the cauchy problem for indefinite improper affine spheres and their hessian equation abstract we give a conformal representation for indefinite improper affine spheres which solve the cauchy problem for their hessian equation as consequences we can characterize their geodesics and obtain a generalized symmetry principle then we classify the helicoidal indefinite improper affine spheres and find a new family with geodesically complete non flat affine metric moreover we present interesting examples with singular curves and isolated singularities

approximate factorization constraint preconditioners for saddle point matrices we consider the application of the conjugate gradient method to the solution of large symmetric indefinite linear systems special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given numerical experiments validate these conclusions

a parameterized deteriorated pss preconditioner and its optimization for nonsymmetric saddle point problems abstract based on the deteriorated positive definite and skew hermitian splitting dpss preconditioner we develop a parameterized dpss pdpss preconditioner for the nonsymmetric saddle point linear systems the spectral properties of the pdpss preconditioned coefficient matrix are analyzed moreover an upper bound of the degree of the minimal polynomial of the pdpss preconditioned matrix is also obtained inasmuch as the efficiency of the pdpss preconditioner depends on the values of its parameters we further derive a fast and effective method to compute the optimal parameter values of the pdpss preconditioner finally numerical examples are employed to illustrate the feasibility and the efficiency of the pdpss preconditioner

isometric immersions of mathbb r 2 into mathbb r 4 and perturbation of hopf tori we produce a new general family of flat tori in mathbb r 4 the first one since bianchi s classical works in the nineteenth century to construct these flat tori obtained via small perturbation of certain hopf tori in mathbb s 3 we first present a global description of all isometric immersions of mathbb r 2 into mathbb r 4 with flat normal bundle

symbolic extensions and smooth dynamical systems let f x x be a homeomorphism of the compact metric space x a symbolic extension of f x is a subshift on a finite alphabet g y which has f as a topological factor we show that a generic c1 non hyperbolic i e non anosov area preserving diffeomorphism of a compact surface has no symbolic extensions for r 1 we exhibit a residual subset mathcal r of an open set mathcal u of cr diffeomorphisms of a compact surface such that if f in mathcal r then any possible symbolic extension has topological entropy strictly larger than that of f these results complement the known fact that any c diffeomorphism has symbolic extensions with the same entropy we also show that cr generically on surfaces homoclinic closures which contain tangencies of stable and unstable manifolds have hausdorff dimension two

a relaxed deteriorated pss preconditioner for nonsymmetric saddle point problems from the steady navier stokes equation for nonsymmetric saddle point problems arising from the steady navier stokes equations pan ng and bai presented a deteriorated positive definite and skew hermitian splitting dpss preconditioner pan et al 2006 to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the unconditional convergence property of the dpss iteration method is proved and a relaxed dpss rdpss preconditioner is proposed the rdpss preconditioner is much closer to the coefficient matrix than the dpss preconditioner in certain norm which straightforwardly results in an rdpss iteration method the convergence conditions of the rdpss iteration are analyzed and the optimal parameter which minimizes the spectral radius of the rdpss iteration matrix is derived using the rdpss preconditioner to accelerate some krylov subspace methods like gmres is also studied the eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the rdpss preconditioner

soficity amenability and dynamical entropy in a previous paper the authors developed an operator algebraic approach to lewis bowen s sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measure preserving transformations on a standard probability space we show here that these measure and topological entropy invariants both coincide with their classical counterparts when the acting group is amenable

on local hermitian and skew hermitian splitting iteration methods for generalized saddle point problems in this paper we first present a local hermitian and skew hermitian splitting lhss iteration method for solving a class of generalized saddle point problems the new method converges to the solution under suitable restrictions on the preconditioning matrix then we give a modified lhss mlhss iteration method and further extend it to the generalized saddle point problems obtaining the so called generalized mlhss gmlhss iteration method numerical experiments for a model navier stokes problem are given and the results show that the new methods outperform the classical uzawa method and the inexact parameterized uzawa method

mathcal c 2 surface diffeomorphisms have symbolic extensions we prove that mathcal c 2 surface diffeomorphisms have symbolic extensions i e topological extensions which are subshifts over a finite alphabet following the strategy of downarowicz and maass invent math 176 617 636 2009 we bound the local entropy of ergodic measures in terms of lyapunov exponents this is done by reparametrizing bowen balls by contracting maps in a approach combining hyperbolic theory and yomdin s theory

