unbounded solutions of the equation asymptotic behavior of solutions of first order differential equation with deviating arguments in the email protected t email protected email protected i t y email protected i y email protected i is discussed for t a criterion for representing solutions in exponential form is proved inequalities for solution estimation are given sufficient conditions for the existence of unbounded solutions are derived a relevant illustrative example is given as well known results are discussed and compared

the autocorrelation function of a pseudointegrable system abstract the autocorrelation function of a pseudointegrable system is considered the system consists of billiards on plane surface formed out of three squares arranged in an l shape this system has the important property of being constructed from copies of an integrable subsystem the single square the motion can be decomposed into a continuous and a discrete part the unpredictability in the system being associated with the latter a discrete autocorrelation function is calculated and its decay properties investigated structure found in this autocorrelation function is associated with the continued fraction expansion of the ratio of velocity components for repeating continued fractions such as the golden mean the autocorrelation function exhibits a selfsimilar structure for the general case of a randomly chosen velocity ratio we derive the time dependence of the number of occurences of large autocorrelation values which differs from the behavior in integrable and chaotic systems

qualitative properties of systems of two complex homogeneous ode s a connection to polygonal billiards a correspondence between the orbits of a system of two complex homogeneous polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed this correspondence is general in the sense that it applies to an open set of systems of ordinary differential equations of the specified kind this allows us to transfer results well known from the theory of polygonal billiards such as ergodicity the existence of periodic orbits the absence of exponential divergence the existence of additional conservation laws and the presence of discontinuities in the dynamics to the corresponding systems of ordinary differential equations it also shows that the considerable intricacy known to exist for polygonal billiards also attends these apparently simpler systems of ordinary differential equations

level statistics and eigenfunctions of square torus billiards manifesting the transition from regular to chaotic behaviors we thoroughly analyze the level statistics and eigenfunctions in concentric as well as nonconcentric square torus billiards we confirm the characteristics of quantum and classical correspondence and the existence of scarred and superscarred modes in concentric square torus billiards furthermore we not only verify that the transition from regular to chaotic behaviors can be manifested in nonconcentric square torus billiards but also develop an analytical distribution to excellently fit the numerical level statistics finally we intriguingly observe that numerous eigenstates commonly exhibit the wave patterns to be an ensemble of classical diamond trajectories as the effective wavelengths are considerably shorter than the size of internal hole

a global code invariant under the higman sims group in this paper we consider certain doubly transitive permutation groups which have monomial representations which are reducible these groups have been studied by hale and shult 14 and by taylor i7 1 the stabilizer h of a vertex has a subgroup l of index two for which elements of l are not conjugate in g to elements of h l the monomial representations have two invariant subspaces which can be reduced modulo any prime to form codes over finite fields the invariant subspaces form global codes this approach is motivated by the work of ward 1 181 who considered the groups rslf2 q and sp 2n 4 the codes for psl 2 q give the classical extended quadratic residue codes in section 2 we construct the monomial representations and determine the two invariant subspaces taylor constructed 2 graphs from the centralizer algebra of these representations the invariant subspaces are spanned by vectors whose entries are parameters in these z graphs we illustrate the construction in the remaining sections with the doubiv transitive permutation representation on 176 vertices of the iiigman sims group this permutation representation was first discovered by graham higman the stabilizer of a vertex is a group h isomorphic to ptu 3 5 it

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

stability and asymptotic properties of a system of functional differential equations with nonconstant delays the asymptotic properties of a real two dimensional differential system x t a t x t k 1 mb k t x q k t h t x t x q 1 t x q m t with unbounded nonconstant delays t q k t 0 satisfying lim t q k t are studied here a b k and h are supposed to be matrix functions and a vector function the conditions for the stability and asymptotic stability of solutions and the conditions under which all solutions tend to zero are given the methods are based on the transformation of the considered real system to one equation with complex valued coefficients asymptotic properties are studied by means of a lyapunov krasovskii functional the results generalize some previous ones where the asymptotic properties for two dimensional systems with one or more constant delays or one nonconstant delay were studied

