knot complexity and the probability of random knotting the probability of a random polygon or a ring polymer having a knot type k should depend on the complexity of the knot k through computer simulation using knot invariants we show that the knotting probability decreases exponentially with respect to knot complexity here we assume that some aspects of knot complexity are expressed by the minimal crossing number c and the rope length of k which is defined by the smallest length of rope with unit diameter that can be tied to make the knot k

elementary function expansions for madelung constants the madelung constant essentially the coulomb energy density of a crystal is usually calculated via ewald error function expansions or for the simpler cubic structures by the cosech series of modern vintage by considering generalised functional equations for multidimensional zeta functions the authors provide explicit expansions for the spatial potential and energy density of three dimensional periodic structures these formulae involving only elementary functions are suitable for systematic calculation of madelung constants of arbitrary point charge crystals they indicate how zeta function relations may be used for dimensional reduction of certain multiple sums arising in the special cubic cases

on the spectrum of second order differential operators with complex coefficients the main objective of this paper is to extend the pioneering work of sims on second order linear differential equations with a complex coefficient in which he obtains an analogue of the titchmarsh weyl theory and classification the generalization considered exposes interesting features not visible in the special case in sims paper from 1957 an m function is constructed which is either unique or a point on a limit circle and the relationship between its properties and the spectrum of underlying m accretive differential operators analysed the paper is a contribution to the study of non self adjoint operators in general the spectral theory of such operators is rather fragmentary and further study is being driven by important physical applications to hydrodynamics electro magnetic theory and nuclear physics for instance

resolvent convergence and spectral approximations of sequences of self adjoint subspaces abstract this paper studies resolvent convergence and spectral approximations of sequences of self adjoint subspaces relations in complex hilbert spaces concepts of strong resolvent convergence norm resolvent convergence spectral inclusion and spectral exactness are introduced fundamental properties of resolvents of subspaces are studied by applying these properties several equivalent and sufficient conditions for convergence of sequences of self adjoint subspaces in the strong and norm resolvent senses are given it is shown that a sequence of self adjoint subspaces is spectrally inclusive under the strong resolvent convergence and spectrally exact under the norm resolvent convergence a sufficient condition is given for spectral exactness of a sequence of self adjoint subspaces in an open interval lacking essential spectral points in addition criteria are established for spectral inclusion and spectral exactness of a sequence of self adjoint subspaces that are defined on proper closed subspaces

a new evaluation of the energy of wigner bcc and fcc crystals new expressions for energies per site for two wigner crystals are given through euler type constants numerical values specify those known previously

j self adjoint extensions of a class of hamiltonian differential systems abstract this paper is concerned with j self adjoint extensions for a class of hamiltonian differential systems the domains of the corresponding minimal and maximal operators are described a classification of limit types of the systems at each of endpoints is given a property of the regularity field of the corresponding minimal operator is derived and an expression of the j defect index of the minimal operator by those of the left and right minimal operators is presented based on them a characterization of all j self adjoint extensions for the hamiltonian system is obtained in terms of square integrable solutions of a certain equation which is generated by this hamiltonian system and its formal adjoint as a consequence characterizations of all j self adjoint extensions are given for the systems in some special cases in particular all j self adjoint extensions are characterized in terms of square integrable solutions of the system itself in certain special cases

asymptotics for minimal discrete energy on the sphere we investigate the energy of arrangements of n points on the surface of the unit sphere sd in rd 1 that interact through a power law potential v 1 rs where s 0 and r is euclidean distance with ed s n denoting the minimal energy for such n point arrangements we obtain bounds valid for all n for ed s n in the cases when 0 s d and 2 d s for s d we determine the precise asymptotic behavior of ed d n as n as a corollary lower bounds are given for the separation of any pair of points in an n point minimal energy configuration when s d 2 for the unit sphere in r3 d 2 we present two conjectures concerning the asymptotic expansion of e2 s n that relate to the zeta function l s for a hexagonal lattice in the plane we prove an asymptotic upper bound that supports the first of these conjectures of related interest we derive an asymptotic formula for the partial sums of l s when 0 s 2 the divergent case

numerical solution of non self adjoint sturm liouville problems and related systems this paper gives the analysis and numerics underlying a shooting method for approximating the eigenvalues of non self adjoint sturm liouville problems we consider even order problems with equally divided separated boundary conditions the method can find the eigenvalues in a rectangle and in a left half plane it combines the argument principle with the compound matrix method using the magnus expansion in some cases the computational cost of compound matrices can be reduced by transforming to a second order vector sturm liouville problem we study the asymptotics of the solutions of the ode for large absolute values of the eigenvalue parameter in order to calculate the eigenvalues in a left half plane the method is applied to the orr sommerfeld equation and other examples

operator formalism for topology conserving crossing dynamics in planar knot diagrams we address here the topological equivalence of knots through the so called reidemeister moves these topology conserving manipulations are recast into dynamical rules on the crossings of knot diagrams this is presented in terms of a simple graphical representation related to the gauss code of knots drawing on techniques for reaction diffusion systems we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded the aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation this necessitates the introduction of composite paulionic operators the formalism is applied to calculate some differential equations for the time evolution of densities and correlators of crossings subject to topology conserving stochastic dynamics we consider here the simplified situation of two dimensional knot projections however we hope that this is a first valuable step towards addressing a number of important questions regarding the role of topological constraints and specifically of topology conservation in dynamics through a variety of solution and approximation schemes further applicability arises in the context of the simulated annealing of knots the methods presented here depart significantly from the invariant based path integral descriptions often applied in polymer systems and in our view offer a fresh perspective on the conservation of topological states and topological equivalence in knots

spectral inclusion and spectral exactness for singular non self adjoint hamiltonian systems we consider the effect on the spectrum of a singular non self adjoint hamiltonian system of regularization by interval truncation for problems where the deficiency indices are not maximal there i

crossings and writhe of flexible and ideal knots the data of ideal knots nature 384 142 1996 are reanalyzed and the average crossing number of the ideal knots x mathrm ideal shows a nonlinear behavior with the essential crossing number c supplemented with our monte carlo simulations using the bond fluctuation model on flexible knotted polymers our analysis indicates that x mathrm ideal varies nonlinearly with both c and the corresponding average crossing number of the flexible knot which is contrary to previous claims our extensive simulation data on the average crossing number of flexible knots suggest that it varies linearly with the square root of c furthermore our data on the average writhe number mathrm wr indicate that various knots are classified into holonomous groups and mathrm wr has a quantized linear increment with c in all four knot groups in our study

a legendre spectral element method for eigenvalues in hydrodynamic stability a legendre polynomial based spectral technique is developed to be applicable to solving eigenvalue problems which arise in linear and nonlinear stability questions in porous media and other areas of continuum mechanics the matrices produced in the corresponding generalised eigenvalue problem are sparse reducing the computational and storage costs where the superimposition of boundary conditions is not needed due to the structure of the method several eigenvalue problems are solved using both the legendre polynomial based and chebyshev tau techniques in each example the legendre polynomial based spectral technique converges to the required accuracy utilising less polynomials than the chebyshev tau method and with much greater computational efficiency

solution of convergence difficulties in the madelung sum problem an extrapolation scheme for sawtooth sequences an extrapolation scheme is presented to handle successfully partial madelung sum sequences with zig zag both convergent and divergent estimates so obtained are found to agree reasonably with the exact results providing in cases even values of superior quality compared to those obtained by invoking the more popular methods of sequence acceleration the strategy is extremely simple quite general and above all capable of physically interpreting the limit points in case of divergent sequences possibilities of further improvements of the parent scheme are discussed remarks on the notion of summability of divergent series in the present context have also been made with a brief analysis demonstrative calculations involve a two dimensional square lattice and a few three dimensional cubic lattices of practical concern

the growth of the mean average crossing number of equilateral polygons in confinement the physical and biological properties of collapsed long polymer chains as well as of highly condensed biopolymers such as dna in all organisms are known to be determined at least in part by their topological and geometrical properties with this purpose of characterizing the topological properties of such condensed systems equilateral random polygons restricted to confined volumes are often used however very few analytical results are known in this paper we investigate the effect of volume confinement on the mean average crossing number acn of equilateral random polygons the mean acn of knots and links under confinement provides a simple alternative measurement for the topological complexity of knots and links in the statistical sense for an equilateral random polygon of n segments without any volume confinement constrain it is known that its mean acn acn is of the order here we model the confining volume as a simple sphere of radius r we provide an analytical argument which shows that acn of an equilateral random polygon of n segments under extreme confinement meaning r n grows as o n2 we propose to model the growth of acn as a r n2 b r nln n under a less extreme confinement condition where a r and b r are functions of r with r being the radius of the confining sphere computer simulations performed show a fairly good fit using this model

