density of states potts zeros and fisher zeros of the q state potts model for continuous q the q state potts model can be extended to noninteger and even complex q by expressing the partition function in the fortuin kasteleyn f k representation in the f k representation the partition function z q a is a polynomial in q and v a ensuremath 1 a e ensuremath beta j and the coefficients of this polynomial ensuremath phi b c are the number of graphs on the lattice consisting of b bonds and c connected clusters we introduce the random cluster transfer matrix to compute ensuremath phi b c exactly on finite square lattices with several types of boundary conditions given the f k representation of the partition function we begin by studying the critical potts model z mathrm cp z q a c q where a c q 1 sqrt q we find a set of zeros in the complex w sqrt q plane that map to or close to the beraha numbers for real positive q we also identify q c l the value of q for a lattice of width l above which the locus of zeros in the complex p v sqrt q plane lies on the unit circle by finite size scaling we find that 1 q c l ensuremath rightarrow 0 as stackrel ensuremath rightarrow l ensuremath infty we then study zeros of the antiferromagnetic af potts model in the complex q plane and determine q c a the largest value of q for a fixed value of a below which there is af order we find excellent agreement with baxter s conjecture q c mathrm af a 1 ensuremath a a 3 we also investigate the locus of zeros of the ferromagnetic potts model in the complex q plane and confirm that q c mathrm fm a a ensuremath 1 2 we show that the edge singularity in the complex q plane approaches q c as q c l ensuremath sim q c al ensuremath y q and determine the scaling exponent y q for several values of q finally by finite size scaling of the fisher zeros near the antiferromagnetic critical point we determine the thermal exponent y t as a function of q in the range 2l ql 3 using data for lattices of size 3l ll 8 we find that y t is a smooth function of q and is well fitted by y t 1 au bu 2 c du where u ensuremath 2 ensuremath pi mathrm cos ensuremath 1 sqrt q 2 for q 3 we find y t ensuremath simeq 0 6 however if we include lattices up to l 12 we find y t ensuremath simeq 0 50 8 in rough agreement with a recent result of ferreira and sokal j stat phys 96 461 1999

exact partition functions for potts antiferromagnets on cyclic lattice strips abstract we present exact calculations of the zero temperature partition function of the q state potts antiferromagnet on arbitrarily long strips of the square triangular and kagome lattices with width l y 2 or 3 vertices and with periodic longitudinal boundary conditions from these in the limit of infinite length we obtain the exact ground state entropy s 0 k b ln w these results are of interest since this model exhibits nonzero ground state entropy s 0 0 for sufficiently large q and hence is an exception to the third law of thermodynamics we also include results for homeomorphic expansions of the square lattice strip the analytic properties of w q are determined and related to zeros of the chromatic polynomial for long finite strips

ground state entropy of the potts antiferromagnet on strips of the square lattice abstract we present exact solutions for the zero temperature partition function chromatic polynomial p and the ground state degeneracy per site w exponent of the ground state entropy for the q state potts antiferromagnet on strips of the square lattice of width ly vertices and arbitrarily great length lx vertices the specific solutions are for a ly 4 fbcy pbcx cyclic b ly 4 fbcy tpbcx mobius c ly 5 6 pbcy fbcx cylindrical and d l y 5 fbc y fbc x open where fbc pbc and tpbc denote free periodic and twisted periodic boundary conditions respectively in the lx limit of each strip we discuss the analytic structure of w in the complex q plane the respective w functions are evaluated numerically for various values of q several inferences are presented for the chromatic polynomials and analytic structure of w for lattice strips with arbitrarily great ly the absence of a nonpathological lx limit for real nonintegral q in the interval 0 0 for strips of the square triangular lattice is discussed

complex temperature phase diagrams for the q state potts model on self dual families of graphs and the nature of the q limit exact calculations of the potts model partition function z g q v have been presented for arbitrary q and temperature like variable v on self dual strip graphs g of the square lattice with fixed width l y and arbitrarily great length lx with two types of boundary conditions letting lx infinity the resultant free energy and complex temperature phase diagram have been computed including the locus b where the free energy is nonanalytic results are analyzed for widths l y 1 2 3 these results have been used to study the approach to the large q limit of b