local instability of orbits in polygonal and polyhedral billiards we classify when local instability of orbits of closeby points can occur for billiards in two dimensional polygons for billiards inside three dimensional polyhedra and for geodesic flows on surfaces of three dimensional polyhedra we sharpen a theorem of boldrighini keane and marchetti we show that polygonal and polyhedral billiards have zero topological entropy we also prove that billiards in polygons are positive expansive when restricted to the set of non periodic points the methods used are elementary geometry and symbolic dynamics

asymptotic convergence criteria of solutions of delayed functional differential equations abstract the convergence of all solutions of the scalar equation x f t x t is studied under the main assumption that every constant is its solution the criterion and the sufficient conditions for convergence are proved for the considered classes of equations it is among others proved that all solutions are convergent if and only if there exists at least one increasing and convergent solution as applications criteria for some classes of linear equations as well as for nonlinear equations are obtained some comparisons with already known results are given too

non periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons this paper proves the existence of non periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons for anyn 3 there exists a corresponding convexn agon forn 3 this will be a right triangle with a small acute angle while in three dimensional space it will be a prism then agon serving as the base

convergence of the solutions of the equation y t β t y t δ y t τ in the critical case we study the asymptotic behavior of the solutions of the first order differential equation containing two delays y t t y t y t with t0 r 0 the convergence of all solutions is characterized by the existence of a strictly increasing bounded solution a critical case is found for the coefficient function for coefficients below the critical function a strictly increasing and bounded solution is constructed and thus the convergence of all solutions is shown relations with known results are discussed too

asymptotic representation of solutions of equationẏ t β t y t y t τ t abstract in this paper the equation t t y t y t t is considered where c i 1 r i 1 t 1 r 0 t t is an increasing function on i 1 t t i 1 0 c i 1 r results about asymptotic structure and behavior of the solutions of this equation are proved a connection with the equation t c t x t t and the asymptotic representation of its solutions is obtained

billiards in polygons survey of recent results in the ten or so years since the publication of ref 19 polygonal billiards have remained an active subject of research in the mathematics and physics literature as a result our understanding of the subject although still far from being complete is much better than it was ten years ago this survey attempts to give a broad overview of the dynamics of billiards in polygons with an emphasis on the material that was not in the literature in 1985 this is not a survey of the publications on polygonal billiards and we apologize to authors whose work has not been included the selection of topics has been strongly influenced by the personal taste of the author and by space limitations thus we do not discuss quantum polygonal billiards in particular quantum chaos see e g refs 41 44 and 45 we hope that somebody will write a survey on this important subject in the near future a fascinating aspect of the subject is the interplay between the geometric shape of the billiard table i e a planar curve and the qualitative features of the billiard dynamics hence it is instructive to compare polygonal billiard tables with other classes of billiard tables in particular the smooth strictly convex tables and the dispersing or sinai tables produce strikingly different types of billiard dynamics we recommend refs 46 and 15 for a general overview of the subject and a comparative study of billiard dynamics for various types of billiard tables

dual polygonal billiards and necklace dynamics we study the orbits of the dual billiard map about a polygonal table using the technique of necklace dynamics our main result is that for a certain class of tables called the quasi rational polygons the dual billiard orbits are bounded this implies that for the subset of rational tables i e polygons with rational vertices the dual billiard orbits are periodic

unbounded increasing solutions of a system of difference equations with delays we consider a homogeneous system of difference equations with deviating arguments in the form y n k 1 q k n y n p k y n r k where n n 0 n 0 z p k r k are integers r k p k 0 q is a positive integer y y 1 y s t y n 0 r n 0 r 1 r s is an unknown discrete vector function s 1 is an integer r max r 1 r q y n y n 1 y n and k n ij k n i j 1 s are real matrices such that ij k n 0 n 0 1 0 and k 1 q j 1 s ij k n 0 for each admissible i and all n n 0 the behavior of solutions of this system is discussed for n the existence of unbounded increasing solutions in an exponential form is proved and estimates of solutions are given the scalar case is discussed as well

