the bogomolov conjecture for totally degenerate abelian varieties we prove the bogomolov conjecture for a totally degenerate abelian variety a over a function field we adapt zhang s proof of the number field case replacing the complex analytic tools by tropical analytic geometry a key step is the tropical equidistribution theorem for a at the totally degenerate place v as a corollary we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v

big line bundles over arithmetic varieties 5 equidistribution theory 26 5 1 a generic equidistribution theorem 27 5 2 equidistribution at infinite places 30 5 3 equidistribution at finite places 32 5 4 equidistribution of small subvarieties 35 5 5 equidistribution over algebraic dynamical systems 37 5 6 equidistribution over multiplicative groups 39 5 7 equidistribution theory 41

optimization over the efficient set abstract in this paper we study the problem of optimizing a linear function over the set of efficient solutions for a vector maximization problem this problem arises whenever a linear function is available which acts as a criterion for measuring the importance of or for discriminating among the efficient alternatives that are available using the concepts of an efficient direction and of a direction of recession we develop necessary and sufficient conditions for this problem to be unbounded we also present necessary and sufficient conditions for efficient and for arbitrary solutions of the underlying vector maximization problem to be optimal solutions for this problem some of these conditions are geometric in nature others algebraic the algebraic conditions suggest potential computational procedures for finding an optimal solution to the problem finally we give conditions under which the set of optimal solutions possesses certain special properties

intersecting subvarieties of g_m n with algebraic subgroups let x subset g m n be an irreducible closed subvariety defined over bar q we bound the height of algebraic points on x that are in a certain sense close to the union of all algebraic subgroup of g m n of dimension m n dim x the bound we obtain is effective and will be expressed as a function of the height of x the degree of x and n we then apply this bound to derive certain finiteness results if m is also strictly less than n dim x

range characterization of the cosine transform on higher grassmannians abstract we characterize the range of the cosine transform on real grassmannians in terms of the decomposition under the action of the special orthogonal group so n we also give a geometric interpretation of this image in terms of valuations in addition we discuss the non archimedean analogues

fibers of tropicalization we use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the zariski topology for subvarieties of tori over fields of generalized power series points in each tropical fiber are obtained constructively using kedlaya s transfinite version of newton s method

the number of maximal torsion cosets in subvarieties of tori we present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus gm n our first main result gives a bound in terms of the degree of the defining polynomials a second result gives a bound in terms of the toric degree of the subvariety as a consequence we prove the conjectures of ruppert and of aliev and smyth on the number of isolated torsion points of a hypersurface these conjectures bound this number in terms of the multidegree and the volume of the newton polytope of a polynomial defining the hypersurface respectively

intertwining operators for the generalized principal series on a symmetric r space three questions about the intertwining operators for the generalized principal series on a symmetric r space are solved description of the functional kernel both in the noncompact and in the compact picture domain of convergence meromorphic continuation a large use is made of the theory of positive jordan triple systems the meromorphic continuation of the intertwining integral is achieved via a bernstein sato identity and a precise description of the poles is obtained 0 2000 mathematics subject classification 22e45 43a80

intersection bodies and generalized cosine transforms the paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis a new concept of intersection body that unifies some known classes of geometric objects is introduced a parallel between trace theorems in function theory restriction onto lower dimensional subspaces of the spherical radon transforms and the generalized cosine transforms and sections of intersection bodies is established new integral formulas for different classes of cosine transforms are obtained and examples of intersection bodies are given we also revisit some known facts in this area and give them new simple proofs

the busemann petty problem via spherical harmonics abstract the busemann petty problem asks whether symmetric convex bodies in r n with smaller central hyperplane sections necessarily have smaller n dimensional volume the solution has recently been completed and the answer is affirmative if n 4 and negative if n 5 in this article we present a short proof of the affirmative result and its generalization using the funk hecke formula for spherical harmonics

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

inversion of fractional integrals related to the spherical radon transform explicit inversion formulas are obtained for the analytic family of fractional integrals t f x n sn xy 1 f y dyon the unit sphere in rn 1 arbitrary complex andn 2 are considered in the ease 0 the integralt fcoincides with the spherical radon transform for 1 1 3 5 such integrals are known as the blaschke levy representations and arise in convex geometry probability and the banach space theory for 1 3 5 the integralt fis defined by continuity as the spherical convolution with the power logarithmic kernel different inversion methods are discussed

on the degenerate principal series of complex symplectic groups we apply techniques introduced by clerc kobayashi orsted and pevzner to study the degenerate principal series of sp n c an explicit description of the k types is provided and knapp stein normalised operators are realised as symplectic fourier transforms and their k spectrum explicitly computed reducibility phenomena are analysed in terms of k types and eigenvalues of intertwining operators we also construct a new model for these representations in which knapp stein intertwiners take an algebraic form

a quantitative version of the absolute subspace theorem the celebrated subspace theorem of w m schmidt 12 says the following subspace theorem let l1 ln be linearly independent linear forms in n variables with real or complex algebraic coefficients suppose 0 consider the inequality 1 1 l1 x ln x 0 put 1 5 q c x r li x qi 1 i n then the parametric subspace theorem can be stated as follows parametric subspace theorem let c be a fixed tuple satisfying 1 3 let 0 then there are finitely many proper linear subspaces t1 tb of q n such that for any q which is sufficiently large there exists a subspace ti t1 tb with 1 6 q q c z ti

on the meromorphic extension of the spherical functions on noncompactly causal symmetric spaces abstract we determine integral formulas for the meromorphic extension in the parameter of the spherical functions on a noncompactly causal symmetric space the main tool is bernstein s theorem on the meromorphic extension of complex powers of polynomials the regularity properties of are deduced in particular the possible poles of are located among the translates of the zeros of the bernstein polynomial the translation parameter depends only on the structure of the symmetric space the expression of the bernstein polynomial is conjectured the relation between the bernstein polynomial and the product formula of the c function is analyzed the conjecture is verified in the rank one case the explicit formulas obtained in this case yield a detailed description of singularities of in the general higher rank case the integral formulas are applied to find asymptotic estimates for the spherical functions in the appendix the spherical functions on noncompactly causal symmetric spaces are regarded as a special instance of harish chandra type expansions associated with roots systems with arbitrary multiplicities we study expansions obtained by taking averages over arbitrary parabolic subgroups of the weyl group of the root system the possible singularities are located in this general context

a computational study for bilevel quadratic programs using semidefinite relaxations abstract in this paper we deal with bilevel quadratic programming problems with binary decision variables in the leader problem and convex quadratic programs in the follower problem for this purpose we transform the bilevel problems into equivalent quadratic single level formulations by replacing the follower problem with the equivalent karush kuhn tucker kkt conditions then we use the single level formulations to obtain mixed integer linear programming milp models and semidefinite programming sdp relaxations thus we compute optimal solutions and upper bounds using linear programming lp and sdp relaxations our numerical results indicate that the sdp relaxations are considerably tighter than the lp ones consequently the sdp relaxations allow finding tight feasible solutions for the problem especially when the number of variables in the leader problem is larger than in the follower problem moreover they are solved at a significantly lower computational cost for large scale instances

processes of flats induced by higher dimensional processes stationary processes of k flats in mathbb e d can be thought of as point processes on the grassmannian mathcal l kd of k dimensional subspaces of mathbb e d if such a process is sampled by a d k j dimensional space f it induces a process of j flats in f in this work we will investigate the possibility of determining the original k process from knowledge of the intensity measures of the induced j processes we will see that this is impossible precisely when 1 k d 1 and j 0 2 r 2 1 where r is the rank of the manifold mathcal l kd we will show how the problem is equivalent to the study of the kernel of various integral transforms these will then be investigated using harmonic analysis on grassmannian manifolds

optimality conditions for minimization over the weakly or properly efficient set abstract the paper deals with the optimization problem of a nonlinear real function over the weakly or properly efficient set associated to a convex multi objective program in banach spaces abstract optimality conditions are obtained via derivatives for set valued functions in the strict convex case these conditions lead to a nonlinear system involving the classical derivatives up to the second order

kinematic formulas on the quaternionic plane we introduce different bases for the vector space of sp 2 sp 1 invariant translation invariant continuous valuations on the quaternionic plane and determine a complete set of kinematic formulas

