l and r localized solvabilities of max separable interval linear equations and its applications in this paper we propose new types of solutions an l localized solution and an r localized solution to interval systems of max separable linear equations a x b where a a a and b b b we provide some small application examples to emphasize the importance of our proposed solutions we give necessary and sufficient conditions for checking l localized and r localized solvabilities and apply them to achieve the closed forms of the sets of all l localized solutions and r localized solutions respectively

strong solvability of interval max plus systems and applications to optimal control abstract this paper considers the strong solvability of interval max plus systems the solvable entry and solvable interval are introduced to establish the necessary and sufficient condition for strong solvability of interval max plus systems it is proved that such a criterion is equivalent to the criterion proposed by cechlarova and cuninghame green the interval strong solution of interval max plus systems is described and the necessary and sufficient condition for uniquely strong solvability is given the method of testing strong solvability of interval max plus systems is constructive and leads to an effective algorithm to illustrate our results a workshop optimal control problem is addressed

brief equivalence of hybrid dynamical models this paper establishes equivalences among five classes of hybrid systems mixed logical dynamical mld systems linear complementarity lc systems extended linear complementarity elc systems piecewise affine pwa systems and max min plus scaling mmps systems some of the equivalences are established under rather mild additional assumptions these results are of paramount importance for transferring theoretical properties and tools from one class to another with the consequence that for the study of a particular hybrid system that belongs to any of these classes one can choose the most convenient hybrid modeling framework

steady states in the scheduling of discrete time systems abstract steady solutions to a two sided max min linear system of equations with real coefficients are considered the problem is motivated by the demand to schedule cyclically repeated activities with deterministic processing times the maximum steady solutions are characterized as min eigenvectors of a special matrix q this condition is a necessary but not a sufficient one it has also been proven that the general problem of the recognition of solvability of this two sided system is np complete

ramanujan s series for 1 π arising from his cubic and quartic theories of elliptic functions using certain representations for eisenstein series we derive several of ramanujan s series for 1 arising from his cubic and quartic theories of elliptic functions

controllability of piecewise linear systems abstract the paper deals with a discontinuous control system that consists of two finite dimensional linear systems the first describes the motion in a given half space the second in the complementary half space necessary and sufficient conditions for local controllability are obtained

interval systems of max separable linear equations for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations we define and characterize several types of solutions and solvability of interval systems of max separable linear equations

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

class number three ramanujan type series for 1 p abstract there is a class of remarkable series for 1 of the form c 3 n 0 a nb c 3n 6n 3n n 3 where a b c are certain algebraic numbers these were first examined by ramanujan these examples arise by computing singular invariants for j and the constants involved will have degree equal to the class number of the associated imaginary quadratic field our intention is compute all such class number three examples the largest such example with discriminant 907 adds 37 additional digit accuracy per term a class number four example with discriminant 1555 gives 50 digits per term

a polynomial algorithm for solving system of inequalities in max plus algebra abstract this paper deals with the system of inequalities a x x in which a is a square matrix x is a column vector and is a max plus composition it provides two algorithms which can be used to find a finite proper supereigenvectors of a corresponding to the eigenvalue 0 in a case of irreducible matrix a with the maximum cycle mean a 0 and a finite solution of the system in a case of reducible matrix a respectively in a polynomial number of steps

control solvability of interval systems of max separable linear equations abstract for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations several types of solvability of interval systems are known we define control solvability weak control solvability and weak universal solvability and give necessary and sufficient condition for an interval system to be solvable according to each definition

ae solutions to interval linear systems over max plus algebra abstract this paper introduces a concept of ae solutions to interval max plus linear systems a rather general concept which includes many known concepts of solutions to interval systems weak solutions strong solutions tolerance solutions and control solutions as its special cases we state full characterizations of ae solutions for the interval max plus systems including both linear inequalities and linear equations moreover a kind of dependency characterized by double appearance of some sub matrix of the constraint matrix is discussed and we prove that which can be relaxed with no change in ae solvability

extending conceptualisation modes for generalised formal concept analysis formal concept analysis fca is an exploratory data analysis technique for boolean relations based on lattice theory its main result is the existence of a dual order isomorphism between two set lattices induced by a binary relation between a set of objects and a set of attributes pairs of dually isomorphic sets of objects and attributes called formal concepts form a concept lattice but actually model only a conjunctive mode of conceptualisation in this paper we augment this formalism in two ways first we extend fca to consider different modes of conceptualisation by changing the basic dual isomorphism in a modal logic motivated way this creates the three new types of concepts and lattices of extended fca viz the lattice of neighbourhood of objects that of attributes and the lattice of unrelatedness second we consider incidences with values in idempotent semirings concretely the completed max plus or schedule algebra email protected m a x and focus on generalising fca to try and replicate the modes of conceptualisation mentioned above to provide a concrete example of the use of these techniques we analyse the performance of multi class classifiers by conceptually analysing their confusion matrices