on the maximal number of exceptional values of gauss maps for various classes of surfaces the main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of gauss maps for several classes of immersed surfaces in space forms for example complete minimal surfaces in the euclidean three space weakly complete improper affine spheres in the affine three space and weakly complete flat surfaces in the hyperbolic three space for this purpose we give an effective curvature bound for a specified conformal metric on an open riemann surface

smooth interval maps have symbolic extensions we prove that if x denotes the interval or the circle then every transformation t x x of class cr where r 1 is not necessarily an integer admits a symbolic extension i e every such transformation is a topological factor of a subshift over a finite alphabet this is done using the theory of entropy structure for such transformations we control the entropy structure by providing an upper bound in terms of lyapunov exponents of local entropy in the sense of newhouse of an ergodic measure near an invariant measure the antarctic theorem this bound allows us to estimate the so called symbolic extension entropy function on invariant measures the main theorem and as a consequence to estimate the topological symbolic extension entropy i e a number such that there exists a symbolic extension with topological entropy arbitrarily close to that number this last estimate coincides in dimension 1 with a conjecture stated by downarowicz and newhouse 13 conjecture 1 2 the passage from the antarctic theorem to the main theorem is applicable to any topological dynamical system not only to smooth interval or circle maps

flat surfaces in the hyperbolic 3 space in this paper we give a conformal representation of flat surfaces in the hyperbolic 3 space using the complex structure induced by its second fundamental form we also study some examples and the behaviour at infinity of complete flat ends

weak expansion properties and large deviation principles for expanding thurston maps abstract in this paper we prove that an expanding thurston map f s 2 s 2 is asymptotically h expansive if and only if it has no periodic critical points and that no expanding thurston map is h expansive as a consequence for each expanding thurston map without periodic critical points and each real valued continuous potential on s 2 there exists at least one equilibrium state for such maps we also establish large deviation principles for iterated preimages and periodic points it follows that iterated preimages and periodic points are equidistributed with respect to the unique equilibrium state for an expanding thurston map without periodic critical points and a potential that is holder continuous with respect to a visual metric on s 2

local entropy theory for a countable discrete amenable group action in the paper we throw the first light on studying systematically the local entropy theory for a countable discrete amenable group action for such an action we introduce entropy tuples in both topological and measure theoretic settings and build the variational relation be tween these two kinds of entropy tuples by establishing a local variational principle for a given finite open cover moreover based the idea of topological entropy pairs we introduce and study two special classes of such an action uniformly positive entropy and completely positive entropy note that in the building of the local variational principle following romagnoli s ideas two kinds of measure theoretic entropy are introduced for finite borel covers these two kinds of entropy turn out to be the same where danilenko s orbital approach becomes an inevitable tool

restrictive preconditioners for conjugate gradient methods for symmetric positive definite linear systems the restrictively preconditioned conjugate gradient rpcg method for solving large sparse system of linear equations of a symmetric positive definite and block two by two coefficient matrix is further studied in fact this rpcg method is essentially the classical preconditioned conjugate gradient pcg method with a specially structured preconditioner within this setting we present algorithmic descriptions of two restrictive preconditioners that respectively employ the block jacobi and the block symmetric gauss seidel matrix splitting matrices as approximations to certain matrices involved in them and give convergence analyses of the correspondingly induced two pcg methods numerical results show that these restrictive preconditioners can lead to practical and effective pcg methods for solving large sparse systems of linear equations of symmetric positive definite and block two by two coefficient matrices

entropy and the variational principle for actions of sofic groups recently lewis bowen introduced a notion of entropy for measure preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy by applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants and we establish the variational principle in this context in the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group c algebra

a modified sor like method for the augmented systems for the augmented system of linear equations by considering a new splitting of the coefficient matrix a modified sor like msor like method is presented in this paper the convergence of this method is discussed under suitable restrictions on iteration parameters the optimal parameters and the corresponding optimal convergence factor are determined numerical examples are reported to confirm the efficiency of the proposed method

preconditioning for the steady state navier stokes equations with low viscosity we introduce a preconditioner for the linearized navier stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero for constant coefficient problems with periodic boundary conditions we show that the preconditioning yields a system with a single eigenvalue equal to 1 so that performance is independent of both viscosity and mesh size for other boundary conditions we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon we also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration with virtually no degradation of performance

