lattice models of branched polymers effects of geometrical constraints the authors consider uniform brushes on subsets of the square and simple cubic lattices they show that in a slab geometry in three dimensions the connective constant of a brush is identical to that of a walk in that slab in two dimensions in a slit geometry the connective constant of a brush is strictly less than that of a walk they also consider the intermediate case of a rectangular prism and show that the connective constant of a brush is strictly less than that of a walk similar results are presented for combs stars and polygons

adsorbing and collapsing directed animals a model of a self interacting directed animal which also interacts with a solid wall is studied as a model of a directed branched polymer which can undergo both a collapse and an adsorption transition the directed animal is confined to a 45 wedge and it interacts with one of the walls of this wedge the existence of a thermodynamic limit in this model shown and the presence of an adsorption transition is demonstrated by using constructive techniques by comparing this model to a process of directed percolation we show that there is also a collapse or transition in this model we examine directed percolation in a wedge to show that there is a collapse phase present for arbitrary large values of the adsorption activity the generating function of adsorbing directed animals in a half space is found next from which we find the tricritical exponents associated with the adsorption transition a full solution for a collapsing directed animal seems intractible so instead we examine the collapse transition of a model of column convex directed animals with a contact activity next

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

recurrence transience and bounded harmonic functions for diffusions in the plane we give conditions for transience and recurrence for certain diffusion processes in the plane which look recurrent in certain sectors and transient in others we also give conditions for the convergence of diffusion paths to paths of the deterministic dynamical system corresponding to the first order part of l the generator of the process this in turn is related to the question of existence of bounded harmonic functions for the operator l conditions are given for the existence and nonexistence of bounded harmonic functions for l

temporal autocorrelation in univariate linear modeling of fmri data in functional magnetic resonance imaging statistical analysis there are problems with accounting for temporal autocorrelations when assessing change within voxels techniques to date have utilized temporal filtering strategies to either shape these autocorrelations or remove them shaping or coloring attempts to negate the effects of not accurately knowing the intrinsic autocorrelations by imposing known autocorrelation via temporal filtering removing the autocorrelation or prewhitening gives the best linear unbiased estimator assuming that the autocorrelation is accurately known for single event designs the efficiency of the estimator is considerably higher for prewhitening compared with coloring however it has been suggested that sufficiently accurate estimates of the autocorrelation are currently not available to give prewhitening acceptable bias to overcome this we consider different ways to estimate the autocorrelation for use in prewhitening after high pass filtering is performed a tukey taper set to smooth the spectral density more than would normally be used in spectral density estimation performs best importantly estimation is further improved by using nonlinear spatial filtering to smooth the estimated autocorrelation but only within tissue type using this approach when prewhitening reduced bias to close to zero at probability levels as low as 1 x 10 5

bond percolation on subsets of the square lattice and the threshold between one dimensional and two dimensional behaviour for each value of pi in 1 2 1 the author constructs a subgraph r pi of the square lattice z2 such that the bond percolation process on r pi has critical probability pi it is shown that the critical probability pi a of the region x y 0 or y or a ln x 1 0 or x infinity depends in a non trivial way upon the choice of the number a pi a is a continuous decreasing function of a and satisfies pi a to 1 2 as a to infinity and pi a to 1 as a down arrow 0 a family of one dimensional bond percolation processes is also constructed whose critical probabilities range over the interval 0 1 all proofs are rigorous

on the convergence of the series expansion analysis for self avoiding walks attached to a surface extensions of exact enumeration data for self avoiding walks attached to the surfaces of face centered cubic and triangular lattices are reported the convergence of the estimates for the critical points pc and exponent gamma 1 obtained by the baker hunter 1973 confluent singularity analysis is discussed and comparison made with estimates of pc from the corresponding bulk series

branched polymers attached in a wedge geometry the authors use exact enumeration and monte carlo techniques to test some recent predictions by duplantier and saleur 1986 of the values of the critical exponent gamma for uniform star branched polymers in a wedge geometry in two dimensions results support their predictions the authors have also estimated the exponent v and amplitude governing the n dependence of the mean square radius of gyration and the mean square end to end branch length in some cases the branches are distinguished and have different mean lengths but the exponent v is equal to the bulk self avoiding walk value in every case

