efficient discrete approximations of quantum gates quantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set it has been shown that this is possible for almost all choices of base sets and furthermore that the number of gates required for precision e is only polynomial in log 1 e here we prove that using certain sets of base gates quantum compiling requires a string length that is linear in log 1 e a result which matches the lower bound from counting volume up to constant factor

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

on the spectral gap for finitely generated subgroups of su 2 we prove the spectral gap property for free subgroups of su 2 generated by elements satisfying a noncommutative diophantine property in particular for free subgroups generated by elements with algebraic entries

random walks in the group of euclidean isometries and self similar measures we study products of random isometries acting on euclidean space building on previous work of the second author we prove a local limit theorem for balls of shrinking radius with exponential speed under the assumption that a markov operator associated to the rotation component of the isometries has spectral gap we also prove that certain self similar measures are absolutely continuous with smooth densities these families of self similar measures give higher dimensional analogues of bernoulli convolutions on which absolute continuity can be established for contraction ratios in an open set

exact synthesis of single qubit unitaries over clifford cyclotomic gate sets we generalize an efficient exact synthesis algorithm for single qubit unitaries over the clifford t gate set which was presented by kliuchnikov maslov and mosca quantum inf comput 13 7 8 607 630 2013 their algorithm takes as input an exactly synthesizable single qubit unitary one which can be expressed without error as a product of clifford and t gates and outputs a sequence of gates which implements it the algorithm is optimal in the sense that the length of the sequence measured by the number of t gates is smallest possible in this paper for each positive even integer n we consider the clifford cyclotomic gate set consisting of the clifford group plus a z rotation by n we present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of n z rotations for the clifford t case n 4 the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring z e i n 1 2 we prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100

on measure preserving mathcal c1 transformations of compact open subsets of non archimedean local fields we introduce the notion of a locally scaling transformation defined on a compactopen subset of a non archimedean local field we show that this class encompasses the haar measure preserving transformations defined by c in particular polynomial maps and prove a structure theorem for locally scaling transformations we use the theory of polynomial approximation on compact open subsets of non archimedean local fields to demonstrate the existence of ergodic markov and mixing markov transformations defined by such polynomial maps we also give simple sufficient conditions on the mahler expansion of a continuous map zp zp for it to define a bernoulli transformation

geometric location of periodic points of 2 ramified power series abstract in this paper we study the geometric location of periodic points of power series defined over fields of positive characteristic p we find a lower bound for the norm of all nonzero periodic points in the open unit disk of 2 ramified power series we prove that this bound is optimal for a large class of power series our main technical result is a computation of the first significant terms of p power iterates of 2 ramified power series as a by product we obtain a self contained proof of the characterization of 2 ramified power series

on 2 generator subgroups of so 3 we classify all subgroups of so 3 that are generated by two elements each a rotation of finite order about axes separated by an angle that is a rational multiple of 7r in all cases we give a presentation of the subgroup in most cases the subgroup is the free product or the amalgamated free product of cyclic groups or dihedral groups the relations between the generators are all simple consequences of standard facts about rotations by 7r and 7r 2 embedded in the subgroups are explicit free groups on 2 generators as used in the banach tarski paradox

asymptotic equivalence of neutral and infinite retarded differential equations staffan a demontre qu une equation differentielle fonctionnelle fde neutre avec d operateur stable peut souvent etre traitee comme une fde a retard avec retardement infini le but principal de l article est d illustrer comment les techniques de staffans peuvent etre employees pour etudier le comportement asymptotique de solutions de certaines fde neutres non lineaires et non autonomes en particulier on considere une equation neutre scalaire de la forme d dt x t cx t r f t x t x t r

asymptotic constancy for pseudo monotone dynamical systems on function spaces abstract a pseudo monotone dynamical system is a dynamical system which preserves the order relation between initial points and equilibrium points the purpose of this paper is to present some convergence oscillation and order stability criteria for pseudo monotone dynamical systems on function spaces for which each constant function is an equilibrium point some applications to neutral functional differential equations and semilinear parabolic partial differential equations with neumann boundary condition are given

