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

preconditioning by fast direct methods for nonself adjoint nonseparable elliptic equations we consider the use of fast direct methods as preconditioners for iterative methods for computing the numerical solution of nonself adjoint elliptic boundary value problems we derive bounds on convergence rates that are independent of discretization mesh size for two dimensional problems on rectangular domains discretized on an n times n grid these bounds lead to asymptotic operation counts of o n 2 log n log varepsilon 1 to achieve relative error varepsilon and o n 2 log n 2 to reach truncation error

maximum number of periodic orbits in parallel dynamical systems abstract for parallel dynamical systems over undirected graphs with a boolean maxterm or minterm functions as global evolution operators it is known that every periodic orbit has period less than or equal to two in fact periodic orbits of different periods cannot coexist and a fixed point theorem based in the uniqueness of a fixed point is also known in this paper we complete the study of the periodic structure of such systems providing a 2 periodic orbit theorem and giving an upper bound for the number of fixed points and also for the number of 2 periodic orbits actually we provide examples where these bounds are attained demonstrating that they are the best possible ones

vertex irf correspondence and factorized l operators for an elliptic r operator as for an ellipticr operator which satisfies the yang baxter equation the incoming and outgoing intertwining vectors are constructed and the vertex irf correspondence for the ellipticr operator is obtained the boltzmann weights of the corresponding irf model satisfy the star traingle relation by means of these intertwining vectors the factorizedl operators for the ellipticr operator are also constructed the vertex irf correspondence and the factorizedl operators for belavin sr matrix are reproduced from those of the ellipticr operator

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

the hard hexagon model and rogers ramanujan type identities abstract in regime ii of baxter s solution of the hard hexagon model baxter r j 1980 j phys a 13 l61 l70 he presents six conjectures identifying certain one dimensional partition functions with infinite products an outline of the proof of these conjectures is given here

isogeometric collocation for elastostatics and explicit dynamics we extend the development of collocation methods within the framework of isogeometric analysis iga to multi patch nurbs configurations various boundary and patch interface conditions and explicit dynamic analysis the methods developed are higher order accurate stable with no hourglass modes and efficient in that they require a minimum number of quadrature evaluations the combination of these attributes has not been obtained previously within standard finite element analysis

optimal order isogeometric collocation at galerkin superconvergent points abstract in this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least squares collocation method presented in anitescu et al 2015 and the variational collocation method presented in gomez and de lorenzis 2016 the focus is on smoothest b splines nurbs approximations i e having global c p 1 continuity for polynomial degree p within the framework of gomez and de lorenzis 2016 we select as collocation points a subset of those considered in anitescu et al 2015 which are related to the galerkin superconvergence theory with our choice that features local symmetry of the collocation stencil we improve the convergence behavior with respect to gomez and de lorenzis 2016 achieving optimal l 2 convergence for odd degree b splines nurbs approximations the same optimal order of convergence is seen in anitescu et al 2015 where however a least squares formulation is adopted further careful study is needed since the robustness of the method and its mathematical foundation are still unclear

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

parallel discrete dynamical systems on independent local functions in this paper we extend the manner of defining the evolution update of discrete dynamical systems on boolean functions without limiting the local functions to being dependent restrictions of a global one then we analyze the cases concerned with parallel dynamical systems with the or and nand and nor functions as independent local functions over undirected and also directed dependency graphs this extension of the update method widely generalizes the traditional one where only a global boolean function is considered for establishing the evolution operator of the system besides our analysis allows us to show a richer dynamics in these new kinds of parallel dynamical systems

a simple algorithm for obtaining nearly optimal quadrature rules for nurbs based isogeometric analysis we develop new quadrature rules for isogeometric analysis based on the solution of a local nonlinear problem a simple and robust algorithm is developed to determine the rules which are exact for important b spline spaces of uniform and geometrically stretched knot spacings we consider both periodic and open knot vector configurations and illustrate the efficiency of the rules on selected boundary value problems we find that the rules are almost optimally efficient but much easier to obtain than optimal rules which require the solution of global nonlinear problems that are often ill posed

elements of a theory of simulation ii sequential dynamical systems we study a class of discrete dynamical systems that is motivated by the generic structure of simulations the systems consist of the following data a a finite graph y with vertex set 1 n where each vertex has a binary state b functions f i f 2 n f 2 n and c an update ordering p the functions f i update the binary state of vertex i as a function of the state of vertex i and its y neighbors and leave the states of all other vertices fixed the update ordering is a permutation of the y vertices by composing the functions f i in the order given by p one obtains the sequential dynamical system sds f y p email protected i 1nf p i f 2 n f 2 n we derive a decomposition result characterize invertible sds and study fixed points in particular we analyse how many different sds that can be obtained by reordering a given multiset of update functions and give a criterion for when one can derive concentration results on this number finally some specific sds are investigated