transfer matrices and partition function zeros for antiferromagnetic potts models iii triangular lattice chromatic polynomial we study the chromatic polynomial pg q for m n triangular lattice strips of widths m 12p 9f with periodic or free transverse boundary conditions respectively and arbitrary lengths n with free longitudinal boundary conditions the chromatic polynomial gives the zero temperature limit of the partition function for the q state potts antiferromagnet we compute the transfer matrix for such strips in the fortuin kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q plane in the limit n we recompute the limiting curve obtained by baxter in the thermodynamic limit m n and find new interesting features with possible physical consequences finally we analyze the isolated limiting points and their relation with the beraha numbers

exact potts model partition functions for strips of the triangular lattice we present exact calculations of the potts model partition function z g q v for arbitrary q and temperature like variable v on n vertex strip graphs g of the triangular lattice for a variety of transverse widths equal to l vertices and for arbitrarily great length equal to m vertices with free longitudinal boundary conditions and free and periodic transverse boundary conditions these partition functions have the form z g q v sum j 1 n z g lambda c z g j z g j m 1 we give general formulas for n z g j and its specialization to v 1 for arbitrary l the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed it is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the potts model on the infinite triangular lattice considering the full generalization to arbitrary complex q and v we determine the singular locus mathcal b arising as the accumulation set of partition function zeros as m in the q plane for fixed v and in the v plane for fixed q explicit results for partition functions are given in the text for l 3 free and l 3 4 cylindrical and plots of partition function zeros and their asymptotic accumulation sets are given for l up to 5 a new estimate for the phase transition temperature of the q 3 potts antiferromagnet on the 2d triangular lattice is given

asymptotic limits and zeros of chromatic polynomials and ground state entropy of potts antiferromagnets we study the asymptotic limiting function w g q lim n to infty p g q 1 n where p g q is the chromatic polynomial for a graph g with n vertices we first discuss a subtlety in the definition of w g q resulting from the fact that at certain special points q s the following limits do not commute lim n to infty lim q to q s p g q 1 n ne lim q to q s lim n to infty p g q 1 n we then present exact calculations of w g q and determine the corresponding analytic structure in the complex q plane for a number of families of graphs g including circuits wheels biwheels bipyramids and cyclic and twisted ladders we study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs all but a finite number of the zeros lie exactly on a unit circle whose position depends on the family using the connection of p g q with the zero temperature potts antiferromagnet we derive a theorem concerning the maximal finite real point of non analyticity in w g q denoted q c and apply this theorem to deduce that q c sq 3 and q c hc 3 sqrt 5 2 for the square and honeycomb lattices finally numerical calculations of w hc q and w sq q are presented and compared with series expansions and bounds

exact potts model partition functions on strips of the honeycomb lattice abstract we present exact calculations of the partition function of the q state potts model on i open ii cyclic and iii mobius strips of the honeycomb brick lattice of width l y 2 and arbitrarily great length in the infinite length limit the thermodynamic properties are discussed the continuous locus of singularities of the free energy is determined in the q plane for fixed temperature and in the complex temperature plane for fixed q values we also give exact calculations of the zero temperature partition function chromatic polynomial and w q the exponent of the ground state entropy for the potts antiferromagnet for honeycomb strips of type iv l y 3 cyclic v l y 3 mobius vi l y 4 cylindrical and vii l y 4 open in the infinite length limit we calculate w q and determine the continuous locus of points where it is nonanalytic we show that our exact calculation of the entropy for the l y 4 strip with cylindrical boundary conditions provides an extremely accurate approximation to a few parts in 10 5 for moderate q values to the entropy for the full 2d honeycomb lattice where the latter is determined by monte carlo measurements since no exact analytic form is known

rank tests for association with right censored data summary rank tests for association between a time variable and a continuous covariate and between two time variables are given in the presence of censoring in the first case these tests are related to the semiparametric tests of cox 1972 and prentice 1978 but differ in that the covariate is replaced by a score computed from its rank in the second case the test statistic is again a simple inner product which in the proportional hazards model reduces to the correlation between the two vectors of log rank scores the loss of efficiency arising from incorrect score functions is also explored when there is no censoring in this paper we consider a general model for association between pairs of observations in which either variable or both variables may be right censored the aim is to develop tests based on the generalized ranks of both variables which have maximal power at alternatives which are near to the null hypothesis of independence attention will be concentrated on two special cases of interest in survival analysis tests for association between a censored survival time and a possibly censored continuous covariate and tests for association between the times to different events beginning with cox 1972 the first situation has been studied extensively in a semiparametric set up in which information based on ranks is used for the time variable but exact values are used for the covariate procedures which use only the ranks of both variables have been developed by brown hollander korwar 1974 and more recently by o brien 1978 however neither of these procedures have been developed with a particular model in mind and one of the achievements of the present paper is to derive rank tests with good local power when a particular model such as the proportional hazards model holds in the case of a proportional hazards model with a continuous covariate the present development may be preferable to the semiparametric methods when the functional form of the relation ship between the hazard and the covariate is not known in advance or when errors or outliers may exist in the observed covariate values thus significance levels may be obtained which are not inflated by multiple regroupings or transformations of the covariate data although it would be expected that parametric model building with variables found to be associated with survival would often be carried out in further analyses

