the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

the fast multipole method fmm for electromagnetic scattering problems the fast multipole method fmm developed by v rokhlin 1990 to efficiently solve acoustic scattering problems is modified and adapted to the second kind integral equation formulation of electromagnetic scattering problems in two dimensions the present implementation treats the exterior dirichlet problem for two dimensional closed conducting objects of arbitrary geometry the fmm reduces the operation count for solving the second kind integral equation from o n sup 3 for gaussian elimination to o n sup 4 3 per conjugate gradient iteration where n is the number of sample points on the boundary of the scatterer a sample technique for accelerating convergence of the iterative method termed complexifying k the wavenumber is also presented this has the effect of bounding the condition number of the discrete system consequently the operation count of the entire fmm all iterations becomes o n sup 4 3 computational results for moderate values of ka where a is the characteristic size of the scatterer are given

a preparation theorem for weierstrass systems it is shown that lion and rolin s preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a weierstrass system

characteristic cycles and wave front cycles of representations of reductive lie groups in the papers v1 and bv vogan and barbasch vogan attach two similar invariants to representations of a reductive lie group one by an algebraic process the other analytic they conjectured that the two invariants determine each other in a deflnite manner here we prove the conjecture our arguments involve two flner invariants the characteristic cycles of representations which are interesting in their own right to describe the invariants we consider a linear reductive lie group g and flx a maximal compact subgroup k g we denote their lie algebras by and and the complexifled lie algebras by an element of the dual space is said to be nilpotent if it corresponds to a nilpotent element of via the isomorphism induced by the killing form via the adjoint action the complexiflcation g of g acts with flnitely many orbits onn the cone of all nilpotents in like all coadjoint orbits each g orbit o n carries a distinguished complex algebraic symplectic structure the intersection ofo with i f 2 jh i i g consists of flnitely many g orbits each of which is lagrangian ino the choice of a maximal compact subgroup determines a cartan decomposition and dually the complexiflcationk of the groupk operates onn with flnitely many orbits and each of these orbits is lagrangian in the g orbit which contains it kr now let be an irreducible admissible representation of g for example an irreducible unitary representation to such a representation one can associate its harish chandra module v which is simultaneously and compatibly a module for the lie algebra and a locally flnite module for the algebraic group k then v admits k invariant good flltrations as module over the universal enveloping algebra u relative to its canonical flltered structure vogan v1 shows that the annihilator of the graded module deflnes an equidi

numerical properties of a gaussian translation operator for the 2d fmm a diagonal gaussian translation operator for the time harmonic fast multipole method fmm in two dimensions is examined numerically the gaussian translation operator depends on a beam parameter that determines its sharpness when the beam parameter is set to zero the gaussian translation operator reduces to the standard fmm translation operator the sampling rate can be determined straightforwardly to achieve any desired accuracy the directionality of the gaussian translation operator makes it possible to reduce the number of plane waves required to achieve a given accuracy the required number of plane waves depends strongly on the actual source receiver locations not just on the diameter of the source and receiver regions

nonlinear second order elliptic equations v the dirichlet problem for weingarten hypersurfaces on etudie le probleme de dirichlet pour une fonction u dans un domaine borne de r n a frontiere lisse strictement convexe en un point x de les courbures principales k k 1 k n du graphe x u x satisfont une relation f k 1 k n x 0 ou est une fonction positive lisse donnee sur de plus u doit satisfaire la condition aux limites de dirichlet u 0 sur sous certaines conditions on demontre qu il existe une solution u unique admissible lisse dans

singular sets for curvature equation of order k abstract we consider the removability of singular sets for the curvature equations of the form h k u which is determined by the kth elementary symmetric function in an n dimensional domain we prove that for 1 k n 1 and a compact set k whose n k dimensional hausdorff measure is zero any generalized solution to the curvature equation on k is always extendable to a generalized solution on the whole domain