hausdorff dimension and conformal dynamics iii computation of dimension this paper presents an eigenvalue algorithm for accurately computing the hausdorff dimension of limit sets of kleinian groups and julia sets of rational maps the algorithm is applied to schottky groups quadratic polynomials and blaschke products yielding both numerical and theoretical results dimension graphs are presented for a the family of fuchsian groups generated by reflections in 3 symmetric geodesics b the family of polynomials f c z z 2 c c 1 1 2 and c the family of rational maps f t z z t 1 z t 0 1 we also calculate h dim 1 305688 for the apollonian gasket and h dim j f 1 3934 for douady s rabbit where f z z 2 c satisfies f 3 0 0

correlations and spectra of triangular billiards abstract we consider the motion of a point particle in right triangular billiards for which acute angles are not rationally connected to we consider in particular the behaviour of correlations for a particular phase space variable the behaviour is compatible with the mixing character of the dynamics even though large fluctuations are exhibited by fourier transforming correlation functions we may as well analyse scaling properties of the spectral measure both dynamical and spectral features are quite similar to what observed in a class of quantum kicked systems

exponential solutions of equation ẏ t β t y t δ y t τ abstract asymptotic behaviour of solutions of first order differential equation with two deviating arguments of the form y t t y t y t is discussed for t a criterion for representing solutions in exponential form is proved as consequences inequalities for such solutions are given connections with known results are discussed and a sufficient condition for existence of unbounded solutions generalizing previous ones is derived an illustrative example is considered too

on riddling and weak attractors abstract we propose general definitions for riddling and partial riddling of a subset v of r m with non zero lebesgue measure and show that these properties are invariant for a large class of dynamical systems we introduce the concept of a weak attractor a weaker notion than a milnor attractor and use this to re examine and classify riddled basins of attractors we find that basins of attraction can be partially riddled but if this is the case then any partial riddling must be evident near the attractor we use these concepts to aid classification of bifurcations of attractors from invariant subspaces in particular our weak attractor is a generalisation of the absorbing area investigated by other authors and we suggest that a transition of a basin to riddling is usually associated with loss of stability of a weak attractor

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

asymptotic convergence of the solutions of a discrete system with delays abstract a system of s discrete equations y n n y n j y n k is considered where k and j are integers k j 0 n is a real s s square matrix defined for n n 0 k n 0 z with non negative elements ij n i j 1 s such that j 1 s ij n 0 y y 1 y 2 y s t n 0 k n 0 k 1 r s and y n y n 1 y n for n n 0 a method of auxiliary inequalities is used to prove that every solution of the given system is asymptotically convergent under some conditions i e for every solution y n defined for all sufficiently large n there exists a finite limit lim n y n moreover it is proved that the asymptotic convergence of all solutions is equivalent to the existence of one asymptotically convergent solution with increasing coordinates some discussion related to the so called critical case known for scalar equations is given as well

periodic orbit theory and spectral rigidity in pseudointegrable systems we calculate numerically the periodic orbits of pseudointegrable systems of low genus numbers g that arise from rectangular systems with one or two salient corners from the periodic orbits we calculate the spectral rigidity delta3 l using semiclassical quantum mechanics with l reaching up to quite large values we find that the diagonal approximation is applicable when averaging over a suitable energy interval comparing systems of various shapes we find that our results agree well with delta3 calculated directly from the eigenvalues by spectral statistics therefore additional terms such as e g diffraction terms seem to be small in the case of the systems investigated in this work by reducing the size of the corners the spectral statistics of our pseudointegrable systems approaches that of an integrable system whereas very large differences between integrable and pseudointegrable systems occur when the salient corners are large both types of behavior can be well understood by the properties of the periodic orbits in the system

