lattice models of branched polymers effects of geometrical constraints the authors consider uniform brushes on subsets of the square and simple cubic lattices they show that in a slab geometry in three dimensions the connective constant of a brush is identical to that of a walk in that slab in two dimensions in a slit geometry the connective constant of a brush is strictly less than that of a walk they also consider the intermediate case of a rectangular prism and show that the connective constant of a brush is strictly less than that of a walk similar results are presented for combs stars and polygons

knotting in stretched polygons the knotting in a lattice polygon model of ring polymers is examined when a stretching force is applied to the polygon by examining the incidence of cut planes in the polygon we prove a pattern theorem in the stretching regime for large applied forces this theorem can be used to examine the incidence of entanglements such as knotting and writhing in particular we prove that for arbitrarily large positive but finite values of the stretching force the probability that a stretched polygon is knotted approaches 1 as the length of the polygon increases in the case of writhing we prove that for stretched polygons of length n and for every function the probability that the absolute value of the mean writhe is less than f n approaches 0 as n for sufficiently large values of the applied stretching force

atmospheres of polygons and knotted polygons in this paper we define two statistics a and a the positive and negative atmospheres of a lattice polygon of fixed length n these statistics have the property that a a pn 2 pn where pn is the number of polygons of length n counted modulo translations we use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres using these data we directly estimate the growth constants of polygons in two and three dimensions we find that where the error bars are 67 confidence intervals and the second error bar in the three dimensional estimate of is an estimated systematic error we also compute atmospheres of polygons of fixed knot type k sampled by the bfacf algorithm we discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n

winding angle distribution for planar random walk polymer ring entangled with an obstacle and all that spitzer edwards prager frisch model revisited using a general green function formulation we re derive both i spitzer and his followers results for the winding angle distribution of the planar brownian motion and ii edwards prager frisch results on the statistical mechanics of a ring polymer entangled with a straight bar in the statistical mechanics part we consider both cases of quenched and annealed topology among new results we compute exactly the expectation value of the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory ring polymer we also consider the generalizations of the problem for the finite diameter disc like obstacle and winding within a cavity

poland scheraga models and the dna denaturation transition poland scheraga models were introduced to describe the dna denaturation transition we give a rigorous and refined discussion of a family of these models we derive possible scaling functions in the neighborhood of the phase transition point and review common examples we introduce a self avoiding poland scheraga model displaying a first order phase transition in two and three dimensions we also discuss exactly solvable directed examples this complements recent suggestions as to how the poland scheraga class might be extended in order to display a first order transition which is observed experimentally

liouville green approximations for a class of linear oscillatory difference equations of the second order abstract an asymptotic approximation theorem is proved for the solutions of linear oscillatory three term recurrence equations in a certain class this result represents a discrete analogue of the well known liouville green or wkbj theorem rigorously proved by olver for second order linear differential equations as in olver s theorem precise estimates are obtained for the error terms and the double asymptotic nature of the approximation is made clear applications to certain orthogonal polynomials are also given

knot localization in adsorbing polymer rings we study by monte carlo simulations a model of a knotted polymer ring adsorbing onto an impenetrable attractive wall the polymer is described by a self avoiding polygon on the cubic lattice we find that the adsorption transition temperature the crossover exponent ensuremath phi and the metric exponent ensuremath nu are the same as in the model where the topology of the ring is unrestricted by measuring the average length of the knotted portion of the ring we are able to show that adsorbed knots are localized this knot localization transition is triggered by the adsorption transition but is accompanied by a less sharp variation of the exponent related to the degree of localization indeed for a whole interval below the adsorption transition one can not exclude a contiuous variation with temperature of this exponent deep into the adsorbed phase we are able to verify that knot localization is strong and well described in terms of the flat knot model

loose flat knots in collapsed polymers we consider single ring polymers which are confined on a plane but maintain a fixed three dimensional knotted topology the equilibrium statistics of such systems is studied on the basis of a model on square lattice in which the configurations are represented by n step polygons with a number of self intersections restricted to the minimum compatible with the topology this allows to define the size s of the flat knots and to study their localization properties due to the presence of both excluded volume and attractive interactions the model undergoes a theta transition accurate monte carlo results show that while in the high temperature swollen regime both prime and composite knot components are localized s n n t with t 0 in the low temperature compact phase they are fully delocalized t 1 right at the theta transition weak localization prevails t 0 44 0 02 part of the results can be interpreted by taking into account a dominance of figure eight shapes for the coarse grained knotted polymer configurations and by applying the scaling theory of polymer networks of fixed topology in particular t 3 7 can be conjectured as an exact exponent characterizing the weak knot localization at the theta point

