mixed boundary value problems for elliptic and parabolic differential equations of second order on considere les problemes aux valeurs limites mixtes suivants lu f 1 dans mu f 2 sur u f 3 sur avec l operateur elliptique et pu f 1 dans 0 t mu f 2 sur 0 t u f 3 sur 0 t 0 pour des operateurs p paraboliques pour des operateurs obliques generaux m et pour des domaines ou et se rencontrent a des angles arbitraires on montre l existence de solutions classiques quand les angles sont non nuls et f 3 est continue de holder

structure of higher spin gauge interactions in a previous paper higher spin gauge field theory was formulated in an abstract way essentially only keeping enough machinery to discuss gauge invariance of an action the approach could be thought of as providing an interface or syntax toward an implementation or semantics yet to be constructed the structure then revealed turned out to be that of a strongly homotopy lie algebra in the present paper the framework will be connected to more conventional field theoretic concepts the fock complex vertex operator implementation of the interactions in the brst bv formulation of the theory will be elaborated the relation between the vertex order expansion and homological perturbation theory will be clarified a formal nonobstruction argument is reviewed the syntactically derived sh lie algebra structure is semantically mapped to the fock complex implementation and it is shown that the equations governing the higher order vertices are reproduced global symmetries and subsidiary conditions are discu

groupe de chow de codimension deux des varietes definies sur un corps de nombres un theoreme de finitude pour la torsion letx be a smooth projective variety defined over a number field and let ch2 x denote the chow group of codimension two cycles modulo rational equivalence we show that if the cohomology grouph2 x ox vanishes then the torsion subgroup of ch2 x is a finite group this result covers all previous results in this direction the hypothesish2 x ox 0 is used to lift line bundles

optimal holder regularity for mixed boundary value problems abstract we continue our study of mixed boundary value problems lu 1 in mu 2 on u 3 on for second order elliptic operators l and first order oblique operators m a best possible estimate of the regularity of u is proved under weak hypotheses on the coefficients of l and m

non commutative batalin vilkovisky algebras homotopy lie algebras and the courant bracket we consider two different constructions of higher brackets first based on a grassmann odd nilpotent operator we define a non commutative generalization of the higher koszul brackets which are used in a generalized batalin vilkovisky algebra and we show that they form a homotopy lie algebra secondly we investigate higher so called derived brackets built from symmetrized nested lie brackets with a fixed nilpotent lie algebra element q we find the most general jacobi like identity that such a hierarchy satisfies the numerical coefficients in front of each term in these generalized jacobi identities are related to the bernoulli numbers we suggest that the definition of a homotopy lie algebra should be enlarged to accommodate this important case finally we consider the courant bracket as an example of a derived bracket we extend it to the big bracket of exterior forms and multi vectors and give closed formulas for the higher courant brackets

potential theory for regular and mach reflection of a shock at a wedge if a plane shock hits a wedge a self similar pattern of reflected shocks travels outward as the shock moves forward in time the nature of the pattern is explored for weak incident shocks strength b and small wedge angles 2 w through potential theory a number of different scalings some study of mixed equations and matching asymptotics for the different scalings the self similar equations are of mixed type a linearization gives a linear mixed flow valid away from a sonic curve near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used the parameter c1 2w 1 b ranges from 0 to here is the polytropic constant and c1 is the sound speed behind the incident shock for 2 regular reflection weak or strong can occur and the whole pattern is reconstructed to lowest order in shock strength for 1 2 mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle with an open elliptic problem and matched the case 0 can be solved for 1 2 2 or even larger the flow behind a mach reflection may be transonic and further investigation must be made to determine what happens the basic pattern of reflection is an almost semi circular shock issuing for regular reflection from the reflection point on the wedge and for mach reflection matched with a local interaction flow assuming their nature choosing the least entropy generation the weak regular reflection will occur for sufficiently large von neumann paradox an accumulation point of vorticity occurs on the wedge above the leading point 1994 john wiley sons inc

