systems of numeration and fractal functions relating to substitutions french abstract let a be a finite alphabet a substitution over a un n n a fixed point for and for each a a a a real number we establish under some assumptions an asymptotic formula concerning the sum s n i n u i n n this result generalizes some previous results from coquet or brillhart erdos and morton moreover relations with self affine functions in a sense which generalizes a definition from kamae are proved the calculi leave over systems of representation of integers and real numbers

constant depth reducibility the purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing and s or s and not s with no bound placed on the fan in of and gates and or gates two such reducibilities are defined and reductions and equivalences among several common problems such as parity sorting integer multiplication graph connectivity bipartite matching and network flow are given certain problems are shown to be complete with respect to these reducibilities in the complexity classes deterministic logarithmic space nondeterministic logarithmic space and deterministic polynomial time new upper bounds on the size depth unbounded fan in circuit complexity of symmetric boolean functions are established

distribution of values of bounded generalized polynomials a generalized polynomial is a real valued function which is obtained from conventional polynomials by the use of the operations of addition multiplication and taking the integer part a generalized polynomial mapping is a vector valued mapping whose coordinates are generalized polynomials we show that any bounded generalized polynomial mapping u zd rl has a representation u n f n x n zd where f is a piecewise polynomial function on a compact nilmanifold x x x and is an ergodic zd action by translations on x this fact is used to show that the sequence u n n zd is well distributed on a piecewise polynomial surface mathcal s subset mathbf r l with respect to the borel measure on mathcal s that is the image of the lebesgue measure under the piecewise polynomial function defining mathcal s as corollaries we also obtain a von neumann type ergodic theorem along generalized polynomials and a result on diophantine approximations extending the work of van der corput and of furstenberg weiss

threshold circuits of small majority depth we investigate the complexity of computations with constant depth threshold circuits such circuits are composed of gates that determine if the sum of their inputs is greater than a certain threshold when restricted to polynomial size these circuits compute exactly the functions in the class tc sp0 these circuits are usually studied by measuring their efficiency in terms of their total depth using this point of view the best division and iterated multiplication circuits have depth three and four respectively in this thesis we propose a different approach since threshold gates are much more powerful than and or gates we allow the explicit use of and or gates and consider the main measure of complexity to be the majority depth of the circuit i e the maximum number of threshold gates on any path in the circuit using this approach we obtain division and iterated multiplication circuits of total depth four and five but of majority depth two and three the technique used is called chinese remaindering we present this technique as a general tool for computing functions with integer values and use it to obtain depth four threshold circuits of majority depth two for other arithmetic problems such as the logarithm and power series approximation we also consider the iterated multiplication problem for integers modulo q and for finite fields the notion of majority depth naturally leads to a hierarchy of subclasses of tc sp0 we investigate this hierarchy and show that it is closely related to the usual depth hierarchy

fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows in this paper we investigate the numerical simulation of particulate flows using a new moving mesh method combined with the multigrid fictitious boundary method fbm s turek d c wan l s rivkind the fictitious boundary method for the implicit treatment of dirichlet boundary conditions with applications to incompressible flow simulations challenges in scientific computing lecture notes in computational science and engineering vol 35 springer berlin 2003 pp 37 68 d c wan s turek l s rivkind an efficient multigrid fem solution technique for incompressible flow with moving rigid bodies numerical mathematics and advanced applications enumath 2003 springer berlin 2004 pp 844 853 d c wan s turek direct numerical simulation of particulate flow via multigrid fem techniques and the fictitious boundary method int j numer method fluids 51 2006 531 566 with this approach the mesh is dynamically relocated through a linear partial differential equation to capture the surface of the moving particles with a relatively small number of grid points the complete system is realized by solving the mesh movement and the partial differential equations of the flow problem alternately via an operator splitting approach the flow is computed by a special ale formulation with a multigrid finite element solver and the solid particles are allowed to move freely through the computational mesh which is adaptively aligned by the moving mesh method in every time step one important aspect is that the data structure of the undeformed initial mesh in many cases a tensor product mesh or a semi structured grid consisting of many tensor product meshes is preserved while only the spacing between the grid points is adapted in each time step so that the high efficiency of structured meshes can be exploited numerical results demonstrate that the interaction between the fluid and the particles can be accurately and efficiently handled by the presented method it is also shown that the presented method significantly improves the accuracy of the previous multigrid fbm to simulate particulate flows with many moving rigid particles

chaotic dynamics of continuous time topological semi flows on polish spaces abstract differently from lyapunov exponents li yorke devaney and others that appeared in the literature we introduce the concept chaos for a continuous semi flow f r x x on a polish space x with a metric d which is useful in the theory of ode and is invariant under topological equivalence of semi flows our definition is weaker than devaney s one since here f may have neither fixed nor periodic elements but it implies repeatedly observable sensitive dependence on initial data there is an 0 such that for any x x there corresponds a dense g set s u x in x satisfying lim sup t d f t x f t y y s u x this sensitivity is obviously stronger than guckenheimer s one that requires only d f t x f t y for some moment t 0 and some y arbitrarily close to x

schreier sets in ramsey theory we show that ramsey theory a domain presently conceived to guaran tee the existence of large homogeneous sets for partitions on k tuples of words for every natural number k over a finite alphabet can be extended to one for partitions on schreier type sets of words of every countable ordinal indeed we establish an extension of the partition theorem of carlson about words and of the more general partition theorem of furstenberg katznelson about combinatorial subspaces of the set of words generating from k tuples of words for any fixed natural number k into a partition theorem about combinatorial subspaces generating from schreier type sets of words of order any fixed countable ordinal furthermore as a result we obtain a strengthening of carlson s infinitary nash williams type and ellentuck type partition theorem about infinite sequences of variable words into a theorem in which an infinite sequence of variable words and a binary partition of all the finite sequences of words one of whose components is in addition a tree are assumed concluding that all the schreier type finite reductions of an infinite reduction of the given sequence have a be havior determined by the cantor bendixson ordinal index of the tree component of the partition falling in the tree component above that index and in its complement below it

