erratic behavior of invariant circles in standard like mappings abstract we study mappings like the standard map but with the sine replaced by another function by varying a parameter in this function it changes smoothly from the sine to a piecewise linear pwl function the function is always analytic except when it is pwl taking the function close enough to the pwl function it is found that kam tori can reappear after they disappeared when the nonlinearity is increased when plotting the boundary around the area where a kam torus with noble winding number exists in the plane of the non linearity versus the distance of the function to the pwl function self similarity is observed

two non holonomic lattice walks in the quarter plane we present two classes of random walks restricted to the quarter plane with non holonomic generating functions the non holonomicity is established using the iterated kernel method a variant of the kernel method this adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions m mishna classifying lattice walks in the quarter plane j combin theory ser a 116 2009 460 477 the method also yields an asymptotic expression for the number of walks of length n

nonlinear lie derivations of triangular algebras in this paper we prove that every nonlinear lie derivation of triangular algebras is the sum of an additive derivation and a map into its center sending commutators to zero

renormalisation in implicit complex maps abstract we study the dynamics of a family of implicit complex maps restricted to an invariant circle the renormalisation analysis for golden mean rotation number is given we find a one parameter family of universality classes and relate this to a line of fixed points of the square of the circle map renormalisation transformation these period 2 points are pairs of fractional linear maps

nonlinear mappings on upper triangular matrices derivable at zero point abstract let t n f be the set of all n n upper triangular matrices over a field f recently by using a technique from graph theory l wang in 8 proved that a bijective mapping on t n f with f a finite field preserves zero product i e x y 0 whenever x y 0 if and only if can be decomposed into the product of an inner automorphism an automorphism of t n f induced by an automorphism of f and a regular automorphism of the zero divisor graph of t n f algebraic group theory shows that derivations of an algebra often go parallel with automorphisms of the algebra this viewpoint encourages us to determine nonlinear mappings on t n f which are derivable at zero point i e x y x y 0 whenever x y 0 it is proved that a mapping on t n f is derivable at zero point if and only if is the sum of an inner derivation an additive derivation of t n f induced by an additive derivation of f and a regular mapping on t n f

operators on triangular algebras abstract we study the algebra of differential operators on the triangular algebras and the upper triangular algebras we further identify all the ideals of the algebra of differential operators on the upper triangular algebras

derivations of upper triangular matrix rings we give a description of the derivations in tn r the ringof upper triangular matrices over a ring r assuming only the existence of an identity element we show that every derivation is the sum of an inner derivation and another one induced from r

automorphisms and derivations of upper triangular matrix rings abstract kezlan proved that for a commutative ring c every c automorphism of the ring of upper triangular matrices over c is inner we generalize this result to rings in which all idempotents are central moreover we show that for a semiprime ring a and central subring c every c automorphism of the ring of upper triangular matrices over c is the composite of an inner automorphism and an automorphism induced from a c automorphism of a by the method of proof we re prove results of s p coelho and c p milies and of mathis stating that a derivation of a ring of upper triangular matrices of a c algebra n n matrices over a is a sum of an inner derivation and a derivation induced from a c derivation of a by an example we show that an extra assumption is needed for proving the above result of automorphisms of upper triangular matrices finally we consider automorphisms of subrings of n n matrices over a commutative ring c where entries over the diagonal are from c and below the diagonal are taken from a nil ideal we prove that all such automorphisms are inner

an iterated implicit complex map abstract iterated complex maps defined by a root of a biquadratic equation g z z 0 are studied numerical studies suggest conjugacy functions with differentiable natural boundaries and maps having island chains occupying much of the riemann sphere an analogy with hamiltonian systems is discussed

n derivations of triangular algebras abstract let a be a triangular algebra let n 2 be an integer a map a a a a is said to be a n derivation if it is a derivation in each argument in this paper we investigate n derivations n 3 for a certain class of triangular algebras the main result is then applied to upper triangular matrix algebras and nest algebras

