notes on a conservative nonlinear oscillator the amplitude frequency relationship is an important mathematical property for a nonlinear oscillator he s amplitude frequency formulation and the max min approach are used to handle the conservative nonlinear oscillator email protected email protected 2 x 0 for the amplitude frequency relationship the obtained result is compared with those in the open literature revealing the effectiveness of the used methods

kazhdan projections random walks and ergodic theorems in this paper we investigate generalizations of kazhdan s property t to the setting of uniformly convex banach spaces we explain the interplay between the existence of spectral gaps and that of kazhdan projections our methods employ markov operators associated to a random walk on the group for which we provide new norm estimates and convergence results this construction exhibits useful properties and flexibility and allows to reclassify the existence of kazhdan projections from a c algebraic phenomenon to a natural object associated to random walks on groups we give a number of applications of these results in particular we address several open questions we give a direct comparison of properties te and fe with lafforgue s reinforced banach property t we obtain shrinking target theorems for orbits of kazhdan groups and apply them to a question of kleinbock and margulis finally we construct non compact ghost projections for warped cones answering a question of willett and yu in this last case we conjecture that such warped cones provide new counterexamples to the coarse baum connes conjecture

frequency analysis of strongly nonlinear generalized duffing oscillators using he s frequency amplitude formulation and he s energy balance method in this paper he s frequency amplitude formulation hfaf and he s energy balance method hebm are employed to solve the generalized duffing equation in the form of x x a 3x 3 a 5x 5 a 7x 7 a nx n 0 for any arbitrary power of n a frequency analysis is carried out and the relationship between the natural frequency and the initial amplitude is obtained in analytical form accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the hfaf hebm and exact integration method numerical simulations are extended for even very strong nonlinearities and very good correlations are achieved between the numerical results

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

he s frequency formulation for nonlinear oscillators using a golden mean location he s frequency formulation derived on the basis of an ancient chinese mathematical method is an effective method for treating nonlinear oscillators there are different approaches for choosing location points in the formulation this paper suggests another location point an example is given and the accuracy is improved

pointwise theorems for amenable groups in this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along folner sequences that obey some restrictions these restrictions are mild enough so that such sequences exist for all amenable groups we also prove a generalization of the shannon mcmillan breiman theorem to all discrete amenable groups

application of the modified frequency formulation to a nonlinear oscillator he s frequency formulation is used to obtain the relationship between the frequency and amplitude of a nonlinear oscillator the general approach is to choose two linear oscillators in this paper however one linear oscillator and the duffing oscillator are chosen as trial equations the solution procedure is of utter simplicity while the result is of high accuracy

ergodic properties of linear actions of 2 2 matrices in this paper we consider the ergodic properties of linear actions on the plane of 2 2 matrices with entries in the complex numbers quaternions or clifford numbers 0 introduction in this paper we study ergodic properties of linear actions of discrete groups of 2 2 matrices with determinant 1 consider the usual linear action on the real plane r2 of a discrete subgroup 0 in the 2 2 matrices with real entries sl 2 r given by x1 x2 7 ax1 bx2 cx1 dx2 where a b c d 0 the action by such matrices clearly preserves the usual lebesgue measure on the plane it follows from work of g hedlund 18 that if 0 is a cocompact group or more generally a lattice then this linear action is ergodic this result was extended by f ledrappier in 23 to the case of normal subgroups 0 g 0 for which 0 0 is an infinite abelian subgroup in this note we extend these ergodicity results to discrete subgroups of 2 2 matrices over other fields the simplest case relates to 2 2 matrices with complex entries in theorem 1 we show the ergodicity of the linear actions on the complex plane c2 by discrete cocompact groups 0 and by normal subgroups 0 g 0 for which 0 0 is an infinite abelian subgroup the next most obvious case deals with discrete subgroups of 2 2 matrices where the entries are quaternions where the nonabelian nature of the entries provides additional features of interest in the analysis the corresponding ergodicity result for the linear actions on the quaternionic plane h 2 is described in theorem 2 finally as we shall see there is a very rich general framework into which many of these results fit involving the study of matrices whose entries are if necessary we simply replace 0 by 0 i we can then identify 0 psl 2 r sl 2 r i i with orientation preserving isometries of h2 duke mathematical journal vol 116 no 2 c 2003 received 7 september 2001 revision received 23 january 2002 2000 mathematics subject classification primary 37c85 37c35 secondary 37a45 28d99

