critical exponents of self avoiding walks on a family of truncated n simplex lattices the authors have studied self avoiding walks saws on a family of truncated n simplex lattices which provide a family of fractals in which the fractal dimension d can be varied to a wide range while the spectral dimension d is held almost fixed by means of exact renormalisation group transformations they have calculated the critical exponents nu alpha and gamma and the connectivity constant mu of the saws for n 5 and 6 they propose an approximate theory for calculating the critical exponents of the saws on fractals which is expected to be accurate at large values of n they show that the theory gives results which are in good agreement with exact values even for small values of n and leads to simple closed relations for the exponents

subword conditions and subword histories this paper introduces the notion of a subword condition and investigates languages defined by them the special case where the language reduces to one word concerns the inference of a sequence from its subsequences we obtain various characterization and decidability results for languages defined by subword conditions the results contribute to the theory of parikh matrices and arithmetizing the study of words an important notion from early automata theory that of a quasi uniform event plays a central role in our characterization

correction to scaling exponents and amplitudes for the correlation length of linear polymers in two dimensions we consider the scaling behaviour of the radius of gyration for a system of dilute linear polymers we focus on pn the mean square end to end distance of an n step self avoiding walk for which an additional two terms were recently calculated for the close packed triangular lattice combining several extrapolation methods we find that a consistent description of the scaling behaviour exists if and only if the correction to scaling exponent a is roughly half as large as commonly believed we conclude that all data ale consistent with the equation pn an2 1 b na c n where y a a 1 j2 abso 21 and c lo

free energy of magnetic model systems at the fractal to euclidian crossover abstract we prove that the free energy per spin of the nearest neighbour classical n vector spin model situated on the sierpinski gasket sg family of fractals tends to the free energy per spin of the same model situated on the standard triangular lattice when the linear size b of the sg fractal generator tends to infinity this finding can be regarded as the first step in the analytical investigation of the spin system thermodynamics at the fractal to euclidean crossover

independence of certain quantities indicating subword occurrences when words are characterized in terms of numerical quantities awkward considerations due to the noncommutativity of words are avoided the numerical quantity investigated in this paper is w u the number of occurrences of a word u as a scattered subword of a word w parikh matrices recently introduced have these quantities as their entries according to the main result in this paper no entry in a parikh matrix no matter how high the dimension can be computed in terms of the other entries consequences concerning various inference problems between numbers w u themselves as well as of the word w from these numbers are obtained

connections between subwords and certain matrix mappings parikh matrices recently introduced have turned out to be a powerful tool in the arithmetizing of the theory of words in particular many inequalities between scattered subword occurrences have been obtained as consequences of the properties of the matrices this paper continues the investigation of parikh matrices and subword occurrences in particular we study certain inequalities as well as information about subword occurrences sufficient to determine the whole word uniquely some algebraic considerations facts about forbidden subwords as well as some open problems are also included

symplectic integration of constrained hamiltonian systems by composition methods recent work reported in the literature suggests that for the long term integration of hamiltonian dynamical systems one should use methods that preserve the symplectic structure of the flow in this paper the symplecticity of numerical integrators is investigated for constrained hamiltonian systems with holonomic constraints the following two results will be derived i it is shown that any first or second order symplectic integrator for unconstrained problems can be generalized to constrained systems such that the resulting scheme is symplectic and preserves the constraints based on this higher order methods can be derived by the same composition methods used for unconstrained problems ii leimkuhler and reich math comp 63 1994 pp 589 605 derived symplectic integrators based on dirac s reformulation of the constrained problem as an unconstrained hamiltonian system however although the unconstrained reformulation can be handled by direct application of any symplectic implicit runge kutta m

