averages and higher moments for the ell torsion in class groups we prove upper bounds for the average size of the ell torsion mathrm cl k ell of the class group of k as k runs through certain natural families of number fields and ell is a positive integer we refine a key argument used in almost all results of this type which links upper bounds for mathrm cl k ell to the existence of many primes splitting completely in k that are small compared to the discriminant of k our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument in conjunction with new counting results for these invariants this leads to significantly improved upper bounds for the average and sometimes even higher moments of mathrm cl k ell for many families of number fields k considered in the literature for example for the families of all degree d fields for d in 2 3 4 5 and non d 4 if d 4 as an application of the case d 2 we obtain the best upper bounds for the number of d p fields of bounded discriminant for primes p 3

unbounded order convergence and application to martingales without probability abstract a net x in a vector lattice x is unbounded order convergent uo convergent to x if x x y o 0 for each y x and is unbounded order cauchy uo cauchy if the net x x is uo convergent to 0 in the first part of this article we study uo convergent and uo cauchy nets in banach lattices and use them to characterize banach lattices with the positive schur property and kb spaces in the second part we use the concept of uo cauchy sequences to extend doob s submartingale convergence theorems to a measure free setting our results imply in particular that every norm bounded submartingale in l 1 f is almost surely uo cauchy in f where f is an order continuous banach lattice with a weak unit

unbounded topologies and uo convergence in locally solid vector lattices abstract suppose x is a vector lattice and there is a notion of convergence x x in x then we can speak of an unbounded version of this convergence by saying that x u x if x x u 0 for every u x in the literature the unbounded versions of the norm order and absolute weak convergence have been studied here we create a general theory of unbounded convergence but with a focus on uo convergence and those convergences deriving from locally solid topologies we will see that not only do the majority of recent results on unbounded norm convergence generalize but they do so effortlessly not only that but the structure of unbounded topologies is clearer without a norm we demonstrate this by removing metrizability completeness and local convexity from nearly all arguments while at the same time making the proofs simpler and more general we also give characterizations of minimal topologies in terms of unbounded topologies and uo convergence

stationary solutions for the ellipsoidal bgk model in a slab abstract we address the boundary value problem for the ellipsoidal bgk model of the boltzmann equation posed in a bounded interval the existence of a unique mild solution is established under the assumption that the inflow boundary data does not concentrate too much around the zero velocity and the gas is sufficiently rarefied

square free values of f p f cubic let f in mathbb z x deg f 3 assume that f does not have repeated roots assume as well that for every prime q f x not equiv 0 mod q2 has at least one solution in mathbb z q 2 mathbb z then under these two necessary conditions there are infinitely many primes p such that f p is square free

rate of convergence to equilibrium for the spatially homogeneous boltzmann equation with hard potentials for the spatially homogeneous boltzmann equation with hard potentials and grad s cutoff e g hard spheres we give quantitative estimates of exponential convergence to equilibrium and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation on which we provide a lower bound our approach is based on establishing spectral gap like estimates valid near the equilibrium and then connecting the latter to the quantitative nonlinear theory this leads us to an explicit study of the linearized boltzmann collision operator in functional spaces larger than the usual linearization setting

ellipsoidal bgk model for polyatomic molecules near maxwellians a dichotomy in the dissipation estimate abstract we consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal bgk model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic maxwellian we observe that the linearized relaxation operator is decomposed into a truly polyatomic part and an essentially monatomic part leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter reaches zero accordingly we employ two different sets of micro macro system to derive the full coercivity and close the energy estimate

periodic cubic hyperbolic schrodinger equation on t2 abstract in this paper we consider the cubic hyperbolic schrodinger equation 1 1 on torus t 2 and get the sharp l 4 strichartz estimate which implies that 1 1 is analytic locally well posed in h s t 2 for s 1 2 meanwhile the ill posedness in h s t 2 for s 1 2 is also obtained