composite models of polygons a composite polygon is composed of a lattice polygon in the square lattice which contains in its interior an internal structure which may also be a lattice polygon or a lattice tree or a lattice animal or a lattice disc or a collection of these the properties of composite polygons are considered in this manuscript in particular i shall consider the growth constants and generating functions of these models as well as the statistical mechanics of interacting models of composite polygons it is shown that there is an adsorption transition of the internal structure on the containing polygon and a transition which corresponds to the inflation of the containing polygon by the internal structure

wave propagation in one dimensional waveguides with slowly varying random spatially correlated variability this paper investigates structural wave propagation in waveguides with randomly varying material and geometrical properties along the axis of propagation more specifically it is assumed that the properties vary slowly enough such that there is no or negligible backscattering due to any changes in the propagation medium this variability plays a significant role in the so called mid frequency region for dynamics and vibration but wave based methods are typically only applicable to homogeneous and uniform waveguides the wkb approximation is used to find a suitable generalization of the wave solutions for finite waveguides undergoing longitudinal and flexural motion an alternative wave formulation approximation with piecewise constant properties is also derived and included so that the internal reflections are taken into account but this requires a discretization of the waveguide moreover a fourier like series the karhunen loeve expansion is used to represent homogeneous and spatially correlated randomness and subsequently the wave propagation approach allows the statistics of the natural frequencies and the forced response to be derived experimental validation is presented using a cantilever beam whose mass per unit length is randomized by adding small discrete masses to an otherwise uniform beam it is shown how the correlation length of the random material properties affects the natural frequency statistics and comparison with the predictions using the wkb approach is given

frequency coalescence and mode localization phenomena a geometric theory physical systems the natural frequencies of which depend parametrically on certain quantities may exhibit the phenomenon of frequency coalescence when two or more natural frequencies become equal or the phenomenon of avoided crossings when two or more frequencies approach rapidly and then avoid crossing each other in the process the natural modes undergo significant variations it is shown that both phenomena can be described in a unified manner from the singularities of the mapping from the complex parameter plane into the complex frequency plane in particular the formation of a saddle point in the frequency plane and a branch point in the parameter space is the basis for explaining the properties associated with these phenomena the question of sensitivity of the system response near such singularities is subsequently examined it is shown that the development of a branch point singularity in the complex parameter plane can cause large or even infinite sensitivity of the system about this point depending on the distance of the branch point from the real axis in the parameter space it is further shown that mode localization is also associated with the appearance of a branch point near the real axis of the complex parameter plane which causes the large sensitivity that characterizes the phenomenon the theory is applied to examples of conservative and dissipative systems undergoing free vibrations

the linking number and the writhe of uniform random walks and polygons in confined spaces random walks and polygons are used to model polymers in this paper we consider the extension of the writhe self linking number and linking number to open chains we then study the average writhe self linking and linking number of random walks and polygons over the space of configurations as a function of their length we show that the mean squared linking number the mean squared writhe and the mean squared self linking number of oriented uniform random walks or polygons of length n in a convex confined space are of the form o n2 moreover for a fixed simple closed curve in a convex confined space we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form our numerical studies confirm those results they also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons of n edges each is of the form o n equilateral random walks and polygons are used to model polymers in conditions we use numerical simulations to investigate how the self linking and linking number of equilateral random walks scale with their length

knot probabilities in random diagrams we consider a natural model of random knotting choose a knot diagram at random from the finite set of diagrams with n crossings we tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type allowing us to compute exact probabilities for knots in this model as expected most diagrams with 10 and fewer crossings are unknots about 78 of the roughly 1 6 billion 10 crossing diagrams for these crossing numbers the unknot fraction is mostly explained by the prevalence of tree like diagrams which are unknots for any assignment of over under information at crossings the data shows a roughly linear relationship between the log of knot type probability and the log of the frequency rank of the knot type analogous to zipf s law for word frequency the complete tabulation and all knot frequencies are included as supplementary data

lattice stick number of knots the minimal number of straight line segments required to construct a polygonal presentation of the knot k in the cubic lattice is called the lattice stick number of the knot k denoted by it is known that if the crossing number of k satisfies and the main result of this paper is to improve this to if furthermore we will show that for and which implies that this lower bound cannot be improved except for knots with higher crossing numbers