forces in motzkin paths in a wedge entropic forces in models of motzkin paths in a wedge geometry are considered as models of forces in polymers in confined geometries a motzkin path in the square lattice is a path from the origin to a point in the line y x while it never visits sites below this line and it is constrained to give unit length steps only in the north and east directions and steps of length in the north east direction motzkin path models may be generalized to ensembles of ne oriented paths above the line y rx where r 0 is an irrational number these are paths giving east north and north east steps from the origin in the square lattice and confined to the r wedge formed by the y axis and the line y rx the generating function gr of these paths is not known but if r 1 then i determine its radius of convergence to be and if r 0 1 then tr 1 3 the entropic force the path exerts on the line y rx may be computed from this an asymptotic expression for the force for large values of r is given by in terms of the vertex angle of the r wedge the moment of the force about the origin has leading terms as 0 and f 0 if 4 2 moreover numerical integration of the force shows that the total work done by closing the wedge is 1 085 07 lattice units an alternative ensemble of ne oriented paths may be defined by slightly changing the generating function gr in this model it is possible to determine closed form expressions for the limiting free energy and the force the leading term in an asymptotic expansions for this force agrees with the leading term in the asymptotic expansion of the above model and the subleading term only differs by a factor of 2

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

lattice models of branched polymers statistics of uniform stars the authors investigate the statistical properties of uniform star polymers with f branches modelled on lattices in two and three dimensions it is shown that the growth constant exists and is equal to mu f where mu is the self avoiding walk limit the f dependence of the corresponding critical exponent gamma f is studied using exact enumeration and monte carlo techniques and the results are compared with the predictions of miyake and freed 1983 obtained using chain conformation space renormalisation group method

the growth constant of uniform star polymers in a slab geometry the authors consider uniform star branched polymer molecules modelled by the corresponding graph weakly embedded in a simple cubic lattice confined to a parallel sided slab in three and higher dimensions they show that the limiting entropy per bond is independent of the number of branches in the star and is equal to the connective constant of a self avoiding walk on that subset of the lattice in two dimensions however the limiting entropy per bond of a three branch star is shown to be strictly less than that for a self avoiding walk and present numerical evidence that this is a general result in two dimensions

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

applying tensor based morphometry to parametric surfaces can improve mri based disease diagnosis many methods have been proposed for computer assisted diagnostic classification full tensor information and machine learning with 3d maps derived from brain images may help detect subtle differences or classify subjects into different groups here we develop a new approach to apply tensor based morphometry to parametric surface models for diagnostic classification we use this approach to identify cortical surface features for use in diagnostic classifiers first with holomorphic 1 forms we compute an efficient and accurate conformal mapping from a multiply connected mesh to the so called slit domain next the surface parameterization approach provides a natural way to register anatomical surfaces across subjects using a constrained harmonic map to analyze anatomical differences we then analyze the full riemannian surface metric tensors which retain multivariate information on local surface geometry as the number of voxels in a 3d image is large sparse learning is a promising method to select a subset of imaging features and to improve classification accuracy focusing on vertices with greatest effect sizes we train a diagnostic classifier using the surface features selected by an l1 norm based sparse learning method stability selection is applied to validate the selected feature sets we tested the algorithm on mri derived cortical surfaces from 42 subjects with genetically confirmed williams syndrome and 40 age matched controls multivariate statistics on the local tensors gave greater effect sizes for detecting group differences relative to other tbm based statistics including analysis of the jacobian determinant and the largest eigenvalue of the surface metric our method also gave reasonable classification results relative to the jacobian determinant the pair of eigenvalues of the jacobian matrix and volume features this analysis pipeline may boost the power of morphometry studies and may assist with image based classification