completeness for vector lattices abstract a net x in a vector lattice x is unbounded order convergent uo convergent to x if x x y is order convergent to 0 for each y x and is unbounded order cauchy uo cauchy if the net x x is unbounded order convergent to 0 the vector space x is unbounded order complete uo complete if every uo cauchy net in x is uo convergent recently li and chen proved that a vector lattice having the countable sup property is universally complete if it is uo complete the main result of this paper states that this equivalence is still true without any further assumption on the vector lattice x which answers an open question asked by li and chen some applications to this are given too

optimal exponential decay for the linearized ellipsoidal bgk model in weighted sobolev spaces this paper deals with the asymptotic behavior of solution to the linearized ellipsoidal bgk model in torus we prove that the solution converges exponentially to the equilibrium in the weighted sobolev spaces with polynomial weight our exponential decay rate e lambda t is optimal in the sense that lambda 0 equals to the spectral gap of the linearized operator in the standard hilbert space our strategy is taking advantage of the quantitative spectral gap estimates in a smaller reference hilbert space the factorization method and the enlargement of the functional space for the associated semigroup

integral points on elliptic curves and 3 torsion in class groups we give new bounds for the number of integral points on elliptic curves the method may be said to interpolate between approaches via diophantine techniques bp hbr and methods based on quasi orthogonality in the mordell weil lattice sil6 gs he we apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the 3 torsion part of the class group of a quadratic field the same ideas can be used to count rational points on curves of higher genus

on the trend to global equilibrium in spatially inhomogeneous entropy dissipating systems the linear fokker planck equation we study the long time behavior of kinetic equations in which transport and spatial confinement in an exterior potential or in a box are associated with a degenerate collision operator acting only in the velocity variable we expose a general method based on logarithmic sobolev inequalities and the entropy to overcome the well known problem due to the degeneracy in the position variable of the existence of infinitely many local equilibria this method requires that the solution be somewhat smooth in this paper we apply it to the linear fokker planck equation and prove decay to equilibrium faster than o t 1 for all 0 2001 john wiley sons inc

conditional expectations on riesz spaces abstract conditional expectations operators acting on riesz spaces are shown to commute with a class of principal band projections using the above commutation property conditional expectation operators on riesz spaces are shown to be averaging operators here the theory of f algebras is used when defining multiplication on the riesz spaces this leads to the extension of these conditional expectation operators to their so called natural domains i e maximal domains for which the operators are both averaging operators and conditional expectations the natural domain is in many aspects analogous to l 1

regularization estimates and cauchy theory for inhomogeneous boltzmann equation for hard potentials without cut off in this paper we investigate the problems of cauchy theory and exponential stability for the inhomogeneous boltzmann equation without angular cut off we only deal with the physical case of hard potentials type interactions with a moderate angular singularity we prove a result of existence and uniqueness of solutions in a close to equilibrium regime for this equation in weighted sobolev spaces with a polynomial weight contrary to previous works on the subject all developed with a weight prescribed by the equilibrium it is the first result in this more physically relevant framework for this equation moreover we prove an exponential stability for such a solution with a rate as close as we want to the optimal rate given by the semigroup decay of the linearized equation let us highlight the fact that a key point of the development of our cauchy theory is the proof of new regularization estimates in short time for the linearized operator thanks to pseudo differential tools

positive line bundles on arithmetic varieties p 5 u integers is defined to count the volume of the lattice i l n of integral sections in the space j7 lon of real sections with supremum norm we want to prove that the leading term of this hilbert function as n f oo is given in terms of the height of x in as 1 this is known as a theorem of gillet and soule gs2 if x has a regular generic fiber beside this known result our proof uses hironaka s theorem on resolutions of singularities and minkowski s theorem on successive minima by hironaka s theorem we may construct

semigroup decay of the linearized boltzmann equation in a torus abstract we study the semigroup decay of the linearized boltzmann equation in general banach spaces including l x l 0 and l x p l q 1 p 1 q in a torus our results reveal that the size of the domain affects the decay rate of the solution

