nodal surface conjectures for the convex quantum billiard stemming from known properties of one dimensional 1 d and 2 d quantum billiards it is conjectured that the nodal surface of the first excited state of the convex 3 d quantum billiard intersects the billiard surface in a single simple closed curve examples of the validity of this conjecture are given for a number of elementary 3 d billiard configurations from these examples a second conjecture is introduced that addresses convex quantum billiards which are figures of rotation and contain one and only one plane of mirror symmetry normal to the axis of rotation two characteristic displacement parameters are defined which are labeled an axis length l and diameter a it is conjectured that a parameter 1 exists whose exact value depends on the properties of the billiard such that for l a prolatelike the nodal surface of the first excited state of a quantum billiard is a plane surface of mirror symmetry which divides the length of the billiard in half for l a oblatelike the nodal surface of the first excited state is a plane surface of mirror symmetry which contains the rotation axis and divides the diameter of the billiard in half arguments are given in support of a third conjecture which addresses the regular polyhedra quantum billiards termed spherical like it is hypothesized that the nodal surface of the first excited state for any of these billiards is any plane of reflection symmetry of the given polyhedron

metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

regularization methods for solving differential algebraic equations in m knorrenschild siam j numer anal 29 1992 1694 1715 a class of regularization methods for solving autonomous index 1 daes differential algebraic equations has been proposed in this paper we generalize it to non autonomous index 1 and higher index daes we will treat a daes as an infinitely stiff odes ordinary differential equations therefore the solutions of a daes can be considered as the limit of the solutions of the corresponding stiff odes in some sense some similar results with m knorrenschild 1992 loc cit are obtained

reducing index method for differential algebraic equations with constraint singularities this paper extends an earlier work appl math comput 140 2003 77 to differential algebraic equations with constraint singularities numerical solution of these problems is considered by pseudospectral method with domain decomposition and by giving a condition under which the general linear form problem can easily be transformed to the index reduced form by a simple formulation furthermore with providing some examples the aforementioned cases are dealt with numerically

a modified spectral method for numerical solution of ordinary differential equations with non analytic solution in this article there is an attempt to introduce generally spectral methods for numerical solution of ordinary differential equations also to focus on those problems in which some coefficient functions or solution function is not analytic then by expressing weak and strong aspects of spectral methods to solve this kind of problems a modified spectral method which is more efficient than other spectral methods is suggested furthermore with providing some examples the aforementioned cases are dealt with numerically

adomian decomposition method for solution of differential algebraic equations solutions of differential algebraic equations is considered by adomian decomposition method in e babolian m m hosseini reducing index and spectral methods for differential algebraic equations j appl math comput 140 2003 77 and m m hosseini an index reduction method for linear hessenberg systems j appl math comput in press an efficient technique to reduce index of semi explicit differential algebraic equations has been presented in this paper adomian decomposition method is applied to reduced index problems the scheme is tested for some examples and the results demonstrate reliability and efficiency of the proposed methods

circular sector quantum billiard and allied configurations the circular sector quantum billiard problem is studied numerical evaluation of the zeros of first order bessel functions finds that there is an abrupt change in the nodal line structure of the first excited state of the system equivalently second eigenstate of the laplacian at the critical sector angle c 0 354 for sector angle 0 in the domain 0 0 c the nodal curve of the first excited state is a circular arc segment for c 0 the nodal curve of the first excited state is the bisector of the sector otherwise nondegenerate first excited states become twofold degenerate at the critical angle c the ground and first excited state energies eg e1 increase monotonically as 0 decreases from its maximum value a graph of e1 vs 0 reveals an inflection point at 0 c which is attributed to the change in bessel function contribution to the development of e1 a proof is given for the existence of a common zero for two bessel functions whose respective orders differ by a noninteger applicat

function mixing hypothesis and quantum chaos abstract billiards are studied whose boundaries comprise two or more segments which allow local but not global separation of the helmholtz equation related local solutions are labeled sep functions such billiards comprise the and sets a definition of quantum chaos for these sets of billiards is presented based on the ratio of the fluctuation length of the wave function nodal pattern to the c diameter of the billiard the function mixing hypothesis states that a sufficient condition for a billiard to be chaotic is that the billiard be an element of one of these sets it further ascribes such chaotic behavior to be due to mixing of dissimilar sep functions examples of the application of this hypothesis are described a set theoretic formalism is introduced to describe perturbation theory for infinite potentials and applied to the concave sided square billiard of sufficiently small concavity it is concluded that the adiabatic theorem of quantum mechanics does not apply to this configuration in the limit of large quantum numbers

reducing index and pseudospectral methods for differential algebraic equations in this paper numerical solution of linear differential algebraic equations is considered by pseudospectral method also an index reduction technique for them is suggested for hessenberg index 2 system the index reduction technique is defined and its use is demonstrated we give a condition under which the general linear form problem can be easily transformed to the index reduced form by a simple formulation furthermore with providing some examples the aforementioned cases are dealt with numerically

minimal surfaces and harmonic functions in the heisenberg group abstract we study the blow up of h perimeter minimizing sets in the heisenberg group h n n 2 we show that the lipschitz approximations rescaled by the square root of excess converge to a limit function assuming a stronger notion of local minimality we prove that this limit function is harmonic for the kohn laplacian in a lower dimensional heisenberg group