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

an algorithm for solving two sided interval system of max plus linear equations we study six types of solutions weak strong tolerance control left localized and right localized solutions to a two sided interval system of max plus linear equations with the same vector of variables on both sides of the equations and obtain their corresponding solvability conditions these conditions are in the form of two sided systems of max plus linear inequalities which can be derived as a union of a number of interval inclusion linear systems therefore we could work on the interval inclusion linear systems instead of directly solving these six types of solutions themselves an optimization problem with the two sided interval system of max plus linear constraints may be solved using the convexity of each solution set of the interval inclusion linear systems

tolerance types of interval eigenvectors in max plus algebra interval eigenvectors of interval matrices corresponding to steady states of systems with inexact data are investigated tolerable strongly tolerable and weakly tolerable interval eigenvectors are described polynomial solutions to the recognition versions of all three types of tolerability are presented max plus algebra plays a key role in the study of discrete event systems in connection with optimization problems such as scheduling or project management in which the objective function depends on the operations maximum and plus the steady states of such systems correspond to the eigenvectors of max plus matrices therefore the investigation of the properties of these eigenvectors is important for applications matrices and vectors with interval coefficients in a max plus algebra are studied in this paper and polynomial algorithms are presented for the recognition problem with three types of tolerance interval eigenvector

a sensitivity analysis of the music algorithm a first order analysis of the sensitivity of the music algorithm to system errors which cause differences between the array manifold used by music and the true array manifold is presented the effect of such errors on the direction of arrival estimates is investigated the ability of the music algorithm to resolve two closely spaced sources in the presence of phase and gain errors in the array elements or the receivers or errors in the element locations is analyzed a formula for computing the failure threshold of music as a function of source separation and other system parameters is derived and tested by simulation the analysis assumes that the exact covariance matrix of array element outputs is known the infinite data case

matrix fitting approach to direction of arrival estimation with imperfect spatial coherence of wavefronts the performance of high resolution direction of arrival doa estimation methods significantly degrades in several practical situations where the wavefronts have imperfect spatial coherence the original solution to this problem was proposed by paulraj and kailath 1988 but their technique requires a priori knowledge of the matrix characterizing the loss of wavefront coherence along the array aperture a novel solution to this problem is proposed which does not require a priori knowledge of the spatial coherence matrix

optimistic optimization for model predictive control of max plus linear systems model predictive control for max plus linear discrete event systems usually leads to a nonsmooth nonconvex optimization problem with real valued variables which may be hard to solve efficiently an alternative approach is to transform the given problem into a mixed integer linear programming problem however the computational complexity of current mixed integer linear programming algorithms increases in the worst case exponentially as a function of the prediction horizon the focus of this paper is on making optimistic optimization suited to solve the given problem optimistic optimization is a class of algorithms that can find an approximation of the global optimum for general nonlinear optimization a key advantage of optimistic optimization is that one can specify the computational budget in advance and guarantee bounds on the suboptimality with respect to the global optimum we prove that optimistic optimization can be applied for the given problem by developing a dedicated semi metric and by proving it satisfies the necessary requirements for optimistic optimization moreover we show that the complexity of optimistic optimization is exponential in the control horizon instead of the prediction horizon hence using optimistic optimization is more efficient when the control horizon is small and the prediction horizon is large

interval max plus matrix equations abstract this paper deals with the solvability of interval matrix equations in max plus algebra max plus algebra is the algebraic structure in which classical addition and multiplication are replaced by and where a b max a b and a b a b the notation a x c b represents an interval max plus matrix equation where a b and c are given interval matrices we define four types of solvability of interval max plus matrix equations i e the tolerance weak tolerance left weak tolerance and right weak tolerance solvability we derive the necessary and sufficient conditions for checking each of them whereby all can be verified in polynomial time

interval systems of max separable linear equations an interval system of equations is weakly solvable if at least one of its subsystems is solvable and it is strongly solvable if all its subsystems are solvable we give necessary and sufficient conditions enabling efficient testing of weak and strong solvability over max plus and max min algebras