degenerations of amoebae and berkovich spaces we prove a continuity result for the fibers of the berkovich analytification of a complex algebraic variety with respect to the maximum of the archimedean norm and the trivial norm as a consequence we obtain generalizations of a result of mikhalkin and rullgard about degenerations of amoebae onto tropical varieties

the complex busemann petty problem on sections of convex bodies abstract the complex busemann petty problem asks whether origin symmetric convex bodies in c n with smaller central hyperplane sections necessarily have smaller volume we prove that the answer is affirmative if n 3 and negative if n 4

standard conjectures for the arithmetic grassmannian g 2 n and racah polynomials we prove the arithmetic hodge index and hard lefschetz conjectures for the grassmannian g g 2 n parametrizing lines in projective space for the natural arithmetic lefschetz operator defined via the plucker embedding of g in projective space the analysis of the hodge index inequality involves estimates on values of certain racah polynomials

calculus on arithmetic surfaces in a2 and a3 arakelov introduces an intersection calculus for arithmetic surfaces that is for stable models of curves over a number field in this paper we intend to show that his intersection product has a lot of useful properties more precisely we show that the following properties from the theory of algebraic surfaces have an analogue in our situation

efficiency and proper efficiency in vector maximization with respect to cones abstract vector maximization problems arise when more than one objective function is to be maximized over a given feasibility region the concepts of efficiency and proper efficiency have played a useful role in the analysis of such problems recently these concepts have been extended to vector maximization problems in which the underlying domination cone is a convex cone in this paper efficient and properly efficient solutions for the vector maximization problem in which the underlying domination cone is any nontrivial closed convex cone are examined differences between properly and improperly efficient solutions are established characterizations of efficient and properly efficient solutions are presented and conditions under which efficient solutions exist and fail to exist are derived

compactifications of subvarieties of tori we study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non archimedean amoebas these compactifications have many nice properties for example any k boundary divisors intersect in codimension k we consider some examples including m 0 n subset overline m 0 n and more generally log canonical models of complements of hyperplane arrangements and compact quotients of grassmannians by a maximal torus

arithmetic height functions over finitely generated fields in this paper we propose a new height function for a variety defined over a finitely generated field over for this height function we prove northcott s theorem and bogomolov s conjecture so that we can recover the original raynaud s theorem manin mumford s conjecture

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

unitary representations of lie groups with reflection symmetry we consider the following class of unitary representations of some real lie groupgwhich has a matched pair of symmetries described as follows i supposeghas a period 2 automorphism and that the hilbert spaceh carries a unitary operatorjsuch thatj j i e selfsimilarity ii an added symmetry is implied ifh further contains a closed subspacek0having a certainorder covarianceproperty and satisfying thek0 restricted positivity v jv 0 v k0 where is the inner product inh from i ii we get an induced dual representation of an associated dual groupgc all three properties selfsimilarity order covariance and positivity are satisfied in a natural context whengis semisimple and hermitean but whengis the ax b group or the heisenberg group positivity is incompatible with the other two axioms for the infinite dimensional irreducible representations we describe a class ofg containing the latter two which admits a classification of the possible spacesk0 h satisfying the axioms of selfsimilarity and order covariance

fourier and poisson transformation associated to a semisimple symmetric space soit g un groupe de lie semisimple connexe contenu dans un groupe de lie semisimple simplement connexe dont l algebre de lie est la complexification de celle de g soit une involution non triviale de g et une involution de cartan commutant avec soit h un groupe point fixe de on etudie les representations h spheriques de dimension finie de g contenant un vecteur k fixe ou k est un sous groupe compact maximal stable de g sous representations de dimension finie de le noyau de poisson la distribution u i la transformation de fourier et poisson formules integrales

weak linear bilevel programming problems existence of solutions via a penalty method we are concerned with a class of weak linear bilevel programs with nonunique lower level solutions for such problems we give via an exact penalty method an existence theorem of solutions then we propose an algorithm

generalizations of the busemann petty problem for sections of convex bodies abstract we present generalizations of the busemann petty problem for dual volumes of intermediate central sections of symmetric convex bodies it is proved that the answer is negative when the dimension of the sections is greater than or equal to 4 for two three dimensional sections both negative and positive answers are given depending on the orders of dual volumes involved and certain cases remain open for bodies of revolution a complete solution is obtained in all dimensions

spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup abstract we present a new method of calculating intertwining operators between principal series representations of semisimple lie groups g working in the compact realization we find the eigenvalues of the operators on the k types and give several examples among the advantages of our method is its applicability to bundle valued cases

the cos λ and sin λ transforms as intertwining operators between generalized principal series representations of sl n 1 k abstract in this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple lie groups by showing that the cos and sin transforms on the grassmann manifolds gr p k su n 1 k s u p k u n 1 p k are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup p p of sl n 1 k the index p indicates the dependence of the parabolic on p the general results of knapp and stein 1971 1980 20 21 and vogan and wallach 1990 44 then show that both transforms have meromorphic extension to c and are invertible for generic c furthermore known methods from representation theory combined with a selberg type integral allow us to determine the k spectrum of those operators

vecteurs distributions h invariants pour les series principales generalisees d espaces symetriques reductifs et prolongement meromorphe d integrales d eisenstein

a note on divisible points of curves let c be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3 we partially answer a question posed by levin on points on c for which a non trivial power lies again on c our results have connections to zilber s conjecture on intersections with tori and yield to methods arising in transcendence theory and the theory of o minimal structures

an exact penalty on bilevel programs with linear vector optimization lower level we are interested in a class of linear bilevel programs where the upper level is a linear scalar optimization problem and the lower level is a linear multi objective optimization problem we approach this problem via an exact penalty method then we propose an algorithm illustrated by numerical examples

computations of bott chern classes on ℙ e we compute the bott chern classes of the metric eulersequence describing the relative tangent bundle of the variety p e of hyperplans of a holomorphic hermitian vector bundle e h on a complex manifold we give applications to the construction of the arithmetic characteristic classes of an arithmetic vector bundle and to the computation of the height of p e with respect to the tautological quotient bundle o e 1

canonical representations related to hyperbolic spaces abstract we extend the notion of canonical representation introduced by a m vershik i m gel fand and m i graev 1982 in representation theory cambridge univ press cambridge u k to all classical rank one semisimple lie groups we determine the spectral decomposition of the canonical representations in detail

structure of the degenerate principal series on symmetric r spaces and small representations abstract let g be a simple real lie group with maximal parabolic subgroup p whose nilradical is abelian then x g p is called a symmetric r space we study the degenerate principal series representations of g on c x in the case where p is not conjugate to its opposite parabolic we find the points of reducibility the composition series and all unitarizable constituents among the unitarizable constituents we identify some small representations having as associated variety the minimal nilpotent k c orbit in p c where k c is the complexification of a maximal compact subgroup k g and g k p the corresponding cartan decomposition

effective bounds of linear series on algebraic varieties and arithmetic varieties in this paper we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes they are effective versions of the hilbert samuel formula and the arithmetic hilbert samuel formula the treatments are high dimensional generalizations of duke math j 162 2013 1723 1770 and relative noether inequality on fibered surfaces preprint 2013 similar results are obtained independently by huayi chen majorations explicites de fonctions de hilbert samuel geometrique et arithmetique preprint 2014 with less explicit error terms

classification of invariant valuations on the quaternionic plane abstract we describe the orbit space of the action of the group sp 2 sp 1 on the real grassmann manifolds gr k h 2 in terms of certain quaternionic matrices of moore rank not larger than 2 we then give a complete classification of valuations on the quaternionic plane h 2 which are invariant under the action of the group sp 2 sp 1

a further improvement of the quantitative subspace theorem in 2002 evertse and schlickewei 11 obtained a quantitative version of the so called absolute parametric subspace theorem this result deals with a parametrized class of twisted heights one of the consequences of this result is a quantitative version of the absolute subspace theorem giving an explicit upper bound for the number of subspaces containing the solutions of the diophantine inequality under consideration in the present paper we further improve evertse s and schlickewei s quantitative version of the absolute parametric subspace theorem and deduce an improved quantitative version of the absolute subspace theorem we combine ideas from the proof of evertse and schlickewei which is basically a substantial refinement of schmidt s proof of his subspace theorem from 1972 22 with ideas from faltings and wustholz proof of the subspace theorem 14 a new feature is an interval result which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the subspace theorem