p adic dynamics the quadratic map overp adic numbers is studied in some detail we prove that near almost all indifferent fixed points it is topologically conjugate to a quasiperiodic linear map we also establish the existence of chaotic behavior and describe it using symbolic dynamics

asymptotic equivalence of differential equations and asymptotically almost periodic solutions abstract in this paper we use rab s lemma m rab uber lineare perturbationen eines systems von linearen differentialgleichungen czechoslovak math j 83 1958 222 229 m rab note sur les formules asymptotiques pour les solutions d un systeme d equations differentielles lineaires czechoslovak math j 91 1966 127 129 to obtain new sufficient conditions for the asymptotic equivalence of linear and quasilinear systems of ordinary differential equations yakubovich s result v v nemytskii v v stepanov qualitative theory of differential equations princeton university press princeton new jersey 1966 v a yakubovich on the asymptotic behavior of systems of differential equations mat sb 28 1951 217 240 on the asymptotic equivalence of a linear and a quasilinear system is developed on the basis of the equivalence the existence of asymptotically almost periodic solutions of the systems is investigated the definitions of biasymptotic equivalence for the equations and biasymptotically almost periodic solutions are introduced theorems on the sufficient conditions for the systems to be biasymptotically equivalent and for the existence of biasymptotically almost periodic solutions are obtained appropriate examples are constructed

asymptotic behavior of solutions for a class of difference equations abstract consider the difference equation x n x n 1 f x n g x n k eqno 1 wherekis a positive integer fandgare continuous andfis increasing onr we show that whenf y g y fory r every solution of 1 tends to either a constant or asn while whenf y g y fory r every solution of 1 tends to either a constant or asn in particular iff y g y fory r then every solution of 1 tends to a constant asn

on the chaotic behavior of a generalized logistic p adic dynamical system abstract in the paper we describe basin of attraction p adic dynamical system g x a x 2 x 1 moreover we also describe the siegel discs of the system since the structure of the orbits of the system is related to the geometry of the p adic siegel discs

compact actions whose orbit equivalence relations are not profinite abstract let x be a measure preserving action of a countable group on a standard probability space x we prove that if the action x is not profinite and satisfies a certain spectral gap condition then there does not exist a countable to one borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action i e an inverse limit of actions on countable sets as a consequence we show that if is a countable dense subgroup of a compact non profinite group g such that the left translation action g has spectral gap then g is antimodular and not orbit equivalent to any not necessarily free profinite action this provides the first such examples of compact actions partially answering a question of kechris and answering a question of tsankov

uniform expansion bounds for cayley graphs of sl2 fp we prove that cayley graphs of sl2 fp are expanders with respect to the projection of any fixed elements in sl 2 z generating a non elementary subgroup and with respect to generators chosen at random in sl2 fp

convergence of a class of discrete time semiflows with application to neutral delay differential equations abstract in this paper we introduce a class of pseudo monotone maps on ordered topological spaces by exploiting monotonicity methods and the invariance of the omega limit set we establish a convergence principle for discrete time semiflows generated by the maps introduced the convergence principle is then applied to a class of periodic neutral delay differential equations which leads to some novel and sharper results

convergence of solutions for a class of neutral difference equations abstract consider the neutral difference equation x n cx n k f n x n x n k where k is a positive integer 0 c y n y n y n 1 f n r 2 r is continuous and decreasing with respect to the second argument we give the results that solutions of the equation convergent to constants

asymptotic behavior of solutions of nonautonomous neutral dynamical systems abstract this paper studies the dynamics of families of monotone nonautonomous neutral functional differential equations with nonautonomous operator of great importance for their applications to the study of the long term behavior of the trajectories of problems described by this kind of equations such us compartmental systems and neural networks among many others precisely more general admissible initial conditions are included in the study to show that the solutions are asymptotically of the same type as the coefficients of the neutral and non neutral part

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

quaquaversal tilings and rotations we construct a hierarchical tiling of 3 dimensional euclidean space based on a triangular prism using repeated rotations about orthogonal axes by angles 2 m and 2 n to analyze the structure of the tiling we are led to determine the group g m n generated by such a pair of rotations for m n 3 and for m 3 n 4

geometry of p adic siegel discs abstract we survey recent advances in the study of regular motions over p adic fields show its varied connections with dynamics and number theory and illustrate its significance to an important class of discrete dynamical systems we also show that mappings supporting quasi periodic motions can be naturally interpreted as flows with p adic time

the general exact bijective continuous solution of feigenbaum s functional equation solutionsf of feigenbaum s functional equationf f x 1 f x where 0 is a fixed real number account for many of the fascinating properties of the behaviour of successive iterates of one parameter families of nonlinear maps in connection with the phenomenon of intermittency interesting families of exact solutions have recently been found for 0 these solutions can all be derived from continuous bijective solutions which are topologically equivalent to translations in this paper thegeneral exact continuous bijective solution is found for any 0 positive or negative in particular it is shown that forany 0 there are solutions which areinequivalent to translations and it is shown that bijective solutions equivalent to translations exist only when 0 1 these results considerably enlarge the stock of available exact solutions of feigenbaum s equation

on the diameter of permutation groups given a finite group g and a set a of generators the diameter diam g g a of the cayley graph g g a is the smallest l such that every element of g can be expressed as a word of length at most l in a a 1 we are concerned with bounding diam g max a diam g g a it has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n but the best previously known upper bound was exponential in nlogn v we give a quasipolynomial upper bound namely diam g exp o logn 4 loglogn exp loglog g o 1 for g sym n or g alt n where the implied constants are absolute this addresses a key open case of babai s conjecture on diameters of simple groups by a result of babai and seress 1992 our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n

arithmetic exponents in piecewise affine planar maps abstract we consider the growth of some indicators of arithmetical complexity of rational orbits of piecewise affine maps of the plane with rational parameters the exponential growth rates are expressed by a set of exponents one exponent describes the growth rate of the so called logarithmic height of the points of an orbit while the others describe the growth rate of the size of such points measured with respect to the p adic metric here p is any prime number which divides the parameters of the map we show that almost all the points in a domain of linearity such as an elliptic island in an area preserving map have the same set of exponents we also show that the convergence of the p adic exponents may be non uniform with arbitrarily large fluctuations occurring arbitrarily close to any point we explore numerically the behaviour of these quantities in the chaotic regions in both area preserving and dissipative systems in the former case we conjecture that wherever the lyapunov exponent is zero the arithmetical exponents achieve a local maximum

a spectral gap theorem in simple lie groups we establish the spectral gap property for dense subgroups generated by algebraic elements in any compact simple lie group generalizing earlier results of bourgain and gamburd for unitary groups

arithmetical properties of strongly chaotic motions abstract the orbits of the generalized arnold sinai cat maps or hyperbolic automorphisms of the two dimensional torus typify purely chaotic anosov motion we transform the dynamics of the periodic orbits of these maps into modular arithmetic in suitable domains of quadratic integers classify all periodic orbits and show how to determine their periods and initial conditions the methods are based on ideal theory in quadratic fields which is reviewed it is shown that the structure of orbits rests upon some basic arithmetical notions such as unique factorization into prime ideals

subgroups ofso 3 associated with tilings abstract we give a thorough analysis of those subgroups ofso 3 generated by rotations about perpendicular axes by 2 pand 2 q a corollary is that such a group is the free product of the cyclic groups of rotations about the separate axes if and only ifp q 3 and are both odd these groups are naturally associated with a family of hierarchical tilings of euclidean 3 space

weak containment rigidity for distal actions abstract we prove that if a measure distal action of a countable group is weakly contained in a strongly ergodic probability measure preserving action of then is a factor of in particular this applies when is a compact action as a consequence we show that the weak equivalence class of any strongly ergodic action completely remembers the weak isomorphism class of the maximal distal factor arising in the furstenberg zimmer structure theorem

convolution of discrete measures on linear groups abstract let p q r such that 1 p 2 and 2 p 1 1 q define f p max x g 1 y g 1 f x y p 1 p where g 1 is taken in some class of subgroups specified later we prove the following two theorems about convolutions theorem 2 let g sl 2 c equipped with the discrete topology then there is a constant p 0 such that for f l p g f f q 1 2 c f p 1 f p where the maximum in is taken over all abelian subgroups g 1 g and x g theorem 3 there is a constant c c p 0 and 1 p 0 such that if f l p sl 3 z then f f q 1 2 c f p 1 f p where the maximum in is taken over all nilpotent subgroups g 1 of sl 3 z and x sl 3 z

growth and generation in mathrm sl _2 mathbb z p mathbb z we show that every subset of sl2 z pz grows rapidly when it acts on itself by the group operation it follows readily that for every set of generators a of sl2 z pz every element of sl2 z pz can be expressed as a product of at most o log p c elements of a a 1 where c and the implied constant are absolute

growth in solvable subgroups of mathrm gl _r mathbb z p mathbb z let k mathbb z p mathbb z and let a be a subset of mathrm gl r k such that langle a rangle is solvable we reduce the study of the growth of a under the group operation to the nilpotent setting fix a positive number c ge 1 we prove that either a grows meaning a 3 ge c a or else there are groups u r and s with u r unlhd s unlhd langle a rangle such that s u r is nilpotent a k cap s is large and u r subseteq a k where k depends only on the rank r of mathrm gl r k here a k x 1 x 2 cdots x k x i in a cup a 1 cup 1 when combined with recent work by pyber and szabo the main result of this paper implies that it is possible to draw the same conclusions without supposing that langle a rangle is solvable

universal small scale structure near the boundary of siegel disks of arbitrary rotation number abstract we present numerical evidence that for analytic maps of the complex plane with a siegel disk of any irrational rotation number and a quadratic critical point on its boundary the small scale structure near the critical point can be described asymptotically by a universal two parameter family we propose an explanation in terms of an ergodic attractor for a renormalisation operator

a neutral fde with stable d operator is retarded abstract a neutral functional differential equation with a linear autonomous and stable d operator can be rewritten as a retarded functional differential equation with infinite delay this fact can be used e g in the study of the stability of the neutral equation and to simplify the proofs of some standard asymptotic results

convergence theorems and asymptotic integration for functional differential equations abstract this paper is devoted to the investigation on convergence of solutions and asymptotic integration of some functional differential equations fdes in the first part of the paper some convergence criteria are obtained and an invariance principle is established for the convergence of solutions of linear fdes under perturbations and disturbances with l 1 integrable coefficients these theorems are applicable to some nonlinear or nonhomogeneous equations in the second part the convergence theorems are applied to investigate the asymptotic integration of some linear fdes this research is made mainly to discuss a situation that was avoided by some previous literature some examples are given to illustrate the theorems

asymptotic behavior of solutions for a class of nonlinear difference equations abstract in this paper we study the asymptotic behavior of solutions for a class of nonlinear difference equations and give the results about the solutions of the equations convergent to a constant our results generalize all conclusions obtained in 1

rigidity of warped cones and coarse geometry of expanders abstract we study the geometry of warped cones over free minimal isometric group actions and related constructions of expander graphs we prove a rigidity theorem for the coarse geometry of such warped cones namely if a group has no abelian factors then two such warped cones are quasi isometric if and only if the actions are finite covers of conjugate actions as a consequence we produce continuous families of non quasi isometric expanders and superexpanders the proof relies on the use of coarse topology for warped cones such as a computation of their coarse fundamental groups

canonical heights transfinite diameters and polynomial dynamics let z be a polynomial of degree at least 2 with coefficients in a number field k iterating gives rise to a dynamical system and a corresponding canonical height function as defined by call and silverman we prove a simple product formula relating the transfinite diameters of the filled julia sets of over various completions of k and we apply this formula to give a generalization of bilu s equidistribution theorem for sequences of points whose canonical heights tend to zero

