affine automata and related techniques for generation of complex images abstract in this paper we introduce probabilistic affine automata paa which are probabilistic finite generators having transitions labeled with affine transformations it is shown that paa are capable of generating highly complex images barnsley s 1988 ifs method to generate fractals is a special case of paa when the automaton happens to have only a single state a number of theoretical results on paa are shown the relationship between paa mutually recursive function systems mrfs and affine regular sets is investigated

tree based picture generation the concept of tree based picture generation is introduced it is shown that there are equivalent tree based definitions of four picture generating devices known from the literature namely collage

tree based generation of languages of fractals the notion of p interpreted top down tree generators is introduced combining the nondeterministic nature of grammars as known from formal language theory with the infinite refinement of pictures studied in fractal geometry

a markov partition that reflects the geometry of a hyperbolic toral automorphism we show how to construct a markov partition that reflects the geometrical action of a hyperbolic automorphism of the n torus the transition matrix is the transpose of the matrix induced by the automorphism in u dimensional homology provided this is non negative here u denotes the expanding dimension that condition is satisfied at least for some power of the original automorphism under a certain non degeneracy condition on the galois group of the characteristic polynomial the nu rectangles are constructed by an iterated function system and they resemble the product of the projection of a u dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal n u dimensional face onto the stable subspace

a sequent calculus for subtyping polymorphic types we present two complete systems for polymorphic types with sub typing one system is in the style of natural deduction while another is a gentzen style sequent calculus system we prove several meta mathematical properties for these systems including cut elimination subject reduction coherence and decidability of type reconstruction following the approach by j mitchell the sequents are given a simple semantics using logical relations over applicative structures the systems are complete with respect to this semantics the logic which emerges from this paper can be seen as a successor to the original hilbert style system proposed by j mitchell in 1988 and to the half way sequent calculus of g longo k milsted and s soloviev proposed in 1995

numerical solution of continuous state dynamic programs using linear and spline interpolation this paper demonstrates that the computational effort required to develop numerical solutions to continuous state dynamic programs can be reduced significantly when cubic piecewise polynomial functions rather than tensor product linear interpolants are used to approximate the value function tensor product cubic splines represented in either piecewise polynomial or b spline form and multivariate hermite polynomials are considered computational savings are possible because of the improved accuracy of higher order functions and because the smoothness of higher order functions allows efficient quasi newton methods to be used to compute optimal decisions the use of the more efficient piecewise polynomial form of the spline was slightly superior to the use of hermite polynomials for the test problem and easier to program in comparison to linear interpolation use of splines in piecewise polynomial form reduced the cpu time to obtain results of equivalent accuracy by a factor of 250 330 for a stochastic

on the l q spectrum of planar self affine measures the author was supported by the epsrc grant ep j013560 1 this work was started whilst the author was an epsrc funded phd student at the university of st andrews and he expresses his gratitude for the support he found there

myopic solutions of homogeneous sequential decision processes an optimum of a markov decision process mdp is myopic if it can be obtained by solving a series of static problems myopic optima are desirable because they can be computed relatively easily we identify new classes of mdps with myopic optima and sequential games with myopic equilibrium points in one of the classes the single period reward is homogeneous with respect to the state variable we illustrate the results with models of revenue management and investment

architectural bias in recurrent neural networks fractal analysis we have recently shown that when initialized with small weights recurrent neural networks rnns with standard sigmoid type activation functions are inherently biased toward markov models even prior to any training rnn dynamics can be readily used to extract finite memory machines hammer tino 2002 tino cernansk benuskova 2002a 2002b following christiansen and chater 1999 we refer to this phenomenon as the architectural bias of rnns in this article we extend our work on the architectural bias in rnns by performing a rigorous fractal analysis of recurrent activation patterns we assume the network is driven by sequences obtained by traversing an underlying finite state transition diagram a scenario that has been frequently considered in the past for example when studying rnn based learning and implementation of regular grammars and finite state transducers we obtain lower and upper bounds on various types of fractal dimensions such as box counting and hausdorff dimensions it turns out that not only can the recurrent activations inside rnns with small initial weights be explored to build markovian predictive models but also the activations form fractal clusters the dimension of which can be bounded by the scaled entropy of the underlying driving source the scaling factors are fixed and are given by the rnn parameters