convergence of lattice sums and madelung s constant the lattice sums involved in the definition of madelung s constant of an nacl type crystal lattice in two or three dimensions are investigated the fundamental mathematical questions of convergence and uniqueness of the sum of these not absolutely convergent series are considered it is shown that some of the simplest direct sum methods converge and some do not converge in particular the very common method of expressing madelung s constant by a series obtained from expanding spheres does not converge the concept of analytic continuation of a complex function to provide a basis for an unambiguous mathematical definition of madelung s constant is introduced by these means the simple intuitive direct sum methods and the powerful integral transformation methods which are based on theta function identities and the mellin transform are brought together a brief analysis of a hexagonal lattice is also given

on the convergence problem for lattice sums three dimensional lattice sums sigma or r s where r is the distance to a or charged lattice point arise in classical lattice point problems s 0 and the madelung problem of physics and chemistry s 1 in the latter case the sum over an infinite lattice is purely formal and the value of the madelung constant must be defined precisely e g via related convergent sums or analytic continuation in s indeed the partial sum over the sphere r r does not converge as r becomes large the authors verify a conjecture of j f delord 1988 that convergence can be obtained by neutralising each sphere with an appropriate surface charge specifically if lr s is the sum over the lattice points in the sphere r r then for neutrally charged lattices they show that as r goes to infinity the difference lr s r slr 0 approaches l s where l s is defined by analytic continuation when s 1 the term l 1 is the madelung constant and lr 0 r is the coulombic correction term

limit point criteria for semi degenerate singular hamiltonian differential systems with perturbation terms abstract some limit point criteria are obtained for higher dimensional semi degenerate singular hamiltonian differential systems with perturbation potential terms by using m theory results in this paper cover many previous results of hartman levinson titchmarsh and read

eigenvalue problems on exterior domains and dirichlet to neumann maps we consider a schroedinger equation on an exterior domain in the case where the potential which may be complex valued has a limit at infinity associated with the problem is a dirichlet to neumann map on the inner boundary we examine approximations to the problem obtained by a truncating the domain b replacing the potential by its value at infinity outside some set c a combination of a and b these approximations give rise to approximate dirichlet to neumann maps on the inner boundary we analyze the convergence of these maps and deduce results on the approximation of the spectrum of the original problem by the spectra of the approximating problems we show that in theory spurious eigenvalues cannot be generated by these procedures numerical experiments show however that the inherent ill conditioning of a problem may cause spurious eigenvalues to appear due to the discretization of the truncated problems even when the potential decays exponentially fast

eigenvalue problems on infinite intervals abstract this paper is concerned with eigenvalue problems for boundary value problems of ordinary differential equations posed on an infinite interval problems of that kind occur for example in fluid mechanics when the stability of laminar flows is investigated characterizations of eigenvalues and spectral subspaces are given and the convergence of approximating problems which are derived by reducing the infinite interval to a finite but large one and by imposing additional boundary conditions at the far end is proved exponential convergence is shown for a large class of problems author

spectral inclusion and spectral exactness for singular non self adjoint sturm liouville problems we consider the effect of regularization by interval truncation on the spectrum of a singular non self adjoint sturm liouville operator we present results on spectral inclusion and spectral exactness for the cases where the singularity is in sims case ii or sims case iii for sims case i we present a test for spectral inexactness which can be used to detect when the interval truncation process is generating spurious eigenvalues numerical results illustrate the effectiveness of this test

non recursive functions knots with thick ropes and self clenching thick hyperspheres we introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n dimensional sphere for n 5 sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem this recursion theoretic approach is applied in this paper to a class of functionals on the space of c1 1 smooth hypersurfaces diffeomorphic to sn in rn 1 where n is any fixed number 5 the simplest of these functionals kv is defined by the formula kv n vol n 1 n r n where r n denotes the radius of injectivity of the normal exponential map for n rn l we prove the existence of an infinite set of distinct locally minimal values of kv on the space of c1 1 smooth topological hyperspheres in rn 1 for any n 5 the functional kv naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of thick circles and more generally thick spheres into euclidean spaces we introduce the notion of knot with thick rope types the theory of knot with thick rope types turns out to be quite different from the classical knot theory because of the following result there exists an infinite set of non trivial knot with thick rope types in codimension one for every dimension greater than or equal to five

