a simple proof of kostant s theorem that u g is free over its center in this paper we present a simple proof of the fundamental result by b kostant which claims that the universal enveloping algebra of a reductive lie algebra g is free over its center we also indicate how this result allows to simplify the proof of another important result of b kostant the description of the algebra of functions on the nilpotent cone we use this technique to prove some generalizations of kostant s theorem we also deduce from it a way to check which subalgebras of g are centrally free

measures for oscillatory sequences abstract the classical concept of oscillation is not sufficient to fully describe the oscillatory behavior of a real sequence by introducing the concepts of frequency measures of subsets of integers and investigating their properties we are able to strengthen the original concept of an oscillatory sequence and thus new oscillation criteria for difference equations can be derived these criteria differ from the classical ones in that they are described by means of the measures of the level sets of the involved parameter sequences a brief description of the theory of frequency measures for multiple sequences is also given

oscillation of nonlinear partial difference equations with delays abstract in this paper we first derive a discrete gaussian formula and then apply the formula to two classes of nonlinear parabolic and hyperbolic partial difference equations with delays to obtain sufficient conditions under which every solution of the two classes of nonlinear partial difference equations with delays is oscillatory

necessary and sufficient conditions for oscillations of partial difference equations with continuous variables abstract this paper is concerned with the delay partial difference equation a 1 y a y a y pa y 0 where p is a real number and are nonnegative real numbers sufficient and necessary conditions for all continuous solutions of this equation to be oscillatory are obtained explicit condition for oscillation in terms of p and is also given

characters for projective modules in the bgg category o for general linear lie superalgebras abstract we determine the verma multiplicities and the characters of projective modules for atypical blocks in the bgg category o for the general linear lie superalgebras gl 2 2 and gl 3 1 we then explicitly determine the composition factor multiplicities of verma modules in the atypicality 2 block of gl 2 2

oscillation for difference equations with continuous variable consider the delay difference equation y t y t p t h y t t t 0 where 0 0 and p c r r f c r r h c r r some oscillation criteria for this equation are obtained

the stability of partial difference systems with retarded arguments abstract for certain partial difference systems with retarded arguments sufficient conditions for the existence of decaying solutions are given the stability of these systems is discussed also

eventually positive and monotonely decreasing solutions of partial difference equations we establish criteria for the nonexistence of eventually positive negative and monotonely decreasing increasing solutions of the partial difference inequality where gi m and hi n 1 i are some deviating arguments several examples are presented to dwell upon the importance of the results obtained

distribution of the distance between opposite nodes of random polygons with a fixed knot we examine numerically the distribution function fk r of the distance r between opposite polygonal nodes for random polygons of n nodes with a fixed knot type k here we consider three knots such as 31 and 3131 in a wide range of r the shape of fk r is well fitted by the scaling form 1 of self avoiding walks the fit yields the gaussian exponents and k 1 furthermore if we re scale the intersegment distance r by the average size rk of random polygons of knot k the distribution function of the variable r rk should become the same gaussian distribution for any large value of n and any knot k we also introduce a fitting formula to the distribution gk r of gyration radius r for random polygons under some topological constraint k

bounded solutions and asymptotic stability of nonlinear difference equations in the complex plane ii abstract a functional analytic method is used to prove a theorem which establishes the existence and the uniqueness of the solution of a nonlinear difference equation in the banach space l1 the boundedness of the solution and the local asymptotic stability of the equilibrium points of the nonlinear difference equation under discussion are derived immediately from this theorem the proof of the theorem has a constructive character which allow us to obtain information about the size of the region of attraction of the equilibrium points some known nonlinear difference equations are studied as particular cases of the theorem

oscillation of a kind of two variables functional equation abstract in this paper we consider the functional equation of the form where a b c are positive constants k l are positive integers function p i i r 0 i 0 is an unbounded set i i x x y y and limx x limy y x y i some oscillation criteria of this equation are obtained

an existence theorem for a nonlinear difference equation where p 1 and s p is a real sequence we establish conditions under which 1 1 has a positive nondecreasing solution here a solution of 1 1 is a real sequence y y which satisfies 1 1 since 1 1 is a recurrence relation given real initial values y and y it is clear that we can inductively obtain y y since p is a real number greater than 1 the sequence yk may not be real that is not a solution note however that any constant multiple of a solution is again a solution the existence question originated in l where it was shown that hardy s inequality for a series 2 can be viewed as a necessary condition for the existence of a positive nondecreasing solution of 1 1 our existence theorem reduces to a nonoscillation criterion in the case p 2 since then 1 1 changes to a second order linear difference equation see e g 3 a brief outline of the paper is as follows section 2 introduces a riccati type transformation and then develops various necessary conditions for the existence of a positive nondecreasing solution section 3 discusses some of the functional analytic background which is needed in order to apply schauder s theorem section 4 contains the proof of our existence theorem and section 5 describes a comparison theorem which allows us to drop the assumption that the coefficients sk in 1 1 are nonnegative

oscillation criteria of certain nonlinear partial difference equations this paper is concerned with the nonlinear partial difference equation with continuous variables a x a y a x y a a x y i 1mhi x y a x i y i 0 where a i i are positive numbers hi x y u c r r r r uhi x y u 0 for u 0 hi is nondecreasing in u i 1 m some oscillation criteria of this equation are obtained

oscillation of partial difference equations with variable coefficients for the partial difference equations am 1 n am n 1 am n pm nam k n 1 0 and am 1 n am n 1 am n i 1upi m n fi am ki n li 0 we shall obtain sufficient conditions for the oscillation of all solutions

symmetric pairs and gorelik elements abstract the second author gave a formula for the elements of the enveloping algebra of a lie superalgebra defined by gorelik under an appropriate unimodularity assumption we show that this formula is a particular case of a formula for the jacobian of the exponential map of a symmetric superspace

on the theory of frobenius extensions and its application to lie superalgebras by using an approach to the theory of frobenius extensions that emphasizes notions related to associative forms we obtain results concerning the trace and corestriction mappings and transitivity these are employed to show that the extension of enveloping algebras determined by a subalgebra of a lie superalgebra is a frobenius extension and to study certain questions in representation theory

the minimal primitive spectrum of the enveloping algebra of the lie superalgebra osp 1 2l abstract a well known theorem of duflo the annihilation theorem claims that the annihilator of a verma module in the enveloping algebra of a complex semisimple lie algebra is centrally generated for the lie superalgebra osp 1 2 l this result does not hold in this article we introduce a correct analogue of the centre for which the annihilation theorem does hold in the case osp 1 2 l this substitute of the centre is the centralizer of the even part of the enveloping algebra this algebra shares some nice properties with the centre as a consequence of the annihilation theorem we obtain the description of the minimal primitive spectrum of the enveloping algebra of the lie superalgebra osp 1 2 l we also deduce a criterium for a osp 1 2 l verma module to be a direct sum of sp 2 l verma modules

solutions of nonlinear delay and advanced partial difference equations in the space ln n1 abstract a functional analytic method is used to prove a general theorem which establishes the existence and the uniqueness of a solution of a class of nonlinear delay and advanced partial difference equations in the banach space l n n 1 the proof of the theorem has a constructive character which enables us to obtain a bound of the solution and a region depending on the initial conditions and the parameters of the equation under consideration where this solution holds some known nonlinear partial difference equations which appear in applications are studied as particular cases of the theorem

asymptotic constancy of solutions of linear parabolic volterra difference equations abstract in this paper we obtain some sufficient conditions under which every solution of a class of linear parabolic volterra difference equations of neutral type tends to a constant as n we also consider the asymptotic constancy of solutions for a class of linear ordinary volterra difference equations of neutral type the results obtained improve and generalize some known results

nonexistence of positive solutions of a class of nonlinear delay partial difference equations sufficient conditions are derived for the nonexistence of positive solutions of a class of positive nonlinear delay partial difference equations results are also obtained for the nonexistence of positive solutions of the dual equation

oscillation of a class of partial difference equations with unbounded delay abstract this paper is concerned with the partial difference equation where and n z are nondecreasing n and n are strictly less than n for n n lim n n lim n n a m n b m n and p m n are three real double sequences some oscillation criteria for this equation are obtained

average size of random polygons with fixed knot topology we have evaluated by numerical simulation the average size r k of random polygons of fixed knot topology k ensuremath emptyset 3 1 3 1 ensuremath sharp 4 1 and we have confirmed the scaling law r k 2 ensuremath sim n 2 ensuremath nu k for the number n of polygonal nodes in a wide range n 100 char21 2200 the best fit gives 2 ensuremath nu k ensuremath simeq 1 11 char21 1 16 with good fitting curves in the whole range of n the estimate of 2 ensuremath nu k is consistent with the exponent of self avoiding polygons in a limited range of n n ensuremath gtrsim 600 however we have another fit with 2 ensuremath nu k ensuremath simeq 1 01 char21 1 07 which is close to the exponent of random polygons

oscillation criteria for nonlinear partial difference equations with delays abstract we offer sufficient conditions for the oscillation of all solutions of the partial difference equations y m 1 n m n y m n 1 m n y m n p m n y m k n l q m n y m k n l and y m 1 n m n y m n 1 m n y m n i 1 p i m n y m k i n l i i 1 q i m n y m k i n l i several examples which dwell upon the importance of our results are also included

stability criteria for parabolic type partial difference equations this paper is concerned with the stability of neutral parabolic type partial difference equations with delays over cylin drical domains see 1 1 1 4 and 3 1 stability criteria involving the spatial euclidean norms of the solutions are derived by means of basic analytic techniques these criteria will likely find applications in the stability of finite difference schemes resulting from discretising parabolic reaction diffusion equations

anomalous finite size effects for the mean squared gyration radius of gaussian random knots anomalously strong finite size effects have been observed for the mean square radius of gyration r2k of gaussian random polygons with a fixed knot k as a function of the number n of polygonal nodes through computer simulations with n 2000 we find that the gyration radius r2k can be approximated by a power law r2k n2 keff for several knots where the effective exponents keff are larger than 0 5 and less than 0 6 furthermore a crossover occurs for the gyration radius of the trivial knot when n is roughly equal to the characteristic length nc of random knotting assuming an asymptotic fitting formula we also discuss possible asymptotic behaviours for rk2 of gaussian random polygons

further oscillation criteria for partial difference equations with variable coefficients in this paper some new oscillation criteria on the oscillation of first order partial delay difference equations with nonnegative variable coefficients which improve the recent ones under some additional conditions are given some examples to illustrate the applicability of our results are also supplied of which solutions are plotted by the mathematical programming language mathematica 7 0

primitive ideals twisting functors and star actions for classical lie superalgebras we study three related topics in representation theory of classical lie superalgebras the first one is classification of primitive ideals i e annihilator ideals of simple modules and inclusions

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

existence of monotone positive solution of neutral partial difference equation abstract this paper is concerned with a class of neutral partial difference equations the conditions for the existence of monotone eventually positive solutions are established which improve and extend some of the criteria existing in the literature comparison theorems are also derived results are obtained on the existence of a monotone eventually positive solution of dual equation

the stability of certain partial difference equations abstract this paper is concerned with the linear delay partial difference equation where and are positive integers a i j b i j p i j are real sequences defined on i j sufficient conditions for this equation to be stable are derived stability of certain nonlinear partial difference equations is studied also

partial difference equations arising in numerical schemes and game theory a functional analytic method is used to prove that some linear and non linear partial difference equations of two three and four variables which arise in numerical schemes and game theory have a unique solution in the spaces l 2 n 2 or l 1 n 2 l 1 n 3 and l 1 n 4 respectively in the case of non linear difference equations a region which depends on the initial conditions and the parameters of the equation where the solution holds is given for both linear and non linear difference equations a bound of the solution is determined

characteristics of shape and knotting in ideal rings we present two descriptions of the local scaling and shape of ideal rings primarily featuring subsegments our focus will be the squared radius of gyration of subsegments and the squared internal end to end distance defined to be the average squared distance between vertices k edges apart we calculate the exact averages of these values over the space of all such ideal rings not just a calculation of the order of these averages and compare these to the equivalent values in open chains this comparison will show that the structure of ideal rings is similar to that of ideal chains for only exceedingly short lengths these results will be corroborated by numerical experiments they will be used to analyze the convergence of our generation method and the effect of knotting on these characteristics of shape

linearized oscillation theorems for certain nonlinear delay partial difference equations this paper is concerned with the nonlinear delay partial difference equation am 1 n am n 1 am n pm nf am k n 1 0 where k and l are nonnegative integers and m n 0 1 some linearized oscillation theorems for this equation are obtained i e we will prove that under some conditions on f this equation has the same oscillatory character as an associated linear equation an existence result for positive solutions of this equation is obtained also

the primitive spectrum of a basic classical lie superalgebra we prove conjecture 5 7 in coulembier and musson math j arxiv 1409 2532 describing all inclusions between primitive ideals for the general linear superalgebra in terms of the rm ext 1 quiver of simple highest weight modules for arbitrary basic classical lie superalgebras we formulate two types of kazhdan lusztig quasi orders on the dual of the cartan subalgebra where one corresponds to the above conjecture both orders can be seen as generalisations of the left kazhdan lusztig order on hecke algebras and are related to categorical braid group actions we prove that the primitive spectrum is always described by one of the orders obtaining for the first time a description of the inclusions we also prove that the two orders are identical if category mathcal o admits enough abstract kazhdan lusztig theories in particular they are identical for the general linear superalgebra concluding the proof of the conjecture

what is the length of a knot in a polymer we give statistical definitions of the length l of a loose prime knot tied into a long fluctuating ring macromolecule monte carlo results for the equilibrium good solvent regime show that l nt where n is the ring length and t 0 75 is independent of the knot type in the collapsed regime below the theta temperature length determinations based on the entropic competition of different knots within the same ring show knot delocalization t 1

asymptotic behavior of recursions via fixed point theory in this paper we provide a formulation of initial value problems for explicit and implicit difference equations in terms of abstract equations in sequence spaces they will be solved using appropriate fixed point theorems and we obtain quantitative attractivity properties

on the center of the enveloping algebra of a classical simple lie superalgebra 0 1 let g be a semisimple lie algebra with cartan subalgebra h weyl group w and enveloping algebra u g several results illustrate the important role played by the center z g of u g first there is the harish chandra homomorphism which shows that spec z g h rw and describes the action of z g on highest weight modules next we w x recall the separation of variables theorem of kostant ko d 8 2 4 which states that there is an ad g invariant subspace k of u g such that the multiplication map k m z g a u g is an isomorphism of ad g modules w kostant s theorem is a key ingredient in the proof of a result of duflo d x 8 4 3 stating that the annihilator of a verma module is generated by its intersection with z g a version of the harish chandra homomorphism has been given by kac for basic classical simple lie superalgebras we recall the details in subsection 1 1 the proof of kostant s theorem depends heavily on the fact that every finite dimensional g module is completely reducible by contrast if g is a simple lie superalgebra such that every finite dimensional g module is completely reducible then either g is a simple lie algebra or g is an w orthosymplectic lie superalgebra g s osp 1 2 r for some r g 1 sch x theorem 1 p 239 in this paper we show that for g s osp 1 2 r we again have u g k m z g for k ad invariant in addition if e is any finite dimensional

strongly typical representations of the basic classical lie superalgebras 1 1 in ps1 i penkov and v serganova show that the category of representations of a basic classical lie superalgebra g of type i with a fixed typical central character is equivalent to the category of representations of the even part go with a suitable central character a similar result for type ii was proven by i penkov in p for generic central characters the aim of this paper is to understand for which central characters such an equivalence holds we also study a simplest example when such an equivalence fails to exist but the corresponding category of representations of a lie superalgebra still has a good description

a mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense abstract let x be a uniformly convex banach space such that its dual x has the kadec klee property c a bounded closed convex subset of x and t t t t 0 a one parameter nonlinear semigroup which is asymptotically nonexpansive in the intermediate sense we show that every continuous almost orbit of t is weakly almost convergent to a common fixed point of t a discrete version of our main result is also included

scaling behavior of random knots using numerical simulations we investigate how overall dimensions of random knots scale with their length we demonstrate that when closed non self avoiding random trajectories are divided into groups consisting of individual knot types then each such group shows the scaling exponent of 0 588 that is typical for self avoiding walks however when all generated knots are grouped together their scaling exponent becomes equal to 0 5 as in non self avoiding random walks we explain here this apparent paradox we introduce the notion of the equilibrium length of individual types of knots and show its correlation with the length of ideal geometric representations of knots we also demonstrate that overall dimensions of random knots with a given chain length follow the same order as dimensions of ideal geometric representations of knots