vector stabilizers and subgroups of leech lattice groups the leech lattice is a 2bdimensional lattice whose automo hism group is the double cover 2 co of conway s group co see 3 p 1801 from it the other conway groups co and co mclaughlin s group mcl and the higman sims group hs may be obtained by fixing certain sets of vectors the lattice played a central role in the determination of the maximal subgroups of co see 10 and co l see 1 i the crucial observation is that if a group fixes a non zero vector in a rational representation then it fixes a non zero vector in any reduction of this representation modulo 2 one of the aims of the present paper is to generalise this result and put it in its proper context in modular representation theory the other main aim is to apply this result to other leech lattice groups and so give a uniform classification of the maximal non local subgroups of mcl and co originally determined by finkelstein in s and hs originally determined by magliveras in s and co see lo as well as the automorphism groups mcl 2 of mcl and hs 2 of hs originally determined in 127 in section a we study the geometry of the leech lattice in detail section b contains the representation theory we need in section c to determine the maximal non local subgroups of the various groups finally in section d we show how to obtain various graphs from the leech lattice and in particular deduce that the groups we are studying really are isomorphic to mcl hs and u 2 it should perhaps be stressed that few if any of the results in this paper are new although a number of them have not appeared in print before my intention has been rather to present a unified picture of the relation between the subgroup structure of the various groups involved and the geometric structure of the leech lattice

exponential energy growth in a fermi accelerator an unbounded energy growth of particles bouncing off two dimensional 2d smoothly oscillating polygons is observed notably such billiards have zero lyapunov exponents in the static case for a special 2d polygon geometry a rectangle with a vertically oscillating horizontal bar we show that this energy growth is not only unbounded but also exponential in time for the energy averaged over an ensemble of initial conditions we derive an a priori expression for the rate of the exponential growth as a function of the geometry and the ensemble type we demonstrate numerically that the ensemble averaged energy indeed grows exponentially at a close to the analytically predicted rate namely the process is controllable

fusion systems and amalgams we study reduced fusion systems from the point of view of their essential subgroups using the classification by goldschmidt and fan of amalgams of prime index to analyze certain pairs of such subgroups our results are applied here to study reduced fusion systems over 2 groups of order at most 64 and also reduced fusion systems over 2 groups having abelian subgroups of index two more applications are given in later papers

some designs and binary codes preserved by the simple group ru of rudvalis abstract the binary codes c n where n 1755 1756 2304 2305 4059 defined by the rank 3 primitive permutation representation of degree 4060 of the simple group ru of rudvalis on the cosets of the ree group 2 f 4 2 are examined these codes are obtained from the row span of the incidence matrices of the neighbourhood designs of graphs r r and s defined by the union of the orbits of 2 f 4 2 we prove that dim c 1756 29 dim c 2304 28 c 1756 c 2304 and ru acts irreducibly on c 2304 furthermore we have c 1755 c 2305 c 4059 v 4060 f 2 aut d 1755 aut d 1756 aut d 2304 aut d 2305 aut c 1756 aut c 2304 ru while aut d 4059 aut c 1755 aut c 2305 aut c 4059 s 4060 we also determine the weight distribution of c 1756 and c 2304 and that of their duals for each word w l of weight l in the codes c 1756 or c 2304 we determine the stabilizer ru w l and some primitive designs held by particular codewords

dimension and measures for a curvilinear sierpinski gasket or apollonian packing in this paper we apply some results about general conformal iterated function systems to a the residual set of a standard apollonian packing or a curvilinear sierpinski gasket within this context it is straightforward to show that h the hausdorff dimension of a is greater than 1 and the packing dimension and the upper and lower box counting dimensions are all the same as the hausdorff dimension among other things we verify sullivan s result that 0 h h a p h a

level statistics and eigenfunctions of pseudointegrable systems dependence on energy and genus number we study the level statistics second half moment i 0 and rigidity ensuremath delta 3 and the eigenfunctions of pseudointegrable systems with rough boundaries of different genus numbers g we find that the levels form energy intervals with a characteristic behavior of the level statistics and the eigenfunctions in each interval at low enough energies the boundary roughness is not resolved and accordingly the eigenfunctions are quite regular functions and the level statistics shows poisson like behavior at higher energies the level statistics of most systems moves from poisson like toward wigner like behavior with increasing g on investigating the wave functions we find many chaotic functions that can be described as a random superposition of regular wave functions the amplitude distribution p ensuremath psi of these chaotic functions was found to be gaussian with the typical value of the localization volume v mathrm loc ensuremath approx 0 33 for systems with periodic boundaries we find several additional energy regimes where i 0 is relatively close to the poisson limit in these regimes the eigenfunctions are either regular or localized functions where p ensuremath psi is close to the distribution of a sine or cosine function in the first case and strongly peaked in the second case an interesting intermediate case between chaotic and localized eigenfunctions also appears

