encounter distribution of two random walkers on a finite one dimensional interval we analyse the first passage properties of two random walkers confined to a finite one dimensional domain for the case of absorbing boundaries at the endpoints of the interval we derive the probability that the two particles meet before either one of them becomes absorbed at one of the boundaries for the case of reflecting boundaries we obtain the mean first encounter time of the two particles our approach leads to closed form expressions that are more easily tractable than a previously derived solution in terms of the weierstrass elliptic function

a complete axiom system for propositional projection temporal logic with cylinder computation model to specify and verify multi core parallel programs in a uniform framework this paper proposes an axiom system for ccm pptl which extends that of pptl by including transformation rules for sequence expressions and axioms as well as inference rules on the ccm construct further the soundness and completeness of the extended axiom system are proved

a family of almost periodic schrodinger operators bellissard et al 1 introduced a one dimensional almost periodic discrete schrodinger operator which is defined by a parameter we allow this parameter to become complex and develop a geometric formalism to control the operator the support of the spectrum of this operator is the julia set of the mapping x x2 we prove that the almost periodicity holds over wide regions of the complex plane even though hermiticity fails

iterated networks and the spectra of renormalizable electromechanical systems beginning with anlrc network with impedance functionz a sequence of iterated networksnk with impedance functionsz k k 1 2 3 is introduced the asymptotic comportment ofz k and the spectra ofnk are analyzed in terms of the julia set ofz an example is given of an iterated network associated with a cascade of period doubling bifurcations

spectra of certain jacobi operators we discuss the spectra and in particular the essential spectra of some bounded self adjoint jacobi operators associated with orthogonal polynomial sequences obtained via polynomial mappings

condensed julia sets with an application to a fractal lattice model hamiltonian on considere l ensemble de julia pour l application rationnelle complexe z z 2 e z ou et e sont des parametres complexes dans la limite e 0

orthogonal and l q extremal polynomials on inverse images of polynomial mappings abstract let t be a polynomial of degree n and let k be a compact set with c first it is shown if zero is a best approximation to f from p n on k with respect to the lq norm q 1 then zero is also a best approximation to f t on t 1 k with respect to the l q t norm where t arises from by the transformation t in particular t is the equilibrium measure on t 1 k if is the equilibrium measure on k for q i e the sup norm a corresponding result is presented in this way polynomials minimal on several intervals on lemniscates on equipotential lines of compact sets etc are obtained special attention is given to lq minimal polynomials on julia sets next based on asymptotic results of widom we show that the minimum deviation of polynomials orthogonal with respect to a positive measure on t 1 k behaves asymptotically periodic and that the orthogonal polynomials have an asymptotically periodic behaviour too some open problems are also given

an explicit theory of heights we consider the problem of explicitly determining the naive height constants for jacobians of hyperelliptic curves for genus 1 it is impractical to apply hilbert s nullstellensatz directly to the defining equations of the duplication law we indicate how this technical difficulty can be overcome by use of isogenies the height constants are computed in detail for the jacobian of an arbitrary curve of genus 2 and we apply the technique to compute generators of j q the mordell weil group for a selection of rank 1 examples

the group law on the jacobian of a curve of genus 2 an explicit description is given of the group law on the jacobian of a curve c of genus 2 the kummer surface provides a useful intermediary stage bilinear forms relating to the kummer surface imply that the global group law may be given projectively by biquadratic forms defined over the same ring as the coefficients of c it is not assumed that c has a rational weierstrass point and the theory presented applies over an arbitrary ground field

two coverings of jacobians of curves of genus 2 given a curve c of genus 2 defined over a field k of characteristic different from 2 with a jacobian variety j we show that the two coverings corresponding to elements of a large subgroup of h1 gal ks k j 2 ks containing the selmer group when k is a global field can be embedded as an intersection of 72 quadrics in 15k just as the jacobian j itself moreover we actually give explicit equations for the models of these twists in the generic case extending the work of gordon and grant which applied only to the case when all weierstrass points are rational in addition we describe elegant equations of the jacobian itself and answer a question of cassels and flynn concerning a map from the kummer surface in 3 to the desingularized kummer surface in 5

orthogonal polynomials and cubic transformations abstract starting from a sequence pn of orthogonal polynomials with respect to a quasi definite or regular linear functional we construct a new sequence qn of orthogonal polynomials such that q3n x pn x3 the basic tool is the three term recurrence relation which such a family satisfies as an application we present some examples concerning chebyshev and hermite polynomials

almost periodic jacobi matrices associated with julia sets for polynomials let be the jacobi matrix associated with polynomialt z of degreen 2 the spectrum of is the julia set associated witht z which in many cases is a cantor set let 1 denote the result of omitting the first row and column ofj then it is shown that the spectrum of 1 may be purely discrete

on a sequence of rational functions abstract we study a sequence of rational functions considered by bessis mehta and moussa 1982 and we show that a conjecture given in this paper is not correct moreover we give a relation between values of these rational functions and a finite automaton

meblin transforms associated with julia sets and physical applications we introduce the mellin transform of the balanced invariant measure associated to the julia set generated by a rational transformation we show that its analytic continuation is a meromorphic function the poles of which are on a semi infinite periodic lattice this allows one to have an understanding of the behavior of the measure near a repulsive fixed point trace identities corresponding to the fact that the analytically continued mellin transform vanishes at negative integers are derived for the polynomial case the quadratic map is first analyzed in detail and the analytic properties of the inverse of the green s function are exhibited of interest is the appearance of a dense set of spikes at dyadic points when the julia set is disconnected these results are used to study the residues of the mellin transform a certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems the excitation spectrum of which displays unusual oscillations the appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion

a ruelle operator for a real julia set letr be an expanding rational function with a real bounded julia set and let left lg right left x right sum limits ry x frac g left y right left r left y right right 2 be a ruelle operator acting in a space of functions analytic in a neighbourhood of the julia set we obtain explicit expressions for the resolvent function e left x z lambda right left i lambda l right 1 frac 1 z x and in particular for the fredholm determinantd det i l it gives us an equation for calculating the escape rate we relate our results to orthogonal polynomials with respect to the balanced measure ofr two examples are considered

geometry and combinatorics of julia sets of real quadratic maps for real a correspondence is made between the julia setb forz z 2 in the hyperbolic case and the set of chains with the aid of cremer s theorem it is shown how a number of features ofb can be understood in terms of chains the structure ofb is determined by certain equivalence classes of chains fixed by orders of visitation of certain real cycles and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of chains the functional equations obeyed by attractive cycles are investigated and their relation to chains is given the first cascade of period doubling bifurcations is described from the point of view of the associated julia sets and chains certain julia sets associated with the feigenbaum function and some theorems of lanford are discussed

powers from products of consecutive terms in arithmetic progression we show that if k is a positive integer then there are under certain technical hypotheses only finitely many coprime positive k term arithmetic progressions whose product is a perfect power if 4 leq k leq 11 we obtain the more precise conclusion that there are in fact no such progressions our proofs exploit the modularity of galois representations corresponding to certain frey curves together with a variety of results classical and modern on solvability of ternary diophantine equations as a straightforward corollary of our work we sharpen and generalize a theorem of sander on rational points on superelliptic curves

orthogonality properties of iterated polynomial mappings we consider a measure defined on a complex contour and its associated orthogonal polynomials the action of a polynomial transformation on the measure and the transformation laws of the corresponding orthogonal polynomials are given iterating the transformation provides an invariant measure whose support is the julia set corresponding to the polynomial transformation it appears that up to a constant the iterated polynomials generated by the initial mapping form a subset of the invariant set of orthogonal polynomials which fulfill a three term recursion relation an algorithm is given to compute the coefficients of this recursion relation which can be interpreted as a linear extension of the iterative procedure

synchronous vs asynchronous dynamics of diffusion controlled reactions abstract an analytical method based on the classical ruin problem is developed to compute the mean reaction time between two walkers undergoing a generalized random walk on a 1d lattice at each time step either both walkers diffuse simultaneously with probability p synchronous event or one of them diffuses while the other remains immobile with complementary probability asynchronous event reaction takes place through same site occupation or position exchange we study the influence of the degree of synchronicity p of the walkers and the lattice size n on the global reaction s efficiency for odd n the purely synchronous case p 1 is always the most effective one while for even n the encounter time is minimized by a combination of synchronous and asynchronous events this new parity effect is fully confirmed by monte carlo simulations on 1d lattices as well as for 2d and 3d lattices in contrast the 1d continuum approximation valid for sufficiently large lattices predicts a monotonic increase of the efficiency as a function of p the relevance of the model for several research areas is briefly discussed

preperiodic points for quadratic polynomials with small cycles over quadratic fields given a number field k and a polynomial f z in k z one can naturally construct a finite directed graph g f k whose vertices are the k rational preperiodic points of f with an edge alpha rightarrow beta if and only if f alpha beta the dynamical uniform boundedness conjecture of morton and silverman suggests that for fixed integers n ge 1 and d ge 2 there are only finitely many isomorphism classes of directed graphs g f k as one ranges over all number fields k of degree n and polynomials f z in k z of degree d in the case n d 1 2 poonen has given a complete classification of all directed graphs which may be realized as g f mathbb q for some quadratic polynomial f z in mathbb q z under the assumption that f does not admit rational points of large period the purpose of the present article is to continue the work begun by the author faber and krumm on the case n d 2 2 by combining the results of the previous article with a number of new results we arrive at a partial result toward a theorem like poonen s with a similar assumption on points of large period but over all quadratic extensions of mathbb q

preperiodic points of polynomials over global fields given a global field k and a polynomial defined over k of degree at least two morton and silverman conjectured in 1994 that the number of k rational preperiodic points of is bounded in terms of only the degree of k and the degree of in 1997 for quadratic polynomials over k q call and goldstine proved a bound which was exponential in s the number of primes of bad reduction of by careful analysis of the filled julia sets at each prime we present an improved bound on the order of s log s our bound applies to polynomials of any degree at least two over any global field k let k be a field and let k z be a rational function let denote the n iterate of under composition that is is the identity function and for n 1 n 1 we will study the dynamics on the projective line p k in particular we say a point x is preperiodic under if there are integers n m 0 such that x x the point y x satisfies n m y y and is said to be periodic of period n m note that x p k is preperiodic if and only if its orbit x n 0 is finite for example let k q and z z 29 16 then 5 4 1 4 7 4 forms a periodic cycle of period 3 and 5 4 1 4 7 4 and 3 4 each land on this cycle after one or two iterations in addition the point is of course fixed these nine q rational points are all preperiodic under meanwhile it is not difficult to see that no other point in p q is preperiodic by showing that the denominator of a rational preperiodic point must be 4 and that the absolute value must be less than 2 in general for any global field k any dimension n 1 and any morphism p p over k of degree at least two northcott proved in 1950 that the number of k rational preperiodic points of is finite 25 more precisely he showed that the preperiodic points form a set of bounded arithmetic height years later by analogy with the theorems of mazur 19 and merel 20 on k rational torsion of elliptic curves morton and silverman proposed the following conjecture 23 uniform boundedness conjecture morton and silverman 1994 given integers d n 1 and d 2 there is a constant d n d with the following property let k be a number field with k q d and let p p be a morphism of degree d defined over k then has at most preperiodic points in p k the analogy between preperiodic points and torsion comes from the fact that the torsion points of an elliptic curve e are precisely the preperiodic points of the multiplication bytwo map 2 e e in fact taking x coordinates the map 2 induces a rational function known to dynamicists as a lattes map p p of degree 4 whose preperiodic points date june 21 2005 revised december 21 2005 2000 mathematics subject classification primary 11g99 secondary 11d45 37f10

abeliants and their application to an elementary construction of jacobians abstract the abeliant is a polynomial rule which to each n n by n 2 array with entries in a commutative ring with unit associates an n n matrix with entries in the same ring the theory of abeliants first introduced in an earlier paper of the author is simplified and extended here now let j be the jacobian of a nonsingular projective algebraic curve defined over an algebraically closed field with the aid of the theory of abeliants we obtain explicit defining equations for j and its group law

efficiency of encounter controlled reaction between diffusing reactants in a finite lattice topology and boundary effects abstract the role of dimensionality euclidean vs fractal spatial extent boundary effects and system topology on the efficiency of diffusion reaction processes involving two simultaneously diffusing reactants is analyzed we present numerically exact values for the mean time to reaction as gauged by the mean walklength before reactive encounter obtained via application of the theory of finite markov processes and via monte carlo simulation as a general rule we conclude that for sufficiently large systems the efficiency of diffusion reaction processes involving two synchronously diffusing reactants two walker case relative to processes in which one reactant of a pair is anchored at some point in the reaction space one walker plus trap case is higher and is enhanced the lower the dimensionality of the system this differential efficiency becomes larger with increasing system size and for periodic systems its asymptotic value may depend on the parity of the lattice imposing confining boundaries on the system enhances the differential efficiency relative to the periodic case while decreasing the absolute efficiencies of both two walker and one walker plus trap processes analytic arguments are presented to provide a rationale for the results obtained the insights afforded by the analysis to the design of heterogeneous catalyst systems is also discussed

computing the mordell weil rank of jacobians of curves of genus two we derive the equations necessary to perform a two descent on the jacobians of curves of genus two with rational weierstrass points we compute the mordell weil rank of the jacobian of some genus two curves defined over the rationals and discuss the practicality of using this method

inverted regions induced by geometric constraints on a classical encounter controlled binary reaction abstract the efficiency of an encounter controlled reaction between two independently mobile reactants on a lattice is characterized by the mean number n of steps to reaction the two reactants are distinguished by their mass with the light walker performing a jump to a nearest neighbor site in each time step while the heavy walker hops only with a probability p we associate p with the temperature of the system to account for geometric exclusion effects in the reactive event two reaction channels are specified for the walkers irreversible reaction occurs either in a nearest neighbor collision or when the two reactants attempt to occupy the same site lattices subject to periodic and to confining boundary conditions are considered for periodic lattices depending on the initial state the reaction time either falls off monotonically with p or displays a local minimum with respect to p occurrence of the latter signals a regime where the efficiency of the reaction effectively decreases with increasing temperature such behavior can also occur when one averages over all initial conditions but it disappears if the jump probability of the light walker falls below a characteristic threshold value even more robust behavior can occur on lattices subject to confining boundary conditions depending on the initial conditions the reaction time as a function of p may increase monotonically decrease monotonically display a single maximum or even a maximum and minimum in the latter case one can identify distinct regimes where the above noted inversion in reaction efficiency can occur we document both numerically and theoretically that these inversion regions are a consequence of a strictly classical interplay between excluded volume effects implicit in the specification of the two reaction channels and the system s dimensionality and spatial extent our results highlight situations where the description of an encounter controlled reactive event cannot be described by a single effective diffusion coefficient we also distinguish between the inversion region identified here and the marcus inverted region which arises in electron transfer reactions

a canonical form based decision procedure and model checking approach for propositional projection temporal logic this paper proposes a canonical form cf for chop formulas of propositional projection temporal logic pptl based on cf an improved algorithm for constructing labeled normal form graph lnfg of a pptl formula is presented this improvement leads to a better decision procedure for pptl with infinite models in addition a transformation from lnfgs to generalized buchi automata gba and then buchi automata ba is formalized thus a spin based model checking approach is generalized for pptl to illustrate how these algorithms work several examples are given

empirical evidence for the birch and swinnerton dyer conjectures for modular jacobians of genus 2 curves this paper provides empirical evidence for the birch and swinnerton dyer conjectures for modular jacobians of genus 2 curves the second of these conjectures relates six quantities associated to a jacobian over the rational numbers one of these six quantities is the size of the shafarevich tate group unable to compute that we computed the five other quantities and solved for the last one in all 32 cases the result is very close to an integer that is a power of 2 in addition this power of 2 agrees with the size of the 2 torsion of the shafarevich tate group which we could compute

on semiclassical orthogonal polynomials via polynomial mappings abstract we consider orthogonal polynomials via polynomial mappings in the framework of the semiclassical class we prove that this class is stable under polynomial transformations several consequences of this fact are deduced as an application we analyze in detail cubic transformations for semiclassical orthogonal polynomials of class at most 2 recovering and extending some results proved recently for class 1 and producing new examples of semiclassical orthogonal polynomials of class 2 in particular we show how to obtain integral representations for the regular functionals with respect to which these new semiclassical families are orthogonal

on the invariant sets of a family of quadratic maps the julia setb for the mappingz z 2 is considered where is a complex parameter for 2 a new upper bound for the hausdorff dimension is given and the monic polynomials orthogonal with respect to the equilibrium measure onb are introduced a method for calculating all of the polynomials is provided and certain identities which obtain among coefficients of the three term recurrence relations are given a unifying theme is the relationship betweenb and chains which is explored for 1 4 2 and for with 1 4 with the aid of the bottcher equation thenb is shown to be a holder continuous curve for 1 4

