a notion of effectiveness for subshifts on finitely generated groups abstract we generalize the classical definition of effectively closed subshift to finitely generated groups we study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of a group this new class of subshifts forms a conjugacy class that contains all sofic subshifts motivated by the question of whether there exists a group where the class of sofic subshifts coincides with that of effective subshifts we show that the inclusion is strict for several groups including recursively presented groups with undecidable word problem amenable groups and groups with more than two ends we also provide an extended model of turing machine which uses the group itself as a tape and characterizes our extended notion of effectiveness as applications of these machines we prove that the origin constrained domino problem is undecidable for any group of the form g z subject to a technical condition on g and we present a simulation theorem which is valid in any finitely generated group

staged and constrained two dimensional guillotine cutting problems an and or graph approach abstract an and or graph approach has been proposed by morabito et al to solve non staged and unconstrained two dimensional guillotine cutting problems basically this approach consists of representing cutting patterns as complete paths in an and or graph where nodes and arcs correspond to rectangles and cuts respectively and choosing a search strategy to traverse or enumerate the nodes of the graph in this present paper the and or graph approach is extended to solve staged and constrained problems computational experiments with examples from the literature are performed and indicate that this approach generates good and fast solutions using a microcomputer in addition to the fact that the and or graph approach can handle important constraints it can be easily extended to solve cutting and packing problems with multiple dimensions

an integrated cutting stock and sequencing problem in this paper an integrated problem formulated as an integer linear programming problem is presented to find an optimal solution to the cutting stock problem under particular pattern sequencing constraints the solution uses a lagrangian approach the dual problem is solved using a modified subgradient method a heuristic for the integrated problem is also presented the computational results obtained from a set of unidimensional instances that use these procedures are reported

two dimensional packing problems a survey we consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste in two dimensional bin packing problems these units are finite rectangles and the objective is to pack all the items into the minimum number of units while in two dimensional strip packing problems there is a single standardized unit of given width and the objective is to pack all the items within the minimum height we discuss mathematical models and survey lower bounds classical approximation algorithms recent heuristic and metaheuristic methods and exact enumerative approaches the relevant special cases where the items have to be packed into rows forming levels are also discussed in detail

a new linear programming approach to the cutting stock problem a new approach to the one dimensional cutting stock problem is described and compared to the classical model for which gilmore and gomory have developed a special column generation technique the new model is characterized by a dynamic use of simply structured cutting patterns nevertheless it enables the representation of complex combinations of cuts it can be advantageous in practical applications where many different stock lengths or a relatively large number of order lengths have to be dealt with the new approach is applied to a real problem where the trim loss is not valueless since it can be used for further demands arising in later planning periods

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

an exact two dimensional non guillotine cutting tree search procedure we consider the two dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut we develop a lagrangean relaxation of a zero one integer programming formulation of the problem and use it as a bound in a tree search procedure subgradient optimization is used to optimize the bound derived from the lagrangean relaxation problem reduction tests derived from both the original problem and the lagrangean relaxation are given incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved

an lp newton method nonsmooth equations kkt systems and nonisolated solutions we define a new newton type method for the solution of constrained systems of equations and analyze in detail its properties under suitable conditions that do not include differentiability or local uniqueness of solutions the method converges locally quadratically to a solution of the system thus filling an important gap in the existing theory the new algorithm improves on known methods and when particularized to kkt systems derived from optimality conditions for constrained optimization or variational inequalities it has theoretical advantages even over methods specifically designed to solve such systems

integer linear programming models for 2 staged two dimensional knapsack problems we are given a unique rectangular piece of stock material s with height h and width w and a list of m rectangular shapes to be cut from s each shape s type i i 1 m is characterized by a height a width a profit and an upper bound ub i indicating the maximum number of items of type i which can be cut we refer to the two dimensional knapsack tdk as the problem of determining a cutting pattern of s maximizing the sum of the profits of the cut items in particular we consider the classical variant of tdk in which the maximum number of cuts allowed to obtain each item is fixed to 2 and we refer to this problem as 2 staged tdk 2tdk for the 2tdk problem we present two new integer linear programming models we discuss their properties and we compare them with other formulations in terms of the lp bound they provide finally both models are computationally tested within a standard branch and bound framework on a large set of instances from the literature by reinforcing them with the addition of linear inequalities to eliminate symmetries