on the oscillation of two partial difference equations abstract this paper is concerned with two partial difference equations of the form a m 1 n a m n 1 a m n p m n a m k n l 0 and a m 1 n a m n 1 a m n i 1 u p i m n f i a m k i n l i 0 respectively we obtain sufficient conditions for the oscillation of all solutions of these equations

oscillation of certain partial difference equations in this paper we consider a class of nonlinear delay partial difference equations and a class of linear delay partial difference equations with variable coefficients which may change sign we obtain oscillation criteria for these equations there are no results for the oscillation of these equations up to now

asymptotic behaviour of solutions of higher order difference and partial difference equations with distributed deviating arguments we shall study the asymptotic behaviour of solutions of the difference equation m y j g j y j s a 1 d m y j y j s b x at j x f y h j x 0 j j 0 where a 0 as an application of the results obtained oscillation theorems are established for the partial difference equation m y j g j y j s b x a i j x f y h j x d j ly i j m l 1d l j ly i t l j i e w j

nonexistence of unbounded nonoscillatory solutions of partial difference equations abstract we develop criteria for the nonexistence of eventually positive negative and nondecreasing nonincreasing solutions of the partial difference equation m n y m n p m n y m k n l q m n y m k n l and m n y m n i 1 p i m n y m k i n l i i 1 q i m n y m k i n l i several examples which dwell upon the importance of our results are also included

gyration radius of a circular polymer under a topological constraint with excluded volume it is nontrivial whether the average size of a ring polymer should become smaller or larger under a topological constraint making use of some knot invariants we numerically evaluate the mean square radius of gyration for ring polymers having a fixed knot type where the ring polymers are given by self avoiding polygons consisting of freely jointed hard cylinders we obtain plots of the gyration radius versus the number of polygonal nodes for the trivial trefoil and figure eight knots we discuss possible asymptotic behaviors of the gyration radius under the topological constraint in the asymptotic limit the size of a ring polymer with a given knot is larger than that of no topological constraint when the polymer is thin and the effective expansion becomes weak when the polymer is thick enough

the oscillation of partial difference equations with continuous arguments abstract for the partial difference equations and we shall obtain sufficient conditions for the oscillation of all solutions of these equations

bernstein gelfand gelfand resolutions for basic classical lie superalgebras abstract we study kostant cohomology and bernstein gelfand gelfand resolutions for finite dimensional representations of basic classical lie superalgebras for each choice of parabolic subalgebra and representation of such a lie superalgebra there is a natural definition of the boundary and coboundary operators which define co homology of the nilradical of the parabolic subalgebra we prove that complete reducibility of the homology groups is a necessary condition to have a resolution of an irreducible module in terms of generalised verma modules every such a resolution is then given by modules induced by these homology groups we also prove that if these homology groups are completely reducible a sufficient condition for the existence of this resolution is the property that these groups are isomorphic to the kernel of the kostant quabla operator which is equivalent with disjointness of the boundary and coboundary operators then we use these results to derive very explicit conditions under which bgg resolutions exist which are particularly useful for the superalgebras of type i for the unitarisable representations of gl m n and osp 2 2 n we derive conditions on the parabolic subalgebra under which the bgg resolutions exist we also apply the obtained theory to construct specific examples of bgg resolutions for osp m 2 n finally we state some results for typical modules of gl m n and osp 2 2 n

scattering functions of knotted ring polymers we discuss the scattering function of a gaussian random polygon with n nodes under a given topological constraint through simulation we evaluate the form factor pk q of a gaussian polygon of n 200 having a fixed knot k for some different knots such as the trivial trefoil and figure eight knots here the gaussian polygons with different knots k have distinct values of the mean square radius of gyration r2 g k we obtain the kratky plots of the form factors i e the plots of qr g k 2pk q versus qr g k for the different topological constraints and discuss nontrivial large q behavior as well as small q behavior for the scattering functions we also find that the distinct values of r2 g k play an important role in the large q and small q properties of the kratky plots