generating self affine fractals by collage grammars abstract self affinity and self similarity are fundamental concepts in fractal geometry in this paper they are related to collage grammars syntactic devices based on hyperedge replacement that generate sets of collages essentially a collage is a picture consisting of geometric parts like line segments circles polygons polyhedra etc the overlay of all collages in a collage language yields a fractal pattern we show that collage grammars of a special type so called increasing generalized sierpinski grammars yield self affine fractals if one replaces the overlay by an intersection of all generated collages the same result holds for decreasing generalized sierpinski grammars here the converse also holds every self affine fractal can be generated by a decreasing generalized sierpinski grammar which provides a characterization of this class of fractals

lipschitz equivalence of fractal triangles abstract given an integer number n 2 and two digit sets a k 1 k 2 k 1 k 2 n 1 and k 1 k 2 n 0 b k 1 k 2 2 k 1 k 2 n and k 1 k 2 n where 1 0 1 2 3 2 there is a self similar set t t a b r 2 satisfying the set equation t t a b t n we call such t a fractal triangle by examining deeper the types of connected components in each step of constructing such fractal triangles we in this paper successfully characterize the lipschitz equivalence of two classes of totally disconnected fractal triangles in it through the number of basic triangles in the first step of construction

multi stage stochastic optimization applied to energy planning this paper presents a methodology for the solution of multistage stochastic optimization problems based on the approximation of the expected cost to go functions of stochastic dynamic programming by piecewise linear functions no state discretization is necessary and the combinatorial explosion with the number of states the well known curse of dimensionality of dynamic programming is avoided the piecewise functions are obtained from the dual solutions of the optimization problem at each stage and correspond to benders cuts in a stochastic multistage decomposition framework a case study of optimal stochastic scheduling for a 39 reservoir system is presented and discussed

propositional lax logic we investigate a peculiar intuitionistic modal logic called propositional lax logic pll which has promising applications to the formal verification of computer hardware the logic has emerged from an attempt to express correctness up to behavioural constraints a central notion in hardware verification as a logical modality as a modal logic it is special since it features a single modal operator that has a flavour both of possibility and of necessity in the paper we provide the motivation for pll and present several technical results we investigate some of its proof theoretic properties presenting a cut elimination theorem for a standard gentzen style sequent presentation of the logic we go on to define a new class of fallible two frame kripke models for pll these models are unusual since they feature worlds with inconsistent information furthermore the only frame condition imposed is that the frame be a subrelation of the frame we give a natural translation of these models into goldblatt s j space models of pll our completeness theorem for these models yields a godel style embedding of pll into a classical bimodal theory of type s4 s4 and underpins a simple proof of the finite model property we proceed to prove soundness and completeness of several theories for specialized classes of models we conclude with a brief exploration of two concrete and rather natural types of model from hardware verification for which the modality models correctness up to timing constraints we obtain decidability of free fragment of the logic of the first type of model which coincides with the stable form of maksimova s intermediate logicl

multiple expansions of real numbers with digits set 0 1 q for q 1 we consider expansions in base q with digits set left 0 1 q right let mathcal u q be the set of points which have a unique q expansion for k 2 3 ldots aleph 0 let mathcal b k be the set of bases q 1 for which there exists x having precisely k different q expansions and for q in mathcal b k let mathcal u q k be the set of all such x s which have exactly k different q expansions in this paper we show that begin aligned mathcal b aleph 0 2 infty quad text and quad mathcal b k q c infty quad text for any quad k ge 2 end aligned where q c approx 2 32472 is the appropriate root of x 3 3x 2 2x 1 0 moreover we show that for any integer k ge 2 and any q in mathcal b k the hausdorff dimensions of mathcal u q k and mathcal u q are the same i e begin aligned dim h mathcal u q k dim h mathcal u q quad text for any quad k ge 2 end aligned finally we conclude that the set of points having a continuum of q expansions has full hausdorff dimension