cauchy problem for the ellipsoidal bgk model of the boltzmann equation ellipsoidal bgk model es bgk is a generalized version of the original bgk model designed to reproduce the physically correct prandtl number in the navier stokes limit in this paper we study the cauchy problem for the es bgk model under the condition of finite initial mass energy and entropy equivalence type estimates for the temperature tensor are crucially used

the difference between the weil height and the canonical height on elliptic curves estimates for the difference of the weil height and the canonical height of points on elliptic curves are used for many purposes both theoretical and computational in this note we give an explicit estimate for this difference in terms of the j invariant and discriminant of the elliptic curve the method of proof suggested by serge lang is to use the decomposition of the canonical height into a sum of local heights we illustrate one use for our estimate by computing generators for the mordell weil group in three examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 x ax b with a and b in the ring of integers of k the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 10 chapter viii 9 or 6 chapter vi the canonical height is determined by this property together with the fact that the difference

counting rational points on hypersurfaces for any n geq 2 let f in mathbb z x 1 ldots x n be a form of degree d geq 2 which produces a geometrically irreducible hypersurface in mathbb p n 1 this paper is concerned with the number n f b of rational points on f 0 which have height at most b for any varepsilon 0 we establish the estimate n f b o b n 2 varepsilon whenever either n leq 5 or the hypersurface is not a union of lines here the implied constant depends at most upon d n and varepsilon

exponential stability of slowly decaying solutions to the kinetic fokker planck equation the aim of the present paper is twofold 1 we carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non symmetric operators in banach spaces as introduced in 10 we extend the method so as to consider the shrinkage of the functional space roughly speaking we consider a class of operators written as a dissipative part plus a mild perturbation and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another smaller or larger banach space under the condition that a certain iterate of the mild perturbation part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way the cornerstone of our approach is a factorization argument reminiscent of the dyson series 2 we apply this method to the kinetic fokker planck equation when the spatial domain is either the torus with periodic boundary conditions or the whole space with a confinement potential we then obtain spectral gap estimates for the associated semigroup for various metrics including lebesgue norms negative sobolev norms and the monge kantorovich wasserstein distance w1

unbounded order convergence in dual spaces abstract a net x in a vector lattice x is said to be unbounded order convergent or uo convergent for short to x x if the net x x y converges to 0 in order for all y x in this paper we study unbounded order convergence in dual spaces of banach lattices let x be a banach lattice we prove that every norm bounded uo convergent net in x is w convergent iff x has order continuous norm and that every w convergent net in x is uo convergent iff x is atomic with order continuous norm we also characterize among order complete banach lattices the spaces in whose dual space every simultaneously uo and w convergent sequence converges weakly in norm

limit laws for martingales in vector lattices abstract we prove limit laws for martingales in vector lattices under suitable technical hypotheses involving the increments of the sequences therein

on the trend to global equilibrium for spatially inhomogeneous kinetic systems the boltzmann equation as part of our study of convergence to equilibrium for spatially inhomogeneous kinetic equations started in 21 we derive estimates on the rate of convergence to equilibrium for solutions of the boltzmann equation like o t our results hold conditionally to some strong but natural estimates of smoothness decay at large velocities and strict positivity which at the moment have only been established in certain particular cases among the most important steps in our proof are 1 quantitative variants of boltzmann s h theorem as proven in 52 60 based on symmetry features hypercontractivity and information theoretical tools 2 a new quantitative version of the instability of the hydrodynamic description for non small knudsen number 3 some functional inequalities with geometrical content in particular the korn type inequality which we established in 22 and 4 the study of a system of coupled differential inequalities of second order by a treatment inspired from 21 we also briefly point out the particular role of conformal velocity fields when they are allowed by the geometry of the problem

spectrum analysis of some kinetic equations we analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory the models include the classical boltzmann equation for hard potentials with or without angular cutoff and the landau equation with gamma geqq 2 as an application we show that the solutions to these two fundamental equations are asymptotically equivalent mod time decay rate t 5 4 as t to infty to that of the compressible navier stokes equations for initial data around an equilibrium state