selective and reduced numerical integrations for nurbs based isogeometric analysis we propose a new approach to construct selective and reduced integration rules for isogeometric analysis based on nurbs elements the notion of an approximation space that approximates the target space is introduced we explore the use of various approximation spaces associated with optimal patch wise numerical quadratures that exactly integrate the polynomials in approximation spaces with the minimum number of quadrature points patch rules exploit the higher continuity of spline basis functions the tendency of smooth spline functions to exhibit numerical locking in nearly incompressible problems when using a full gauss legendre quadrature is alleviated with selective or reduced integration stability and accuracy of the schemes are examined analyzing the discrete spectrum in a generalized eigenvalue problem we propose a local algorithm which is robust and computationally efficient to compute element by element the quadrature points and weights in patch rules the performance of the methods is assessed on several numerical examples in two dimensional elasticity and reissner mindlin shell structures

parallel dynamical systems over directed dependency graphs abstract in this work we analyze the orbit structure of parallel discrete dynamical systems over directed dependency graphs with boolean functions as global functions in this sense for the cases corresponding to the simplest boolean functions and and or it is proved that only fixed or eventually fixed points appear as occurs over undirected dependency graphs however for general boolean functions it is shown that any period can appear so breaking the pattern found for the undirected case where only eventually fixed points or 2 periodic orbits can exist

reduced bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis abstract we explore the use of various element based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis the rules studied encompass tensor product gauss and gauss lobatto rules and certain so called monomial rules that do no possess a tensor product structure the objective of the study is to determine quadrature strategies which enjoy the same accuracy and stability behavior as full gauss quadrature but with significantly fewer quadrature points several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard c 0 continuous finite elements of the same order

the cost of continuity performance of iterative solvers on isogeometric finite elements in this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized galerkin weak form specifically we compare performance of linear solvers when discretizing using c 0 b splines which span traditional finite element spaces and c p 1 b splines which represent maximum continuity we provide theoretical estimates for the increase in cost of the matrix vector product as well as for the construction and application of black box preconditioners we accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size h and polynomial order of approximation p in addition to the aforementioned continuity of the basis finally we present timing results for a range of preconditioning options for the laplace problem we conclude that the matrix vector product operation is at most left 33p 2 middle 8 right times more expensive for the more cont

zn n 1 generalization of the chiral potts model we show that ther matrix which intertwines twon by nn 1 state cyclicl operators related with a generalization ofuq sl n algebra can be considered as a boltzmann weight of four spin box for a lattice model with two spin interaction just as ther matrix of the checkerboard chiral potts model the rapidity variables lie on the algebraic curve of the genusg n2 n 1 n 1 n n 1 defined by 2n 3 independent moduli this curve is a natural generalization of the curve which appeared in the chiral potts model factorization properties of thel operator and its connection to the sos models are also discussed

periods in xor parallel dynamical systems over directed dependency graphs abstract we analyze the periodic structure of parallel dynamical systems over directed dependency graphs whose evolution operator is the boolean function xor we prove that such systems can present periodic orbits of any period moreover we demonstrate that any kinds of periods can coexist at the same time in view of these results we study the periodic structure of these dynamical systems over complete digraphs complete bipartite digraphs acyclic digraphs and out trees

predecessors and garden of eden configurations in parallel dynamical systems on maxterm and minterm boolean functions abstract in this work we solve the classical predecessor problems for parallel dynamical systems on maxterm and minterm boolean functions actually we solve analytically the predecessor existence problem by giving a characterization to have a predecessor for any given configuration as a consequence we also get a characterization of the garden of eden configurations of these systems moreover the structure of the predecessors found out allows us to give a solution to the unique predecessor problem the coexistence of predecessors problem and the number of predecessors problem

on periods and equilibria of computational sequential systems abstract in this paper we show that sequential systems with boolean maxterms and minterms as global evolution operators can present orbits of any period besides we prove that periodic orbits with different periods greater than or equal to 2 can coexist nevertheless when a sequential dynamical system has fixed points we demonstrate that periodic orbits of other periods cannot appear finally we provide conditions to obtain a fixed point theorem in this context this work provides a relevant advance in the knowledge of the dynamics of such systems which constitute one of the most effective mathematical tools to model computational processes and other phenomena from other sciences moreover the ideas developed here could help to obtain similar results for other related systems