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

on a conjecture about parikh matrices based on salomaa s characterization of m equivalence atanasiu conjectured that a certain natural generalization of me equivalence solves the injectivity problem of parikh matrices for the ternary alphabet this paper refutes his conjecture but continues to study the interesting proposed thue system characterization of certain irreducible elementary transformations under this system is obtained furthermore these transformations are further scrutinized in terms of their replaceability by simpler ones

conservative integration of rigid body motion by quaternion parameters with implicit constraints summary an angular momentum and energy conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four component conjugate momentum vector via hamilton s equations the introduction of an extended mass matrix leads to a symmetric set of eight state space equations of motion the extra inertial parameter serves as a multiplier on the kinematic constraint and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments external loads enter these equations via the set of momentum equations initially the normalization of the quaternion array is introduced via a lagrange multiplier however this lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential and elimination of the lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient an algorithm is developed by forming a finite increment of the hamiltonian this procedure identifies the proper selection of increments and mean values and leads to an algorithm with conservation of momentum and energy implementation conservation properties and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top copyright 2012 john wiley sons ltd

parikh matrices and amiable words using the fact that the parikh matrix mapping is not an injective mapping the paper investigates some properties of the set of words with the same parikh matrix these words are called amiable the presented results extend the results obtained in a atanasiu binary amiable words int j found comput sci 18 2 2007 387 400 for the binary case in particular it is shown that all the words having the same parikh matrix can be obtained one from another by applying only two types of transformations moreover the mirrors of two amiable words are also amiable thus forming a symmetrical class of words

simple and efficient integration of rigid rotations suitable for constraint solvers simple and efficient way of integrating rigid rotations is presented the algorithm is stable second order accurate and in its explicit version involves evaluation of only two exponential maps per time step the semi explicit version of the proposed scheme improves upon the long term stability while it retains the explicitness in the force evaluation the algebraic structure of both schemes makes them suitable forthe analysis of constrained multi body systems the explicit algorithm is specifically aimed at the analysis involving small incremental rotations where its modest computational cost becomes the major advantage the semi explicit scheme naturally broadens the scope of possible applications copyright 2009 john wiley sons ltd

subword histories and associated matrices the basic numerical quantity investigated in this paper is w u the number of occurrences of a word u as a scattered subword of a word w arithmetical combinations of such quantities yield a so called subword history we investigate the information content of subword histories reducing subword histories to linear ones as well as the recently introduced parikh matrices will be important tools simple polynomial formulas for computing the value of a subword history for arbitrary powers of a word are obtained

conformal invariance and self avoiding walks in restricted geometries the prediction of conformal invariance for the statistics of self avoiding random walks restricted to both semi infinite and wedge shaped geometries are tested by extrapolating exact enumerations close agreement is found both for the angular distribution of the end to end vector and for the dependence of the critical exponent gamma 2 on the opening angle of the wedge

explicit momentum conserving integrator for dynamics of rigid bodies approximating the midpoint lie algorithm we reformulate the midpoint lie algorithm which is implicit in the torque calculation to achieve explicitness in the torque evaluation this is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step thus we obtain two related variants both of which are explicit in the torque calculation but only first order in the time step both variants are momentum conserving and both are symplectic consequently drawing on the properties of the composition of maps we introduce another algorithm that combines the two variants in a single time step the resulting algorithm is explicit momentum conserving symplectic and second order its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators

renormalisation on sierpinski type fractals presents a family of deterministic fractals which generalise the d dimensional sierpinski gaskets and establishes their order of ramification and their fractal and spectral dimensions random walks on these fractals are renormalisable and lead to rational not necessarily polynomial mapping

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

critical exponents of the self avoiding walks on a family of finitely ramified fractals the authors have studied the self avoiding walks saw on a family of finitely ramified fractals the first member b 2 of the family is the two dimensional sierpinski gasket while the last member b infinity appears to be a wedge of the homogeneous triangular lattice by means of the exact renormalisation group transformations they have calculated the critical exponents alpha nu and gamma and the connectivity constant mu of saw on each member of a sequence 2 or b or 8 of the studied fractal family the obtained exact results are compared with the recent phenomenological proposals and with the results believed to be exact in the case of a homogeneous two dimensional lattice

