discretizations of the numerical entropy flux associated with such schemes are studied clarifying the importance of this discretization with regard to the production of numerical entropy using a numerical entropy flux constructed in a canonical way we prove that a wide class of finite difference methods cannot satisfy a discrete entropy inequality together with a well known result of schonbek concerning lax wendroff type schemes our result indicates a strong relationship between entropy production and oscillations in numerical solutions

fully discrete entropy conservative schemes of arbitrary order we consider weak solutions of hyperbolic or hyperbolic elliptic systems of conservation laws in one space dimension and their approximation by finite difference schemes in conservative form the systems under consideration are endowed with an entropy entropy flux pair we introduce a general approach to construct second and third order accurate fully discrete in both space and time entropy conservative schemes in general these schemes are fully nonlinear implicit but in some important cases can be explicit or linear implicit furthermore semidiscrete entropy conservative schemes of arbitrary order are presented the entropy conservative schemes are used to construct a numerical method for the computation of weak solutions containing nonclassical regularization sensitive shock waves finally specific examples are investigated and tested numerically our approach extends the results and techniques by tadmor in numerical methods for compressible flows finite difference element and volume techniques asme new york 1986 pp 149 158 lefloch and rohde siam j numer anal 37 2000 pp 2023 2060

generalized lyapunov criteria on finite time stability of stochastic nonlinear systems abstract this paper considers the problem of finite time stability for stochastic nonlinear systems a general lyapunov theorem of stochastic finite time stability is proposed and an important corollary is obtained some comparisons with the existing results are also given this theorem not only generalizes the classical stochastic finite time theorem but also somewhat shows that brownian noise takes an active role in stochastic finite time stability in addition more general conditions on stochastic finite time stability via multiple lyapunov functions are given to further relax the constraint of the differential operator l v moreover with the help of this criterion it shows that an unstable deterministic system can even be finite time stabilized by brownian noise some concrete examples are constructed to show the significant features of the proposed results and simulations are presented to verify the theoretical analysis

the moduli space of cubic fourfolds via the period map we characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space it follows then that the geometric invariant theory git compactification of the moduli space of cubic fourfolds is isomorphic to the looijenga compactification associated to this arrangement this paper builds on and is a natural continuation of our previous work on the git compactification of the moduli space of cubic fourfolds

on the monodromy of the moduli space of calabi yau threefolds coming from eight planes in mathbb p 3 it is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as zariski open subsets of locally hermitian symmetric domains in the present work we prove that the moduli space of calabi yau threefolds coming from eight planes in mathbb p 3 does not have this property we show furthermore that the monodromy group of a good family is zariski dense in the corresponding symplectic group moreover we study a natural sublocus which we call hyperelliptic locus over which the variation of hodge structures is naturally isomorphic to wedge product of a variation of hodge structures of weight one it turns out the hyperelliptic locus does not extend to a shimura subvariety of type iii siegel space within the moduli space besides general hodge theory representation theory and computational commutative algebra one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of hodge structures

stochastic versions of the lasalle theorem the main aim of this paper is to establish stochastic versions of the well known lasalle stability theorem from these stochastic versions follow many classical results on stochastic stability this shows clearly the power of our new results

nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations in this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls with a proof that works equally well in a more general equivariant setting for a non equidimensional surjective holomorphic map between compact ball quotients our method applies to show that the set of critical values must be nonempty and of codimension 1 in the equivariant setting the line of arguments extend to holomorphic mappings of maximal rank into the complex projective space or the complex euclidean space yielding in the latter case a lower estimate on the dimension of the singular locus of certain holomorphic maps defined by integrating holomorphic 1 forms in another direction we extend the nonexistence statement on holomorphic submersions to the case of ball quotients of finite volume provided that the target complex unit ball is of dimension m 2 giving in particular a new proof that a local biholomorphism between noncompact m ball quotients of finite volume must be a covering map whenever m 2 finally combining our results with hermitian metric rigidity we show that any holomorphic submersion from a bounded symmetric domain into a complex unit ball equivariant with respect to a lattice must factor through a canonical projection to yield an automorphism of the complex unit ball provided that either the lattice is cocompact or the ball is of dimension at least 2