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

groups and dynamical systems discrete time dynamics on the e 2 group abstract it is shown that discrete time dynamical systems can be derived from defined for a lie group g the equation governing the evolution of discrete time dynamical systems induced by the shimizu leutbecher sequence is solved explicitly the solution is expressed in terms of a special binary sequence the notions of macro and micro dynamics are also introduced the obtained explicit formulae are used for the group e 2 to describe micro dynamics generated by the binary sequence as well as macro dynamics induced by the shimizu leutbecher sequence fractals induced by the action of the group e 2 in the plane are described and are related to rayleigh pearson type random walks

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

an exact algorithm for the subset sum problem abstract the subset sum problem ssp is defined as given n positive integers w1 wn find a combination amongst them such that their sum is the closest to but not exceeding a positive integer c we suggest an exact algorithm by introducing a new type of core problem and also by using an improved version of bellman s recursion we show that the resulting algorithm is bounded in time and space resource utilizations respectively by o max n log 2 c 2 c c log 2 c and o n c in addition to the sharp memory requirement decrease in comparison with any dynamic programming based algorithm the search space for a vast range of instances is restricted only to feasible states

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

lexicographic differentiation of nonsmooth functions we present a survey on the results related to the theory of lexicographic differentiation this theory ensures an efficient computation of generalized lexicographic derivative of a nonsmooth function belonging to a special class of lexicographically smooth functions this class is a linear space which contains all differentiable functions all convex functions and which is closed with respect to component wise composition of the members in order to define lexicographic derivative in a unique way it is enough to fix a basis in the space of variables lexicographic derivatives can be used in black box optimization methods we give some examples of applications of these derivatives in analysis of nonsmooth functions it is shown that the system of lexicographic derivatives along a fixed basis correctly represents corresponding nonsmooth function newton leibnitz formula we present nonsmooth versions of standard theorems on potentiality of nonlinear operators on differentiation of parametric integrals and on differentiation of functional sequences finally we show that an appropriately defined lexicographic subdifferential ensures a more rigorous selection of a candidate optimal solution than the subdifferential of clarke

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

bifurcations of lattice structures equilibrium configurations of a model for a crystal in one spatial dimension with anharmonic and up to third neighbour harmonic interactions are related to orbits in a two or four dimensional space under a nonlinear symplectic mapping in this way a connection is given with dynamical systems with two or three degrees of freedom depending on the parameters the orbits are smooth or stochastic their characterisation in terms of their fixed points and their ljapunov exponents is given the mapping shows a large number of bifurcations of which one can distinguish various types and these are discussed

sensitivity analysis of limit cycle oscillating hybrid systems a theory is developed for local first order sensitivity analysis of limit cycle oscillating hybrid systems which are dynamical systems exhibiting both continuous state and discrete state dynamics whose state trajectories are closed isolated and time periodic methods for the computation of initial condition sensitivities and parametric sensitivities are developed to account exactly for any jumps in the sensitivities at discrete transitions and to exploit the time periodicity of the system it is shown that the initial condition sensitivities of any limit cycle oscillating hybrid system can be represented as the sum of a time decaying component and a time periodic component so that they become periodic in the long time limit a method is developed for decomposition of the parametric sensitivities into three parts characterizing the influence of parameter changes on period state variable amplitudes and relative phases respectively the computation of parametric sensitivities of period amplitudes and different types of phases is also described the methods developed in this work are applied to particular models for illustration including models exhibiting state variable jumps

an exact algorithm for orthogonal 2 d cutting problems using guillotine cuts abstract we consider the two dimensional cutting problem which requires cutting a number of small rectangular pieces each of a given size and value from a large reatangular stock plate the objective is to maximise the value of pieces cut or for a slightly simpler problem to minimise the wastage we consider the case where there is a maximum number of times that a piece may be used in a cutting pattern we present a tree search algorithm for this problem in which the size of the tree search is limited by using a bound derived from a state space relaxation of a dynamic programming formulation of the problem a state space ascent method is used to optimise this bound the computational performance of the algorithm is investigated by tests performed on randomly generated problems with constraints of varying tightness the results indicate that the algorithm is an effective procedure capable of solving optimally practical two dimensional cutting problems of medium size

sensitivity analysis of composite piecewise smooth equations this paper is a contribution to the sensitivity analysis of piecewise smooth equations a piecewise smooth function is a lipschitzian homeomorphism near a given point if and only if it is coherently oriented and has an invertible b derivative at this point we emphasise the role of functions of the typef g h whereg is piecewise smooth andh is smooth and present verifiable conditions which ensure that the functionf g tilde h is a lipschitzian homeomorphism near a given point for every tilde h sufficiently close toh with respect to thec1 topology