tests for distributed nonfocal brain activations abstract most approaches to detecting changes in functional brain images assume that activations are focal or very localized however the brain s response to cognitive of sensorimotor challenge may be spatially or anatomically distributed in this paper we consider the use fulness of a test based on the mean sum of squares of statistical parametric maps the performance of this test is evaluated using simulated and real data and is compared to the 2 test a test of the size of the activated region and a focal activation test based on the intensity of local maxima we demonstrate that the mean sum of squares test is more sensitive to nonfocal signals and propose that it could be used to complement approaches that are more sensitive to focal activations

linked independent component analysis for multimodal data fusion in recent years neuroimaging studies have increasingly been acquiring multiple modalities of data and searching for task or disease related changes in each modality separately a major challenge in analysis is to find systematic approaches for fusing these differing data types together to automatically find patterns of related changes across multiple modalities when they exist independent component analysis ica is a popular unsupervised learning method that can be used to find the modes of variation in neuroimaging data across a group of subjects when multimodal data is acquired for the subjects ica is typically performed separately on each modality leading to incompatible decompositions across modalities using a modular bayesian framework we develop a novel linked ica model for simultaneously modelling and discovering common features across multiple modalities which can potentially have completely different units signal and contrast to noise ratios voxel counts spatial smoothnesses and intensity distributions furthermore this general model can be configured to allow tensor ica or spatially concatenated ica decompositions or a combination of both at the same time linked ica automatically determines the optimal weighting of each modality and also can detect single modality structured components when present this is a fully probabilistic approach implemented using variational bayes we evaluate the method on simulated multimodal data sets as well as on a real data set of alzheimer s patients and age matched controls that combines two very different types of structural mri data morphological data grey matter density and diffusion data fractional anisotropy mean diffusivity and tensor mode

exact enumeration and monte carlo results for self avoiding walks in a slab we analyse exact enumeration data and monte carlo simulation results for a self avoiding walk model of a polymer confined between two parallel attractive walls plates we use the exact enumeration data to establish the regions where the polymer exerts an effective attractive force between the plates and where the polymer exerts an effective repulsive force by estimating the boundary zero force curve while the phase boundaries of the phase diagram have previously been conjectured we delineate this further by establishing the order of the phase transitions for the so called infinite slab that is when the plates are a macroscopic distance apart we conclude that the adsorption transitions associated with either plate are similar in nature to the half space situation even when a polymer is attached to the opposite wall the transition between the two adsorbed phases is established as first order importantly we conjecture a scaling theory valid in the desorbed and critically adsorbed regions of the phase diagram and demonstrate the consistency of the monte carlo data with these hypotheses by estimating the corresponding scaling functions

self avoiding walks with geometrical constraints we consider self avoiding walks on ad dimensional hypercubic lattice confined to a slab geometry and confined to a half space we present a proof of the existence of a connective constant for the slab geometry and review some corresponding results for the half space we also discuss the way in which scaling arguments can be used to give stronger but nonrigorous results

forces in square lattice directed paths in a wedge a square lattice directed path confined to a wedge with vertex angle exerts an entropic force f on the wedge we show that this is a repulsive force of magnitude this force is determined by examining the combinatorial properties of the directed path and by determining the exact entropic contribution to the free energy in the limit as the path length goes to infinite

adsorption of uniform lattice animals with specified topology the author studies the effects of topology on the free energy of uniform lattice animals interacting with a surface topology is specified by fixing an abstract graph tau and the lattice animals considered are embeddings of tau in the square and simple cubic lattice he proves that such embeddings in the simple cubic lattice and interacting with a plane have the same free energy as self avoiding walks independent of the choice of tau and independent of whether or not the embeddings are restricted to be uniform for embeddings in the square lattice and interacting with a line he proves that the free energy may differ from that for walks depending on whether tau has a cut edge further restricting the embeddings to be uniform forces the free energy to be different from that for walks for all tau except the tau corresponding to walks and we obtain bounds on the free energy which depend on the number of branches cycles and vertices of degree 3 and 4 in the graph

critical behaviour at an edge the critical behaviour of a magnetic system with o n spin symmetry bounded by two d 1 dimensional hyperplanes meeting at an angle alpha is studied within mean field theory and in d 4 epsilon dimensions new exponents emerge for correlation functions and magnetisation and susceptibilities for spins close to the edge they can be expressed in terms of known bulk and surface exponents together with a single new edge exponent which depends however on the angle alpha this exponent is computed to first order in epsilon

