kalman decomposition for implicit linear systems the kalman decomposition for a general class of implicit systems is introduced and studied the properties of strong reachability observability acceptance of all inputs and outer sequences output uniqueness uniqueness and trajectory observability of the original system are related to the properties of its subsystems obtained via the kalman decomposition the problems of transfer relation regularizability by feedback output feedback and output injection are studied by means of this decomposition

limit sets of automatic sequences abstract we show that for every k automatic sequence there exists a natural number p 0 such that the sequences of the form kpn j n 0 with j 0 p 1 are scaling sequences for f moreover we demonstrate that every limit set is the union of certain basic limit sets

singular robustly chain transitive sets are singular volume partial hyperbolic for diffeomorphisms or for non singular flows there are many results relating properties persistent under mathcal c 1 perturbations and a global structures for the dynamics such as hyperbolicity partial hyperbolicity dominated splitting however a difficulty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point in bonatti star flows and multisingular hyperbolicity arxiv 1705 05799 2017 with christan bonatti we propose a a general procedure for adapting the usual hyperbolic structures to the singularities using this tool we recover the results in bonatti et al ann math 158 2 355 418 2003 for flows showing that robustly chain transitive sets have a weak form of hyperbolicity allowing us to conclude as well the kind of hyperbolicity carried by the examples in bonatti et al j inst math jussieu 12 3 449 501 2013 a robust chain transitive singular attractor with periodic orbits of different indexes along with the results in 8 this shows that the way we propose to interpret the effect of singularities has the potential to adapt to other settings in which there is coexistence of singularities and regular orbits with the goal of re obtaining the results that we already know for diffeomorphisms

automatic sets and delone sets automatic sets are characterized by having a finite number of decimations they are equivalently generated by fixed points of certain substitution systems or by certain finite automata as examples two dimensional versions of the thue morse baum sweet rudin shapiro and paperfolding sequences are presented we give a necessary and sufficient condition for an automatic set to be a delone set in the result is then extended to automatic sets that are defined as fixed points of certain substitutions the morphology of automatic sets is discussed by means of examples

correlation functions of higher dimensional automatic sequences a procedure for calculating the auto correlation function of an m dimensional complex valued automatic sequence is presented this is done by deriving a recursion for the vector correlation function ker f k whose components are the cross correlation functions between all sequences in the finite set ker f the so called kernel of f which contains all properly defined decimations of f the existence of ker f k which is defined as a limit for all is shown to depend only on the existence of ker f 0 this is illustrated for the higher dimensional thue morse paper folding and rudin shapiro sequences

the star systems x and a proof of the c1ω stability conjecture for flows abstract in this paper we give a new proof for the c 1 stability conjecture for flows our approach is considerably different from that of hayashi or wen and a good deal of liao s obstruction sets theory is used

area efficient multipliers based on multiple radix representations in this paper we shall introduce several new algorithms for integer multiplication that are based on specific multiple radix representation of one of the multiplicands we provide extensive theoretical analysis and experimental results for multipliers based on the new representations on 0 18 m cmos technology they provide a clear picture about the advantages of the new method in 64 bit hardware implementations compared to array based classical multiplier and radix 8 based multiplier the proposed multipliers have better area and power consumption compared to reference multipliers

disturbance decoupling of linear time varying singular systems the problem of disturbance decoupling by state feedback is defined for a linear time varying singular system it is required that the closed loop system has a unique impulse free solution and its output is not affected by disturbances an algorithm namely disturbance decoupling algorithm is proposed it is proved that the feasibility of the disturbance decoupling algorithm is invariant under any regular feedback control law based on the disturbance algorithm a constructive method is provided to design a disturbance decoupling feedback control law sufficient conditions for the solvability of the disturbance decoupling problem are derived it is proved that one of the sufficient conditions is also necessary provided that other conditions are satisfied

new redundant representations of complex numbers and vectors the authors present a redundant representation for complex numbers called polygonal representation this representation enables fast carry free addition in a way quite similar to the carry free addition in signed digit number systems and is convenient for multiplication the technique is generalized in order to handle n dimensional vectors

