two classes of exactly solvable quantum models with moving boundaries it is shown here that the non stationary schrodinger equation can be solved exactly for two quantum models subject to dirichlet boundary conditions one of them is a modified problem of a quantum bouncer i e the problem of a particle falling down in the gravitational field on a moving oscillating platform such as a loudspeaker the second model is a cuff off oscillator with a moving infinite potential wall and a time dependent frequency in both cases exact solutions are given in closed forms easy to use their possible applications are also indicated in each of the models extra coordinate and time dependent phase factors are generated by moving boundaries in the former case giving rise to a non local effect in quantum mechanics

the dynamics of maps tangent to the identity and with nonvanishing index let f be a germ of a holomorphic self map of c 2 at the origin o tangent to the identity and with o as a nondicritical isolated fixed point a parabolic curve for f is a holomorphic f invariant curve with o on the boundary attracted by o under the action of f it has been shown by m abate 2001 that if the characteristic direction v p t o c 2 has residual index not belonging to q then there exist parabolic curves for f tangent to v in this paper we prove using a different method that the conclusion still holds just assuming that the residual index is not vanishing at least when f is regular along v

keeping track of limit cycles this article uses analytic geometry methods to bound the number of limit cycles of bounded period for a planar polynomial vector fields it introduces the notion of limit periodic sets and shows that the hilbert s 16th problem can be reduced to the existence of a uniform bound on its cyclicity it also shows the existence of a universal bound on the number of limit cycles in the perturbation of a linear focus as a consequence of the noetherian property of the bautin ideal

a classification of taut stein surfaces with a proper r action we present a classification of 2 dimensional taut stein manifolds with a proper mathbb r action for such manifolds the globalization with respect to the induced local mathbb c action turns out to be stein as an application we determine all 2 dimensional taut non complete hartogs domains over a riemann surface

topological properties of self similar fractals with one parameter abstract in this paper we study two classes of planar self similar fractals t e with a shifting parameter e the first one is a class of self similar tiles by shifting x coordinates of some digits we give a detailed discussion on the disk likeness i e the property of being a topological disk in terms of e we also prove that t e determines a quasi periodic tiling if and only if e is rational the second one is a class of self similar sets by shifting diagonal digits we give a necessary and sufficient condition for t e to be connected

approximation of biholomorphic mappings by automorphisms of cn in this paper we prove several results on approximation of biholomorphic mappings between domains in c n 2 by holomorphic automorphisms of c recall that the group of holomorphic automorphisms of c denoted aut c consists of those holomorphic mappings c c that have a holomorphic inverse 4 1 c c with the topology of uniform convergence on compact sets aut c is a topological group while the group aut c 1 of automorphisms of the complex plane consists only of the linear mappings az b a b e c a 4 0 the group aut c is very large and complicated when n 2 let us choose a linear coordinate system on c and write the coordinates as z z w where z z1 zn i c c n 1 and w z c then autc n contains mappings of the form

disklikeness of planar self affine tiles we consider the disklikeness of the planar self affine tile t generated by an integral expanding matrix a and a consecutive collinear digit set v 0 v 2v q 1 v z 2 let f x x 2 px q be the characteristic polynomial of a we show that the tile t is disklike if and only if 2 p g 2 moreover t is a hexagonal tile for all the cases except when p 0 in which case t is a square tile the proof depends on certain special devices to count the numbers of nodal points and neighbors of t and a criterion of bandt and wang 2001 on disklikeness

a characterization of connected self affine fractals arising from collinear digits abstract let a be an expanding integer matrix with characteristic polynomial f x x 2 p x q and let d 0 1 q 2 q m v be a collinear digit set where m 0 v z 2 it is well known that there exists a unique self affine fractal t satisfying a t t d in this paper we give a complete characterization of connectedness of t that generalizes the previous result for q 3

embeddings of submanifolds and normal bundles this paper studies the embeddings of a complex submanifold s inside a complex manifold m in particular we are interested in comparing the embedding of s in m with the embedding of s as the zero section in the total space of the normal bundle ns of s in m we explicitly describe some cohomological classes allowing to measure the difference between the two embeddings in the spirit of the work by grauert griffiths and camacho movasati and sad we are also able to explain the geometrical meaning of the separate vanishing of these classes our results hold for any codimension but even for curves in a surface we generalize previous results due to laufert and camacho movasati and sad

analytical and formal classifications of quasi homogeneous foliations in c2 0 we prove a result of classification for germs of formal and convergent quasi homogeneous foliations in c 2 with fixed separatrix basically we prove that the analytical and formal class of such a foliation depend respectively only on the analytical and formal class of its representation of projective holonomy

residual indices of holomorphic maps relative to singular curves of fixed points on surfaces let m be a two dimensional complex manifold and let f m to m be a holomorphic map that fixes pointwise a possibly singular compact reduced and globally irreducible curve c subset m we give a notion of degeneracy of f at a point of c it turns out that f is non degenerate at one point if and only if it is non degenerate at every point of c when f is non degenerate on c we define a residual index for f at each point of c then we prove that the sum of the indices is equal to the self intersection number of c

topological invariants and equidesingularization for holomorphic vector fields on etudie l invariance du nombre de milnor et ses applications on demontre un theoreme d equidesingularisation

exact time dependence of solutions to the time dependent schrodinger equation solutions of the schrodinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time independent operators using time dependent unitary transformations exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time dependent hamiltonians in d dimensions including non hermitean symmetric hamiltonians hamiltonians are constructed using time dependent unitary spatial transformations comprising dilatations translations and rotations and solutions are found in several forms as eigenfunctions of a quadratic invariant as coherent state eigenfunctions of boson operators as plane wave solutions from which the general solution is obtained as an integral transform by means of the fourier transform and as distributional solutions for which the initial wavefunction is the dirac function for the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d 1 including barut girardello and perelomov coherent states i e vector coherent states which are shown to be related to eigenfunctions of the quadratic invariant by the transformation this transformation which leaves the ermakov equation invariant implements su 1 1 transformations on linear dynamical invariants coherent states are derived also for the time dependent linear potential exact solutions are found for hamiltonians with electromagnetic interactions in which the time dependent magnetic and electric fields are not necessarily spatially uniform as an example it is shown how to find exact solutions of the time dependent schrodinger equation for the dirac magnetic monopole in the presence of time dependent magnetic and electric fields of a specified form

germs of holomorphic vector fields in c3 without a separatrix oblatum 28 xii 1988 26 iii 1990 10 x 1990 13 vi 1991 30 ix 1991 supported by universite de toulouse conacyt cnrs and caicyt 87 336 fulltext preview

remarks on solving the one dimensional time dependent schrodinger equation on the interval the case of a quantum bouncer it is shown that the 1d hamiltonian which is a sum of operators which generate a finite nilpotent lie algebra and depends explicitly on time existing closed form solutions of the time dependent schrodinger equation cannot fulfil in general boundary and normalization conditions on a positive semi axis an explanation of the controversy surrounding the solutions of the quantum bouncer model which appeared recently in the literature is given

suppression of period doubling in the dynamics of a bouncing ball abstract we present a theoretical and experimental study of a bouncing ball system where the effect of a near resonant perturbation is to suppress the onset of the first period doubling bifurcation near the bifurcation point the full dynamical equations are reduced to a discrete time map governing the dynamics on a slowly oscillating center manifold the derivation emphasizes the geometry of the phase space dynamics and serves to clarify several points of a recently proposed theory regarding the effects of strong near resonant perturbations our experimental results agree with the simple theory for moderate perturbations for larger perturbations higher order effects must be included

polynomial flows in the plane soit f r n r un champ vectoriel c 1 et on considere le systeme d equations differentielles x f x avec x 0 x 0 la solution x t x 0 depend de t et x 0 pour n 2 sur r et c on donne le f tel que le flot depende polynomialement de la condition initiale x 0

polynomial completion of symplectic jets and surfaces containing involutive lines motivated by work of dragt and abell on accelerator physics we study the completion of symplectic jets by polynomial maps of low degrees we use andersen lempert theory to prove that symplectic completions always exist and we prove the degree bound conjectured by dragt and abell in the physically relevant cases however we disprove the degree bound for 3 jets in dimension 4 this follows from the fact that if sigma is the disjoint union of r 7 involutive lines in mathbb p 3 then sigma is contained in a degree d 4 hypersurface i e the restriction morphism iota h 0 mathbb p 3 mathcal o mathbb p 3 4 rightarrow h 0 sigma mathcal o sigma 4 has a nontrivial kernel todd we give two new proofs of this fact and finally we show that if r d ne 7 4 then the map iota has maximal rank

topological structure of a class of planar self affine sets abstract let t t a d be a disk like z 2 tile generated by an expanding 2 2 matrix a and a digit set d z 2 we study the subset f of t defined by a f f d where d d is a sub digit set by studying a periodic extension h f z 2 we classify f into three types according to their topological properties which generalizes a result of lau et al 13 we also provide some simple criteria for such classification

connectedness of a class of two dimensional self affine tiles associated with triangular matrices abstract we study the connectedness of planar self affine sets t a d generated by a matrix of the form a p 0 a q together with nonconsecutive and noncollinear digit sets of the form d l 0 l 1 l p 1 m 0 m 1 m q 1 where l 0 l 1 l p 1 and m 0 m 1 m q 1 are residue systems for p and q respectively we give a necessary and sufficient condition for t a d to be connected and extend some results by deng and lau 2011 5 to nonconsecutive digit sets

darboux integrating factors inverse problems abstract we discuss planar polynomial vector fields with prescribed darboux integrating factors in a nondegenerate affine geometric setting we establish a reduction principle which transfers the problem to polynomial solutions of certain meromorphic linear systems and show that the space of vector fields with a given integrating factor modulo a subspace of explicitly known standard vector fields has finite dimension for several classes of examples we determine this space explicitly

connectedness of planar self affine sets associated with non consecutive collinear digit sets abstract in the paper we focus on the connectedness of planar self affine sets t a d generated by an integer expanding matrix a with det a 3 and a collinear digit set d 0 1 b v where b 1 and v r 2 such that v a v is linearly independent we discuss the domain of the digit b to determine the connectedness of t a d especially a complete characterization is obtained when we restrict b to be an integer some results on the general case of det a 3 are obtained as well

invariant curves and topological invariants for real plane analytic vector fields abstract let z be a germ of a singular real analytic vector field at o r 2 we give conditions on the multiplicity and the milnor number of z which imply that the foliation defined by z has a characteristic orbit or an analytic invariant curve with the hypothesis that z is a real generalized curve then it is proved that for a non dicritical real generalized curve the multiplicity mod 2 is invariant under bilipshitz homeomorphisms preserving foliations

extendable cohomologies for complex analytic varieties we introduce a cohomology called extendable cohomology for abstract complex singular varieties based on suitable differential forms beside a study of the general properties of such a cohomology we show that given a complex vector bundle one can compute its topological chern classes using the extendable chern classes defined via a chern weil type theory we also prove that the localizations of the extendable chern classes represent the localizations of the respective topological chern classes thus obtaining an abstract residue theorem for compact singular complex analytic varieties as an application of our theory we prove a camacho sad type index theorem for holomorphic foliations of singular complex varieties

the generalized harmonic oscillator and the infinite square well with a moving boundary for the one dimensional generalized harmonic oscillator we obtain in this paper its wavefunctions in closed form by means of two independent methods which are based respectively in i the algebraic properties of the dynamical symmetry of the system and ii the construction of an invariant operator the total equivalence of these two formulations is shown and quantal properties are described in terms of a classical solution of the equations of motion two possible reductions for the system exist the static harmonic oscillator and the free particle in the latter case the quantum system becomes a fermi oscillator or equivalently it can describe a free particle in a well with one moving boundary which in turn follows certain classical rules the time dependent boundary conditions in the well play the role of an effective interaction acting on the particle the formalism is shown to be compatible with the gauge principle of minimal coupling and several different gauges are constructed and analysed

self affine tiles in ℝn abstract a self affine tile in r n is a set t of positive measure with a t d d t d where a is an expanding n n real matrix with det a m an integer and d d d 2 d m r n is a set of m digits it is known that self affine tiles always give tilings of r n by translation this paper extends known characterizations of digit sets d yielding self affine tiles it proves several results about the structure of tilings of r n possible using such tiles and gives examples showing the possible relations between self replicating tilings and general tilings which clarify results of kenyon on self replicating tilings

index theorems for holomorphic self maps let m be a complex manifold and s subset m a possibly singular subvariety of m let f colon m to m be a holomorphic map such that f restricted to s is the identity we show that one can associate to f a holomorphic section x f of a sheaf related to the embedding of s in m and that such a section reads the dynamical behavior of f along s in particular we prove that under generic hypotheses the canonical section x f induces a holomorphic action in the sense of bott on the normal bundle of the regular part of s in m and this allows to obtain for holomorphic self maps with non isolated fixed points index theorems similar to camacho sad baum bott and variation index theorems for holomorphic foliations finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self maps of surfaces with a compact curve of fixed points

globalizations fiber bundles and envelopes of holomorphy we describe a relationship between globalizations of local holomorphic actions on stein manifolds induced by global actions of certain non compact lie groups and holomorphic fiber bundles with smooth stein base and fiber and connected structure group to this end we prove a univalence result for particular stein riemann domains with a free and properly discontinuous action of a discrete group of biholomorphisms we then derive some consequences on the existence of stein globalizations

on the solvability of the bouncer model two new exact solutions for the bouncer model are found and their relations to some solvable and non solvable lie algebras are shown

polynomially convex hulls of singular real manifolds we obtain local and global results on polynomially convex hulls of lagrangian and totally real submanifolds of c n with self intersections and open whitney umbrella points

on the solution of hill s equation using milne s method milne s method is applied to the hill equation i e the schrodinger equation for a periodic one dimensional potential the method allows a compact discussion of the organization and parameter dependence of the stability bands as well as an efficient numerical computation of the band edges as an example a class of potentials which show a transition from two to one minimum per period is studied numerically furthermore periodic solutions of the milne equation are discussed and constructed

a new notion of tameness abstract we generalize the notion of tame discrete sets introduced by rosay and rudin from complex euclidean space to arbitrary complex manifolds and establish their basic properties we show that complex linear algebraic groups different from the complex line or the punctured complex line contain tame discrete sets

meromorphic vector fields with single valued solutions on complex surfaces abstract we study ordinary differential equations in the complex domain given by meromorphic vector fields on compact complex kahler surfaces we prove that if such an equation has a maximal single valued solution in particular an entire one whose image is not contained in a proper analytic subset then up to a bimeromorphic transformation either the vector field is holomorphic or it preserves a fibration

complete algebraic vector fields on affine surfaces part ii in this work we study the singularities at infinity of algebraic vector fields in dimension 2 these singularities will be classified under a mild assumption the general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in part ii our main results are local and therefore can be carried over more general surfaces whereas we deal with complete vector fields the results also apply to complete ones thanks to a theorem of forstneric 7

self similar measures and their fourier transforms ii a self similar measure on r n was defined by hutchinson to be a probability measure satisfying formule here s j x j r j x b j is a contractive similarity 0 j 1 r j orthogonal and the weights a j satisfy 0 a j 1 j 1 m a j 1 by analogy we define a self similar distribution by the same identity but allowing the weights a j to be arbitrary complex numbers we give necessary and sufficient conditions for the existence of a solution to among distributions of compact support and show that the space of such solutions is always finite dimensional if f denotes the fourier transformation of a self similar distribution of compact support let formule where is defined by the equation j 1 m j a j 2 1

local levi flat hypersurfaces invariants by a codimension one holomorphic foliation in this paper we study codimension one holomorphic foliations leaving invariant real analytic hypersurfaces in particular we prove that a germ of real analytic levi flat hypersurface with sufficiently small singular set is given by the zeroes of the imaginary part of a holomorphic function

basic digit sets for radix representation abstract the use of a negative base did not appear until the 1950s when several authors independently introduced the concept complement representation also became much discussed in this period as an alternative to sign magnitude for designing the arithmetic unit of a computer the arithmetic of numbers represented in positional notation has a firm foundation derived from the theory of polynomial arithmetic that readily allows these extensions to negative bases and or negative digit values complement representation and digit values in excess of the base our primary concern in this paper is the characterization and computation of those integral valued base and digit set pairs that provide complete and unique finite radix representation of the integers