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

some new results in multiplicative and additive ramsey theory there are several notions of largeness that make sense in any semigroup and others such as the various kinds of density that make sense in sucien tly well behaved semigroups including n and n it is known that sets with positive multiplicative density must contain arbitrarily large geoarithmetic progressions that is sets of the form fr j a id i j 2 f0 1 kgg we establish some combined additive and multiplicative ramsey theoretic consequences of known algebraic results in the semigroups n and n derive some new algebraic results and derive consequences of them involving geoarithmetic progressions for example we show that in any nite partition of n there must be for each k sets of the form fb a id j i j 2 f0 1 kgg together with d the arithmetic progression fa id i 2 f0 1 kgg and the geometric progression fbd j j 2 f0 1 kgg in one cell of the partition

efficient threshold circuits for power series abstract we show that functions with convergent real power series can be well approximated by two classes of polynomial size small weight threshold circuits depth three circuits with threshold gates on all levels and depth four circuits with threshold gates on the first two levels and and or gates on the last two this is done without restricting the input to a fixed closed subinterval of the interval of convergence of the series we also point out that rational functions and the logarithm of x in base b can be well approximated by the same classes of circuits when both x and b are given as input

a fast parallel algorithm for determining all roots of a polynomial with real roots given a polynomial p z of degree n with m bit integer coefficients and an integer mu the problem of determining all its roots with error less than 2 mu is considered it is shown that this problem is in the class nc if p z has all real roots some very interesting properties of a sturm sequence of a polynomial with distinct real roots are proved and used in the design of a fast parallel algorithm for this problem using newton identities and a novel numerical integration scheme for evaluating a contour integral to high precision this algorithm determines good approximations to the linear factors of p z

regular article direct simulations of 2d fluid particle flows in biperiodic domains we propose a method to simulate the motion of 2d rigid particles in a viscous incompressible fluid within the arbitrary lagrangian eulerian framework momentum equations for both the fluid and the particles are discretized and a coupled variational formulation is established by introducing an appropriate finite element approximation a symmetric linear system is obtained this system is solved by an inexact uzawa algorithm the main interest of such simulations lies in the average behaviour of a high number of particles we therefore introduced a biperiodic formulation of the problem which makes it possible to represent many body mixtures at a reasonable computational cost in order to model realistic situations an extra term must be added to the pressure this extra term is shown to be the lagrange multiplier associated with the vertical volume conservation constraint we developed an appropriate unstructured mesh generator so that the biperiodicity of the fields can be treated in a strong sense the question of particle contact is addressed and a simple technique to overcome numerical problems is proposed the influence of periodic lengths is investigated through different simulations the same case is simulated for different sizes of the window and the behaviour of some space averaged quantities is studied

revisiting the nilpotent polynomial hales jewett theorem abstract answering a question posed by bergelson and leibman in 6 we establish a nilpotent version of the polynomial hales jewett theorem that contains the main theorem in 6 as a special case important to the formulation and the proof of our main theorem is the notion of a relative syndetic set relative with respect to a closed non empty subsets of g 25 as a corollary of our main theorem we prove an extension of the restricted van der waerden theorem to nilpotent groups which involves nilprogressions

particulate flows with the subspace projection method a novel finite element method for the 3d simulation of many particles in a newtonian carrier liquid is presented the methods features the celebrated one domain approach a subspace projection method to account for the rigid body motion within the particles and operator splitting combined with local mesh refinement the method results in a fast and accurate though conceptually simple to implement algorithm validation is achieved using the sedimentation of one particle and comparing the resulting drag coefficient with theoretical and experimental results furthermore a viscometer is considered where the effective viscosity of a particle laden fluid is compared with analytic results

regular languages defined with generalized quantifiers we study an extension of first order logic obtained by adjoining quantifiers that count with respect to an integer modulus it is shown that the languages definable in this framework are precisely the regular languages whose syntactic monoids contain only solvable groups we obtain an analogous result for regular languages and establish some connections with complexity theory for fixed depth families of circuits

root finding with threshold circuits we show that for any constant d complex roots of degree d univariate rational or gaussian rational polynomials given by a list of coefficients in binary can be computed to a given accuracy by a uniform tc 0 algorithm a uniform family of constant depth polynomial size threshold circuits the basic idea is to compute the inverse function of the polynomial by a power series we also discuss an application to the theory v tc 0 of bounded arithmetic

devaney chaos li yorke chaos and multi dimensional li yorke chaos for topological dynamics abstract let t x x written t x be a topological semiflow flow on a uniform space x with t a multiplicative topological semigroup group not necessarily discrete we then prove if t x is non minimal topologically transitive with dense almost periodic points then it is sensitive to initial conditions as a result of this devaney chaos sensitivity to initial conditions for this very general setting let r x be a c 0 semiflow on a polish space then we show if r x is topologically transitive with at least one periodic point p and there is a dense orbit with no nonempty interior then it is multi dimensional li yorke chaotic that is there is a uncountable set x such that for any k 2 and any distinct points x 1 x k one can find two time sequences s n t n with s n x 1 x k x 1 x k x k and t n x 1 x k p p x k consequently devaney chaos multi dimensional li yorke chaos moreover let x be a non singleton polish space then we prove any weakly mixing c 0 semiflow r x is densely multi dimensional li yorke chaotic any minimal weakly mixing topological flow t x with t abelian is densely multi dimensional li yorke chaotic any weakly mixing topological flow t x is densely li yorke chaotic we in addition construct a completely li yorke chaotic minimal sl 2 r acting flow on the compact metric space r our various chaotic dynamics are sensitive to the choices of the topology of the phase semigroup group t