classical solutions for the ellipsoidal bgk model with fixed collision frequency abstract we establish the existence of global in time smooth solutions for the ellipsoidal bgk model which is a variant of the bgk model for the boltzmann equation designed to yield the correct prandtl number in the hydrodynamic approximation at the navier stokes level for this we carefully design a function space which captures the growth of the solution in a weighted sobolev norm and show that the ellipsoidal relaxation operator is lipschitz continuous in the induced metric this approach is restricted to the case when the collision frequency does not depend on the macroscopic field but no smallness on the initial data is required

from the boltzmann equation to the incompressible navier stokes equations on the torus a quantitative error estimate we investigate the boltzmann equation depending on the knudsen number in the navier stokes perturbative setting on the torus using hypocoer civity we derive a new proof of existence and exponential decay for solutions close to a global equilibrium with explicit regularity bounds and rates of convergence these results are uniform in the knudsen number and thus allow us to obtain a strong derivation of the incompressible navier stokes equations as the knudsen number tends to 0 moreover our method is also used to deal with other kinetic models finally we show that the study of the hydrodynamical limit is rather different on the torus than the one already proved in the whole space as it requires averaging in time unless the initial layer conditions are satisfied

unbounded norm topology in banach lattices abstract a net x in a banach lattice x is said to un converge to a vector x if x x u 0 for every u x in this paper we investigate un topology i e the topology that corresponds to un convergence we show that un topology agrees with the norm topology iff x has a strong unit un topology is metrizable iff x has a quasi interior point suppose that x is order continuous then un topology is locally convex iff x is atomic an order continuous banach lattice x is a kb space iff its closed unit ball b x is un complete for a banach lattice x b x is un compact iff x is an atomic kb space we also study un compact operators and the relationship between un convergence and weak convergence

metrizability of minimal and unbounded topologies abstract in 1987 i labuda proved a general representation theorem that as a special case shows that the topology of local convergence in measure is the minimal topology on orlicz spaces and l minimal topologies connect with the recent and actively studied subject of unbounded convergences in fact a hausdorff locally solid topology on a vector lattice x is minimal iff it is lebesgue and the and unbounded topologies agree in this paper we study metrizability submetrizability and local boundedness of the unbounded topology u associated to on x regarding metrizability we prove that if is a locally solid metrizable topology then u is metrizable iff there is a countable set a with i a x we prove that a minimal topology is metrizable iff x has the countable sup property and a countable order basis in line with the idea that uo convergence generalizes convergence almost everywhere we prove relations between minimal topologies and uo convergence that generalize classical relations between convergence almost everywhere and convergence in measure

weighted l bounds and uniqueness for the boltzmann bgk model we prove rather general l bounds for hydrodynamical fields in terms of weighted l norms of the kinetic density we use these estimates to prove l bounds and uniqueness for the solution of the bgk equation which is a simple relaxation model introduced by bhatnagar gross krook to mimic boltzmann flows

geometric postulation of a smooth function and the number of rational points this paper is devoted to giving refinements and extensions of some of the results of bombieri and the author 1 obtaining upper bounds for the number of integral lattice points on the graphs of functions consider a sufficiently smooth function f x with graph and a positive integer d the main device of that paper was to consider integral points on that do not lie on any real algebraic curve of degree d the main lemma of 1 shows that such points cannot be too close together relative to certain norms of the function we pursue here two different goals relative to the main lemma the first is to obtain local conditions on the function f x that control the multiplicity of the intersection of with any algebraic curve of degree d this is essentially an investigation into the hypotheses of the main lemma and constitutes the geometric postulation of the title indeed in sections 2 and 3 we will obtain such conditions for any linear space of real algebraic curves the applications to integral points are given in section 5 for example we show that if f x c on 0 1 and w f 2 f f 3f 0 f iv 4f 6f f 5f iv 20f is nowhere zero then for every e 0