incompleteness of first order temporal logic with until the results presented in this paper concern the axiomatizability problem of first order temporal logic with linear and discrete time we show that the logic is incomplete i e it cannot be provided with a finitistic and complete proof system we show two incompleteness theorems although the first one is weaker it assumes some first order signature we decided to present it for its proof is much simpler and contains an interesting fact that finite sets are characterizable by means of temporal formulas the second theorem shows that the logic is incomplete independently of any particular signature

cycles of quadratic polynomials and rational points on a genus 2 curve it has been conjectured that for n sufficiently large there are no quadratic polynomials in q z with rational periodic points of period n morton proved there were none with n 4 by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to x 1 16 whose rational points had been previously computed we prove there are none with n 5 here the relevant curve has genus 14 but it has a genus 2 quotient whose rational points we compute by performing a 2 descent on its jacobian and applying a refinement of the method of chabauty and coleman we hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves we also describe the three possible galois stable 5 cycles and show that there exist galois stable n cycles for infinitely many n furthermore we answer a question of morton by showing that the genus 14 curve and its quotient are not modular finally we mention some partial results for n 6

block orthogonal systems for symmetric p forms abstract to study polynomials orthogonal with respect to the logarithmic equilibrium measure on the julia set of a nonlinear polynomial p bessis and his coworkers introduced a symmetric bilinear form the symmetric p form it is a non hermitian variant of a genuine inner product the hermitian p form the main reason for doing so was the existence of a three term recurrence relation between the polynomials of successive degree of an orthogonal system however this is only true if the moment matrix of the symmetric p form is strongly regular in this paper we investigate what happens if this condition is not met we distinguish between the case where an orthogonal system still exists and the one where it does not and illustrate them by z z 2 and p z z 2 1 respectively

a practical decision procedure for propositional projection temporal logic with infinite models this paper presents a practical decision procedure for propositional projection temporal logic with infinite models first a set prop l of labels l i 0 i n n 0 is used to mark nodes of an lnfg of a formula and a node with l i is treated as an accepting state as in an automaton further the generalized buchi accepting condition for automata is employed to identify a path resulting a word in an lnfg as a model of the formula in addition the implementation details of the decision procedure and relevant algorithms including pre processing lnfg circle finding algorithms are presented as a matter of fact all algorithms are implemented by c programs

inverse images of polynomial mappings and polynomials orthogonal on them let j be a polynomial with complex coefficients first we study the inverse images of the real and imaginary axes under a polynomial mapping j in detail then for an arbitrary polynomial and a sequence pn of orthogonal polynomials the orthogonality behaviour of the sequence of polynomials pn j n n is investigated in particular necessary and sufficient conditions are given such that pn j n n is a subsequence of polynomials orthogonal with respect to a positive measure supported on a compact subset of the real line

a decision procedure and complete axiomatization for projection temporal logic abstract to specify and verify the concurrent and reactive systems with the theorem proving approach a complete axiomatization is formalized for first order projection temporal logic ptl with both finite and infinite time to this end ptl is restricted to a finite domain and the syntax semantics as well as the axiomatization of ptl are presented further the techniques of labeled normal form and labeled normal form graph of ptl formulas are introduced respectively with which a decision procedure for quantifier free ptl qfptl formulas is given moreover a generalized labeled normal form graph is defined and employed to transform a quantified ptl formula into its equivalent qfptl formula finally a decision procedure for ptl is formalized and the completeness of the axiomatic system is proved based on the decidability of ptl formulas

efficiency of encounter controlled reaction between diffusing reactants in a finite lattice non nearest neighbor effects abstract the influence of non nearest neighbor displacements on the efficiency of diffusion reaction processes involving one and two mobile diffusing reactants is studied an exact analytic result is given for dimension d 1 from which for large lattices one can recover the asymptotic estimate reported 30 years ago by lakatos lindenberg and shuler for dimensions d 2 3 we present numerically exact values for the mean time to reaction as gauged by the mean walklength before reactive encounter obtained via the theory of finite markov processes and supported by monte carlo simulations qualitatively different results are found between processes occurring on d 1 versus d 1 lattices and between results obtained assuming nearest neighbor only versus non nearest neighbor displacements