ipr recurrence and nilsystems abstract by a result due to furstenberg a homeomorphism t of a compact space is distal if and only if it possesses the property of ip recurrence meaning that for any x 0 x for any open neighborhood u of x 0 and for any sequence n i in z the set r u x 0 n z t n x 0 u has a non trivial intersection with the set of finite sums n i 1 n i 2 n i s i 1 i 2 i s s n we show that translations on compact nilmanifolds which are known to be distal are characterized by a stronger property of ip r recurrence which asserts that for any x 0 x and any neighborhood u of x 0 there exists r n such that for any r element sequence n 1 n r in z the set r u x 0 has a non trivial intersection with the set n i 1 n i 2 n i s i 1 i 2 i s s r we also show that the property of ip r recurrence is equivalent to an ostensibly much stronger property of polynomial ip r recurrence this should be juxtaposed with the fact that for general distal transformations the polynomial ip recurrence is strictly stronger than the ip recurrence

simulating threshold circuits by majority circuits we prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial size depth 2 majority circuit in general we show that a polynomial size depth d threshold circuit can be simulated uniformly by a polynomial size majority circuit of depth d 1 goldmann hastad and razborov showed in comput complexity 2 1992 pp 277 300 that a nonuniform simulation exists our construction answers two open questions posed by them we give an explicit construction whereas they use a randomized existence argument and we show that such a simulation is possible even if the depth d grows with the number of variables n their simulation gives polynomial size circuits only when d is constant

the expressiveness of a family of finite set languages abstract in this paper we characterize exactly the complexity of a set based database language called srl which presents a unified framework for queries and updates by imposing simple syntactic restrictions on it we are able to express exactly the classes p and logspace we also discuss the role of ordering in database query languages and show that the hom operator of machiavelli language in ohori et al 1989 does not capture all the order independent properties

bounds for the computational power and learning complexity of analog neural nets it is shown that high order feedforward neural nets of constant depth with piecewise polynomial activation functions and arbitrary real weights can be simulated for boolean inputs and outputs by neural nets of a somewhat larger size and depth with heaviside gates and weights from 1 0 1 this provides the first known upper bound for the computational power of the former type of neural nets it is also shown that in the case of first order nets with piecewise linear activation functions one can replace arbitrary real weights by rational numbers with polynomially many bits without changing the boolean function that is computed by the neural net in order to prove these results we introduce two new methods for reducing nonlinear problems about weights in multilayer neural nets to linear problems for a transformed set of parameters these transformed parameters can be interpreted as weights in a somewhat larger neural net as another application of our new proof technique we show that neural nets with piecewise polynomial activation functions and a constant number of analog inputs are probably approximately correct pac learnable in valiant s model for pac learning comm assoc comput mach 27 1984 pp 1134 1142

adaptive grid generation based onthe least squares finite element method approximate solutions of a partial differential equation become inaccurate if they arecomputed on a fixed grid that is not sufficiently fine in regions of the domain where the variables change rapidly for time dependent problems special features of a partial differential equation and their location could change in time as well thus adaptive grid methods are necessary in this paper we develop an adaptive deformation method based on the least squares finite element method lsfem a main advantage of this method as compared to the existing deformation method is its ability to generate adaptive grids on domains with moving boundary it computes the node velocity from a div curl system according to an error indicator monitor function and then moves the nodes to new locations so that the size of the new grid cells can be directly controlled in this method the connectivity of the nodes is unchanged if the grid quality is acceptable otherwise various optimization procedures can be applied after node movements to improve grid quality the grid formed becomes refined in regions where the solution error is large

practical aspects of formulation and solution of moving mesh partial differential equations moving mesh partial differential equations mmpdes are used in the mmpde moving mesh method to generate adaptive moving meshes for the numerical solution of time dependent problems how mmpdes are formulated and solved is crucial to the efficiency and robustness of the method in this paper several practical aspects of formulating and solving mmpdes are studied they include spatial balance scaling invariance effective control of mesh concentration bounds on time steps multiple sub steps and two level mesh movement numerical results are also given c 2001 academic press

distributional chaos and irregular recurrence abstract for a continuous map x x of a compact metric space we study relations between distributional chaos and the existence of a point which is quasi weakly almost periodic but not weakly almost periodic we provide an example showing that the existence of such a point does not imply the strongest version of distributional chaos dc1 using this we prove that even in the class of triangular maps of the square there are no relations to dc1 this result among others contributes to the solution of a problem formulated by a n sharkovsky in the eighties

depth size tradeoffs for neural computation the tradeoffs between the depth i e the time for parallel computation and the size i e the number of threshold gates in neural networks are studied the authors focus the study on the neural computations of symmetric boolean functions and some arithmetic functions it is shown that a significant reduction in the size is possible for symmetric functions and some arithmetic functions at the expense of a small constant increase in depth in the process several neural networks which have the minimum size among all the known constructions have been developed results on implementing symmetric functions can be used to improve results about arbitrary boolean functions in particular it is shown that any boolean function can be computed in a depth 3 neural network with o 2 sup n sup 2 threshold gates it is also proven that at least omega 2 sup n sup 3 threshold gates are required