class numbers of quadratic fields and shimura s correspondence it is well known see fueter 8 and aigner 1 that fermat s equation x y 3 1 has no solutions in a quadratic field k q d for d 1 mod 3 provided the class number of k is not divisible by 3 this subject was further investigated by frey 6 who gave estimates for the 3 rank of the class group of k in terms of 3 descent on a curve y 2 x d in this paper we consider curves ed dy 2 4x 27 which are quadratic twists of the fermat s curve x y 3 1 we give a precise formula for the rank of the selmer group corresponding to the complex multiplication 3 ed e 3d in terms of the 3 rank of the class group of q d resp q 3d this result may be considered as a quantitative version of 1 and 8 we discuss also birch and swinnerton dyer s conjecture for curves ed according to a theorem of waldspurger 22 23 natural rational factor of l ed q 1 may be expressed in terms of coefficients of certain modular forms of weight 3 2 we identify these forms explicitely and verify mod 3 part of birch and swinnertondyer s conjecture for ed in most cases for d of density 5 6

on the concentration of points of polynomial maps and applications for a polynomial f in mathbb f p x we obtain upper bounds on the number of points x f x modulo a prime p which belong to an arbitrary square with the side length h our results in particular are based on the vinogradov mean value theorem using these estimates we obtain results on the expansion of orbits in dynamical systems generated by nonlinear polynomials and we obtain an asymptotic formula for the number of visible points on the curve f x equiv y rm mod p where f in mathbb f p x is a polynomial of degree d 2 we also use some recent results and techniques from arithmetic combinatorics to study the values x f x in more general sets

spaces of regular abstract martingales abstract in 15 10 the authors introduced and studied the space m r of regular martingales on a vector lattice and the space m r of bounded regular martingales on a banach lattice in this note we study these two spaces from the vector lattice point of view we show in particular that these spaces need not be vector lattices however if the underlying space is order complete then m r is a vector lattice and m r is a banach lattice under the regular norm

hermitian vector bundles and extension groups on arithmetic schemes i geometry of numbers we define and investigate extension groups in the context of arakelov geometry the arithmetic extension groups ext xi f g we introduce are extensions by groups of analytic types of the usual extension groups extxi f g attached to ox modules f and g over an arithmetic scheme x in this paper we focus on the first arithmetic extension group ext x1 f g the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over x and we especially consider the case when x is an arithmetic curve namely the spectrum specok of the ring of integers in some number field k then the study of arithmetic extensions over x is related to old and new problems concerning lattices and the geometry of numbers namely for any two hermitian vector bundles f and g over x specok we attach a logarithmic sizesf g to any element of ext x1 f g and we give an upper bound on sf g in terms of slope invariants of f and g we further illustrate this notion by relating the sizes of restrictions to points in p1 z of the universal extension over pz1 to the geometry of psl2 z acting on poincare s upper half plane and by deducing some quantitative results in reduction theory from our previous upper bound on sizes finally we investigate the behaviour of size by base change i e under extension of the ground field k to a larger number field k when the base field k is q we establish that the size which cannot increase under base change is actually invariant when the field k is an abelian extension of k or when f g is a direct sum of root lattices and of lattices of voronoi s first kind the appendices contain results concerning extensions in categories of sheaves on ringed spaces and lattices of voronoi s first kind which might also be of independent interest

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

torsion anomalous points and families of elliptic curves we prove that there are at most finitely many complex lambda neq 0 1 such that two points on the legendre elliptic curve y 2 x x 1 x lambda with coordinates x 2 3 both have finite order this is a very special case of some conjectures on unlikely intersections in semiabelian schemes

explicit uniform estimation of rational points ii hypersurface coverings by using the slope method we obtain an explicit uniform estimation for the density of rational points in an arithmetic projective variety with given degree and dimension embedded in a given arithmetic projective space