periodic orbits and spectral statistics of pseudointegrable billiards we demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than l increases as frac ensuremath pi b 0 l 2 a l where b 0 is a constant and a l is the average area occupied by these families we also find that a l increases with l before saturating finally we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies

on the ground state of regular polygonal billiards the helmholtz equation for regular polygons is investigated by means of the conformal mapping from the circle which provides an expansion parameter for the approximate evaluation of the lowest eigenvalue and the corresponding eigenvector

asymptotic convergence of the solutions of a discrete equation with several delays abstract a discrete equation y n i 1 s i n y n j i y n k i with integer delays k i and j i k i j i 0 is considered it is assumed that i z n 0 k 0 n 0 z k max k 1 k 2 k s z p p p 1 p z i 1 s i n 0 s z 1 is a fixed integer and n z n 0 criteria of asymptotic convergence of solutions are expressed in terms of inequalities for the functions i i 1 s some general properties of solutions are derived as well it is e g proved that for the asymptotical convergence of all solutions the existence of a strictly monotone and asymptotically convergent solution is sufficient a crucial role in the analysis of convergence is played by an auxiliary inequality derived from the form of a given discrete equation

dynamics of kaleidoscopic maps abstract first we introduce a certain class of piecewise affine elliptic rotation maps on t 2 called the kaleidoscopic maps and describe its importance and then we concentrate our efforts on a special case when the rotation angle of a kaleidoscopic map is n n z for the special case we answer the conjectures regarding the periodicity and the singularity structure of such kaleidoscopic dynamics in the process we prove the partial riddling of the regular orbits that gives rise to the classification of periodic sets and estimate the hausdorff dimension of the singular set finally we study the dynamics of such kaleidoscopic maps restricted within the singular set and answer conjectures concerning the chaos the local chaos and the ergodicity with respect to the normalized hausdorff measure of the singular set

some designs and codes invariant under the simple group co 2 we examine some designs di and associated binary codes ci constructed from a primitive permutation representation of degree 2300 of the sporadic simple group co2

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

velocity dependent invariants for hard particles in one dimension the relative motion of n impenetrable mass points in one dimension is shown to be equivalent to a billiard in a simplex of n 1 dimension the orientation of the confining hyperplanes being fixed by the mass ratios the discrete group generated by reflections in these planes and its closure in so n 1 are used to show that the only invariant functions of the particle velocities are total momentum and total energy if n 3 and at least one mass is different from the others

the asymptotic bounds of solutions of linear delay systems abstract our aim is to obtain conditions under which every solutionyof the vector functional differential equation formula can be estimated in terms of a solution of a scalar functional equation formula wherea x is a suitable real positive function andbis a suitable real negative constant moreover ifb x is diagonal matrix we derive more precise estimates ofyin terms of a solution of a vector functional equation formula where the elements ofa x are formed by the absolute values of the corresponding elements ofa x these asymptotic results enable us to obtain sharp estimates of solutions of certain linear delay systems

the alperin and dade conjectures for ree groups2f4 q2 in non defining characteristics abstract this paper is part of a program to study alperin s weight conjecture and dade s ordinary conjecture on counting characters in blocks for several finite groups the local structures of radical subgroups and certain radical 3 chains of a ree group of typef4are given and the conjectures and a conjecture of olsson are confirmed for the group when the characteristic of the modular representation is distinct from the defining characteristic of the group

a rational billiard flow is uniquely ergodic in almost every direction on demontre qu un flot de billard rationnel est uniquement ergodique dans presque chaque direction

convergence in asymptotically autonomous functional differential equations in this paper we consider linear and nonlinear perturbations of a linear autonomous functional differential equation which has infinitely many equilibria we give sufficient conditions under which the solutions of the perturbed equation tend to the equilibria of the unperturbed equation at infinity as a consequence we obtain sufficient conditions for systems of delay differential equations to have asymptotic equilibrium

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces