on the invertible cellular automata 150 over f p in this work the reversibility problem for cellular automata with rule number 150 over the finite field f p is tackled it is shown that when null boundary conditions are stated the reversibility is independent of the state set f p and it appears when the number of cells of the cellular space satisfies some conditions furthermore the explicit expressions of the inverse cellular automata are computed

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

tracking while zooming using affine transfer and multifocal tensors this paper presents algorithms for tracking unknown objects in the presence of zoom since prior models are unavailable point and line matches in affine views are used to characterize the structure and to transfer a fixation point into new images in a sequence because any affine projection matrix is permitted the intrinsic camera parameters such as focal length may change freely also since the techniques do not require long feature tracks a further desirable property is insensitivity to partial occlusion caused for instance by part of the object falling off the image plane while zooming in if only point matches are available a previous method based on factorization is applied when also incorporating lines the affine trifocal and quadrifocal tensors are used for tracking in monocular and stereo systems respectively methods for computing the tensors minimizing algebraic error are developed in comparison with their projective counterparts the affine tensors offer significant advantages in terms of computation time and convenience of parameterization and the relations between the different tensors are shown to be much simpler successful tracking is demonstrated on several real image sequences

reversible one dimensional cellular automata with one of the two welch indices equal to 1 and full shifts reversible cellular automata are invertible discrete dynamical systems which have been widely studied both for analysing interesting theoretical questions and for obtaining relevant practical applications for instance simulating invertible natural systems or implementing data coding devices an important problem in the theory of reversible automata is to know how the local behaviour which is not invertible is able to yield a reversible global one in this sense symbolic dynamics plays an important role for obtaining an adequate representation of a reversible cellular automaton in this paper we prove the equivalence between a reversible automaton where the ancestors only differ at one side technically with one of the two welch indices equal to 1 and a full shift we represent any reversible automaton by a de bruijn diagram and we characterize the way in which the diagram produces an evolution formed by undefined repetitions of two states by means of amalgamations we prove that there is always a way of transforming a de bruijn diagram into the full shift finally we provide an example illustrating the previous results

invertible cellular automata a review abstract in the light of recent developments in the theory of invertible cellular automata we attempt to give a unified presentation of the subject and discuss its relevance to computer science and mathematical physics

efficient exhaustive listings of reversible one dimensional cellular automata this paper looks at an algebraic formulation of one dimensional cellular automata using the formulation connections to combinatorial structures and graph theory become clear strong results about uniqueness and isomorphism allows us to outline effective algorithms for the generation of exhaustive lists of reversible one dimensional cellular automata and to count the number of distinct examples that exist these algorithms use the orderly algorithm methods to avoid the pitfalls of brute force searches

structure from motion using line correspondences a theory is presented for the computation of three dimensional motion and structure from dynamic imagery using only line correspondences the traditional approach of corresponding microfeatures interesting points highlights corners high curvature points etc is reviewed and its shortcomings are discussed then a theory is presented that describes a closed form solution to the motion and structure determination problem from line correspondences in three views the theory is compared with previous ones that are based on nonlinear equations and iterative methods

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

graph theoretical characterization of invertible cellular automata abstract a cellular automaton is defined to be invertible if its induced global rule is surjective on the set of all finite configurations it is shown that all one dimensional invertible cellular automata are obtainable from invertible nearest pair ones i e cellular automata given by a rule matrix for these an algorithm is proposed which runs in a time which is a polynomial function of the number of letters of the alphabet the result is expressed in a graph theoretical way the cellular automaton is invertible if and only if two graphs constructed by the algorithm have no edges in common it follows that if the number of letters in the alphabet is a prime the only invertible cellular automata are those for which every row or column of the rule matrix is a permutation of the alphabet for composite alphabet cardinalities several methods analytical and computational ones to obtain more complicated invertible cellular automata are proposed and used to construct examples all four letter invertible automata are constructed

self calibration of an affine camera from multiple views a key limitation of all existing algorithms for shape and motion from image sequences under orthographic weak perspective and para perspective projection is that they require the calibration parameters of the camera we present in this paper a new approach that allows the shape and motion to be computed from image sequences without having to know the calibration parameters this approach is derived with the affine camera model introduced by mundy and zisserman 1992 which is a more general class of projections including orthographic weak perspective and para perspective projection models the concept of self calibration introduced by maybank and faugeras 1992 for the perspective camera and by hartley 1994 for the rotating camera is then applied for the affine camera this paper introduces the 3 intrinsic parameters that the affine camera can have at most the intrinsic parameters of the affine camera are closely related to the usual intrinsic parameters of the pin hole perspective camera but are different in the general case based on the invariance of the intrinsic parameters methods of self calibration of the affine camera are proposed it is shown that with at least four views an affine camera may be self calibrated up to a scaling factor leading to euclidean similarity shape reconstruction up to a global scaling factor another consequence of the introduction of intrinsic and extrinsic parameters of the affine camera is that all existing algorithms using calibrated affine cameras can be assembled into the same framework and some of them can be easily extented to a batch solution experimental results are presented and compared with other methods using calibrated affine cameras

predicting nonlinear cellular automata quickly by decomposing them into linear ones we show that a wide variety of nonlinear cellular automata cas can be decomposed into a quasidirect product of linear ones these cas can be predicted by parallel circuits of depth o log2 t using gates with binary inputs or o log t depth if sum mod p gates with an unbounded number of inputs are allowed thus these cas can be predicted by idealized parallel computers much faster than by explicit simulation even though they are nonlinear this class includes any ca whose rule when written as an algebra is a solvable group we also show that cas based on nilpotent groups can be predicted in depth o log t or o 1 by circuits with binary or sum mod p gates respectively we use these techniques to give an efficient algorithm for a ca rule which like elementary ca rule 18 has diffusing defects that annihilate in pairs this can be used to predict the motion of defects in rule 18 in o log2 t parallel time

physics like models of computation abstract reversible cellular automata are computer models that embody discrete analogues of the classical physics notions of space time locality and microscopic reversibility they are offered as a step towards models of computation that are closer to fundamental physics

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

motion and structure from line correspondences closed form solution uniqueness and optimization this work discusses estimating motion and structure parameters from line correspondences of a rigid scene the authors present a closed form solution to motion and structure parameters from line correspondences through three monocular perspective views the algorithm makes use of redundancy in the data to improve the accuracy of the solutions the uniqueness of the solution is established and necessary and sufficient conditions for degenerate spatial line configurations are given optimization has been employed to further improve the accuracy of the estimates in the presence of noise simulations have shown that the errors of the optimized estimates are close to the theoretical lower error bound

lines and points in three views and the trifocal tensor this paper discusses the basic role of the trifocal tensor in scene reconstruction from three views this 3 times 3 times 3 tensor plays a role in the analysis of scenes from three views analogous to the role played by the fundamental matrix in the two view case in particular the trifocal tensor may be computed by a linear algorithm from a set of 13 line correspondences in three views it is further shown in this paper that the trifocal tensor is essentially identical to a set of coefficients introduced by shashua to effect point transfer in the three view case this observation means that the 13 line algorithm may be extended to allow for the computation of the trifocal tensor given any mixture of sufficiently many line and point correspondences from the trifocal tensor the camera matrices of the images may be computed and the scene may be reconstructed for unrelated uncalibrated cameras this reconstruction will be unique up to projectivity thus projective reconstruction of a set of lines and points may be carried out linearly from three views

fuzzy multiple objective programming and compromise programming with pareto optimum abstract a fuzzy multiple objective decision making approach is proposed based on the desirable features of compromise programming and the fuzzy set theory the proposed two phase approach guarantees both nondominated and balanced solutions for solving both the crisp and the fuzzy multiple objective decision making problems furthermore it is shown that the original compromise programming does not guarantee nondominated solutions if the distance parameter is assumed to be infinite and the resulting programming problem s solution is not unique

an integrated approach for deriving priorities in analytic network process a multiple objective programming approach for the analytic network process anp is proposed to obtain all local priorities for crisp or interval judgments at one time even in an inconsistent situation the weakness of the anp and fuzzy anp fanp is that the complexity of generating priorities is equal to the number of comparison matrices in the proposed approach all sets of crisp priorities for each pairwise comparison matrix can be obtained directly moreover from the outcomes of three examples the power to reach a limiting supermatrix is less than or equal to the power of the fanp thus the proposed approach can be regarded as an efficient alternative of the fuzzy anp

reversibility of linear cellular automata in this paper the reversibility problem for linear cellular automata defined by a characteristic matrix of the form of a pentadiagonal matrix is tackled specifically a criterion for the reversibility in terms of the number of cells is stated and in these cases the inverse cellular automata are explicitly computed

exact results for deterministic cellular automata with additive rules deterministic cellular automata ca with additive rules are studied by exploiting the properties of circulant matrices on finite fields complete state transition diagrams for higher order and multidimensional ca on finite lattices are analyzed conditions on the rules which make them reversible are obtained it is shown that all state transition diagrams of the ca have identical trees rooted on cycles general formulae for cycle lengths and multiplicities are given

approximate articulation of preference and priority derivation a comment abstract in a recent paper by a arbel 1989 the standard analytic hierarchy process ahp model is extended to the case where preference strength is expressed in terms of a range of scale values rather than a single value while this extension holds and is natural for the consistent case it is shown that the proposed solution for the more common inconsistent case is erroneous

some applications of information theory to cellular automata abstract in this paper general deterministic one dimensional cellular automata are identified with mappings of the unit interval into itself this allows the machinery of dynamical systems analysis to be employed the emphasis of the paper however is on applications of existing concepts and techniques of information theory to these automata a basic paper by w m conner is utilized to obtain equality of the capacity and hausdorff dimension of each line of the automata and existence of limiting values of these quantities is established assuming a probability measure on the initial line that is stationary and ergodic for the shift a consistent ergodic theory is derived for any finite or infinite collection of lines of the automata a body of related work by russian authors on probabilistic automata is briefly examined important questions about the existence and properties of limiting distributions remain unresolved

classifying circular cellular automata we introduce a hierarchy of linear cellular automata based on their limiting behavior on spatially periodic configurations we show that it is undecidable to which class in the hierarchy a cellular automaton belongs in particular it is undecidable whether all spatially periodic configurations evolve to a fixed point furthermore there is no computable bound on the period lengths of these configurations our arguments are based on a non standard simulation of turing machines on circular cellular automata

affine structure from line correspondences with uncalibrated affine cameras this paper presents a linear algorithm for recovering 3d affine shape and motion from line correspondences with uncalibrated affine cameras the algorithm requires a minimum of seven line correspondences over three views the key idea is the introduction of a one dimensional projective camera this converts 3d affine reconstruction of line directions into 2d projective reconstruction of points in addition a line based factorization method is also proposed to handle redundant views experimental results both on simulated and real image sequences validate the robustness and the accuracy of the algorithm

algebraic properties of linear cellular automata abstract cellular automata are systems evolving on lattices according to a local transition rule in this paper we present an algebraic formalism for dealing with cellular automata whose local transition rule satisfies an additivity property we discuss the phenomenon of self replication and its connection with higher order cellular automata and the state transition graph

the structure of reversible one dimensional cellular automata abstract an algorithm is presented for determining reversibility characteristics of 1 dimensional cellular automaton laws the concept of local reversibility is defined each locally reversible automaton is shown to be isomorphic to a member of the class of center reversible automata algorithms are described for generating the set of center reversible laws and the set of center reversible additive laws

inverse rules of eca with rule number 150 the explicit expressions of local transition functions of the inverse cellular automata of elementary cellular automata with rule number 150 and periodic boundary conditions are given the procedure to obtain such local transition functions is based on the algebraic properties of elementary cellular automata

extraction of mth roots in matrix rings over fields abstract a matrix a m n f f an arbitrary field with characteristic p not necessarily positive is an m th power in m n f if and only if each of its p x primary components is where p x runs through the irreducible factors of the minimal or characteristic polynomial of a this paper establishes necessary and sufficient criteria for determining when such a p x primary matrix is an m th power the criteria fall into three cases 1 p x x if e 1 e 2 is the sequence of exponents of p x which form the elementary divisors of a extended by adding infinitely many 0 terms the criterion states that for all i 1 e i 1 m 1 e im 0 or 1 2 p x x m not divisible by p if is a root of the separable core q x of p x and e f then the multiplicity of each elementary divisor of a must be representable as a sum of integers not necessarily distinct each of which is the degree of some irreducible factor over e of the polynomial x m 3 p x x m a power of p in this case the criterion combines the criteria in the first two cases if e 1 e 2 is the sequence of exponents of p x which form the elementary divisors of a extended by adding infinitely many 0 terms then for all i 1 e i 1 m 1 e im 0 or 1 and the multiplicity of each e i must be representable as a sum of integers not necessarily distinct each of which is the degree of some irreducible factor over e of the polynomial x m

quasilinear cellular automata abstract simulating a cellular automaton ca for t time steps into the future requires t 2 serial computation steps or t parallel ones however certain cas based on an abelian group such as addition mod 2 are termed linear because they obey a principle of superposition this allows them to be predicted efficiently in serial time o t robinson 1987 or o log t in parallel in this paper we generalize this by looking at cas with a variety of algebraic structures including quasigroups non abelian groups steiner systems and other structures we show that in many cases an efficient algorithm exists even though these cas are not linear in the previous sense we term them quasilinear we find examples which can be predicted in serial time proportional to t t log t t log 2 t and t for t log t log log t and log 2 t we also discuss what algebraic properties are required or implied by the existence of scaling relations and principles of superposition and exhibit several novel vector valued cas

