a korovkin type theory for finite toeplitz operators via matrix algebras preconditioned conjugate gradients pcg are widely and successfully used methods for solving a toeplitz linear system a n vec x vec b 59 9 20 5 34 62 6 10 28 45 44 46 49 frobenius optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the frobenius distance from a n the convergence features of these pcg have been naturally studied by means of the weierstrass jackson theorem 17 36 45 owing to the profound relationship between the spectral features of the matrices a n generated by the fourier coefficients of a continuous function f and the analytical properties of the symbol f itself in this paper we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f on the other hand by modifying the korovkin theorem to study the frobenius optimal preconditioning problem we provide a new and unifying tool for analyzing all frobenius optimal preconditioners in any generic matrix algebra related to trigonometric transforms finally the multilevel case is sketched and discussed by showing that a korovkin type theory also holds in a multivariate sense

riemannian geometry of hamiltonian chaos hints for a general theory we aim at assessing the validity limits of some simplifying hypotheses that within a riemmannian geometric framework have provided an explanation of the origin of hamiltonian chaos and have made it possible to develop a method of analytically computing the largest lyapunov exponent of hamiltonian systems with many degrees of freedom therefore a numerical hypotheses testing has been performed for the fermi pasta ulam beta model and for a chain of coupled rotators these models for which analytic computations of the largest lyapunov exponents have been carried out in the mentioned riemannian geometric framework appear as paradigmatic examples to unveil the reason why the main hypothesis of quasi isotropy of the mechanical manifolds sometimes breaks down the breakdown is expected whenever the topology of the mechanical manifolds is nontrivial this is an important step forward in view of developing a geometric theory of hamiltonian chaos of general validity

factorized banded inverse preconditioners for matrices with toeplitz structure in this paper we study factorized banded inverse preconditioners for matrices with toeplitz structure we show that if a toeplitz matrix t has certain off diagonal decay property then the factorized banded inverse preconditioner approximates t 1 accurately and the spectra of these preconditioned matrices are clustered around 1 in nonlinear image restoration applications toeplitz related systems of the form i t d t are required to solve where d is a positive nonconstant diagonal matrix we construct inverse preconditioners for such matrices numerical results show that the performance of our proposed preconditioners are superior to that of circulant preconditioners a two dimensional nonlinear image restoration example is also presented to demonstrate the effectiveness of the proposed preconditioner

non affine geometrization can lead to non physical instabilities geometrization of dynamics consists of representing trajectories by geodesics on a configuration space with a suitably defined metric previously efforts were made to show that the analysis of dynamical stability can also be carried out within geometrical frameworks by measuring the broadening rate of a bundle of geodesics two known formalisms are via jacobi and eisenhart metrics we find that this geometrical analysis measures the actual stability when the length of any geodesic is proportional to the corresponding time interval we prove that the jacobi metric is not always an appropriate parametrization by showing that it predicts chaotic behavior for a system of harmonic oscillators furthermore we show by explicit calculation that the correspondence between dynamical and geometrical spread is ill defined for the jacobi metric we find that the eisenhart dynamics corresponds to the actual tangent dynamics and is therefore an appropriate geometrization scheme

weak and pointwise compactness in the space of bounded continuous functions let t be a completely regular hausdorff space cb t the space of bounded continuous real valued functions on t m t the banach space dual of cb t let sc denote the family of subsets of cb t which are uniformly bounded and relatively compact for the topology p of pointwise convergence the basic question considered here is what is the largest subspace z of m t such that every member of sc is relatively a cb z compact classical results of grothendieck and ptak show that z m t if t is pseudocompact in general m1 c z c m3 assuming martin s axiom a deep result of talagrand improves the lower bound to mt it is frequently but not always true that z ms counterexamples are given which use banach spaces in their weak topologies to construct the underlying t s a well known theorem of grothendieck 28 asserts that if t is a compact hausdorff space and h is a uniformly bounded pointwise compact set of real valued continuous functions on t then h is compact in the weak topology of the banach space c t thus in this case the weak and pointwise topologies agree on h this article is concerned with extensions of grothendieck s theorem to the setting t is a completely regular hausdorff space and cb t is the banach space of bounded continuous real valued functions on t with the supremum norm ptak 51 and tomasek 64 have shown that grothendieck s theorem as stated above holds if and only if t is pseudocompact however one can pursue the matter as follows main problem for a given completely regular t what is the largest linear subspace z of m t the banach space dual of cb t such that the pointwise topology and the weak topology a cb z agree on uniformly bounded pointwise compact sets of continuous functions now z is easily seen to be a norm closed band in the banach lattice m t the thrust of this work is to position z with respect to the subspaces of m t commonly encountered in topological measure theory the spaces m c mt c ms c ma of tight t additive separable and a additive baire measures respectively it is an easy consequence of grothendieck s theorem that m c z for all t but one can do better a deep result of talagrand 62 which assumes martin s axiom leads to a proof that mt c z received by the editors july 29 1980 1980 mathematics subject classification primary 46a50 28c15 secondary 46e10 54d60 1 partially supported by nsf grant mcs 8002523 1981 american mathematical society 0002 9947 81 0000 0360 05 00