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

transfer matrices for the partition function of the potts model on cyclic and mobius lattice strips abstract we present a method for calculating transfer matrices for the q state potts model partition functions z g q v for arbitrary q and temperature variable v on cyclic and mobius strip graphs g of the square sq triangular tri and honeycomb hc lattices of width l y vertices and of arbitrarily great length l x vertices for the cyclic case we express the partition function as z l y l x q v d 0 l y c d tr t z l y d m where denotes lattice type c d are specified polynomials of degree d in q t z l y d is the transfer matrix in the degree d subspace and m l x l x 2 for sq tri hc respectively an analogous formula is given for mobius strips we exhibit a method for calculating t z l y d for arbitrary l y explicit results for arbitrary l y are given for t z l y d with d l y and l y 1 in particular we find very simple formulas the determinant det t z l y d and trace tr t z l y corresponding results are given for the equivalent tutte polynomials for these lattice strips and illustrative examples are included we also present formulas for self dual cyclic strips of the square lattice

general structural results for potts model partition functions on lattice strips abstract we present a set of general results on structural features of the q state potts model partition function z g q v for arbitrary q and temperature boltzmann variable v for various lattice strips of arbitrarily great width ly vertices and length lx vertices including i cyclic and mobius strips of the square and triangular lattices and ii self dual cyclic strips of the square lattice we also present an exact solution for the chromatic polynomial for the cyclic and mobius strips of the square lattice with width ly 5 the greatest width for which an exact solution has been obtained so far for these families in the lx limit we calculate the ground state degeneracy per site w q and determine the boundary b across which w q is singular in the complex q plane

exact potts model partition functions on wider arbitrary length strips of the square lattice abstract we present exact calculations of the partition function of the q state potts model for general q and temperature on strips of the square lattice of width l y 3 vertices and arbitrary length l x with periodic longitudinal boundary conditions of the following types i fbc y pbc x cyclic ii fbc y tpbc x mobius iii pbc y pbc x toroidal and iv pbc y tpbc x klein bottle where fbc and t pbc refer to free and twisted periodic boundary conditions results for the l y 2 torus and klein bottle strips are also included in the infinite length limit the thermodynamic properties are discussed and some general results are given for low temperature behavior on strips of arbitrarily great width we determine the submanifold in the c 2 space of q and temperature where the free energy is singular for these strips our calculations are also used to compute certain quantities of graph theoretic interest

partition function zeros of a restricted potts model on lattice strips and effects of boundary conditions we calculate exactly the partition function z g q v of the q state potts model with temperature like boltzmann variable v for strip graphs g of the square and triangular lattices of various widths ly and arbitrarily great lengths lx with a variety of boundary conditions and with q and v restricted to satisfy conditions corresponding to the ferromagnetic phase transition on the associated two dimensional lattices from these calculations in the limit lx we determine the continuous accumulation loci of the partition function zeros in the v and q planes strips of the honeycomb lattice are also considered we discuss some general features of these loci

on uniqueness for some non lipschitz sde abstract we study the uniqueness in the path by path sense i e by of solutions to stochastic differential equations with additive noise and non lipschitz autonomous drift the notion of path by path solution involves considering a collection of ordinary differential equations and is in principle weaker than that of a strong solution since no adaptability condition is required we use results and ideas from the classical theory of ode s together with probabilistic tools like girsanov s theorem to establish the uniqueness property for some classes of noises including brownian motion and some drift functions not necessarily bounded nor continuous

transfer matrices and partition function zeros for antiferromagnetic potts models vi square lattice with extra vertex boundary conditions we study using transfer matrix methods the partition function zeros of the square lattice q state potts antiferromagnet at zero temperature square lattice chromatic polynomial for the boundary conditions that are obtained from an m n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column we provide numerical evidence that the partition function zeros are becoming dense everywhere in the complex q plane outside the limiting curve mathcal b infty mathrm sq for this model with ordinary e g free or cylindrical boundary conditions despite this the infinite volume free energy is perfectly analytic in this region