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

a performance analysis of subspace based methods in the presence of model errors i the music algorithm application of subspace based algorithms to narrowband direction of arrival doa estimation requires that both the array response in all directions of interest and the spatial covariance of the noise must be known in practice however neither of these quantities is known precisely depending on the degree to which they deviate from their nominal values serious performance degradation can result the performance of the music algorithm is examined for situations in which the noise covariance and array response are perturbed from their assumed values theoretical expressions for the error in the music doa estimates are derived and compared with simulations performed for several representative cases and with the appropriate cramer rao bound an optimally weighted version of music is proposed for a particular class of array errors

efficient quadratic detection in perturbed arrays via fourier transform techniques matched field beamforming used in combination with a generalized likelihood ratio test is the most common detector structure in array processing situations unfortunately various array perturbations caused by phase calibration propagation effects or modeling errors can cause the sensor observations to become only partially correlated limiting the performance of traditional matched field beamformers that assume perfect coherence of the signal wavefronts quadratic array processing is optimal for many perturbed array problems however direct implementation poses a significant computational burden we show that under certain conditions the optimal quadratic detector for dealing with perturbed arrays can be approximately realized efficiently and robustly employing only discrete fourier transforms to deal with spatial processing in addition we show that the proposed spatial processing allows for convenient integration of conventional frequency domain methods for angle of arrival searches our proposed array detection structure provides the robustness and performance benefits of complicated quadratic processing at a computational cost comparable with that of traditional matched field beamforming

the equation a x b y over max for the two sided homogeneous linear equation system a x b y over max with no infinite rows or columns in a or b an algorithm is presented which converges to a finite solution from any finite starting point whenever a finite solution exists if the finite elements of a b are all integers convergence is in a finite number of steps for which a precise bound can be calculated if moreover one of a b has only finite elements the algorithm is thus pseudopolynomial in complexity

localized solutions to interval linear equations in this paper we propose a new type of solutions localized solutions to interval linear equations ax b the new solutions are based on l localized solutions and r localized solutions the concepts of the three solutions are given and the characteristics of the new solutions to the interval linear equations are described by means of nonlinear inequality as is similar to the well known oettli prager inequality which is related to the mid point and interval radius some necessary conditions of the three types of solutions to the equations are given

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

direction of arrival estimation by eigenstructure methods with imperfect spatial coherence of wave fronts eigenstructure subspace methods for direction of arrival estimation are only applicable when each wave front contributes a rank one component to the array covariance this implies that the wave fronts need to have perfect spatial coherence in several practical situations where the wave fronts may only have partial spatial coherence eigenstructure algorithms show poor performance such as overestimation of the number of sources bias and reduced accuracy and resolution in estimating the directions of arrival in this article the performance degradation due to partial spatial coherence is studied and a new approach is proposed that exploits the underlying coherence model to yield asymptotically exact estimates for finite data this method provides improved results only when the number of snapshots exceeds a certain threshold results of computer simulations are also presented

max plus algebra based wavelet transforms and their fpga implementation for image coding in the present paper we propose new image coding methods based on max plus algebra based wavelet transforms mp wavelets max plus algebra is an algebraic structure on the set of integers endowed with maximum minimum and standard addition as operations the proposed wavelet decomposition schemes are novel classes of morphological wavelets which in contrast to the existing approaches involve only integers and only max min and addition in their computation since they do not involve floating point calculations and multiplications mp wavelets have a very low computational complexity and they are suitable for efficient hardware implementation through image coding experiments using 10 000 images extracted from low resolution web crawled misc database the characteristics of the proposed mp wavelets are presented furthermore we show a computational complexity analysis to compare mp wavelets to one of the fastest wavelet transforms the conventional haar wavelet the applicability of mp wavelets in video coding is also confirmed by image coding experiments using residual frames finally the field programmable gate arrays fpga implementation of mp wavelets is presented to confirm suitability of mp wavelets for hardware implementation

