dirac equation with external potential and initial data on cauchy surfaces with this paper we provide a mathematical review on the initial value problem of the one particle dirac equation on space like cauchy hypersurfaces for compactly supported external potentials we first discuss the physically relevant spaces of solutions and initial values in position and mass shell representation second review the action of the poincare group as well as gauge transformations on those spaces third introduce generalized fourier transforms between those spaces and prove convenient paley wiener and sobolev type estimates these generalized fourier transforms immediately allow the construction of a unitary evolution operator for the free dirac equation between the hilbert spaces of square integrable wave functions of two respective cauchy surfaces with a picard lindelof argument this evolution map is generalized to the dirac evolution including the external potential for the latter we introduce a convenient interaction picture on cauchy surfaces these tools immediately provide another proof of the well known existence and uniqueness of classical solutions and their causal structure

hermitian analogues of hilbert s 17 th problem abstract we pose and discuss several hermitian analogues of hilbert s 17 th problem we survey what is known offer many explicit examples and some proofs and give applications to cr geometry we prove one new algebraic theorem a non negative hermitian symmetric polynomial divides a non zero squared norm if and only if it is a quotient of squared norms we also discuss a new example of putinar scheiderer

balanced line bundles on fano varieties a conjecture of batyrev and manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants in particular counting functions defined by metrized ample line bundles and the corresponding asymptotics of rational points of bounded height are interpreted in terms of cones of effective divisors and certain thresholds with respect to these cones this framework leads to the notion of balanced line bundles whose counting functions conjecturally capture generic distributions of rational points we investigate balanced line bundles in the context of the minimal model program with special regard to the classification of fano threefolds

homogeneous asymptotic limits of haar measures of semisimple linear groups and their lattices we show that haar measures of connected semisimple groups embedded via a representation into a matrix space have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled this is still true if the haar measure of the semisimple group is replaced by the haar measure of a irreducible lattice of the group and the asymptotic measure is the same in the case of an almost simple group of rank greater than 2 a remainder term is also obtained this extends and makes precise anterior results of duke rudnick and sarnak and eskin mcmullen in the case of a group variety

varieties with too many rational points we investigate fano varieties defined over a number field that contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety

height uniformity for integral points on elliptic curves we recall the result of d abramovich and its generalization by p pacelli on the uniformity for stably integral points on elliptic curves it says that the lang vojta conjecture on the distribution of integral points on a variety of logarithmic general type implies the uniformity for the numbers of stably integral points on elliptic curves in this paper we will investigate its analogue for their heights under the assumption of the vojta conjecture basically we will show that the vojta conjecture gives a naturally expected simple uniformity for their heights

inhomogeneous cubic congruences and rational points on del pezzo surfaces for given non zero integers a b q we investigate the density of solutions x y 2 z 2 to the binary cubic congruence ax 2 by 3 0 mod q and use it to establish the manin conjecture for a singular del pezzo surface of degree 2 dened over q

nombre de points de hauteur bornee sur les surfaces de del pezzo de degre 5 we state manin s conjecture in the particular case of the split del pezzo s surfaces of degree 5 over q we show that for an open set u v nu q b card p u q h p b c b log b 4 b the constant c is the one conjectured by e peyre sommaire

leclate du plan projectif en quatre points dont deux conjugues we state manin s conjecture in the particular case of the blow up of the projective plan in four points where two are conjugated we show that for an open set u v nu q b card p 2 u q h p 6 b cb log b b 1 the constant c is the one conjectured by peyre

rigidity of proper holomorphic maps between bounded symmetric domains our first main result gives assumptions guaranteeing that proper holomorphic maps between cartan type i bounded symmetric domains have simple block matrix shape answering positively a question of mok the proof is based on the second main result establishing similar phenomenon for local cr maps between arbitrary boundary components of two bounded symmetric domains of the above type since boundary components other than shilov boundaries are levi degenerate our analysis is based on their 2 nondegeneracy combining levi forms with higher order tensors

linear growth for châtelet surfaces an upper bound of the expected order of magnitude is established for the number of mathbb q rational points of bounded height on ch telet surfaces defined over mathbb q

on the distribution of points of bounded height on equivariant compactifications of vector groups we prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space

consistency of multi time dirac equations with general interaction potentials in 1932 dirac proposed a formulation in terms of multi time wave functions as candidate for relativistic many particle quantum mechanics a well known consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles it was conjectured by petrat and tumulka that interactions described by multiplication operators are generally excluded by this condition and they gave a proof of this claim for potentials without spin coupling under suitable assumptions on the differentiability of possible solutions we show that there are potentials which are admissible give an explicit example however show that none of them fulfills the physically desirable poincare invariance we conclude that in this sense dirac s multi time formalism does not allow to model interaction by multiplication operators and briefly point out several promising approaches to interacting models one can instead pursue

universal torsors of del pezzo surfaces and homogeneous spaces let cox sr be the homogeneous coordinate ring of the blow up sr of in r general points i e a smooth del pezzo surface of degree 9 r we prove that for r 6 7 proj cox sr can be embedded into gr pr where gr is an algebraic group with root system given by the primitive picard lattice of sr and pr gr is a certain maximal parabolic subgroup

on the question of current conservation for the two body dirac equations of constraint theory the two body dirac equations of constraint theory are of special interest not only in view of applications for phenomenological calculations of mesonic spectra but also because they avoid no go theorems about relativistic interactions furthermore they provide a quantum mechanical description in a manifestly lorentz invariant way using the concept of a multi time wave function in this paper we place them into the context of the multi time formalism of dirac tomonaga and schwinger for the first time a general physical and mathematical framework is outlined and the mechanism which permits relativistic interaction is identified the main requirement derived from the general framework is the existence of conserved tensor currents with a positive component which can play the role of a probability density we analyze this question for a general class of two body dirac equations thoroughly and comprehensively while the free dirac current is not conserved it is possible to find replacements improving on previous research we achieve definite conclusions whether restrictions of the function space or of the interaction terms can guarantee the positive definiteness of the currents and whether such restrictions are physically adequate the consequences of the results are drawn with respect to both applied and foundational perspectives

the equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics we consider the class of levi nondegenerate hypersurfaces m in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i that admit a local cr transversal embedding near a point p e m into a standard nondegenerate hyperquadric in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i with codimension k n n small compared to the cr dimension n of m we show that for hypersurfaces in this class there is a normal form which is closely related to the embedding such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric we also show that if the signature of m and that of the standard hyperquadric in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i are the same then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric

counting algebraic numbers with large height ii let denote the field of rational numbers open image in new window an algebraic closure of and h open image in new window the absolute multiplicative weil height for each positive integer d and real number mathcal h geqslant 1 it is well known that the number open image in new window of points in open image in new window having degree d over and satisfying h left alpha right leqslant mathcal h is finite this is the one dimensional case of northcott s theorem 8 see also 5 page 59 the systematic study of the counting function open image in new window and that of related functions in higher dimensions was begun by schmidt 10 it is relatively easy to prove the existence of a positive constant c c d such that open image in new window 1 and also the existence of positive constants c c d and mathcal h 0 mathcal h 0 left d right such that open image in new window 2

density of rational points on a quadric bundle in mathbb p 3 times mathbb p 3 we establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain zariski dense subset of the biprojective hypersurface x1y12 x4y42 0 in p3 p3 this confirms the modified manin conjecture for this variety in which the removal of a thin set of rational points is allowed

higher order hamilton dynamics and hamilton jacobi divergence pde abstract this paper using a non standard legendrian duality investigates the hamiltonian dynamics and formulates a hamilton jacobi type divergence pde governed by higher order lagrangians

rational points of bounded height on general conic bundle surfaces a conjecture of manin predicts the asymptotic distribution of rational points of bounded height on fano varieties in this paper we use conic bundles to obtain correct lower bounds for a wide class of surfaces over number fields for which the conjecture is still far from being proved for example we obtain the conjectured lower bound of manin s conjecture for any del pezzo surface whose picard rank is sufficiently large or for arbitrary del pezzo surfaces after possibly an extension of the ground field of small degree