optimal and nearly optimal algorithms for approximating polynomial zeros abstract we substantially improve the known algorithms for approximating all the complex zeros of an n th degree polynomial p x our new algorithms save both boolean and arithmetic sequential time versus the previous best algorithms of schonhage 1 pan 2 and neff and reif 3 in parallel nc implementation we dramatically decrease the number of processors versus the parallel algorithm of neff 4 which was the only nc algorithm known for this problem so far specifically under the simple normalization assumption that the variable x has been scaled so as to confine the zeros of p x to the unit disc x x 1 our algorithms which promise to be practically effective approximate all the zeros of p x within the absolute error bound 2 b by using order of n arithmetic operations and order of b n n 2 boolean bitwise operations in both cases up to within polylogarithmic factors the algorithms allow their optimal work preserving nc parallelization so that they can be implemented by using polylogarithmic time and the orders of n arithmetic processors or b n n 2 boolean processors all the cited bounds on the computational complexity are within polylogarithmic factors from the optimum in terms of n and b under both arithmetic and boolean models of computation in the boolean case under the additional realistic assumption that n o b

an infinitary extension of the graham rothschild parameter sets theorem the graham rothschild parameter sets theorem is one of the most powerful results of ramsey theory the hales jewett theorem is its most trivial instance using the algebra of s the stone cech compactification of a discrete semigroup we derive an infinitary extension of the graham rothschild parameter sets theorem even the simplest finite instance of this extension is a significant extension of the original the original theorem says that whenever k m in n and the k parameter words are colored with finitely many colors there exist a color and an m parameter word w with the property that whenever a k parameter word of length m is substituted in w the result is in the specified color the simplest finite instance referred to above is that given finite colorings of the k parameter words for each k m there is one m parameter word which works for each k some additional ramsey theoretic consequences are derived we also observe that unlike any other ramsey theoretic result of which we are aware central sets are not necessarily good enough for even the k 1 and m 2 version of the graham rothschild parameter sets theorem

geometric theory of unimodular pisot substitutions we are concerned with the tiling flow t associated to a substitution over a finite alphabet our focus is on substitutions that are unimodular pisot i e their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the perron eigenvalue 1 the motivation is provided by the still open conjecture asserting that t has pure discrete spectrum for any such we develop a number of necessary and sufficient conditions for pure discrete spectrum including injectivity of the canonical torus map the geometric realization geometric coincidence condition partial commutation of t and the dual r d 1 action measure and tiling properties of rauzy fractals and concrete algorithms some of these are original and some have already appeared in the literature as sufficient conditions only but they all emerge from a unified approach based on the new device the strand space f of the proof of the necessity hinges on determination of the discrete spectrum of t as that of the associated kronecker toral flow

ramsey theory in noncommutative semigroups by utilizing ultrafilters we give a general version of the central sets theorem 6 proposition 8 21 this enables us to derive noncommutative versions of van der waerden s theorem and several of its generalizations we also derive some standards results including the hales jewett theorem

topological multiple recurrence for polynomial configurations in nilpotent groups abstract we establish a general multiple recurrence theorem for an action of a nilpotent group by homeomorphisms of a compact space this theorem can be viewed as a nilpotent version of our recent polynomial hales jewett theorem ann math 150 1999 33 and contains nilpotent extensions of many known abelian results as special cases

the complexity of searching implicit graphs abstract the standard complexity classes of complexity theory do not allow for direct classification of most of the problems solved by heuristic search algorithms the reason is that almost always these are defined in terms of implicit graphs of state or problem reduction spaces while the standard definitions of all complexity classes are specifically tailored to explicit inputs to allow for more precise comparisons with standard complexity classes we introduce here a model for the analysis of algorithms on graphs given by vertex expansion procedures it is based on previously studied concepts of succinct representation techniques and allows us to prove pspace completeness or exptime completeness of specific natural problems on implicit graphs such as those solved by a ao and other best first search strategies

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

set polynomials and polynomial extension of the hales jewett theorem an abstract hales jewett type extension of the polynomial van der waerden theorem bl is established theorem let r d q n there exists n n such that for any r coloring of the set of subsets of v 1 n d 1 q there exist a set a v and a nonempty set 1 n such that a d 1 q and the subsets a a d 1 a d 2 a d q are all of the same color this polynomial hales jewett theorem contains refinements of many combinatorial facts as special cases the proof is achieved by introducing and developing the apparatus of set polynomials polynomials whose coefficients are finite sets and applying the methods of topological dynamics

a dual form of ramsey s theorem abstract let k where is the set of all natural numbers ramsey s theorem deals with colorings of the k element subsets of our dual form deals with colorings of the k element partitions of let k respectively be the set of all partitions of having exactly k respectively infinitely many blocks given x let x k be the set of all y k such that y is coarser than x dual ramsey theorem if k c 0 c t 1 where each c i is borel then there exists x such that x k c i for some i l dual galvin prikry theorem same as before with k replaced by we also obtain dual forms of theorems of ellentuck and mathias our results also provide an infinitary generalization of the graham rothschild parameter set theorem trans amer math soc 159 1971 257 292 and a new proof of the halpern lauchli theorem trans amer math soc 124 1966 360 367

a class of cartesian grid embedded boundary algorithms for incompressible flow with time varying complex geometries abstract we present a class of numerical algorithms for simulating viscous fluid problems of incompressible flow interacting with moving rigid structures the proposed cartesian grid embedded boundary algorithms employ a slightly different idea from the traditional direct forcing immersed boundary methods the proposed algorithms calculate and apply the force density in the extended solid domain to uphold the solid velocity and hence the boundary condition at the rigid body surface the principle of the embedded boundary algorithm allows us to solve the fluid equations on a cartesian grid with a set of external forces spread onto the grid points occupied by the rigid structure the proposed algorithms use the mac marker and cell algorithm to solve the incompressible navier stokes equations unlike projection methods the mac scheme incorporates the gradient of the force density in solving the pressure poisson equation so that the dipole force due to the jump of pressure across the solid fluid interface is directly balanced by the gradient of the force density we validate the proposed algorithms via the classical benchmark problem of flow past a cylinder our numerical experiments show that numerical solutions of the velocity field obtained by using the proposed algorithms are smooth across the solid fluid interface finally we consider the problem of a cylinder moving between two parallel plane walls numerical solutions of this problem obtained by using the proposed algorithms are compared with the classical asymptotic solutions we show that the two solutions are in good agreement

