common hypercyclic vectors for multiples of backward shift we prove that l 2 contains a dense set of vectors which are hypercyclic simultaneously for all multiples of the backward shift oper ator by constants of absolute value greater than 1

complete systems of partial derivatives of entire functions and frequently hypercyclic operators abstract we find some sufficient conditions for a system of partial derivatives of an entire function to be complete in the space h c d of all entire functions of d variables as an application of this result we describe new classes of frequently hypercyclic operators on h c d

discrete aggregation operators with annihilator abstract mas et al have investigated a special kind of commutative and associative aggregation operators with annihilator on the unit interval 0 1 which includes s uninorms s nullnorms t uninorms t nullnorms bi uninorms and bi nullnorms this paper is devoted to extending this kind of operators from the unit interval 0 1 to a finite chain a characterization of this kind of operators on a finite chain is given including some examples this class of operators can be viewed as a generalization of both uninorms and nullnorms on a finite chain

some questions about subspace hypercyclic operators abstract a bounded linear operator t on a banach space x is called subspace hypercyclic for a subspace m if orb t x m is dense in m for a vector x m we show examples that answer some questions posed by h rezaei 2013 7 in particular we provide an example of an operator t such that orb t x m is somewhere dense in m but it is not everywhere dense in m

hypercyclic weighted shifts an operator t acting on a hilbert space is hypercyclic if for some vector x in the space the orbit tnx n 0 is dense in this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic as a consequence we show within the class of weighted shifts that multi hypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic for general hypercyclic operators the corresponding questions were posed by d a herrero and they still remain open using a different technique we prove that i t is hypercycic whenever t is a unilateral backward weighted shift thus answering in more generality a question recently posed by k c chan and j h shapiro

algebraic genericity of frequently universal harmonic functions on trees abstract we show that the set of frequently universal harmonic functions on a tree t contains a vector space except 0 which is dense in the space of harmonic functions on t seen as subset of c t in order to prove this we replace the complex plane c by any separable frechet space e and we repeat all the theory

genericity of wild holomorphic functions and common hypercyclic vectors abstract let t be the translation operator by in the space of entire functions h c defined by t f z f z we prove that there is a residual set g of entire functions such that for every f g and every c 0 the sequence t n f is dense in h c that is g is a residual set of common hypercyclic vectors functions for the family t c 0 also we prove similar results for many families of operators as multiples of differential operator multiples of backward shift weighted backward shifts

characterization of idempotent n uninorms abstract the structure of idempotent n uninorms is studied we show that each idempotent 2 uninorm can be expressed as an ordinal sum of an idempotent uninorm possibly also of a countable number of idempotent semigroups with operations min and max and a 2 uninorm from class 1 possibly restricted to open or half open unit square similar results are shown also for idempotent n uninorms further it is shown that idempotent n uninorms are in one to one correspondence with special lower semi lattices defined on the unit interval the z ordinal sum construction for partially ordered semigroups is also defined

hypercyclic operators that commute with the bergman backward shift the backward shift b on the bergman space of the unit disc is known to be hypercyclic meaning it has a dense orbit here we ask which operators that commute with b inherit its hypercyclicity we show that the problem reduces to the study of operators of the form 9p b where so is a holomorphic self map of the unit disc that multiplies the dirichlet space into itself and that the question of hypercyclicity for such an operator depends on how freely 9p z is allowed to approach the unit circle as lz s 1

s linear and convex structures in function families abstract in this paper the notion of s lineability originally coined by vladimir i gurariy is introduced and developed in a general abstract setting this new notion is then applied to specific situations as for instance classes of differentiable nowhere monotone functions as well as families of vectors having dense orbit with respect to an operator large convex structures are also shown to exist inside the family of topologically mixing continuous selfmaps of a real compact interval

hypercyclic orbits intersect subspaces in wild ways abstract let t x x be a hypercyclic operator of an infinite dimensional separable banach space x by modifying a construction of grivaux we will show that the t orbit of a hypercyclic vector can intersect certain closed vector subspaces of x in many strange ways moreover the set of visiting times of the orbit to the subspace can also be quite exotic especially when the operator satisfies a stronger form of hypercyclicity as a by product we will improve and or supply new proofs to some of the recent results about subspace hypercyclicity

on the structure of 2 uninorms abstract in this paper we describe the structure of 2 uninorms defined on the unit interval we divide the class of 2 uninorms into five mutually exclusive subclasses based on their boundary behaviour as determined by the absorbing element and the conjunctive disjunctive nature of the underlying uninorms which every 2 uninorm is known to have for each of these subclasses we fully characterize the structure of its members albeit for two subclasses only under an additional continuity assumption a common practice in the study of associative operations

weak uninorm aggregation operators abstract by weakening the identity condition of uninorm a general class of associative aggregation operators called weak uninorm is introduced it is proved that continuous weak uninorms are the combination of continuous t norm and continuous t conorm through a median operator the representation theorem for archimedean weak uninorm is also given finally the counting number of the identity elements of weak uninorm is also investigated

mean li yorke chaos in banach spaces abstract we investigate the notion of mean li yorke chaos for operators on banach spaces we show that it differs from the notion of distributional chaos of type 2 contrary to what happens in the context of topological dynamics on compact metric spaces we prove that an operator is mean li yorke chaotic if and only if it has an absolutely mean irregular vector as a consequence absolutely cesaro bounded operators are never mean li yorke chaotic dense mean li yorke chaos is shown to be equivalent to the existence of a dense or residual set of absolutely mean irregular vectors as a consequence every mean li yorke chaotic operator is densely mean li yorke chaotic on some infinite dimensional closed invariant subspace a dense mean li yorke chaos criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained moreover we construct an example of an invertible hypercyclic operator t such that every nonzero vector is absolutely mean irregular for both t and t 1 several other examples are also presented finally mean li yorke chaos is also investigated for c 0 semigroups of operators on banach spaces

topologically transitive extensions of bounded operators let x be any banach space and t a bounded operator on x an extensionopen image in new window of the pair x t consists of a banach space open image in new window in which x embeds isometrically through an isometry i and a bounded operatoropen image in new window on open image in new window such that open image in new window when x is separable it is additionally required that open image in new window be separable we say that open image in new window is a topologically transitive extension of x t when open image in new window is topologically transitive on open image in new window i e for every pair open image in new window of non empty open subsets of open image in new window there exists an integer n such that open image in new window is non empty we show that any such pair x t admits a topologically transitive extension open image in new window and that when h is a hilbert space h t admits a topologically transitive extension open image in new window where open image in new window is also a hilbert space we show that these extensions are indeed chaotic

hereditarily hypercyclic operators abstract we show that a continuous linear operator t on a frechet space satisfies the so called hypercyclicity criterion if and only if it is hereditarily hypercyclic and if and only if the direct sum t t is hypercyclic in particular hypercyclic operators with either a dense generalized kernel or a dense set of periodic points i e chaotic in the sense of r l devaney 1989 an introduction to chaotic dynamical systems addison wesley reading ma must satisfy the criterion finally we provide a characterization of those weighted shifts t that are hereditarily hyper cyclic with respect to a given sequence n k of positive integers as well as conditions under which t and t n k k 1 share the same set of hypercyclic vectors

hypercyclic and cyclic vectors abstract let x denote an arbitrary separable banach space over the field of complex numbers and b x the banach algebra of all bounded linear operators on x we prove the following results 1 an element of the space x is hypercyclic supercyclic for all positive powers tn of an operator t in b x if it is hypercyclic supercyclic for t 2 under some condition on the spectrum of the adjoint of a cyclic operator the set of all cyclic vectors of the operator is dense this result extends to any cyclic commutative subset of b x 3 under a mild condition on the spectrum of a cyclic operator t the set of all separating vectors for the commutant t of t is dense this also extends to any cyclic commutative subset of b x 4 a slightly stronger version of a theorem of k f clancey and d d rogers on cyclic vectors finally we define and discuss hereditarily hypercyclic operators

van melle s combining function in mycin is a representable uninorm an alternative proof abstract recently tsadiras and margaritis have shown that the function implemented in the expert system mycin for combining certainty factors is a uninorm in this paper we present a compact alternative to the given direct and lengthy proof based on the construction of representable uninorms by means of an additive generator consequently we show that van melle s combining function is a representable uninorm in fact it is a rescaled version of the representable uninorm on the unit interval constructed from the algebraic product and the probabilistic sum

the role of the angle in supercyclic behavior abstract a bounded operator t acting on a hilbert space h is said to be supercyclic if there is a vector f h such that the projective orbit t n f n 0 and c is dense in h we use a new method based on a very simple geometric idea that allows us to decide whether an operator is supercyclic or not the method is applied to obtain the following result a composition operator acting on the hardy space whose inducing symbol is a parabolic linear fractional map of the disk onto a proper subdisk is not supercyclic this result finishes the characterization of the supercyclic behavior of composition operators induced by linear fractional maps and thus completes previous work of bourdon and shapiro

existence of hypercyclic operators on topological vector spaces the present paper introduces a very simple but very useful notion of the so called quasi extension ofl1 operators and proves that a large class of topological vector spaces admit continuous hypercyclic operators in particular it answers in the affirmative a question of s rolewicz posed in 1969 whether or not every infinite dimensional separable banach space admits a continuous hypercyclic operator

hypercyclic operators are subspace hypercyclic abstract in this short note we prove that for a dense set a x x is a banach space there is a non trivial closed subspace m x such that a m is dense in m we use this result to answer a question posed in madore and martinez avendano 2011 9 in particular we show that every hypercyclic operator is subspace hypercyclic

invariant gaussian measures for operators on banach spaces and linear dynamics we give conditions for an operator t on a complex separable banach space x with sufficiently many eigenvectors associated to eigenvalues of modulus 1 to admit a non degenerate invariant gaussian measure with respect to which it is weak mixing the existence of such a measure depends on the geometry of the banach space and on the possibility of parametrizing the mathbb t eigenvector fields of t in a regular way we also investigate the connection with frequent hypercyclicity

sums of hypercyclic operators abstract every bounded operator on a complex infinite dimensional separable hilbert space can be written as the sum of two hypercyclic operators and also as the sum of two chaotic operators

an advanced interpretable fuzzy neural network model based on uni nullneuron constructed from n uninorms abstract this paper formulates a fuzzy logic neuron that uses n uninorms to construct uni nullneurons a fuzzy neural network fnn composed of these neurons is easy to operate with nullnorms and uninorms at different times which results in high accuracy of the model outputs and increases the flexibility in connecting the rule antecedents enabling and and or connections within one rule this in turn may allow experts operators to extract more knowledge from data the fnn uses a one versus rest classifier learning scheme for multi class classification problems where neuron activation levels construct the indicator regression matrix this results in a non linear regression by indicator which can resolve the inherent class masking problem apparent in the linear case we propose a specific neuron selection strategy in the learning stage that applies lasso to bootstrap replications in order to ensure that the rule base is as compact as possible and induced by a low number of neurons to evaluate the new neuron acting in fnns we performed pattern classification and regression tests compared with traditional fnn in the literature our variant showed improved model accuracies for several high dimensional real world datasets in binary and multi class classification and regression problems combined with the ability to generate human readable rules this offers the ability to generate parsimonious responses with a high degree of confidence

frequently hypercyclic operators with irregularly visiting orbits abstract we prove that a bounded operator t on a separable banach space x satisfying a strong form of the frequent hypercyclicity criterion which implies in particular that the operator is universal in the sense of glasner and weiss admits frequently hypercyclic vectors with irregularly visiting orbits i e vectors x x such that the set n t x u n 1 t n x u of return times of x into u under the action of t has positive lower density for every non empty open set u x but there exists a non empty open set u 0 x such that n t x u 0 has no density

hypercyclicity and unimodular point spectrum abstract we show that an operator on a separable complex banach space with sufficiently many eigenvectors associated to eigenvalues of modulus 1 is hypercyclic we apply this result to construct hypercyclic operators with prescribed k unimodular point spectrum we show how eigenvectors associated to unimodular eigenvalues can be used to exhibit common hypercyclic vectors for uncountable families of operators and prove that the family of composition operators c on h 2 d where is a disk automorphism having 1 as attractive fixed point has a residual set of common hypercyclic vectors

two remarks on frequent hypercyclicity abstract we show that if t x x is a continuous linear operator on an f space x 0 then the set of frequently hypercyclic vectors of t is of first category in x and this answers a question of a bonilla and k g grosse erdmann we also show that if t x x is a bounded linear operator on a banach space x 0 and if t is frequently hypercyclic or more generally syndetically transitive then the t orbit of every non zero element of x is bounded away from 0 and in particular t is not hypercyclic

linear structure of hypercyclic vectors abstract a vector x in a banach space b is called hypercyclic for a bounded linear operator t b b if the orbit t n x n 1 is dense in b our main result states that if t is a compact perturbation of an operator of norm 1 and satisfies an appropiate extra hypothesis then there is an infinite dimensional closed subspace consisting except for zero entirely of hypercyclic vectors for t in particular the result applies to compact perturbations of the identity we also include applications to some weighted backward shifts and compact perturbations of the identity by weighted backward shifts this last result in combination with a recent one that states that every banach space admits an operator with a hypercyclic vector proves that in all banach space there is an operator t with an infinite dimensional closed subspace consisting except for zero of hypercyclic vectors the main result also applies to the differentiation operator and the translation operator t f z f z 1 on certain hilbert spaces consisting of entire functions

distributivity and conditional distributivity of a uninorm and a continuous t conorm the open problem recalled by klement in the linz2000 closing session related to distributivity and conditional distributivity of a uninorm and a continuous t conorm is solved for the most usual known classes of uninorms from the obtained results it is deduced that distributivity and conditional distributivity are equivalent for these cases it is remarkable that solutions appear involving not only strict t conorms but also ordinal sums of the maximum with a strict t conorm conversely the distributivity of a t conorm over a uninorm is also studied leading only to already known solutions moreover the dual case of distributivity and conditional distributivity involving uninorms and continuous t norms is also solved proving again the equivalence of both kinds of distributivities

frequently hypercyclic operators we investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators t on separable complex f spaces t is frequently hypercyclic if there exists a vector x such that for every nonempty open subset u of x the set of integers n such that t n x belongs to u has positive lower density we give several criteria for frequent hypercyclicity and this leads us in particular to study linear transformations from the point of view of ergodic theory several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting

optimal growth of entire functions frequently hypercyclic for the differentiation operator we solve a problem posed by bonilla and grosse erdmann 2007 7 by constructing an entire function f which is frequently hypercyclic with respect to the differentiation operator and satisfies mf r cerr 1 4 where c 0 may be chosen arbitrarily small this growth rate is sharp we also obtain optimal results for minimal growth in terms of average lp norms among other tools the proof uses the rudin shapiro polynomials and heat kernel estimates

operators with the specification property abstract we study a version of the specification property for linear dynamics operators having the specification property are investigated and relationships with other well known dynamical notions such as mixing devaney chaos and frequent hypercyclicity are obtained

nullnorms on bounded lattices derived from t norms and t conorms abstract nullnorms with an annihilator a in any point of a bounded lattice are generalizations and unifications of t norms and t conorms this study continues to investigate the construction of nullnorms on bounded lattices we propose some methods to construct nullnorms derived from t norms and t conorms on bounded lattices where some sufficient and necessary conditions on theirs annihilator are required as a by product of these constructions idempotent nullnorms on bounded lattices are obtained further we provide some illustrative examples of the new classes of nullnorms idempotent nullnorms on bounded lattices

invertible composition operators on hp abstract the weakly closed algebras generated by certain sets of composition operators are shown to be reflexive a structure theorem for invertible composition operators on h2 is obtained and used to show that such operators are reflexive the structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal thus the general invariant subspace problem for hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators

strong mixing measures for linear operators and frequent hypercyclicity abstract we construct strongly mixing invariant measures with full support for operators on f spaces which satisfy the frequent hypercyclicity criterion for unilateral backward shifts on sequence spaces a slight modification shows that one can even obtain exact invariant measures

structure of n uninorms in this paper we study binary operators on 0 1 which are associative monotone non decreasing in both variables and commutative amc with neutral element in this work we generalize the concept of neutral element and this generalization gives rise to a new class of amc binary operators on 0 1 called n uninorms n uninorms are denoted as u n where n comes from the generalization of the neutral element we study the structure of n uninorms the structure resembles an ordinal sum structure made up of n uninorms we characterize some special cases of them based on some continuity considerations and show that t norms t conorms uninorms and nullnorms t operators are special cases of n uninorms we also show that given n there are n 1 classes of operators in u n and each of them has many subclasses we also study the frank equation involving n uninorms and show that we need to consider only n uninorms for the study finally we show that the total number of subclasses of operators in u n follows the famous series called catalan numbers

operators with dense invariant cyclic vector manifolds abstract we study a class of banach space operators patterned after the weighted backward shifts on hilbert space and show that any non scalar operator in the commutant of one of these generalized backward shifts has a dense invariant linear manifold whose non zero members are cyclic vectors under appropriate extra hypotheses on the commuting operator stronger forms of cyclicity are possible the most extreme being hypercyclicity density of an orbit motivated by these results we examine the cyclic behavior of two seemingly unrelated classes of operators adjoint multiplications on hilbert spaces of holomorphic functions and differential operators on the frechet space of entire functions we show that each of these operators other than the scalar multiples of the identity possesses a dense invariant linear submanifold each of whose non zero elements is hypercyclic finally we explore some connections with dynamics many of the hypercyclic operators discussed here are in at least one of the commonly accepted senses of the word chaotic

transplantation and jacobians of sunada isospectral riemann surfaces abstract we consider relations among the jacobians of isospectral compact riemann surfaces constructed using sunada s theorem we use a simple algebraic formulation of transplantation of holomorphic 1 forms and singular 1 cycles to obtain two main results first we obtain a geometric proof of a result of prasad and rajan that sunada isospectral riemann surfaces have isogenous jacobians second we determine a relationship weaker than isogeny that holds among the jacobians of sunada isospectral riemann surfaces when the jacobians extra structure as principally polarized abelian varieties is taken into account we also show all sunada isospectral manifolds have isomorphic real cohomology algebras finally we exhibit transplantation of cycles explicitly in a concrete example of a pair of isospectral riemann surfaces constructed by brooks and tse

a single point characterization of representable uninorms we study uninorms of which both the underlying t norm and underlying t conorm are strict such uninorms are the only candidates for being representable by an additive generator we prove that the representability of such a uninorm depends solely on its value at a single arbitrary point in the remaining open part of the unit square more explicitly such a uninorm turns out to be representable if and only if this single value is located strictly between the minimum and the maximum of the corresponding arguments if this single value coincides with one of these bounds then the value of the uninorm at any point in the remaining open part is determined by the same bound

the principal measure of a quantum harmonic oscillator at the end of his paper oprocha 2006 j phys a math gen 39 14 559 65 oprocha said that it is difficult and interesting to know the exact value of the principal measure of an annihilation operator of the unforced quantum harmonic oscillator motivated by this in this paper we prove that this exact value is equal to 1 besides we point out that exhibits distributional chaos for any 0 2

notes on subspace hypercyclic operators a bounded linear operator t on a banach space x is called subspace hypercyclic for a subspace m if orb t x m is dense in m for a vector x m in this case we show that p t has a relatively dense range for every real or complex polynomial p which in turn answers a question posed in madore and martinez avendano 2011 8 as a consequence the algebraic structure of the set of subspace hypercyclic vectors can be described

uninorms on bounded lattices with the underlying t norms and t conorms abstract uninorms are a generalization of the concepts of triangular norms t norms and triangular conorms t conorms that admit a neutral element as any element in a bounded lattice in this study we discuss the characterization of uninorms on bounded lattices we introduce some new constructions for uninorms on bounded lattices which are derived from both t norms and t conorms in particular two idempotent uninorms on bounded lattices are obtained we also provide some corresponding examples to help understand the structure of these new classes of uninorms in addition we provide an answer to the open problem proposed by cayli 2019 14