valuation theory of exponential hardy fields i we describe the valuation theoretic properties of the hardy fields associated to models of t exp where t is the theory of a polynomially bounded o minimal expansion of the reals and exp is the real exponential function we deduce that t exp has levels with parameters and is exponentially bounded we establish a maximality property of h mathbb r rm an powers the hardy field of the expansion by the restricted analytic functions and power functions

on the transmittance between oam antennas a quasi analytical solution is derived for the transmittance between two oam antennas it makes use of the properties of the modified circle polynomials related to zernike functions and of integration in the complex plane of the radial spectral coordinate the resulting formulation can be estimated extremely fast and is shown to exhibit several expected properties an asymptotic expression is proposed and validated

definable sets in ordered structures i this paper introduces and begins the study of a well behaved class of linearly ordered structures the minimal structures the definition of this class and the corresponding class of theories the strongly dlminlmal theories is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories theorems 2 1 and 2 3 respectively provide characterizations of 41 minimal ordered groups and rings several other simple results are collected in 3 the primary tool in the analysis of minimal structures is a strong analogue of ii forking symmetry given by theorem 4 2 this result states that any parametri cally definable unary function in an yminimal structure is piecewise either constant or an ore er preserving or reversing bijection of intervals the results that follow include the existence and uniqueness of prime models over sets theorem 5 1 and a characterization of all bto categorical aminimal structures theorem 6 1 1 introduction the class of linearly ordered structures has long been an im portant subject of concern to model theorists impressive results have been obtained in the study of models of several particular theories that extend the theory of linear order among those theories that have been approached successfully are peano arithmetic the theory of ordered abelian groups that of real closed fields and that of linear order itself yet very little has been done in the way of developing a general model theory for ordered structures in this paper we develop the model theory for a class of linearly ordered structures that we isolate by demanding that a structure in this class satisfy a condition whose effect is that the linear ordering and the algebraic part of the structure behave quite well with respect to one another let l be a finitary first order language and w an l structure a set of n tuples a c x is said to be parametrically definable if there is some l formula

generalized solutions of curvature equations abstract we introduce a notion of generalized solutions for the so called curvature equations it is an extension of the definition of generalized solutions of the gauss curvature equation and gives a new framework for the study of more general curvature equations we prove that if the curvature equation has a convex solution then the inhomogeneous term must be a borel measure we also discuss basic properties of generalized solutions among other things we show that our class of generalized solutions is wider than that of weak solutions

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

smooth functions in o minimal structures abstract fix an o minimal expansion of the real exponential field that admits smooth cell decomposition we study the density of definable smooth functions in the definable continuously differentiable functions with respect to the definable version of the whitney topology this implies that abstract definable smooth manifolds are affine moreover abstract definable smooth manifolds are definably c diffeomorphic if and only if they are definably c 1 diffeomorphic

complements of subanalytic sets and existential formulas for analytic functions we show that the complement of a subanalytic set defined by real analytic functions from any subalgebra closed under differentiation is a subanalytic set defined by the functions from the same subalgebra this result has an equivalent formulation in logic consider an expression built from functions as above using equalities and inequalities as well as existential and universal quantifiers such an expression is equivalent to an existential expression involving functions from the same class provided that the variables approach neither infinity nor the boundary of the domain

ilyashenko algebras based on transserial asymptotic expansions abstract we construct a hardy field that contains ilyashenko s class of germs at of almost regular functions found in 12 as well as all log exp analytic germs this implies non oscillatory behaviour of almost regular germs with respect to all log exp analytic germs in addition each germ in this hardy field is uniquely characterized by an asymptotic expansion that is an le series as defined by van den dries et al 7 as these series generally have support of order type larger than the notion of asymptotic expansion itself needs to be generalized

the real field with convergent generalized power series we construct a model complete and o minimal expansion of the field of real numbers in which each real function given on 0 1 by a series cnxn with 0 n and cn r n 1 is definable this expansion is polynomially bounded