stationary measures and equidistribution for orbits of nonabelian semigroups on the torus let be a semigroup of d d nonsingular integer matrices and consider the action of on the torus t we assume throughout that the action is strongly irreducible there is no subtorus invariant under a finite index subsemigroup of the strong irreducibility assumption in particular implies that acts ergodically on t equipped with the lebesgue measure m therefore the orbit of lebesgue almost every x t is dense and in an appropriate sense even becomes equidistributed however when is cyclic there is a set of full hausdorff dimension of exceptional points x for which x fails to be dense when is bigger the distribution of individual orbits can be expected to be much more restrictive an important result in this direction is due to furstenberg who showed for d 1 in which case 1 an abelian semigroup never satisfies condition 1 indeed the group generated by a semigroup satisfying 1 is nonamenable assumption 2 is a technical condition which is in particular satisfied when is a zariski dense semigroup of sld z 16

aperiodic order and spherical diffraction i auto correlation of regular model sets we study uniform and non uniform model sets in arbitrary locally compact second countable lcsc groups which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by meyer and used as mathematical models of quasi crystals we then define a notion of auto correlation for subsets of finite local complexitiy in arbitrary lcsc groups which generalizes hof s classical definition beyond the class of amenable groups and prov ide a formula for the auto correlation of a regular model set along the way we show that the punctured hull of an arbitrary regular model set admits a unique invariant probability measure even in the case where the punctured hull is non compact and the group is non amenable in fact this measure is also the unique stationary measure with respect to any admissible probability measure

an ancient chinese mathematical algorithm and its application to nonlinear oscillators an ancient chinese mathematical method is briefly introduced and its application to nonlinear oscillators is elucidated where he s amplitude frequency formulation is outlined three examples are given to show the extremely simple solution procedure and remarkably accurate solutions

approximate solutions for nonlinear oscillation of a mass attached to a stretched elastic wire in this paper the approximate solutions of the mathematical model of a mass attached to a stretched elastic wire are presented at the beginning of the study the equation of motion is derived in a detailed way he s max min approach he s frequency amplitude method and the parameter expansion method are implemented to solve the established model the numerical results are further compared with the approximate analytical solutions for both a small and large amplitude of oscillations and a very good agreement is observed the relative errors are computed to illustrate the strength of agreement between the numerical and approximate analytical results

the role of the ramanujan conjecture in analytic number theory we discuss progress towards the ramanujan conjecture for the group gln and its relation to various other topics in analytic number theory

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

pointwise ergodic theorems for radial averages on the heisenberg group leth hn cn r denote the heisenberg group and let rdenote the normalized lebesgue measure on the sphere z 0 z r let x b m be a standard borel probability space on whichhacts measurably and ergodically by measure preserving transformations and let r denote the operator canonically associated with ronlp x we prove maximal and pointwise ergodic theorems inlp for radial averages ron the heisenberg grouphn n 1 the results are best possible for actions of the reduced heisenberg group the method of proof is to use the spectral theory of the banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods in particular littlewood paley stein square functions and analytic interpolation

knot invariants from symbolic dynamical systems if g is the group of an oriented knot k then the set hom k j of representations of the commutator subgroup k g g into any finite group e has the structure of a shift of finite type d a special type of dynamical system completely described by a finite directed graph invariants of d such as its topological entropy or the number of its periodic points of a given period determine invariants of the knot when s is abelian r gives information about the infinite cyclic cover and the various branched cyclic covers of k similar techniques are applied to oriented links

diophantine approximation for negatively curved manifolds let m be a geometrically finite pinched negatively curved riemannian manifold with at least one cusp inspired by the theory of diophantine approximation of a real or complex number by rational ones we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it we define a new invariant for m thehurwitz constant of m it measures how well all geodesic lines starting from the cusp are approximated by ones returning to it in the case of constant curvature we express the hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood using the cut locus of the cusp we define an explicit approximation sequence for a geodesic line starting from the cusp and explore its properties we prove that the modular once punctured hyperbolic torus has the minimum hurwitz constant in its moduli space

the affine sieve we establish the main saturation conjecture connected with executing a brun sieve in the setting of an orbit of a group of affine linear transformations this is carried out under the condition that the zariski closure of the group is levi semisimple it is likely that this condition is also necessary for such saturation to hold 2013 american mathematical society