on chaotic minimal center of attraction of a lagrange stable motion for topological semi flows abstract let f r x x be a topological semi flow on a polish space x in 1977 karl sigmund conjectured that if there is a point x in x such that the motion f x has just x as its minimal center of attraction then all such x form a residual subset of x in this paper we first present a positive solution to this conjecture and then apply it to the study of chaotic dynamics occurring inside or near the minimal center of attraction of a motion f x

nonmetrizable topological dynamics and ramsey theory applying ideas from topological dynamics in compact metric spaces to the stone cech compactification of a discrete semigroup several new proofs of old results and some new results in ramsey theory are obtained in particular two ultrafilter proofs of van der waerden s theorem are given an ultrafilter approach to central sets sets which are combinatorially rich is developed this enables us to show that for any partition of the positive integers one cell is both additively and multiplicatively central also a fortuitous answer to a question of ellis is obtained

logarithmic depth circuits for algebraic functions this paper describes circuits for computation of a large class of algebraic functions on polynomials power series and integers for which it has been a long standing open problem to compute in depth less than omega log n 2 algebraic circuits assume unit cost for elemental addition and multiplication this paper describes o log n depth algebraic circuits which given as input the coefficients of n degree polynomials over an appropriate ring compute the product of n o 1 polynomials the symmetric functions as well as division and interpolation of real polynomials also described are o log n depth algebraic circuits which are given as input the first n coefficients of a power series over an appropriate ring compute the product of n o 1 power series as well as division reciprocal and reversion of real power series furthermore this paper describes boolean circuits of depth o log n log log n which given n bit binary numbers compute the product of n numbers and integ

poincare recurrence and number theory thirty years later hillel furstenberg s 1981 article in the bulletin gives an elegant introduction to the interplay between dynamics and number theory summarizing the major developments that occurred in the few years after his landmark paper 21 the field has evolved over the past thirty years with major advances on the structural analysis of dynamical systems and new results in combinatorics and number theory furstenberg s article continues to be a beautiful introduction to the subject drawing together ideas from seemingly distant fields furstenberg s article 21 gave a general correspondence between regularity properties of subsets of the integers and recurrence properties in dynamical systems now dubbed the furstenberg correspondence principle he then showed that such recurrence properties always hold proving what is now referred to as the multiple recurrence theorem combined these results gave a new proof of szemeredi s theorem 45 if s z has positive upper density then s contains arbitrarily long arithmetic progressions this proof lead to an explosion of activity in ergodic theory and topological dynamics beginning with new proofs of classic results of ramsey theory and ultimately leading to significant new combinatorial and number theoretic results the full implications of these connections have yet to be understood the approach harks back to the earliest results on recurrence in the measurable setting and in the topological setting a measure preserving system is a quadruple x b t where x denotes a set b is a algebra on x is a probability measure on x b and t x x is a measurable transformation such that t 1 a a for all a b poincare recurrence states that if x b t is a measure preserving system and a b with a 0 there exists n n such that a t na 0 a topological dynamical system is a pair x t where x is a compact metric space and t x x is a continuous map one can show that any such topological space admits a borel probability measure that preserves t in particular poincare recurrence implies recurrence in the topological setting birkhoff 13 gave a direct proof of

an arbitrary lagrangian eulerian formulation for solving moving boundary problems with large displacements and rotations an arbitrary lagrangian eulerian ale formulation based on the unstructured finite volume method is proposed for solving moving boundary problems with large displacements and rotations the numerical method is based on the side centered arrangement of the primitive variables that does not require any ad hoc modifications in order to enhance pressure coupling the continuity equation is satisfied within each element at machine precision and the summation of the continuity equations can be exactly reduced to the domain boundary which is important for the global mass conservation a special attention is given to construct an ale algorithm obeying the discrete geometric conservation law dgcl the mesh deformation algorithm is based on the indirect radial basis function rbf algorithm at each time level while avoiding remeshing in order to enhance numerical robustness for the parallel solution of resulting large scale algebraic equations in a fully coupled form a matrix factorization is introduced similar to that of the projection method for the whole system and the parallel algebraic multigrid solver boomeramg is used for the scaled discrete laplacian provided by the hypre library which we access through the petsc library the present numerical algorithm is initially validated for the decaying taylor green vortex flow the flow past an oscillating circular cylinder in a channel and the flow induced by an oscillating sphere in a cubic cavity then the numerical algorithm is applied to the numerical simulation of flow field around a pair of flapping drosophila wings in hover flight the time variation of the eulerian coherent structures in the near wake is shown along with the aerodynamic loads

multiplicative richness of additively large sets in zd abstract in their proof of the ip szemeredi theorem a far reaching extension of the classic theorem of szemeredi on arithmetic progressions furstenberg and katznelson 14 introduced an important class of additively large sets called ip r sets which underlies recurrence aspects in dynamics and is instrumental to enhanced formulations of combinatorial results the authors recently showed that additive ip r subsets of z d are multiplicatively rich with respect to every multiplication on z d without zero divisors e g multiplications induced by degree d number fields in this paper we explain the relationships between classes of multiplicative largeness with respect to different multiplications on z d we show for example that in contrast to the case for z there are infinitely many different notions of multiplicative piecewise syndeticity for subsets of z d when d 2 this is accomplished by using the associated algebra representations to prove the existence of sets which are large with respect to some multiplications while small with respect to others in the process we give necessary and sufficient conditions for a linear transformation to preserve a class of multiplicatively large sets one consequence of our results is that additive ip r sets are multiplicatively rich in infinitely many genuinely different ways we conclude by cataloging a number of sources of additive ip r sets from combinatorics and dynamics