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

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

brownian motion and harmonic functions on rotationally symmetric manifolds we consider brownian motion x on a rotationally symmetric manifold m g mathbb r n ds 2 ds 2 dr 2 g r 2 d theta 2 an integral test is presented which gives a necessary and sufficient condition for the nontriviality of the invariant sigma field of x hence for the existence of nonconstant bounded harmonic functions on m g conditions on the sectional curvatures are given which imply the convergence or the divergence of the test integral

polygons and stars in a slit geometry the authors consider self avoiding polygons uniform 3 stars and uniform 4 stars weakly embeddable in the square lattice and confined between two parallel lines y 0 and y l they show rigorously that the connective constants of these three structures provided that the corresponding limits exist are all strictly less than the connective constant kappa l of self avoiding walks with the same geometrical constraint they also derive lower bounds on the connective constants of uniform 3 stars and 4 stars these bounds appear to be strong at least for small l

nonergodicity of local length conserving monte carlo algorithms for the self avoiding walk it is proved that every dynamic monte carlo algorithm for the self avoiding walk based on a finite repertoire of local n conserving elementary moves is nonergodic heren is the number of bonds in the walk indeed for largen each ergodic class forms an exponentially small fraction of the whole space this invalidates at least in principle the use of the verdier stockmayer algorithm and its generalizations for high precision monte carlo studies of the self avoiding walk

critical behaviour at an edge for the saw and ising model the authors have studied the behaviour of two and three dimensional self avoiding walks confined to a wedge of wedge angle alpha series have been obtained and analysed for the angular dependent critical exponents characterising various edge susceptibilities in terms of a general scaling form for the edge free energy fe approximately t d 2 nu psi e ht y0 nu h1t y1 nu h2t y2 nu they find for the two dimensional case the following scaling indices y0 91 48 y1 3 8 y2 alpha 5 pi 8 alpha they argue that these results are exact from which follow all exponents for the bulk surface and edge problem in three dimensions the authors obtain y0 approximately 2 488 y1 0 65 or 0 02 y2 alpha alpha b pi alpha where a 0 51 or 0 04 b 0 847 or 0 017 which for y2 is precisely of the functional form given by mean field theory y2 alpha 1 2 pi 2 they argue that a 1 2 for all three dimensional o n models this simple angular dependence of y2 is different from that suggested by cardy s one loop epsilon expansion 1983 for the square lattice they have also studied the case in which the wedge is rotated through an angle of pi 4 and find that the various exponents are unchanged for the three dimensional ising model in a wedge analogy with the saw results plus mean field results in conjunction with rg and series work yield y0 approximately 2 485 y1 0 71 or 0 02 and y2 a b pi alpha with a 1 2 and b 0 79 or 0 02

criticality of self avoiding walks with an excluded infinite needle the authors study the effect of repulsion for self avoiding walks and random walks from excluded sets they show in particular that the mean displacement away from an excluded infinite needle of self avoiding random walks in three dimensions has to diverge along the privileged axis as nsigma where n is the number of steps and sigma is a sub leading critical exponent for the two point function this exponent has been determined by using a high precision monte carlo simulation sigma 0 370 or 0 011 its knowledge is used to improve the measure of universal quantities like the exponent nu nu 0 5867 or 0 0025 in agreement with the in expansion estimate and with experimental data and amplitude ratios they verify also that for simple random walks the excluded needle introduces instead logarithmic violations to scaling

branched polymers in a wedge geometry in three dimensions we investigate the statistical and dimensional properties of uniform star polymers attached by the branching vertex of degreef in a wedge geometry in three dimensions and described by the wedge angles and we show that the growth constant is equal to f where is the self avoiding walk limit thef and dependences of the corresponding critical exponent f are studied using monte carlo techniques in the casef 1 our results are compared with existing predictions obtained from series expansion and renormalization group methods we have also estimated the amplitudes for the mean square radius of gyration and the mean square end to end branch length our results for the ratio of the mean square radius of gyration of anf star to that of a linear polymer of the same degree of polymerization attached in a similar wedge and the analogous ratio for the mean square end to end branch length are consistent with these ratios being lattice independent quantities

decomposing eeg data into space time frequency components using parallel factor analysis abstract finding the means to efficiently summarize electroencephalographic data has been a long standing problem in electrophysiology a popular approach is identification of component modes on the basis of the time varying spectrum of multichannel eeg recordings in other words a space frequency time atomic decomposition of the time varying eeg spectrum previous work has been limited to only two of these dimensions principal component analysis pca and independent component analysis ica have been used to create space time decompositions suffering an inherent lack of uniqueness that is overcome only by imposing constraints of orthogonality or independence of atoms conventional frequency time decompositions ignore the spatial aspects of the eeg framing of the data being as a three way array indexed by channel frequency and time allows the application of a unique decomposition that is known as parallel factor analysis parafac each atom is the tri linear decomposition into a spatial spectral and temporal signature we applied this decomposition to the eeg recordings of five subjects during the resting state and during mental arithmetic common to all subjects were two atoms with spectral signatures whose peaks were in the theta and alpha range these signatures were modulated by physiological state increasing during the resting stage for alpha and during mental arithmetic for theta furthermore we describe a new method source spectra imaging or ssi to estimate the location of electric current sources from the eeg spectrum the topography of the theta atom is frontal and the maximum of the corresponding ssi solution is in the anterior frontal cortex the topography of the alpha atom is occipital with maximum of the ssi solution in the visual cortex we show that the proposed decomposition can be used to search for activity with a given spectral and topographic profile in new recordings and that the method may be useful for artifact recognition and removal

double coupled canonical polyadic decomposition for joint blind source separation joint blind source separation j bss is an emerging data driven technique for multi set data fusion in this paper j bss is addressed from a tensorial perspective we show how by using second order multi set statistics in j bss a specific double coupled canonical polyadic decomposition dc cpd problem can be formulated we propose an algebraic dc cpd algorithm based on a coupled rank 1 detection mapping this algorithm converts a possibly underdetermined dc cpd to a set of overdetermined cpds the latter can be solved algebraically via a generalized eigenvalue decomposition based scheme therefore this algorithm is deterministic and returns the exact solution in the noiseless case in the noisy case it can be used to effectively initialize optimization based dc cpd algorithms in addition we obtain the deterministic and generic uniqueness conditions for dc cpd which are shown to be more relaxed than their cpd counterpart we also introduce optimization based dc cpd methods including alternating least squares and structured data fusion based methods experiment results are given to illustrate the superiority of dc cpd over standard cpd based bss methods and several existing j bss methods with regards to uniqueness and accuracy

probabilistic independent component analysis for functional magnetic resonance imaging we present an integrated approach to probabilistic independent component analysis ica for functional mri fmri data that allows for nonsquare mixing in the presence of gaussian noise in order to avoid overfitting we employ objective estimation of the amount of gaussian noise through bayesian analysis of the true dimensionality of the data i e the number of activation and non gaussian noise sources this enables us to carry out probabilistic modeling and achieves an asymptotically unique decomposition of the data it reduces problems of interpretation as each final independent component is now much more likely to be due to only one physical or physiological process we also describe other improvements to standard ica such as temporal prewhitening and variance normalization of timeseries the latter being particularly useful in the context of dimensionality reduction when weak activation is present we discuss the use of prior information about the spatiotemporal nature of the source processes and an alternative hypothesis testing approach for inference using gaussian mixture models the performance of our approach is illustrated and evaluated on real and artificial fmri data and compared to the spatio temporal accuracy of results obtained from classical ica and glm analyses