words and special factors in this paper we consider sets of factors of a given finite word over a finite alphabet which permit us to reconstruct the entire word this analysis is based on the notion of special factor a factor u of a finite word w is called right resp left special if there exist two distinct letters x and y such that ux uy resp xu yu are factors of w a factor is bispecial if it is right and left special a proper box of w is any factor of w of the kind asb with a b letters and s a bispecial factor of w the initial resp terminal box of w is the shortest prefix resp suffix of w which is an unrepeated factor a box is called maximal if it is not a proper factor of another box the main result of the paper is the following theorem maximal box theorem any finite word w is uniquely determined by the initial box the terminal box and the set of maximal boxes a consequence is that a finite word w is uniquely determined by the knowledge of its factors up to the length n max rw kw 1 where kw is the length of the terminal box and rw is the minimal natural number for which there is no right special factor of length rw some structural properties of boxes are studied another important combinatorial notion is that of superbox a superbox is any factor of w of the kind asb with a b letters and such that s is a repeated factor whereas as and sb are unrepeated factors a theorem for superboxes similar to the maximal box theorem is proved some algorithms allowing us to construct boxes and superboxes and conversely to reconstruct the word are given in this combinatorial frame we give an upper and a lower bound to the number of states of a minimal deterministic automaton recognizing the set of the factors of w these bounds are sharper than the known bounds

connective constant of saws on the sierpinski gasket family using a graph counting technique suitable for regular fractals an exact evaluation of the total number of embeddings of self avoiding walks on the generalized sierpinski gasket is obtained numerical estimates for the connective constants are quoted for the first time where b is the generation parameter of the gaskets it is shown that the number of distinct n step saws per site converges to the triangular lattice values when our analysis indicates that converges to the euclidean value in the same limit and an asymptotic expression is given

extending parikh matrices we introduce the notion of parikh matrix induced by a word a natural extension to the notion of parikh matrix and prove a set of properties for this kind of matrices we also study the relation between these two notions we show that combining properties from both we obtain a more powerful tool for proving algebraic properties of words

semi implicit reversible algorithms for rigid body rotational dynamics this paper presents two semi implicit algorithms based on splitting methodology for rigid body rotational dynamics the first algorithm is a variation of partitioned runge kutta prk methodology that can be formulated as a splitting method the second algorithm is akin to a multiple time stepping scheme and is based on modified crouch grossman mcg methodology which can also be expressed as a splitting algorithm these algorithms are second order accurate and time reversible however they are not poisson integrators i e non symplectic these algorithms conserve some of the first integrals of motion but some others are not conserved however the fluctuations in these invariants are bounded over exponentially long time intervals these algorithms exhibit excellent long term behavior because of their reversibility property and their approximate poisson structure preserving property the numerical results indicate that the proposed algorithms exhibit superior performance compared to some of the currently well known algorithms such as the simo wong algorithm newmark algorithm discrete moser veselov algorithm lewis simo algorithm and the liemid ea algorithm

criteria for the matrix equivalence of words this paper investigates the criteria for deciding whether two words are matrix equivalent certain upper diagonal matrices generally referred to as the parikh matrices have been widely investigated because of their usefulness in computing the numbers of subword occurrences and thereby characterizing words by numbers however apart from the binary alphabet not much is known about the properties of the matrix equivalent words that is words possessing the same matrix the paper investigates both the general criteria as well as the criteria valid in natural special cases an exhaustive solution is obtained for ternary alphabets

numerical integration of lie poisson systems while preserving coadjoint orbits and energy in this paper we apply geometric integrators of the rkmk type to the problem of integrating lie poisson systems numerically by using the coadjoint action of the lie group g on the dual lie algebra mbox normalsize mathfrak g to advance the numerical flow we devise methods of arbitrary order that automatically stay on the coadjoint orbits first integrals known as casimirs are retained to machine accuracy by the numerical algorithm within the proposed class of methods we find integrators that also conserve the energy these schemes are implicit and of second order nonlinear iteration in the lie algebra and linear error growth of the global error are discussed numerical experiments with the rigid body and a finite dimensional truncation of the euler equations for a two dimensional 2d incompressible fluid are used to illustrate the properties of the algorithm

