periodic schwarz christoffel mappings with multiple boundaries per period we present an extension to the theory of schwarzchristoffel sc mappings by permitting the target domain to be a single period window of a periodic configuration having multiple polygonal straigh

theories and ultra efficient computation of joint spectral radius for estimating first passage time distribution of markov set chain this technical note is concerned with the tail distribution of the first passage time of markov set chains msc an original two part idea a more progressive relation and a sortedness test is conceived to characterize such chains the theoretical construction based on this idea further results in an algorithm that can compute the tightest exponent bound of the tail distribution for high dimensional problem instances with surprising ease to understand the computational implication of this algorithm note that the problem is equivalent to computing the joint spectral radius jsr of a special independent column polytope one that defines markov set chains of nonnegative matrices in this context the reported algorithm can compute the exact jsr value for cases of 100 t 100 matrices in less than a second in matlab problems of this size is far beyond the scope of known jsr techniques it is worth noting that the fields of msc and jsr have not had significant overlap as one may expect despite their conceptual akiness meanwhile the present technical note is a contribution that belongs to both fields

nonstationary multisplittings with general weighting matrices for mildly nonlinear systems abstract in this paper we present a new nonstationary multisplitting iterative method for solving the mildly nonlinear equations ax f x where a rn n is an n by n sparse real symmetric positive definite matrix and f rn rn is a general nonlinear mapping and establish the convergence theories with general weighting matrices by more than one inner iteration meanwhile we also give the lower bound of the inner iteration number in this nonstationary multisplitting iterative method

convergence theorems for block splitting iterative methods for linear systems in this paper we will present the block splitting iterative methods with general weighting matrices for solving linear systems of algebraic equations ax b when the coefficient matrix a is symmetric positive definite of block form and establish the convergence theories with respect to the general weighting matrices but special splittings finally a numerical example shows the advantage of this method

ends of maps ii versions of the finiteness obstruction and simple homotopy theory within e overx are developed this provides a setting for obstructions to the map analogs of the end ands cobordism theorems for manifolds these are applied to study equivariant mapping cylinder neighborhoods in topological group actions triangulations of locally triangulable spaces and block bundle structures on approximate fibrations

on the perron eigenvalue of a block cocyclic pair of nonnegative matrices abstract upper and lower bounds for the ratio between the spectral radius of a product of nonnegative matrices and the product of their spectral radii have been given by johnson and bru we study the case of equality of the upper bound for a block cocyclic pair of nonnegative matrices

shape calculus for vortex patch equilibria and its application to lattice configurations abstract a new shape derivative formula for singular contour integrals with logarithmic kernels is proposed this formula yields a simple numerical scheme for computing vortex patch equilibria owing to its simplicity any steady configuration of point vortices can be extended to that of vortex patches as a test problem a doubly periodic array of vortex patches is considered to show the efficiency of the new formula several non trivial families of stationary vortex patch lattices are also presented

generalizations of the nonstationary multisplitting iterative method for symmetric positive definite linear systems in this paper we generalize the nonstationary parallel multisplitting iterative method for solving the symmetric positive definite linear systems with several choices of variable weighting matrices the convergence properties of these generalized methods can be improved finally the numerical comparison of several nonstationary parallel multisplitting methods are shown

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

distributions of singular values for some random matrices the singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data such as complex space time images obtained in both physical and biological systems in this paper we examine the distribution of singular values of low rank matrices corrupted by additive noise past studies have been limited to uniform uncorrelated noise using diagrammatic and saddle point integration techniques we extend these results to heterogeneous and correlated noise sources we also provide perturbative estimates of error bars on the reconstructed low rank matrix obtained by truncating a singular value decomposition 1999 the american physical society

finite rank bratteli diagrams structure of invariant measures we consider bratteli diagrams of finite rank not necessarily simple and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces it is shown that every ergodic invariant measure finite or regular infinite is obtained by an extension from a simple subdiagram we further investigate quantitative properties of these measures which are mainly determined by the asymptotic behavior of products of incidence matrices a number of sufficient conditions for unique ergodicity are obtained one of these is a condition of exact finite rank which parallels a similar notion in measurable dynamics several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them we then prove that the vershik map on the path space of an exact finite rank diagram cannot be strongly mixing independent of the ordering on the other hand for the so called consecutive ordering the vershik map is not strongly mixing on all finite rank diagrams msc 37b05 37a25 37a20

parallel chaotic extrapolated jacobi like methods parallel chaotic schemes based on the extrapolated jacobi method and a second degree stationary method are studied sufficient conditions for the convergence of the above methods for the synchronous and asynchronous models are given these sufficient conditions are related to properties of the block jacobi matrix the schemes are illustrated by computational results on a shared memory multiprocessor vector computer

on the convergence of a new splitting iterative method for non hermitian positive definite linear systems in this paper we present a new splitting method for solving a linear systems with non hermitian positive definite coefficient matrix this splitting overcomes the computation complexity of hss the spectral radius and some norm properties of the iteration matrix are discussed with the results obtained we study the reasonable choices of the parameter and introduce a preconditioner moreover an accelerated algorithm is proposed finally the numerical examples show the new method is much more efficient than the hss or the nss and the pss iteration method