analogs of wiener s ergodic theorems for semisimple lie groups ii let g be a connected finite center simple lie group of real rank one and let k be a maximal compact subgroup of g let a at i t c r be a 1 parameter subgroup of hyperbolic translations such that g ka k is a cartan decomposition denote by ot the probability measure on g given by ot mk 6at mk where mk is the normalized haar measure on k let it denote the probablity measure with density vol bt 1xb g where bt g d gk k t and d denotes the canonical g invariant riemannian metric on g k let x b a be a standard borel probability space on which g acts measurably and ergodically by measure preserving transformations let 7r g iso lp x be the associated isometric representation and let 7r ut and 7r 3t denote the operators canonically associated with ct and it let n g dimr g k we prove maximal and pointwise ergodic theorems for the ball averages it on simple real rank one lie groups which are analogous to n wiener s maximal and pointwise ergodic theorems for ball averages on in the general case of ball averages on semisimple groups is completed in m n s our main result is the following

prime and almost prime integral points on principal homogeneous spaces we develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space in particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors in many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers when the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number and thus establish the zariski density of matrices all of whose entries are prime numbers among the key tools used are explicit approximations to the generalized ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms

counting lattice points for a locally compact second countable group g and a lattice subgroup we give an explicit quantitative solution of the lattice point counting problem in general domains in g provided that i g has finite upper local dimension and the domains satisfy a basic regularity condition ii the mean ergodic theorem for the action of g on g holds with a rate of convergence the error term we establish matches the best current result for balls in symmetric spaces of simple higher rank lie groups but holds in much greater generality a significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap in particular the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property we discuss a number of applications including counting lattice points in general domains in semisimple s algebraic groups counting rational points on group varieties with respect to a height function and quantitative angular or conical equidistribution of lattice points in symmetric spaces and in affine symmetric varieties we note that the mean ergodic theorems which we establish are based on spectral methods including the spectral transfer principle and the kunze stein phenomenon we formulate and prove appropriate analogues of both of these results in the set up of adele groups and they constitute a necessary step in our proof of quantitative results in counting rational points

analysis of a nonlinear oscillator with discontinuity in this paper a nonlinear oscillator with discontinuity is analyzed using he s amplitude frequency formulation the natural frequency of the oscillator system can be obtained using concise and straightforward calculation steps good agreement between the results from the proposed and other classical approaches is reached

analytic continuation of resolvent kernels on noncompact symmetric spaces let x g k be a symmetric space of noncompact type and let be the laplacian associated with a g invariant metric on x we show that the resolvent kernel of admits a holomorphic extension to a riemann surface depending on the rank of the symmetric space this riemann surface is a branched cover of the complex plane with a certain part of the real axis removed it has a branching point at the bottom of the spectrum of it is further shown that this branching point is quadratic if the rank of x is odd and is logarithmic otherwise in case g has only one conjugacy class of cartan subalgebras the resolvent kernel extends to a holomorphic function on a branched cover of with the only branching point being the bottom of the spectrum

he s frequency amplitude formulation for nonlinear oscillators with an irrational force in this paper he s frequency amplitude formulation is applied to determine the periodic solution for a nonlinear oscillator system with an irrational force comparison with the exact solution shows that the result obtained is of high accuracy

affine linear sieve expanders and sum product let mathcal o be an orbit in n of a finitely generated subgroup of gl n whose zariski closure zcl is suitably large e g isomorphic to sl2 we develop a brun combinatorial sieve for estimating the number of points on mathcal o at which a fixed integral polynomial is prime or has few prime factors and discuss applications to classical problems including pythagorean triangles and integral apollonian packings a fundamental role is played by the expansion property of the congruence graphs that we associate with mathcal o this expansion property is established when zcl sl2 using crucially sum product theorem in q for q square free

variational iteration method some recent results and new interpretations this paper is an elementary introduction to the concepts of variational iteration method first the main concepts in variational iteration method such as general lagrange multiplier restricted variation correction functional are explained heuristically subsequently the solution procedure is systematically addressed in particular for nonlinear oscillators particular attention is paid throughout the paper to give an intuitive grasp for the method the main motivation is to put things together in a convenient form for later reference and systematic use