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

feigenbaum theory for unimodal maps with asymmetric critical point rigorous results we apply the methods of h epstein to prove the existence of a line of period 2 solutions of the feigenbaum period doubling renormalisation transformation these solutions govern the universal behaviour of families of unimodal maps with asymmetric critical points of degree d for which the d th derivative has differing left and right limits

walks confined in a quadrant are not always d finite we consider planar lattice walks that start from a prescribed position take their steps in a given finite subset of z2 and always stay in the quadrant x 0 y 0 we first give a criterion which guarantees that the length generating function of these walks is d finite that is satisfies a linear differential equation with polynomial coefficients this criterion applies among others to the ordinary square lattice walks then we prove that walks that start from 1 1 take their steps in 2 1 1 2 and stay in the first quadrant have a non d finite generating function our proof relies on a functional equation satisfied by this generating function and on elementary complex analysis

asymptotics of scaling parameters for period doubling in unimodal maps with asymmetric critical points the universal period doubling scaling of a unimodal map with an asymmetric critical point is governed by a period 2 point of a renormalization operator the period 2 point is parametrized by the degree of the critical point and the asymmetry modulus in this paper we study the asymptotics of period 2 points and their associated scaling parameters in the singular limit of degree tending to 1

biderivations of triangular algebras abstract let a be a triangular algebra a bilinear map a a a is called a biderivation if it is a derivation with respect to both arguments in this paper we define the concept of an extremal biderivation and prove that under certain conditions a biderivation of a triangular algebra a is a sum of an extremal and an inner biderivation the main result is then applied to block upper triangular matrix algebras and nest algebras we also consider the question when a derivation of a triangular algebra is an inner derivation

automorphism groups of algebras of triangular matrices abstract if a is an algebra of n x n matrices over an integral domain we can associate with a a graph whose edges are pairs i j such that the i j entry of every element of a is zero the graph in turn defines a group of permutations and automorphisms of a which are conjugations by permutations matrices for a suitably restricted class of algebras a we show that the automorphism group of a is the semidirect product of this group of permutation matrices with the subgroup of inner automorphisms

universal properties of maps of the circle with ɛ singularities following the work of collet eckmann and lanford on the feigenbaum conjecture we study the structure of the renormalization transformation introduced in 12 upon maps of the circle with critical points of the formx x

derivations of tensor products of nonassociative algebras abstract let r and s be nonassociative unital algebras assuming that either one of them is finite dimensional or both are finitely generated we show that every derivation of r s is the sum of derivations of the following three types a ad u where u belongs to the nucleus of r s b l z f where f is a derivation of s and z lies in the center of r and c g l w where g is a derivation of r and w lies in the center of s

on functional equations associated with the renormalization of non commuting circle mappings we investigate the functional equations associated with critical non commuting circle homeomorphisms using epstein s herglotz function methods we show that there are analytic solutions of a class of functional equations closely related to period two points of the circle map renormalization transformations for rotation number with continued fraction

jordan derivations of full matrix algebras abstract let a be a unital associative ring and m be a 2 torsion free a bimodule using an elementary and constructive method we show that every jordan derivation from m n a into m n m is a derivation

higher derivations of triangular algebras and its generalizations abstract let r be a commutative ring with identity a and b be unital algebras over r and m be a unital it a it b bimodule let t a m 0 b be the triangular algebra consisting of a it band m motivated by the work of cheung 14 we mainly consider the question whether every higher derivation on a triangular algebra is an inner higher derivation we also give some characterizations on generalized jordan triple higher derivations of triangular algebras

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

basic analytic combinatorics of directed lattice paths this paper develops a unified enumerative and asymptotic theory of directed two dimensional lattice paths in half planes and quarter planes the lattice paths are specified by a finite set of rules that are both time and space homogeneous and have a privileged direction of increase they are then essentially one dimensional objects the theory relies on a specific kernel method that provides an important decomposition of the algebraic generating functions involved as well as on a generic study of singularities of an associated algebraic curve consequences are precise computable estimates for the number of lattice paths of a given length under various constraints bridges excursions meanders as well as a characterization of the limit laws associated to several basic parameters of paths

feigenbaum theory for unimodal maps with asymmetric critical point we extend the renormalization analysis for period doubling in unimodal maps to the case of asymmetric critical points universal scaling phenomena are governed by period two points of a renormalization operator