the number of integral points on arcs and ovals integral lattice points and that the exponent and constant are best possible however swinnerton dyer 10 showed that the preceding result can be substantially improved if we start with a fixed c strictly convex arc and consider the number of lattice points on t the dilation of by a factor t t 1 this of course is the same as asking for rational points mn n n on as n in fact swinnerton dyer proves a bound of type t zz c e t 3 5 e

smooth solutions to the bgk equation and the es bgk equation with infinite energy abstract the bgk model and the es bgk model of the boltzmann equation are of great importance in the kinetic theory of rarefied gases for the cauchy problems with general initial data smooth solutions were obtained only in bounded domains for both models in this paper we establish the existence and uniqueness of smooth solutions in the whole space more important some smooth solutions can permit infiniteness of energy

the density of rational points on curves and surfaces let c be an irreducible projective curve of degree d in mathbb p 3 defined over overline mathbb q it is shown that c has o varepsilon d b 2 d varepsilon rational points of height at most b for any varepsilon 0 uniformly for all curves c this result extends an estimate of bombieri and pila duke math j 59 1989 337 357 to projective curves for a projective surface s in mathbb p 3 of degree d ge 3 it is shown that there are o varepsilon d b 2 varepsilon rational points of height at most b of which at most o varepsilon d b 52 27 varepsilon do not lie on a rational line in s for non singular surfaces one may reduce the exponent to 4 3 16 9d for d 4 or 5 or max 1 3 sqrt d 2 d 1 for d ge 6 even for the surface x 1 d x 2 d x 3 d x 4 d this last result improves on the previous best known as a further application it is shown that almost all integers represented by an irreducible binary form f x y in mathbb z x y have essentially only one such representation this extends a result of hooley j reine angew math 226 1967 30 87 which concerned cubic forms only the results are not restricted to projective surfaces and as an application of other results in the paper it is shown that x 1 x 2 x 3 in mathbb n 3 x 1 d x 2 d x 3 d n ll varepsilon d n theta d varepsilon with theta frac 2 sqrt d frac 2 d 1 when d ge 8 this provides the first non trivial bound for the number of representations as a sum of three d th powers

on the density of rational and integral points on algebraic varieties let x n be a projective geometrically integral variety over of dimension r and degree d 4 suppose that there are only finitely many r 1 planes over on x the main result of this paper is a proof of the fact that the number n x b of rational points on xwhich have height at most b satisfies for any 0 the implied constant depends at most on d n and

rational points of bounded height on projective surfaces we give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces

explicit uniform estimation of rational points i estimation of heights by using the slope method in arakelov geometry we study the complexity of the singular locus of an arithmetic projective variety and explicit estimations of the arithmetic hilbert samuel function

enumerative galois theory for cubics and quartics abstract we show that there are o e h 1 5 e monic cubic polynomials with integer coefficients bounded by h in absolute value whose galois group is a 3 we also show that the order of magnitude for d 4 quartics is h 2 log h 2 and that the respective counts for a 4 v 4 c 4 are o h 2 91 o h 2 log h o h 2 log h our work establishes that irreducible non s 3 cubic polynomials are less numerous than reducible ones and similarly in the quartic setting these are the first two solved cases of a 1936 conjecture made by van der waerden

an effective chebotarev density theorem for families of number fields with an application to formula see text torsion in class groups we prove a new effective chebotarev density theorem for galois extensions l mathbb q that allows one to count small primes even as small as an arbitrarily small power of the discriminant of l this theorem holds for the galois closures of almost all number fields that lie in an appropriate family of field extensions previously applying chebotarev in such small ranges required assuming the generalized riemann hypothesis the error term in this new chebotarev density theorem also avoids the effect of an exceptional zero of the dedekind zeta function of l without assuming grh we give many different appropriate families including families of arbitrarily large degree to do this we first prove a new effective chebotarev density theorem that requires a zero free region of the dedekind zeta function then we prove that almost all number fields in our families yield such a zero free region the innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non cuspidall functions this builds on and greatly generalizes the applicability of work of kowalski and michel on the average density of zeroes of a family of cuspidall functions a surprising feature of this new method which we expect will have independent interest is that we control the number of zeroes in the family of l functions by bounding the number of certain associated fields with fixed discriminant as an application of the new chebotarev density theorem we prove the first nontrivial upper bounds for ell torsion in class groups for all integers ell ge 1 applicable to infinite families of fields of arbitrarily large degree