classification of non self adjoint singular sturm liouville difference equations abstract this paper is concerned with non self adjoint singular sturm liouville difference equations by introducing a new spectral parameter we rewrite the sturm liouville difference equation as a formally self adjoint hamiltonian difference system applying the theory of the limit point and limit circle cases for this difference system we classify the considered equation into cases i ii and iii two examples are illustrated to show the dependence of cases ii and iii on the corresponding half planes furthermore the exact dependence of cases ii and iii on the corresponding half planes is obtained

on the solvability complexity index the n pseudospectrum and approximations of spectra of operators we show that it is possible to compute spectra and pseudospectra of linear operators on separable hilbert spaces given their matrix elements the core in the theory is pseudospectral analysis and in particular the n pseudospectrum and the residual pseudospectrum we also introduce a new classification tool for spectral problems namely the solvability complexity index this index is an indicator of the difficultness of different computational spectral problems

regular approximations of spectra of singular second order symmetric linear difference equations abstract this paper is concerned with regular approximations of spectra of singular second order symmetric linear difference equations for any given self adjoint subspace extension of the corresponding minimal subspace its spectrum can be approximated by eigenvalues of a sequence of regular self adjoint subspace extensions generated by the same difference expression on smaller finite intervals

energy of static electron lattices a new approach to evaluating the static lattice energy of any given wigner lattice is proposed the method is much simpler than the traditional one and the results are much faster to evaluate the new approach is applied to two dimensional triangular lattices where it is shown that the triangular lattice is more stable than the square lattice three dimensional hexagonal lattices are also investigated in addition the authors place many of their considerations on a rigorous mathematical footing

titchmarsh sims weyl theory for complex hamiltonian systems the paper extends to complex hamiltonian systems previous work of the authors on the sims extension of the titchmarsh weyl theory for sturm liouville equations with complex potentials and analyses the spectral properties of associated non self adjoint operators 2000 mathematics subject classification 34b20 34lxx

a fast ergodic algorithm for generating ensembles of equilateral random polygons knotted structures are commonly found in circular dna and along the backbone of certain proteins in order to properly estimate properties of these three dimensional structures it is often necessary to generate large ensembles of simulated closed chains i e polygons of equal edge lengths such polygons are called equilateral random polygons however finding efficient algorithms that properly sample the space of equilateral random polygons is a difficult problem currently there are no proven algorithms that generate equilateral random polygons with its theoretical distribution in this paper we propose a method that generates equilateral random polygons in a step wise uniform way we prove that this method is ergodic in the sense that any given equilateral random polygon can be generated by this method and we show that the time needed to generate an equilateral random polygon of length n is linear in terms of n these two properties make this algorithm a big improvement over the existing generating methods detailed numerical comparisons of our algorithm with other widely used algorithms are provided

on classification of second order differential equations with complex coefficients this paper is concerned with the deficiency index problem of second order differential equations with complex coefficients it is known that this class of equations is classified into cases i ii and iii according to the number of linearly independent solutions in suitable weighted square integrable spaces in this study the original equation is reformulated into a new formally self adjoint differential system by introducing a new spectral parameter and the relationship between the classifications of the equation and the system is obtained moreover the exact dependence of cases ii and iii on the corresponding half planes is given and some criteria of the three cases are established

on the zero distribution of epstein zeta functions we prove that the mean value of the real parts of the nontrivial zeros of the epstein zeta function open image in new window associated with a positive definite quadratic form open image in new window in n variables is equal to open image in new window furthermore we show that epstein zeta functions in general have an asymmetric zero distribution with respect to the critical line re open image in new window

asymptotics of knotted lattice polygons we use monte carlo methods to investigate the asymptotic behaviour of the number and mean square radius of gyration of polygons in the simple cubic lattice with fixed knot type let be the number of n edge polygons of a fixed knot type in the cubic lattice and let be the mean square radius of gyration of all the polygons counted by if we assume that where is the growth constant of polygons of knot type and is the entropic exponent of polygons of knot type then our numerical data are consistent with the relation where is the unknot and is the number of prime factors of the knot if we assume that then our data are consistent with both and being independent of these results support the claims made in janse van rensburg and whittington 1991a 24 3935 and orlandini et al 1996 29 l299 1998 topology and geometry in polymer science ima volumes in mathematics and its applications berlin springer