the transonic shock in a nozzle 2 d and 3 d complete euler systems abstract in this paper we study a transonic shock problem for the euler flows through a class of 2 d or 3 d nozzles the nozzle is assumed to be symmetric in the diverging or converging part if the supersonic incoming flow is symmetric near the divergent or convergent part of the nozzle then as indicated in section 147 of r courant k o friedrichs supersonic flow and shock waves interscience publ new york 1948 there exist two constant pressures p 1 and p 2 with p 1 p 2 such that for given constant exit pressure p e p 1 p 2 a symmetric transonic shock exists uniquely in the nozzle and the position and the strength of the shock are completely determined by p e moreover it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting furthermore we establish the global existence stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure p e when the initial unsteady shock lies in the symmetric diverging part of the 2 d or 3 d nozzle on the other hand it is shown that an unsteady symmetric transonic shock is structurally unstable in a global in time sense if it lies in the symmetric converging part of the nozzle

stability of transonic shocks in supersonic flow past a wedge abstract in this paper we study the stability of transonic shocks in steady supersonic flow past a wedge we take the potential flow equation as the mathematical model to describe the compressible flow it is known that in generic case such a problem admits two possible location of shock connecting the flow ahead it and behind it they can be distinguished as supersonic supersonic shock and supersonic subsonic shock or transonic shock both these possible shocks satisfy the rankine hugoniot conditions and entropy condition in this paper we prove that the transonic shock is also stable under perturbation of the coming flow provided the pressure at infinity is well controlled

rational points on pencils of conics and quadrics with many degenerate fibres for any pencil of conics or higher dimensional quadrics over q with all degenerate bres dened over q we show that the brauer manin obstruction controls weak approximation the proof is based on the hasse principle and weak approximation for some special intersections of quadrics over q which is a consequence of recent advances in additive combinatorics

varietes unirationnelles non rationnelles au dela de l exemple d artin et mumford keywords non rational but unirational variety grothendieck invariants rationality brauer invariants reference cma article 1989 001doi 10 1007 bf01850658 record created on 2008 12 16 modified on 2016 08 08

transonic shock in a nozzle i 2d case in this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two dimensional nozzle with variable sections the flow is governed by the inviscid potential equation and is supersonic upstream has no flow boundary conditions on the nozzle walls and an appropriate boundary condition at the exit of the exhaust section the transonic shock is a free boundary dividing two regions of c 1 1 0 flow in the nozzle the potential equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region in particular our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry this problem is motivated by the conjecture of courant and friedrichs on the transonic phenomena in a nozzle 10 furthermore the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation the main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and comer singularities

on the global existence and stability of a multi dimensional supersonic conic shock wave we establish the global existence and stability of a three dimensional supersonic conic shock wave for a compactly perturbed steady supersonic flow past an infinitely long circular cone with a sharp angle the flow is described by a 3 d steady potential equation which is multi dimensional quasilinear and hyperbolic with respect to the supersonic direction making use of the geometric properties of the pointed shock surface together with the rankine hugoniot conditions on the conic shock surface and the boundary condition on the surface of the cone we obtain a global uniform weighted energy estimate for the nonlinear problem by finding an appropriate multiplier and establishing a new hardy type inequality on the shock surface based on this we prove that a multi dimensional conic shock attached to the vertex of the cone exists globally when the mach number of the incoming supersonic flow is sufficiently large moreover the asymptotic behavior of the 3 d supersonic conic shock solution which is shown to approach the corresponding background shock solution in the downstream domain for the uniform supersonic constant flow past the sharp cone is also explicitly given