effects of confinement on the statistics of encounter times exact analytical results for random walks in a partitioned lattice we study the effects of temporarily and permanently confining domains on the statistics of first passage times in finite lattices in one and two dimensions we present exact results for the mean and variance of the first passage time between arbitrary sites in the following 1 a finite one dimensional lattice partitioned into temporarily confining domains and 2 a finite two dimensional lattice with reflecting boundaries for a single random walker and an immobile target in the one dimensional case we also present the full first passage time distribution via numerical inversion of laplace transforms

the power of temporal proofs abstract some methods for reasoning about concurrent programs and hardware devices have been based on proof systems for temporal logic unfortunately all effective proof systems for temporal logic are incomplete for the standard semantics in the sense that some formulas hold in every intended model but cannot be proved we evaluate and compare the power of several proof systems for temporal logic specifically we relate temporal systems to classical systems with explicit time parameters a typical temporal system turns out to be incomplete in a strong sense we exhibit a short valid formula it fails to prove we suggest the addition of new rules to define auxiliary predicates with these rules we obtain nonstandard soundness and completeness results in particular one of the simple temporal systems we describe is as powerful as peano arithmetic

orthogonal polynomials on the unit circle via a polynomial mapping on the real line let f n n 0 be a sequence of monic orthogonal polynomials on the unit circle opuc with respect to a symmetric and finite positive borel measure email protected on 0 email protected and let 1 a 0 a 1 a 2 be the associated sequence of verblunsky coefficients in this paper we study the sequence email protected n n 0 of monic opuc whose sequence of verblunsky coefficients is 1 b 1 b 2 b n 1 a 0 b n 1 b 2 b 1 a 1 b 1 b 2 b n 1 a 2 b n 1 b 2 b 1 a 3 where b 1 b 2 b n 1 are n 1 fixed real numbers such that b email protected 1 1 for all j 1 2 n 1 so that email protected n n 0 is also orthogonal with respect to a symmetric and finite positive borel measure email protected on the unit circle we show that the sequences of monic orthogonal polynomials on the real line oprl corresponding to f n n 0 and email protected n n 0 by szego s transformation are related by some polynomial mapping giving rise to a one to one correspondence between the monic opuc email protected n n 0 on the unit circle and a pair of monic oprl on a subset of the interval 1 1 in particular we prove email protected q email protected n 1 q email protected q n q email protected q n q q n q supported on a subset of the union of 2n intervals contained in 0 email protected such that any two of these intervals have at most one common point and where up to an affine change in the variable z n 1 and email protected n are algebraic polynomials in email protected of degrees n 1 and n respectively defined only in terms of a 0 b 1 b n 1 this measure induces a measure on the unit circle supported on the union of 2n arcs pairwise symmetric with respect to the real axis the restriction to symmetric measures or real verblunsky coefficients is needed in order that szego s transformation may be applicable

orthogonal polynomials on several intervals via a polynomial mapping starting from a sequence pn x no of orthogonal polynomials with an orthogonality measure yurj supported on eo c 1 1 we construct a new sequence p x fi of orthogonal polynomials on t 1 eq t is a polynomial of degree tv with an orthogonality measure i that is related to no if eo 1 1 then e t 1 l l will in general consist of tv intervals we give explicit formulas relating pn x fi and pn x fio and show how the recurrence coefficients in the three term recurrence formulas for these orthogonal polynomials are related if one chooses t to be a chebyshev polynomial of the first kind then one gets sieved orthogonal polynomials

analytic expression for the mean time to absorption for a random walker on the sierpinski gasket the exact analytic expression for the mean time to absorption or mean walk length for a particle performing a random walk on a finite sierpinski gasket with a trap at one vertex is found to be t n 3 n 5 n 1 4 5 n ensuremath 3 n 3 n 1 1 where n denotes the generation index of the gasket and the mean is over a set of starting points of the walk distributed uniformly over all the other sites of the gasket in terms of the number n n of sites on the gasket and the spectral dimension stackrel ifmmode tilde else fi d of the gasket the precise asymptotic behavior for large n n is t n ensuremath rightarrow 1 3 2n n 2 stackrel ifmmode tilde else fi d ensuremath sim n 1 464 this serves as a partial check on our result as it is a intermediate between the known results t ensuremath sim n 2 d 1 and t ensuremath sim n mathrm ln n d 2 for random walks on d dimensional euclidean lattices and b consistent with the known result for the asymptotic behavior of the mean number of distinct sites visited in a random walk on a fractal lattice