volumes of complex analytic subvarieties of hermitian symmetric spaces we give lower bounds of volumes of k dimensional complex analytic subvarieties of certain naturally defined domains in n dimensional complex space forms of constant positive zero or negative holomorphic sectional curvature for each 1 k n the lower bounds are sharp in the sense that these bounds are attained by k dimensional complete totally geodesic complex submanifolds such lower bounds are obtained by constructing singular potential functions corresponding to blow ups of the kahler metrics involved similar lower bounds are also obtained in the case of hermitian symmetric spaces of noncompact type in this case the lower bounds are sharp for those values of k at which the hermitian symmetric space contains k dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature

holomorphic isometries of mathbb b m into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals we study general properties of images of holomorphic isometric embeddings of complex unit balls mathbb b m into irreducible bounded symmetric domains varomega of rank at least 2 in particular we show that such holomorphic isometries with the minimal normalizing constant arise from linear sections varlambda of the compact dual x c of varomega the question naturally arises as to which linear sections z varlambda cap varomega are actually images of holomorphic isometries of complex unit balls we study the latter question in the case of bounded symmetric domains varomega of type iv alias lie balls i e bounded symmetric domains dual to hyperquadrics we completely classify images of all holomorphic isometric embeddings of complex unit balls into such bounded symmetric domains varomega especially we show that there exist holomorphic isometric embeddings of complex unit balls of codimension 1 incongruent to the examples constructed by mok proc am math soc 144 4515 4525 2016 from varieties of minimal rational tangents and that moreover any holomorphic isometric embedding f mathbb b m rightarrow varomega extends to a holomorphic isometric embedding f mathbb b n 1 rightarrow varomega dim varomega n the case of lie balls is particularly relevant because holomorphic isometric embeddings of complex unit balls of sufficiently large dimensions into an irreducible bounded symmetric domain other than a type iv domain are expected to be more rigid

polarized variation of hodge structures of calabi yau type and characteristic subvarieties over bounded symmetric domains in this paper we extend the construction of the canonical polarized variation of hodge structures over tube domain considered by gross math res lett 1 1 9 1994 to bounded symmetric domain and introduce a series of invariants of infinitesimal variation of hodge structures which we call characteristic subvarieties we prove that the characteristic subvariety of the canonical polarized variations of hodge structures over irreducible bounded symmetric domains are identified with the characteristic bundles defined by mok ann math 125 1 105 152 1987 we verified the generating property of gross for all irreducible bounded symmetric domains which was predicted in gross math res lett 1 1 9 1994

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

a discontinuous galerkin finite element method for a 1d prototype of the boltzmann equation to develop and analyze new computational techniques for the boltzmann equation based on model or approximation adaptivity it is imperative to have disposal of a compliant model problem that displays the essential characteristics of the boltzmann equation and that admits the extraction of highly accurate reference solutions for standard collision processes the boltzmann equation itself fails to meet the second requirement for d 2 3 spatial dimensions on account of its setting in 2d while for d 1 the first requirement is violated because the boltzmann equation then lacks the convergence to equilibrium property that forms the substructure of simplified models in this article we present a numerical investigation of a new one dimensional prototype of the boltzmann equation the underlying molecular model is endowed with random collisions which have been fabricated such that the corresponding boltzmann equation exhibits convergence to maxwell boltzmann equilibrium solutions the new boltzmann model is approximated by means of a discontinuous galerkin dg finite element method to validate the one dimensional boltzmann model we conduct numerical experiments and compare the results with monte carlo simulations of equivalent molecular dynamics models

on discontinuous action of monodromy groups on the complex n ball in dm 3 10 3 15 there is defined for each n 3 tuple 1u n 3 of nonintegral real numbers with ei2 i integral a subgroup i acting on a complex n 1 dimensional vector space vj which preserves a hermitian form of signature zi p 1 ei 1 ui 1 dm 2 21 where for any real number r r denotes its fractional part i e 0 0 we will be concerned principally with the case that the signature of the hermitian form is 1 positive n negative in which case the image of v in the projective space pn associated to v is the complex n ball bn also called the n disc we deal almost exclusively with the action of f on bn and pn we call an n 3 tuple u satisfying 0 pi 1 for i 1 n 3 and eiz i 2 a disc n 3 tuple in dm theorem 10 19 it is proved let u be a disc n 3 tuple satisfying condition int for all i a j such that ui uj 1