identities in moufang sets moufang sets were introduced by jacques tits as an axiomatization of the buildings of rank one that arise from simple algebraic groups of relative rank one these fascinating objects have a simple definition and yet their structure is rich while it is rigid enough to allow for at least partial classification in this paper we obtain two identities that hold in any moufang set these identities are closely related to the axioms that define a quadratic jordan algebra we apply them in the case when the moufang set is so called special and has abelian root groups in addition we push forward the theory of special moufang sets

torsion of polygons in the torsion of polygons and self avoiding walks in the cubic lattice is a measure of the self entanglement of these objects we consider several definitions of torsion in polygons and introduce a fugacity conjugate to the torsion in our models we study the thermodynamic behaviour of these models using probabilistic methods and rigorous methods from statistical mechanics in particular we prove that at least one of our models has a non analyticity in its free energy corresponding to a transition between phases with high and low torsion

the linking number in systems with periodic boundary conditions periodic boundary conditions pbc are often used for the simulation of complex physical systems using the gauss linking number we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing pbc in the case of closed chains in pbc the periodic linking number is an integer topological invariant that depends on a finite number of components in the periodic system for open chains the periodic linking number is an infinite series that accounts for all the topological interactions in the periodic system in this paper we give a rigorous proof that the periodic linking number is defined for the infinite system i e that it converges for one two and three pbc models it gives a real number that varies continuously with the configuration and gives a global measure of the geometric complexity of the system of chains similarly for a single oriented chain we define the periodic self linking number and prove that it also is defined for open chains in addition we define the cell periodic linking and self linking numbers giving localizations of the periodic linking numbers these can be used to give good estimates of the periodic linking numbers in infinite systems we also define the local periodic linking number associated to chains in the immediate cell neighborhood of a chain in order to study local linking measures in contrast to the global linking measured by the periodic linking numbers finally we study and compare these measures when applied to a pbc model of polyethylene melts

polygons in restricted geometries subjected to infinite forces we consider self avoiding polygons in a restricted geometry namely an infinite l m tube in these polygons are subjected to a force f parallel to the infinite axis of the tube when the force stretches the polygons while when the force is compressive we obtain and prove the asymptotic form of the free energy in both limits we conjecture that the asymptote is the same as the limiting free energy of hamiltonian polygons polygons which visit every vertex in a box we investigate such polygons and in particular use a transfer matrix methodology to establish that the conjecture is true for some small tube sizes

on the topology of a polymer ring first we survey the literature on knots and links in theoretical physics next we report a numerical study in which equilibrium configurations of ring polymers in an infinite space or confined to the interior of a sphere are generated by using a new algorithm the a priori probability for the occurrence of a knot is determined numerically the results are compatible with scaling laws of striking simplicity we also study the mutual entanglement of links comparing the gauss invariant with the alexander polynomial

energy flow variability in a pair of coupled stochastic rods abstract random variations in the transfer functions describing energy flows in a pair of coupled axially vibrating stochastic rods are considered the material properties of the rods are modelled either as random variables or as random processes and the consequent effects on the nature of the power receptance functions are studied by using monte carlo simulation techniques the relevance of these results is discussed in the context of variability in average estimates obtained when using statistical energy analysis sea formalisms the response statistics are shown to vary significantly with the details of the statistical modelling of the system especially in the low and medium frequency ranges moreover the confidence limits show no tendency to converge toward the mean with increasing frequency critical systems the responses of which deviate maximally from the average values are identified and attempts made to understand their nature

algebraic structures related to universal covers of special rank one groups abstract in this note we give a presentation of special rank one groups with abelian unipotent subgroups abbreviated aus depending on some algebraic structure k and then show that each special rank one group with aus is a center factor group of a so presented group as shown in t de medts r weiss moufang sets and jordan division algebras math ann 335 2006 415 433 one example of such an algebraic structure is unital quadratic jordan division algebras in subsequent work my student sebastian weis studied these structures called r structures systematically and showed among other things that the problem of classifying such special rank one groups with aus is equivalent to the classification of these r structures up to isotopy

entanglement complexity of self avoiding walks self avoiding walks on three dimensional lattices are flexible linear objects which can be self entangled the authors discuss several ways to measure entanglement complexity for n step walks and prove that these complexity measures tend to infinity with n for small n they use monte carlo methods to estimate and compare the n dependence of two of these complexity measures