a directed walk model of a long chain polymer in a slit with attractive walls we present the exact solutions of various directed walk models of polymers confined to a slit and interacting with the walls of the slit via an attractive potential we consider three geometric constraints on the ends of the polymer and concentrate on the long chain limit apart from the general interest in the effect of geometrical confinement this can be viewed as a two dimensional model of steric stabilization and sensitized flocculation of colloidal dispersions we demonstrate that the large width limit admits a phase diagram that is markedly different from the one found in a half plane geometry even when the polymer is constrained to be fixed at both ends on one wall we are not able to find a closed form solution for the free energy for finite width at all values of the interaction parameters but we can calculate the asymptotic behaviour for large widths everywhere in the phase plane this allows us to find the force between the walls induced by the polymer and hence the regions of the plane where either steric stabilization or sensitized flocculation would occur

directed paths in a wedge models of directed paths have been used extensively in the scientific literature to model linear polymers in this paper we examine a directed path model of a linear polymer in a confining geometry a wedge we are particularly interested in cn the number of directed lattice paths of length n steps which take steps in the north east and south east directions and are confined in the wedge y x p where p is an integer we examine the case p 2 in detail and show that the generating function satisfies a functional equation with quartic kernel we give a formal solution by using the kernel method show that this model is not d finite and determine asymptotics to leading order for cn in particular the number of paths of length n in the wedge y x 2 is given by where the constant 0 67874 can be determined to arbitrary accuracy with little computational effort

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

critical points and intermediate phases on wedges of zd the authors examine the phase structure of self avoiding walks the ising magnet and bond percolation defined on subsets of zd d or 2 which have the geometry of wedges they prove that if the cross sectional area of the wedge diverges with its width then the high temperature critical point of any of these models defined on the wedge coincides with that of the corresponding model on the full lattice for the ising magnet and bond percolation they show that there is a non trivial low temperature critical point if the cross sectional area of the wedge diverges logarithmically with its width moreover for bond percolation they show that the low temperature critical point of a logarithmic wedge may be made arbitrarily close to that of the full lattice by taking the coefficient of the logarithm large enough corollaries to these theorems include the existence of an intermediate phase for the ising magnet and bond percolation on logarithmic wedges and the existence of a first order transition for percolation models on a subclass of these wedges

on the asymptotic behaviour of first passage times for discussions we investigate the asymptotic behaviour of first passage times of diffusions or birth death processes necessary and sufficient conditions in terms of moments and the speed and scale functions are given for convergence of the first passage times to a normal or an exponential distribution

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

explicit and almost explicit spectral calculations for diffusion operators abstract the diffusion operator h d 1 2 d d x a d d x b d d x 1 2 exp 2 b d d x a exp 2 b d d x where b x 0 x b a y d y defined either on r 0 with the dirichlet boundary condition at x 0 or on r can be realized as a self adjoint operator with respect to the density exp 2 q x d x the operator is unitarily equivalent to the schrodinger type operator h s 1 2 d d x a d d x v b a where v b a 1 2 b 2 a b we obtain an explicit criterion for the existence of a compact resolvent and explicit formulas up to the multiplicative constant 4 for the infimum of the spectrum and for the infimum of the essential spectrum for these operators we give some applications which show in particular how inf h d scales when a a 0 and b b 0 where and are parameters and a 0 and b 0 are chosen from certain classes of functions we also give applications to self adjoint multi dimensional diffusion operators