hyperfinite transversal theory a measure theoretic version of a well known p hall s theorem about the existence of a system of distinct representatives of a finite family of finite sets has been proved for the case of the loeb space of an internal uniformly distributed hyperfinite counting space we first prove hall s theorem for 0 1 graphs after which we develop the version of discrete transversal theory we then prove a new version of hall s theorem in the case of 0 1 monotone graphs and give an example of a 0 1 graph which satisfies hall s condition and which does not possess an internal a e matching

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

a primal dual active set method for non negativity constrained total variation deblurring problems this paper studies image deblurring problems using a total variation based model with a non negativity constraint the addition of the non negativity constraint improves the quality of the solutions but makes the solution process a difficult one the contribution of our work is a fast and robust numerical algorithm to solve the non negatively constrained problem to overcome the nondifferentiability of the total variation norm we formulate the constrained deblurring problem as a primal dual program which is a variant of the formulation proposed by chan golub and mulet for unconstrained problems here dual refers to a combination of the lagrangian and fenchel duals to solve the constrained primal dual program we use a semi smooth newton s method we exploit the relationship between the semi smooth newton s method and the primal dual active set method to achieve considerable simplification of the computations the main advantages of our proposed scheme are no parameters need significant adjustment a standard inverse preconditioner works very well quadratic rate of local convergence theoretical and numerical numerical evidence of global convergence and high accuracy of solving the optimality system the scheme shows robustness of performance over a wide range of parameters a comprehensive set of numerical comparisons are provided against other methods to solve the same problem which show the speed and accuracy advantages of our scheme

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

an efficient tvl1 algorithm for deblurring multichannel images corrupted by impulsive noise we extend the alternating minimization algorithm recently proposed in y wang j yang w yin and y zhang siam j imag sci 1 2008 pp 248 272 j yang w yin y zhang and y wang siam j imag sci 2 2009 pp 569 592 to the case of recovering blurry multichannel color images corrupted by impulsive rather than gaussian noise the algorithm minimizes the sum of a multichannel extension of total variation and a data fidelity term measured in the ell 1 norm and is applicable to both salt and pepper and random valued impulsive noise we derive the algorithm by applying the well known quadratic penalty function technique and prove attractive convergence properties including finite convergence for some variables and q linear convergence rate under periodic boundary conditions the main computational requirements of the algorithm are fast fourier transforms and a low complexity gaussian elimination procedure numerical results on images with different blurs and impulsive noise are presented to demonstrate the efficiency of the algorithm in addition it is numerically compared to the least absolute deviation method h y fu m k ng m nikolova and j l barlow siam j sci comput 27 2006 pp 1881 1902 and the two phase method j f cai r chan and m nikolova aims j inverse problems and imaging 2 2008 pp 187 204 for recovering grayscale images we also present results of recovering multichannel images

baire class 1 selectors for upper semicontinuous set valued maps let t be a metric space and x a banach space let f t x be a set valued map assuming arbitrary values and satisfying the upper semicontinuity condition t t f t c o is closed for each weakly closed set c in x then there is a sequence of norm continuous functions converging pointwise in the norm to a selection for f we prove a statement of similar precision and generality when x is a metric space

densely defined selections of multivalued mappings rather general suficient conditions are found for a given multivalued mapping f x y to possess an upper semicontinuous and compactvalued selection g which is defined on a dense g subset of the domain of f the case when the selection g is single valued and continuous is also investigated the results are applied to prove some known as well as new results concerning generic differentiability of convex functions lavrentieff type theorem generic well posedness of optimization problems and generic nonmultivaluedness of metric projections and antiprojections

the inverse jacobi problem the paper presents a discussion of the relation between the dynamics of a mechanical system based upon a lagrangian admitting energy conservation and the dynamics based upon its jacobi lagrangian which determines the space trajectories of the system the basic result found in the paper is the general solution of the inverse problem i e how to determine the full lagrangian when as a starting point an arbitrary homogeneous lagrangian which is used to determine the space trajectories of a system and an arbitrarily assigned energy function which specifies the interaction of the system are given

universally meager sets and principles of generic continuity and selection in banach spaces abstract in this paper we use large cardinals to address some problems about generic continuity and generic selection that occur in the study of fragmentability and differentiability in the context of banach spaces

radon nikodým indexes and measures of weak noncompactness abstract the aim of this paper is to present a quantitative version of the radon nikod m property and some other results related to it this approach gives an extra insight to the classical results we introduce two indexes an index of representability of measures and an index of dentability we review classic results in order to obtain relationships between these and other indexes

borel selectors for upper semi continuous multi valued functions on presente un theoreme de selection plus fin qui s applique dans plusieurs espaces de banach non separables

geometry of spaces with the jacobi metric the generalized maupertuis principle is formulated for systems with the natural lagrangian and an indefinite form of the kinetic energy the generalization is applied to the theory of gravity and cosmology for such systems the metric determined by the kinetic energy form has a lorentz signature the theorem is proved concerning the behavior of trajectories in a neighborhood of the boundary of the region admissible for motion this region is not a smooth manifold but turns out to be a differential space of constant differential dimension this fact allows us to use geometric methods analogous to those elaborated for smooth manifolds it is shown that singularities of the jacobi metric are not dangerous for the motion its trajectories are smooth in the sense of the theory of differential spaces

boundaries of asplund spaces we study the relationship between the classical combinatorial inequalities of simons and the more recent i property of fonf and lindenstrauss we obtain a characterization of strong boundaries for asplund spaces using the new concept of finitely self predictable set strong properties for w k analytic boundaries are established as well as a sup lim sup theorem for baire maps

borel measure extensions of measures defined on sub σ algebras we develop a new approach to the measure extension problem based on nonstandard analysis the class of thick topological spaces which includes all locally compact and all k analytic spaces is introduced in this paper and measure extension results of the following type are obtained if x t is a regular lindelof and thick space a t is a algebra and is a finite measure on a inner regular with respect to the closed sets in a then has a radon extension the methods developed here allow us to improve on previously known extension results

extensions of tight set functions with applications in topological measure theory let x1 j f2 be lattices of subsets of a set x with x1 c 2 the main result of this paper states that every semifinite tight set function on s1 can be extended to a semifinite tight set function on 2 furthermore conditions assuring that such an extension is uniquely determined or a smooth at f are given since a semifinite tight set function defined on a lattice s and being a smooth at f can be identified with a semifinite sregular content measure on the algebra generated by x the general results are applied to various extension problems in abstract and topological measure theory introduction the construction of a finitely or countably additive measure usually proceeds in such a way that one first defines a set function on a small family of sets and thereafter applies a suitable measure extension procedure one condition which guarantees a measure extension is the concept of tightness that has been studied by several authors see 18 32 33 17 2 for the case of 0 x valued set functions to which this paper is restricted the central subject of this paper is the extension of a given tight set function defined on a lattice y1 of subsets of a set x to a tight set function on a lattice 2 2 1 the main result 2 5 states that every semifinite tight set function a on y1 can be extended to a y2 x regular tight set function p on 2 x where 2 denotes the family of all y2 sets which are subsets of y1 sets of finite a measure in addition we give conditions assuring that such an extension p is uniquely determined 2 7 or a smooth at 2 8 these results are applied to various extension problems in topological measure theory among others we obtain a generalization of the main extension theorem of bachman and sultan 3 5 see 3 2 some further applications deal with the existence of a preimage measure the extension of the product of two a finite baire borel measures to a baire borel measure on the product space and the extension of baire to borel measures see 3 8 3 10 3 13 3 14 in particular we obtain a variant of a measure extension theorem due to marik 25 this result enables us to introduce a class of topological spaces x including in particular all normal metacompact ones for which the following two statements are valid x is measure compact iff each closed discrete subset of x has cardinal of measure zero x is realcompact iff each closed discrete subset of x has nonmeasurable cardinal this result see 3 16 leads to a common generalization of results in 16 26 13 7 8 received by the editors february 17 1983 and in revised form september 8 1983 1980 mathematics subject classification primary 28a10 28a12 28c15 secondary 54a25 1984 american mathematical society 0025 5726 84 1 00 25 per page

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

asplund spaces characterized by rich families and separable reduction of frechet subdifferentiability abstract asplund property of a banach space x is characterized by the existence of a rich family in the product x x consisting of some carefully chosen separable subspaces this structural result is then used to add a lot of precision and simplicity to the known separable reductions of frechet subdifferentials

compactness in spaces of inner regular measures and a general portmanteau lemma abstract this paper may be understood as a continuation of topsoe s seminal paper f topsoe compactness in spaces of measures studia math 36 1970 195 212 to characterize within an abstract setting compact subsets of finite inner regular measures w r t the weak topology the new aspect is that neither assumptions on compactness of the inner approximating lattices nor nonsequential continuity properties for the measures will be imposed as a providing step also a generalization of the classical portmanteau lemma will be established the obtained characterizations of compact subsets w r t the weak topology encompass several known ones from literature the investigations rely basically on the inner extension theory for measures which has been systemized recently by konig h konig measure and integration springer berlin 1997 h konig on the inner daniell stone and riesz representation theorems doc mat 5 2000 301 315 h konig measure and integration an attempt at unified systematization rend istit mat univ trieste 34 2002 155 214

on the integrability of bianchi cosmological models in this paper we investigate the problem of the integrability of bianchi class a cosmological models this class of systems is reduced to the form of hamiltonian systems with exponential potential forms the dynamics of bianchi class a models is investigated through the euler lagrange equations and geodesic equations in the jacobi metric on this basis we have come to some general conclusions concerning the evolution of the volume of 3 space of constant time the general form of this function has been found it can serve as a controller during numerical calculations of the dynamics of cosmological models the integrability of cosmological models is also discussed from the points of view of different integrability criteria we show that the dimension of the phase space of bianchi class a hamiltonian systems can be reduced by two we prove that the vector field of the reduced system is polynomial and that it does not admit any analytic or even formal first integral

optimal quasi optimal and superlinear band toeplitz preconditioners for asymptotically ill conditioned positive definite toeplitz systems in this paper we are concerned with the solution of n x n hermitian toeplitz systems with nonnegative generating functions f the preconditioned conjugate gradient pcg method with the well known circulant preconditioners fails in the case where f has zeros in this paper we consider as preconditioners band toeplitz matrices generated by trigonometric polynomials g of fixed degree l we use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n for each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy we obtain different estimates of l for which the total cost of the proposed pcg methods is optimal and the related rates of convergence are superlinear finally for the most economical strategy we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems

universal loeb measurability of sets and of the standard part map with applications it is shown in this paper that for k saturated models many important external sets of nonstandard analysis such as monadic sets or the set of ail near standard points or all pre near standard points or all compact points are universally loeb measurable i e loeb measurable with respect to every internal content we furthermore obtain universal loeb measurability of the standard part map for topological spaces which are not covered by previous results in this dlrectlon moreover the standard part map can be used as a measure preserving transformation for all smooth measures and not only for radon measures as known up to now applications of our results lead to simple new proofs for theorems of classical measure theory we obtain e g the extension of smooth baire measures to t smooth borel measures the decomposition theorems for smooth baire measures and smooth borel measures and kakutanis theorem for product measures

curvature of gravitationally bound mechanical systems in the present work mathematical aspects of determining the local instability parameters are focused on by using invariant characteristics of the internal riemannian geometry with the jacobi metric in principle for hamiltonian dynamical systems with the natural lagrangian first it is shown that the ricci scalar indeed measures the sectional curvature averaged upon all two directions second necessary and sufficient criteria for non negativity and of nonpositivity of the sectional curvature for any system with the natural lagrangian are given third analytical formulas allowing us to compute the separation rate of nearby trajectories are given fourth it is shown that for any collisionless problem of n gravitationally bounded bodies the sectional curvature in every direction is negative if n tends to infinity

four short stories about toeplitz matrix calculations the stories told in this paper are dealing with the solution of finite infinite and biinfinite toeplitz type systems a crucial role plays the off diagonal decay behavior of toeplitz matrices and their inverses classical results of gelfand et al on commutative banach algebras yield a general characterization of this decay behavior we then derive estimates for the approximate solution of bi infinite toeplitz systems by the finite section method showing that the approximation rate depends only on the decay of the entries of the toeplitz matrix and its condition number furthermore we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems finally some quantitative results on the construction of preconditioners via circulant embedding are derived which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems

efficient minimization methods of mixed l 2 l 1 and l 1 l 1 norms for image restoration image restoration problems are often solved by finding the minimizer of a suitable objective function usually this function consists of a data fitting term and a regularization term for the least squares solution both the data fitting and the regularization terms are in the ell 2 norm in this paper we consider the least absolute deviation lad solution and the least mixed norm lmn solution for the lad solution both the data fitting and the regularization terms are in the ell 1 norm for the lmn solution the regularization term is in the ell 1 norm but the data fitting term is in the ell 2 norm since images often have nonnegative intensity values the proposed algorithms provide the option of taking into account the nonnegativity constraint the lmn and lad solutions are formulated as the solution to a linear or quadratic programming problem which is solved by interior point methods at each iteration of the interior point method a structured linear system must be solved the preconditioned conjugate gradient method with factorized sparse inverse preconditioners is employed to solve such structured inner systems experimental results are used to demonstrate the effectiveness of our approach we also show the quality of the restored images using the minimization of mixed ell 2 ell 1 and ell1 ell 1 norms is better than that using only the ell 2 norm

desingularization of jacobi metrics and chaos in general relativity it has been proposed by wheeler and dewitt to look at the evolution of three metrics as a geodesic flow on the superspace since then a lot of attention has been paid towards better understanding the geometric structure of the superspace in particular it has been appreciated that the minisuperspace can in a natural way be equipped with the jacobi metric however the jacobi metric is degenerate on certain codimension one hypersurfaces boundary sets leading to severe difficulties the dynamics of minisuperspace models is a special case of dynamics of simple indefinite mechanical systems it is proved that trajectories of an indefinite mechanical system m g v with the natural lagrangian are pregeodesics with respect to the jacobi metric ge 2 e v g where m is the configuration space g the metric defined by the kinetic energy form v a potential function and e the total energy of the system in this paper we also propose to use eisenhart s principle as an alternative geometrical construction on minis

on structure of upper semicontinuity abstract the refinement of a choquet theorem on strong upper semi continuity and its relation to the vainstein lemma are dealt with here relevance of subcontinuity is discussed consequently an improvement of a characterization theorem of dolecki and rolewicz is achieved

first class selectors for upper semi continuous multifunctions by a multtfunction x y we mean a function defined on the space x and whose values are subsets of the space y a point x0 in x is a point of upper semi continuity for provided x0 belongs to the interior of x q x c g whenever g is an open set in y containing x0 q is said to be upper semi continuous if it is upper semi continuous at ach point of x in this case x g x c g is open in x whenever g is open in y and x x nf is closed in x w h enever f is closed in y interchanging the roles of open and closed sets in this definition we get the concept of a lower semi continuous multifunction a selector for is a single valued function cp x y with q x e cd x for each x in x in this paper we are primarily concerned with the problem of determin ing the best possible selector cp for an upper semi continuous multifunction defined on a metric space x and taking only non empty values in a second metric space y very elementary examples show that such multi functions will in general fail to have continuous selectors e g take 4p x 0 foro x o 1 and x 1 for x 1 hence an optimal selector in our case would be one that is the pointwise limit of a sequence of continuous functions or more generally a first class bore1 function i e cp g is an 9 set in x for each open set g c y here we will prove

time metric equivalence and dimension change under time reparameterizations we study the behavior of dynamical systems under time reparameterizations which is important not only to characterize chaos in relativistic systems but also to probe the invariance of dynamical quantities we first show that time transformations are locally equivalent to metric transformations a result that leads to a transformation rule for all lyapunov exponents on arbitrary riemannian phase spaces we then show that time transformations preserve the spectrum of generalized dimensions d q except for the information dimension d1 which interestingly transforms in a nontrivial way despite previous assertions of invariance the discontinuous behavior at q 1 can be used to constrain and extend the formulation of the kaplan yorke conjecture

an approximate inverse preconditioner for toeplitz systems with multiple right hand sides abstract we are interested in preconditioning techniques for symmetric positive definite but ill conditioned toeplitz systems with multiple right hand sides tx b here the right hand sides are available simultaneously our main idea is to use an approximate inverse of t as a preconditioner for t the approximate inverse of t is generated by the inverse formula given by lv and huang x g lv t z huang a note on inversion of toeplitz matrices appl math lett 20 2007 1189 1193 the construction of the preconditioner is of low computational cost requires only the entries of t and does not require explicit knowledge of the generating function f of t we show that if given a proper parameter then the approximate inverse preconditioner approximates t 1 accurately and thus the spectra of these preconditioned matrices are clustered around 1 it follows that the conjugate gradient method converges very quickly when applied to solve the preconditioned systems numerical results are given to demonstrate the effectiveness of our preconditioner

on a theorem of choquet and dolecki abstract the results of this paper clarify and extend slightly the previous work of dolecki and lechicki c r acad sci paris 293 1981 219 221 j math anal appl 88 1982 547 584 and hansell jayne rogers and the author math z 189 1985 297 318 let x y be hausdorff spaces and f x y an upper semicontinuous set valued map a subset k of f x is said to be a peak of f at x if for every open set v containing k there exists a neighbourhood u of x such that f u f x t v criteria choquet dolecki theorems are given in order that f has the smallest possible peak it turns out that in unexpectedly general situations an upper semicontinuous map f has for every x in x a peak which is the smallest possible at x and moreover compact

superintegrability on curved spaces orbits and momentum hodographs revisiting a classical result by hamilton the equation of the orbits in the configuration space and of the hodographs in the momentum plane for the curved kepler and harmonic oscillator systems living in a configuration space of any constant curvature and either signature type are derived by purely algebraic means this result extends to the curved kepler or harmonic oscillator for the classical hamilton derivation of the orbits of the euclidean kepler problem through its hodographs in both cases the fundamental property allowing these derivations to work is the superintegrability of the curved kepler and harmonic oscillator no matter whether the constant curvature of the configuration space is zero or not or whether the configuration space metric is riemannian or lorentzian in the curved case the basic result does not refer to the velocity hodograph but to the momentum hodograph both coincide in a euclidean configuration space but only the latter is unambiguously defined in all curved spaces

lyapunov exponents from geodesic spread in configuration space the exact form of the jacobi char21 levi civita jlc equation for geodesic spread is here explicitly worked out at arbitrary dimension for the configuration space manifold m e q ensuremath in r n v q le of a standard hamiltonian system equipped with the jacobi or kinetic energy metric g j as the hamiltonian flow corresponds to a geodesic flow on m e g j the jlc equation can be used to study the degree of instability of the hamiltonian flow it is found that the solutions of the jlc equation are closely resembling the solutions of the standard tangent dynamics equation which is used to compute lyapunov exponents therefore the instability exponents obtained through the jlc equation are in perfect quantitative agreement with usual lyapunov exponents this work completes a previous investigation that was limited only to two degrees of freedom systems

relativistic invariance of lyapunov exponents in bounded and unbounded systems the study of chaos in relativistic systems has been hampered by the observer dependence of lyapunov exponents les and of conditions such as orbit boundedness invoked in the interpretation of les as indicators of chaos here we establish a general framework that overcomes both difficulties and apply the resulting approach to address three fundamental questions how les transform under lorentz and rindler transformations and under transformations to uniformly rotating frames the answers to the first and third questions show that inertial and uniformly rotating observers agree on a characterization of chaos based on les the second question on the other hand is an ill posed problem due to the event horizons inherent to uniformly accelerated observers

homoclinic heteroclinic and periodic orbits for a class of indefinite hamiltonian systems si e est l espace euclidien de produit interieur 1 on etudie les orbites d une classe de systemes hamiltoniens dans e e quand la fonctionnelle d energie est donnee par h q p 1 2 sp p v q q p e e et la forme quadratique sp p est indefinie mais non degeneree