a nilpotent roth theorem let t and s be invertible measure preserving transformations of a probability measure space x we prove that if the group generated by t and s is nilpotent then lim n ras infty frac 1 n sum n 1 n u t n x v s n x exists in l 2 norm for any u v l x we also show that for a with a 0 one has lim n ras infty frac 1 n sum n 1 n mu a cap t n a cap s n a 0 by the way of contrast we bring examples showing that if measure preserving transformations t s generate a solvable group then i the above limits do not have to exist ii the double recurrence property fails that is for some a a 0 one may have a t n a s n a 0 for all n finally we show that when t and s generate a nilpotent group of class c lim n ras infty frac 1 n sum n 1 n u t n x v s n x int ud mu int vd mu in l 2 x for all u v l x if and only if t s is ergodic on x x and the group generated by t 1 s t 2 s 2 t c s c acts ergodically on x

nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear boundary condition abstract this paper studies the nonisotropic chaotic oscillations of the initial boundary value problem of one dimensional wave equation with a mixing transport term it separately considers that the boundary condition at the right end of the wave equation is a superlinear type and linear perturbation of such type each causing the total energy of the underlying system to rise and fall due to the interaction with a mixing transport term for each type of boundary condition the occurrence of nonisotropic chaotic oscillations is rigorously proved numerical examples verify the effectiveness of theoretical prediction

a disjoint unions theorem for trees abstract we prove an infinitary disjoint unions theorem for level products of trees to implement the proof we develop a hales jewett type result for words indexed by a level product of trees

an adaptive grid method and its application to steady euler flow calculations an adaptive remeshing procedure based on a cell volume deformation method is presented starting with an initial grid this method offers direct cell volume control through the specification of the transformation jacobian grid points are moved with appropriate grid velocities so that the specified cell volume distribution can be achieved at the end of the grid movement without adding or removing grid points the grid velocities are determined by solving a scalar poisson equation this method is applied to solving the compressible euler equations computational test cases of transonic flow over an airfoil are presented and demonstrate the desired control of grid sizes across shock waves

the complexity of planarity testing we clarify the computational complexity of planarity testing by showing that planarity testing is hard for l and lies in sl this nearly settles the question since it is widely conjectured that l sl the upper bound of sl matches the lower bound of l in the context of non uniform circuit complexity since l poly is equal to sl poly similarly we show that a planar embedding when one exists can be found in flsl previously these problems were known to reside in the complexity class ac1 via the o log n time crcw pram algorithm of ramachandran and reif although planarity checking for degree three graphs had been shown to be in sl chicago j theoret comput sci 1995 j acm 31 2 1984 401

properties of words and recognizability of fixed points of a substitution resume soit u un mot infini non periodique point fixe d une substitution primitive nous montrons que pour n assez grand u ne contient aucune puissance n eme de mot nous donnons ensuite une condition necessaire et suffisante pour que soit reconnaissable au sens de host et queffelec et nous montrons qu ill suffit de bilateraliser cette definition pour obtenir une propriete verifiee cette fois par toutes les substitutions primitives a point fixe non periodique

central sets and substitutive dynamical systems abstract in this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of pisot type central sets first introduced by furstenberg using notions from topological dynamics constitute a special class of subsets of n possessing strong combinatorial properties each central set contains arbitrarily long arithmetic progressions and solutions to all partition regular systems of homogeneous linear equations we give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the stone cech compactification n this provides a new arithmetical approach to an outstanding conjecture in tiling theory the pisot substitution conjecture the results in this paper rely on interactions between different areas of mathematics some of which had not previously been directly linked they include the general theory of combinatorics on words abstract numeration systems tilings topological dynamics and the algebraic topological properties of stone cech compactification of n

invariant linear functionals on l r abstract we consider a continuous version of the classical notion of banach limits i e normalized positive linear functionals on l r invariant under translations f x f x s of l r for every s 0 we give one of its characterizations in terms of the invariance under the operation of a certain linear transformation on l r we also deal with invariant linear functionals under dilations f x f r x r 1 and give a similar characterization via the hardy operator applications to summability methods are presented in the last section

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

one way functions and circuit complexity abstract a finite function f is a mapping of 0 1 n into 0 1 m where is a symbol to be thought of as undefined a family of finite functions is said to be one way in a circuit complexity sense if it can be computed with polynomial size circuits but every family of inverses of these functions cannot in this paper we show that provided functions that are not one to one are allowed one way functions exist if and only if the satisfiability problem sat does not have polynomial size circuits a family of functions fi x can be checked if some family of polynomial size circuits with inputs x and y can determine if fi x y a family of functions fi x can be evaluated if some family of polynomial size circuits with input x can compute fi x can all families of total functions that can be checked also be evaluated we show that this is true if and only if the nonuniform versions of the complexity classes p and up co up are equal a family of functions fi is one way for constant depth circuits if fi can be computed with unbounded famin circuits of polynomial size and constant depth but every family of inverses fi 1 cannot we give two provably one way functions in fact permutaions for constant depth circuits the second example has the stronger property that no bit of its inverse can be computed in polynomial size and constant depth

flows with suspended and floating particles abstract the evolution of the configuration of a set of particles dispersed in a flowing liquid is crucial in many applications such as sedimentation slurry transport rheology and structured arrays of micro and nano particles direct simulation based on what is called fictitious domain method coupled with finite element method has been used to study particulate flows and sedimentation process here we extend the previously proposed formulations to naturally include buoyancy force and the capillary driven attraction or repulsion of particles located at fluid interfaces the set of differential equations is discretized using a fully implicit fully coupled fictitious domain finite element approach avoiding numerical instabilities that may arise from explicit integration the proposed formulation and implementation are validated by comparing the predictions of simple 2d flows to available numerical or analytical solutions the method is then used to analyze the flotation of 2d particles and capillary driven aggregation at fluid interfaces

higher composition laws i a new view on gauss composition and quadratic generalizations two centuries ago in his celebrated work disquisitiones arithmeticae of 1801 gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow even today two centuries later this law of composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders it is hence only natural to ask whether higher analogues of this composition law exist that could shed light on the structure of other algebraic number rings and fields this article forms the first of a series of four articles in which our aim is precisely to develop such higher composition laws in fact we show that gauss s law of composition is only one of at least fourteen composition laws of its kind which yield information on number rings and their class groups in this paper we begin by deriving a general law of composition on 2 2 2 cubes of integers from which we are able to obtain gauss s composition law on binary quadratic forms as a simple special case in a manner reminiscent of the group law on plane elliptic curves we also obtain from this composition law on 2 2 2 cubes four further new laws of composition these laws of composition are defined on 1 binary cubic forms 2 pairs of binary quadratic forms 3 pairs of quaternary alternating 2 forms and 4 senary six variable alternating 3 forms more precisely gauss s theorem states that the set of sl2 z equivalence classes of primitive binary quadratic forms of a given discriminant d has an inherent group structure the five other spaces of forms mentioned above including the space of 2 2 2 cubes also possess natural actions by special linear groups over z and certain products thereof we prove that just like gauss s space of binary quadratic forms each of these group actions has the following remarkable properties first each of these six spaces possesses only a single polynomial invariant for the corresponding group action which we call the discriminant this discriminant invariant is found to take only values that

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

on discrete projection methods for the incompressible navier stokes equations an algorithmical approach abstract we derive a general class of iteration schemes for the incompressible navier stokes equations which contains fully coupled solution techniques as well as operator splitting projection methods we combine the advantages of both namely accuracy stability and efficiency and obtain a special form of discrete projection schemes in combination with a nonlinear iteration of quasi newton type one may use these schemes analogously to the well known pure projection schemes e g of chorin and van kan or apply them as preconditioners in a defect correction approach to obtain the fully coupled galerkin solution the corresponding complexity analysis shows that in combination with certain nonconforming finite element discretizations a huge gain in efficiency may be obtained particularly in the highly nonstationary case our theoretical results are confirmed by comparative numerical tests for both types of schemes it turns out that the appropriate time steps for the pure projection approach are only moderately smaller than those for the fully implicitly coupled schemes but that the work to obtain comparative results with the discrete projection methods as solvers is much lower an interesting observation is that in the case of higher reynolds numbers no significant pressure boundary layers occur even for the pure projection schemes these considerations and first numerical tests in 3d give hope to obtain a powerful cfd tool

an efficient multigrid fem method for the simulation of solid liquid two phase flows an efficient multigrid fem method for the detailed simulation of solid liquid two phase flows with large number of moving particles is presented an explicit fictitious boundary method based on a fem background grid which covers the whole computational domain and can be chosen independently from the particles of arbitrary shape size and number is used to deal with the interactions between the fluid and the particles since the presented method treats the fluid part the calculation of forces and the movement of particles in a subsequent manner it is potentially powerful to efficiently simulate real particulate flows with huge number of particles the presented method is first validated using a series of simple test cases and then as an illustration simulations of three big disks plunging into 2000 small particles and of sedimentation of 10 000 particles in a cavity are presented

an interface fitted subspace projection method for finite element simulations of particulate flows abstract a novel finite element method for the direct numerical simulation of particles in a newtonian carrier fluid is presented the proposed method is based on a fictitious or one domain formulation and a subspace projection method to account for the rigid body motion of the particles underlying equations are posed in an ale arbitrary lagrangian eulerian formulation allowing for moving computational meshes the mesh is adapted to the particles by a novel mesh smoothing approach guaranteeing both mesh optimality and a sharp representation of the particles boundaries we show that by using quadratic taylor hood finite elements for the discretization of the navier stokes equations and isoparametric elements to represent the geometry second order convergence with respect to the energy norm can be achieved as opposed to a sharp h 1 2 result which holds if no mesh adaptation is performed we present numerical examples in 2d confirming the theoretical convergence results furthermore we validate the approach by simulating the sedimentation process of a single particle and show the potential of the method by simulating a lid driven cavity flow with 100 particles

on interpolation and automatization for frege systems the interpolation method has been one of the main tools for proving lower bounds for propositional proof systems loosely speaking if one can prove that a particular proof system has the feasible interpolation property then a generic reduction can usually be applied to prove lower bounds for the proof system sometimes assuming a usually modest complexity theoretic assumption in this paper we show that this method cannot be used to obtain lower bounds for frege systems or even for tc0 frege systems more specifically we show that unless factoring of blum integers is feasible neither frege nor tc0 frege has the feasible interpolation property in order to carry out our argument we show how to carry out proofs of many elementary axioms theorems of arithmetic in polynomial sized tc0 frege as a corollary we obtain that tc0 frege as well as any proof system that polynomially simulates it is not automatizable under the assumption that factoring of blum integers is hard we also show under the same hardness assumption that the k provability problem for frege systems is hard

a fictitious domain parallel numerical method for rigid particulate flows in this paper we develop a fictitious domain parallel numerical method for the direct numerical simulation of the flow of rigid particles in an incompressible viscous newtonian fluid a simultaneous directions implicit algorithm is employed which gives the model a high level of parallelization the projection of the fluid velocity onto rigid motion on the particles is based on a fast computational technique which relies on the conservation of linear and angular momenta numerical results are presented which confirm the ability of the proposed method to simulate the sedimentation of one and many particles the parallel efficiency of the algorithm is also assessed

multiplication division and shift instructions in parallel random access machines abstract we prove that polynomial time on a parallel random access machine pram with unit cost multiplication and division or on a pram with unit cost shifts is equivalent to polynomial space on a turing machine pspace this extends the result that polynomial time on a basic pram is equivalent to pspace to hold when the pram is allowed multiplication or division or unrestricted shifts it also extends to the pram the results that polynomial time on a random access machine ram with multiplication is equivalent to pspace and that polynomial time on a ram with shifts that is a vector machine is equivalent to pspace we obtain simulations of uniform circuits by rams with enhanced instruction sets and use the enhanced rams to simulate prams with enhanced instruction sets

existence of chaotic oscillations in second order linear hyperbolic pdes with implicit boundary conditions abstract this paper establishes rigorously mathematical theorems that guarantee the existence of chaotic oscillations in the systems of second order linear hyperbolic pdes it separately considers the systems with nonlinear explicit boundary conditions ebcs and nonlinear implicit boundary conditions ibcs as well as those with such ibcs subjected to small perturbations where ibcs include ebcs as special cases but the latter cannot in general be expressed by the former numerical examples are demonstrated to illustrate the effectiveness of theoretical results

sph simulations of swimming linked bodies in this paper we describe how the swimming of linked rigid bodies can be simulated using smoothed particle hydrodynamics sph the fluid is assumed to be viscous and weakly compressible though with a speed of sound which ensures the mach number is 0 1 and the density fluctuation relative to the average density is typically 0 01 the motion is assumed to be two dimensional the boundaries of the rigid bodies are replaced by boundary particles and the forces between these particles and the fluid particles determine the forces and torques on the rigid bodies the links between the bodies are described by constraint equations which are taken into account by lagrange multipliers we integrate the equations by a second order method which conserves linear momentum exactly and in the absence of viscosity is reversible we test our method by simulating the motion of a cylinder moving in a viscous fluid both under forced oscillation and when tethered to a spring we then apply our method to three problems involving three linked bodies the first of these consists of three linked diamonds in a periodic domain and shows the convergence of the algorithm the second is the system considered by kanso et al e kanso j e marsden c w rowley j b melli huber locomotion of articulated bodies in a perfect fluid j nonlinear sci 15 2005 255 289 referred to in the text as k05 and our results show similar behaviour for the same gait the third example clarifies the conservation of angular momentum by simulating the motion of three linked ellipses moving within and through the surface of an initially circular patch of fluid our method can be easily extended to linked bodies swimming near surfaces or to swimming near fixed boundaries of arbitrary shape or to swimming in fluids which are stratified some of these systems have biological significance and others are applicable to the study of undersea vessels which move because of shape changes

chaotic dynamics of minimal center of attraction of discrete amenable group actions abstract let g be a discrete infinite amenable group which acts from the left on a compact metric space x in this paper we study the chaotic dynamics exhibited inside and near a minimal center of attraction of g x relative to any folner net in g

mathematical and numerical analysis of a robust and efficient grid deformation method in the finite element context among a variety of grid deformation methods the method proposed by bochev liao and de la pena methods partial differential equations 12 1996 pp 489 506 cai et al comput math appl 48 2004 pp 1077 1086 and liao and anderson appl anal 44 1992 pp 285 298 is one of the most favorable because it prevents mesh tangling and offers precise control over the element volumes its numerical realization requires only solving a poisson problem and a system of fully decoupled initial value problems many other deformation methods in contrast involve the solution of complicated nonlinear partial differential equations pdes in this article we introduce a generalization of liao s method which allows for generating a desired mesh size distribution for quite arbitrary grids without giving rise to mesh tangling we elaborate on its numerical realization and prove the convergence of our method our results are confirmed by numerical experiments

computing with noisy information this paper studies the depth of noisy decision trees in which each node gives the wrong answer with some constant probability in the noisy boolean decision tree model tight bounds are given on the number of queries to input variables required to compute threshold functions the parity function and symmetric functions in the noisy comparison tree model tight bounds are given on the number of noisy comparisons for searching sorting selection and merging the paper also studies parallel selection and sorting with noisy comparisons giving tight bounds for several problems

on threshold circuits and polynomial computation a threshold circuit consists of an acyclic digraph of unbounded fanin where each node computes a threshold function or its negation this paper investigates the computational power of threshold circuits a surprising relationship is uncovered between threshold circuits and another class of unbounded fanin circuits which are denoted finite field z p n circuits where each node computes either multiple sums or products of integers modulo a prime p n in particular it is proved that all functions computed by threshold circuits of size s n x2265 n and depth d n can also be computed by z p n circuits of size o s n log s n np n log p n and depth o d n furthermore it is shown that all functions computed by z p n circuits of size s n and depth d n can be computed by threshold circuits of size o 1 x2208 2 s n log p n 1 x2208 and depth o 1 x2208 5 d n these are the main results of this paper there are many useful and quite surprising consequences of this result for example an integer reciprocal can be computed in size n o 1 m and depth o 1 more generally any analytic function with a convergent rational polynomial power series such as sine cosine exponentiation square root and logarithm can be computed within accuracy 2 nc for any constant c by threshold circuits of polynomial size and constant depth in addition integer and polynomial division fff polynomial interpolation chinese remaindering all the elementary symmetric functions banded matrix inverse and triangular toeplitz matrix inverse can be exactly computed by threshold circuits of polynomial size and constant depth all these results and simulations hold for polytime uniform circuits this paper also gives a corresponding simulation oflogspace uniform z p n circuits by logspace uniform threshold circuits requiring an additional multiplying factor of o log log log p n depth finally purely algebraic methods forlowerbounds for zp n circuits are developed using degree arguments a depth hierarchy theorem for z p n circuits is proved for any s n x2265 n d n o s n c for some constant c p n where 6 s n d n d n p n c 2 n there exist explicitly constructible functions computable by z p n circuits of size s n and depth d n but provably not computable by z p n circuits of size s n c and depth o d n for any constant c x2265 1

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use