a new approach to the martin boundary via diffusions conditioned to hit a compact set let l generate a transient diffusion x t on rd and let d be an exterior domain let h be the smallest positive solution of lh 0 in d and h 1 on dd define xh t to be the process x t conditioned to hit dd by doob s h transform theory xh t is also a markov diffusion and its generator lh is defined by lhf l h l hf letting rd be the hitting time of dd define the harmonic measure for xh t on dd starting from x e d by i4x dy p r xh trd e dy let xn n 1 c d be a sequence satisfying limp xn i for which i4h converges weakly call two such sequences xn 1 and x 1 equivalent if lim k limn o x we call the set of equivalence classes thus generated the harmonic measure boundary at infinity for lh this boundary is independent of the particular exterior domain d we prove that the harmonic measure boundary at infinity for lh coincides with the martin boundary for l on rd the formal adjoint of the operator l on rd in the case that l generates a reversible diffusion the martin boundaries of l and l coincide and hence the harmonic measure boundary of lh coincides with the martin boundary for l on rd a similar probabilistic description of the martin boundary for l on rd can be given in the nonreversible case these results are then used to give explicit representations of the martin boundaries of l and l for several classes of diffusion processes

lattice models of branched polymers uniform combs in two dimensions the authors establish rigorously the relationship between the growth constant of uniform trees with two branch points and that of self avoiding walks for combs they estimate the corresponding exponent and study the mean square lengths of the internal and external branches

probabilistic inference of regularisation in non rigid registration a long standing issue in non rigid image registration is the choice of the level of regularisation regularisation is necessary to preserve the smoothness of the registration and penalise against unnecessary complexity the vast majority of existing registration methods use a fixed level of regularisation which is typically hand tuned by a user to provide nice results however the optimal level of regularisation will depend on the data which is being processed lower signal to noise ratios require higher regularisation to avoid registering image noise as well as features and different pairs of images require registrations of varying complexity depending on their anatomical similarity in this paper we present a probabilistic registration framework that infers the level of regularisation from the data an additional benefit of this proposed probabilistic framework is that estimates of the registration uncertainty are obtained this framework has been implemented using a free form deformation transformation model although it would be generically applicable to a range of transformation models we demonstrate our registration framework on the application of inter subject brain registration of healthy control subjects from the nirep database in our results we show that our framework appropriately adapts the level of regularisation in the presence of noise and that inferring regularisation on an individual basis leads to a reduction in model over fitting as measured by image folding while providing a similar level of overlap

self avoiding polygons and walks in slits a polymer in a confined geometry may be modeled by a self avoiding walk or a self avoiding polygon confined between two parallel walls in two dimensions this model involves self avoiding walks or self avoiding polygons in the square lattice between two parallel confining lines interactions of the polymer with the confining walls are introduced by energy terms associated with edges in the walk or polygon which are at or near the confining lines we use transfer matrix methods to investigate the forces between the walk or polygon and the confining lines as well as to investigate the effects of the confining slit s width and of the energy terms on the thermodynamic properties of the walks or polygons in several models the phase diagram found for the self avoiding walk models is qualitatively similar to the phase diagram of a directed walk model confined between two parallel lines as was previously conjectured however the phase diagram of one of our polygon models is found to be significantly different and we present numerical data to support this for that particular model we prove that for any finite values of the energy terms there are an infinite number of slit widths where a polygon will induce a steric repulsion between the confining lines

independent factor analysis we introduce the independent factor analysis ifa method for recovering independent hidden sources from their observed mixtures ifa generalizes and unifies ordinary factor analysis fa principal component analysis pca and independent component analysis ica and can handle not only square noiseless mixing but also the general case where the number of mixtures differs from the number of sources and the data are noisy ifa is a two step procedure in the first step the source densities mixing matrix and noise covariance are estimated from the observed data by maximum likelihood for this purpose we present an expectation maximization em algorithm which performs unsupervised learning of an associated probabilistic model of the mixing situation each source in our model is described by a mixture of gaussians thus all the probabilistic calculations can be performed analytically in the second step the sources are reconstructed from the observed data by an optimal nonlinear estimator a variational approximation of this algorithm is derived for cases with a large number of sources where the exact algorithm becomes intractable our ifa algorithm reduces to the one for ordinary fa when the sources become gaussian and to an em algorithm for pca in the zero noise limit we derive an additional em algorithm specifically for noiseless ifa this algorithm is shown to be superior to ica since it can learn arbitrary source densities from the data beyond blind separation ifa can be used for modeling multidimensional data by a highly constrained mixture of gaussians and as a tool for nonlinear signal encoding