collapsing and adsorbing polygons the adsorption transition in the phase diagram of a self interacting lattice polygon is examined the polygon has a nearest neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall the partition function of this model is z c y z d p v cp c v c y v z c where p c v c is the number of polygons with c nearest neighbour contacts v visits to the wall and n edges and counted up to translations parallel to the wall the limiting free energy of this model is f c y z d limn 1 1 logz c y z and it is known to be a non analytic function of y for each z 1 for all z2 0 1

the number of smallest knots on the cubic lattice it has been shown that the smallest knots on the cubic lattice are all trefoils of length 24 in this paper we show that the number of such unrooted knots on the cubic lattice is 3496

localization of flexural waves in a disordered periodic piezoelectric beam abstract localization of bending waves in a disordered periodic piezoelectric beam is studied in this paper the equation of the wave motion for a piezoelectric beam is derived on the assumption of an euler bernoulli beam and the harmonic solution is presented the transfer matrix between two consecutive unit cells in the structures is obtained by using the continuity conditions the expression of the localization factor is given by wolf s algorithm numerical examples are presented and the effects of several disordered parameters on the localization factor are analyzed the results show that piezoelectricity has obvious effects on the passbands and stopbands of the periodic piezoelectric beam the behavior of wave propagation and localization in disordered periodic piezoelectric beams can be altered by tuning different structural parameters

the dimensions of knotted polygons the authors study the dimensions mean square radius of gyration and mean span of self avoiding polygons on the simple cubic lattice with fixed knot type the approach used is a monte carlo algorithm which is a combination of the bfacf algorithm and the pivot algorithm so that the polygons are studied in the grand canonical ensemble but the autocorrelation time is not too large they show that although the dimensions of polygons are sensitive to knot type the critical exponent v and the leading amplitude are independent of the knot type of the polygon the knot type influences the confluent correction to the scaling term and hence the rate of approach to the limiting behaviour

entropic exponents of lattice polygons with specified knot type ring polymers in three dimensions can be knotted and the dependence of their critical behaviour on knot type is an open question we study this problem for polygons on the simple cubic lattice using a novel grand canonical monte carlo method and present numerical evidence that the entropic exponent depends on the knot type of the polygon we conjecture that the exponent increases by unity for each additional factor in the knot factorization of the polygon

localization of vibrations by structural irregularity an investigation of the localization phenomenon is presented for disrodered structural systems consisting of weakly coupled component systems emphasis is placed on the development of a perturbation method that allows one to obtain the localized modes of vibration of the disordered system from the modes of the individual subsystems the principal feature of this perturbation method is that it not only considers the variations of the parameters as perturbations but it also treats the small coupling between subsystems as a perturbation such a perturbation analysis is cost effective as compared to a global eigenvalue analysis of the entire system and proves to be very accurate moreover it provides physical insight into the localization phenomenon and allows one to formulate a criterion that predicts the occurrence of strongly localized modes results are obtained for a chain of single degree of freedom coupled oscillators but these can be readilt generalized to deal with chains of multi degree of freedom component systems

vibration isolation from irregularity in a nearly periodic structure theory and measurements this article describes the theory and a simple experiment carried out to demonstrate the phenomenon of anderson localization in an acoustical context this is an effect whereby the propagation of vibration in a structure which is not entirely regular is impeded by the irregularities giving rise on the average to an exponential decay of vibration level away from the driving point even in the absence of any dissipation the structure used in the experiment was a stretched string with masses attached to it this string was studied with regular spacings of the masses and after the masses had been moved in a controlled way to provide a small degree of irregularity in both cases the transmission of energy from end to end of the string was measured as a function of frequency and also the mode shapes in the second and fourth passbands were measured so as to demonstrate the underlying physics of the localization phenomenon in which the individual modes making up each passband change from being extended throughout the structure in the regular case to being localized in specific areas of the structure in the presence of irregularity all measurements yielded satisfactory agreement with the theoretical predictions

on the mean and variance of the writhe of random polygons we here address two problems concerning the writhe of random polygons first we study the behavior of the mean writhe as a function length second we study the variance of the writhe suppose that we are dealing with a set of random polygons with the same length and knot type which could be the model of some circular dna with the same topological property in general a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons if the mean writhe is non zero then the knot is chiral how accurate is this method for example if for a specific knot type k the mean writhe decreased to zero as the length of the polygons increased then this method would be limited in the case of long polygons furthermore we conjecture that the sign of the mean writhe is a topological invariant of chiral knots this sign appears to be the same as that of an ideal conformation of the knot we provide numerical evidence to support these claims and we propose a new nomenclature of knots based on the sign of their expected writhes this nomenclature can be of particular interest to applied scientists the second part of our study focuses on the variance of the writhe a problem that has not received much attention in the past in this case we focused on the equilateral random polygons we give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons of length n behaves as a linear function of the length of the equilateral random polygon