polarized mixed hodge structures and the local monodromy of a variation of hodge structure a variation of polarized hodge structures on the complement of a divisor with normal crossings gives rise locally on the divisor to a commuting set ni of nilpotent endomorphisms of the vector space underlying the variation the logarithms of the unipotent parts of the various picard lefschetz transformations these reflect properties of the singularities of the period map associated to the variation cf 5 7 9 and sect 3 for example a variation depending on a single parameter defines asymptotically a mixed hodge structure whose weight filtration is the monodromy weight filtration of the corresponding endomorphism n 9 11 this paper is mainly concerned with the properties of the set n arising in the several parameters case in particular theorem 3 3 asserts that all the elements in the open polyhedral cone c spanned over r by the n s determine the same monodromy weight filtration it also describes the relationship between this common filtration and those associated to the faces of the cone this statement was conjectured by deligne based on his analogous result 4 1 9 2 for the q cone associated to those variations which arise from families of polarized non singular algebraic varieties in w 1 we discuss some general properties of commuting sets of nilpotent endomorphisms and their associated filtrations section2 focuses on the case when each element in such set defines in the sense of 2 4 a polarized mixed hodge structure with a fixed hodge filtration f as is the case for the monodromy cone c of a variation these conditions imply cf 2 16 a generic uniqueness for the monodromy weight filtration we also consider those pola

asymptotic stability and boundedness of delay switching diffusions multiple lyapunov functions are used to establish sufficient conditions for locating the limit sets of switching diffusions the conditions lead to a number of useful results on stochastic asymptotic stability and boundedness for this class of stochastic hybrid systems

affordable robust moment closures for cfd based on the maximum entropy hierarchy the use of moment closures for the prediction of continuum and moderately non equilibrium flows offers modelling and numerical advantages over other methods the maximum entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations however their are two issues that limit their practical implementation firstly for closures that have a treatment for heat transfer fluxes cannot be written in closed form and a very expensive iterative procedure is required at every flux evaluation secondly for these same closures there are physically possible moment states for which the entropy maximization problem has no solution and the entire framework breaks down this paper demonstrates that affordable closed form moment closures that are inspired by the maximum entropy framework can be proposed it is known that closing fluxes in the maximum entropy hierarchy approach a singularity as the region of non solvability is approached this paper shows that far from a disadvantage this singularity allows for smooth and accurate prediction of shock wave structure even for high mach numbers the presence of unphysical sub shocks within shock profile predictions of traditional closures has long been regarded as an unfortunate limitation of the entire moment closure technique the realization that smooth shock profiles are in fact possible for moment methods with a moderate number of moments greatly increases the method s applicability to high speed flows in this paper a 5 moment system for a simple one dimensional gas and a 14 moment system for realistic gases are developed and examined numerical solution for shock waves at a variety of incoming flow mach numbers demonstrate both the robustness and the accuracy of the closures

regularity of mappings into classical domains we study the regularity and algebraicity for mappings into classical domains among other things we establish everywhere regularity and algebraicity results for c 2 cr maps into the set of smooth boundary points of a classical domain where the codimension can be arbitrarily large

stabilization of linear systems by noise it is proved that the biggest lyapunov number lambda max of the system dot x a f t x where a is a fixed d times d matrix and f t is a zero mean strictly stationary matrix valued stochastic process satisfies 1 d trace a leqq lambda max on the other hand for each varepsilon 0 there is a process f t for which lambda max leqq 1 d trace a varepsilon in particular the system dot x ax can be stabilized by zero mean stationary parameter noise if and only if trace a 0 the stabilization can be accomplished by a one dimensional noise source the results carry over to the case where a is a stationary process they are also true for f t white noise

second order conservative schemes and the entropy condition we consider numerical approximations to solutions of systems of hyperbolic conservation laws of the form au at af u ax 0 u e r andf r r smooth we show that conservative three point second order accurate methods cannot satisfy a local entropy inequality

infinitesimal variation of hodge structures and the weak global torelli theorem for complete intersections in this paper we will study the weak global torelli theorem for complete intersections of hypersurfaces in a projective space let x be a non singular projective variety of dimension r with an ample line bundle l over the complex number field c then the primitive part hrm x z of the cohomology of x has a polarized hodge structure if the moduli space x of x exists we can define the global period map ho d r from the moduli space x to the classifying space d r of hodge structures attaching the point p in x to the hodge structure of the primitive part hrim xp z of the cohomology group of xp p e 4

the monodromy groups of dolgachev s cy moduli spaces are zariski dense abstract let m n 2 n 2 be the coarse moduli space of cy manifolds arising from a crepant resolution of double covers of p n branched along 2 n 2 hyperplanes in general position we show that the monodromy group of a good family for m n 2 n 2 is zariski dense in the corresponding symplectic or orthogonal group if n 3 in particular the period map does not give a uniformization of any partial compactification of the coarse moduli space as a shimura variety whenever n 3 this disproves a conjecture of dolgachev as a consequence the fundamental group of the coarse moduli space of m ordered points in p n is shown to be large once it is not a point similar zariski density result is obtained for moduli spaces of cy manifolds arising from cyclic covers of p n branched along m hyperplanes in general position a classification towards the geometric realization problem of b gross for type a bounded symmetric domains is given

motives with exceptional galois groups and the inverse galois problem we construct motivic l adic representations of textup gal overline mathbb q mathbb q into exceptional groups of type e 7 e 8 and g 2 whose image is zariski dense this answers a question of serre the construction is uniform for these groups and is inspired by the langlands correspondence for function fields as an application we solve new cases of the inverse galois problem the finite simple groups e 8 mathbb f ell are galois groups over mathbb q for large enough primes l

hyperbolic turbulence models for moment closures abstract the ten moment equations of gaskinetic theory provide a first order hyperbolic model for the treatment of viscous compressible gas flows that is valid both in and out of local thermodynamic equilibrium until now the practical applicability of this model to real world continuum regime problems has been limited by the lack of appropriate turbulence models in this paper the reynolds averaging concept is applied to the ten moment model to close the new unknowns generated by this process a turbulence model is required however instead of coupling to traditional rans models which require second order derivatives a first order turbulence model is proposed it is developed by hyperbolizing prandtl s existing one equation rans model the resulting system is shown to recover the same solution as prandtl s original model in the appropriate limit this new first order turbulence model is in hyperbolic relaxation form and includes one extra equation per spacial dimension unlike the traditional rans model this model has the same hyperbolic balance law form as the ten moment equations solutions of such hyperbolic relaxation systems exhibit dispersive wave behaviour to study this an eigenstructure and dispersion analysis is presented for the new turbulence model coupled to the ten moment system the model is then validated by solving a canonical two dimensional turbulent plane mixing layer problem the numerical solution in steady state is compared to experimental measurements at various locations in the flow field

on a cell entropy inequality for discontinuous galerkin methods we prove a cell entropy inequality for a class of high order dis continuous galerkin finite element methods approximating conservation laws which implies convergence for the one dimensional scalar convex case

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

discontinuous galerkin method for navier stokes equations using kinetic flux vector splitting kinetic schemes for compressible flow of gases are constructed by exploiting the connection between boltzmann equation and the navier stokes equations this connection allows us to construct a flux splitting for the navier stokes equations based on the direction of molecular motion from which a numerical flux can be obtained the naive use of such a numerical flux function in a discontinuous galerkin dg discretization leads to an unstable scheme in the viscous dominated case stable schemes are constructed by adding additional terms either in a symmetric or non symmetric manner which are motivated by the dg schemes for elliptic equations the novelty of the present scheme is the use of kinetic fluxes to construct the stabilization terms in the symmetric case interior penalty terms have to be added for stability and the resulting schemes give optimal convergence rates in numerical experiments the non symmetric schemes lead to a cell energy entropy inequality but exhibit sub optimal convergence rates these properties are studied by applying the schemes to a scalar convection diffusion equation and the 1 d compressible navier stokes equations in the case of navier stokes equations entropy variables are used to construct stable schemes

a complex ball uniformization of the moduli space of cubic surfaces via periods of k3 surfaces in this paper we show that the moduli space of nodal cubic surfaces is iso morphic to a quotient of a 4 dimensional complex ball by an arithmetic subgroup of the unitary group this complex ball uniformization uses the periods of certain k3 surfaces which are naturally associated to cubic surfaces a similar uniformization is given for dif ferent covers of the moduli space corresponding to geometric markings of the picard group or a choice of a line on the surface we also give a detailed description of the boundary com ponents corresponding to singular surfaces

