universal amplitude ratios for self avoiding walks on the kagome lattice we have calculated exactly the number the mean square end to end distance the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on the kagome lattice up to 30 steps we have estimated the connective constant and the critical amplitudes our numerical results are consistent with the theoretical prediction by cardy and saleur 1989 on the universality for certain amplitude ratios

existence of nonoscillatory solutions of higher order neutral delay difference equations with variable coefficients abstract in this paper we consider the following higher order neutral delay difference equations with positive and negative coefficients m x n cx n k p n x n r q n x n l 0 n n 0 where c r m 1 k 1 r l 0 are integers and p n n n 0 and q n n n 0 are sequences of nonnegative real numbers we obtain the global results with respect to c which are some sufficient conditions for the existences of nonoscillatory solutions

universal amplitude combinations for self avoiding walks and polygons on the honeycomb lattice abstract we have calculated exactly the mean square end to end distance the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on the honeycomb lattice up to 46 steps we also calculated the mean square radius of gyration and moments of area for n step self avoiding polygons on the honeycomb lattice up to 60 steps we estimated the critical amplitudes and our numerical results are consistent with the theoretical predictions by the universality for certain amplitude combinations

nonoscillatory solutions of second order nonlinear difference equations abstract in this paper we will study the nonoscillatory solution of the second order nonlinear difference equation x n 1 1 x n 1 q n x n 1 x n 0 where q n n 0 is a nonnegative sequence and 0 new sufficient conditions are established for the existence of nonoscillatory solutions of the equation which generalize and complement the main result in y g sun oscillation and nonoscillation for second order linear difference equations appl math comput 170 2005 1095 1103

oscillation for higher order neutral superlinear delay difference equations with unstable type first we prove that the following even order neutral superlinear delay difference equation with unstable type d m x n p nx n t q nx n s a n n 0 has always unbounded and nonoscillatory solution then some bounded oscillatory criteria for equation are obtained

universal distance ratios for 2d saws series results the mean square distance of a monomer to the origin and the mean square distance of the centre of mass to the origin are studied by exact enumeration on the square and triangular lattices the numerical results are inconsistent with the theoretical prediction of cardy and saleur 1989 by more than fifteen times the numerical uncertainty in the authors calculation

fixed sign solutions of second order difference equations with neumann boundary conditions in this paper we study the existence of solution of the nonlinear second order difference problem with neumann boundary conditions assuming the existence of a pair of ordered lower and upper solutions and we obtain optimal existence results for the case and even for to this end we do an exhaustive study of the values of the real parameters and for which the associated green s function to the linear operator l u u 2 2 u 1 u with neumann boundary conditions has fixed sign

oscillation for second order nonlinear neutral delay difference equations in this paper we obtain sufficient conditions for the oscillation of all solutions of the second order nonlinear delay difference email protected a n d x n p nx g n a f n x s n 0 where dx n x n 1 x n is the forward difference operator p n is a sequence of nonnegative real numbers and a 1 is a quotient of positive odd integers our results extend and improve the known results in the literature an example that dwell up the advantage of our results is included

time domain analysis of the sommerfeld vmd problem based on the exact image theory a novel time domain solution of the sommerfeld half space problem is constructed through the exact image theory the radiation from a vertical magnetic dipole vmd above the planar interface of two lossless and dispersionless dielectric media is analyzed the reflection field is given as a laplace transform when the refraction index of the lower medium is greater the converse case of refraction indexes is also studied and the reflection field is written as originating from an image source located in homogeneous space the present theory is seen to lead to a numerically efficient result which is demonstrated with examples

error analysis for far field approximate expression of sommerfeld type integral the error of the sommerfeld type integral approximate expression given by chew 1988 in the far field is discussed it is demonstrated that chew s method is correct not only intuitively but also rigorously

universal amplitude ratios for three dimensional self avoiding walks we have calculated exactly the number the mean square end to end distance the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on the simple cubic diamond body centred cubic and face centred cubic lattices respectively up to 20 30 16 and 13 steps by a computer two universal amplitude ratios are estimated

on existence of positive solutions and bounded oscillations for neutral difference equations abstract a criterion for the existence of a positive solution for the first order difference equation xn cxn m pnxn k 0 pn 0 is established results are also obtained for the oscillation and nonoscillation of solutions of a second order difference equation 2 xn cxn m pnxn k where the pn are either of constant sign or are oscillatory some of the results are sharp

comment on the universality class of self avoiding walks on a two layer square lattice abstract we have calculated exactly the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on a two layer square lattice up to 24 steps we estimated two universal amplitude ratios and the results provide additional evidence that self avoiding walks on a two layer lattice belongs to the same universality class as the self avoiding walk on a planar lattice

universal amplitude combinations for self avoiding walks and polygons on directed lattices two directed lattices the l lattice and manhattan lattice are studied we have calculated exactly the number the mean square end to end distance the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on the l lattice and manhattan lattice for up to 60 and 50 steps respectively we have also computed the number and mean square radius of gyration for self avoiding polygons on the l lattice and manhattan lattice for up to 80 and 60 steps respectively we have estimated the critical amplitudes and our numerical results are consistent with the conjecture of universality for certain amplitude combinations however some amplitude combinations have values different from the corresponding values for undirected lattices

oscillation and comparison results for delay difference equations abstract we establish a necessary and sufficient condition for the existence of positive solutions and obtain sturm type comparison theorem for the difference equation x n 1 x n i 1 m pi n x n ki o n 0 we also obtain the following comparison result equation oscillates if and only if x n 1 x n i 1 m qi n x n ki o n 0 oscillates provided n lim pi n qi i 1 2 m

some computational aspects of pocklington s electric field integral equation for thin wires a procedure is presented for extracting the singularity from the kernel in pocklington s electric field integral equation for thin wires the modified kernel which results may be used to dramatically improve the computational efficiency of certain moment method formulations a simple example is considered in which pulse basis functions and point matching are used for this case a convenient approximation of the self impedance term is also derived

enumeration of self avoiding walks on the square lattice we describe a new algorithm for the enumeration of self avoiding walks on the square lattice using up to 128 processors on a hp alpha server cluster we have enumerated the number of self avoiding walks on the square lattice to length 71 series for the metric properties of mean square end to end distance mean square radius of gyration and mean square distance of monomers from the end points have been derived to length 59 an analysis of the resulting series yields accurate estimates of the critical exponents and confirming predictions of their exact values likewise we obtain accurate amplitude estimates yielding precise values for certain universal amplitude combinations finally we report on an analysis giving compelling evidence that the leading non analytic correction to scaling exponent 1 3 2

approximative solutions to difference equations of neutral type asymptotic properties of solutions to difference equations of the form m x n u n x n k a n f x n b n are studied replacing the sequence u by its limit and the right side of the equation by zero we obtain an equation which we call the fundamental equation first we investigate the space of all solutions of the fundamental equation we show that any such solution is a sum of a polynomial sequence and a product of a geometric sequence and a periodic sequence next using a new version of the krasnoselski fixed point theorem and the iterated remainder operator we establish sufficient conditions under which a given solution of the fundamental equation is an approximative solution to the above equation our approach based on the iterated remainder operator allows us to control the degree of approximation in this paper we use o ns for a given nonpositive real s as a measure of approximation

accurate evaluation of sommerfeld integrals using the fast fourier transform it is shown that the fast fourier transform fft combines naturally with simpson s rule for sommerfeld type integral computation the principal advantage of using the fft is that a single subroutine call yields a set of sample values of an integral i e the integral for various values of an integrand parameter such samples could be useful in themselves in other applications sommerfeld integrals represent green s functions nested within other spatial integrals so samples from the fft might be useful in approximating the outernested integral several examples are provided to illustrate the process

asymptotic properties of solutions of second order neutral difference equations abstract in this paper we study asymptotic behavior of solutions of second order neutral difference equation 2 x n px n k f n x n 0 we present conditions under which all nonoscillatory solutions of the above equation have the property x n cn o n for some c r as well as sufficient conditions under which all nonoscillatory solutions are asymptotically linear

oscillation and nonoscillation for second order linear differential equations abstract new oscillation and nonoscillation theorems are obtained for the second order linear differential equation u p t u 0 where p t c 0 and p t 0 conditions only about the integrals of p t on every interval 2 n t 0 2 n 1 t 0 n 1 2 for some fixed t 0 0 are used in the results

explicit conditions for the oscillation of difference equations we obtain sufficient conditions for the oscillation of all solutions of the difference equation an 1 an i 1mpian ki 0 n 0 1 2 where the pi s are real numbers and the ki s are integers the conditions are given explicitly in terms of the pi s and the ki s

on the oscillation of solutions and existence of positive solutions of neutral difference equations abstract we obtain sufficient conditions for the oscillation of all solutions and existence of positive solutions of the neutral difference equation x n cx n m p n x n k 0 n 0 1 2 where c and p n are real numbers m and k are integers and p n m and k are nonnegative

new oscillation criteria for second order linear difference equations with positive and negative coefficients the oscillation of second order neutral difference equations with positive and negative coefficients of the form d 2 x n la nx n t p nx n d q nx n s 0 l 1 is investigated we obtain many new results using the comparison between both first order and second order difference equations an example is given to show the strength of the obtained results

comparison theorems for second order nonlinear difference equations abstract new oscillation results are obtained for the second order nonlinear difference equation r n f x n 1 g n x n 0 and its functional form r n f x n 1 g n x n 0 the role played by the argument n on the oscillation of the functional equation is explored in particular we characterize a class of sequences n which have a harmless effect on the oscillation of this type of equations some of our results generalize improve or unify known fundamental oscillation results for several particular cases of the above equations

remarks on oscillation of second order linear difference equation in this remark we shall show that the lemma 1 and parts of the proof of theorem 1 in the paper oscillation criteria for second order linear difference equations applied mathematics and computation 145 2003 591 601 are incorrect

self avoiding rings on the triangular lattice the finite lattice method of series expansion is used to extend the enumeration of self avoiding polygons on the triangular lattice through to rings of 35 steps the authors also give the enumeration of these triangular polygons grouped by perimeter and area of up to 21 unit triangles for perimeters up to 24 steps

universal amplitudes ratio at the collapse transition of polymers in two dimensions first values of the universal ratio of the mean square radius of gyration and the mean square end to end distance for 2d linear polymers at their collapse transition are calculated by the scanning simulation method two models are considered i self avoiding walks saws with nearest neighbour attractions for which this ratio is found to be 0 179 or 0 003 at the theta point ii self intersecting trails at their tricritical point for which the value 0 166 or 0 002 is obtained this provides yet another indication that these models may belong to different tricritical universality classes the authors have also computed this ratio for trails at the high temperature limit with non interacting intersections and found 0 141 or 0 001 in excellent agreement with earlier estimates of this ratio for saws in their swollen phase

oscillation of second order perturbed nonlinear difference equations using the riccati transformation techniques we will establish some new oscillation criteria for the second order perturbed nonlinear difference equation d a n 1 dx n 1 c f n x n g n x n dx n n 1where c 0 is a quotient of odd positive integers some comparison between our theorems and those previously known results are indicated examples are interested in the text to illustrate the relevance of our results

oscillations of delay difference equations in a critical state consider the first order delay difference equation xn i 1m pi n xn ki 0 n 0 1 2 in the case where lim infn pi n pi 0 ki 0 i 1 2 a necessary and sufficient condition for the oscillation of all solutions of the above equation is established in the critical state that the corresponding limiting equation xn i 1m pixn ki 0 n 0 1 2 admits a nonoscillatory solution it is to be pointed out that there is no result in this critical state for a delay difference equation with more than one delay argument