a proof of existence of perturbed steady transonic shocks via a free boundary problem we prove the existence of a solution of a free boundary problem for the transonic small disturbance equation the free boundary is the position of a transonic shock dividing two regions of smooth flow assuming inviscid irrotational flow as modeled by the transonic small disturbance equation the equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region to study the stability of a uniform planar transonic shock we consider perturbation by a steady c1 upstream disturbance if the upstream disturbance is small in a c1 sense then there is a steady solution in which the downstream flow and the transonic shock are holder continuous perturbations of the uniform configuration this result provides a new use of inviscid perturbation techniques to demonstrate in two dimensions the stability of transonic shock waves of the type that appear for example over the wing of an airplane along an airfoil or as bow shocks in a flow with a supersonic free stream velocity 2000 john wiley sons inc

groupe de brauer non ramifie des espaces homogenes a stabilisateur fini we develop different formulas of algebraic and or arithmetic nature allowing an explicit calculation both over a finite field and over a field of characteristic 0 of the algebraic part of the unramified brauer group of a homogeneous space g g under a semisimple simply connected linear group g with finite stabilizer g we also give examples of the calculations that can be done with these formulas

a free boundary problem for a quasi linear degenerate elliptic equation regular reflection of weak shocks we prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance utsd model for shock reflection by a wedge in weak regular reflection the state immediately behind the reflected shock is supersonic and constant the flow becomes subsonic further downstream the equation in self similar coordinates is degenerate at the sonic line the reflected shock becomes transonic and begins to curve there its position is the solution to a free boundary problem for the degenerate equation using the rankine hugoniot conditions along the reflected shock we derive an evolution equation for the transonic shock and an oblique derivative boundary condition at the unknown shock position by regularizing the degenerate problem we construct uniform bounds we apply local compactness arguments to extract a limit that solves the problem the solution is smooth in the interior and continuous up to the degenerate boundary this work completes a stage in our program to construct self similar solutions of two dimensional riemann problems in a series of papers we developed techniques for solving the degenerate elliptic equations that arise in self similar reductions of hyperbolic conservation laws in other papers especially in joint work with gary lieberman we developed techniques for solving free boundary problems of the type that arise from rankine hugoniot relations for the first time in this paper we combine these approaches and show that they are compatible although our construction is limited to a finite part of the unbounded subsonic region it suggests that this approach has the potential to solve a variety of problems in weak shock reflection including mach and von neumann reflection in the utsd equation and the analogous problems for the unsteady full potential equation 2002 john wiley sons inc

on transonic shocks in a conic divergent nozzle with axi symmetric exit pressures abstract in this paper we establish the existence and stability of a 3 d transonic shock solution to the full steady compressible euler system in a class of de laval nozzles with a conic divergent part when a given variable axi symmetric exit pressure lies in a suitable scope thus for this class of nozzles we have solved such a transonic shock problem in the axi symmetric case described by courant and friedrichs 1948 in section 147 of 8 given the appropriately large exit pressure p e x if the upstream flow is still supersonic behind the throat of the nozzle then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e x

hasse principle for pencils of curves of genus one whose jacobians have rational 2 division points une serie d articles exploite une nouvelle technique qui mene a des conditions suffisantes d existence et de densite des points rationnels sur certaines surfaces fibrees en courbes de genre un au dessus de la droite projective dans les premiers articles de cette serie plusieurs articles de swinnerton dyer un article en collaboration de skorobogatov swinnerton dyer et l auteur la jacobienne de la fibre generique des surfaces considerees a tous ses points d ordre 2 rationnels un article recent de bender et swinnerton dyer traite de cas ou cette jacobienne possede seulement un point d ordre 2 non trivial pour que la methode fonctionne il semble necessaire que la jacobienne possede un point de torsion rationnel non trivial le present article est une reecriture de celui de bender et swinnerton dyer la principale contribution est une reformulation plus abstraite des hypotheses principales des theoremes la premiere hypothese est formulee de facon entierement algebrique certains groupes de selmer algebrico geometriques sont supposes petits et la seconde hypothese est simplement il n y a pas d obstruction de brauer manin verticale comme dans la plupart des articles de cette serie les resultats dependent de deux conjectures difficiles l hypothese de schinzel et la finitude des groupes de tate shafarevich