linking probabilities of off lattice self avoiding polygons and the effects of excluded volume we evaluate numerically the probability of linking i e the probability of a given pair of self avoiding polygons saps being entangled and forming a nontrivial link type l in the simulation we generate pairs of saps of n spherical segments of radius rd such that they have no overlaps among the segments and each of the saps has the trivial knot type we evaluate the probability of a self avoiding pair of saps forming a given link type l for various link types with fixed distance r between the centers of mass of the two saps we define normalized distance r by where denotes the square root of the mean square radius of gyration of sap of the trivial knot 01 we introduce formulae expressing the linking probability as a function of normalized distance r which gives good fitting curves with respect to 2 values we also investigate the dependence of linking probabilities on the excluded volume parameter rd and the number of segments n quite interestingly the graph of linking probability versus normalized distance r shows no n dependence at a particular value of the excluded volume parameter rd 0 2

oscillation in nonlinear difference equations abstract in this paper we shall discuss oscillatory behavior of the solutions of difference equations including the self adjoint second order linear equation and the discrete version of the nonlinear wave equation our work is to give sufficient conditions such that every nontrivial solution of the equations oscillates

further oscillation criteria for delay partial difference equations this paper is concerned with the partial difference equation a m 1 n a m n 1 a m n p m na m k n l 0 m n 0 1 2 where k and l are two positive integers p m n is a real double sequence some new oscillation criteria for this equation are obtained

a reduction method for atypical representations of classical lie superalgebras abstract we introduce a reduction method for studying representations of classical lie superalgebras with atypical central character we show that the atypical quotient of universal enveloping algebra has a non trivial jacobson radical the factor by this radical has a new center which is calculated for sl 1 n and psl 2 2 using this center we obtain new character formulae generalization of borel weil bott and beilinson bernstein localization theorems

frequent oscillatory solutions of a nonlinear partial difference equation this paper is concerned with a class of nonlinear delay partial difference equations with variable coefficients which may change sign by making use of frequency measures some new oscillatory criteria are established

stability criteria for a class of linear delay partial difference equations this paper is concerned with the linear delay partial difference equation a i j u i 1 j u i j 1 b i j u i j p i j u i j 0 where a i j b i j p i j i j n0 are real sequences sufficient conditions for this equation to be stable are derived some conditions for this equation to be unstable are obtained also

exponents of intrachain correlation for self avoiding walks and knotted self avoiding polygons we show numerically that critical exponents for two point intrachain correlation of an infinite chain characterize those of finite chains in self avoiding walk saw and self avoiding polygon sap under a topological constraint we evaluate short distance exponents i j through the probability distribution functions of the distance between the ith and jth vertices of n step saw or sap with a knot for all pairs 1 i j n we construct the contour plot of i j and express it as a function of i and j we suggest that it has quite a simple structure here exponents i j generalize des cloizeaux s three critical exponents for short distance intrachain correlation of saw and we show the crossover among them we also evaluate the diffusion coefficient of knotted sap for a few knot types which can be calculated with the probability distribution functions of the distance between two nodes

stability criteria for two partial difference equations abstract this paper is concerned with two classes of linear partial difference equations with constant coefficients explicit conditions are derived which are sufficient and or necessary for these equations to have stable solutions

finite size and asymptotic behaviors of the gyration radius of knotted cylindrical self avoiding polygons several nontrivial properties are shown for the mean square radius of gyration r k 2 of ring polymers with a fixed knot type k through computer simulation we discuss both finite size and asymptotic behaviors of the gyration radius under the topological constraint for self avoiding polygons consisting of n cylindrical segments with radius r we find that the average size of ring polymers with the knot k can be much larger than that of no topological constraint the effective expansion due to the topological constraint depends strongly on the parameter r that is related to the excluded volume the topological expansion is particularly significant for the small r case where the simulation result is associated with that of random polygons with the knot k

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

strong and weak oscillatory behavior in time varying linear difference equations of arbitrary order in the presence of unmodeled dynamics and nominal parametrical errors abstract this paper is devoted to investigate the presence of oscillating solutions in time varying difference equations of arbitrary n th order even in the case when there are parametrical errors i e errors in the sequences defining their coefficients or unmodeled dynamics i e the current order n is unknown with n n 0 the nominal known order the problem is formulated linked to the concepts of conjugacy disconjugacy positivity and generalized zeros and very general conditions of oscillation are obtained both over particular intervals and for the whole solution some results concerned with the presence of stable asymptotically stable oscillatory solutions are also presented