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

universal distance ratios for two dimensional polymers by using a quantitative version of the c theorem in conformal theories the authors determine some universal geometrical features of two dimensional critical systems with emphasis on the ratios of mean square distances for polymers

semicycles of solutions of nonlinear difference equations with several delays abstract in this paper we study the semicycles of oscillatory solutions of nonlinear difference equation with several delays where pi n is a real sequence with pi n 0 for all large n l mi kij are positive integers ij i 1 2 l j 1 2 mi are nonnegative real numbers with mij 1 ij 1 i 1 2 l upper bound of numbers of terms of semicycles are determined

existence for nonoscillatory solutions of higher order nonlinear neutral difference equations abstract in this paper we consider the following forced higher order nonlinear neutral difference equation m x n c n x n k s 1 u p s n f s x n r s q n n n 0 where m u 1 k 0 and rs 0 are integers cn p s n s 1 2 u and qn are sequence of real numbers and f s c r r s 1 2 u by using krasnoselskii s fixed point theorem and some new techniques we obtain sufficient conditions for the existence of nonoscillatory solutions for general p s n s 1 2 u and qn which means that we allow oscillatory p s n s 1 2 u and qn

new numerical approach in the analysis of a thin wire radiating over a lossy half space the finite element formulation for a half space electric field integral equation is described sommerfeld integrals appearing in the kernel of the integral equation are calculated by means of exponential approximations this approach shows advantages over the usual techniques obtained results are compared with other results available

application of the fast fourier transform to the computation of the sommerfeld integral for a vertical electric dipole above a half space the fast fourier transform fft is used in conjunction with analytical techniques to obtain an approximate expression for the sommerfeld integral associated with a vertical electric dipole above a half space the resulting expression which is an analytic function of the source and observation locations can be used in lieu of a numerical integration of the sommerfeld integral it can be used in either the near zone or the far zone and it can even be applied for observation angles close to grazing the accuracy of the expansion directly depends on the number of terms used in the fft results obtained using the expansion are compared with those obtained both from a direct numerical integration of the sommerfeld integral and the reflection coefficient method

oscillation and non oscillation for second order linear difference equations abstract we prove new oscillation and non oscillation theorems for the second order linear difference equation 2xn 1 pnxn 0 where p n n 1 is a real sequence with pn 0 these results are extensions of earlier results of zhang and zhou comput math appl 39 2000 1 7

classification of neutral difference equations of any order with respect to the asymptotic behavior of their solutions the asymptotic behavior of the solutions to a neutral difference equation of the email protected mx n q n x t n p n x s n 0 email protected n n 0 where t n is a general retarded argument s n is a general deviated argument p n n 0 and q n n 0 are sequences of real numbers d denotes the forward difference operator dx n x n 1 x n and d j denotes the jth forward difference operator d jx n email protected d j 1x n for j 2 3 m is studied examples illustrating the results are also given

nonoscillation and oscillation of second order half linear difference equations this paper considers the oscillation problem for the second order half linear difference equation xn 1 r 1 xn 1 qn xn r 1xn 0 where qn n 0 is a nonnegative sequence and r 0 we establish new criteria for this equation to be nonoscillatory and oscillatory respectively when r 1 our results are complete extensions of work by zhang and zhou oscillation and nonoscillation for second order linear difference equations comp math appl 39 2000 1 7 and reduce to the results in y g sun oscillation and nonoscillation for second order linear difference equations appl math comput 170 2005 1095 1103 when r 0 our results generalize and complement sun s results in y g sun oscillation and nonoscillation for half linear second order difference equations indian j pure appl math 35 2004 133 142

oscillations of higher order linear difference equations abstract consider the higher order linear difference equation where m 1 is an odd integer and p n is a sequence of nonnegative real numbers we obtain several new sufficient conditions for the oscillation of all solutions of equation examples which dwell upon the importance of our results are also included

asymptotic behaviour of bounded solutions of a class of higher order neutral dynamic equations in this paper necessary and sufficient conditions for the asymptotic behaviour of all bounded solutions of a class of nonlinear higher order neutral dynamic equations are stated related to various ranges of the coefficient associated with the neutral part in one of which the coefficient is allowed to oscillate and not tend to zero asymptotically the new results improve and generalize the known results stated for difference equations and differential equations and answer some problems left open for the case t r for arbitrary time scales some certain examples which the results in the literature fail to deliver an answer to are given to illustrate the applicability and significance of the new results

oscillation and nonoscillation for second order linear difference equations new oscillation and nonoscillation theorems are obtained for the second order linear difference equation 2xn 1 pnxn 0 where if pn n 1 is a real sequence with pn 0