predecessors of cellular automata states ii pre images of finite sequences abstract graph theoretic techniques are used to derive algorithms which determine all pre images of any given finite sequence s under one dimensional cellular automata rules defined over z 2 it is also shown that for a given rule q the number of pre images of any finite sequence is determined by an algebraic form in the components of q this extends results of jen on the enumeration of pre images we also show that the set of k site rules for which a given sequence s has no pre images is determined by the solution set of a non linear diophantine equation

3d motion recovery via affine epipolar geometry algorithms to perform point based motion estimation under orthographic and scaled orthographic projection abound in the literature a key limitation of many existing algorithms is that they operate on the minimum amount of data required often requiring the selection of a suitable minimal set from the available data to serve as a local coordinate frame such approaches are extremely sensitive to errors and noise in the minimal set and forfeit the advantages of using the full data set furthermore attention is seldom paid to the statistical performance of the algorithms

reversibility of 1d cellular automata with periodic boundary over finite fields pmb mathbb z _ p the reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied recently over the binary field 2 del rey and rodriguez sanchez in appl math comput 2011 doi 10 1016 j amc 2011 03 033 in this paper we study one dimensional linear cellular automata with periodic boundary conditions over any finite field p for any given p 2 we show that the reversibility problem can be reduced to solving a recurrence relation depending on the number of cells and the coefficients of the local rules defining the one dimensional linear cellular automata more specifically for any given values from any fixed field p of the coefficients of the local rules we outline a computer algorithm determining the recurrence relation which can be solved by testing reversibility of the cellular automaton for some finite number of cells as an example we give the full criteria for the reversibility of the one dimensional linear cellular automata over the fields 2 and 3

two dimensional cellular automata a largely phenomenological study of two dimensional cellular automata is reported qualitative classes of behavior similar to those in one dimensional cellular automata are found growth from simple seeds in two dimensiona cellular automata can produce patterns with complicated boundaries characterized by a variety of growth dimensions evolution from disordered states can give domains with boundaries that execute effectively continuous motions some global properties of cellular automata can be described by entropies and lyapunov exponents others are undecidable

active tracking of foveated feature clusters using affine structure we describe a novel method of obtaining a fixation point on a moving object for a real time gaze control system the method makes use of a real time implementation of a corner detector and tracker and reconstructs the image position of the desired fixation point from a cluster of corners detected on the object using the affine structure available from two or three views the method is fast reliable viewpoint invariant and insensitive to occlusion and or individual corner dropout or reappearance we compare two and three dimensional forms of the algorithm present results for the method in use with a high performance head eye platform and compare the results with two naive fixation methods

linear reversible second order cellular automata and their first order matrix equivalents linear or one dimensional reversible second order cellular automata exemplified by three cases named as rca1 3 are introduced displays of their evolution in discrete time steps from their simplest initial states and on the basis of updating rules in modulo 2 arithmetic are presented in these shaded and unshaded squares denote cells whose cell variables are equal to one and zero respectively this paper is devoted to finding general formulas for and explicit numerical evaluations of the weights n t of the states or configurations of rca1 3 i e the total number of shaded cells in tth line of their displays this is achieved by means of the replacement of rca1 3 by the equivalent linear first order matrix automata mca1 3 for which the cell variables are matrices instead of just numbers as for rca1 3 mca1 3 are tractable because it has been possible to generalize to them the heavy duty methods already well developed for ordinary first order cellular automata like those of wolfram s rules 90 and 150 while the automata mca1 3 are thought to be of genuine interest in their own right with untapped further mathematical potential their treatment has been applied here to expediting derivation of a large body of general and explicit results for n t for rca1 3 amongst explicit results obtained are formulas also for each of rca1 3 for the total weight of the configurations of the first times

affine structure and motion from points lines and conics in this paper several new methods for estimating scene structure and camera motion from an image sequence taken by affine cameras are presented all methods can incorporate both point line and conic features in a unified manner the correspondence between features in different images is assumed to be known three new tensor representations are introduced describing the viewing geometry for two and three cameras the centred affine epipoles can be used to constrain the location of corresponding points and conics in two images the third order or alternatively the reduced third order centred affine tensors can be used to constrain the locations of corresponding points lines and conics in three images the reduced third order tensors contain only 12 components compared to the 16 components obtained when reducing the trifocal tensor to affine cameras a new factorization method is presented the novelty lies in the ability to handle not only point features but also line and conic features concurrently another complementary method based on the so called closure constraints is also presented the advantage of this method is the ability to handle missing data in a simple and uniform manner finally experiments performed on both simulated and real data are given including a comparison with other methods

welch sets for random generation and representation of reversible one dimensional cellular automata abstract reversible one dimensional cellular automata are studied from the perspective of welch sets this paper presents an algorithm to generate random welch sets that define a reversible cellular automaton then properties of welch sets are used in order to establish two bipartite graphs describing the evolution rule of reversible cellular automata the first graph gives an alternative representation for the dynamics of these systems as block mappings and shifts the second graph offers a compact representation for the evolution rule of reversible cellular automata both graphs and their matrix representations are illustrated by the generation of random reversible cellular automata with 6 and 18 states

algebraic functions for recognition in the general case a trilinear relationship between three perspective views is shown to exist the trilinearity result is shown to be of much practical use in visual recognition by alignment yielding a direct reprojection method that cuts through the computations of camera transformation scene structure and epipolar geometry moreover the direct method is linear and sets a new lower theoretical bound on the minimal number of points that are required for a linear solution for the task of reprojection the proof of the central result may be of further interest as it demonstrates certain regularities across homographics of the plane and introduces new view invariants experiments on simulated and real image data were conducted including a comparative analysis with epipolar intersection and the linear combination methods with results indicating a greater degree of robustness in practice and a higher level of performance in reprojection tasks

algebraic properties of cellular automata cellular automata are discrete dynamical systems of simple construction but complex and varied behaviour algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata the complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities the systems are usually irreversible and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations

computation theoretic aspects of cellular automata abstract cellular automata may be viewed as computational models of complex systems they also can be seen as continuous functions on a particular family of compact metric spaces this paper surveys results largely motivated by an attempt to answer questions posed by the latter approach by means of techniques from the former

motion of points and lines in the uncalibrated case in the present paper we address the problem of computing structure and motion given a set point and or line correspondences in a monocular image sequence when the camera is not calibrated

shape and motion from image streams under orthography a factorization method inferring scene geometry and camera motion from a stream of images is possible in principle but is an ill conditioned problem when the objects are distant with respect to their size we have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step an image stream can be represented by the 2fxp measurement matrix of the image coordinates of p points tracked through f frames we show that under orthographic projection this matrix is of rank 3 based on this observation the factorization method uses the singular value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively two of the three translation components are computed in a preprocessing stage the method can also handle and obtain a full solution from a partially filled in measurement matrix that may result from occlusions or tracking failures the method gives accurate results and does not introduce smoothing in either shape or motion we demonstrate this with a series of experiments on laboratory and outdoor image streams with and without occlusions

procedures for calculating reversible one dimensional cellular automata we describe two algorithms for calculating reversible one dimensional cellular automata of neighborhood size 2 we explain how this kind of automaton represents all the other cases using two basic properties of reversible automata such as uniform multiplicity of ancestors and welch indices these algorithms only require matrix products and transitive closures of binary relations to classify all the possible reversible automata of neighborhood size 2 we expose the features advantages and differences with other well known methods finally we present results for reversible automata from three to six states and neighborhood size 2

an effective markov based approach for calculating the limit matrix in the analytic network process abstract analytic network process is a multiple criteria decision analysis mcda method that aids decision makers to choose among a number of possible alternatives or prioritize the criteria for making a decision in terms of importance it handles both qualitative and quantitative criteria that are compared in pairs in order to forge a best compromise answer according to the different criteria and influences involved the method has been widely applied and the literature review reveals a rising trend of anp related articles the power matrix method a procedure necessary for the stability of the decision system is one of the critical calculations in the mathematical part of the method the present study proposes an alternative mathematical approach that is based on markov chain processes and the well known gauss jordan elimination the new approach obtains practically the same results as the power matrix method requires slightly less time and number of calculations and handles effectively cyclic supermatrices optimizing thus the whole procedure