critical behaviour at an edge for the saw and ising model the authors have studied the behaviour of two and three dimensional self avoiding walks confined to a wedge of wedge angle alpha series have been obtained and analysed for the angular dependent critical exponents characterising various edge susceptibilities in terms of a general scaling form for the edge free energy fe approximately t d 2 nu psi e ht y0 nu h1t y1 nu h2t y2 nu they find for the two dimensional case the following scaling indices y0 91 48 y1 3 8 y2 alpha 5 pi 8 alpha they argue that these results are exact from which follow all exponents for the bulk surface and edge problem in three dimensions the authors obtain y0 approximately 2 488 y1 0 65 or 0 02 y2 alpha alpha b pi alpha where a 0 51 or 0 04 b 0 847 or 0 017 which for y2 is precisely of the functional form given by mean field theory y2 alpha 1 2 pi 2 they argue that a 1 2 for all three dimensional o n models this simple angular dependence of y2 is different from that suggested by cardy s one loop epsilon expansion 1983 for the square lattice they have also studied the case in which the wedge is rotated through an angle of pi 4 and find that the various exponents are unchanged for the three dimensional ising model in a wedge analogy with the saw results plus mean field results in conjunction with rg and series work yield y0 approximately 2 485 y1 0 71 or 0 02 and y2 a b pi alpha with a 1 2 and b 0 79 or 0 02

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

statistical mechanics of polymer chains grafted to adsorbing boundaries of fractal lattices embedded in three dimensional space abstract we study the adsorption problem of linear polymers immersed in a good solvent when the container of the polymer solvent system is taken to be a member of the sierpinski gasket sg family of fractals embedded in the three dimensional euclidean space members of the sg family are enumerated by an integer b 2 b and it is assumed that one side of each sg fractal is impenetrable adsorbing boundary we calculate the surface critical exponents 11 1 and s which within the self avoiding walk model saw of polymer chain are associated with the numbers of all possible saws with both one and no ends grafted to the adsorbing surface adsorbing boundary respectively by applying the exact renormalization group method for 2 b 4 we have obtained specific values for these exponents for various types of polymer conformations to extend the obtained sequences of exact values for surface critical exponents we have applied the monte carlo renormalization group method for fractals with 2 b 40 the obtained results show that all studied exponents are monotonically increasing functions of the parameter b for all possible polymer states we discuss mutual relations between the studied critical exponents and compare their values with those found for other types of lattices in order to attain a unified picture of the attacked problem

criticality of two and three spin ising model in an external field on a fractal family abstract we study the ising model with pair and alternate triplet interactions subjected to an external magnetic field on a family of infinitely ramified fractal lattices with a triangular topology the three dimensional phase diagram and correlation length critical exponents are calculated within an exact real space renormalization group framework the zero field results for the ferromagnetic model show that although the pure triplet case and the pure nearest neighbor pair interaction model are in different universality classes there is no crossover phenomenon since the system becomes paramagnetic in the mixed case in the pure nearest neighbor antiferromagnetic model the appearance of an unusual berker and kadanoff s phase type with a power law decay of correlations when the fractal dimension is sufficiently high is destroyed by the application of a magnetic field or a triplet interaction field

symplectic integration of constrained hamiltonian systems a hamiltonian system in potential form h q p ptmp 2 f q subject to smooth constraints on q can be viewed as a hamiltonian system on a manifold but numerical computations must be performed in rn in this paper methods which reduce hamiltonian differential algebraic equations to odes in euclidean space are examined the authors study the construction of canonical parametrizations or local charts as well as methods based on the construction of ode systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold in each case a hamiltonian system of ordinary differential equations is produced the stability of the constraint invariants and the behavior of the original hamiltonian along solutions are investigated both numerically and analytically

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

transfer matrix studies of branching fractals abstract we introduce the method of transfer matrices to study physical models defined on branching fractal lattices in this context we derive matricial recurrence relations connecting the matrices associated with successive fractal generations these relations depend only on the geometry of the fractal and may be used for any model explicit results are obtained for ferromagnetic and antiferromagnetic ising models on branching koch curves and sierpinski gasket

a numerical test of long time stability for rigid body integrators summary in the context of hamiltonian odes a necessary condition for an integrator to be symplectic or conjugate symplectic is that it nearly preserves the exact hamiltonian this paper introduces a numerical test of this necessity for rigid body methods it turns out that several rigid body integrators proposed in literature fail this test hence these integrators should be used with caution for long time simulation copyright 2011 john wiley sons ltd

adsorption of a flexible self avoiding polymer chain exact results on fractal lattices the large scale behavior of surface interacting self avoiding polymer chains placed on finitely ramified fractal lattices is studied using exact recursion relations it is shown how to obtain surface susceptibility critical indices and how to modify a scaling relation for these indices in the case of fractal lattices we present the exact results for critical exponents at the point of adsorption transition for polymer chains situated on a class of sierpinski gasket type fractals we provide numerical evidence for a critical behavior of the type found recently in the case of bulk self avoiding random walks at the fractal to euclidean crossover

core words and parikh matrices parikh matrices have been widely investigated due to their applicability in arithmetizing words by numbers this paper introduces the core of a binary word which captures the essential part of a word from the perspective of its parikh matrix additionally the stronger notion of core m unambiguity is introduced and the characterization of core m unambiguous binary words is obtained finally a generalization of the core of a binary word and some of its interesting properties are investigated

on strongly m unambiguous prints and şerbǎnuţǎ s conjecture for parikh matrices abstract in the combinatorial study of words the parikh matrix mapping was introduced by mateescu et al in 2001 as a natural expansion of the classical parikh mapping solving the general injectivity problem of parikh matrices remains as one of the most sought after triumph among researchers in this area of study in this paper we tackle this problem by extending erb nu s work regarding prints and m unambiguity to the context of strong m equivalence consequently we obtain results on the finiteness of strongly m unambiguous prints for any finite alphabet finally a related conjecture by erb nu is conclusively addressed

end to end distance of linear polymers in two dimensions a reassessment recent exact enumeration studies of the two dimensional self avoiding walk have produced evidence in support of non analytic scaling corrections newly extended series for the triangular and square lattices to n 19 and 25 respectively are examined in the light of recent monte carlo results suggesting the absence of corrections of this type using a modified approach to series analysis it is concluded that the end to end distance can best be understood in terms of analytic corrections alone in view of the lack of theoretical support there is little reason to introduce additional kinds of correction in attempting to explain the data

asymptotic form of the spectral dimension at the fractal to lattice crossover the authors study the spectral dimension of the sierpinski gasket family of fractals each member of the family is labelled by an integer b 2 or b infinity so that when b to infinity both the fractal and spectral dimension approach the euclidean value 2 the asymptotic law for the spectral dimension was recently suggested to be ds approximately 2 b 1nb beta with b and beta being some constants here the authors demonstrate that this form should be replaced by the asymptotic law ds approximately 2 1n 1nb constant 1nb their analysis is based on the exact calculation of the electric resistances rb for all members of the family up to b 650

self avoiding walks attached to triangular and face centred cubic lattice surfaces extended exact enumeration data the analysis of exact enumeration data of self avoiding walks attached to the surfaces of triangular and face centred cubic lattices is consistent with gamma 1 0 718 or 0 008 136 delta pc where delta pc pc 0 0995 in three dimensions and 0 9530 or gamma 1 or 0 9548 in two dimensions where pc for the triangular lattice is assumed to lie in the range 0 240 915 or pc or 0 240 93

explicit newmark verlet algorithm for time integration of the rotational dynamics of rigid bodies we reformulate the traditional velocity based vector space newmark algorithm for the rotational dynamics of rigid bodies that is for the setting of the so 3 lie group we show that the most naive re write of the vector space algorithm possesses the properties of symplecticity and almost momentum conservation thus we obtain an explicit algorithm for rigid body dynamics that matches or exceeds performance of existing algorithms but which curiously does not seem to have been considered in the open literature so far copyright 2005 john wiley sons ltd

flory approximant for self avoiding walks on fractals a flory approximant for the exponent describing the end to end distance of a self avoiding walk saw on fractals is derived the approximant involves the fractal dimensionalities of the backbone and of the minimal path and the exponent describing the resistance of the fractal the approximant yields values which are very close to those available from exact and numerical calculations

true self avoiding walks on a percolation cluster the authors investigate by monte carlo simulations the critical behaviour of true self avoiding walks tsaws on a percolation cluster performed very close to the percolation threshold specifically they generate tsaws on a site percolated incipient infinite cluster for various values of the self avoidance parameter g 0 they found that such walks exhibit critical behaviour different from that of ordinary self avoiding walks and also from that of random walks of no constraint the flory exponent obtained was v 0432 or 0 005 for all g 0 which agrees well with the flory type formula suggested by rammal 1984

subword occurrences weighted automata and iterated morphisms especially the fibonacci morphism the paper investigates the number of subword occurrences in a word in particular in a word obtained by iterating a morphism associated graphs or automata are used as a tool special attention is paid to the fibonacci morphism explicit formulas in terms of n are obtained for the number of occurrences of subwords of length at most 2 in the nth iteration of the fibonacci morphism

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

mechanical integrators derived from a discrete variational principle many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion in this paper we present a procedure to construct time stepping algorithms that approximate the flow of continuous odes for mechanical systems by discretizing hamilton s principle rather than the equations of motion the discrete equations share similarities to the continuous equations by preserving invariants including the symplectic form and the momentum map we first present a formulation of discrete mechanics along with a discrete variational principle we then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of noether s theorem we then use the discrete mechanics formulation to develop a procedure for constructing symplectic momentum mechanical integrators for lagrangian systems with holonomic constraints we apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators

self avoiding walks on percolation clusters the author studies self avoiding walks saws on percolation clusters a scaling function representation for r the mean end to end distance is proposed which describes a crossover from ordinary saws to saws on fractals he distinguishes between saws on a single cluster for which r approximately nx where x nu 1 and saws on all clusters for which r approximately nz where z nu 2 and n is the number of monomers in the walk he estimates nu 1 d 2 approximately 1 285 and the correction to scaling exponent omega d 2 approximately 1 3 and nu 1 d 3 approximately 1 38 two plausible generalisations of the flory 1953 approximation for nu 1 are investigated and it is argued that none of them provides a satisfactory approximation for nu 1 at all dimensions

geometric integration on spheres and some interesting applications geometric integration theory can be employed when numerically solving odes or pdes with constraints in this paper we present several one step algorithms of various orders for odes on a collection of spheres to demonstrate the versatility of these algorithms we present representative calculations for reduced free rigid body motion a conservative ode and a discretization of micromagnetics a dissipative pde we emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions

hamiltonian walks on sierpinski and n simplex fractals we study hamiltonian walks hws on sierpinski and n simplex fractals via numerical analysis of exact recursion relations for the number of hws we calculate the connectivity constant and find the asymptotic behaviour of the number of hws depending on whether or not the polymer collapse transition is possible on a studied lattice different scaling relations for the number of hws are obtained these relations are in general different from the well known form characteristic of homogeneous lattices which has thus far been assumed to also hold for fractal lattices

order of weak m relation and parikh matrices abstract the classical inference problem on words studies whether a word can be uniquely inferred from some given binomial coefficients the entries of the parikh matrix are certain canonical binomial coefficients depending on the ordered alphabet the wide open injectivity problem asks when two words are m equivalent that is sharing the same parikh matrix two words are weakly m related if and only if they are m equivalent with respect to some ordering on the alphabet in this work we introduce a function that measures how hard it is for two words to become weakly m related we characterize this function and solve various connected combinatorial questions

diffusion on fractal lattices and the fractal einstein relation finitely ramified fractal lattices show anomalous diffusion with r2 varies as t2h there is a hierarchy of transit times which are shown by monte carlo simulation to satisfy tau n varies as alpha n for large n where alpha b1h and b is the base of the lattice the lattice resistivity scales with size as rho varies as rn where r is characteristic of the lattice exact renormalisation group methods yield alpha and r analytically and verify the einstein relation for fractals alpha rn

momentum conserving symplectic integrators abstract in this paper we show that symplectic partitioned runge kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of hamiltonian differential equations by extended point transformation we also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of lie poisson systems on certain submanifolds of the general matrix group gl n

critical exponent γ for a class of directed walks at the fractal to euclidean crossover abstract an exact evaluation of the critical exponent associated with the average number of n step walks of the piecewise directed random walks is presented in the case of the complete family of the sierpinski gasket type of fractals

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells

statistics of equally weighted random paths on a class of self similar structures we study the statistics of equally weighted random walk paths on a family of sierpinski gasket lattices whose members are labelled by an integer b 2 b 2 mean path end to end distance grows more slowly than any power of its length n we provide arguments for the emergence of usual power law critical behaviour in the limit b when fractal lattices become almost compact