on a conjecture of hammersley and whittington concerning bond percolation on subsets of the simple cubic lattice the author verifies the truth of a conjecture of hammersley and whittington 1985 concerning bond percolation on certain subsets of the simple cubic lattice z3 let f and g be non decreasing non negative functions on 0 infinity and let z3 f g denote the f g wedge of z3 being the set of points x y z such that 0 or 0 the author shows that the condition 1 f x 1 g x to infinity as x to infinity is sufficient for the critical probability of the bond percolation process on z3 f g to be less than or equal to 1 2

the statistical mechanics of random copolymers random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness we review the statistical mechanics of random copolymers focusing on self avoiding walk models where there are two types of monomers a and b which are randomly distributed along the polymer chain theoretical approximate and numerical results are reviewed for models of the random copolymer adsorption localization and collapse phase transitions we concentrate on what is known about the existence of phase transitions the morita approximation and results about self averaging we also discuss in less detail the replica trick and numerical methods including monte carlo methods exact enumeration and transfer matrix methods important open problems are identified throughout and highlighted in the conclusions

motzkin path models of long chain polymers in slits we consider motzkin path models for polymers confined to a slit the path interacts with each of the two confining lines and we define parameters a and b to characterize the strengths of the interactions with the two lines we consider the cases where i each vertex in a confining line contributes to the energy and ii where each edge in a confining line contributes to the energy for the vertex and edge versions of motzkin paths we can find the generating functions and give rigorous explicit expressions for the free energy at some special points in the a b plane and asymptotically i e for large slit widths elsewhere we find regions where the force between the lines is long range and repulsive short range and repulsive and short range and attractive our results indicate that the general form of the phase diagram is model independent and similar to previous results for a dyck path model although the details do depend on the underlying configurational model we also contrast the method used here to find the generating function with the transfer matrix and heap methods

the asymptotic behavior of the solution of the exterior dirichlet problem for brownian motion perturbed by a small parameter drift let l 2a eb v in rd d 2 3 generate a recurrent diffusion for each e 0 where b ei ca rd and let d c rd be an exterior domain then by the recurrence assumption for each f eis c ad there exists a unique solution in the class of bounded solutions to the dirichlet problem leue 0 in d and ue 4i on ad on the other hand by the transience of d dimensional brownian motion there is no uniqueness in the class of bounded solutions for the dirichlet problem 1 au 0 in d and u f on ad since the martin boundary at oo for brownian motion consists of a single point uniqueness is obtained by adding the condition lim1xj u x c we show that u0 x lime o u x exists and satisfies 1 au0 0 in d u0 fi on dd and limb1os uo x c where c is given as follows let pxh denote the measure associated with doob s conditioned brownian motion conditioned to exit d at ad rather than at oo let x inflt 2 0 x t e ad and define the harmonic measure u dy pxh x t e dy then ah lim 4 exists and c jadq y 4 dy we also show that the energy integral jdivui dx when varied over all bounded functions u es w 10o2 d which satisfy u ip on dd takes on its minimum uniquely at u0

variational bayes for generalized autoregressive models we describe a variational bayes vb learning algorithm for generalized autoregressive gar models the noise is modeled as a mixture of gaussians rather than the usual single gaussian this allows different data points to be associated with different noise levels and effectively provides robust estimation of ar coefficients the vb framework is used to prevent overfitting and provides model order selection criteria both for ar order and noise model order we show that for the special case of gaussian noise and uninformative priors on the noise and weight precisions the vb framework reduces to the bayesian evidence framework the algorithm is applied to synthetic and real data with encouraging results

free energy of self interacting uniform stars we consider uniform star polymers on a variety of lattices with an additional contact interaction between pairs of vertices which are unit distance apart and are not joined by an edge of the star we present some rigorous results and other evidence which indicate that these systems have the same limiting free energy as self interacting self avoiding walks we discuss the extension of these results to other homeomorphism types and to systems with an additional surface interaction term