euler s de partitio numerorum chapter 16 of euler s introductio in analysin infinitorum is titled de partitio numerorum our goal is to provide a survey of the impact of euler s chapter on the study of partitions in the following 250 plus years we ask the natural question do we still have anything to learn from euler the answer is yes

r matrix quantization of the elliptic ruijsenaars schneider model it is shown that the classical l operator algebra of the elliptic ruijsenaars schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions it is governed by two dynamic and bar r matrices satisfying a closed system of equations the corresponding quantum r and bar r matrices are found as solutions to quantum analogues of these equations we present the quantum l operator algebra and show that the system of equations for r and bar r arises as the compatibility condition for this algebra it turns out that the r matrix is twist equivalent to the felder elliptic rf matrix with bar r playing the role of the twist the simplest representation of the quantum l operator algebra corresponding to the elliptic ruijsenaars schneider model is obtained the connection of the quantum l operator algebra to the fundamental relation rb ll llrb with the belavin elliptic r matrix is established as a by product of our construction we find a new n parameter elliptic solution to the classical yang baxter equation

factorized representation of felder s elliptic quantum group we constructed a factorized representation of the felder s elliptic quantum group by using the intertwining vectors and further derived its explicit form which is in a special case identical with what bo yu hou et al gave i e a cyclic representation of the sklyanin algebra up to a factor

the effect of number of hamiltonian paths on the complexity of a vertex coloring problem a generalization of dobkin and lipton s element uniqueness problem is introduced for any fixed undirected graph g on vertex set v 1 v 2 cdots v n the problem is to determine given n real numbers x 1 x 2 cdots x n whether x i ne x j for every edge upsilon i upsilon j in g this problem is shown to have upper and lower bounds of theta n log n linear comparisons if g is any dense graph the proof of the lower bound involves showing that any dense graph must contain a subgraph with many hamiltonian paths and demonstrating the relevance of these hamiltonian paths to a geometric argument in addition we exhibit relatively sparse graphs for which the same lower bound holds and relatively dense graphs for which a linear upper bound holds

enumerating periodic orbits in sequential dynamical systems over graphs abstract it is well known that periodic orbits with any period can appear in sequential dynamical systems over undirected graphs with a boolean maxterm or minterm function as global evolution operator indeed fixed points cannot coexist with periodic orbits of greater periods while periodic orbits with different periods greater than 1 can coexist additionally a fixed point theorem about the uniqueness of a fixed point is known in this paper we provide an m periodic orbit theorem for m 1 and we give an upper bound for the number of fixed points and periodic orbits of period greater than 1 so completing the study of the periodic structure of such systems we also demonstrate that these bounds are the best possible ones by providing examples where they are attained

neutrality in fitness landscapes the interplay of ruggedness and neutrality in fitness landscapes plays an important role in explaining the dynamics of evolutionary adaptation while various measures of ruggedness correlation functions adaptive walks or the density of local optima are reasonably well understood and models for constructing landscapes with a desired degree of ruggedness are readily available very little is known about neutrality we introduce the notion of additive random landscapes as a framework for tuning both neutrality and ruggedness at once and we develop a formalism that allows the explicit computation of the most salient parameters that are associated with neutrality in landscapes of this type

l operator for belavin s r matrix acting on the space of theta functions for the z nz symmetric elliptic r matrix l operators which act on the space of a 1 n 1 integrable characters or the affine weyl group invariants of the fixed level l are constructed for l 1 the matrix elements of the l operator and those of the r matrix are shown to be the same

crossing symmetry in elliptic solutions of the yang baxter equation and a new l operator for belavin s solution some algebraic structures in elliptic solutions of the yang baxter equations are investigated the author proves the crossing symmetry in belavin s model 1981 as well as in the an 1 1 face model and constructs a new family of l operators for belavin s r matrix as an application

robust and optimal multi iterative techniques for iga galerkin linear systems we consider fast solvers for the large linear systems coming from the isogeometric analysis iga collocation approximation based on b splines of full elliptic d dimensional partial differential equations pdes we are interested in designing iterative algorithms which are optimal and robust the former property implies that the computational cost is linear with respect to the number of degrees of freedom i e the matrix size the latter property means that the convergence rate is completely independent of or only mildly dependent on all the relevant parameters in our setting we can mention the coefficients of the pde the dimensionality d the geometric map g describing the physical domain the matrix size related to the fineness parameters and the spline degrees associated with the iga approximation order our approach is based on the spectral symbol which describes the global eigenvalue behavior of the iga collocation matrices it is precisely the spectral information contained in the symbol that is exploited for the design of an optimal and robust multi iterative solver of multigrid type several numerical experiments are presented and discussed in view of recent theoretical findings concerning the symbol and its properties