fuglede kadison determinants and entropy for actions of discrete amenable groups consider a discrete group t and an element in the integral group ring zt then t acts from the left on the discrete additive group zt ztf by automorphisms of groups dualizing we obtain a left t action on the compact pontrjagin dual group xf zt ztf by continuous automorphisms of groups by definition xf is a closed subshift of s1 1 for measurable or topological actions of amenable groups a good entropy theory is available ow m thus from now on let t be a finitely generated discrete amenable group we are interested in determining the topological entropy hf of the t action on xf it agrees with the measure theoretic entropy of the t action on xf equipped with its haar probability measure in their paper fk from 1952 fuglede and kadison introduced a determinant for units in finite factor von neumann algebras more recently l ck has general ized their theory to nonunits and to group von neumann algebras which are not necessarily factors l2 he needed this to define l2 torsion for general covering spaces the von neumann algebra a t of the amenable group t is a certain completion of cr and hence we may consider the fuglede kadison l ck determinant detjvr of we refer to section 6 for the definition of a log strong f0lner sequence it is known that finitely generated virtually nilpotent groups have log strong f0lner sequences our main result is

complete flat surfaces with two isolated singularities in hyperbolic 3 space abstract we construct examples of flat surfaces in h 3 which are graphs over a two punctured horosphere and classify complete embedded flat surfaces in h 3 with only one end and at most two isolated singularities

local variational principle concerning entropy of a sofic group action abstract recently lewis bowen introduced a notion of entropy for measure preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy and then david kerr and hanfeng li developed an operator algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space and established the global variational principle concerning measure theoretic and topological entropy in this sofic context by localizing these two kinds of entropy in this paper we prove a local version of the global variational principle for any finite open cover of the space and show that these local measure theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group

rotation hypersurfaces in spaces of constant curvature rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed a local characterization of such hypersurfaces with dimensions greater than two is given in terms of principal curvatures some special cases of rotation hypersurfaces with constant mean curvature in hyperbolic space are studied in particular it is shown that the well known conjugation between the helicoid and the catenoid in euclidean three space extends naturally to hyperbolic three space h3 in the latter case catenoids are of three different types and the explicit correspondence is given it is also shown that there exists a family of simply connected complete embedded nontotally geodesic stable minimal surfaces in h3

tail variational principle for a countable discrete amenable group action abstract in this paper we define the measure theoretic tail entropy for a countable discrete amenable group action and prove the tail variational principle when there exists a refining sequence of essential partitions additionally some properties of local entropy are investigated

on a galerkin lagrange multiplier method for the stationary navier stokes equations a galerkin lagrange multiplier formulation is used for the numerical solution of the stationary navier stokes equations in order to avoid the construction of zero divergence elements the formulation is based on different approximating spaces for the velocity field and the pressure optimal rate of convergence estimates are derived moreover a galerkin newton scheme for the solution of the nonlinear equations is shown to be quadratically locally convergent another scheme is shown to be linearly globally convergent

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

entropy and mixing for amenable group actions for i a countable amenable group consider those actions of i as measurepreserving transformations of a standard probability space written asft g 2i acting on x f we sayft g 2i has completely positive entropy or simply cpe for short if for any flnite and nontrivial partition p of x the entropy h t p is not zero our goal is to demonstrate what is well known for actions of and even d that actions of completely positive entropy have very strong mixing properties let si be a list of flnite subsets of i we say the si spread if any particular 6 id belongs to at most flnitely many of the sets sis i1 i theorem 0 1 for ft g 2i an action of i of completely positive entropy and p any flnite partition for any sequence of flnite sets si i which spread we have

the cauchy problem for improper affine spheres and the hessian one equation we give a conformal representation for improper affine spheres which is used to solve the cauchy problem for the hessian one equation with this representation we characterize the geodesics of an improper affine sphere study its symmetries and classify the helicoidal ones finally we obtain the complete classification of the isolated singularities of the hessian one monge ampere equation

a relaxed positive definite and skew hermitian splitting preconditioner for saddle point problems for saddle point linear system whose coefficient matrix is non hermitian of positive definite hermitian part pan et al 2006 proposed a positive definite and skew hermitian splitting pss preconditioner in this paper based on the pss preconditioner we present a relaxed positive definite and skew hermitian splitting rpss preconditioner we analyze the spectral properties of the new preconditioned matrix and conclude that the rpss preconditioner has much better properties than pss numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the new preconditioner

a practical formula for computing optimal parameters in the hss iteration methods in the hss iteration methods proposed by bai golub and ng z z bai g h golub m k ng hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems siam j matrix anal appl 24 2003 603 626 the determination of the optimal parameter is a tough task when solving a non hermitian positive definite linear system in this paper a new and simple strategy for obtaining the optimal parameter is proposed which computes the optimal parameter by solving a cubic polynomial equation the coefficients of this polynomial are determined by several traces of some matrices related to the symmetric and skew symmetric parts of the coefficient matrix of the real linear system numerical experiments show that our new strategy is very effective for approximating the optimal parameter in the hss iteration methods as it leads to fast convergence of the method

the generalized hss method for solving singular linear systems for the singular non hermitian and positive semidefinite linear systems we propose an alternating direction iterative method with two parameters based on the hermitian and skew hermitian splitting the semi convergence analysis and the quasi optimal parameters of the proposed method are discussed moreover the corresponding preconditioner based on the splitting is given to improve the semi convergence rate of the gmres method numerical examples are given to illustrate the theoretical results and the efficiency of the generalized hss method either as a solver or a preconditioner for gmres

the entropy theory of symbolic extensions fix a topological system x t with its space k x t of t invariant borel probabilities if y s is a symbolic system subshift and y s x t is a topological extension factor map then the function h ext on k x t which assigns to each the maximal entropy of a measure on y mapping to is called the extension entropy function of the infimum of such functions over all symbolic extensions is called the symbolic extension entropy function and is denoted by h sex in this paper we completely characterize these functions in terms of functional analytic properties of an entropy structure on x t the entropy structure is a sequence of entropy functions h k defined with respect to a refining sequence of partitions of x or of x z for some auxiliary system z r with simple dynamics whose boundaries have measure zero for all the invariant borel probabilities we develop the functional analysis and computational techniques to produce many dynamical examples for instance we resolve in the negative the question of whether the infimum of the topological entropies of symbolic extensions of x t must always be attained and we show that the maximum value of h sex need not be achieved at an ergodic measure we exhibit several characterizations of the asymptotically h expansive systems of misiurewicz which emerge as a fundamental natural class in the context of the entropy structure the results of this paper are required for the downarowicz newhouse results dn on smooth dynamical systems

ribaucour transformations for flat surfaces in the hyperbolic 3 space abstract we consider ribaucour transformations for flat surfaces in the hyperbolic 3 space h 3 we show that such transformations produce complete embedded ends of horosphere type and curves of singularities which generically are cuspidal edges moreover we prove that these ends and curves of singularities do not intersect we apply ribaucour transformations to rotational flat surfaces in h 3 providing new families of explicitly given flat surfaces h 3 which are determined by several parameters for special choices of the parameters we get surfaces that are periodic in one variable and surfaces with any even number or an infinite number of embedded ends of horosphere type

the cauchy problem for the liouville equation and bryant surfaces abstract we give a construction that connects the cauchy problem for the 2 dimensional elliptic liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3 space h 3 and solve both of them we construct the unique mean curvature one surface in h 3 that passes through a given curve with a given unit normal along it and provide diverse applications in particular topics such as period problems symmetries finite total curvature planar geodesics rigidity etc are treated for these surfaces

symmetry and uniqueness of parabolic affine spheres the proof of a variety of problems about symmetry properties in partial differential equations and differential geometry is based on the maximum principle for elliptic linear equations see a1 gnn and s in this paper we apply a similar technique to a locally strongly convex parabolic affine sphere in short pa sphere embedded in the unimodular affine real 3 space j 3 the study of pa spheres is locally equivalent see c1 c2 to the study of convex solutions of the monge amp re equation

sofic mean dimension abstract we introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces these generalize the gromov lindenstrauss weiss mean dimensions for actions of countable amenable groups and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy

homoclinic points of algebraic ℤ actions an algebraic zd action is an action of zd by continuous automorphisms of a compact abelian group the dynamics of a single group automorphism have been investigated in great detail over the past several decades more recently the study of algebraic zd actions for d 2 has revealed a striking interplay between these actions and commutative algebra in 2 we summarize those parts of this interaction needed here the purpose of this paper is to study the homoclinic points of algebraic zd actions let al be an algebraic zd action on the compact abelian group x and let ox denote the additive identity of x a point x e x is homoclinic for ce if c nx ox as llnll oo the set ai x of all homoclinic points for oc is clearly a subgroup of x which we call the homoclinic group of oz in 3 we discuss some elementary properties of the homoclinic group including counltability of za x whenever oz is expansive our two main results are contained in 4 these are that if oz is an expansive algebraic ed action then 1 a x is nontrivial if and only if oz has strictly positive entropy and 2 a x is nontrivial and dense in x if and only if oz has completely positive entropy the second result is proved by first establishing in lemma 4 5 the density of a x for certain principal expansive actions by use of fourier analysis for these actions a x is generated by a single fundamental homoclinic point which can be computed explicitly this lemma is then combined with some commutative algebra to prove 2 recent work of kaminker and putnam 3 11 has suggested a general duality in the k theory of c algebras for a principal expansive action al on x we show that a x is isomorphic to the dual group of x providing a class of examples to which their duality theory applies ruelle has investigated expansive topological zd actions which satisfy an orbit tracing property called specification see 12 and 13 showing that there is a thermodynamic formalism for such actions in 5 we show that expansive algebraic zd actions with completely positive entropy always satisfy very strong specification properties thereby providing an extensive class of examples to which the thermodynamic formalism applies

surfaces of constant curvature in r3 with isolated singularities abstract we prove that finite area isolated singularities of surfaces with constant positive curvature k 0 in r 3 are removable singularities branch points or immersed conical singularities we describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally strictly convex curves in s 2 with admissible cusp singularities characterizing when the singularity is actually embedded in the global setting we describe the space of peaked spheres in r 3 i e compact convex surfaces of constant curvature k 0 with a finite number of singularities and give applications to harmonic maps and constant mean curvature surfaces

complete surfaces of constant mean curvature 1 in the hyperbolic 3 space in the study of minimal surfaces in the euclidean 3 space the weierstrass representation plays an important role bryant br showed that an analogue of the weierstrass representation formula holds for surfaces of constant mean curvature i in the hyperbolic 3 space x3 in this article we abbreviate the term constant mean curvature i as cmc 1 like minimal surfaces in the euclidean space the hyperbolic gauss map of cmc 1 surfaces is defined as a holomorphic map to c u oo however in contrast to the euclidean case the hyperbolic gauss map of a cmc 1 surface may not be extended across the ends even if the total gaussian curvature is finite we call a complete cmc 1 surface whose gauss map can be extended across all of its ends a cmc 1 surface of regular ends in this article we produce an explicit tool to construct cmc 1 surfaces of regular ends in section 2 we show that such surfaces are constructed by solving some ordinary differential equations with regular singularity in our cmc 1 category ossermann s inequality is not expected and the cohn vossen inequality is the best possible one we show in section 4 that the equality of the cohn vossen inequality never holds for complete cmc 1 surfaces in xh3 in section 5 we give a necessary and sufficient condition that a regular end of a cmc 1 surface be embedded in section 6 we classify complete cmc 1 surfaces of genus 0 with two regular ends our classification contains new examples furthermore in section 7 we construct several new cmc 1 surfaces with regular embedded ends each of these examples has a nontrivial deformation which is mentioned in section 3 it should be remarked that our construction does not work for surfaces with irregular ends but there is another construction by perturbing minimal surfaces in the euclidean 3 space the authors constructed cmc 1 surfaces all of whose ends are irregular see uy1

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

shift splitting preconditioners for saddle point problems in this paper we first present a shift splitting preconditioner for saddle point problems the preconditioner is based on a shift splitting of the saddle point matrix resulting in an unconditional convergent fixed point iteration based on the idea of the splitting we further propose a local shift splitting preconditioner some properties of the local shift splitting preconditioned matrix are studied these preconditioners extend those studied by bai yin and su for solving non hermitian positive definite linear systems bai et al 2006 finally numerical experiments of a model stokes problem are presented to show the effectiveness of the proposed preconditioners

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

complete linear weingarten surfaces of bryant type a plateau problem at infinity in this paper we study a large class of weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3 space we show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the euclidean 3 space and we use it to study their completeness we also establish some existence and uniqueness theorems by studing the plateau problem at infinity when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family and how many embedded solutions are there

motivations and realizations of krylov subspace methods for large sparse linear systems we briefly introduce typical and important direct and iterative methods for solving systems of linear equations concretely describe their fundamental characteristics in viewpoints of both theory and applications and clearly clarify the substantial differences among these methods in particular the motivations of searching the solution of a linear system in a krylov subspace are described and the algorithmic realizations of the generalized minimal residual gmres method are shown and several classes of state of the art algebraic preconditioners are briefly reviewed all this is useful for correctly deeply and completely understanding the application scopes theoretical properties and numerical behaviors of these methods and is also helpful in designing new methods for solving systems of linear equations

entropy determinants and l 2 torsion we show that for any amenable group and any z module m of type fl with vanishing euler characteristic the entropy of the natural action on the pontryagin dual of m is equal to the l 2 torsion of m as a particular case the entropy of the principal algebraic action associated with the module z z f is equal to the logarithm of the fuglede kadison determinant of f whenever f is a non zero divisor in z this conrms a conjecture of deninger as a key step in the proof we provide a general szeg o type approximation theorem for the fuglede kadison determinant on the group von neumann algebra of an amenable group as a consequence of the equality between l 2 torsion and entropy we show that the l 2 torsion of a non trivial amenable group with nite classifying space vanishes this was conjectured by luck finally we establish a milnor turaev formula for the l 2 torsion of a nite acyclic chain complex

homoclinic groups ie groups and expansive algebraic actions we give algebraic characterizations for expansiveness of algebraic actions of countable groups the notion of p expansiveness is introduced for algebraic actions and we show that for countable amenable groups a finitely presented algebraic action is 1 expansive exactly when it has finite entropy we also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms and show that the ie group determines the pinsker factor for such actions for an expansive algebraic action of a polycyclic by finite group on x it is shown that the entropy of the action is equal to the entropy of the induced action on the pontryagin dual of the homoclinic group the homoclinic group is a dense subgroup of the ie group the homoclinic group is nontrivial exactly when the action has positive entropy and the homoclinic group is dense in x exactly when the action has completely positive entropy

weak expansiveness for actions of sofic groups in this paper we shall introduce h expansiveness and asymptotical h expansiveness for actions of sofic groups by definition each h expansive action of a sofic group is asymptotically h expansive we show that each expansive action of a sofic group is h expansive and for any given asymptotically h expansive action of a sofic group the entropy function with respect to measures is upper semi continuous and hence the system admits a measure with maximal entropy observe that asymptotically h expansive property was first introduced and studied by misiurewicz for z actions using the language of tail entropy and thus in the remaining part of the paper we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of misiurewicz s ideas when the group is amenable it turns out that these two definitions are equivalent in this setting

topological r entropy and topological entropy of free semigroup actions abstract we introduce the notion of topological r entropy for free semigroup actions on a compact metric space and provide some properties of it by using the skew product transformation as bridge we get the following two main results 1 we extend the result that the topological entropy is the limit of topological r entropy in 15 to free semigroup actions r 0 2 let f i i 0 1 m 1 be homeomorphisms on a compact metric space we verify that the topological entropy of f 0 f m 1 equals the topological entropy of f 0 1 f m 1 1

new choices of preconditioning matrices for generalized inexact parameterized iterative methods for large sparse saddle point problems chen and jiang recently studied a class of generalized inexact parameterized iterative methods see f chen y l jiang a generalization of the inexact parameterized uzawa methods for saddle point problems appl math comput 206 2008 765 771 in this paper the methods are modified and some choices of preconditioning matrices are given these preconditioning matrices have advantages in solving large sparse linear system numerical experiments of a model stokes problem are presented

on the generalized weyl problem for flat metrics in the hyperbolic 3 space abstract the paper deals with the study of complete embedded flat surfaces in h 3 with a finite number of isolated singularities we give a detailed information about its topology conformal type and metric properties we show how to solve the generalized weyl s problem of realizing isometrically any complete flat metric with euclidean singularities in h 3 which gives the existence of complete embedded flat surfaces with a finite arbitrary number of isolated singularities