lebesgue measure and integration theory on non archimedean real closed fields with archimedean value group given a non archimedean real closed field with archimedean value group which contains the reals we establish for the category of semialgebraic sets and functions a full lebesgue measure and integration theory such that the main results from the classical setting hold the construction involves methods from model theory o minimal geometry and valuation theory we set up the construction in such a way that it is determined by a section of the valuation if the value group is isomorphic to the group of rational numbers the construction is uniquely determined up to isomorphism the range of the measure and integration is obtained in a controlled and tame way from the real closed field we start with the main example is given by the case of the field of puiseux series where the range is the polynomial ring in one variable over this field

removable singularities for fully nonlinear elliptic equations we obtain a removability result for the fully nonlinear uniformly elliptic equations f d 2 u f u 0 the main theorem states that every solution to the equation in a punctured ball without any restrictions on the behaviour near the centre of the ball is extendable to the solution in the entire ball provided the function f satisfies certain sharp conditions depending on f previously such results were known for linear and quasilinear operators f in comparison with the semi or quasilinear theory the techniques for the fully nonlinear equations are new and based on the use of the viscosity notion of generalised solution rather than the distributional or the weak solutions

ax lindemann for mathcal a _g we prove the ax lindemann theorem for the coarse moduli space a g of principally polarized abelian varieties of dimension g 1 we affirm the andre oort conjecture unconditionally for a g for g 6 and under grh for all g

isolated singularities for some types of curvature equations abstract we consider the removability of isolated singularities for the curvature equations of the form h k u 0 which is determined by the k th elementary symmetric function in an n dimensional domain we prove that for 1 k n 1 isolated singularities of any viscosity solutions to the curvature equations are always removable provided the solution can be extended continuously at the singularities we also consider the class of generalized solutions and prove the removability of isolated singularities

harmonics based inhomogeneous plane wave method hipw a formulation is presented for fast interactions between subdomains in two dimensional 2 d scattering problems the formulation combines inhomogeneous plane waves with cylindrical harmonic decompositions of fields radiated by the subdomains it is shown that the complexity of interactions naturally decays with the distance between subdomains and that very few elementary operations are involved at the lowest level an example of iterative solution for scattering by a collection of cylinders validates the proposed approach

lebesgue classes and preparation of real constructible functions abstract we call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively valued globally subanalytic functions for any q 0 and constructible functions f and on e r n we prove a theorem describing the structure of the set x p e 0 f x l p x q where x q is the positive measure on r n whose radon nikodym derivative with respect to the lebesgue measure is x q y x y q we also prove a closely related preparation theorem for f and these results relate analysis the study of l p spaces with geometry the study of zero loci

the elementary theory of restricted analytic fields with exponentiation numbers with exponentiation is model complete when we combine this with hovanskii s finiteness theorem 9 it follows that the real exponential field is o minimal in o minimal expansions of the real field the definable subsets of r share many of the nice structural properties of semialgebraic sets for example definable subsets have only finitely many connected components definable sets can be stratified and triangulated and continuous definable maps are piecewise trivial see 5 in this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions and use this result to obtain o minimality we were led to this while studying work of ressayre 13 and several of his ideas emerge here in simplified form however our treatment is formally independent of the results of 16 17 9 and 13

hessian measures ii in our previous paper on this topic we introduced the notion of k hessian measure associated with a continuous k convex function in a domain q in euclidean n space k 1 n and proved a weak continuity result with respect to local uniform convergence in this paper we consider k convex functions not necessarily continuous and prove the weak continuity of the associated k hessian measure with respect to convergence in measure the proof depends upon local integral estimates for the gradients of k convex functions

randomizing a model abstract a randomization of a first order structure m is a new structure with certain closure properties whose universe is a setkof random elements of m randomizations assign probabilities to sentences of the language of m with new constants fromk our main theorem shows that all randomizations of m are models of the same first order theoryt which has a nice set of axioms and admits elimination of quantifiers moreover the class of substructures of models oftis characterized by a natural setvof universal axioms oft so thattis the model completion ofv

compactness theorems for differential forms we prove compactness properties of various sets of differential forms with bounds on their exterior derivatives this gives simple proofs of the federer fleming result on normal currents and of compensated compactness lemmas