an isogeometric collocation method using superconvergent points abstract we develop an iga collocation method modified by collocating at points other than the standard greville abscissae the method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory therefore we refer to this method as super collocation iga sc by carefully choosing the collocation points it can be seen that the iga sc converges in the first derivative energy norms at rates similar to that of the galerkin solution this is different from the collocation at greville abscissae iga c where the convergence in energy norm for odd polynomial degrees is typically suboptimal the method is tested on 1d 2d and 3d numerical examples in which it is compared to iga c and galerkin s method iga g the comparison includes a detailed cost vs accuracy analysis which shows an improved efficiency of the proposed method in particular for odd polynomial degrees

ruijsenaars macdonald type difference operators from belavin model with open boundary conditions antisymmetric fusions of the belavin model with integrable open boundary conditions are studied the centre of the corresponding operator algebra the sklyanin determinant is constructed we prove that the transfer matrices of the fusion hierarchies mutually commute utilizing the factorized l operators this commuting family is expressed as mutually commuting difference operators acting on a function space this gives ruijsenaars macdonald type commuting difference operators which describe the conservation laws of an integrable system

sand piles from physics to cellular automata models we analyze the dynamical behavior of the usual one dimensional sand pile model which actually describes the physical situation in which the pile is submitted to the uniform blow of a unidirectional wind in the first step the lagrangian formalism is investigated showing that the stationary action principle does not select in a unique way the path which satisfies either the minimal or the maximal action principle this drawback is solved making use of the information shannon entropy which enables one to determine the unique path in which at any time step the entropy variation is minimal adiabatic or maximal anti adiabatic a cellular automata ca model describing this sand pile behavior is introduced and the consequent deterministic dynamic is compared with the entropy results showing that also in this case there are some drawbacks moreover it is shown that our ca local rule is a particular case of some standard ca sand pile models present in literature

new rough approximations for n cycles and n paths in this paper we continue the study of a finite simple undirected graph through the rough set techniques introduced by pawlak in the early 80 s in this sense we focus on the boolean information system induced from a circle or a line graph with n vertices and for such graphs we provide a complete description of the rough membership function and some results on the positive regions and the attribute dependency function

parallel and sequential dynamics of two discrete models of signed integer partitions abstract in this paper we complete and generalize some previous results concerning the computing of the sequential and parallel convergent time for two discrete dynamical system of signed integer partitions we also refine the concept of parallel convergent time for a finite graded partially ordered set briefly poset x which is also a discrete dynamical model to this aim we define the concept of fundamental sequence of x and we compute this sequence in two particularly important cases in the first case when x is the finite lattice s n r of all the signed integer partitions a r a 1 b 1 b n r such that r a r a 1 0 b 1 b n r n r where n r 0 and the unique part that can be repeated is 0 in the second case when x is the sub lattice s n d r of all the signed integer partitions of s n r having exactly d non zero parts the relevance of the previous lattices as discrete dynamical models is related to their link with some unsolved extremal combinatorial sum problems

modeling and analyzing social network dynamics using stochastic discrete graphical dynamical systems motivated by applications such as the spread of epidemics and the propagation of influence in social networks we propose a formal model for analyzing the dynamics of such networks our model is a stochastic version of discrete graphical dynamical systems using this model we formulate and study the computational complexity of two fundamental problems called reachability and predecessor existence problems which arise in the context of social networks we also address other problems that deal with the time evolution of such stochastic dynamical systems further we point out the implications of our results to problems for other computational models such as hopfield networks communicating finite state machines and systolic arrays in particular our polynomial time algorithms for the predecessor existence problem for stochastic dynamical systems imply similar results for one dimensional finite cellular automata

isogeometric preconditioners based on fast solvers for the sylvester equation we consider large linear systems arising from the isogeometric discretization of the poisson problem on a single patch domain the numerical solution of such systems is considered a challenging task particularly when the degree of the splines employed as basis functions is high we consider a preconditioning strategy which is based on the solution of a sylvester like equation at each step of an iterative solver we show that this strategy which fully exploits the tensor structure that underlies isogeometric problems is robust with respect to both mesh size and spline degree although it may suffer from the presence of complicated geometry or coefficients we consider two popular solvers for the sylvester equation a direct one and an iterative one and we discuss in detail their implementation and efficiency for two dimensional 2d and three dimensional 3d problems on single patch or conforming multipatch nonuniform rational b splines nurbs geometries we present numerical experiments for problems