on the poincare problem abstract we derive some restrictions on the possible degrees of algebraic invariant curves and on the possible form of algebraic integrating factors for plane polynomial vector fields whose stationary points at infinity satisfy a certain genericity condition the method is elementary and we show by example that it also yields strong results for certain non generic vector fields

holomorphic dynamics near germs of singular curves let m be a two dimensional complex manifold p m and open image in new window a germ of holomorphic foliation of m at p letopen image in new window be a germ of an irreducible possibly singular curve at p in m which is a separatrix for open image in new window we prove that if the camacho sad suwa index ind open image in new window then there exists another separatrix for open image in new window at p a similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points

invariant curves and analytic integrability of a planar vector field abstract we give an expression of the irreducible invariant curves at the singular point for analytically integrable systems we provide an expression of its primitive first integral this fact allows us to obtain necessary conditions of analytic integrability at degenerate singular points

the connectedness of some two dimensional self affine sets abstract in the paper we mainly discuss the connectedness of two kinds of self affine sets one is generated by matrix a p 0 a q and digit set d i s j t t i 0 1 q 1 j 0 1 p 1 where s t 0 and p q z with 3 p 1 q 2 p 1 the other is generated by matrix a p 0 a q and digit set d i s d i j t t i 0 1 p 1 j 0 1 q 1 where s t 0 and p q d z with p q 2 the sufficient or necessary conditions for their connectedness are revealed

connectedness of a class of planar self affine tiles we consider a class of planar self affine tiles t that are generated by the lower triangular expanding matrices and the product form digit sets we give necessary and sufficient conditions for t to be connected and disk like also for the disconnect case we give a condition that enumerates the number of connected components of t

rigidity of planar tilings aperturbation of a tiling of a region inr n is a set of isometries one applied to each tile so that the images of the tiles tile the same region

the quantum square well with moving boundaries a numerical analysis we present some numerical discussions concerning the infinite square well in one dimension with moving boundaries our results show that if the speed of displacement is small objects of physical relevance like probability density averaged position or mean value of the energy have a smooth behavior on the contrary if this speed becomes large many irregularities arise which has a difficult qualitative explanation these irregularities manifest themselves as sharp bumps on the probability distribution or a chaotic shape on the averaged values of position and energy none of these patterns is the result of numerical errors and therefore we conclude that an unknown and very nontrivial effect is produced at high speeds of the moving wall

holomorphic flexibility properties of complex manifolds we obtain results on approximation of holomorphic maps by algebraic maps the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds and the homotopy principle for holomorphic submersions of stein manifolds to certain algebraic manifolds

liouvillian integration and bernoulli foliations analytic foliations in the 2 dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable it is shown that such a condition leads to the existence of a liouville type first integral and under generic extra conditions it is proven that these foliations can be defined by bernoulli equations

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

on invariance of plurigenera for foliations on surfaces let x be a smooth algebraic surface a foliation f on x is roughly speaking a subline bundle t f of the tangent bundle of x the dual of t f is called the canonical bundle of the foliation k f in the last few years birational methods have been successfully used in order to study foliations more precisely geometric properties of the foliation are translated into properties of the canonical bundle of the foliation one of the most important invariants describing the properties of a line bundle l is its kodaira dimension kappa l which measures the growth of the global sections of l and its tensor powers the kodaira dimension of a foliation f is defined as the kodaira dimension of its canonical bundle kappa k f in their fundamental works brunella and mcquillan give a classfication of foliations on surfaces on the model of enriques kodaira classification of surfaces the next step is the study of the behaviour of families of foliations brunella proves that for a family of foliations x t f t of dimension one on surfaces satisfying certain hypotheses of regularity the kodaira dimension of the foliation does not depend on t by analogy with siu s invariance of plurigenera it is natural to ask whether for a family of foliations x t f t the dimensions of global sections of the canonical bundle and its powers depend on t in this talk we will discuss to which extent an invariance of plurigenera for foliations is true and under which hypotheses on the family of foliations it holds