establishing the optimality of sequencing heuristics for cutting stock problems cutting stock problems typically involve the generation of feasible cutting patterns whilst minimizing waste in some instances the sequence of cutting these patterns is important one objective of the sequencing stage is to minimize the maximum queue of partially cut orders this paper presents two exact methods to meet this objective and to determine the optimality of previously introduced sequencing heuristics the first and more simple method is based on the lower bound the other a more general method is an implicit exhaustive search procedure various enhancements are introduced which allow sets of up to thirty pattern types to be successfully searched aspects of graph theory have been utilised to minimise the size of sets of patterns to be searched computing experience is reported for determining optimality

two dimensional cutting stock problem with sequence dependent setup times motivated by a firm in the technical textile industry we study a two dimensional cutting stock problem with sequence dependent setup times and permissible tolerances we provide a sequential heuristic with feedback loop based on the approach of gilmore and gomory and formulate the sequencing problem as a mixed integer program we derive a lower bound algorithm and demonstrate the near optimal performance of our heuristic finally we use real data to test our heuristic and illustrate its applicability to a problem of realistic size

quantum kicked top lie algebraic approach abstract dynamical system describing nonlinear dynamics of a three dimensional angular momentum j is analysed it is shown that the dynamics can be linearised by suitable embedding in a larger linear space of 2j 1 lie algebra consequences of the linearisation are analysed and especially possibility of defining dual classical dynamics is considered

lyapunov based sufficient conditions for stability of hybrid systems with memory hybrid systems with memory are dynamical systems exhibiting both hybrid and delay phenomena in this note we study the asymptotic stability of hybrid systems with memory using generalized concepts of solutions these generalized solutions motivated by studying robustness and well posedness of such systems are defined on hybrid time domains and parameterized by both continuous and discrete time we establish lyapunov based sufficient conditions for asymptotic stability using both lyapunov razumikhin functions and lyapunov krasovskii functionals examples are provided to illustrate these conditions

exact solutions for constrained two dimensional cutting problems abstract the constrained two dimensional cutting problem is concerned with the way a set of pieces should be cut from a plate taking into account that the cuts are of the guillotine type and that the number of times a piece can be cut is limited both plate and pieces are rectangular shaped yr 1983 wang proposed an algorithm for incremental development of rectangles later oliveira and ferreira not only proposed but numerically tested an improvement to the algorithm as a generalization of these methods is proposed it will be stated that such methods are particular cases of uninformed search methods widely studied in the area of artificial intelligence knowing that more efficient methods exist namely informed methods diffren alternatives to orientate the search process while reducing the required computer memory and time are defined using an adequate selection of an heuristic function it is possible to guarantee the determination of a best solution such ideas are numerically tested in this work the results indicate the superiority of the proposed algorithm

generalized derivatives for hybrid systems established sensitivity results for hybrid discrete continuous dynamic systems are generalized by relaxing smoothness assumptions on the functions governing the systems continuous evolution and discrete event handling the new results only require l smoothness of these functions in the sense of nesterov instead of continuous differentiability parametric lexicographic derivatives for such a hybrid system provide useful local first order sensitivity information and are described as the unique solutions of auxiliary hybrid systems this sensitivity analysis framework permits generalized derivative evaluation even for certain hybrid systems in which small changes in parameters can change the sequence of discrete modes visited to handle parametric sensitivities of event times that are not known explicitly conditions are provided under which a local inverse function or implicit function is l smooth with lexicographic derivatives that are described as the unique solutions of certain equation systems these equation systems are readily solved when the functions involved are piecewise differentiable

exponential stability of impulsive stochastic functional differential equations in this paper we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using lyapunov method several stability theorems of impulsive stochastic functional differential equations with finite delay are derived these new results are employed to impulsive stochastic equations with bounded time varying delays and stochastically perturbed equations meanwhile an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay

generalized sensitivity analysis of nonlinear programs this paper extends classical sensitivity results for nonlinear programs to cases in which parametric perturbations cause changes in the active set this is accomplished using lexicographic directional derivatives a recently developed tool in nonsmooth analysis based on nesterov s lexicographic differentiation a nonsmooth implicit function theorem is augmented with generalized derivative information and applied to a standard nonsmooth reformulation of the parametric kkt system it is shown that the sufficient conditions for this implicit function theorem variant are implied by a kkt point satisfying the linear independence constraint qualification and strong second order sufficiency mirroring the classical theory the resulting sensitivity system is a nonsmooth equation system which admits primal and dual sensitivities as its unique solution practically implementable algorithms are provided for calculating the nonsmooth sensitivity system s unique solution which is then used to furnish b subdifferenti