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

definitions of quasiconformality summary we establish that the infinitesimal h definition for quasiconforreal mappings on carnot groups implies global quasisymmetry and hence the absolute continuity on almost all lines our method is new even in r where we obtain that the limsup condition in the h definition can be replaced by a liminf condition this leads to a new removability result for quasi conformal mappings in euclidean spaces an application to parametrizations of chord arc surfaces is also given

the polygon quantum billiard problem the present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates the polygon ground state theorem is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the elemental polygons defined in the text the polygon first excited state theorem is established which states that for every n sided regular polygon n equivalent first excited states exist each of which contains a nodal curve that is a line of mirror symmetry of the related polygon a vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the n sided regular polygon is spanned by any two of these n nodal diagonal eigenstates i e the first excited state is twofold degenerate at various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations thus for example correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive nonoverlapping classes of polygons stemming from the preceding conclusions a discussion is included on the possibility of employing transverse magnetic tm modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic

area stationary and stable surfaces of class c1 in the sub riemannian heisenberg group h1 abstract we consider surfaces of class c 1 in the 3 dimensional sub riemannian heisenberg group h 1 assuming the surface is area stationary i e a critical point of the sub riemannian perimeter under compactly supported variations we show that its regular part is foliated by horizontal straight lines in case the surface is complete and oriented without singular points and stable i e a second order minimum of perimeter we prove that the surface must be a vertical plane this implies the following bernstein type result a complete locally area minimizing intrinsic graph of a c 1 function in h 1 is a vertical plane

sequential regularization methods for higher index daes with constraint singularities the linear index 2 case standard stabilization techniques for higher index differential algebraic equations daes often involve elimination of the algebraic solution components this may not work well if there are singularity points where the constraint jacobian matrix becomes rank deficient this paper proposes instead a sequential regularization method srm a functional iteration procedure for solving problems with isolated singularities which have smooth differential solution components for linear index 2 daes we consider both initial value problems ivps and boundary value problems bvps the convergence of the srm is described and proved in detail we believe that this is the first convergence proof for any method for daes with this type of constraint singularities moreover the regularization parameter in our method is not necessarily very small so the srm is an important improvement over usual regularization methods various aspects of the subsequent numerical discretization of the regularized problems are discussed as well and some numerical verifications are carried out

singular solutions homogeneous norms and quasiconformal mappings in carnot groups in any carnot nilpotent stratified lie group g of homogeneous dimension q green s function u for the q laplace equation exists and is unique we prove that there exists a constant gamma gamma g so that n e gamma u is a homogeneous norm in g then the extremal lengths of spherical ring domains measured with respect to n can be computed and used to give estimates for the extremal lengths of ring domains measured with respect to the carnot caratheodory metric applications include regularity properties of quasiconformal mappings and a geometric characterization of bi lipschitz mappings

sub riemannian calculus on hypersurfaces in carnot groups we develop a sub riemannian calculus for hypersurfaces in graded nilpotent lie groups we introduce an appropriate geometric framework such as horizontal levi civita connection second fundamental form and horizontal laplace beltrami operator we analyze the relevant minimal surfaces and prove some basic integration by parts formulas using the latter we establish general first and second variation formulas for the horizontal perimeter in the heisenberg group such formulas play a fundamental role in the sub riemannian bernstein problem

adomian decomposition method for solution of nonlinear differential algebraic equations in m m hosseini adomian decomposition method with chebyshev polynomials appl math comput in press an efficient modification of the adomian decomposition method was presented by using chebyshev polynomials also in m m hosseini adomian decomposition method for solution of differential algebraic equations j comput appl math in press solution of linear differential algebraic equations was considered by adomian decomposition method in this paper standard and modified adomian decomposition methods are applied to non linear differential algebraic equations the schemes are tested for some examples and the results demonstrate reliability and efficiency of the proposed methods

hardy littlewood inequality for quasiregular maps on carnot groups abstract we generalize a hardy littlewood inequality for l p norms of conjugate harmonic functions to horizontal components of quasiregular mappings on carnot groups

an index reduction method for linear hessenberg systems in e babolian m m hosseini reducing index and pseudospectral methods for differential algebraic equations appl math comput 140 2003 77 90 a reducing index method has been proposed for some cases of semi explicit daes differential algebraic equations in this paper this method is generalized to more cases also it is focused on hessenberg index 2 systems and proposed reduction index method will be illustrated for this problem the hessenberg system and its obtained reduced index system are numerically solved through pseudospectral method in addition aforementioned methods will be considered by one example

ode methods for the solution of differential algebraic systems in this paper we study the numerical solution of the differential algebraic systems f t y y 0 many of these systems can be solved conveniently and economically using a range of ode methods others can be solved only by a small subset of ode methods and still others present insurmountable difficulty for all current ode methods we examine the first two groups of problems and indicate which methods we believe to be best for them then we explore the properties of the third group which cause the methods to fail we describe a reduction technique which allows systems to be reduced to ones that can be solved it also provides a tool for the analytical study of the structure of systems

quasilinear equations with source terms on carnot groups in this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of lane emden type with measure data on a carnot group g of arbitrary step the quasilinear part involves operators of the p laplacian type g p 1 p these results are based on new a priori estimates of solutions in terms of nonlinear potentials of th wolff s type as a consequence we characterize completely removable singularities and prove a liouville type theorem for supersolutions of quasilinear equations with source terms which has been known only for equations involving the sub laplacian p 2 on the heisenberg group