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

relational expressive power of constraint query languages the expressive power of first order query languages with several classes of equality and inequality constraints is studied in this paper we settle the conjecture that recursive queries such as parity test and transitive closure cannot be expressed in the relational calculus augmented with polynomial inequality constraints over the reals furthermore noting that relational queries exhibit several forms of genericity we establish a number of collapse results of the following form the class of generic boolean queries expressible in the relational calculus augmented with a given class of constraints coincides with the class of queries expressible in the relational calculus with or without an order relation we prove such results for both the natural and active domain semantics as a consequence the relational calculus augmented with polynomial inequalities expresses the same classes of generic boolean queries under both the natural and active domain semantics in the course of proving these results for the active domin semantics we establish ramsey type theorems saying that any query involving certain kinds of constraints coincides with a constraint free query on databases whose elements come from a certain infinite subset of the domain to prove the collapse results for the natural semantics we make use of techniques from nonstandard analysis and from the model theory of ordered structures

counting special points logic diophantine geometry and transcendence theory we expose a theorem of pila and wilkie on counting rational points in sets definable in o minimal structures and some applications of this theorem to problems in diophantine geometry due to masser peterzil pila starchenko and

inclusion dynamics hybrid automata hybrid systems are dynamical systems with the ability to describe mixed discrete continuous evolution of a wide range of systems consequently at first glance hybrid systems appear powerful but recalcitrant neither yielding to analysis and reasoning through a purely continuous time modeling as with systems of differential equations nor open to inferential processes commonly used for discrete state transition systems such as finite state automata a convenient and popular model called hybrid automata was introduced to model them and has spurred much interest on its tractability as a tool for inference and model checking in a general setting intuitively a hybrid automaton is simply a finite state automaton with each state augmented by continuous variables which evolve according to a set of well defined continuous laws each specified separately for each state this article investigates both the notion of hybrid automaton and the model checking problem over such a structure in particular it relates first order theories and analysis results on multivalued maps and reduces the bounded reachability problem for hybrid automata whose continuous laws are expressed by inclusions x f x t to a decidability problem for first order formulaeover the reals furthermore the paper introduces a class of hybrid automata for which the reachability problem can be decided and shows that the problem of deciding whether a hybrid automaton belongs to this class can be again decided using first order formulaeover the reals despite the fact that the bisimulation quotient for this class of hybrid automata can be infinite we show that our techniques permit effective model checking for a nontrivial fragment of ctl

hessian measures iii abstract in this paper we continue previous investigations into the theory of hessian measures we extend our weak continuity result to the case of mixed k hessian measures associated with k tuples of k convex functions on domains in euclidean n space k 1 2 n applications are given to capacity quasicontinuity and the dirichlet problem with inhomogeneous terms continuous with respect to capacity or combinations of dirac measures

o minimality and the andre oort conjecture for mathbb c n we give an unconditional proof of the andre oort conjecture for arbitrary products of modular curves we establish two generalizations the first includes the manin mumford conjecture for arbitrary products of elliptic curves defined over q as well as lang s conjecture for torsion points in powers of the multiplicative group the second includes the manin mumford conjecture for abelian varieties defined over q our approach uses the theory of o minimal structures a part of model theory and follows a strategy proposed by zannier and implemented in three recent papers a new proof of the manin mumford conjecture by pila zannier a proof of a special but new case of pink s relative manin mumford conjecture by masser zannier and new proofs of certain known results of andre oort manin mumford type by pila 2010 mathematics subject classification 11g18 03c64