the dh kd algorithm a hybrid approach for unconstrained two dimensional cutting problems abstract we study both weighted and unweighted unconstrained two dimensional guillotine cutting problems we develop a hybrid approach which combines two heuristics from the literature the first one dh uses a tree search procedure introducing two strategies depth first search and hill climbing the second one kd is based on a series of one dimensional knapsack problems using dynamic programming techniques the dh kd algorithm starts with a good initial lower bound obtained by using the kd algorithm at each level of the tree search the proposed algorithm uses also the kd algorithm for constructing new lower bounds and uses another one dimensional knapsack for constructing refinement upper bounds the resulting algorithm can be seen as a generalization of the two heuristics and solves large problem instances very well within small computational time our algorithm is compared to morabito et al s algorithm the unweighted case and to beasley s 2 approach the weighted case on some examples taken from the literature as well as randomly generated instances

an and or graph approach to the solution of two dimensional non guillotine cutting problems this work is concerned with unconstrained two dimensional non guillotine cutting problems where a rectangular plate is cut into a number of rectangular pieces in such a way as to optimize an objective function we reduce the state space of the problem by considering non guillotine cutting patterns which are combinations of guillotine and simple non guillotine cuts these cutting patterns are represented as complete paths in an and or graph and a branch and bound method is described we provide rules and heuristics that reduce the graph search and present computational results from randomly generated examples as well as from the literature

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

heuristic and exact methods for the cutting sequencing problem cutting stock problems deal with the generation of a set of cutting patterns that minimizes waste sometimes it is also important to find the processing sequence of this set of patterns to minimize the maximum queue of partially cut orders in such instances a cutting sequencing problem has to be solved this paper presents a new mathematical model and a three phase approach for the cutting sequencing problem in the first phase a greedy algorithm produces a good starting solution that is improved in the second phase by a tabu search or a generalized local search procedure while in the last phase the problem is optimally solved by an implicit enumeration procedure that uses the best solution previously found as an upper bound computing experience based on 300 randomly generated problems shows the good performance of the heuristic methods presented

dependence of solutions of nonsmooth differential algebraic equations on parameters abstract the well posedness of nonsmooth differential algebraic equations daes is investigated more specifically semi explicit daes with caratheodory style assumptions on the differential right hand side functions and local lipschitz continuity assumptions on the algebraic equations the daes are classified as having differential index one in a generalized sense solution regularity is formulated in terms of projections of generalized clarke jacobians existence of solutions is derived under consistency and regularity of the initial data uniqueness of a solution is guaranteed under analogous caratheodory ordinary differential equation uniqueness assumptions the continuation of solutions is established and sufficient conditions for continuous and lipschitzian parametric dependence of solutions are also provided to accomplish these results a theoretical tool for analyzing nonsmooth daes is provided in the form of an extended nonsmooth implicit function theorem the findings here are a natural extension of classical results and lay the foundation for further theoretical and computational analyses of nonsmooth daes

predictive control of switched nonlinear systems with scheduled mode transitions in this work a predictive control framework is proposed for the constrained stabilization of switched nonlinear systems that transit between their constituent modes at prescribed switching times the main idea is to design a lyapunov based predictive controller for each constituent mode in which the switched system operates and incorporate constraints in the predictive controller design which upon satisfaction ensure that the prescribed transitions between the modes occur in a way that guarantees stability of the switched closed loop system this is achieved as follows for each constituent mode a lyapunov based model predictive controller mpc is designed and an analytic bounded controller using the same lyapunov function is used to explicitly characterize a set of initial conditions for which the mpc irrespective of the controller parameters is guaranteed to be feasible and hence stabilizing then constraints are incorporated in the mpc design which upon satisfaction ensure that 1 the state of the closed loop system at the time of the transition resides in the stability region of the mode that the system is switched into and 2 the lyapunov function for each mode is nonincreasing wherever the mode is reactivated thereby guaranteeing stability the proposed control method is demonstrated through application to a chemical process example

on determining the dimension of chaotic flows abstract we describe a method for determining the approximate fractal dimension of an attractor our technique fits linear subspaces of appropriate dimension to sets of points on the attractor the deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques we show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time varying quantity is available for analysis these methods are applied to several dissipative dynamical systems