sequential dynamical systems over words this paper is motivated by the theory of sequential dynamical systems sds developed as a basis for a mathematical theory of computer simulation a sequential dynamical system is a collection of symmetric boolean local update functions with the update order determined by a permutation of the boolean variables in this paper the notion of sds is generalized to allow arbitrary functions over a general finite field with the update schedule given by an arbitrary word on the variables the paper contains generalizations of some of the known results about sds with permutation update schedules in particular an upper bound on the number of different sds over words of a given length is proved and open problems are discussed

optimal quadrature for univariate and tensor product splines abstract numerical integration is a core subroutine in many engineering applications including the finite element method fem isogeometric analysis is a fem technology that uses smooth b spline and nurbs basis functions traditionally gaussian quadrature was used for numerical integration but this yields suboptimal performance this is attributed to the fact that gaussian quadrature rules do not take inter element smoothness of the spline basis into account resulting in over integration the equations for exact quadrature are well known but prove notoriously hard to solve due to their nonlinear nature these generalized gauss rules were first introduced in hughes et al 2010 in the context of isogeometric analysis where newton iteration was utilized to study and tabulate several cases later homotopy continuation barto and calo 2015 was used as an alternative strategy for finding these rules in this paper we describe an algorithm to generalize on both of these techniques it is optimal in the sense that it will integrate a space of dimension n using no more than n 1 2 quadrature points the algorithm works on arbitrary nonuniform knot vectors of any polynomial degree and continuity and is demonstrated on polynomial orders up to 15 it extends to 2d and 3d integrals by tensor product

ets iv sequential dynamical systems fixed points invertibility and equivalence sequential dynamical systems sdss are discrete dynamical systems that are obtained from the following data a a finite labeled graph y with vertex set 1 n where each vertex has a binary state b a vertex labeled sequence of functions f i y f 2 n f 2 n i and c a permutation email protected s n the function f i y updates the binary state of vertex i as a function of the states of vertex i and its y neighbors and leaves the states of all other vertices fixed the permutation p represents a y vertex ordering according to which the functions f i y are applied by composing the functions f i y in the order given by p we obtain the sdsf y email protected i 1 nf p i y f 2 n f 2 n in this paper we will generalize a class of results on sds that have been proven for symmetric boolean local functions to quasi symmetric local functions further we completely classify invertible sds and investigate fixed points of sequential and parallel cellular automata ca finally we show sharpness of a combinatorial upper bound for the number of non equivalent sds that can be obtained through rescheduling for a certain class of graphs

overlapping schwarz methods for isogeometric analysis based on generalized b splines abstract generalized b splines gb splines are a special class of tchebycheff b splines that are smooth piecewise function with sections in more general spaces gb splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation in this paper we present overlapping schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on gb splines an h analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes numerical results in two and three dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters the domain deformation and the jumping coefficients

covariant and equivariant formality theorems abstract we give a proof of kontsevich s formality theorem for a general manifold using fedosov resolutions of algebras of polydifferential operators and polyvector fields the main advantage of our construction of the formality quasi isomorphism is that it is based on the use of covariant tensors unlike kontsevich s original proof which is based on jets of polydifferential operators and polyvector fields using our construction we prove that if a group g acts smoothly on a manifold m and m admits a g invariant affine connection then there exists a g equivariant quasi isomorphism of formality this result implies that if a manifold m is equipped with a smooth action of a finite or compact group g or equipped with a free action of a lie group g then m admits a g equivariant formality quasi isomorphism in particular this gives a solution of the deformation quantization problem for an arbitrary poisson orbifold

optimal randomized algorithms for local sorting and set maxima randomized algorithms for two sorting problems are presented in the local sorting problem a graph is given in which each vertex is assigned an element of a total order and the task is to determine the relative order of every pair of adjacent vertices in the set maxima problem a collection of sets whose elements are drawn from a total order is given and the task is to determine the maximum element in each set lower bounds for the problems in the comparison model are described and it is shown that the algorithms are optimal within a constant factor