additive cellular automata and global injectivity abstract we employ algebraic methods to investigate the global injectivity of additive cellular automata when the state alphabet is a finite commutative ring the set of local rules of such additive cellular automata in d 1 dimensions itself forms a ring the invertible elements in this ring correspond to the globally injective additive cellular automata we indicate how one determines the invertible elements and thus the globally injectibe cellular automata in particular we show that in contrast to the general situation the global injectivity of additive cellular automata in d 1 dimensions is decidable construction of local rules for the inverse cellular automata is illustrated through examples a strict upper bound on the radius of the inverse cellular automata is obtained in the special case of linear cellular automata with state alphabet the ring of integers modulo m we count the globally injective cellular automata of up to a given radius we show that in one dimension a non globally injective additive cellular automaton is either not injective on spatially periodic configurations of any period n n or the set of n 0 such that the cellular automation is injective on spatially periodic configurations of period n has positive density in n

learning to track the visual motion of contours abstract a development of a method for tracking visual contours is described given an untrained tracker a training motion of an object can be observed over some extended time and stored as an image sequence the image sequence is used to learn parameters in a stochastic differential equation model these are used in turn to build a tracker whose predictor imitates the motion in the training set tests show that the resulting trackers can be markedly tuned to desired curve shapes and classes of motions

deriving priorities from fuzzy pairwise comparison judgements a new approach for deriving priorities from fuzzy pairwise comparison judgements is proposed based on cuts decomposition of the fuzzy judgements into a series of interval comparisons the assessment of the priorities from the pairwise comparison intervals is formulated as an optimisation problem maximising the decision maker s satisfaction with a specific crisp priority vector a fuzzy preference programming method which transforms the interval prioritisation task into a fuzzy linear programming problem is applied to derive optimal crisp priorities aggregating the optimal priorities which correspond to different cut levels enables overall crisp scores of the prioritisation elements to be obtained a modification of the linear fuzzy preference programming method is also proposed to derive priorities directly from fuzzy judgements without applying cut transformations the formulation of the prioritisation problem as an optimisation task is similar to the previous approach but it requires the solution of a non linear optimisation program the second approach also derives crisp priorities and has the advantage that it does not need additional aggregation and ranking procedures both proposed methods are illustrated by numerical examples and compared to some of the existing fuzzy prioritisation methods

predecessors of cellular automata states i additive automata abstract let e be the space of right half infinite sequences with entries in z 2 and let q be the global transition rule for the one dimensional cellular automaton q e for which e is the state space the predecessor problem for q e is to find all solutions of the equation q where is an arbitrary element of e in this paper we consider the case in which the operator q represents an additive rule the predecessor equation is solved in closed form for all two and three site rules in particular for rules 90 and 150 and a technique is introduced which generates a closed form solution for many higher site values this is demonstrated with several examples

classification of triangular and honeycomb cellular automata abstract cellular automata on a two dimensional triangular and honeycomb lattice are classified according to their fixed points and periods the results are compared with other lattices in one of three dimensions we also compare the totalistic rules for all mentioned lattices

reduced multilinear constraints theory and experiments this paper deals with the problem of reconstructing the locations of n points in space from m different images without camera calibration we will show how these reconstruction problems for different n and m can be put into a similar theoretical framework this will be done using a special choice of coordinates both in the object and in the images called reduced affine coordinates this choice of coordinates simplifies the analysis of the multilinear geometry and gives simpler forms of the multilinear tensors in particular we will investigate the cases which can be solved by linear methods i e 8 points in 2 images 7 points in 3 images and 6 points in 4 images a new concept the reduced fundamental matrix is introduced which gives bilinear expressions in the image coordinates it has six nonzero elements which depend on just four parameters and can be used to make reconstruction from 2 images we also introduce the concept of the reduced trifocal tensor which gives trilinear expressions in the image coordinates in 3 images it has 15 nonzero elements and depends on nine parameters and can be used to make reconstruction from 3 images finally the reduced quadfocal tensor is introduced which describes the relations between points in 4 images and gives quadlinear expressions in the image coordinates this tensor has 36 nonzero elements which depend on 14 independent parameters and can be used to make reconstruction from 4 images these tensors give the possibility to calculate linear solutions from 8 points in 2 images 7 points in 3 images and also from 6 points in 4 images furthermore a canonical form of the camera matrices in a sequence is presented and it is shown that the quadlinear constraints can be calculated from the trilinear ones and that in general the trilinear constraints can be calculated from the bilinear ones

predecessors of cellular automata states iii garden of eden classification of cellular automata abstract a procedure is given which determines all finite sequences without pre images for any one dimensional cellular automata with entries in z 2 on this basis we determine some properties of such sequences and classify all three site rules categories defined by this classification are compared to other phenomenological classifications introduced by wolfram gutowitz and jen