sieving rational points on varieties an upper bound sieve for rational points on suitable varieties isdeveloped together with applications tocounting rational points in thin sets to local solubility in families and to the notion of friable rational pointswith respect to divisors in the special case of quadrics sharper estimates areobtained by developing a version of the selberg sieve for rational points

multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type we are concerned with the existence and stability of multidimensional transonic shocks for the euler equations for steady potential compressible fluids the euler equations consisting of the conservation law of mass and the bernoulli law for the velocity can be written as the following second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential r r 1 1 div d d 0 where the density function q is

local global principle for 0 cycles on fibrations over rationally connected bases abstract we study the brauer manin obstruction to the hasse principle and to weak approximation for 0 cycles on algebraic varieties that possess a fibration structure the issue is to establish the exactness of a local to global sequence e of chow groups of 0 cycles recently harpaz and wittenberg proved the exactness of e for rationally connected fibrations whose smooth fibres verify the exactness of e and whose base is either the projective space or a curve verifying the exactness of e in the present paper we prove the exactness of e for fibrations whose bases are ch telet surfaces or smooth proper models of homogeneous spaces of connected linear algebraic groups with connected stabilizers we require that either the fibration is valuatively split in codimension 1 and most closed fibres satisfy weak approximation for 0 cycles or the generic fibre has a 0 cycle of degree 1 and e is exact for most fibres

global supersonic conic shock wave for the steady supersonic flow past a cone polytropic gas abstract in this paper we establish the global existence and stability of a steady conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body as long as the vertex angle is less than a critical value the flow is assumed to be polytropic isentropic and described by a steady potential equation based on the delicate asymptotic expansion of the background solution one can verify that the boundary conditions on the shock and the conic surface satisfy the dissipative property from this property by use of the reflected characteristics method and the special form of the shock equation we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic coming flow is appropriately large on the other hand we remove the smallness restriction on the sharp vertex angle in order to establish the global existence of a shock or a global weak solution moreover our proof approach is different from that in shuxing chen zhouping xin huicheng yin global shock wave for the supersonic flow past a perturbed cone comm math phys 228 2002 47 84 and zhouping xin huicheng yin global multidimensional shock wave for the steady supersonic flow past a three dimensional curved cone anal appl 4 2 2006 101 132

boundary values as hamiltonian variables ii graded structures it is shown that the new formula for the field theory poisson brackets arises naturally in the proposed extension of the formal variational calculus incorporating divergences the linear spaces of local functionals evolutionary vector fields functional forms multi vectors and differential operators become graded with respect to divergences the bilinear operations such as the action of vector fields onto functionals the commutator of vector fields the interior product of forms and vectors and the schouten nijenhuis bracket are compatible with the grading a definition of the adjoint graded operator is proposed and antisymmetric operators are constructed with the help of boundary terms the fulfilment of the jacobi identity for the new poisson brackets is shown to be equivalent to vanishing of the schouten nijenhuis bracket of the poisson bivector with itself

on the fibration method for zero cycles and rational points conjectures on the existence of zero cycles on arbitrary smooth projective varieties over number elds were proposed by colliot th el ene sansuc kato and saito in the 1980 s we prove that these conjectures are compatible with brations for brations into rationally connected varieties over a curve in particular they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers we prove the analogous result for rational points conditionally on a conjecture on locally split values of polynomials which a recent work of matthiesen establishes in the case of linear polynomials over the rationals

the sh lie structure of poisson brackets in field theory a general construction of an sh lie algebra l algebra from a homological resolution of a lie algebra is given it is applied to the space of local functionals equipped with a poisson bracket induced by a bracket for local functions along the lines suggested by gel fand dickey and dorfman in this way higher order maps are constructed which combine to form an sh lie algebra on the graded differential algebra of horizontal forms the same construction applies for graded brackets in field theory such as the batalin fradkin vilkovisky bracket of the hamiltonian brst theory or the batalin vilkovisky antibracket