energy and laplacian on hanoi type fractal quantum graphs this article studies potential theory and spectral analysis on compact metric spaces which we refer to as fractal quantum graphs these spaces can be represented as a possibly infinite union of one dimensional intervals and a totally disconnected possibly uncountable compact set which roughly speaking represents the set of junction points classical quantum graphs and fractal spaces such as the hanoi attractor are included among them we begin with proving the existence of a resistance form on the hanoi attractor and go on to establish heat kernel estimates and upper and lower bounds on the eigenvalue counting function of laplacians corresponding to weakly self similar measures on the hanoi attractor these estimates and bounds rely heavily on the relation between the length and volume scaling factors of the fractal we then state and prove a necessary and sufficient condition for the existence of a resistance form on a general fractal quantum graph finally we extend our spectral results to a large class of weakly self similar fractal quantum graphs

efficient implementation of regular languages using reversed alternating finite automata alternating finite automata afa provide a natural and succinct way to denote regular languages we introduce a bit wise representation of reversed afa r afa transition functions and describe an efficient implementation method for r afa and their operations using this representation experiments have shown that this implementation is much more efficient than for instance the grail dfa implementation in both space and time

exact and approximate solutions of affine reservoir models we formulate an optimization model of a multiple reservoir water resource system that encompasses interbasin transfers among two or more river basins autocorrelated inflows are modeled with a linear autoregressive stochastic process benefits for each period are assumed to depend separably on storage levels and discharges with the dependence on discharges being linear for the important special case of a single river basin a myopic policy hence computed easily is optimal when the model includes the possibility of interbasin transfers a myopic policy is optimal if the deterministic and stochastic portions of the inflow process are always nonnegative even if this assumption is not valid myopic policies yield useful bounds

a reservoir hydroelectric system exactly and approximately optimal policies abstract we analyze a discrete time model of a single reservior whose discharges generate hydroelectric power inflows in successive periods are random variables and the structure of the model is invariant over time revenue from hydroelectric production is assumed to follow a piecewise linear function we characterize an optimal discharge policy and describe numerical results for the computation of approximately optimal policies

gradient dynamic programming for stochastic optimal control of multidimensional water resources systems a new computational algorithm is presented for the solution of discrete time linearly constrained stochastic optimal control problems decomposable in stages the algorithm designated gradient dynamic programming is a backward moving stagewise optimization the main innovations over conventional discrete dynamic programming ddp are in the functional representation of the cost to go function and the solution of the single stage problem the cost to go function assumed to be of requisite smoothness is approximated within each element defined by the discretization scheme by the lowest order polynomial which preserve its values and the values of its gradient with respect to the state variables at all nodes of the discretization grid the improved accuracy of this hermitian interpolation scheme reduces the effect of discretization error and allows the use of coarser grids which reduces the dimensionality of the problem at each stage the optimal control is determined on each node of the discretized state space using a constrained newton type optimization procedure which has quadratic rate of convergence the set of constraints which act as equalities is determined from an active set strategy which converges under lenient convexity requirements this method of solving the single stage optimization is much more efficient than the conventional way based on enumeration or iterative methods with linear rate of convergence once the optimal control is determined the cost to go function and its gradient with respect to the state variables is calculated to be used at the next stage the proposed technique permits the efficient optimization of stochastic systems whose high dimensionality does not permit solution under the conventional ddp framework and for which successive approximation methods are not directly applicable due to stochasticity results for a four reservoir example are presented the purpose of this paper is to present a new computational algorithm for the stochastic optimization of sequential decision problems one important and extensively studied class of such problems in the area of water resources is the discrete time optimal control of multireservoir systems under stochastic inflows other applications include the optimal design and operation of sewer systems e g mays and wenzel 1976 labadie et al 1980 the optimal conjunctive utilization of surface and groundwater resources e g buras 1972 and the minimum cost water quality maintenance in rivers e g dracup and fogarty 1974 chang and yeh 1973 to mention only a few of the water resources applications and pertinent references an extensive review of dynamic programming applications in water resources can be found in the works by yakowitz 1982 and yeh 1985 before we proceed with the

a binary modal logic for the intersection types of lambda calculus abstract intersection types discipline allows to define a wide variety of models for the type free lambda calculus but the curry howard isomorphism breaks down for this kind of type systems in this paper we show that the correspondence between types and suitable logical formulas can still be recovered appealing to the fact that there is a strict connection between the semantics for lambda calculus induced by the intersection types and a kripke style semantics for modal and relevant logics indeed we present a modal logic hinted by the analysis of the sub typing relation for intersection types and we show that the deduction relation for such a modal system is a conservative extension of the relation of sub typing then we define a kripke style semantics for the formulas of such a system present suitable sequential calculi prove a completeness theorem and give a syntactical proof of the cut elimination property finally we define a decision procedure for theorem hood and we show that it yields the finite model property and cut redundancy

type inference with recursive types syntax and semantics abstract in this paper we study type inference systems for calculus with a recursion operator over types the main syntactical properties notably the existence of principal type schemes are proved to hold when recursive types are viewed as finite notations for infinite regular type expressions representing their infinite unfoldings exploiting the approximation structure of a model for the untyped language of terms types are interpreted as limits of sequences of their approximations we show that the interpretation is essentially unique and that two types have equal interpretation if and only if their infinite unfoldings are identical finally a completeness theorem is proved to hold w r t the specific model we consider for a natural infinitary extension of the type inference system

absolute continuity of vector valued self affine measures this paper is devoted to the absolute continuity of scalar valued or vector valued self affine measures and their properties on the boundary of an invariant set we first extend the definition of wsc to self affine ifs and then obtain a necessary and sufficient condition for the vector valued self affine measures to be absolutely continuous with respect to the lebesgue measure in addition we prove that for any ifs and any invariant open set v the corresponding scalar valued or vector valued invariant measure is supported either in v or in v

iterated function systems and control languages valuations morphisms from e to 0 1 are a generalization of bernoulli morphisms introduced by eilenberg automata languages and machines academic press new york 1974 here we show how to generalize the notion of entropy of a language in order to obtain new formulas to determine the hausdorff dimension of fractal sets also in euclidean spaces especially defined via regular languages by doing this we can sharpen and generalize earlier results in two ways first we treat the case where the underlying basic iterated function system contains noncontractive mappings and second we obtain results valid for nonregular languages as well

polymorphic type inference and containment abstract type expressions may be used to describe the functional behavior of untyped lambda terms we present a general semantics of polymorphic type expressions over models of untyped lambda calculus and give complete rules for inferring types for terms some simplified typing theories are studied in more detail and containments between types are investigated

ifs on a metric space with a graph structure and extensions of the kelisky rivlin theorem abstract we develop the hutchinson barnsley theory for finite families of mappings on a metric space endowed with a directed graph in particular our results subsume a classical theorem of j e hutchinson j e hutchinson fractals and self similarity indiana univ math j 30 1981 713 747 on the existence of an invariant set for an iterated function system of banach contractions and a theorem of l mate l mate the hutchinson barnsley theory for certain non contraction mappings period math hungar 27 1993 21 33 concerning finite families of locally uniformly contractions introduced by edelstein also they generalize recent fixed point theorems of a c m ran and m c b reurings a c m ran m c b reurings a fixed point theorem in partially ordered sets and some applications to matrix equations proc amer math soc 132 2004 1435 1443 j j nieto and r rodriguez lopez j j nieto r rodriguez lopez contractive mapping theorems in partially ordered sets and applications to ordinary differential equations order 22 2005 223 239 j j nieto r rodriguez lopez existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations acta math sin engl ser 23 2007 2205 2212 and a petrusel and i a rus a petrusel i a rus fixed point theorems in ordered l spaces proc amer math soc 134 2006 411 418 for contractive mappings on an ordered metric space as an application we obtain a theorem on the convergence of infinite products of linear operators on an arbitrary banach space this result yields new generalizations of the kelisky rivlin theorem on iterates of the bernstein operators on the space c 0 1 as well as its extensions given recently by h oruc and n tuncer h oruc n tuncer on the convergence and iterates of q bernstein polynomials j approx theory 117 2002 301 313 and h gonska and p pi ul h gonska p pi ul remarks on an article of j p king comment math univ carolin 46 2005 645 652

optimization of a large scale water reservoir network by stochastic dynamic programming with efficient state space discretization a numerical solution to a 30 dimensional water reservoir network optimization problem based on stochastic dynamic programming is presented in such problems the amount of water to be released from each reservoir is chosen to minimize a nonlinear cost or maximize benefit function while satisfying proper constraints experimental results show how dimensionality issues given by the large number of basins and realistic modeling of the stochastic inflows can be mitigated by employing neural approximators for the value functions and efficient discretizations of the state space such as orthogonal arrays latin hypercube designs and low discrepancy sequences

monotone comparative statics the authors derive a necessary and sufficient condition for the solution set of an optimization problem to be monotonic in the parameters of the problem in addition they develop practical methods for checking the condition and demonstrate its applications to the classical theories of the competitive firm the monopolist the bertrand oligopolist consumer and growth theory game theory and general equilibrium analysis copyright 1994 by the econometric society

dimensions of multitype moran sets with lower limit of the contractions being zero abstract multitype moran sets were studied by liu and wen 2005 9 under the assumption of primitivity they proved that for any multitype moran set e with a positive lower bounded condition on contracting ratios dim h e s s dim p e dim b e where s and s are the lower and upper pre dimension according to the natural coverings in this paper we permit that the lower limit of contraction ratios is zero and study the two classes of them for one of the two by assuming some stronger condition than primitivity we show that the above formula still holds for the other under no assumption of primitivity we prove that dim h e s resp dim b e s by assuming a mild condition and show that the above formula also holds provided primitivity and these two mild conditions are satisfied

intersecting nonhomogeneous cantor sets with their translations abstract a scheme is given to compute the hausdorff dimensions for the intersection of a class of nonhomogeneous cantor sets with their translations

logic of subtyping we introduce new modal logical calculi that describe subtyping properties of cartesian product and disjoint union type constructors as well as mutually recursive types defined using those type constructors basic logic of subtyping s extends classical propositional logic by two new binary modalities and an interpretation of s is a function that maps standard connectives into set theoretical operations intersection union and complement and modalities into cartesian product and disjoint union type constructors this allows s to capture many subtyping properties of the above type constructors we also consider logics s and s that incorporate into s mutually recursive types over arbitrary and well founded universes correspondingly the main results are completeness of the above three logics with respect to appropriate type universes in addition we prove cut elimination theorem for s and establish decidability of s and s

higher order and average reward myopic affine dynamic models the dynamics and rewards in a higher order model depend directly on states and actions in earlier periods here the expected values of the single period reward and of the state in the next period are affine functions of the states in the current and earlier periods it is shown that computations in higher order infinite horizon models with myopic solutions are not essentially harder than in the corresponding first order models these results are obtained for discounted and average reward criteria and similar conclusions are indicated for sequential games the results are applied to an aquaculture harvesting model and an advertising model with delayed responses