π formulas with free parameters abstract in terms of the hypergeometric method we establish ten general formulas with free parameters which include several known results as special cases

data based identification and control of nonlinear systems via piecewise affine approximation the piecewise affine pwa model represents an attractive model structure for approximating nonlinear systems in this paper a procedure for obtaining the pwa autoregressive exogenous arx autoregressive systems with exogenous inputs models of nonlinear systems is proposed two key parameters defining a pwarx model namely the parameters of locally affine subsystems and the partition of the regressor space are estimated the former through a least squares based identification method using multiple models and the latter using standard procedures such as neural network classifier or support vector machine classifier having obtained the pwarx model of the nonlinear system a controller is then derived to control the system for reference tracking both simulation and experimental studies show that the proposed algorithm can indeed provide accurate pwa approximation of nonlinear systems and the designed controller provides good tracking performance

a summation formula and ramanujan type series abstract using some properties of the general rising shifted factorial and the gamma function we derive a variant form of dougall s f 4 5 summation for the classical hypergeometric functions this variant form allows us to derive easily many ramanujan type series for 1 and ramanujan type series for some other constants

on visualization scaling subeigenvectors and kleene stars in max algebra abstract the purpose of this paper is to investigate the interplay arising between max algebra convexity and scaling problems the latter which have been studied in nonnegative matrix theory are strongly related to max algebra one problem is that of strict visualization scaling defined as for a given nonnegative matrix a a diagonal matrix x such that all elements of x 1 ax are less than or equal to the maximum cycle geometric mean of a with strict inequality for the entries which do not lie on critical cycles in this paper such scalings are described by means of the max algebraic subeigenvectors and kleene stars of nonnegative matrices as well as by some concepts of convex geometry

hypergeometric series summations and π formulas abstract by two slater s hypergeometric series identities we establish two general summation formulas with six free parameters specializing certain parameters in these formulas we obtain a list of ramanujan type series formulas for 2 and 3

controllable solution set to interval static systems for interval systems of algebraic equations the concept of a controllable solution set is introduced and its practical interpretation is given the main results of the work concern interval linear systems and afford simple sufficient criteria for the controllable solution set to be nonempty and for the intersection of the controllable solution set with some orthants to be empty

mpc for continuous piecewise affine systems abstract a large class of hybrid systems can be described by a max min plus scaling mmps model i e using the operations maximization minimization addition and scalar multiplication first we show that continuous piecewise affine systems are equivalent to mmps systems next we consider model predictive control mpc for these systems in general this leads to nonlinear nonconvex optimization problems we present a new mpc method for mmps systems that is based on canonical forms for mmps functions in case the mpc constraints are linear constraints in the inputs only this results in a sequence of linear optimization problems such that the mpc control can often be computed in a much more efficient way than by just applying nonlinear optimization as was done in previous work

the generators of the solution space for a system of inequalities the generators of the solution space for a system of inequalities a x x are considered in this paper where a is a square matrix over r r x is a column vector and is a max plus composition this paper presents some properties of solutions to the system of inequalities and proposes an algorithm to find a set of generators for the solution space of the system of inequalities with a r n n

on max plus linear dynamical system theory a class of timed discrete event systems can be modeled by using timed event graphs a class of timed petri nets that can have its firing dynamic described by using an algebra called max plus algebra for this kind of systems it may be desirable to enforce some timing constraints in steady state in this paper this problem is called a max plus regulation problem in this context we show a necessary condition for solving these regulation problems and in addition that this condition is sufficient for a large class of problems the obtained controller is a simple linear static state feedback and can be computed using efficient pseudo polynomial algorithms simulation results will illustrate the applicability of the proposed methodology

recognizing weakly stable matrices a max plus matrix a is called weakly stable if the sequence orbit x a otimes x a 2 otimes x ldots does not reach an eigenvector of a for any x unless x is an eigenvector this is in contrast to previously studied strongly stable robust matrices for which the orbit reaches an eigenvector with any nontrivial starting vector max plus matrices are used to describe multiprocessor interactive systems for which reachability of a steady regime is equivalent to reachability of an eigenvector by a matrix orbit we prove that an irreducible matrix is weakly stable if and only if its critical graph is a hamiltonian cycle in the associated graph we extend this condition to reducible matrices these criteria can be checked in polynomial time

closed form of l localized solution set of max plus interval linear system and its application on optimization problem a closed form of the l localized solution set of an interval linear system with max plus algebra is a finite union of interval vectors this paper analyzes and proves some properties to reduce number of the union sets to be a smaller number of the union of interval vectors that are not subsets of each other this result helps simplify the feasible set of an optimization problem with max plus linear equation constraints