multiplicative complexity of polynomial multiplication over finite fields let m q n denote the number of multiplications required to compute the coefficients of the product of two polynomials of degree n over a q element field by means of bilinear algorithms it is shown that m q n n 3 n o n in particular if q 2 n l q 1 we establish the tight bound m q n 3 n 1 q 2 the technique we use can be applied to analysis of algorithms for multiplication of polynomials modulo a polynomial as well

on the complexity of multiplication in finite fields abstract in this paper we study the bilinear complexity of multiplying two arbitrary elements from an nth degree extension of a finite field f and the related problem of multiplying over f two polynomials of degree n 1 with indeterminate coefficients we derive a new linear lower bound and we describe an algorithm leading to a quasi linear upper bound

on the number of distinct functional graphs of affine linear transformations over finite fields abstract we study the number of non isomorphic functional graphs of affine linear transformations from f q n to itself and we prove upper and lower bounds on this quantity as n as a corollary to our result we prove bounds on the number of conjugacy classes in the symmetric group s q n that intersect agln q

a novel partitioning method for block structured adaptive meshes we propose a novel partitioning method for block structured adaptive meshes utilizing the meshless lagrangian particle concept with the observation that an optimum partitioning has high analogy to the relaxation of a multi phase fluid to steady state physically motivated model equations are developed to characterize the background mesh topology and are solved by multi phase smoothed particle hydrodynamics in contrast to well established partitioning approaches all optimization objectives are implicitly incorporated and achieved during the particle relaxation to stationary state distinct partitioning sub domains are represented by colored particles and separated by a sharp interface with a surface tension model in order to obtain the particle relaxation special viscous and skin friction models coupled with a tailored time integration algorithm are proposed numerical experiments show that the present method has several important properties generation of approximately equal sized partitions without dependence on the mesh element type optimized interface communication between distinct partitioning sub domains continuous domain decomposition which is physically localized and implicitly incremental therefore it is particularly suitable for load balancing of high performance cfd simulations

controlling chaos in high dimensions theory and experiment naval surface warfare center carderock division silver spring maryland 20903 received 27 november 1995 the main contribution of this work is the development of a high dimensional chaos control method that iseffective robust against noise and easy to implement in experiment assuming no knowledge of the modelequations the method achieves control by stabilizing a desired unstable periodic orbit with any number ofunstable directions using small time dependent perturbations of a single system parameter speci cally ourmajor results are as follows first we derive explicit control laws for time series produced by discrete maps second we show how to apply this control law to continuous time problems by introducing straightforwardways to extract from a continuous time series a discrete time series that measures the dynamics of somepoincare map of the original system third we illustrate our approach with two examples of high dimensionalordinary differential equations one autonomous and the other periodically driven fourth we present the resulton our successful control of chaos in a high dimensional experimental system demonstrating the viability ofthe method in practical applications pacs number s 05 45 1bi introduction

delay differential equation versus 1d map application to chaos control abstract theoretical and experimental results on the control of chaotic motion in an electronic circuit modeled by a delay differential equation are presented the model is derived from a nonlinear optical system with time delay feedback our control method is based on pyragas approach and applied to the infinite dimensional system generalized mean floquet multipliers and mean lyapunov exponents related to delay differential equations are introduced in order to compare a discrete time approximation with the continuous time equation this control method is also used to find the so called isomers related to harmonic periodic solutions predicted by ikeda and matsumoto

minimal dirichlet energy partitions for graphs motivated by a geometric problem we introduce a new nonconvex graph partitioning objective where the optimality criterion is given by the sum of the dirichlet eigenvalues of the partition components a relaxed formulation is identified and a novel rearrangement algorithm is proposed which we show is strictly decreasing and converges in a finite number of iterations to a local minimum of the relaxed objective function our method is applied to several clustering problems on graphs constructed from synthetic data mnist handwritten digits and manifold discretizations the model has a semisupervised extension and provides a natural representative for the clusters as well

efficient sampling set selection for bandlimited graph signals using graph spectral proxies we study the problem of selecting the best sampling set for bandlimited reconstruction of signals on graphs a frequency domain representation for graph signals can be defined using the eigenvectors and eigenvalues of variation operators that take into account the underlying graph connectivity smoothly varying signals defined on the nodes are of particular interest in various applications and tend to be approximately bandlimited in the frequency basis sampling theory for graph signals deals with the problem of choosing the best subset of nodes for reconstructing a bandlimited signal from its samples most approaches to this problem require a computation of the frequency basis i e the eigenvectors of the variation operator followed by a search procedure using the basis elements this can be impractical in terms of storage and time complexity for real datasets involving very large graphs we circumvent this issue in our formulation by introducing quantities called graph spectral proxies defined using the powers of the variation operator in order to approximate the spectral content of graph signals this allows us to formulate a direct sampling set selection approach that does not require the computation and storage of the basis elements we show that our approach also provides stable reconstruction when the samples are noisy or when the original signal is only approximately bandlimited furthermore the proposed approach is valid for any choice of the variation operator thereby covering a wide range of graphs and applications we demonstrate its effectiveness through various numerical experiments

on the optimality of the ott grebogi yorke control scheme abstract some of the characteristics of the ott grebogi yorke ogy control technique are presented as applied to nonlinear flows as distinct from nonlinear maps specifically we consider the case where the magnitude of the control parameter varies in time within each control cycle in proportion to a given function referred to as a basis function the choice of the basis function is shown to influence the basin of convergence for a given level of parameter variation in the ogy controller an algorithm for designing the optimal basis function is presented the optimal basis function is shown to be defined by a step function with potentially several jumps thus revealing the intrinsic power of a standard ogy technique that uses a single step function as a basis function two numerical applications of the optimal design technique to a duffing oscillator are also presented to show that the standard ogy technique may be significantly improved by making an optimal choice of a basis function copyright

controllability of spatiotemporal systems using constant pinnings abstract most natural spatiotemporal systems are an organized ensemble of dynamical subsystems whose behaviour is regulated by nonlinearly coupled multi variable processes the response of each of these variables and or combinations of them to any single or composite external perturbation can influence its spatiotemporal behaviour in a complex and non intuitive manner this paper attempts to study the dynamical response of two coupled map lattice systems having local dynamics described by a coupled discrete maps and b coupled differential equations to external perturbation or pinning applied to each variable individually or simultaneously we show that the response of different variables to external pinning is quite different our results indicate that though this pinning approach is useful in controlling complex dynamics both globally and locally enhancing complexity in dynamics anti control is variable dependent thus complete controllability i e control and anti control of dynamics in these spatially extended systems having coupled multi variable processes does not only depend on the sign and strength of the perturbation but also on the specific variable being pinned this also implies that external manipulation of state variables in natural systems can lead to quite different responses depending at the level of action of the perturbation

symmetry and control spatially extended chaotic systems in a recent paper phys rev e 57 1998 1550 it was demonstrated that the symmetries of the evolution equation and the target state have a profound effect on controlling the chaotic behavior in the present paper we extend these results to the cases of time periodic target trajectories and inexact symmetries and apply the developed formalism to the problem of controlling spatiotemporal chaos we use the example of a lattice dynamical system in arbitrary spatial dimension to show that there exists an intimate relationship between the geometry of an extended system and the geometry of feedback control 2000 elsevier science b v all rights reserved

multiplicative complexity of vector valued boolean functions abstract we consider the multiplicative complexity of boolean functions with multiple bits of output studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity for so called circuits we show that there is a tight connection between error correcting codes and circuits computing functions with high nonlinearity combining this with known coding theory results we show that functions with n inputs and n outputs with the highest possible nonlinearity must have at least 2 32n and gates we further show that one cannot prove stronger lower bounds by only appealing to the nonlinearity of a function we show a bilinear circuit computing a function with almost optimal nonlinearity with the number of and gates being exactly the length of such a shortest code additionally we provide a function which for general circuits has multiplicative complexity at least 2 n 3 finally we study the multiplicative complexity of almost all functions we show that every function with n bits of input and m bits of output can be computed using at most 2 5 1 o 1 m 2 n and gates

multiplication of polynomials modulo xn let n be positive integers with email protected 2n 1 let r be an arbitrary nontrivial ring not necessarily commutative and not necessarily having a multiplicative identity and r x be the polynomial ring over r in this paper we give improved upper bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most n 1 modulo x n over r moreover we introduce a new complexity notion the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most n 1 modulo x over r this new complexity notion provides improved bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most n 1 modulo x n over r

controlling effect of geometrically defined local structural changes on chaotic hamiltonian systems an effective characterization of chaotic conservative hamiltonian systems in terms of the curvature associated with a riemannian metric tensor derived from the structure of the hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric the geodesic equations reproduce the hamilton equations of the original potential model through an inverse map in the tangent space the second covariant derivative of the geodesic deviation in this space generates a dynamical curvature resulting in energy dependent criteria for unstable behavior different from the usual lyapunov criteria we show here that this criterion can be constructively used to modify locally the potential of a chaotic hamiltonian model in such a way that stable motion is achieved since our criterion for instability is local in coordinate space these results provide a minimal method for achieving control of a chaotic system

controlling dynamics in spatially extended systems spatially extended systems exhibit a variety of spatiotemporal dynamics from stable to chaotic these dynamics can change under pathological conditions and impair normal functions thus having the ability to control the altered dynamics for improved functioning has the potential for wide ranging applications in real and artificial systems here we propose a simple and general method that can be used to target the spatiotemporal dynamics both globally and in spatially localized regions in either direction i e towards the stable or unstable manifold by simply changing the strength and the sign of an externally applied perturbation or pinning the method is applicable to both chaotic and nonchaotic systems with discrete and continuous local dynamics and for different topologies of interactions we also apply it to simulate an experiment on epileptogenic neuronal activity in rat hippocampal tissue b j gluckman et al j neurophys 76 6202 1996 this unified approach for differential targeting of global and local dynamics promises to be useful for systems spanning large spatial scales and having structural and functional heterogeneity

multiplicative complexity of direct sums of quadratic systems abstract we consider the quadratic complexity of certain sets of quadratic forms we study classes of direct sums of quadratic forms for these classes of problems we show that the complexity of one direct sum is the sum of the complexities of the summands and that every minimal quadratic algorithm for computing the direct sums is a direct sum algorithm

control of stochastic chaos using sliding mode method stabilizing unstable periodic orbits of a deterministic chaotic system which is perturbed by a stochastic process is studied in this paper the stochastic chaos is modeled by exciting a deterministic chaotic system with a white noise obtained from derivative of a wiener process which eventually generates an ito differential equation it is also assumed that the chaotic system being studied has some model uncertainties which are not random the sliding mode controller with some modifications is used for stochastic chaos suppression it is shown that the system states converge to the desired orbit in such a way that the error covariance converges to an arbitrarily small bound around zero as some case studies the stabilization of 1 cycle and 2 cycle orbits of chaotic duffing and f 6 van der pol systems is investigated by applying the proposed method to their corresponding stochastically perturbed systems simulation results show the effectiveness of the method and the accuracy of the statements proved in the paper

optimal multi dimensional ogy controller abstract a technique which generalizes the standard ogy control scheme and may be applied to higher dimensional systems to stabilize both saddle nodes and fully unstable fixed points is presented the proposed method is designed for nonlinear flows rather than maps and also provides a powerful technique to assess the controllability of nonlinear systems with ogy type controllers an optimal ogy control scheme designed for larger dimensional systems having improved convergence properties compared to the standard ogy controller is also presented numerical examples for the well known van der pol duffing oscillator are presented to illustrate the proposed control schemes

route to high dimensional chaos we present a route to high dimensional chaos that is chaos with more than one positive lyapunov exponent in this route as a system parameter changes a subsystem becomes chaotic through say a cascade of period doubling bifurcations after which the complementary subsystem becomes chaotic leading to an additional positive lyapunov exponent for the whole system a characteristic feature of this route as suggested by numerical evidence is that the second largest lyapunov exponent passes through zero continuously three examples are presented a discrete time map a continuous time flow and a population model for species dispersal in evolutionary ecology

a diffusion generated method for computing dirichlet partitions abstract a dirichlet k partition of a closed d dimensional surface is a collection of k pairwise disjoint open subsets such that the sum of their first laplace beltrami dirichlet eigenvalues is minimal in this paper we develop a simple and efficient diffusion generated method to compute dirichlet k partitions for d dimensional flat tori and spheres for the two dimensional flat torus for most values of k 3 9 11 12 15 16 and 20 we obtain hexagonal honeycombs for the three dimensional flat torus and k 2 4 8 16 we obtain the rhombic dodecahedral honeycomb the weaire phelan honeycomb and kelvin s tessellation by truncated octahedra for the four dimensional flat torus for k 4 we obtain a constant extension of the rhombic dodecahedral honeycomb along the fourth direction and for k 8 we obtain a 24 cell honeycomb for the two dimensional sphere we also compute dirichlet partitions for k 3 7 9 10 12 14 20 our computational results agree with previous studies when a comparison is available as far as we are aware these are the first results for dirichlet partitions of the four dimensional flat torus

the method of fundamental solutions applied to boundary eigenvalue problems we develop methods based on fundamental solutions to compute the steklov wentzell and laplace beltrami eigenvalues in the context of shape optimization in the class of smooth simply connected two dimensional domains the numerical method is accurate and fast a theoretical error bound is given along with comparisons with mesh based methods we illustrate the use of this method in the study of a wide class of shape optimization problems in two dimensions we extend the method to the computation of the laplace beltrami eigenvalues on surfaces and we investigate some spectral optimal partitioning problems

an upper bound on the complexity of multiplication of polynomials modulo a power of an irreducible polynomial let q2 n k denote the minimum number of multiplications required to compute the coefficients of the product of two degree n k 1 polynomials modulo the kth power of an irreducible polynomial of degree n over the q2 element field bbf q2 it is shown that for all odd q and all n 1 2 liminfk q2 n k k n 2 1 1 q 2 for the proof of this upper bound we show that for an odd prime power q all algebraic function fields in the garcia stichtenoth tower over bbf q2 have places of all degrees and apply a chudnovsky like algorithm for multiplication of polynomials modulo a power of an irreducible polynomial

maximal rank of m x n x mn k tensors it is shown that the maximal rank of m times n times mn k tensors with k leqq min m 1 2 2 n 1 2 2 is greater than mn 4 sqrt 2k o 1

the control of higher dimensional chaos comparative results for the chaotic satellite attitude control problem abstract using simulations this paper reports the success and failure of three experiments each using a different technique to control the same higher dimensional chaotic system the three techniques are a simple delayed feedback control method originally suggested by pyragas the otani jones technique and a higher dimensional variation of the ogy method the three methods are applied to a six dimensional system which describes the attitude dynamics of a satellite rigid body subjected to deterministic external perturbations which induce chaotic motion when no control is effected the attitude of the satellite is controlled by three orthogonal pairs of thrusters aligned with the principal axes and the system can be described by a modified set of euler equations the three control methods are compared in terms of the prior calculations required the real time computational requirements and the effectiveness of the method in stabilizing the system the results show that the method of pyragas compares very favourably with the other two techniques requiring no prior calculation and having a very low real time computational overhead pyragas method also provides the most satisfactory control solution the problems in presenting a theoretical justification of pyragas method are briefly discussed and the connection between the method and a recent technique for the location of unstable fixed points in chaotic systems is highlighted