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

special moufang sets their root groups and their μ maps we prove timmesfeld s conjecture that special abstract rank one groups are quasisimple we give two characterizations of the root groups in special moufang sets a normal subgroup of the point stabilizer is a root group if it is either regular or nilpotent and transitive we prove that if a root group of a special moufang set contains an involution then it is of exponent 2 we also show that the root groups are abelian if and only if the so called maps are involutions

bounds for the minimum step number of knots confined to slabs in the simple cubic lattice volume confinement is a key determinant of the topology and geometry of a polymer however the direct relationship between the two is not fully understood for instance recent experimental studies have constructed p4 cosmids i e p4 bacteriophages whose genome sequence and length have been artificially engineered and have shown that upon extraction their dna knot distribution differs from that of wild type bacteriophage p4 in particular it was observed that the complexity of the knots decreases sharply with the length of the packed genome this problem is the motivation of this paper here a polymer is modeled as a self avoiding polygon on the simple cubic lattice and the confining condition is such that the polygon is bounded between two parallel planes i e bounded within a slab we estimate the minimum length required for such a polygon to realize a knot type our numerical simulations show that in order to realize a prime knot with up to ten crossings in a 1 slab i e a slab of height 1 one needs a polygon of length strictly longer than the minimum length needed to realize the same knot when there is no confining condition in the case of the trefoil knot we can in fact establish this result analytically by proving that the minimum length required to tie a trefoil in the 1 slab is 26 which is greater than 24 the known minimum length required to tie a trefoil without a confinement condition additionally we find that in the 1 slab not all geometrical realizations of a given knot type are equivalent under bfacf moves this suggests that in certain confined volumes knowing the topology of a polymer is not enough to describe all its states

upper bounds on the minimum length of cubic lattice knots knots have been considered to be useful models for simulating molecular chains such as dna and proteins one quantity that we are interested in is the molecular knot and the minimum number of monomers necessary to realize a knot in this paper we consider every knot in the cubic lattice in particular the minimum length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice diao introduced this term he used minimum edge number instead and proved that the minimum length of the trefoil knot 31 is 24 also the minimum lengths of the knots 41 and 51 are known to be 30 and 34 respectively in this paper we find a general upper bound of the minimum length of a nontrivial knot k except the trefoil knot in terms of the minimum crossing number c k the upper bound is moreover if k is a non alternating prime knot then the upper bound is our work can be considered a direct consequence of the results obtained by the authors in hong et al 2012 arxiv 1209 0048 furthermore if k is n 1 n torus knot then the upper bound is

entanglement complexity of lattice ribbons we consider a discrete ribbon model for double stranded polymers where the ribbon is constrained to lie in a three dimensional lattice the ribbon can be open or closed and closed ribbons can be orientable or nonorientable we prove some results about the asymptotic behavior of the numbers of ribbons withn plaquettes and a theorem about the frequency of occurrence of certain patterns in these ribbons we use this to derive results about the frequency of knots in closed ribbons the linking of the boundary curves of orientable closed ribbons and the twist and writhe of ribbons we show that the centerline and boundary of a closed ribbon are both almost surely knotted in the infinite n limit for an orientable ribbon the expectation of the absolute value of the linking number of the two boundary curves increases at least as fast as n and similar results hold for the twist and writhe

localization of elastic waves in periodic rib stiffened rectangular plates under axial compressive load based on the theory of elastic dynamics the problem of wave localization in disordered periodic multi span rib stiffened plates is investigated the transfer matrix method is employed to obtain the transfer matrix of the system and the method for calculating the lyapunov exponents in continuous dynamical systems presented by wolf is used to determine the localization factors in discrete dynamical systems as examples the numerical results of the localization factors are given for a disordered periodic multi span rib stiffened plate under axial compressive load the effects of the degree of disorder of span length and the structural parameters on the elastic wave localization are analyzed the larger the degree of disorder the larger the degree of localization the larger the dimensionless torsional and flexural rigidities of the rib the larger the degree of localization