ideal orbits of toral automorphisms abstract ideal orbits of toral automorphisms are the simplest and arguably the most interesting among the periodic orbits of fully chaotic maps we show that for an ideal orbit the coordinate and momentum evolve independently and that their dynamical evolution is determined by multiplication by a fixed integer modulo a prime their spectrum is found to be the product of a regular and an irregular function the former is computed explicitly while the latter is a sum of dirichlet characters whose irregular nature is the source of spectral fluctuations we evaluate some statistical properties of these sums and estimate the average spectral density of the phase coordinates we show how to select toral automorphisms and initial conditions corresponding to ideal orbits of arbitrarily large period having certain prescribed properties these results may be relevant to the problem of generating pseudorandom sequences

cocycle superrigidity for translation actions of product groups let g be either a profinite or a connected compact group and be finitely generated dense subgroups assuming that the left translation action of on g is strongly ergodic we prove that any cocycle for the left right translation action of on g with values in a countable group is virtually cohomologous to a group homomorphism moreover we prove that the same holds if g is a not necessarily compact connected simple lie group provided that contains an infinite cyclic subgroup with compact closure we derive several applications to oe and w superrigidity in particular we obtain the first examples of compact actions of f2 f2 which are w superrigid

additive cellular automata and global injectivity abstract we employ algebraic methods to investigate the global injectivity of additive cellular automata when the state alphabet is a finite commutative ring the set of local rules of such additive cellular automata in d 1 dimensions itself forms a ring the invertible elements in this ring correspond to the globally injective additive cellular automata we indicate how one determines the invertible elements and thus the globally injectibe cellular automata in particular we show that in contrast to the general situation the global injectivity of additive cellular automata in d 1 dimensions is decidable construction of local rules for the inverse cellular automata is illustrated through examples a strict upper bound on the radius of the inverse cellular automata is obtained in the special case of linear cellular automata with state alphabet the ring of integers modulo m we count the globally injective cellular automata of up to a given radius we show that in one dimension a non globally injective additive cellular automaton is either not injective on spatially periodic configurations of any period n n or the set of n 0 such that the cellular automation is injective on spatially periodic configurations of period n has positive density in n

the size of quadratic p adic linearization disks we find the exact radius of linearization disks at indifferent fixed points of quadratic maps in cp we also show that the radius is invariant under power series perturbations localizing all periodic orbits of these quadratic like maps we then show that periodic points are not the only obstruction for linearization in so doing we provide the first known examples in the dynamics of polynomials over cp where the boundary of the linearization disk does not contain any periodic point

asymptotic behavior of linear impulsive integro differential equations asymptotic equilibria of linear integro differential equations and asymptotic relations between solutions of linear homogeneous impulsive differential equations and those of linear integro differential equations are established a new gronwall bellman type lemma for integro differential inequalities is proved an example is given to demonstrate the validity of one of the results

quantum mechanics on p adic fields a formulation of quantum mechanics on p adic number fields is presented quantum amplitudes are taken as complex functions of p adic variables and it is shown how the weyl approach to quantum mechanics can be generalized to the p adic case the p adic analogs of simple one dimensional systems free particle compact and noncompact oscillators are defined by a group of motion which is an abelian subgroup of sl 2 qp in each case the evolution operator is a unitary representation of the appropriate group its spectrum is given by characters and its eigenstates are calculated

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

nonarchimedean green functions and dynamics on projective space let varphi mathbb p n k to mathbb p n k be a morphism of degree d 2 defined over a field k that is algebraically closed field and complete with respect to a nonarchimedean absolute value we prove that a modified green function hat g varphi associated to varphi is holder continuous on mathbb p n k and that the fatou set mathcal f varphi of varphi is equal to the set of points at which hat g phi is locally constant further hat g varphi vanishes precisely on the set of points p such that varphi has good reduction at every point in the forward orbit mathcal o varphi p of p we also prove that the iterates of varphi are locally uniformly lipschitz on mathcal f varphi

random walks in euclidean space fix a probability measure on the space of isometries of euclidean space r d let y 0 0 y 1 y 2 r d be a sequence of random points such that y l 1 is the image of y l under a random isometry of the previously fixed probability law which is independent of y l we prove a local limit theorem for y l under necessary nondegeneracy conditions moreover under more restrictive but still general conditions we give a quantitative estimate which describes the behavior of the law of y l on scales e cl 1 4