global existence to the bgk model of boltzmann equation abstract we present an existence and a stability proof for solutions to the bgk model of boltzmann equation f v x f f m f t 0 x r n v r n m f p 2 t n 2 exp v u 2 2t p pu p u 2 pt 1 x r n 1 v v 2 f t x v dv it relies on the strong compactness of u t and on a new estimate on the third moment of v dv we also prove the entropy relation for 1

boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting in this paper we consider the spatially inhomogeneous diffusively driven inelastic boltzmann equation in different cases the restitution coefficient can be constant or can depend on the impact velocity which is a more physically relevant case including in particular the case of viscoelastic hard spheres in the weak thermalization regime i e when the diffusion parameter is sufficiently small we prove existence of global solutions considering both the close to equilibrium and the weakly inhomogeneous regimes we also study the long time behavior of these solutions and prove a convergence to equilibrium with an exponential rate the basis of the proof is the study of the linearized equation we obtain a new result on it we prove existence of a spectral gap in weighted stretched exponential and polynomial sobolev spaces more precisely there is a one dimensional eigenvalue which is negative and which tends to 0 when the diffusion parameter tends to 0 to do that we take advantage of the recent paper cite gmm where the spatially inhomogeneous equation for elastic hard spheres has been considered as far as the case of a constant coefficient is concerned the present paper improves similar results obtained in cite mm2 in a spatially homogeneous framework concerning the case of a non constant coefficient this kind of results is new and we use results on steady states of the linearized equation from cite al3

cauchy problem for the ellipsoidal bgk model for polyatomic particles abstract we establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal bgk model which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level

derivation of the two dimensional nonlinear schrodinger equation from many body quantum dynamics we derive rigorously for both r 2 and l l 2 the cubic nonlinear schrodinger equation in a suitable scaling limit from the two dimensional many body bose systems with short scale repulsive pair interactions we first prove convergence of the solution of the bbgky hierarchy corresponding to the many body systems to a solution of the infinite gross pitaevskii hierarchy corresponding to the cubic nls and then we prove uniqueness for the infinite hierarchy which requires number theoretical techniques in the periodic case

the density of rational points on non singular hypersurfaces ii for any integers d n geq 2 let x subset mathbb p n be a non singular hypersurface of degree d that is defined over the rational numbers the main result in this paper is a proof that the number of rational points on x which have height at most b is o b n 1 varepsilon for any varepsilon 0 the implied constant in this estimate depends at most upon d varepsilon and n

lp solutions to the cauchy problem of the bgk equation the bgk model of the boltzmann equation plays an important role in the kinetic theory of rarefied gases some existence and uniqueness results of solutions to its cauchy problem were established for large initial data under various circumstances see for example perthame b global existence to the bgk model of boltzmann equation j differ equations 82 191 205 1989 in this paper by establishing some weighted lp estimates of the hydrodynamical quantities of a gas we prove the existence theorem of the lp solutions to the cauchy problem and establish the propagation properties of the lp moments for this kind of solutions

the polynomial method over varieties we establish sharp estimates that adapt the polynomial method to arbitrary varieties these include a partitioning theorem estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic varieties as a first application we provide a general incidence estimate that is tight in its dependence on the size degree and dimension of the varieties involved

quadratic class numbers and character sums we present an algorithm for computing the class number of the quadratic number field of discriminant d the algorithm terminates unconditionally with the correct answer and under the grh executes in o d 1 4 steps the technique used combines algebraic methods with burgess theorem on character sums to estimate l 1 x d we give an explicit version of burgess theorem valid for prime discriminants and as an application we compute the class number of a 32 digit discriminant