knotting in stretched polygons the knotting in a lattice polygon model of ring polymers is examined when a stretching force is applied to the polygon by examining the incidence of cut planes in the polygon we prove a pattern theorem in the stretching regime for large applied forces this theorem can be used to examine the incidence of entanglements such as knotting and writhing in particular we prove that for arbitrarily large positive but finite values of the stretching force the probability that a stretched polygon is knotted approaches 1 as the length of the polygon increases in the case of writhing we prove that for stretched polygons of length n and for every function the probability that the absolute value of the mean writhe is less than f n approaches 0 as n for sufficiently large values of the applied stretching force

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

a combinatorial approach to topological quantum field theories and invariants of graphs the combinatorial state sum of turaev and viro for a compact 3 manifold in terms of quantum 6j symbols is generalized by introducing observables in the form of coloured graphs they satisfy braiding relations and allow for surgeries and a discussion of cobordism theory application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any riemann surface with arbitrary coloured punctures

spectra of wess zumino witten models with arbitrary simple groups we consider the wess zumino witten two dimensional sigma models with fields taking values in an arbitrary connected but not necessarily simply connected simple lie groupg the quantum states of the theory are realized geometrically as sections of a line bundle over the loop grouplg the action of the current algebra of the theory is decomposed into highest weight representations by explicit construction of the highest weight states this solves for the spectrum of the models as a by product we obtain modular invariant partition functions of the theory on tori the present paper extends the results of 7 where the casesg su 2 andso 3 were treated

knot probability for lattice polygons in confined geometries we study the knot probability of polygons confined to slabs or prisms considered as subsets of the simple cubic lattice we show rigorously that almost all sufficiently long polygons in a slab are knotted and we use monte carlo methods to investigate the behaviour of the knot probability as a function of the width of the slab or prism and the number of edges in the polygon in addition we consider the effect of solvent quality on the knot probability in these confined geometries

censored regression local linear approximations and their applications abstract various statistical tools are available for modeling the relationship between response and covariate if the data are fully observable in the situation of censored data however those tools are no longer directly applicable this article provides an easily implemented methodology for modeling the association based on censored data the form of the regression relationship will be completely determined by the data no assumptions are made about this form basic ideas behind the methodology are to transform the observed data in an appropriate simple way and then to apply a locally weighted least squares regression the proposed estimator involves a variable bandwidth that automatically adapts to the design of the data points that the methodology is very easy to implement is illustrated by several examples including simulation studies and an analysis of the stanford heart transplant data and the primary biliary cirrhosis data several theoretical considerations are reflected in the examples fina

braid representations from unitary braided vector spaces we investigate braid group representations associated with unitary braided vector spaces focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite order we verify this conjecture for the two infinite families of gaussian and group type braided vector spaces as well as the generalization to quasi braided vector spaces of group type

knot probability of polygons subjected to a force a monte carlo study we use monte carlo methods to study the knot probability of lattice polygons on the cubic lattice in the presence of an external force f the force is coupled to the span of the polygons along a lattice direction say the z direction if the force is negative polygons are squeezed the compressive regime while positive forces tend to stretch the polygons along the z direction the tensile regime for sufficiently large positive forces we verify that the pincus scaling law in the force extension curve holds at a fixed number of edges n the knot probability is a decreasing function of the force for a fixed force the knot probability approaches unity as 1 exp 0 f n o n where 0 f is positive and a decreasing function of f we also examine the average of the absolute value of the writhe and we verify the square root growth law known for f 0 for all values of f

nonparametric regression estimates with censored data local linear smoothers and their applications for censored time to event data beran unpublished manuscript followed by doksum and yandell 1982 properties of regression estimates based on censored survival data and dabrowska 1987 scandinavian journal of statistics 14 181 197 suggested a nonparametric regression technique based on the kernel method to estimate the conditional survival function we propose to improve the bias term and boundary effects of their procedure by using local linear fits this is illustrated in this paper with simulated and real data moreover consistency results for the local linear based estimates of the conditional survival cumulative hazard mean and median functions are also established

state sum models and simplicial cohomology we study a class of subdivision invariant lattice models based on the gauge groupz p with particular emphasis on the four dimensional example this model is based upon the assignment of field variables to both the 1 and 2 dimensional simplices of the simplicial complex the property of subdivision invariance is achieved when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod p flatness condition by explicit computation of the partition function for the manifoldrp3 s1 we establish that the theory has a quantum hilbert space which differs from the classical one

determinants torsion and strings we apply the results of bf1 bf2 on determinants of dirac operators to string theory for the bosonic string we recover the holomorphic factorization of belavin and knizhik witten s global anomaly formula is used to give sufficient conditions for anomaly cancellation in the heterotic string for arbitrary background spacetimes to prove the latter result we develop certain torsion invariants related to characteristic classes of vector bundles and to index theory

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

self avoiding walks interacting with a surface discusses two models of polymer adsorption in one model a self avoiding walk on a d dimensional lattice interacts with a d 1 dimensional hyperplane in the other model this walk must also lie in or on one side of this hyperplane both models exhibit non analytic behaviour corresponding to a phase transition but these phase transitions do not occur at the same point the authors give numerical estimates of the locations of these transitions

polygons in restricted geometries subjected to infinite forces we consider self avoiding polygons in a restricted geometry namely an infinite l m tube in these polygons are subjected to a force f parallel to the infinite axis of the tube when the force stretches the polygons while when the force is compressive we obtain and prove the asymptotic form of the free energy in both limits we conjecture that the asymptote is the same as the limiting free energy of hamiltonian polygons polygons which visit every vertex in a box we investigate such polygons and in particular use a transfer matrix methodology to establish that the conjecture is true for some small tube sizes

local maximum likelihood estimation and inference local maximum likelihood estimation is a nonparametric counterpart of the widely used parametric maximum likelihood technique it extends the scope of the parametric maximum likelihood method to a much wider class of parametric spaces associated with this nonparametric estimation scheme is the issue of bandwidth selection and bias and variance assessment this paper provides a unified approach to selecting a bandwidth and constructing confidence intervals in local maximum likelihood estimation the approach is then applied to least squares nonparametric regression and to nonparametric logistic regression our experiences in these two settings show that the general idea outlined here is powerful and encouraging

collapsing and adsorbing polygons the adsorption transition in the phase diagram of a self interacting lattice polygon is examined the polygon has a nearest neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall the partition function of this model is z c y z d p v cp c v c y v z c where p c v c is the number of polygons with c nearest neighbour contacts v visits to the wall and n edges and counted up to translations parallel to the wall the limiting free energy of this model is f c y z d limn 1 1 logz c y z and it is known to be a non analytic function of y for each z 1 for all z2 0 1

preadjusted non parametric estimation of a conditional distribution function summary the paper deals with non parametric estimation of a conditional distribution function we suggest a method of preadjusting the original observations non parametrically through location and scale to reduce the bias of the estimator we derive the asymptotic properties of the estimator proposed a simulation study investigating the finite sample performances of the estimators discussed is provided and reveals the gain that can be achieved it is also shown how the idea of the preadjusting opens the path to improved estimators in other settings such as conditional quantile and density estimation and conditional survival function estimation in the case of censored data

nonparametric density estimation with a parametric start the traditional kernel density estimator of an unknown density is by construction completely nonparametric in the sense that it has no preferences and will work reasonably well for all shapes the present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities for example the normal while not losing much in precision when the true density is far from the parametric class the idea is to multiply an initial parametric density estimate with a kernel type estimate of the necessary correction factor this works well in cases where the correction factor function is less rough than the original density itself extensive comparisons with the kernel estimator are carried out including exact analysis for the class of all normal mixtures the new method with a normal start wins quite often even in many cases where the true density is far from normal procedures for choosing the smoothing parameter of the estimator are also discussed the new estimator should be particularly useful in higher dimensions where the usual nonparametric methods have problems the idea is also spelled out for nonparametric regression

self avoiding walks in a slab with attractive walls we consider a self avoiding walk confined between two parallel planes or lines with an energy term associated with each vertex of the walk in the confining planes we allow the energy terms to be different for the top and bottom planes we use exact enumeration and monte carlo methods to investigate the force between the confining planes and how it depends on the width of the slab and on the interaction energy terms the phase diagram is qualitatively similar to that found for a directed walk model

local linear quantile regression abstract in this article we study nonparametric regression quantile estimation by kernel weighted local linear fitting two such estimators are considered one is based on localizing the characterization of a regression quantile as the minimizer of e pp y a x x where p is the appropriate check function the other follows by inverting a local linear conditional distribution estimator and involves two smoothing parameters rather than one our aim is to present fully operational versions of both approaches and to show that each works quite well although either might be used in practice we have a particular preference for the second our automatic smoothing parameter selection method is novel the main regression quantile smoothing parameters are chosen by rule of thumb adaptations of state of the art methods for smoothing parameter selection for regression mean estimation the techniques are illustrated by application to two datasets and compared in simulations

some twisted self dual solutions for the yang mills equations on a hypertorus thesu n yang mills equations are considered in a four dimensional euclidean box with periodic boundary conditions hypertorus gauge invariant twists can be introduced in these boundary conditions to be labeled with integersn n defined modulon the pontryagin number in this space is often fractional whenever this number is zero there are solutions to the equationsg 0 hereg is the covariant curl when this number is not zero we find a set of solutions to the equations g mu nu tilde g mu nu provided that the periodsa of the box satisfy certain relations

locally parametric nonparametric density estimation this paper develops a nonparametric density estimator with parametric overtones suppose f x is some family of densities indexed by a vector of parameters we define a local kernel smoothed likelihood function which for each x can be used to estimate the best local parametric approximant to the true density this leads to a new density estimator of the form f x o x thus inserting the best local parameter estimate for each new value of x when the bandwidth used is large this amounts to ordinary full likelihood parametric density estimation while for moderate and small bandwidths the method is essentially nonparametric using only local properties of data and the model alternative ways more general than via the local likelihood are also described the methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class properties of this new semiparametric estimator are investigated our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias the new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework the methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression

a simple bias reduction method for density estimation a new method for bias reduction in nonparametric density estimation is proposed the method is a simple two stage multiplicative bias correction its theoretical properties are investigated and simulations indicate its practical potential the method is easy to compute and to analyse and extends simply to multivariate and other estimation problems

self avoiding walks in wedges the authors consider the number of self avoiding walks confined to a subset zd f of the d dimensional hypercubic lattice zd such that the coordinates x1 x2 xd of each vertex in the walk satisfy x1 or 0 and 0 or xk or fk x1 for k 2 3 d they show that if fk x to infinity as x to infinity the connective constant of walks in zd f is identical to the convective constant of walks in zd they also explore conditions on fk which lead to a smaller connective constant for walks in zd f and in particular consider walks between two parallel d 1 dimensional hyperplanes finally they contrast some of these results with recent work by grimmett on percolation on subsets of the square lattice

transfer of tail information in censored regression models consider a heteroscedastic regression model y m x sigma x varepsilon where the functions m and sigma are smooth and varepsilon is independent of x the response variable y is subject to random censoring but it is assumed that there exists a region of the covariate x where the censoring of y is light under this condition it is shown that the assumed nonparametric regression model can be used to transfer tail information from regions of light censoring to regions of heavy censoring crucial for this transfer is the estimator of the distribution of varepsilon based on nonparametric regression residuals whose weak convergence is obtained the idea of transferrring tail information is applied to the estimation of the conditional distribution of y given x x with information on the upper tail borrowed from the region of light censoring and to the estimation of the bivariate distribution p x leq x y leq y with no regions of undefined mass the weak convergence of the two estimators is obtained by products of this investigation include the uniform consistency of the conditional kaplan meier estimator and its derivative the location and scale estimators and the estimators of their derivatives

stretched polygons in a lattice tube we examine the topological entanglements of polygons confined to a lattice tube and under the influence of an external tensile force f the existence of the limiting free energy for these so called stretched polygons is proved and then using transfer matrix arguments a pattern theorem for stretched polygons is proved note that the tube constraint allows us to prove a pattern theorem for any arbitrary value of f while without the tube constraint it has so far only been proved for large values of f the stretched polygon pattern theorem is used first to show that the average span per edge of a randomly chosen n edge stretched polygon approaches a positive value non decreasing in f as n we then show that the knotting probability of an n edge stretched polygon confined to a tube goes to one exponentially as n thus as n when polygons are influenced by a force f no matter its strength or direction topological entanglements as defined by knotting occur with high probability

non crossing non parametric estimates of quantile curves in this paper a new nonparametric estimate of conditional quantiles is proposed that avoids the problem of crossing quantile curves calculated for various p 2 0 1 the method uses an initial estimate of the conditional distribution function in a flrst step and solves the problem of inversion and monotonization with respect to p 2 0 1 simultaneously it is demonstrated that the new estimates are asymptotically normal distributed and asymptotically flrst order equivalent to quantile estimates obtained by local constant or local linear smoothing of the conditional distribution function the performance of the new procedure is illustrated by means of a simulation study and some comparisons with the currently available procedures which are similar in spirit with the proposed method are presented

the writhe of a self avoiding polygon we discuss the writhe of a self avoiding polygon on a lattice as a geometrical measure of its entanglement complexity we prove a rigorous result about the dependence of the absolute value of the writhe on the number n of edges in the polygon and use monte carlo methods to estimate the distribution of the writhe both for all polygons with n edges and for the subset of polygons that are trefoils

strong uniform consistency for nonparametric survival curve estimators from randomly censored data let p x t f t p y t g t and p z t h t h t f t g t an important problem of survival analysis is the estimation of the distribution function f recently the properties of two types of estimators were investigated one of them is the product limit p l estimator of kaplan and meier 8 the other is the nonparametric bayesian estimator introduced by v susarla and j van ryzin 11 in an earlier paper 13 b b winter and the authors proved the uniform consistency of the p l in a second joint paper 6 there was proved the uniform consistency of the p l with rate of o log n l12 nll4 the results of the above two papers were proved without any continuity conditions on f and g in case of arbitrary f and discrete g having finitely many jump points the uniform consistency of the p l with rate factor 0 log nl t was proved in 5 in 10 susarla and van rjzin proved for the bayesian estimator the pointwise consistency with rate factor o log n and the pointwise mean square consistency with rate factor