local embeddability of cr manifolds into spheres in this paper we solve local cr embeddability problem of smooth cr manifolds into spheres under a certain nondegeneracy condition on the chern moser s curvature tensor we state necessary and sufficient conditions for the existence of cr embeddings as finite number of equations and rank conditions on the chern moser s curvature tensors and their derivatives we also discuss the rigidity of those embeddings

manin s conjecture for quartic del pezzo surfaces with a conic fibration an asymptotic formula is established for the number of q rational points of bounded height on a non singular quartic del pezzo surface with a conic bundle structure

tamagawa numbers of diagonal cubic surfaces numerical evidence a refined version of manin sconjecture aboutthe asymptotics of points of bounded height on fano varieties has been developped by batyrev and the authors we test numerically this refined conjecture for some diagonal cubic surfaces

counting rational points on biquadratic hypersurfaces abstract an asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables the proof uses the hardy littlewood circle method

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

embedding strictly pseudoconvex domains into balls every relatively compact strictly pseudocc nvex domain d with c2 boundary in a stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball however for each n 2 there exist bounded strictly pseudoconvex domains d in cn with real analytic boundary such that no proper holomorphic map from d into any finite dimensional ball extends smoothly to d 0 introduction in this paper we study the representations of bounded strictly pseudoconvex domains d c cn if the boundary of d is of class ck k e 2 3 oo then by a theorem of fornaess 8 and khenkin 13 d can be mapped biholomorphically onto the intersection xn q of a bounded strictly convex domain q c cn with ck boundary and a closed complex submanifold x defirxed in a neighborhood of q in cn x intersecting the boundary of q transversally moreover the map f d x n q extends to a holomorphic map on a neighborhood of d the convex domain q depends on d hence a natural question is whether a similar result holds with q replaced by the unit ball b z z1l zn e cn z 2 e zjl2 n that intersects bbn transversally such that d is biholomorphically equivalent to xn e3ff this question has been mentioned by lempert 14 pinduk 20 bedford 2 and others our main result is that the answer to this question is negative in general if d is as above then every biholomorphism of d onto xne3n extends smoothly to d according to 3 however we will show that not all such domains d admit a proper holomorphic map into a finite dimensional ball that is smooth on d theorem 1 1 a similar local result was obtained independently by faran 7 we shall show that the answer to the question q is positive if we allow the intersections of complex submanifolds with strictly convex domains q c cn with real analytic boundaries theorem 1 2 thetheoremsoffornaess 8 andkhenkin 13 onlygiveanqwith smooth boundary received by the editors april 4t 1985 1980 mathematics subject classification primary 32h05 1research supported by a fellowship from the alfred p sloan foundation t 1986 american mathematical society 0002 9947 86 1 00 25 per page

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

rational points on quartic hypersurfaces let x be a projective non singular quartic hypersurface of dimension 39 or more which is defined over we show that x is non empty provided that x is non empty and x has p adic points for every prime p 2009 walter de gruyter

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

super rigidity for cr embeddings of real hypersurfaces into hyperquadrics we consider holomorphic mappings sending a given levi nondegenerate pseudoconcave hypersurface m in into a nondegenerate hyperquadric of the same signature in and show that if m is sufficiently close to a hyperquadric in a certain sense then any two such mappings differ only by an automorphism of the hyperquadric

maps from the two ball to the three ball in 4 webster proved the following theorem let b z e c 7 2 3 is a proper holomorphic map which is c a up to the boundary of b then f is linear fractional the purpose of this paper is to extend this result to the case n 2 however the theorem is false when n 2 f need not be linear fractional so instead of trying to prove this false theorem we shall instead try to make a list of maps f bz b a which are c 3 up to the boundary of course given such an f we can get another such by composing f with an automorphism of b 2 and an automorphism of b a so we make the

height zeta functions of equivariant compactifications of unipotent groups we prove manin s conjecture for bi equivariant compactifications of unipotent groups 2015 wiley periodicals inc

eisenstein series on loop groups maass selberg relations 3 in this paper we prove the maass selberg relations for the inner product of truncated pseudo eisenstein series one can then extract the maass selberg relations for truncated eisenstein series themselves from those for truncated pseudo eisenstein series a task which we shall complete in a subsequent paper