overlapping schwarz preconditioners for isogeometric collocation methods isogeometric collocation methods are very recent and promising numerical schemes that preserve the advantages of isogeometric analysis but often exhibit better performances than their galerkin counterparts in the present paper an additive overlapping schwarz method for isogeometric collocation discretizations is introduced and studied the resulting preconditioner accelerated by gmres is shown to be scalable with respect to the number of subdomains and very robust with respect to the isogeometric discretization parameters such as the mesh size and polynomial degree as well as with respect to the presence of discontinuous elliptic coefficients and domain deformations

fast formation of isogeometric galerkin matrices by weighted quadrature abstract in this paper we propose an algorithm for the formation of matrices of isogeometric galerkin methods the algorithm is based on three ideas the first is that we perform the external loop over the rows of the matrix the second is that we calculate the row entries by weighted quadrature the third is that we exploit the local tensor product structure of the basis functions while all ingredients have a fundamental role for computational efficiency the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature the test function is incorporated in the integration weight while the trial function the geometry parametrization and the pdes coefficients form the integrand function this approach is very effective in reducing the computational cost while maintaining the optimal order of approximation of the method analysis of the cost is confirmed by numerical testing where we show that for p large enough the time required by the floating point operations is less than the time spent in unavoidable memory operations the sparse matrix allocation and memory write the proposed algorithm allows significant time saving when assembling isogeometric galerkin matrices for all the degrees of the test spline space and paves the way for a use of high degree k refinement in isogeometric analysis

predecessor existence problems for finite discrete dynamical systems we study the predecessor existence problem for finite discrete dynamical systems given a finite discrete dynamical system s and a configuration c the predecessor existence or pre problem is to determine whether there is a configuration c such that s has a transition from c to c in addition to the decision version we also study the following variants the predecessor existence or pre problem counting the number of predecessors the unique predecessor existence or upre problem deciding whether there is a unique predecessor and the ambiguous predecessor existence or apre problem given a configuration c and a predecessor c of c deciding whether there is a different predecessor c of c general techniques are presented for simultaneously characterizing the computational complexity of the pre problem and its three variants our hardness results are based on the concept of simultaneous reductions single transformations that can be used to simultaneously prove the hardness of the different variants of the pre problem for their respective complexity classes our easiness results are based on dynamic programming and they extend the previous results on pre problem for one dimensional cellular automata the hardness results together with the easiness results provide a tight separation between easy and hard instances further the results imply similar bounds for other classes of finite discrete dynamical systems including discrete hopfield and recurrent neural networks concurrent state machines systolic networks and one and two dimensional cellular automata our results extend the earlier results of green sutner and orponen on the complexity of the predecessor existence problem and its variants

matrix method for linear sequential dynamical systems on digraphs in this paper we introduce the concept of sequential dynamical systems sds on digraphs we focus on the discussion of linear sequential dynamical systems lsds matrix method is given in their analysis two special lsds or sds and par sds are particularly analyzed some structural properties on the image spaces of or d p and par d p are obtained the asymptotic behavior of or d p is described in terms of the properties of the digraph d with respect to the ordering p our results show that lsds on digraphs have much more interesting properties than those on undirected graphs

a preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysis abstract in the context of isogeometric analysis we consider two discretization approaches that make the resulting stiffness matrix nonsymmetric even if the differential operator is self adjoint these are the collocation method and the recently developed weighted quadrature for the galerkin discretization in this paper we are interested in the solution of the linear systems arising from the discretization of the poisson problem using one of these approaches in sangalli and tani 2016 a well established direct solver for linear systems with tensor structure was used as a preconditioner in the context of galerkin isogeometric analysis yielding promising results in particular this preconditioner is robust with respect to the mesh size h and the spline degree p in the present work we discuss how a similar approach can be applied to the considered nonsymmetric linear systems the efficiency of the proposed preconditioning strategy is assessed with numerical experiments on two dimensional and three dimensional problems

commuting difference operators arising from the elliptic c2 1 face model we study a pair of commuting difference operators arising from the elliptic c2 1 face model the operators whose coefficients are expressed in terms of the jacobi s elliptic theta function act on the space of meromorphic functions on the weight space of the c2 type simple lie algebra we show that the space of functions spanned by the level one characters of the affine lie algebra sp 4 c is invariant under the action of the difference operators

linear sequential dynamical systems incidence algebras and mobius functions abstract a sequential dynamical system sds consists of a graph a set of local functions and an update schedule a linear sequential dynamical system is an sds whose local functions are linear in this paper we derive an explicit closed formula for any linear sds as a synchronous dynamical system we also show constructively that any synchronous linear system can be expressed as a linear sds i e it can be written as a product of linear local functions furthermore we study the connection between linear sds and the incidence algebras of partially ordered sets posets specifically we show that the mobius function of any poset can be computed via an sds whose graph is induced by the hasse diagram of the poset finally we prove a cut theorem for the mobius functions of posets with respect to certain chain decompositions

low rank tensor methods in galerkin based isogeometric analysis abstract the global patch wise geometry map which describes the computational domain is a new feature in isogeometric analysis this map has a global tensor structure inherited from the parametric spline geometry representation the use of this global structure in the discretization of partial differential equations may be regarded as a drawback at first glance as opposed to the purely local nature of high order classical finite elements in this work we demonstrate that it is possible to exploit the regularity of this structure and to identify the great potential for the efficient implementation of isogeometric discretizations first we formulate tensor product b spline bases as well as the corresponding mass and stiffness matrices as tensors in order to reveal their intrinsic structure second we derive an algorithm for the separation of variables in the integrands arising in the discretization this is possible by means of low rank approximation of the integral kernels we arrive at a compact separated representation of the integrals the separated form implies an expression of galerkin matrices as kronecker products of matrix factors with small dimensions this representation is very appealing due to the reduction in both memory consumption and computation times our benchmarks performed using the c library g smo demonstrate that the use of tensor methods in isogeometric analysis possesses significant advantages

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

dichotomy results for fixed point counting in boolean dynamical systems we present dichotomy theorems regarding the computational complexity of counting fixed points in boolean discrete dynamical systems i e finite discrete dynamical systems over the domain 0 1 for a class f of boolean functions and a class g of graphs an f g system is a boolean dynamical system with local transitions functions lying in f and graphs in g we show that if local transition functions are given by lookup tables then the following complexity classification holds let f be a class of boolean functions closed under superposition and let g be a graph class closed under taking minors if f contains all min functions all max functions or all self dual and monotone functions and g contains all planar graphs then it is p complete to compute the number of fixed points in an f g system otherwise it is computable in polynomial time we also prove a dichotomy theorem for the case that local transition functions are given by formulas over logical bases this theorem has a significantly more complicated structure than the theorem for lookup tables a corresponding theorem for boolean circuits coincides with the theorem for formulas

ruijsenaars commuting difference operators as commuting transfer matrices for belavin s elliptic quantum r matrix we construct an l operator as a set of difference operators acting on functions on the type a weight space according to the fundamental relation rll llr taking the trace of the l operator gives a set of commuting difference operators we show that for the above mentioned l operator this approach gives macdonald type operators with elliptic theta function coefficient actually equivalent to ruijsenaars operators the relationship between the difference l operator and krichever s lax matrix is given and an explicit formula for elliptic commuting differential operators is derived we also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space

spectral analysis of hermite cubic spline collocation systems in this paper we present an eigenvalue analysis of the first order and second order hermite cubic spline collocation differentiation matrices with arbitrary collocation points some important features are explored and compared with some other discrete methods such as finite difference methods

elements of a theory of simulation iii equivalence of sds in the development of mathematical foundations for a theory of simulation a certain class of discrete sequential dynamical systems sds is of particular importance these systems which we refer to as sds consist of a a graph y with vertex set 1 2 n where each vertex has associated a binary state b a vertex labeled set of functions f i y f 2 n f 2 n and c a permutation p s n the function f i y update the state of vertex i as a function of the states of vertex i and its y neighbors and leaves all other states invariant by composing these functions in the order given by p we obtain the sds f y p i 1 nf p i y f 2 n f 2 n in this paper we give a combinatorial upper bound for the number of non equivalent sds for a given graph and we compute this bound explicitly for certain classes of graphs we give a full characterization of invertible sds and analyze the set of fixed points of sequential and parallel cellular automata further we introduce the concept of maj type sds and show that systems of this class only have fixed points as attractors finally we analyze sds that are fixed point free for arbitrary base graphs

isogeometric collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations abstract we compare isogeometric collocation with isogeometric galerkin and standard c 0 finite element methods with respect to the cost of forming the matrix and residual vector the cost of direct and iterative solvers the accuracy versus degrees of freedom and the accuracy versus computing time on this basis we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric galerkin and standard c 0 finite element methods when a specified level of accuracy is to be achieved with minimum computational cost we then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of nurbs basis functions and collocation points derived from the corresponding multi level greville abscissae we introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two scale relation of b splines using weighted collocation in the transition regions between hierarchical levels we are able to reliably handle coincident collocation points that naturally occur for multi level greville abscissae the resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency local adaptivity robustness and straightforward implementation which we illustrate by numerical examples in one two and three dimensions

optimal equivalent preconditioners preconditioning strategies for elliptic partial differential operators and corresponding difference operators are compared by way of a model problem the technique considered here makes use of a selfadjoint positive definite operator b as the preconditioner for the nonselfadjoint convection diffusion operator a preconditioning by a s leading term plus a positive zeroth order term sigma i and optimizing over sigma is proposed the analysis of a model problem indicates that when a has large first order terms both the spectral and norm condition of b 1 a are minimized by sigma gg 0

elements of a theory of computer simulation i sequential ca over random graphs this paper is a first step in the development of mathematical foundations for a theory of simulation we employ sequentially updated cellular automata sca over arbitrary graphs as a paradigmatic framework in the development of the theory we focus on the properties of causal dependencies among local mappings in a simulation let y be a graph in which a each vertex i has a state x i0 1 and b there exists a local map i defined on the states of the y neighbors and x i that returns the state of y i0 1 suppose i 1 i n is a permutation of the y vertices then the order of application of the local maps i k induces a sequential or asynchronous cellular automaton sca over y in this paper we introduce a mapping between the base graph y over which the sca is defined and which represents the mutual dependencies of the local maps and the update graph u y whose vertices are permutations of all y vertices two permutations p 1 p 2 are adjacent in u y if they differ in exactly two consecutive coordinates i k i k 1 and i k i k 1 ey u y represents a conceptual framework which allows to determine equivalence classes of sca for a given graph x independent of the family of local maps i 1 i n we consider a u as a random variable over g n p and analyze the induced random graph u g n p the update graph and show b that u exhibits properties of a covariant functor

the domination numbers of cylindrical grid graphs abstract let p m c n denote the domination number of the cylindrical grid graph formed by the cartesian product of the graphs p m the path of length m m 2 and the graph c n the cycle of length n n 3 in this paper methods to find the domination numbers of graphs of the form p m c n with n 3 and m 2 3 and 4 are proposed moreover bounds on domination numbers of the graphs p 5 c n n 3 are found the methods that are used to prove that results readily lead to algorithms for finding minimum dominating sets of the above mentioned graphs

the dynamical twisting and nondynamical r matrix structure of the elliptic ruijsenaars schneider model from the dynamical twisting of the classical r matrix we obtain a new lax operator for the elliptic ruijsenaars schneider model cf ruijsenaars the corresponding r matrix is shown to be the classical zn symmetric elliptic r matrix which is the same as that obtained in the study of the nonrelativistic version the an 1 calogero moser model

the variational collocation method we propose the variational collocation method for the numerical solution of partial differential equations the conceptual basis is the establishment of a direct connection between the galerkin method and the classical collocation methods with the perspective of achieving the accuracy of the former with a computational cost of one point evaluation per degree of freedom as in the latter variational collocation requires a discrete space constructed by smooth and pointwise non negative basis functions which makes the approach immediately applicable to isogeometric analysis and some meshfree methods in this paper we concentrate on isogeometric analysis and demonstrate that there exists a set of points such that collocation of the strong form at these points produces the galerkin solution exactly we provide an estimate of these points and show that applying isogeometric collocation at the estimated points completely solves the well known odd even discrepancy in the order of spatial convergence we demonstrate the potential of variational collocation with examples of linear and non linear elasticity as well as kirchhoff plates

the non dynamical r matrix structure for the elliptic calogero moser model in this paper we construct a new lax operator for the elliptic calogero moser model with general from the classical dynamical twisting in which the corresponding r matrix is purely numerical non dynamic the non dynamical r matrix structure of this lax operator is obtained which is the elliptic symmetric r matrix

computational complexity studies of synchronous boolean finite dynamical systems on directed graphs abstract a finite dynamical system is a system consisting of some finite number of objects that take upon a value from some domain as a state in which after initialization the states of the objects are updated based upon the states of the other objects and themselves according to a certain update schedule this paper studies a subclass of finite dynamical systems the synchronous boolean finite dynamical system synchronous bfds for short where the states are boolean and the state update takes place in discrete time and at the same on all objects the paper is concerned with three problems convergence path intersection and cycle length of the synchronous bfds in which the state update functions or the local state transition functions are chosen from a predetermined finite basis of boolean functions b the paper results characterize their computational complexity