correcting for the dependent competing risk of treatment using inverse probability of censoring weighting and copulas in the estimation of natural conception chances when estimating the probability of natural conception from observational data on couples with an unfulfilled child wish the start of assisted reproductive therapy art is a competing event that cannot be assumed to be independent of natural conception in clinical practice interest lies in the probability of natural conception in the absence of art as this probability determines the need for therapy we thus want to estimate the marginal cumulative pregnancy distribution without assumptions on the dependence structure between the two competing events this marginal distribution is not identifiable we first use inverse probability of censoring weighting assuming that the factors influencing the choice to start art are known then we parameterize the event distributions for conception and for start of art and use copulas to account for the dependency between both events by using these two ways of correcting for the dependent risk of treatment we obtain a plausible estimation region for the cumulative pregnancy curve and for the prognostic effect of tubal tests

transfer matrices for the partition function of the potts model on lattice strips with toroidal and klein bottle boundary conditions abstract we present a method for calculating transfer matrices for the q state potts model partition functions z g q v for arbitrary q and temperature variable v on strip graphs g of the square sq triangular tri and honeycomb hc lattices of width l y vertices and of arbitrarily great length l x vertices subject to toroidal and klein bottle boundary conditions for the toroidal case we express the partition function as z l y l x q v d 0 l y j b j d z l y d j m where denotes lattice type b j d are specified polynomials of degree d in q z l y d j are eigenvalues of the transfer matrix t z l y d in the degree d subspace and m l x l x 2 for sq tri hc respectively an analogous formula is given for klein bottle strips we exhibit a method for calculating t z l y d for arbitrary l y in particular we find some very simple formulas for the determinant det t z l y d and trace tr t z l y illustrative examples of our general results are given including new calculations of transfer matrices for potts model partition functions on strips of the square triangular and honeycomb lattices with toroidal or klein bottle boundary conditions

white noise driven parabolic spdes with measurable drift abstract we prove existence and uniqueness of the solution of a white noise driven parabolic spde in case the drift is measurable and satisfies a one sided linear growth condition and the diffusion coefficient is nondegenerate has a locally lipschitz derivative and satisfies a linear growth condition the proof combines arguments similar to those of gyongy and pardoux together with an estimate of the density of the solution of the equation without drift which is obtained with the help of the malliavin calculus

zeros of jones polynomials for families of knots and links abstract we calculate jones polynomials v l t for several families of alternating knots and links by computing the tutte polynomials t g x y for the associated graphs g and then obtaining v l t as a special case of the tutte polynomial for each of these families we determine the zeros of the jones polynomial including the accumulation set in the limit of infinitely many crossings a discussion is also given of the calculation of jones polynomials for non alternating links

dependent competing risks and summary survival curves summary in many contexts where there is interest in inferring the marginal distribution of a survival time t subject to censoring embodied in a latent waiting time c the times t and c may not be independent this paper presents a new class of nonparametric assumptions on the conditional distribution of t given c and shows how they lead to consistent generalizations of the kaplan meier 1958 survival curve estimator the new survival curve estimators are used under weak assumptions to construct bounds on the marginal survival which can be much narrower than those of peterson 1976 in stratified populations where t and c are independent only within strata examples indicate that the kaplan meier estimator is often approximately consistent

bounds on net survival probabilities for dependent competing risks bounds on the marginal survival function based on data from a competing risks experiment are obtained these bounds require an investigator to specify a range of possible concordances for the times to occurrences of the competing risks these bounds are tighter than those of peterson 1976 proceedings of the national academy of sciences 73 11 13

on the use of cause specific failure and conditional failure probabilities examples from clinical oncology data abstract nonparametric maximum likelihood estimation of the probability of failing from a particular cause by time t in the presence of other acting causes i e the cause specific failure probability is discussed a commonly used incorrect approach is to take 1 minus the kaplan meier km estimator 1 km whereby patients who fail of extraneous causes are treated as censored observations examples showing the extent of bias in using the 1 km approach are presented using clinical oncology data this bias can be quite large if the data are uncensored or if a large percentage of patients fail from extraneous causes prior to the occurrence of failures from the cause of interest each cause specific failure probability is mathematically defined as a function of all of the cause specific hazards therefore nonparametric estimates of the cause specific failure probabilities may not be able to identify categorized covariate effects on the cause specific hazards these effects would be correctly identified

flow polynomials and their asymptotic limits for lattice strip graphs we present exact calculations of flow polynomials f g q for lattice strips of various fixed widths l y 4 and arbitrarily great lengths l x with several different boundary conditions square honeycomb and triangular lattice strips are considered we introduce the notion of flows per face fl in the infinite length limit we study the zeros of f g q in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite length limit for the various families of strips the function fl is nonanalytic on this locus the loci are found to be noncompact for many strip graphs with periodic or twisted periodic longitudinal boundary conditions and compact for strips with free longitudinal boundary conditions we also find the interesting feature that aside from the trivial case l y 1 the maximal point q cf where crosses the real axis is universal on cyclic and mobius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q cf 3 for the infinite square lattice

a class of multivanate failure time distributions summary a class of continuous multivariate lifetime distributions is proposed the dependence between individuals in a group is modelled by a group specific quantity which can be interpreted as an unobserved covariate common to the individuals in the group and assumed to follow a positive stable distribution it is possible to include covariates in the model and discuss whether the dependence is still present after specific covariates are taken into account if the conditional hazards are proportional then the hazards in the marginal distributions are also proportional but with different constants of proportionality also the hazard for the minimum in a group is proportional to the marginal hazards if the conditional distributions given the group quantity are weibull then the marginal distributions are also weibull this class can be used to test the hypothesis of independence of litter mates in the proportional hazards model

tutte polynomial of pseudofractal scale free web the tutte polynomial of a graph is a 2 variable polynomial which is quite important in both combinatorics and statistical physics it contains various numerical invariants and polynomial invariants such as the number of spanning trees the number of spanning forests the number of acyclic orientations the reliability polynomial chromatic polynomial and flow polynomial in this paper we study and obtain a recursive formula for the tutte polynomial of pseudofractal scale free web psfw and thus logarithmic complexity algorithm to calculate the tutte polynomial of the psfw is obtained although it is np hard for general graph by solving the recurrence relations derived from the tutte polynomial the rigorous solution for the number of spanning trees of the psfw is obtained therefore an alternative approach to determine explicitly the number of spanning trees of the psfw is given furthermore we analyze the all terminal reliability of the psfw and compare the results with those of the sierpinski gasket which has the same number of nodes and edges as the psfw in contrast with the well known conclusion that inhomogeneous networks e g scale free networks are more robust than homogeneous networks i e networks in which each node has approximately the same number of links with respect to random deletion of nodes the sierpinski gasket which is a homogeneous network as our results show is more robust than the psfw which is an inhomogeneous network with respect to random edge failures

a concordance test for independence in the presence of censoring an extension of the definition of kendall s coefficient of concordance is proposed for testing independence in bivariate censored survival times the test statistic counts the number of known concordances minus the number of known discordances the variance of the statistic is calculated for an random censorship model which however does not require independence of the components of the censoring distribution two examples are given

testing the assumption of independence of truncation time and failure time summary for data subject to random censorship it is well known that the assumption of independence between censoring time and failure time cannot be tested nonparametrically for data subject to random truncation however it is possible to test independence of truncation time and failure time a statistic generalizing kendall s tau is proposed for testing the independence assumption and asymptotic properties under the null hypothesis are derived a conditional permutation test is also derived for small data sets generalization to allow right censoring and an application to truncated linear regression are also briefly discussed

methods for bounding the marginal survival distribution for time to event data with many potential failure types one cannot uniquely determine the distribution of time to a specific event type or marginal survival distribution in the case where event types are mutually exclusive in this paper we discuss several methods for estimating functions that bound the non identifiable marginal survival distribution in the competing risks problem we compute and compare bounds for data simulated from two bivariate survival distributions results show that the methods provide a suitable estimate of the marginal survival probability when one has specified dependence correctly data from a large clinical trial for breast cancer illustrate the methods

how to determine the law of the solution to a stochastic partial differential equation driven by a levy space time noise we consider a stochastic partial differential equation on a lattice tx m2 x xp where is a space time levy noise a perturbative in the sense of formal power series strong solution is given by a tree expansion whereas the correlation functions of the solution are given by a perturbative expansion with coefficients that are represented as sums over a certain class of graphs called parisi wu graphs the perturbative expansion of the truncated connected correlation functions is obtained via a linked cluster theorem as sums over connected graphs only the moments of the stationary solution can be calculated as well in all these solutions the cumulants of the single site distribution of the noise enter as multiplicative constants to determine them e g by comparison with an empirical correlation function one can fit these constants e g by the methods of least squares and thereby one approximately determines laws of the solution and the driving noise