disturbance decoupling for descriptor systems by state feedback we study the disturbance decoupling problem for linear time invariant descriptor systems we give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability via a proportional and or derivative feedback that also makes the resulting closed loop system regular and or of index at most one all results are proved constructively based on condensed forms that can be computed using orthogonal matrix transformations i e transformations that can be implemented in a numerically stable way

disturbance decoupling with stability and impulse free response in singular systems abstract the problem of disturbance decoupling with stability and impulse free response in square singular systems e x ax bu gq with output y dx by applying a state feedback u fx is considered by employing van dooren s recursive algorithm for the computation of the subspace supremal a b invariant subspace in kernel d and using kalman s decomposition theorem the paper s approach presents a complete solution to the underlying problem regarding state space systems it is shown that the solution to the underlying problem is obtained almost directly from van dooren s recursive algorithm

correlation and spectral properties of higher dimensional paperfolding and rudin shapiro sequences we consider higher dimensional generalizations of the classical one dimensional 2 automatic paperfolding and rudin shapiro sequences on this is done by considering the same automaton structure as in the one dimensional case but using binary number systems in instead of in the correlation function and the diffraction spectrum for the resulting m dimensional paperfolding and rudin shapiro point sets are calculated through the corresponding sequences with values 1 they are shown to be quasi independent of the dimension m and of the particular binary number system under consideration it is shown that any paperfolding sequence thus obtained has a discrete spectrum the rudin shapiro sequences have an absolutely continuous lebesgue spectral measure

on the growth rate of periodic orbits for vector fields abstract we establish the relationship between the growth rate of periodic orbits and the topological entropy for c 1 generic vector fields this extends a classical result of katok for c 1 0 surface diffeomorphisms to c 1 generic vector fields of any dimension the main difficulty comes from the existence of singularities and the shear of the flow

a proof of stability conjecture for three dimensional flows we give a proof of the c1 stability conjecture for three dimensional flows i e prove that there exists a hyperbolic structure over the q set for the structurally stable three dimensional flows mafn s proof for the discrete case motivates our proof and we find his perturbation techniques crucial in proving this conjecture we have overcome several new difficulties e g the change of period after perturbation the ergodic closing lemma for flows the existence of dominated splitting over q y where 3 is the set of singularities for the flow the discontinuity of the contracting rate function on singularities etc based on these we finally succeed in separating the singularities from the other periodic orbits for the structurally stable systems i e we create unstable saddle connections if there are accumulations of periodic orbits on the singularities

vector fields with topological stability in this paper we give a characterization of the structurally stable vector fields by making use of the notion of topological stability more precisely it is proved that the c1 interior of the set of all topologically stable c1 vector fields coincides with the set of all vector fields satisfying axiom a and the strong transversality condition

self affine tiles in ℝn abstract a self affine tile in r n is a set t of positive measure with a t d d t d where a is an expanding n n real matrix with det a m an integer and d d d 2 d m r n is a set of m digits it is known that self affine tiles always give tilings of r n by translation this paper extends known characterizations of digit sets d yielding self affine tiles it proves several results about the structure of tilings of r n possible using such tiles and gives examples showing the possible relations between self replicating tilings and general tilings which clarify results of kenyon on self replicating tilings

the disturbance decoupling problem for implicit linear discrete time systems the disturbance decoupling problem for implicit linear discrete time systems is studied in detail necessary and sufficient conditions for the problem to be solved are given the results are obtained with the aid of the concepts of almost invariant sliding and coasting subspaces for implicit systems

disturbance and input output decoupling of singular systems the disturbance decoupling and the simultaneous disturbance and input output decoupling problems for singular systems are considered in the context of the matrix fraction description mfd of the system solvability conditions are obtained in terms of the composite matrix of a column reduced mfd of the system a characterisation of the fixed poles of both problems is given and it is shown that the remaining poles can be arbitrarily assigned

symmetries of decimation invariant sequences and digit sets a bi infinite sequence is called p decimation invariant if all p decimations of it reproduce the sequence albeit with a shift in this paper we discuss symmetry properties of decimation invariant sequences a symmetry is a composition of a translation and a reflection we establish the existence of translation invariant i e periodic decimation invariant sequences moreover we prove that there exist decimation invariant sequences which are left periodic and right periodic i e they are partially translation invariant we present several criteria for the existence of decimation invariant sequences with additional periodicity properties finally we discuss the existence of decimation invariant sequences that are invariant under reflections moreover in passing we demonstrate that properties of decimation invariant sequences are linked with properties of certain digit sets

invariant subspaces and invertibility properties for singular systems the general case abstract open loop definitions and properties of several subspaces for general singular systems are characterized by means of a fully algebraic distributional framework simple recursive algorithms for producing these spaces as well as related duality aspects turn out to follow directly from these definitions next we provide definitions and conditions for two notions of left right invertibility of a general singular system in terms of our distributions subspaces and rosenbrock s system matrix and we show which conditions represent the gap between our invertibility concepts finally we prove that in many cases left right invertibility is equivalent to left right invertibility of the system matrix

vector fields in the interior of kupka smale systems satisfy axiom a abstract in 1994 s hu conjectured in his paper 1994 trans amer math soc 342 753 772 that axiom a systems are dense in the interior of kupka smale systems we will show that if x is in the interior of kupka smale systems then x satisfies axiom a

a case study for the open question disturbance decoupling problem for singular systems by output feedback we study the disturbance decoupling problem for singular systems ex spl dot ax bu gd y cx z hx with left invertibility we give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability by output feedback that also makes the resulting closed loop system regular and or of index at most one all results are proved based on a condensed form that can be computed using orthogonal transformations i e can be computed in a numerically stable way

a tutorial on the geometric analysis of linear time invariant implicit systems abstract a background of recent work in nonregular and regular implicit systems of the form e x ax bu is provided then the output nulling unknown input and composite subspaces are defined for singular systems with recursions given for their computation inner subspaces that is those in the domain of e and a are distinguished from outer subspaces those in the codomain of e and a the notions of composite preimage and composite image are introduced the result is a framework that unifies some of the recent work in the geometric theory of implicit systems system properties are discussed for both regular and nonregular systems and duality is investigated

disturbance decoupling by behavioral feedback for linear differentialalgebraic systems we study disturbance decoupling for linear differentialalgebraic systems which are not necessarily regular compared to previous approaches where state feedback is used we use the concept of behavioral feedback which allows to study a larger class of systems we derive geometric characterizations for solvability of the disturbance decoupling problem following the classical approach exploiting the freedom in the choice of the behavioral feedback we show that whenever disturbance decoupling can be achieved by behavioral feedback we may additionally achieve autonomous zero dynamics finally we solve lebrets twenty year old open problem concerning disturbance decoupling with output uniqueness using behavioral feedback

disturbance decoupled observer design for descriptor systems in this paper we study the observer design problem for descriptor systems with partly unknown inputs we give necessary and sufficient conditions for the existence of a solution to the disturbance decoupled estimation problem with or without stable error spectrum requiring at the same time that the resulting combined system is regular and of index at most one all results are proved based on a condensed form that can be computed using orthogonal matrix transformations i e transformations that can be implemented in a numerically stable way

basic digit sets for radix representation abstract the use of a negative base did not appear until the 1950s when several authors independently introduced the concept complement representation also became much discussed in this period as an alternative to sign magnitude for designing the arithmetic unit of a computer the arithmetic of numbers represented in positional notation has a firm foundation derived from the theory of polynomial arithmetic that readily allows these extensions to negative bases and or negative digit values complement representation and digit values in excess of the base our primary concern in this paper is the characterization and computation of those integral valued base and digit set pairs that provide complete and unique finite radix representation of the integers

c 1 stably expansive flows abstract in this paper the c 1 interior of the set of vector fields whose integrated flows are expansive is characterized as the set of vector fields without singularities satisfying both axiom a and the quasi transversality condition and it is proved that the above vector fields possessing the shadowing property must be structurally stable as a corollary there exists a non empty c 1 open set of vector fields whose integrated flows do not have the shadowing property

redundant radix representations of rings this paper presents an analysis of radix representations of elements from general rings in particular we study the questions of redundancy and completeness in such representations mappings into radix representations as well as conversions between such are discussed in particular where the target system is redundant results are shown valid for normed rings containing only a finite number of elements with a bounded distance from zero essentially assuring that the ring is discrete with only brief references to the more usual representations of integers the emphasis is on various complex number systems including the classical complex number systems for the gaussian integers as well as the eisenstein integers concluding with a summary on properties of some low radix representations of such systems

automaticity of double sequences generated by one dimensional linear cellular automata abstract we give a complete answer to the question whether a double sequence that is generated by a one dimensional linear cellular automaton and whose states are integers modulo m is k automatic or not

a solution to the disturbance decoupling problem in singular systems via analogy with state space systems abstract this paper presents an approach for solving the disturbance decoupling problem in a square linear time invariant singular system a simple criterion and an efficient algorithm for solving the underlying problem by applying a state feedback which preserves the uniqueness property of the closed loop singular system has been established the present algorithm is based on a well established approach borrowed from state space system theory

averages of automatic sequences we give some sufficient criteria for the existence of certain averages mean correlation functions of generalized higher dimensional automatic sequences and show how to calculate these averages then follows an exploration of the nature of necessary and sufficient conditions for the existence of averages some of these criteria are applied to averages which play a central role in the determination of the correlation function of an automatic sequence

disturbance decoupling with closed loop modes stability in switched linear systems this work deals with the problem of rendering the output of a switched linear system insensitive to a disturbance input with the requirement that the dynamics of the closed loop system be exponentially stable for all switching signals with a sufficiently large dwell time a necessary and sufficient condition for the problem to have a solution is shown a complete computational procedure to check the solvability condition and to synthesize the compensation scheme is provided a worked out example is presented with the aim of showing the effectiveness of the devised method

proportional and derivative state feedback decoupling of linear systems we consider the row by row decoupling of linear time invariant systems by proportional and derivative state feedback our contribution with respect to previous results is that our procedure is based on simple operator say matrices manipulations without any need to use a canonical form the only assumptions for applying such a decoupling strategy are that the system is right invertible which is a necessary condition to ensure solvability and minimum phase an illustrative example is proposed

decimation invariant sequences and their automaticity this paper deals with one dimensional bidirectional sequences a z v v a finite set such that any p decimation p 2 of the sequence reproduces the sequence modulo a certain shift we develop a procedure for solving the underlying decimation invariance di equations and find that the number of solutions is always finite conditions for equivalency among solutions of differently parametrized di problems and for possible periodicity and symmetry of solutions are derived it is shown that the set of all possible p based decimations of a such a di sequence the so called full kernel of the sequence is finite this implies finiteness of the kernel for the separate right and left parts of the sequence and also p automaticity of these parts an algorithm is presented that constructs the kernel and associated p automaton of a di sequence explicitly

a simple procedure for the stabilization of a class of uncontrollable generalized systems this paper gives a simple procedure for the stabilization with a prescribed rate of convergence of the free motions of a class of possibly single input uncontrollable generalized continuous time time invariant linear systems preserving the impulse free property if any

on thec1stability conjecture for flows we give a proof of thec1stability conjecture of palis and smale for flows which has reduced to proving thatc1structural stability implies axiom a the proof is based on the fundamental work of liao and mane and on the recent powerfulc1connecting lemma of hayashi

secondary radix recodings for higher radix multipliers for progressively higher radices the reduction in partial products obtained by the well known modified booth multiplier recoding is offset by the need to precompute a rapidly increasing store of odd multiples of the multiplicand as inputs to each partial product generator ppg we propose secondary radix multiplier recoding schemes reducing the number of odd multiples required in the store for very high radix recodings e g radix 2 sup r for 5 spl les r spl les 16 the proposed recoding schemes allow reduction of the number of partial products in the implementation by factors between and beyond the reduction factors of 2 3 and 4 that can be achieved by traditional booth recodings to radices 4 8 and 16 respectively we develop the theory of these recodings and provide methodology for secondary radix selection finally we summarize latency and cost evaluations of selected implementations indicating potential cost and performance cost advantages for practical operand sizes