moufang sets and jordan division algebras we give a simple criterion which determines when a permutation group u and one additional permutation give rise to a moufang set we apply this criterion to show that every jordan division algebra gives rise in a very natural way to a moufang set to provide sufficient conditions for a moufang set to arise from a jordan division algebra and to give a characterization of the projective moufang sets over a commutative field of characteristic different from 2

entanglement complexity of semiflexible lattice polygons we use monte carlo methods to study knotting in polygons on the simple cubic lattice with a stiffness fugacity we investigate how the knot probability depends on stiffness and how the relative frequency of trefoils and figure eight knots changes as the stiffness changes in addition we examine the effect of stiffness on the writhe of the polygons

monte carlo results for projected self avoiding polygons a two dimensional model for knotted polymers we introduce a two dimensional lattice model for the description of knotted polymer rings a polymer configuration is modelled by a closed polygon drawn on the square diagonal lattice with possible crossings describing pairs of strands of polymer passing on top of each other each polygon configuration can be viewed as the two dimensional projection of a particular knot we study numerically the statistics of large polygons with a fixed knot type using a generalization of the bfacf algorithm for self avoiding walks this new algorithm incorporates both the displacement of crossings and the three types of reidemeister transformations preserving the knot topology its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favour of this ergodicity are given together with a tentative sketch of proof assuming this ergodicity we obtain numerically the following results for the statistics of knotted polygons in the limit of a low crossing fugacity we find a localization along the polygon of all the primary factors forming the knot increasing the crossing fugacity gives rise to a transition from a self avoiding walk to a branched polymer behaviour

stretched polygons in a lattice tube we examine the topological entanglements of polygons confined to a lattice tube and under the influence of an external tensile force f the existence of the limiting free energy for these so called stretched polygons is proved and then using transfer matrix arguments a pattern theorem for stretched polygons is proved note that the tube constraint allows us to prove a pattern theorem for any arbitrary value of f while without the tube constraint it has so far only been proved for large values of f the stretched polygon pattern theorem is used first to show that the average span per edge of a randomly chosen n edge stretched polygon approaches a positive value non decreasing in f as n we then show that the knotting probability of an n edge stretched polygon confined to a tube goes to one exponentially as n thus as n when polygons are influenced by a force f no matter its strength or direction topological entanglements as defined by knotting occur with high probability

writhe and mutual entanglement combine to give the entanglement length we propose a method to estimate n e the entanglement length that incorporates both local and global topological characteristics of chains in a melt under equilibrium conditions this estimate uses the writhe of the chains the writhe of the primitive paths and the number of kinks in the chains in a melt an advantage of this method is that it works for both linear and ring chains works under all periodic boundary conditions does not require knowing the contour length of the primitive paths and does not rely on a smooth set of data we apply this method to linear finitely extendable nonlinear elastic chains and we observe that our estimates are consistent with those from other studies

mode localization and eigenvalue loci veering phenomena in disordered structures an investigation of the effects of disorder on the modes of vibration of nearly periodic structures is presented it is shown that in structures with close eigenvalues small structural irregularities result in both strong localization of the mode shapes and abrupt veering away or mutual repulsion of the loci of the eigenvalues when these are plotted against a parameter representing the disorder in the system perturbation methods for the eigenvalue problem are applied to predict the occurrence of strong localization and eigenvalue loci veering which are shown to be two manifestations of the same phenomenon also perturbation methods that handle the dramatic effects of small disorder are developed to analyze eigenvalue loci veering and strong localization two representative disordered nearly periodic structures are studied a mistuned assembly of coupled oscillators and a multi span beam with irregular spacing of the supports

counting closed 2 manifolds in tubes in hypercubic lattices we consider closed 2 manifolds 2 manifolds without boundary embedded in tubes in the hypercubic lattice z d for orientable 2 manifolds with fixed genus we prove that the exponential growth rate is independent of the genus and we use this to prove a pattern theorem for manifolds with fixed genus we prove a similar theorem for the non orientable case for if the genus is not fixed then we prove a pattern theorem and use this to show that 2 manifolds with genus less than any fixed number are exponentially rare so the typical genus increases with the size of the manifold in four and higher dimensions we prove that orientable manifolds are exponentially rare and are dominated by non orientable manifolds in four dimensions all except exponentially few 2 manifolds both orientable and non orientable contain a local knotted ball pair