extremal bounded holomorphic functions and an embedding theorem for arithmetic varieties of rank 2 let be a bounded symmetric domain of rank 2 and aut be a torsion free irreducible cocompact lattice x on the projective manifold x there is the canonical kahler einstein metric which is of nonpositive holomorphic bisectional curvature in mok m1 2 we established a hermitian metric rigidity theorem for such projective manifolds x which in the case when is irreducible says that any hermitian metric of nonpositive curvature in the sense of griffiths is necessarily a constant multiple of the kahler einstein metric as a consequence we proved in the latter case that any nontrivial holomorphic mapping f x z into a hermitian manifold z of nonpositive curvature in the sense of griffiths is necessarily an isometric immersion totally geodesic with respect to the hermitian connection on z the hermitian metric rigidity theorem can be taken as a tool in establishing the statement that any f x z is necessarily a holomorphic immersion a statement which only concerns the complex structure of x and z the hermitian metric rigidity theorem and its consequences were generalized by to to to be applicable to any torsion free lattice aut where x may be noncompact

the gaussian moment closure for gas dynamics the moment closure method of levermore applied to the boltzmann equation for rarefied gas dynamics leads to a hierarchy of symmetric hyperbolic systems of partial differential equations the euler system is the first member of this hierarchy of closures in this paper we investigate the next member the 10 moment gaussian closure we first reduce the collision term to an integral which may be explicitly evaluated for the special case of maxwell molecular interaction the resulting collision term for this case is shown to be equivalent to the term obtained by replacing the boltzmann collision operator with the bhatnagar gross and krook bgk approximation we then analyze the gaussian system applied to the canonical flow problem of a stationary planar shock an analytic shock profile for the gaussian closure is derived and compared with the numerical solutions of the boltzmann and navier stokes equations the results show reasonable agreement for weak shocks and close agreement between the downstream ga

on the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws we extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size with this term present we prove a maximum norm bound for finite element solutionsof burgers equation an thus complete an earlier convergence proof for this equation we further prove using entropy variables that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables results of some numerical experiments for the time dependent compressible euler equations in two dimensions are also reported

new lyapunov conditions of stochastic finite time stability and instability of nonlinear time varying sdes in this paper we further consider the problem of stochastic finite time stability and instability of nonlinear stochastic differential equations sdes with time varying drift and diffusion terms

attraction stability and robustness for stochastic functional differential equations with infinite delay this paper establishes the stochastic lasalle theorem to locate limit sets for stochastic functional differential equations with infinite delay from which some criteria on attraction boundedness stability and robustness are obtained to illustrate the applications of our results clearly this paper considers a scalar stochastic integro differential equation with infinite delay as an example

a characterization of certain shimura curves in the moduli stack of abelian varieties let f x y be a semistable family of complex abelian varieties over a curve y of genus g y and smooth over the com

symmetric differentials kahler groups and ball quotients while margulis superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real lie groups almost nothing is known concerning the representation theory of complex hyperbolic lattices the main result of this paper theorem 1 3 is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices it relies on the following two ingredients theorem 1 6 showing that the representations of the topological fundamental group of a compact kahler manifold x are controlled by the global symmetric differentials on x an arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms in particular deep results of clozel on base change theorem 1 11

an entropy stable discontinuous galerkin finite element moment method for the boltzmann equation this paper presents a numerical approximation technique for the boltzmann equation based on a moment system approximation in velocity dependence and a discontinuous galerkin finite element approximation in position dependence the closure relation for the moment systems derives from minimization of a suitable divergence this divergence based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the boltzmann equation the approximation in position dependence is based on the discontinuous galerkin finite element method the resulting combined discontinuous galerkin moment method corresponds to a galerkin approximation of the boltzmann equation in renormalized form the new momentclosure formulation engenders a new upwind numerical flux function based on half space integrals of the approximate distribution we establish that the proposed upwind flux ensures entropy dissipation of the discontinuous galerkin finite element approximation numerical results are presented for a one dimensional test case

characterization of calabi yau variations of hodge structure over tube domains by characteristic forms sheng and zuo s characteristic forms are invariants of a variation of hodge structure we show that they characterize gross s canonical variations of hodge structure of calabi yau type over hermitian symmetric tube domains

variations of hodge structure and orbits in flag varieties abstract period domains the classifying spaces for pure polarized hodge structures and more generally mumford tate domains arise as open g r orbits in flag varieties g p we investigate hodge theoretic aspects of the geometry and representation theory associated with these flag varieties in particular we relate the griffiths yukawa coupling to the variety of lines on g p under a minimal homogeneous embedding construct a large class of polarized g r orbits in g p and compute the associated hodge theoretic boundary components an emphasis is placed throughout on adjoint flag varieties and the corresponding families of hodge structures of levels two and four

explicit conditions for asymptotic stability of stochastic lienard type equations with markovian switching this paper deals with the boundedness in probability and convergence for solutions of stochastic lienard type equations some explicit conditions to ensure that all solution is bounded in probability and convergent to some fixed sets are obtained we also show that white noise can affect the boundedness of corresponding deterministic lienard equations

uniqueness theorems of hermitian metrics of seminegative curvature on quotients of bounded symmetric domains in the theory of locally symmetric spaces of negative ricci curvature it has been a classical problem to study the extent to which the topology of the manifold determines the geometry and in the hermitian case the complex structure the rigidity theorem of mostow 26 asserts that for a compact locally symmetric riemannian manifold of negative ricci curvature the manifold is determined up to isometry and normalizing constants by its fundamental group among the class of such manifolds with the obvious exceptions involving compact riemann surfaces the same theorem for locally symmetric riemannian manifolds of finite volume and negative ricci curvature was proved by prasad 27 in case of rank 1 and included in the super rigidity theorem of margulis 13 in the case of rank 2 in the class of compact locally symmetric hermitian manifolds of negative ricci curvature calabi vesentini 5 and borel 3 proved vanishing theorems of certain cohomology groups which imply in particular that for x locally irreducible of complex dimension 2 there exists no non trivial deformation of x as a complex manifold in this direction siu 32 33 proved the strong rigidity of the kdhler manifold x in the sense that any compact kkhler manifold m homotopic to x is necessarily biholomorphic or conjugate biholomorphic to x if m is also locally symmetric by assumption then it follows from the uniqueness of kihler einstein metrics of negative ricci curvature yau 37 that the conjugate biholomorphism is in fact an isometry up to a normalizing constant it is thus natural to ask

on holomorphic isometric embeddings of the unit n ball into products of two unit m balls we study holomorphic isometric embeddings of the complex unit n ball into products of two complex unit m balls with respect to their bergman metrics up to normalization constants the isometric constant there are two trivial holomorphic isometric embeddings for m n given by f1 z 0 in m z with the isometric constant equal to m 1 n 1 and f2 z in m z in m z with the isometric constant equal to 2 m 1 n 1 here i n m mathbb c n longrightarrow mathbb c m is the canonical embedding we prove that when m 2n these are the only holomorphic isometric embeddings up to unitary transformations

on discontinuous galerkin approximations of boltzmann moment systems with levermore closure abstract this work considers the discontinuous galerkin dg finite element discretization of first order systems of conservation laws derivable as moments of the kinetic boltzmann equation with levermore c d levermore moment closure hierarchies for kinetic theories j statist phys 83 5 6 1996 1021 1065 closure using standard energy analysis techniques a new class of energy stable numerical flux functions are devised for the dg discretization of boltzmann moment systems simplified energy stable numerical fluxes are then constructed which replace exact state space integration in the numerical flux with gauss lobatto quadrature numerical results for supersonic flow over a cylinder geometry in the continuum and transitional regimes using 5 and 10 moment approximations are presented using the newly devised dg discretizations

boltzmann type schemes for gas dynamics and the entropy property let h be a convex function a boltzmann equation is built following the b g k model for which h is an entropy in the fluid limit the compressible euler equations are obtained with a convex entropy naturally associated to h the motivation is to define numerical boltzmann schemes with finite speed of propagation corresponding to an equilibrium density function with bounded support the classical scheme obtained with h f f log f is based on the equilibrium density function exp v 2 2 which has infinite support

some contributions to stochastic asymptotic stability and boundedness via multiple lyapunov functions most of the existing results on stochastic stability use a single lyapunov function but we shall instead use multiple lyapunov functions in this paper to establish some sufficient criteria for locating the limit sets of solutions of stochastic differential equations from them follow many useful results on stochastic asymptotic stability and boundedness which enable us to construct the lyapunov functions much more easily in applications in particular the well known classical theorem on stochastic asymptotic stability is a special case of our more general results these show clearly the power of our new results