transonic shocks for the full compressible euler system in a general two dimensional de laval nozzle in this paper we study the transonic shock problem for the full compressible euler system in a general two dimensional de laval nozzle as proposed in courant and friedrichs supersonic flow and shock waves interscience new york 1948 given the appropriately large exit pressure pe x if the upstream flow is still supersonic behind the throat of the nozzle then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe x we solve this problem completely for a general class of de laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible euler system the problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic elliptic system one of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted holder space with low regularities for a second order quasilinear elliptic equation with a free parameter the position of the shock curve at one wall of the nozzle and non local terms involving the trace on the shock of the first order derivatives of the unknown function

existence of stationary supersonic flows past a pointed body in this paper we study the mathematical aspects of the stationary supersonic flow past a non axisymmetric curved pointed body the flow is described by a steady potential flow equation which is a quasilinear hyperbolic equation of second order we prove the local existence of the solution to this problem with a pointed shock attached at the tip of the pointed body provided the pointed body is a perturbation of a circular cone and the vertex angle of the approximate cone of the pointed body is less than a critical value the solution is smooth in between the shock and the surface of the body consequently such a structure of flow near the tip of the pointed body and its stability is verified mathematically

six dimensional 1 0 superconformal models and higher gauge theory we analyze the gauge structure of a recently proposed superconformal field theory in six dimensions we find that this structure amounts to a weak courant dorfman algebra which in turn can be interpreted as a strong homotopy lie algebra this suggests that the superconformal field theory is closely related to higher gauge theory describing the parallel transport of extended objects indeed we find that under certain restrictions the field content and gauge transformations reduce to those of higher gauge theory we also present a number of interesting examples of admissible gauge structures such as the structure lie 2 algebra of an abelian gerbe differential crossed modules the 3 algebras of m2 brane models and string lie 2 algebras

unramified brauer groups of finite simple groups of lie type a l we study the subgroup b 0 g of h 2 g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i consisting of all elements which have trivial restrictions to every abelian subgroup of g the group b 0 g serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties v g where v is a faithful complex linear representation of the group g we prove that b 0 g is trivial for finite simple groups of lie type a l

frobenius groups and retract rationality abstract let k be any field g be a finite group acting on the rational function field k x g g g by h x g x h g for any h g g define k g k x g g g g noether s problem asks whether k g is rational purely transcendental over k a weaker notion retract rationality introduced by saltman is also very useful for the study of noether s problem we prove that if g is a frobenius group with abelian frobenius kernel then k g is retract k rational for any field k satisfying some mild conditions as an application we show that for any algebraic number field k for any frobenius group g with frobenius complement isomorphic to s l 2 f 5 there is a galois extension field k over k whose galois group is isomorphic to g i e the inverse galois problem is valid for the pair g k the same result is true for any non solvable frobenius group if k 8 is a cyclic extension of k

on the instability problem of a 3 d transonic oblique shock wave abstract in this paper we are concerned with the instability problem of a 3 d transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge the flow is assumed to be isentropic and irrotational it was indicated on p 317 of 7 that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge then it follows from the rankine hugoniot conditions and the physical entropy condition that there possibly appears a weak shock or a strong shock attached at the edge of the sharp wedge which corresponds to a supersonic shock or a transonic shock respectively the question arises which of the two actually occurs it has frequently been stated that the strong one is unstable and that therefore only the weak one could occur however a convincing proof of this instability has apparently never been given the aim of this paper is to understand such a longstanding open question we will show that the attached 3 d transonic oblique shock problem is overdetermined with respect to the periodic perturbation which implies that the 3 d transonic shock is unstable in general

global shock waves for the supersonic flow past a perturbed cone we prove the global existence of a shock wave for the stationary supersonic gas flow past an infinite curved and symmetric cone the flow is governed by the potential equation as well as the boundary conditions on the shock and the surface of the body it is shown that the solution to this problem exists globally in the whole space with a pointed shock attached at the tip of the cone and tends to a self similar solution under some suitable conditions our analysis is based on a global uniform weighted energy estimate for the linearized problem combining this with the local existence result of chen li 1 we establish the global existence and decay rate of the solution to the nonlinear problem

nonlinear stability of a self similar 3 dimensional gas flow we show that the 3 dimensional supersonic gas flow past an infinite cone is nonlinear staple upon the perturbation of the obstacle the perturbed flow exists globally in space and tends to the self similar flow downstream there is a thin layer of concentration of vorticities and entropy variation our analysis is based on an approximation scheme using local self similar solutions as building blocks this enables us to obtain global estimates of the nonlinear interactions of waves needed for the stability analysis

global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone in this paper we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body the flow is assumed to be polytropic isentropic and described by a steady potential equation theoretically as indicated in r courant k o friedrichs supersonic flow and shock waves interscience publishers inc new york 1948 it follows from the rankine hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface which correspond to the supersonic shock and the transonic shock respectively in the references shuxing chen zhouping xin huicheng yin global shock wave for the supersonic flow past a perturbed cone comm math phys 228 2002 47 84 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone polytropic case preprint 2006 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone isothermal case pacific j math 233 2 2007 257 289 and zhouping xin huicheng yin global multidimensional shock wave for the steady supersonic flow past a three dimensional curved cone anal appl 4 2 2006 101 132 the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow at present for the supersonic symmetric incoming flow we will study the global transonic shock problem when the pressure at infinity is appropriately large

strongly homotopy lie algebras of one even and two odd dimensions abstract we classify strongly homotopy lie algebras also called l algebras of one even and two odd dimensions which are related to 2 1 dimensional z 2 graded lie algebras what makes this case interesting is that there are many nonequivalent l examples in addition to the z 2 graded lie algebra or superalgebra structures yet the moduli space is simple enough that we can give a complete classification up to equivalence

towards the brauer manin obstruction on varieties fibred over the projective line abstract recently dasheng wei proved that the brauer manin obstruction is the only obstruction to the hasse principle for 0 cycles of degree 1 on some fibrations over the projective line defined by bi cyclic normic equations in the present paper we prove the exactness of the global to local sequence for chow groups of 0 cycles of such varieties this signifies that the brauer manin obstruction is also the only obstruction to weak approximation for 0 cycles of arbitrary degree our main theorem also generalizes several existing results

descent on fibrations over the projective line we formulate a general set up for the descent method of j l colliot thelene and j j sansuc applied to varieties fibred over the projective line over a number field this makes it possible to prove that the manin obstruction to the hasse principle and weak approximation is the only one provided there are only few degenerate fibres usually two three in some cases and that sufficiently many smooth fc fibres satisfy the hasse principle and weak approximation we introduce a new concept of what should be called a degenerate fibre the so called split fibres the property which depends only on the generic fibre and not on the choice of a particular model this yields a simpler and more general approach to the previous results of that kind 0 introduction motivated by hasse s theorem on quadratic forms one says that the hasse principle holds for a family of geometrically integral smooth and proper varieties over a number field k9 if for every member of this family the existence of rational points over all the completions of k implies the existence of a rational point the fact that the hasse principle holds for some family signif icantly simplifies the study of its arithmetic because the problem of determining whether or not there is a rational point is then reduced to a finite number of

deformation theory of infinity algebras abstract this work explores the deformation theory of algebraic structures in a very general setting these structures include associative algebras lie algebras and the infinity versions of these structures the strongly homotopy associative and lie algebras in all these cases the algebraic structure is determined by an element of a certain graded lie algebra which determines a differential on the lie algebra we work out the deformation theory in terms of the lie algebra of coderivations of an appropriate coalgebra structure and construct a universal infinitesimal deformation as well as a miniversal formal deformation by working at this level of generality the main ideas involved in deformation theory stand out more clearly