the writhe of a self avoiding walk the writhe of a self avoiding walk in a three dimensional space is the average over all projections onto a plane of the sum of the signed crossings we compute this number using a monte carlo simulation our results suggest that the average of the absolute value of the writhe of self avoiding walks increases as nalpha where n is the length of the walks and alpha approximately 0 5 the mean crossing number of walks is also computed and found to have a power law dependence on the length of the walks in addition we consider the effects of solvent quality on the writhe and mean crossing number of walks

random linking of lattice polygons we study the linking probability of polygons on the simple cubic lattice in particular we consider two polygons each having n edges confined to a cube of side l and ask for the linking probability as a function of n and l we also consider other situations in which the polygons are restricted to be not too far apart but not necessarily confined to a cube we prove several rigorous results and use monte carlo methods to address some questions which we are unable to answer rigorously an interesting feature is that the linking probability is a function of l nv where v is the exponent characterizing the radius of gyration of a polygon

lyapunov exponents and localization phenomena in multi coupled nearly periodic systems abstract lyapunov exponents of a system wave transfer matrix are employed to analyze localization in multi coupled and as a special case mono coupled disordered periodic linear systems an algorithm due to wolfet al 1 is used to calculate the lyapunov exponents numerically perturbation techniques are used to find approximate lyapunov exponents for the case of weak disorder two examples are presented the largest lyapunov exponent is calculated for a representative mono coupled system and compared with localization factors found by monte carlo methods as well as with the approximated localization factors this example is used to discuss computational issues the lyapunov exponents are also calculated for an example of a bi coupled system and compared with wave amplitude decay found in a single realization of the disordered system the physical significance of the lyapunov exponents for a multi coupled nearly periodic system is discussed

asymptotic analysis of wave propagation along weakly non uniform repetitive systems we consider monochromatic wave propagation along a long finite one dimensional slightly non uniform waveguide whose ends are connected to uniform semi infinite waveguides the non uniformity in the system parameters which is assumed slowly varying and deterministic can be tuned to produce a desired scattered wave field or reflection transmission properties for a broad range of incident wave fields with this objective in mind we obtain an analytic solution for wave propagation along repetitive systems asymptotic in the slowness of the variation of the system parameters we consider systems governed by a second order finite difference equation and apply the wkb method allowing the index variable to be complex this allows complex turning points to be considered the coefficients of the difference equation are represented by their discrete fourier modes for complex turning points we obtain exponentially small reflection a new result in the context of difference equations the asymptotic solution besides revealing how the non uniformity in the parameters affects wave propagation furnishes an analytic expression for the system scattering matrix as a function of the system parameters it also sheds light on the mechanism of localization phenomena for this class of repetitive systems we also compare the asymptotic results with numerical experiments for large finite one dimensional non uniform chains of coupled pendula

wave transmission through one dimensional near periodic structures optimum and to random disorder abstract a general analysis is presented of wave transmission through a one dimensional near periodic structure which is embedded in an otherwise uniform system the occurrence of frequencies of perfect transmission is discussed and the inverse problem of design for minimum transmission is addressed this part of the work is aimed at providing design techniques for structural filters which might either block or allow wave transmission depending upon the intended application attention is then turned to the case of wave transmission through a randomly disordered system and a simple and exact method for the calculation of the expected value of the inverse squared transmission coefficient is presented the localization measure which may be deduced from this result is compared with the conventional definition of the localization factor and it is found that quantitative results for the latter may be computed by using the present approach the effects of damping are included and the interaction of disorder and damping is investigated this part of the work is aimed at providing a simple technique for the assessment of vibration localization effects of this type

the knot probability in lattice polygons the incidence of knots in lattice polygons in the face centred cubic lattice is investigated numerically the authors generate a sample of polygons using a pivot algorithm and detect knotted polygons by calculating the alexander polynomial if p0n phi is the probability that the polygon with n edges is the unknot then it is known that lim supn to infinity p0n phi 1n e varies as 0 1 they find that varies as phi 7 6 or 0 9 10 6 the effect of the solvent quality on p0n phi is considered the data show that the probability of a polygon being knotted increases rapidly as the quality of the solvent deteriorates

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow