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

soliton solutions for generalized quasilinear schrodinger equations abstract by introducing a new variable replacement we study the existence of nontrivial solutions for generalized quasilinear schrodinger equations which appear from plasma physics as well as high power ultrashort laser in matter

universality class of trails in two dimensions a trail is a walk on a lattice that may visit a site more than once but a bond at most once we have carried out transfer matrix studies of trails on the square lattice and of hybrid walks that interpolate between self avoiding walks and trails the results are in agreement with the same universal exponents as self avoiding walks however the finite size corrections are much larger than for self avoiding walks an explanation in terms of an irrelevant variable with scaling index is given

monte carlo studies of three dimensional two step restricted self avoiding walks two step restricted walk tsrw models are a class of restricted self avoiding walk saw where in addition to the self avoidance constraint certain restrictions are placed upon each pair of successive steps in this paper we explore the relationship between the restrictions and the scaling of the average size of walks in three dimensional models we use the pruned enriched rosenbluth method algorithm to perform monte carlo studies in five representative tsrw models in three dimensions the results present strong numerical evidence that all non trivial tsrw models in three dimensions have the same size scaling behaviour as unrestricted saws this is in contrast to two dimensions where several universality classes are accepted to exist in particular we find no rule analogous to the spiral walk of two dimensions

multiplicity results for some nonlinear schrodinger equations with potentials multiple semiclassical standing waves for a class of nonlinear schrodinger equations with potentials are found by means of a perturbative variational method

multiplicity and concentration of solutions for a quasilinear choquard equation in this paper we study a quasilinear choquard equation involving the p laplacian operator and a potential function v under suitable assumptions on v and the nonlinearity we prove the existence multiplicity and concentration of solutions for the equation by variational methods

iterated function system and ruelle operator abstract we consider a generalized iterated function system where the weights are variable functions by using the ruelle operator and a dynamical system consideration we prove that if the system is contractive and the weights are strictly positive functions and satisfy the dini condition then there exists a unique eigenmeasure corresponding to the ruelle operator on the attractor if in addition the maps are conformal and satisfy the open set condition then we prove that they satisfy the strong open set condition and by using this we can give a description of the l p scaling spectrum and the multifractal structure of the eigenmeasure the work extends some results of proc london math soc 73 1996 105 154 adv appl math 19 1997 486 513 j statist phys 86 1997 233 275 indiana univ math j 42 1993 367 411

singularly perturbed critical choquard equations abstract in this paper we study the semiclassical limit for the singularly perturbed choquard equation e 2 u v x u e 3 r 3 q y g u y x y d y q x g u in r 3 where 0 3 e is a positive parameter v q are two continuous real function on r 3 and g is the primitive of g which is of critical growth due to the hardy littlewood sobolev inequality under suitable assumptions on g we first establish the existence of ground states for the critical choquard equation with constant coefficients next we establish existence and multiplicity of semi classical solutions and characterize the concentration behavior by variational methods

vector valued ruelle operators abstract the scalar ruelle operators have been extensively studied both in dynamical systems and iterated function systems and the ruelle perron frobenius theorem for scalar ruelle operator was well known we mainly set up a vector analogue theory in this paper

multiple solutions for quasilinear elliptic equations with a finite potential well abstract we consider multiplicity of solutions for a class of quasilinear problems which has received considerable attention in the past including the so called modified nonlinear schrodinger equations by combining a new variational approach via q laplacian regularization and the compactness arguments from 4 we establish infinitely many bound state solutions for the quasilinear schrodinger type equations extending the earlier work of 4 for semilinear equations

localized nodal solutions for quasilinear schrodinger equations abstract in this paper we study the existence of localized nodal solutions for a class of semiclassical quasilinear schrodinger equations including as a special case the modified nonlinear schrodinger equation mnls e 2 v 1 2 v v 2 v x v v q 2 v 0 in r n v x 0 as x we establish for small e the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function by developing new variational perturbation method to treat this class of non smooth variational problems the new method allows the perturbed variational functionals to share critical points with the original functional this method allows us to avoid any limiting process from the perturbed problems to the original problem and it is effective in dealing with multiple existence of solutions

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

lattice trails i exact results considers the problem of lattice trails introduced by malakis 1975 1976 the author proves the existence of a finite connective constant and establishes a result for the growth of n step trails tn analogous to the best known result for self avoiding walks tn lambda n expt o n for the honeycomb d 2 and lave s lattice d 3 the author establishes a counting theorem from which one can deduce the exact value of the connective constant lambda for the honeycomb lattice lambda 2 2 square root 2 further it follows that the trail problem is in the same universality class as the self avoiding walk problem for those lattices an exact amplitude relation between trails and self avoiding walks and between dumb bells and trails and self avoiding walks is established the non existence of a counting theorem for arbitrary lattices is established an inequality for the triangular lattice connective constant is proved a high density expansion for lambda for the d dimensional hybercubic lattice is also obtained the author argues that contrary to recent suggestions the model is in the same universality class as the self avoiding walk model

scaling of self avoiding walks and self avoiding trails in three dimensions motivated by recent claims of a proof that the length scale exponent for the end to end distance scaling of self avoiding walks is precisely 7 12 0 5833 we present results of large scale simulations of self avoiding walks and self avoiding trails with repulsive contact interactions on the simple cubic lattice we find no evidence to support this claim our estimate 0 5874 2 is in accord with the best previous results from simulations

pruned enriched rosenbluth method simulations of θ polymers of chain length up to 1 000 000 we present an algorithm for simulating flexible chain polymers it combines the rosenbluth rosenbluth method with recursive enrichment although it can be applied also in more general situations it is most efficient for three dimensional theta polymers on the simple cubic lattice there it allows high statistics simulations of chains of length up to n 10 sup 6 for storage reasons this is feasable only for polymers in a finite volume for free theta polymers in infinite volume we present very high statistics runs with n 10000 these simulations fully agree with previous simulations made by hegger and grassberger j chem phys bold 102 6681 1995 with a similar but less efficient algorithm showing that logarithmic corrections to mean field behavior are much stronger than predicted by field theory but the finite volume simulations show that the density inside a collapsed globule scales with the distance from the theta point as predicted by mean field theory in contrast to claims in the work mentioned above in addition to the simple cubic lattice we also studied two versions of the bond fluctuation model but with much shorter chains finally we show that our method can be applied also to off lattice models and illustrate this with simulations of amore model studied in detail by freire ital et al macromolecules bold 19 452 1986 and later work copyright ital 1997 ital the american physical society less

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

link lengths and their growth powers for a certain infinite family of knots or links we study the growth power ratios of their stick number lattice stick number minimum lattice length and minimum ropelength compared with their minimum crossing number c k for every it is known that the stick number and lattice stick number grow between the and linear power of the crossing number and minimum lattice length and minimum ropelength grow with at least the power of crossing number which is called the four thirds power law furthermore the minimal lattice length and minimum ropelength grow at most as o but it is unknown whether any family exhibits superlinear growth for any real number r between and 1 we give an infinite family of non splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number furthermore for any real number r between and 1 we give another infinite family of non splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number

vector valued submarkov operators and recurrent iterated function systems abstract we consider a recurrent ifs with place dependent transition weights by using the quasi compact operator theory and the spectral theory of positive operators the eigen problem of the given system is studied this is the vector analogue of fan and lau s results 1999 j math anal appl 231 319 344

lattice trails ii numerical results for pt i see ibid vol 18 p 567 1985 a numerical study of the properties of lattice trails on the honeycomb square triangular and simple cubic lattices is made critical points are estimated for all lattices and upper and lower bounds established extensive series have been obtained and series analysis of both trail generating functions and mean square end to end distance series are not inconsistent with the conclusion that the problem is in the same universality class as the self avoiding walk problem a pseudo star triangle transformation is defined and the analyticity properties of that function coupled with previous exact results clearly supports that conclusion for the triangular lattice as well as providing excellent unbiased critical point estimates the author also shows that the connective constant for d or 2 dimensional hypercubic trails is strictly greater than the corresponding quantity for saws

scaling of self avoiding walks in high dimensions we examine self avoiding walks in dimensions 4 to 8 using high precision monte carlo simulations up to length n 16 384 providing the first such results in dimensions d 4 on which we concentrate our analysis we analyse the scaling behaviour of the partition function and the statistics of nearest neighbour contacts as well as the average geometric size of the walks and compare our results to 1 d expansions and to excellent rigorous bounds that exist in particular we obtain precise values for the connective constants 5 8 838 544 3 6 10 878 094 4 7 12 902 817 3 8 14 919 257 2 and give a revised estimate of 4 6 774 043 5 all of these are by at least one order of magnitude more accurate than those previously given from other approaches in d 4 and all approaches in d 4 our results are consistent with most theoretical predictions though in d 5 we find clear evidence of anomalous n 1 2 corrections for the scaling of the geometric size of the walks which we understand as a non analytic correction to scaling of the general form n 4 d 2 not present in pure gaussian random walks

vector valued ruelle operator with weakly contractive ifs abstract the vector valued ruelle operator defined by contractive iterated function systems ifs was discussed by the author y l ye vector valued ruelle operator j math anal appl 299 2004 341 356 in this paper we consider vector valued ruelle operators defined by weakly contractive ifs and a vector valued analogue of the ruelle perron frobenius theorem for the scalar ruelle operator is set up our theorem gives a sufficient condition for the vector valued ruelle operator to be quasi compact under this sufficient condition we prove that the rate of convergence of the iterated operators is exponential

the cauchy problem for quasi linear schrodinger equations with x rn t r with x1 xn and summation convention one may think of the equation in 1 1 as a non linear schrodinger equation where the operator modeling the dispersion relation is non isotropic and depends also on the unknown function its conjugate and their gradients in the space variables under appropriate assumptions on the function f the method presented here extends to fully non linear schrodinger equations of the form

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

ground state solutions for the quasilinear schrodinger equation abstract this paper is concerned with the quasilinear schrodinger equation u a x 1 u 1 2 u 2 u u p u 0 x r n with a x 0 having a potential well and 4 p 1 2 2 by using the variational method and the concentration compactness method in orlicz space we prove the existence of a ground state solution which localizes near the potential well int a 1 0 for large enough

multifractal measures and a weak separation condition abstract we define a new separation property on the family of contractive similitudes that allows certain overlappings this property is weaker than the open set condition of hutchinson it includes the well known class of infinite bernoulli convolutions associated with the p v numbers and the solutions of the two scale dilation equations our main purpose in this paper is to prove the multifractal formalism under such condition

concentrating standing waves for the fractional nonlinear schrodinger equation abstract we consider the semilinear equation e 2 s s u v x u u p 0 u 0 u h 2 s r n where 0 s 1 1 p n 2 s n 2 s v x is a sufficiently smooth potential with inf r v x 0 and e 0 is a small number letting w be the radial ground state of s w w w p 0 in h 2 s r n we build solutions of the form u e x i 1 k w i x i e e where i v i e and the i e approach suitable critical points of v via a lyapunov schmidt variational reduction we recover various existence results already known for the case s 1 in particular such a solution exists around k nondegenerate critical points of v for s 1 this corresponds to the classical results by floer and weinstein 13 and oh 24 25

self avoiding walk enumeration via the lace expansion we introduce a new method for the enumeration of self avoiding walks based on the lace expansion we also introduce an algorithmic improvement called the two step method for self avoiding walk enumeration problems we obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3 we enumerate 32 step self avoiding polygons in d 3 26 step self avoiding polygons in d 4 30 step self avoiding walks in d 3 and 24 step self avoiding walks and polygons in all dimensions d 4 we analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8 we also provide major extensions of 1 d expansions for the connective constant and for two critical amplitudes

variational perturbative methods and bifurcation of bound states from the essential spectrum this paper consists of two main parts the first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results the second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum

on the location and profile of spike layer nodal solutions to nonlinear schrodinger equations abstract the existence and concentration behavior of a nodal solution are established for the equation e 2 u v x u f u in where is a domain in r n not necessarily bounded v is a positive holder continuous function and f c 1 r is an odd function having subcritical and superlinear growth under additional local condition on v it is proved that such a solution has just one positive and one negative peaks which are located around local minima of v

configuration spaces transfer and 2 nodal solutions of a semiclassical nonlinear schrodinger equation we establish new lower bounds for the number of nodal bound states for the semiclassical nonlinear schrodinger equation varepsilon 2 delta u a x u u p 2 u with bounded and uniformly continuous potential a the solutions we obtain have precisely two nodal domains and their positive and negative parts concentrate near the set of minimum points of a our approach is independent of penalization techniques and yields in some cases the existence of infinitely many nodal solutions for fixed varepsilon via a dynamical systems approach we exhibit positively invariant sets of sign changing functions for the negative gradient flow of the associated energy functional we analyze these sets on the cohomology level with the help of dold s fixed point transfer in particular we estimate their cuplength in terms of the cuplength of equivariant configuration spaces of subsets of mathbb r n we also provide new estimates of the cuplength of configuration spaces

existence and multiplicity of solutions for fourth order elliptic equations of kirchhoff type with critical growth in ℝn in this paper we deal with the existence and multiplicity of solutions for fourth order elliptic equations of kirchhoff type with critical nonlinearity e 4 2 u e 4 a b r n u 2 d x u v x u u 2 2 u h x u t x n by using lions second concentration compactness principle and concentration compactness principle at infinity to prove that ps condition holds locally and by variational method we prove that it has at least one solution and for any m it has at least m pairs of solutions

multifractal decompositions of digraph recursive fractals we prove that the multifractal decomposition behaves as expected for a family of sets k known as digraph recursive fractals using measures of markov type for each value of a parameter between a minimum min and maximum max we define multifractal components k of k and show that they are fractals in the sense of taylor the dimension f of k is computed from the data of the problem the typical concave multifractal f dimension spectrum curve results

global existence of small solutions to a relativistic nonlinear schrodinger equation on etudie le probleme de cauchy associea une equation de schrodinger non lineaire modelisant une impulsion laser ultra intense et ultra courte dans un plasma les termes non lineaires nouveaux sont dus aux effets relativistes et a la force ponderomotrice nous prouvons l existence locale et l unicite de solutions petites lorsque la dimension de l espace transverse est egale a 2 ou 3 ainsi que l existence locale pour des donnees initiales de taille arbitraire en dimension un d espace transverse

renormalized four point coupling constant in the three dimensional o n model with n 0 we simulate self avoiding walks on a cubic lattice and determine the second virial coefficient for walks of different lengths this allows us to determine the critical value of the renormalized four point coupling constant in the three dimensional n vector universality class for n 0 we obtain g 1 4005 5 where g is normalized so that the three dimensional field theoretical beta function behaves as beta g g g 2 for small g as a byproduct we also obtain precise estimates of the interpenetration ratio psi psi 0 24685 11 and of the exponent nu nu 0 5876 2

solutions for a quasilinear schrodinger equation a dual approach abstract we consider quasilinear stationary schrodinger equations of the form 1 u u 2 u g x u x r n introducing a change of unknown we transform the search of solutions u x of 1 into the search of solutions v x of the semilinear equation 2 v 1 1 2f 2 v g x f v x r n where f is suitably chosen if v is a classical solution of 2 then u f v is a classical solution of 1 variational methods are then used to obtain various existence results

standing waves with a critical frequency for nonlinear schrodinger equations this paper is concerned with the existence and qualitative property of standing wave solutions for the nonlinear schrodinger equation with e being a critical frequency in the sense that we show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of v and whose amplitude goes to 0 as moreover depending upon the local behaviour of the potential function v x near the minimum points the limiting profile of the standing wave solutions will be shown to exhibit quite different characteristic features this is in striking contrast with the non critical frequency case which has been extensively studied in recent years

exact and monte carlo study of adsorption of a self interacting polymer chain for a family of three dimensional fractals we study the problem of adsorption of self interacting linear polymers situated in fractal containers that belong to the three dimensional 3d sierpinski gasket sg family of fractals each member of the 3d sg fractal family has a fractal impenetrable 2d adsorbing surface which is in fact 2d sg fractal and can be labelled by an integer b 2 b by applying the exact and monte carlo renormalization group mcrg method we calculate the critical exponents associated with the mean squared end to end distance of polymers and associated with the number of adsorbed monomers for a sequence of fractals with 2 b 4 exactly and 2 b 40 monte carlo we find that both and monotonically decrease with increasing b that is with increase of the container fractal dimension df and the interesting fact that both functions b and b cross the estimated euclidean values besides we establish the phase diagrams for fractals with b 3 and b 4 which reveal the existence of six different phases that merge together at a multi critical point whose nature depends on the value of the monomer energy in the layer adjacent to the adsorbing surface

positive and sign changing clusters around saddle points of the potential for nonlinear elliptic problems we study the existence and asymptotic behavior of positive and sign changing multipeak solutions for the equation varepsilon 2 delta v v x v f v quad rm in mathbb r n where e is a small positive parameter f a superlinear subcritical and odd nonlinearity v a uniformly positive potential no symmetry on v is assumed it is known kang and wei in adv differ equ 5 899 928 2000 that this equation has positive multipeak solutions with all peaks approaching a local maximum of v it is also proved that solutions alternating positive and negative spikes exist in the case of a minimum see d aprile and pistoia in ann inst h poincare anal non lineaire 26 1423 1451 2009 the aim of this paper is to show the existence of both positive and sign changing multipeak solutions around a nondegenerate saddle point of v

high precision estimate of the hydrodynamic radius for self avoiding walks the universal asymptotic amplitude ratio between the gyration radius and the hydrodynamic radius of self avoiding walks is estimated by high resolution monte carlo simulations by studying chains of length of up to n 2 25 34 10 6 monomers we find that the ratio takes the value r g r h 1 5803940 45 which is several orders of magnitude more accurate than the previous state of the art this is facilitated by a sampling scheme which is quite general and which allows for the efficient estimation of averages of a large class of observables the competing corrections to scaling for the hydrodynamic radius are clearly discernible we also find improved estimates for other universal properties that measure the chain dimension in particular a method of analysis which eliminates the leading correction to scaling results in a highly accurate estimate for the flory exponent of 0 58759700 40

multi bump solutions and multi tower solutions for equations on rn let 0 be a small parameter in this paper we study existence of multiple multi bump positive solutions for the semilinear schrodinger equation u u 1 a x u p 2u u h1 rn where n 1 2 2 if n 1 or n 2 a c rn a x 0 for x rn and lim x a x 0 we also study existence of multiple multi tower positive solutions for the prescribed scalar curvature equation u 1 k x un 2n 2 u d1 2 rn where n 3 k c 0 k r 0 for r 0 limr 0k r 0 and limr k r 0