t 0 partition functions for potts antiferromagnets on square lattice strips with twisted periodic boundary conditions we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet or equivalently the chromatic polynomial for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and i periodic and ii twisted periodic boundary conditions in the longitudinal direction so that the strip graphs are embedded on i a torus and ii a klein bottle in the limit of infinite length we calculate the exponent of the entropy w q show it to be the same for i and ii and determine its analytic structure

kaplan meier marginal or conditional probability curves in summarizing competing risks failure time data in the context of competing risks the kaplan meier estimator is often unsuitable for summarizing failure time data we discuss some alternative descriptive methods including marginal probability and conditional probability estimators two sample test statistics are also presented

exact potts model partition functions on ladder graphs abstract we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex ladder graphs i e strips of the square lattice with width ly 2 and arbitrary length lx with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of these ladder graphs and the thermodynamics is discussed by comparison with strip graphs of other widths we analyze how the singularities at the zero temperature critical point of the ferromagnet on infinite length finite width strips depend on the width we point out and study the following noncommutativity at certain special values q s lim n lim q q s z 1 n lim q q s lim n z 1 n it is shown that the potts antiferromagnet on both the infinite length line and ladder graphs with cyclic or mobius boundary conditions exhibits a phase transition at finite temperature if 0 b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros certain properties of the complex temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice showing that exact solutions on infinite length strips provide a way of gaining insight into these complex temperature phase diagrams

potts model partition functions for self dual families of strip graphs abstract we consider the q state potts model on families of self dual strip graphs g d of the square lattice of width l y and arbitrarily great length l x with periodic longitudinal boundary conditions the general partition function z and the t 0 antiferromagnetic special case p chromatic polynomial have the respective forms j 1 n f l y c f l y j f l y j l x with f z p for arbitrary l y we determine i the general coefficient c f l y j in terms of chebyshev polynomials ii the number n f l y d of terms with each type of coefficient and iii the total number of terms n f l y we point out interesting connections between the n z l y d and temperley lieb algebras and between the n f l y and enumerations of directed lattice animals exact calculations of p are presented for 2 l y 4 in the limit of infinite length we calculate the ground state degeneracy per site exponent of the ground state entropy w q generalizing q from z to c we determine the continuous locus b in the complex q plane where w q is singular we find the interesting result that for all l y values considered the maximal point at which b crosses the real q axis denoted q c is the same and is equal to the value for the infinite square lattice q c 3 this is the first family of strip graphs of which we are aware that exhibits this type of universality of q c

stochastic generalized burgers equations driven by fractional noises abstract the existence and uniqueness are proved for the solutions to a class of stochastic generalized burgers equations driven by multi parameter fractional noises in addition the existence and moment estimate are also obtained for the density of such a solution

chromatic polynomials and representations of the symmetric group abstract the chromatic polynomials considered in this paper are associated with graphs constructed in the following way take n copies of a complete graph k b and for i 1 2 n join each vertex in the ith copy to the same vertex in the i 1 th copy taking n 1 1 by convention previous calculations for b 2 and b 3 suggest that the chromatic polynomial contains terms that occur in levels in the present paper the levels are explained by using a version of the sieve principle and it is shown that the terms at level l correspond to the irreducible representations of the symmetric group sym l in the case of the two linear representations the terms can be calculated explicitly by methods based on the theory of distance regular graphs for the nonlinear representations the calculations are more complicated an illustration is given in section 10 where the complete chromatic polynomial for the case b 4 is obtained

exact potts model partition functions for strips of the square lattice we present exact calculations of the potts model partition function z g q v for arbitrary q and temperature like variable v on n vertex square lattice strip graphs g for a variety of transverse widths l t and for arbitrarily great length l l with free longitudinal boundary conditions and free and periodic transverse boundary conditions these have the form z g q v sum j 1 n z g lambda c z g j lambda z g j l ell we give general formulas for n z g j and its specialization to v 1 for arbitrary l t for both types of boundary conditions as well as other general structural results on z the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed it is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the potts model on the infinite square lattice considering the full generalization to arbitrary complex q and v we determine the singular locus b arising as the accumulation set of partition function zeros as l l in the q plane for fixed v and in the v plane for fixed q

transfer matrices for the zero temperature potts antiferromagnet on cyclic and mobius lattice strips abstract we present transfer matrices for the zero temperature partition function of the q state potts antiferromagnet equivalently the chromatic polynomial on cyclic and mobius strips of the square triangular and honeycomb lattices of width l y and arbitrarily great length l x we relate these results to our earlier exact solutions for square lattice strips with l y 3 4 5 triangular lattice strips with l y 2 3 4 and honeycomb lattice strips with l y 2 3 and periodic or twisted periodic boundary conditions we give a general expression for the chromatic polynomial of a mobius strip of a lattice and exact results for a subset of honeycomb lattice transfer matrices both of which are valid for arbitrary strip width l y new results are presented for the l y 5 strip of the triangular lattice and the l y 4 and l y 5 strips of the honeycomb lattice using these results and taking the infinite length limit l x we determine the continuous accumulation locus of the zeros of the above partition function in the complex q plane including the maximal real point of nonanalyticity of the degeneracy per site w as a function of q

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

applications of crude incidence curves crude incidence curves display the cumulative number of failures of interest as a function of time with competing causes of failure they are distinct from cause specific incidence curves that treat secondary types of failures as censored observations after briefly reviewing their definition and estimation we present five applications of crude incidence curves to show their utility in a broad range of studies in some of these applications it is helpful to model survival time distributions with use of two different time metameters for example time from diagnosis and age of the patient we describe how one can incorporate published vital statistics into the models when secondary types of failure correspond to common causes of death

exact chromatic polynomials for toroidal chains of complete graphs abstract we present exact calculations of the partition function of the zero temperature potts antiferromagnet equivalently the chromatic polynomial for graphs of arbitrarily great length composed of repeated complete subgraphs kb with b 5 6 which have periodic or twisted periodic boundary condition in the longitudinal direction in the lx limit the continuous accumulation set of the chromatic zeros b is determined we give some results for arbitrary b including the extrema of the eigenvalues with coefficients of degree b 1 and the explicit forms of some classes of eigenvalues we prove that the maximal point where b crosses the real axis qc satisfies the inequality qc b for 2 b the minimum value of q at which b crosses the real q axis is q 0 and we make a conjecture concerning the structure of the chromatic polynomial for klein bottle strips

spanning forests and the q state potts model in the limit q 0 we study the q state potts model with nearest neighbor coupling v e j 1 in the limit q v 0 with the ratio w v q held fixed combinatorially this limit gives rise to the generating polynomial of spanning forests physically it provides information about the potts model phase diagram in the neighborhood of q v 0 0 we have studied this model on the square and triangular lattices using a transfer matrix approach at both real and complex values of w for both lattices we have computed the symbolic transfer matrices for cylindrical strips of widths 2 l 10 as well as the limiting curves b of partition function zeros in the complex w plane for real w we find two distinct phases separated by a transition point w w0 where w0 1 4 resp w0 0 1753 0 0002 for the square resp triangular lattice for w w0 we find a non critical disordered phase that is compatible with the predicted asymptotic freedom as w for w w0 our results are compatible with a massless berker kadanoff phase with central charge c 2 and leading thermal scaling dimension xt 1 2 marginally irrelevant operator at w w0 we find a first order critical point the first derivative of the free energy is discontinuous at w0 while the correlation length diverges as w w0 and is infinite at w w0 the critical ehavior at w w0 seems to be the same for both lattices and it differs from that of the berker kadanoff phase our results suggest that the central charge is c 1 the leading thermal scaling dimension is xt 1 0 and the critical exponents are 1 d 1 2 and 1

comparing the small sample performance of several variance estimators under competing risks we examine several variance estimators for cumulative incidence estimators that have been proposed over time some of which are derived from asymptotic martingale or counting process theory and some of which are developed from the moments of the multinomial distribution there is little published work comparing these variance estimators largely because the variance estimators are algebraically complex and difficult to interpret and all but one have yet to be programmed for a standard statistical package through simulation and application to real data we compare the performance of six variance estimators in relation to each other and the bootstrap in order to confirm earlier reports of their performance and to provide future direction toward their application we find that the multinomial moment based estimators have performance close to that of the bootstrap and are quite accurate for estimating the variance even in samples of 20 subjects all but one of the martingale theory based estimators tend to perform poorly in small samples tending to either overestimate or underestimate the empirical variance in samples of fewer than 100 subjects

estimates of absolute cause specific risk in cohort studies in this paper we study methods for estimating the absolute risk of an event c1 in a time interval t1 t2 given that the individual is at risk at t1 and given the presence of competing risks we discuss some advantages of absolute risk for measuring the prognosis of an individual patient and some difficulties of interpretation for comparing two treatment groups we also discuss the importance of the concept of absolute risk in evaluating public health measures to prevent disease variance calculations permit one to gauge the relative importance of random and systematic errors in estimating absolute risk efficiency calculations were also performed to determine how much precision is lost in estimating absolute risk with a nonparametric approach or with a flexible piecewise exponential model rather than a simple exponential model and other calculations indicate the extent of bias that arises with the simple exponential model when that model is invalid such calculations suggest that the more flexible models will be useful in practice simulations confirm that asymptotic methods yield reliable variance estimates and confidence interval coverages in samples of practical size

t 0 partition functions for potts antiferromagnets on lattice strips with fully periodic boundary conditions abstract we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet equivalently the chromatic polynomial for families of arbitrarily long strip graphs of the square and triangular lattices with width ly 4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation i e of torus or klein bottle type these boundary conditions have the advantage of removing edge effects in the limit of infinite length we calculate the exponent of the entropy w q and determine the continuous locus b where it is singular we also give results for toroidal strips involving crossing subgraphs these make possible a unified treatment of torus and klein bottle boundary conditions and enable us to prove that for a given strip the locus b is the same for these boundary conditions

what is the probability of connecting two points the two terminal reliability known as the pair connectedness or connectivity function in percolation theory may actually be expressed as a product of transfer matrices in which the probability of operation of each link and site is exactly taken into account when link and site probabilities are p and it obeys an asymptotic power law behaviour for which the scaling factor is the transfer matrix s eigenvalue of largest modulus the location of the complex zeros of the two terminal reliability polynomial exhibits structural transitions as 0 1

computing the tutte polynomial of archimedean tilings abstract we describe an algorithm to compute the tutte polynomial of large fragments of archimedean tilings by squares triangles hexagons and combinations thereof our algorithm improves a well known method for computing the tutte polynomial of square lattices we also address the problem of obtaining tutte polynomial evaluations from the symbolic expressions generated by our algorithm improving the best known lower bound for the asymptotics of the number of spanning forests and the lower and upper bounds for the asymptotics of the number of acyclic orientations of the square lattice

transfer matrices and partition function zeros for antiferromagnetic potts models i general theory and square lattice chromatic polynomial we study the chromatic polynomials zero temperature antiferromagnetic potts model partition functions p g q for m n rectangular subsets of the square lattice with m 8 free or periodic transverse boundary conditions and n arbitrary free longitudinal boundary conditions using a transfer matrix in the fortuin kasteleyn representation in particular we extract the limiting curves of partition function zeros when n which arise from the crossing in modulus of dominant eigenvalues beraha kahane weiss theorem we also provide evidence that the beraha numbers b2 b3 b4 b5 are limiting points of partition function zeros as n whenever the strip width m is 7 periodic transverse b c or 8 free transverse b c along the way we prove that a noninteger beraha number except perhaps b10 cannot be a chromatic root of any graph

structural properties of potts model partition functions and chromatic polynomials for lattice strips abstract the q state potts model partition function equivalent to the tutte polynomial for a lattice strip of fixed width l y and arbitrary length l x has the form z g q v j 1 n z g c z g j z g j l x where v is a temperature dependent variable the special case of the zero temperature antiferromagnet v 1 is the chromatic polynomial p g q using coloring and transfer matrix methods we give general formulas for c x g j 1 n x g c x g j for x z p on cyclic and mobius strip graphs of the square and triangular lattice combining these with a general expression for the unique coefficient c z g j of degree d in q c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we determine the number of z g j s with coefficient c d in z g q v for these cyclic strips of width l y to be n z l y d 2d 1 l y d 1 1 2l y l y d for 0 d l y and zero otherwise for both cyclic and mobius strips of these lattices the total number of distinct eigenvalues z g j is calculated to be n z l y 2l y l y results are also presented for the analogous numbers n p l y d and n p l y for p g q we find that n p l y 0 n p l y 1 1 m l y 1 motzkin number n z l y 0 c l y the catalan number and give an exact expression for n p l y our results for n z l y and n p l y apply for both the cyclic and mobius strips of both the square and triangular lattices we also point out the interesting relations n z l y 2 n da tri l y and n p l y 2 n da sq l y where n da n denotes the number of directed lattice animals on the lattice we find the asymptotic growths n z l y l y 1 2 4 l y and n p l y l y 1 2 3 l y as l y some general geometric identities for potts model partition functions are also presented