representations of higher rank graph c algebras associated to λ semibranching function systems abstract in this paper we discuss a method of constructing separable representations of the c algebras associated to strongly connected row finite k graphs we begin by giving an alternative characterization of the semibranching function systems introduced in an earlier paper with an eye towards constructing such representations that are faithful our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions we present a variety of new examples relying on this characterization we then use some of these methods and a direct limit procedure to identify a faithful separable representation for any row finite source free k graph

the n vortex problem on a rotating sphere ii heterogeneous platonic solid equilibria we describe a new method of constructing point vortex equilibria on a sphere made up of n vortices with different strengths such equilibria called heterogeneous equilibria are obtained for the five platonic solid configurations hence for graphic 1 the method is based on calculating a basis set for the nullspace of a matrix obtained by enforcing the necessary and sufficient condition that the mutual distances between each pair of vortices remain constant by symmetries inherent in the platonic solid configurations this matrix is reduced for each case and we call the dimension of the nullspace the degree of heterogeneity of the structure for the tetrahedron n 4 and octahedron n 6 the degree of heterogeneity is 4 and 6 respectively hence we are free to choose each of the vortex strengths independently for the cube n 8 the degree of heterogeneity is 5 for the icosahedron n 12 it is 7 while for the dodecahedron n 20 it is 4 thus the entire set of equilibria based on the platonic solid configurations is obtained including substructures associated with each configuration constructed by taking different linear combinations of the basis elements 1 embed inline graphic 1 gif

fixed equilibria of point vortices in symmetric multiply connected domains abstract the paper provides symmetric fixed configurations of point vortices in multiply connected domains in the unit circle with many circular obstacles when the circular domain is invariant with respect to rotation around the origin by a degree of 2 m a regular m polygonal ring configuration of identical point vortices becomes a fixed equilibrium on the other hand when we assume a special symmetry called the folding symmetry on the circular domain we find a fixed equilibrium in which m point vortices with the positive unit strength and m point vortices with the negative unit strength are arranged alternately at the vertices of a 2 m polygon we also investigate the stability of these fixed equilibria and their bifurcation for a special circular domain with the rotational symmetry as well as the folding symmetry furthermore we discuss fixed equilibria in non circular multiply connected domains with the same symmetries we give sufficient conditions for the conformal mappings by which fixed equilibria in the circular domains are mapped to those in the general multiply connected domains some examples of such conformal mappings are also provided

analytical formulae for the kirchhoff routh path function in multiply connected domains explicit formulae for the kirchhoff routh path functions or hamiltonians governing the motion of n point vortices in multiply connected domains are derived when all circulations around the holes in the domain are zero the method uses the schottky klein prime function to find representations of the hydrodynamic green s function in multiply connected circular domains the green s function is then used to construct the associated kirchhoff routh path function the path function in more general multiply connected domains then follows from a transformation property of the path function under conformal mapping of the canonical circular domains illustrative examples are presented for the case of single vortex motion in multiply connected domains

analyzing glauber dynamics by comparison of markov chains a popular technique for studying random properties of a combinatorial set is to design a markov chain monte carlo algorithm for many problems there are natural markov chains connecting the set of allowable configurations which are based on local moves or glauber dynamics typically these single site update algorithms are difficult to analyze so often the markov chain is modified to update several sites simultaneously recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems in this work we use the comparison technique of diaconis and saloff coste to show that several of the natural single point update algorithms are efficient the strategy is to relate the mixing rate of these algorithms to the corresponding nonlocal algorithms which have already been analyzed this allows us to give polynomial time bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of convex polygons we also survey several other comparison techniques along with specific applications which have been used in the context of estimating mixing rates of markov chains

construction of point vortex equilibria via brownian ratchets a theory capable of producing equilibrium configurations of point vortices in the plane along with a numerical scheme to compute them is described the theory is formulated as a problem in linear algebra where one must find solutions to the matrix equation agz 0 where a is the 1 2 n nk1 n non normal configuration matrix obtained by requiring that all intervortical distances remain fixed and g2r n are the n vortex strengths for existence of an equilibrium a must have a non trivial nullspace we consider the singular values of a when this has one or more zero singular values the nullspace of a is non empty and an equilibrium exists for some choice of g new equilibrium configurations are found numerically by randomly depositing n points in the plane which generically gives rise to a configuration matrix a with empty nullspace using the sum of squares of the k smallest singular values of a as a ratchet we thermally fluctuate the configuration allowing each point to execute a random walk in the plane retaining only those configurations which reduce this quantity at the next step the configuration is thus driven to one with nullspace a zko0 these converged states are not necessarily nearby their initial configurations typically they are asymmetric and often we can drive the same initial state to several different equilibria a reverse ratchet method is also described which can produce initial conditions that would evolve to a specified equilibrium state once a converged final state is achieved the full singular value decomposition of a is used to calculate an optimal basis set for the nullspace of a and thus all allowable g the distribution of the singular values gives important information on the size of each equilibrium state as measured by frobenius norm their distance from each other spacing and density and how far a randomly chosen system of n points in the plane is from the nearest equilibrium configuration with a specified rank as well as its shannon entropy

on block cocyclic pairs of nonnegative matrices abstract in this paper we introduce the concept of an n 1 n k block cocyclic pair of nonnegative matrices which is a generalization of the concept of a cocyclic pair given by johnson and bru linear algebra appl 141 1990 227 240 as well as of the concept of a block cocyclic pair given by bru et al linear algebra appl 236 1996 231 143 then we generalize the results of johnson and bru concerning the cocyclic pair and also the results of bru et al concerning the block cocyclic pair

robust control of markov decision processes with uncertain transition matrices optimal solutions to markov decision problems may be very sensitive with respect to the state transition probabilities in many practical problems the estimation of these probabilities is far from accurate hence estimation errors are limiting factors in applying markov decision processes to real world problems we consider a robust control problem for a finite state finite action markov decision process where uncertainty on the transition matrices is described in terms of possibly nonconvex sets we show that perfect duality holds for this problem and that as a consequence it can be solved with a variant of the classical dynamic programming algorithm the robust dynamic programming algorithm we show that a particular choice of the uncertainty sets involving likelihood regions or entropy bounds leads to both a statistically accurate representation of uncertainty and a complexity of the robust recursion that is almost the same as that of the classical recursion hence robustness can be added at practically no extra computing cost we derive similar results for other uncertainty sets including one with a finite number of possible values for the transition matrices we describe in a practical path planning example the benefits of using a robust strategy instead of the classical optimal strategy even if the uncertainty level is only crudely guessed the robust strategy yields a much better worst case expected travel time

the homology core of matchbox manifolds and invariant measures we consider the topology and dynamics associated with a wide class of matchbox manifolds including spaces of aperiodic tilings and suspensions of higher rank potentially nonabelian group actions on zero dimensional spaces for such a space we introduce a topological invariant the homology core built using an expansion of it as an inverse sequence of simplicial complexes the invariant takes the form of a monoid equipped with a representation which in many cases can be used to obtain a finer classification than is possible with the previously developed invariants when the space is obtained by suspending a topologically transitive action of the fundamental group gamma of a closed orientable manifold on a zero dimensional compact space z this invariant corresponds to the space of finite borel measures on z which are invariant under the action of gamma this leads to connections between the rank of the core and the number of invariant ergodic borel probability measures for such actions we illustrate with several examples how these invariants can be calculated and used for topological classification and how it leads to an understanding of the invariant measures

analytical solutions for an escape problem in a disc with an arbitrary distribution of exit holes along its boundary we analytically construct solutions for the mean first passage time and splitting probabilities for the escape problem of a particle moving with continuous brownian motion in a confining planar disc with an arbitrary distribution i e of any number size and spacing of exit holes absorbing sections along its boundary the governing equations for these quantities are poisson s equation with a non zero constant forcing term and laplace s equation respectively and both are subject to a mixture of homogeneous neumann and dirichlet boundary conditions our solutions are expressed as explicit closed formulae written in terms of a parameterising variable via a conformal map using special transcendental functions that are defined in terms of an associated schottky group they are derived by exploiting recent results for a related problem of fluid mechanics that describes a unidirectional flow over no slip no shear surfaces as well as results from potential theory all of which were themselves derived using the same theory of schottky groups they are exact up to the determination of a finite set of mapping parameters which is performed numerically their evaluation also requires the numerical inversion of the parameterising conformal map computations for a series of illustrative examples are also presented

on asynchronous iterations asynchronous iterations arise naturally on parallel computers if one wants to minimize idle times this paper reviews certain models of asynchronous iterations using a common theoretical framework the corresponding convergence theory and various domains of applications are presented these include nonsingular linear systems nonlinear systems and initial value problems

a new splitting and preconditioner for iteratively solving non hermitian positive definite systems a new iteration method for solving a linear system with coefficient matrix being non hermitian positive definite is presented in this note we study the spectral radius and contraction properties of the iteration matrix and then analyze the best possible choice of the parameter with the results obtained we show that the new method is convergent for a non hermitian positive definite linear system and propose a preconditioner to improve the condition number of the system the numerical examples show that the new method is much more efficient than the hss or pss iteration method

calculating the lift on a finite stack of cylindrical aerofoils the classic exact solution due to lagally lagally m 1929 die reibungslose stromung im aussengebiet zweier kreise z angew math mech 9 299 305 for streaming flow past two cylindrical aerofoils or obstacles is generalized to the case of an arbitrary finite number of cylindrical aerofoils given the geometry of the aerofoils the speed and direction of the oncoming uniform flow and the individual round aerofoil circulations the complex potential associated with the flow is found in analytical form in a parametric pre image region that can be conformally mapped to the fluid region a complete determination of the flow then follows from knowledge of the conformal mapping between the two regions in the special case where the aerofoils are all circular the conformal mapping from the parametric pre image region to the fluid domain is a mobius mapping the solution for the complex potential in such a case can then be used in combination with the blasius theorem to compute the distribution of hydrodynamic forces on the multi aerofoil configuration

parallel multisplitting methods with optimal weighting matrices for linear systems in this paper the parallel multisplitting iterative methods with optimal weighting matrices are presented to solve a linear system of equations in which the coefficient matrix is a symmetric positive definite matrix the zero pattern in weighting matrices is determined by preset set the non zero entries of weighting matrices in overlap multisplitting methods are determined optimally by finding the optimal point in hyperplane or convex combination of m points generated by parallel multisplitting iterations several schemes of finding the optimal weighting matrices are given especially the nonnegative assumption in the weighting matrices is eliminated the convergence properties are discussed for these parallel multisplitting methods finally our numerical examples show that these parallel multisplitting methods with the optimal weighting matrices are effective

models of parallel chaotic iteration methods abstract we consider two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations ax b in the first model each processor can carry out an arbitrary number of local iterations before the next global approximation to the solution is formed in the second model any processor can update the global approximation which resides in the central processor at any time this model is a generalization of a sequential iterative scheme due to ostrowski called the free steering group jacobi iterative scheme and a chaotic relaxation point iterative scheme due to chazan and miranker we show that when a is a monotone matrix and all the splittings are weak regular both models lead to convergent schemes

existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods given a nonsingular matrix a and a matrix t of the same order under certain very mild conditions there is a unique splitting a b c such that t b 1 c moreover all properties of the splitting are derived directly from the iteration matrix t these results do not hold when the matrix a is singular in this case given a matrix t and a splitting a b c such that t b 1 c there are infinitely many other splittings corresponding to the same matrices a and t and different splittings can have different properties for instance when t is nonnegative some of these splittings can be regular splittings while others can be only weak splittings analogous results hold in the symmetric positive semidefinite case given a singular matrix a not for all iteration matrices t there is a splitting corresponding to them necessary and sufficient conditions for the existence of such splittings are examined as an illustration of the theory developed the convergence of certain alternating iterations is analyzed different cases where the matrix is monotone singular and positive semi definite are studied

a note on comparison theorems for splittings and multisplittings of hermitian positive definite matrices abstract we discuss iterative methods for the solution of the linear system ax b which are based on a single splitting or a multisplitting of a in order to compare different methods it is common to compare the spectral radius of the iterative matrix for m matrices a and weak regular splittings there exist well known comparison theorems here we give a comparison theorem for splittings of hermitian positive definite matrices furthermore we establish a comparison theorem for multisplittings of a hermitian positive definite matrix

equation of motion for point vortices in multiply connected circular domains the paper gives the equation of motion for n point vortices in a bounded planar multiply connected domain inside the unit circle that contains many circular obstacles called the circular domain the velocity field induced by the point vortices is described in terms of the schottky klein prime function associated with the circular domain the explicit representation of the equation enables us not only to solve the euler equations through the point vortex approximation numerically but also to investigate the interactions between localized vortex structures in the circular domain as an application of the equation we consider the motion of two point vortices with unit strength and of opposite signs when the multiply connected domain is symmetric with respect to the real axis the motion of the two point vortices is reduced to that of a single point vortex in a multiply connected semicircle which we investigate in detail

rigidity and crystallographic groups i this is the first in a series of articles about rigidity theorems for crystallographic groups the main theorem is 1 1 it is an h cobordism rigidity statement it implies see 1 6 that any crystallographic manifold with odd order holonomy which is h cobordant and simply homotopically equivalent to the standard one is actually homeomorphic to it in the second part of this work 9 we will exploit this result to prove that one can remove the word h cobordant from the previous sentence the main theorem concerns an involution on the group wh p mr this is the group of g h cobordisms of the flat torus mr of a crystallographic group f with holonomy group g definitions of these terms appear in 1 4 we prove a vanishing result of the tate cohomology

numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions this paper presents a boundary integral equation method with the adjoint generalized neumann kernel for computing conformal mapping of unbounded multiply connected regions and its inverse onto several classes of canonical regions for each canonical region two integral equations are solved before one can approximate the boundary values of the mapping function cauchy s type integrals are used for computing the mapping function and its inverse for interior points this method also works for regions with piecewise smooth boundaries three examples are given to illustrate the effectiveness of the proposed method

word representation of streamline topologies for structurally stable vortex flows in multiply connected domains let us consider incompressible and inviscid flows in two dimensional domains with multiple obstacles the instantaneous velocity field becomes a hamiltonian vector field defined from the stream function and it is topologically characterized by the streamline pattern that corresponds to the contour plot of the stream function the present paper provides us with a procedure to construct structurally stable streamline patterns generated by finitely many point vortices in the presence of the uniform flow starting from some basic structurally stable streamline patterns in a disc of low genus i e a disc with a small number of holes we repeat some fundamental operations that append a streamline pattern by increasing one genus to them owing to the inductive procedure one can assign a sequence of operations as a representing word to each structurally stable streamline pattern we also give the canonical expression for the word representation which allows us to make a catalogue of all possible structurally stable streamline patterns in a combinatorial manner as an example we show all streamline patterns in the discs of genus 1 and 2

the schwarz christoffel mapping to bounded multiply connected polygonal domains a formula for the generalized schwarz christoffel mapping from a bounded multiply connected circular domain to a bounded multiply connected polygonal domain is derived the theory of classical schottky groups is employed the formula for the derivative of the mapping function contains a product of powers of schottky klein prime functions associated with a schottky group relevant to the circular pre image domain the formula generalizes in a natural way the known mapping formulae for simply and doubly connected polygonal domains

convergence of non stationary parallel multisplitting methods for hermitian positive definite matrices non stationary multisplitting algorithms for the solution of linear systems are studied convergence of these algorithms is analyzed when the coefficient matrix of the linear system is hermitian positive definite asynchronous versions of these algorithms are considered and their convergence investigated

spectral gradient flow and equilibrium configurations of point vortices we formulate the problem of finding equilibrium configurations of n point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew symmetric matrix m whose nullspa

an algorithm for finding extremal polytope norms of matrix families in this paper the problem of the computation of the joint spectral radius of a finite set of matrices is considered we present an algorithm which under some suitable assumptions is able to check if a certain product in the multiplicative semigroup is spectrum maximizing the algorithm proceeds by attempting to construct a suitable extremal norm for the family namely a complex polytope norm as examples for testing our technique we first consider the set of two 2 dimensional matrices recently analyzed by blondel nesterov and theys to disprove the finiteness conjecture and then a set of 3 dimensional matrices arising in the zero stability analysis of the 4 step bdf formula for ordinary differential equations

comparison of partitioning techniques for two level iterative solvers on large sparse markov chains experimental results for large sparse markov chains especially the ill conditioned nearly completely decomposable ncd ones are few we believe there is need for further research in this area specifically to aid in the understanding of the effects of the degree of coupling of ncd markov chains and their nonzero structure on the convergence characteristics and space requirements of iterative solvers the work of several researchers has raised the following questions that led to research in a related direction how must one go about partitioning the global coefficient matrix into blocks when the system is ncd and a two level iterative solver such as block sor is to be employed are block partitionings dictated by the ncd form of the stochastic one step transition probability matrix necessarily superior to others is it worth investigating alternative partitionings better yet for a fixed labeling and partitioning of the states how does the performance of block sor or even that of point sor compare to the performance of the iterative aggregation disaggregation iad algorithm finally is there any merit in using two level iterative solvers when preconditioned krylov subspace methods are available we seek answers to these questions on a test suite of 13 markov chains arising in 7 applications

periodic solutions of the n vortex hamiltonian system in planar domains abstract we investigate the existence of collision free nonconstant periodic solutions of the n vortex problem in domains c these are solutions z t z 1 t z n t n of the first order hamiltonian system z k t i z k h z t k 1 n where the hamiltonian h has the form h z 1 z n 1 2 j k 1 j k n log 1 z j z k f z the function f n r depends on the regular part of the hydrodynamic green s function and is unbounded from above f z if z k for some k the hamiltonian is unbounded from above and below it is singular not integrable energy surfaces are not compact and not known to be of contact type using singular perturbation techniques and conley index theory we prove the existence of a family of periodic solutions z r t 0 r r 0 with arbitrarily small minimal period t r 0 as r 0 the solutions are close to the singular set of h our result applies in particular to generic bounded domains which may be simply or multiply connected it also applies to certain unbounded domains depending on the domain there are multiple such families

exact number of ergodic invariant measures for bratteli diagrams abstract for a bratteli diagram b we study the simplex m 1 b of probability measures on the path space of b which are invariant with respect to the tail equivalence relation equivalently m 1 b is formed by probability measures invariant with respect to a homeomorphism of a cantor set we study relations between the number of ergodic measures from m 1 b and the structure and properties of the diagram b we prove a criterion and find sufficient conditions of unique ergodicity of a bratteli diagram in which case the simplex m 1 b is a singleton for a finite rank k bratteli diagram b having exactly l k ergodic invariant measures we explicitly describe the structure of the diagram and find the subdiagrams which support these measures we find sufficient conditions under which i a bratteli diagram has a prescribed number finite or infinite of ergodic invariant measures and ii the extension of a measure from a uniquely ergodic subdiagram gives a finite ergodic invariant measure several examples including stationary bratteli diagrams pascal bratteli diagrams and toeplitz flows are considered

isomorphism conjectures in algebraic k theory duff in this paper we are concerned with the four functors y y3 if and which map from the category of topological spaces x to the category diff of q spectra the functor 3 or if maps the space x to the qspectrum of stable topological or smooth pseudoisotopies of x the spaces diff dif 3i x id9 x i 0 are hatcher s deloopings of 970 x yd x cf 29 the functor j maps a path connected space x to the algebraic ktheoretic s2 spectrum for the integral group ring z7r1x the spaces x x i 0 are the gersten wagoner deloopings of z o x cf 27 52 5 x is also defined on a nonpath connected space x to be x1 where xi i e i are the path components of x and ii xi indicates the direct limit of all finite products of the i xi i e i if x is path connected then the functor maps x to the l surgery classifying spaces for oriented surgery problems with fundamental group r i1x identified by their fourfold periodicity k 90 x k 4j x cf 35 39 44 47 in addition if x is not path connected then we set y77 x tiei5 fj xi where xi i e i are the path components of x results obtained by the authors over the past five years cf 15 25 and in particular 19 16 a 1 a 18 together with many earlier results and conjectures which are reviewed in 1 6 1 1 6 6 suggest that for any connected cwcomplex x and for 5 denoting any of the above functors the q spectrum x should be computable in a simple way from the values taken by 9 on covering spaces of x which have very simple fundamental groups in more detail we let 9 x denote the collection of all subgroups h c ir ix which are either finite or virtually infinite cyclic a group h is virtually infinite cyclic if there is a short exact sequence of groups 1 z h f 1 with f equal to a finite group and z equal to the infinite cyclic group for each such group we let xh x denote the covering space projection corresponding to h then the q spectrum 5 x should be computable in a simple way from the q spectra 5 xh h e 9 x the purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any x

on characters of inductive limits of symmetric groups abstract in the paper we completely describe characters central positive definite functions of simple locally finite groups that can be represented as inductive limits of products of symmetric groups under block diagonal embeddings each such group g defines an infinite graded graph that encodes the embedding scheme the group g acts on the space x of infinite paths of the associated graph by changing initial edges of paths assuming the finiteness of the set of ergodic measures for the system x g we establish that each non regular indecomposable character g c is uniquely determined by the formula g 1 fix g 1 k fix g k where 1 k are g ergodic measures fix g x x g x x and 1 k 0 1 we illustrate our results on the group of rational permutations of the unit interval

parallel relaxed multisplitting methods for a symmetric positive definite matrix we first study convergence of both the relaxed multisplitting method and the relaxed nonstationary two stage multisplitting method using incomplete factorizations for solving a linear system whose coefficient matrix is a large sparse symmetric positive definite matrix we next study a parallel implementation of the relaxed two stage multisplitting method using incomplete factorizations as inner splittings and an application of the relaxed two stage multisplitting method to parallel preconditioner of the cg conjugate gradient method lastly we provide parallel performance results of both the relaxed two stage multisplitting method and the cg method with a parallel preconditioner which is derived from the relaxed two stage multisplitting method on the ibm p690 supercomputer

global continua of periodic solutions of singular first order hamiltonian systems of n vortex type the paper deals with singular first order hamiltonian systems of the form begin aligned gamma k dot z k t j nabla z k h z t quad z k t in omega subset mathbb r 2 k 1 dots n end aligned where j in mathbb r 2 times 2 defines the standard symplectic structure in mathbb r 2 gamma 1 dots gamma n in mathbb r setminus 0 are given and the hamiltonian h is of n vortex type begin aligned h z 1 dots z n frac 1 2 pi sum genfrac 0 0pt j k 1 j ne k n gamma j gamma k log left z j z k right f z end aligned this is defined on the configuration space z 1 ldots z n in omega n z j ne z k text for j ne k of n different points in the domain omega subset mathbb r 2 the function f omega n rightarrow mathbb r may have additional singularities near the boundary of omega n we prove the existence of a global continuum of periodic solutions z t z 1 t dots z n t in omega n that emanates after introducing a suitable singular limit scaling from a relative equilibrium z t in mathbb r 2n of the n vortex problem in the whole plane where f 0 examples for z include thomson s vortex configurations or equilateral triangle solutions the domain omega need not be simply connected a special feature is that the associated action integral is not defined on an open subset of the space of 2 pi periodic h 1 2 functions the natural form domain for first order hamiltonian systems this is a consequence of the singular character of the hamiltonian our main tool in the proof is a degree for s 1 equivariant gradient maps that we adapt to this class of potential operators

asynchronous two stage iterative methods parallel block two stage iterative methods for the solution of linear systems of algebraic equations are studied convergence is shown for monotone matrices and for h matrices two different asynchronous versions of these methods are considered and their convergence investigated

genus n algebraic reductions of the benney hierarchy within a schottky model by exploiting a new theoretical connection between reductions of the benney hierarchy and the dirichlet problem for laplace s equation the solution to a spectral problem associated with n parameter algebraic reductions of the benney hierarchy is found explicitly the solutions can be written in terms of the modified green s function associated with reflectionally symmetric n connected planar domains whose holes are all centred on the symmetry axis explicit formulae for the modified green s function in a canonical class of circular domains are constructed using a schottky model of the schottky double of these domains uniformizations of the spectral problem associated with two different types of reductions then follow from these formulae

on the convergence of iterative methods for solving singular linear systems in this paper we first show that for the stationary iterative methods for solving consistent singular linear systems the convergence and the quotient convergence are equivalent then we use this assertion to analyze the convergence of the multisplitting algorithms for the solution of linear systems when the coefficient matrices of the linear systems are singular hermitian positive semidefinite and the linear systems are consistent

vortex dynamics in complex domains on a spherical surface we consider the motion of both point vortices and uniform vortex patches in arbitrary possibly multiply connected regions bounded by impenetrable walls on the surface of a sphere by exploiting knowledge of the functional form of the relevant green s function in a pre image circular domain that is conformally equivalent to a stereographic projection of the fluid domain on the spherical surface we first generalize kirchhoff routh theory for point vortex motion in the plane to point vortex motion on a spherical shell next we study vortex patch motion and show that there is a contour dynamics formulation for the evolution of uniform vortex patches in any finitely connected domain on a spherical shell bounded by impenetrable walls we describe a novel numerical scheme whereby this motion can be computed some illustrative calculations are shown

secondary schottky klein prime functions associated with multiply connected planar domains in recent years a general mathematical framework for solving applied problems in multiply connected domains has been developed based on use of the schottky klein s k prime function of an underlying compact riemann surface known as the schottky double of the domain in this paper we describe additional function theoretic objects that are naturally associated with planar multiply connected domains and which we refer to as secondary s k prime functions the basic idea develops and extends an observation of burnside dating back to 1892 applications of the new functions to represent conformal slit maps of mixed type that have been a topic of recent interest in the literature are given other possible applications are also surveyed

on the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data abstract convergence of iterative processes in c k of the form x i r i j i x 1 r i 1 1 p j i x i where j i 1 2 n i 1 2 is analyzed it is shown that if the matrices p 1 p n are paracontracting in the same smooth strictly convex norm and if the sequence j i i 1 has certain regularity properties then the above iterates converge this result implies the convergence of a parallel asynchronous implementation of the algebraic reconstruction technique art algorithm often used in tomographic reconstruction from incomplete data

computing the joint spectral radius this paper presents algorithms for finding an arbitrarily small interval that contains the joint spectral radius of a finite set of matrices it also presents a numerical criterion for verifying in certain cases that the joint spectral radius is the maximum of the spectral radii of the given matrices error bounds are derived for the case where calculations are done with finite precision and the matrices are not known exactly the algorithms are implemented and applied to estimate holder exponents of the orthonormal wavelets n constructed by daubechies for 3 n 8

convergence theorems for parallel multisplitting two stage iterative methods for mildly nonlinear systems abstract for the system of mildly nonlinear equations ax f x where a r n n is an n by n sparse real matrix and f r n r n a general nonlinear mapping both local and global convergence properties of the multisplitting two stage iterative method numer algorithms 15 1997 347 are further studied in depth when a r n n is symmetric positive definite and its multiple splittings are symmetric p regular

n vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh poisson and the lane emden fowler equations we prove the existence of equilibria of the n vortex hamiltonian in a bounded domain omega subset mathbb r 2 which is not necessarily simply connected on an arbitrary bounded domain we obtain new equilibria for n 3 or n 4 if has an axial symmetry we obtain a symmetric equilibrium for each n in mathbb n we also obtain new stream functions solving the sinh poisson equation delta psi rho sinh psi in with dirichlet boundary conditions for 0 small the stream function psi rho induces a stationary velocity field v rho solving the euler equation in on an arbitrary bounded domain we obtain velocitiy fields having three or four counter rotating vortices if has an axial symmetry we obtain for each n a velocity field v rho that has a chain of n counter rotating vortices analogous to the mallier maslowe row of counter rotating vortices in the plane our methods also yield new nodal solutions for other semilinear dirichlet problems in particular for the lane emden fowler equation delta u u p 1 u in with p large

approximation of the joint spectral radius using sum of squares we provide an asymptotically tight computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares sos programming the approach is based on a search for an sos polynomial that proves simultaneous contractibility of a finite set of matrices we provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices additionally we present a comparison between our approximation scheme and earlier techniques including the use of common quadratic lyapunov functions and a method based on matrix liftings theoretical results and numerical investigations show that our approach yields tighter approximations

numerical conformal mapping of multiply connected regions by fornberg like methods we develop a new algorithm for computing conformal maps from regions exterior to non overlapping disks to unbounded multiply connected regions exterior to non overlapping smoothly bounded jordan regions the method is an extension of fornberg s original newton like method for mapping of the disk to simply connected regions a fortran program based on the algorithm has been developed and tested for the 2 and 3 disk case numerical examples are reported

a converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

two stage waveform relaxation method for the initial value problems of differential algebraic equations in this paper we consider a two stage strategy for waveform relaxation wr iterations applied to initial value problems for differential algebraic equations daes in the form a y t b y t f t outer iterations of tswr are defined by m a y k 1 t m 1 y k 1 t n 1 y k t n a y k t f t where a m a n a b m 1 n 1 and each iteration y k 1 t is computed using an inner iterative process based on another splitting m 1 m 2 n 2 meanwhile by the means of the theta method the discretized tswr of daes is realized furthermore when m a is an hermitian positive semi definite matrix with p regular splittings the convergence and the comparison theorems of tswr are analyzed finally the numerical experiments are presented

on the convergence of nonstationary multisplitting two stage iteration methods for hermitian positive definite linear systems convergence properties of the nonstationary multisplitting two stage iteration methods for solving large sparse system of linear equations are further studied when the coefficient matrices are hermitian positive definite matrices

generalization of the schwarz christoffel mapping to multiply connected polygonal domains a generalization of the schwarz christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains namely a rectilinear slit domain and a bounded circular domain the conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the schottky klein prime functions which is the same irrespective of the preimage slit domain and a prefactor function that depends on the choice of the rectilinear slit domain a detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half plane with radial slits representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain

quantitative recurrence results letx be a probability measure spacex x endowed with a compatible metricd so that x d has a countable base it is well known that ift x x is measure preserving then almost all pointsx x are recurrent i e lim begin array 20 c inf n geqq 1 end array d x t n x 0 we show that under the additional assumption that the hausdorff measureh x ofx is finite for some 0 this result can be strengthened lim begin array 20 c inf n geqq 1 end array left n 1 alpha d x t n x right infty for almost all pointsx x a number of applications are considered

point vortex dynamics a classical mathematics playground the idealization of a two dimensional ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics several examples are given including the integrability of the three vortex problem the interplay of relative equilibria of identical vortices and the roots of certain polynomials addition formulas for the cotangent and the weierstras function projective geometry and other topics the hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community

a condition of boshernitzan and uniform convergence in the multiplicative ergodic theorem this paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts if such a subshift satisfies a certain condition originally introduced by boshernitzan every locally constant sl 2 r valued cocycle is uniform as a consequence the corresponding schrodinger operators exhibit cantor spectrum of lebesgue measure zero an investigation of boshernitzan s condition then shows that these results cover all earlier results of this type and moreover provide various new ones in particular boshernitzan s condition is shown to hold for almost all circle maps and almost all arnoux rauzy subshifts

the spectral radius of a product of nonnegative matrices abstract for nonnegative n by n matrices al ak with perron eigenvectors xl ak respectively we give simple upper and lower bounds for the ratio p al ak p al p ak which imply that p al ak is approximately p al p ak if the eigenvectors vary slowly in addition several aspects of the bounds such as implications cases of equality and what to do if eigenvectors have some zero components are explored

ancestors graph and an upper bound for the subword complexity function a general upper bound for the subword complexity function of a fixed point of a primitive substitution is presented the approach relies on the introduction and discussion of the ancestors graph of a substitution more precisely we characterize the language of a fixed point of a primitive substitution as the strongly connected component of the letters in its ancestors graph we propose algorithms for the local construction of this graph we give an upper bound for the subword complexity function in terms of the hilbert series of some quotient of a noncommutative polynomial ring by a sum of automatic bilateral ideals

optimal norms and the computation of joint spectral radius of matrices the notion of spectral radius of a set of matrices is a natural extension of spectral radius of a single matrix the finiteness conjecture fc claims that among the infinite products made from the elements of a given finite set of matrices there is a certain periodic product made from the repetition of the optimal product whose rate of growth is maximal fc has been disproved in this paper it is conjectured that fc is almost always true and an algorithm is presented to verify the optimality of a given product the algorithm uses optimal norms as a special subset of extremal norms several conjectures related to optimal norms and non decomposable sets of matrices are presented the algorithm has successfully calculated the spectral radius of several parametric families of pairs of matrices associated with compactly supported multi resolution analyses and wavelets the results of related numerical experiments are presented

force enhancing vortex equilibria for two parallel plates in uniform flow a two dimensional potential flow in an unbounded domain with two parallel plates is considered we examine whether two free point vortices can be trapped near the two plates in the presence of a uniform flow and observe whether these stationary point vortices enhance the force on the plates the present study is an extension of previously published work in which a free point vortex over a single plate is investigated the flow problem is motivated by an airfoil design problem for the double wings moreover it also contributes to a design problem for an efficient wind turbine with vertical blades in order to obtain the point vortex equilibria numerically we make use of a linear algebraic algorithm combined with a stochastic process called the brownian ratchet scheme the ratchet scheme allows us to capture a family of stationary point vortices in multiply connected domains with ease as a result we find that stationary point vortices exist around the two plates and they enhance the downward force and the counter clockwise rotational force acting on the two plates

on the k theory of crystallographic groups for any crystallographic group f we show that the groups ki t are isomorphic via the forget control map to the controlled a groups ki r c for all 0

convergence of relaxed parallel multisplitting methods abstract multisplitting methods are parallel methods for the solution of a linear system ax b it has been observed that the convergence of multisplitting methods is often improved if some kind of relaxation is used we investigate two different variants of relaxed multisplitting methods if a is an h matrix these methods converge if the relaxation parameter is from an interval 0 0 with 0 1