uniform pointwise bounds for matrix coefficients of unitary representations and applications to kazhdan constants let k be a local field and g the group of k rational points of a connected reductive linear algebraic group over k with k semisimple rank g 2 let k be a good maximal compact subgroup of g for a unitary representation of g a vector v in is called kfinite if the subspace spanned bykv is finite dimensional we will use the termk matrix coefficients resp k finite matrix coefficients of to refer to its matrix coefficients with respect to k invariant resp k finite unit vectors following bt we denote by g the subgroup generated by the unipotent k split subgroups of g the main goal of the present paper is to construct a class of uniform pointwise bounds for the k finite matrix coefficients of all infinite dimensional irreducible unitary representations of g or more generally of all unitary representations of g without a non zero g invariant vector let a be a maximal k split torus and a the closed positive weyl chamber of a such that the cartan decomposition g ka k holds where is a finite subset of the

regularized theta lifts for orthogonal groups over totally real fields we define a regularized theta lift from sl2 to orthogonal groups over totally real fields it takes harmonic whittaker forms to automorphic green functions and weakly holomorphic whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors generalizing borcherds work on automorphic products to prove our results we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of kudla and millson

resolvent and lattice points on symmetric spaces of strictly negative curvature we study the asymptotics of the lattice point counting function n x y r gamma in gamma d x gamma y for a riemannian symmetric space x obtained from a semisimple lie group of real rank one and a discontinuous group gamma of motions in x such that gamma backslash x has finite volume we show that as r rightarrow infty for each varepsilon 0 the constant 2 rho corresponds to the sum of the positive roots of the lie group associated to x and n dimx the sum in the main term runs over a system of orthonormal eigenfunctions varphi j in l 2 gamma backslash x of the laplacian such that the eigenvalues rho 2 nu j 2 are less than 4n rho 2 n 1 2

new applications of variational iteration method abstract the variational iteration method is used for solving three types of nonlinear partial differential equations such as coupled schrodinger kdv generalized kdv and shallow water equations the exact and numerical solutions obtained by variational iteration method are compared with that obtained using adomian decomposition method the comparison shows that the two solutions obtained are excellent agreement the method is made showing that the former is more effective than the latter in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

approximate periodic solutions for the helmholtz duffing equation approximate periodic solutions for the helmholtz duffing oscillator are obtained in this paper he s energy balance method hebm and he s frequency amplitude formulation hfaf are adopted as the solution methods oscillation natural frequencies are analytically analyzed error analysis is carried out and accuracy of the solution methods is evaluated

diophantine equations and ergodic theorems let x be a probability measure space and t 1 t n be a family of commuting measure preserving invertible transformations on x let q m 1 m n be a homogeneous positive polynomial with integer coefficients and let s m e z n q m a denote the set of integer solutions m m 1 m n of the diophantine equation q m a we prove that under a certain nondegeneracy condition on the polynomial q m and an ergodic condition on the family of transformations t t 1 t n the images of the solution sets q x t m1 1 t m2 2 t m n n x m s become uniformly distributed on x w r t for a e x e x as that is the pointwise ergodic theorem holds when the standard averages are replaced by the ones where the exponents satisfy a diophantine equation the proof uses a variant of the hardy littlewood method of exponential sums developed by birch and davenport and techniques from harmonic analysis a key point is the corresponding maximal theorem which is a discrete analogue of a maximal theorem on r n corresponding to the level surfaces of the polynomial q x

flows on homogeneous spaces and diophantine approximation on manifolds we present a new approach to metric diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces this approach yields a new proof of a conjecture of mahler originally settled by v g sprindzuk in 1964 we also prove several related hypotheses of baker and sprindzuk formulated in 1970s the core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices

flows on homogeneous spaces and diophantine properties of matrices we generalize the notions of badly approximable resp singular systems of m linear forms in n variables and relate these generalizations to certain bounded resp divergent trajectories in the space of lattices in rm n

harmonic analysis and pointwise ergodic theorems for noncommuting transformations let fk denote the free group on k generators 1 where s n w i n and 1 denotes the word length on fk induced by s let x m m be a probability space on which fk acts ergodically by measure preserving transformations we prove a pointwise ergodic theorem for the sequence of operators ij i a a n acting on l 2 x namely a f x fx fdm almost everywhere for each f in l 2 x we also show that the sequence a2n converges to a conditional expectation operator with re spect to a a algebra which is invariant under fk the proof is based on the spectral theory of the commutative convolution subalgebra of e i fk gener ated by the elements an n 0 we then generalize the discussion to algebras arising as a gelfand pair associated with the group of automorphisms g ri r2 of a semi homogeneous tree t r r2 where r 2 r2 2 r r2 4 this content downloaded from 207 46 13 114 on thu 26 may 2016 06 27 50 utc all use subject to http about jstor org terms

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov