low firing rates in a compartmental model neuron a nonlinear relationship is obtained between the membrane potential of a compartmental model neuron as a function of input stimulation this is used to show how shunting effects can lead to low output firing rates in associative memory networks

inertia tensor and size of a polymer chain guided by an approach used in the study of membranes we construct the partition function for a discrete gaussian chain model of polymers the hamiltonian is where describes the chain connectivity is the inertia tensor which gives the shape and size of the polymer chain and tr denotes the trace here we consider only the size which is obtained from the partition function by differentiating with respect to the conjugate variable the partition function is expressed in terms of the hypergeometric function an orthogonal polynomial where n is the number of monomers and is the ratio of and the coefficient in the limit corresponds to the random coil while the limit describes a compact object we also study the excluded volume problem by discretizing edwards continuous self avoidance term we obtain dimensionally regularized expressions for the radius of gyration and the end to end distance the short distance cut off dependence of the continuous model is reproduced

lattice green functions and calabi yau differential equations by making the connection between four dimensional lattice green functions lgfs and picard fuchs ordinary differential equations of calabi yau manifolds we have given explicit forms for the coefficients of the four dimensional lgfs on the simple cubic and body centred cubic lattices in terms of finite sums of products of binomial coefficients and have shown that the corresponding four dimensional face centred cubic lgf satisfies a fourth order ode of degree 7 which we have found but not solved the fact that the picard fuchs equations that appear here are similar to those that appear in string theory perhaps makes the first connection between these two apparently unrelated topics

fractal and nonfractal shapes in two dimensional vesicles abstract ongoing collaborative work on the statistical mechanics of two dimensional vesicles is reviewed with emphasis on the range of fractal and nonfractal shapes exhibited by the bead and tether model of a closed membrane which also embodies an osmotic pressure difference p and a bending rigidity modulus monte carlo simulations and scaling analyses are described flaccid vesicles with p 0 represent closed self avoiding rings their mean area a scales with the mean square size r g 2 their mean shape is close to elliptical inflated p 0 vesicles become circular deflated p branched polymers the vesicle shapes appear to vary continuously with the scaling combination p n 2 where n is the number of monomers beads and 3 4 for p k b t large relative to n 3 a range of characteristic nonfractal shapes or cytotypes appears

moments of inertia and the shapes of brownian paths we compute the joint probability law of the principal moments of inertia of brownian paths open or closed using constrained path integrals and random matrix theory the case of two dimensional paths is discussed in detail in particular we show that the ratio of the average values of the largest and smallest moments is equal to 4 99 open paths and 3 07 closed paths we also present results of numerical simulations which include investigation of the relationships between the moments of inertia and the arithmetic area enclosed by a path

shapes and sizes of arbitrary random walks at o 1 d3 ii asphericity and prolateness parameters abstract for arbitrary random walks in any d dimensional space expansions in powers of 1 d of asphericity and prolateness parameters and moments of the inverse size ratio have been developed which at o 1 d 3 yield very good approximations to exact values of the parameters for chains rings dumbbells and 3 and 5 arm regular stars the 1 d expansions have also been used to obtain an estimate of these shape asymmetry parameters for 3d edwards chains rings dumbbells and 3 arm stars and to give a mathematical proof that infinitely large random nets such as bethe lattice or starburst and mckay s net exhibit spherical symmetry for arbitrary random walks at d it is proved that these parameters coincide with their corresponding factors while for an end looped self avoiding walk it is found that its shape asymmetry is even larger than that of an open saw an 1 f expansion of the parameters for f arm regular stars has also been obtained and a comparison of the dimensionality dependence of the parameters with that of the corresponding factors has been made for the four types of random walks

spherically symmetric random walks in noninteger dimension a previous article proposed a new kind of random walk on a spherically symmetric lattice in arbitrary noninteger dimension d such a lattice avoids the problems associated with a hypercubic lattice in noninteger dimension this article examines the nature of spherically symmetric random walks in detail a large time asymptotic analysis of these random walks is performed and the results are used to determine the hausdorff dimension of the process exact results are obtained in terms of hurwitz functions incomplete zeta functions for the probability of a walker going from one region of the spherical lattice to another finally it is shown that the probability that the paths of k independent random walkers will intersect vanishes in the continuum limit if d 2k k 1

canonical partition functions of freely jointed chains the freely jointed chain is a simple off lattice ideal model of a heteropolymer we show how to compute the canonical partition function of this model exactly for all physical primary structures degree of polymerization and space dimensions greater than or equal to 2 the canonical partition function of this model of a heteropolymer has an analytical expression involving a complicated multiple hypergeometric function to obtain some accurate approximations we develop and use the independent motion approximation ima

power laws from randomly sampled continuous time random walks abstract it has been shown by reed that random sampling a wiener process x t at times t chosen out of an exponential distribution gives rise to power laws in the distribution p x t x t we show both theoretically and numerically that this power law behaviour also follows by random sampling levy flights as continuous time random walks having fourier distribution w k e k with the exponent

amenable isometry groups of hadamard spaces one of the ever recurring themes in the theory of nonpositively curved spaces is the relation between geometry and topology since the universal covering space of a complete space of nonpositive curvature is contractible this amounts to relations between the geometry of the space or its universal covering space and the algebraic structure of its fundamental group a paradigm is the well known result of avez av that the fundamental group of a compact riemannian manifold m of nonpositive sectional curvature has exponential growth if and only if m is not flat gromov observed that the proof of avez yields a somewhat stronger statement the fundamental group of m is amenable if and only if m is flat see g1 p 93 zimmer generalized this to the case where m is complete and of finite volume zi further generalizations were obtained by anderson an and burger and schroeder bs the objective of this paper is the discussion of these results under the weaker assumption that the underlying spaces are alexandrov spaces of nonpositive curvature for the investigation of relations between the fundamental group and the geometry of a space of nonpositive curvature it is convenient to study the isometric action of the fundamental group on the universal covering space in many situations it is not necessary to assume that a group action arises in this particular way and one studies isometric actions on complete simply connected spaces of nonpositive curvature this is the case in most of our arguments below let x be a hadamard space that is a complete simply connected geodesic space of nonpositive curvature in the sense of alexandrov a flat in x is a

on the application of geometric probability theory to polymer networks and suspensions ii the theory of stochastic processes and integral geometry is applied to polymer networks and suspensions averaged monomer density is calculated for adsorbed and depleted layers of single chain polymers modelled as brownian paths near plane and spherical surfaces and exact solutions are given for an adsorbed layer with a force field asymptotic expressions are deduced for the available space fraction for a small solute in such layers a measure of the typical asymmetry of a single chain polymer modelled as a brownian path is obtained by calculating the averaged integral of the mean curvature surface area and volume of its convex hull the averaged integral of the mean curvature and surface area also give the first two terms of an asymptotic expression for the depletion near a smooth slightly curved surface and an upper bound on the averaged capacitance these quantities are also calculated for model polymer aggregates brownian rings and brownian paths with drift from the latter we deduce averages of powers of the end to end distance of a brownian path weighted by the integral of the mean curvature and surface area of its convex hull the depletion due to a lamina with a smooth slightly curved boundary in a solution of single chain polymers is calculated and the result is interpreted in terms of averaged geometrical properties of brownian paths

geometrical properties of aggregates with tunable fractal dimension we have computed geometrical characteristics of large clusters up to 32 768 particles obtained by a hierarchical cluster cluster aggregation computer model in three dimensions the off lattice variable d model using a box counting method we have calculated the fractal dimensions of the surface and the perimeter of their two dimensional projections as a function of their fractal dimension d by diagonalizing the radius of gyration tensor we have obtained numerical estimates for the intrinsic anisotropy coefficients ratios of the eigenvalues and we have proposed analytical expressions to describe their behaviour as a function of the fractal dimension

the wave equation with wentzell robin boundary conditions on l p spaces abstract let a w 1 0 1 a x 0 b c l 0 1 and consider the differential operator a formally given by a u a u b u c u we prove in the first part that a realization of a with wentzell robin boundary conditions on l p 0 1 c 2 generates a cosine function for p 1 in particular we obtain that this realization of a generates a holomorphic c 0 semigroup of angle 2 on the space l 1 0 1 c 2 this solves an open problem by a favini g r goldstein j a goldstein s romanelli and w arendt of crucial importance is the formulation of the boundary conditions we show in the second part that if 0 1 n n 1 is the cube in r n then the laplacian with pure wentzell boundary conditions generates an times integrated cosine function on c for any n 1 2

singular perturbation for volterra equations of convolution type under the assumption that a is the generator of a twice integrated cosine family and k is a scalar valued kernel we solve the singular perturbation problem e e e 2u e t u e t au e t k au e t f e t t 0 e 0 when e 0 for the integrodifferential equation e w t aw t k aw t f t t 0 on a banach space if the kernel k verifies some regularity conditions then we show that problem e e has a unique solution u e t for each small e 0 moreover u e t converges as e 0 to the unique solution u t of equation e

the aspherity of random walks the asphericity a of a random walk is defined and calculated for non self avoiding walks the definition of a generalises to self avoiding walks percolating clusters and other fractal objects and thus provides a generalised quantitative measure of the departure from spherical symmetry of the gross shape of these objects

meshless membrane model based on the moving least squares method a meshless particle based membrane model is proposed the particles possess no internal degree of freedom and interact via a potential which has three different contributions a short range repulsive pair potential an attractive multibody potential and a curvature potential based on the moving least squares method brownian dynamics simulations are employed to demonstrate that the particles self assemble into a membrane and to study equilibrium properties such as bending rigidity surface tension line tension and diffusion constant the bending rigidity and line tension are shown to depend on different potential parameters and can therefore be varied independently the finite size effects of nearly planar membranes are investigated this model is well suited to study the membrane dynamics with topological changes

a simple dynamical model of two dimensional vesicles two dimensional vesicles are modelled as pressurized self intersecting rings the fluctuating motion of the bounding walls of the vesicle when it is subjected to the random collisions of fluid molecules is treated along the lines suggested by rouse 1953 in his study of dilute polymer solutions within the model exact solutions of the dynamical equation governing the probability distribution of shapes when expressed in an appropriate set of normal coordinates is obtained numerical simulations of the dynamical evolution of the distribution of the principal components of the square of the radius of gyration are also presented

study of the shape of random walks a comprehensive study of the shape of discrete random walks considering the general case of arbitrary length and any space dimension is presented the probability distributions for several magnitudes such as the principal inertia moments the asphericity and the angle between the principal axis of inertia and the end to end vector are evaluated numerically in most cases and especially in low dimensional spaces these probability distributions spread widely and possess a non zero skewness this implies that any description of the shapes of random walks which is based only on mean values of related magnitudes is incomplete this situation does nor vary when random walks of large lengths are considered since the relative fluctuations ratio between the standard deviation and the mean value do not go to zero in that limit on the other hand when the space dimension grows the distributions become more peaked reaching the expected asymptotic limit for infinite dimension the present study also indicates that the probability distributions for the principal inertia moments have approximately the form of chi squared distributions analysing the probability distribution for the asphericity it is shown that the distributions for the different moments are not completely independent

the shapes and sizes of two dimensional pressurized self intersecting rings as models for two dimensional vesicles two dimensional vesicles are modelled as pressurized unrestricted closed random walks the model though suffering from undesirable physical characteristics has the decided advantage of yielding exact analytical statistical analysis of the size and shapes of these objects closed form expressions for the average of the square of the radius of gyration and the asphericity are derived in addition exact expressions for the full probability distribution are also obtained numerical computations of the probability function for several pressures are presented the authors arrive at a new exact formula for the distribution function p r12 r22 of the principal components of the square of the radius of gyration for two dimensional rings

the shapes of open and closed random walks a 1 d expansion a new technique for calculating the shapes of random walks is presented the method is used to derive an exact analytical expression for the asphericity of an unrestricted closed or ring walk embedded in d spatial dimensions a graphical procedure is developed to systematise a 1 d series expansion for the individual principal radii of gyration and their respective probability distribution functions p r2i 1 or i or d the average principal radii of gyration are calculated to o 1 d2 for both open and closed walks and selected terms in the 1 d expansion are summed to all orders in 1 d in the determination of p r2i this leads to an explicit analytical form for p r2i for open walks the distribution of the largest eigenvalue is compared with a distribution obtained from numerical simulations of walks in three dimensions the agreement between the two is extremely good other predictions for various parameters that characterise the average shape of open and closed walks in three dimensions are also found to agree remarkably well with the results of simulations the error being of the order of 5

shapes and sizes of arbitrary random walks at o 1 d3 i most probable size ratio abstract for arbitrary random walks in any d dimensional space a 1 d expansion of the most probable size ratio i e squared radius of gyration s 2 divided by s 2 of open random walks has been developed which at o 1 d 3 yields a very good approximation to the exact value for chains d 2 and rings d 1 and for the first time gives an estimate of the most probable size ratio for end looped random walks asymptotic distribution functions for large and small size ratio have also been investigated analytically for open and closed random walks with explicit results given up to the fourth order for any values of d for random walks at d it has been proved that the most probable size coincides with mean size and the th shape factor is inversely proportional to the th eigenvalue of the architecture matrix for the walks

percolation and cluster structure parameters the radius of gyration the squared radius of gyration of percolation clusters are determined in terms of the clusters second spatial moments using the enhanced hoshen kopelman algorithm for square and triangular lattices containing sites correlation length exponents and related exponents as well as their corresponding amplitudes are calculated in terms of the squared radius of gyration above and below the percolation threshold a coefficient for cluster compactness that is based on the squared radius of gyration of a cluster is introduced that coefficient is compared with a similar coefficient of compactness that is based on the cluster cyclomatic number

the shapes of high dimensional random walks the individual principal radii of gyration are computed for unrestricted random walks in a large number of spatial dimensions a low order expansion in one over the dimensionality yields useful information regarding the distribution and average of these measures of the extent and spatial anisotropy of high dimensional random walks such an expansion may well prove useful in the study of the shapes of other random fractal objects

two dimensional polymer networks near percolation we report an extensive finite size study of polymer networks near the percolation threshold using numerical techniques the polymers are modeled by random walks occupying the bonds of a two dimensional square lattice we measure the percolation threshold and critical exponents of the networks for various polymer lengths we find that the critical occupation probability is a decreasing function of the polymer length and the percolation of polymers with a fixed polymer length belongs to the same universality class as ordinary bond percolation by adding particles to the lattice cells we can study the diffusion process in polymer networks measuring the current near percolation we observe that its critical exponent is also independent of polymer length

a measure of the symmetry of random walks we derive the probability density for a simple measure of the asymmetry of a one dimensional random walk namely the ratio of the minimum of the two maximum displacements in the positive and negative directions to the maximum we show that in the diffusion limit the asymmetry is independent of time these results are shown to apply to random walks in which individual steps have a stable law distribution as well as to multidimensional random walks

singular perturbations of integro differential equations abstract we study the singular perturbation problem e 2 u t u t au t k au t f t t 0 0 for the integro differential equation e w t aw t k aw t f t t 0 in a banach space when 0 we assume that a is the generator of a strongly continuous cosine family then under some regularity conditions on the scalar valued kernel k we show that problem e has a unique solution u t for each small 0 moreover u t converges to u t as 0 the unique solution of equation e

universality and time scale invariance for the shape of planar levy processes for a broad class of planar markov processes viz l evy processes satisfying certain conditions valid e g in the case of brownian motion and l evy flights we establish an exact universal formula describing the shape of the convex hull of sample paths we show indeed that the average number of edges joining paths points separated by a time lapse ensuremath delta ensuremath tau ensuremath in left ensuremath delta ensuremath tau 1 ensuremath delta ensuremath tau 2 right is equal to 2 phantom rule 0 16em 0ex text ln left ensuremath delta ensuremath tau 2 ensuremath delta ensuremath tau 1 right regardless of the specific distribution of the process s increments and regardless of its total duration t the formula also exhibits invariance when the time scale is multiplied by any constant apart from its theoretical importance our result provides insights regarding the shape of two dimensional objects e g polymer chains modeled by the sample paths of stochastic processes generally more complex than brownian motion in particular for a total time or parameter duration t the average number of edges on the convex hull cut off to discard edges joining points separated by a time lapse shorter than some ensuremath delta ensuremath tau lt will be given by 2 phantom rule 0 16em 0ex text ln frac t ensuremath delta ensuremath tau thus it will only grow logarithmically rather than at some higher pace

the asphericity of star polymers a renormalization group study the asphericity of a flexible monodisperse f star polymer in d 3 is calculated in the framework of the standard two parameter model in the asymptotic limit of infinite molecular weight in contrast to previous analytical investigations we employ an asphericity measure which takes into account that the size and shape of the polymer coils are strongly correlated we consider ideal non interacting nev star polymers as well as molecules with excluded volume ev interactions by means of renormalized perturbation theory we show that in both cases the mean asphericity parameter takes a universal numerical value when the molecular weight tends to infinity the same holds for the corresponding asymptotic distribution functions which are also proved to be universal functions these universal properties depend only on the topology i e the number of arms f for nev stars we obtain a decrease of the asphericity with increasing number of arms the numerical values are in excellent agreement with results of previous computer simulations from the extrapolation of the epsilon expansion results for ev stars we conclude that the influence of the ev on the shape asymptotics qualitatively depends on the number of arms for f 1 and j 2 i e a linear chain we reproduce a previous finding that the ev enlarges the asphericity of the chains by about 5 for f 3 however we observe a decrease of the asphericity due to the ev which is intuitively appealing because of the strong steric repulsion for large values of f these results are also in good agreement with previous computer experiments

random walks in noninteger dimension one can define a random walk on a hypercubic lattice in a space of integer dimension d for such a process formulas can be derived that express the probability of certain events such as the chance of returning to the origin after a given number of time steps these formulas are physically meaningful for integer values of d however these formulas are unacceptable as probabilities when continued to noninteger d because they give values that can be greater than 1 or less than 0 in this paper a different kind of random walk is proposed which gives acceptable probabilities for all real values of d this d dimensional random walk is defined on a rotationally symmetric geometry consisting of concentric spheres the exact result is given for the probability of returning to the origin for all values of d in terms of the riemann zeta function this result has a number theoretic interpretation

study of the shape of random walks ii inertia moment ratios and the two dimensional asphericity a study of some features of the quantities used to describe the shape of random walks focusing on the probability distribution of the asphericity of discrete random walks in two dimensions is presented a connection is established between the asphericity distribution and the probability distribution corresponding to the ratio of the principal inertia moments the probability distributions of arbitrary inertia moment ratios are analysed for varying spatial dimension and an analytic expression for them is found which presents excellent agreement with monte carlo data in all the cases considered this function is then used to obtain the corresponding two dimensional asphericity distribution which also proved to be a very good approximation to the true distribution obtained from an independent monte carlo simulation

shapes of random walks at order 1 d2 the authors calculate various parameters which characterize the size and shape of random walks utilizing a recently developed 1 d expansion technique where d is the spatial dimension in which the random walk takes place a new procedure for extracting the averages of the principal radii of gyration is presented and the calculation is carried out to order 1 d2 which is one order higher than previous work comparison with the results of numerical simulations provides new insights regarding the accuracy of the 1 d expansion procedure

new approaches to shapes of arbitrary random walks abstract the problem of the shape of a random object such as a flexible polymer chain was first tackled by kuhn nearly thirty years after the answer to the probability distribution of its size was publicly sought for by pearson in 1905 since then significant progress in the field has been made but the important task of evaluatin both analytically and accurately averaged individual principal components of the shape or inertia tensor for a walk of a certain architectural type remains unfinished we have recently developed a new and general formalism for both exact and approximate calculations of these and other averages such as asphericity and prolateness parameters which is illustrated here for an end looped random walk and a self avoiding or edwards chain we find that this combined open and closed random walks has surprisingly larger shape asymmetry than a simply open walk despite its smaller size

lattice green function at 0 for the 4d hypercubic lattice the generating function for recurrent polya walks on the four dimensional hypercubic lattice is expressed as a kampe de feriet function various properties of the associated walks are enumerated

the two dimensional shapes of simple three and four junction ideal comb polymers we redesign and apply a scheme originally proposed by wei 1995 2 3 to produce numerical shape parameters with high precision for arbitrary tree branched polymers based on their kirchhoff matrix eigenvalue spectrum this algorithm and a monte carlo growth method on square and triangular lattices are employed to investigate the shapes of ideal three and four junction two dimensional comb polymers we find that the extrapolated values obtained by all of these methods are in excellent agreement with each other and the available theory we confirm that polymers with a complete set of interior branches display a more circular shape

on the c chevalley s theorem the group of orthogonal matrices k so n acts by conjugation on the vector space p of all n x ii symmetric matrices and the spectral theorem implies that every orbit intersects the subspace a c p of the diagonal matrices in fact every orbit k x is orthogonal to a under the invariant inner product x y tr xy and the intersection is the orbit of the permutation group on n letters w s which permutes the eigenvalues of x if we let

polymers as self avoiding walks a brief overview is presented of the relation of the properties of real polymers to the problem of self avoiding random walks the self consistent field method is discussed wherein the non markovian continuous self avoiding polymer is replaced by a self consistent markovian approximation an outline is presented of the method of solution of the resultant nonlinear integrodifferential equations a description is also presented of the scaling theories which provide a means for deducing some exponents in the asymptotic dependence of walk properties on the length of the walk in the limit of infinite length walks

the shape of self avoiding walks an exhaustive analysis of the shape properties of discrete self avoiding random walks is presented the dependence of the main parameters of the probability distributions from the length of the walk and the differences between these distributions for conventional and self avoiding walks are studied by means of high precision monte carlo simulations it is shown that the characteristics of the shape of self avoiding random walks when regarded as a function of the walk length present the expected asymptotic behaviour the differences with conventional random walks depend upon the observable considered the probability distributions of the principal inertia eigenvalues of self avoiding walks spread around the most probable value more widely than the corresponding distributions for unrestricted walks while in the cases of the asphericity or ratios of inertia eigenvalues the distributions for self avoiding walks are somewhat more peaked than their counterparts for conventional random walks analytical expressions for the probability distributions of inertia moment ratios and the two dimensional asphericity are given for common random walks there is an excellent agreement between these analytical distributions and the monte carlo data in this work it is established that this concordance is maintained if the same analytical distributions are applied to the self avoiding case this means that within the precision of the simulations the functional form of the mentioned distributions does not vary when passing from conventional to self avoiding walks

rank 1 perturbations of cosine functions and semigroups abstract let a be the generator of a cosine function on a banach space x in many cases for example if x is a umd space a b generates a cosine function for each b l d a 1 2 x if a is unbounded and 1 2 1 then we show that there exists a rank 1 operator b l d a x such that a b does not generate a cosine function the proof depends on a modification of a baire argument due to desch and schappacher it also allows us to prove the following if a b generates a distribution semigroup for each operator b l d a x of rank 1 then a generates a holomorphic c 0 semigroup if a b generates a c 0 semigroup for each operator b l d a x of rank 1 where 0 1 then the semigroup t generated by a is differentiable and t t o t as t 0 for any 1 this is an approximate converse of a perturbation theorem for this class of semigroups

universal shape ratios for open and closed random walks exact results for all d the mean asphericities ad of open and closed gaussian polymer chains are computed exactly and analytically up to simple quadratures for arbitrary space dimensions d excellent agreement is found with existing simulation data for d 2 3 and 4 the authors technique can also be used to compute other averages of ratios of fluctuating variables as well as extended to include the effects of self avoiding walk interactions

statistical properties of the set of sites visited by the two dimensional random walk we study the support i e the set of visited sites of a t step random walk on a two dimensional square lattice in the large t limit a broad class of global properties m t of the support is considered including for example the number s t of its sites the length of its boundary the number of islands of unvisited sites that it encloses the number of such islands of given shape size and orientation and the number of occurrences in space of specific local patterns of visited and unvisited sites on a finite lattice we determine the scaling functions that describe the averages on appropriate lattice size dependent time scales on an infinite lattice we first observe that the all increase with t as where k is an m dependent positive integer we then consider the class of random processes constituted by the fluctuations around average we show that to leading order as t gets large these fluctuations are all proportional to a single universal random process normalized to for the probability law of tends to that of varadhan s renormalized local time of self intersections an implication is that in the long time limit all are proportional to

convoluted c cosine functions and semigroups relations with ultradistribution and hyperfunction sines abstract convoluted c cosine functions and semigroups in a banach space setting extending the classes of fractionally integrated c cosine functions and semigroups are systematically analyzed structural properties of such operator families are obtained relations between convoluted c cosine functions and analytic convoluted c semigroups introduced and investigated in this paper are given through the convoluted version of the abstract weierstrass formula which is also proved in the paper ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups several examples of operators generating convoluted cosine functions analytic convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors as an application it is proved that the polyharmonic operator 2 n n n acting on l 2 0 with appropriate boundary conditions generates an exponentially bounded k n convoluted cosine function and consequently an exponentially bounded analytic k n 1 convoluted semigroup of angle 2 for suitable exponentially bounded kernels k n and k n 1

asymptotic behavior of inflated lattice polygons we study the inflated phase of two dimensional lattice polygons with fixed perimeter n and variable area associating a weight exp pa jb to a polygon with area a and b bends for convex and column convex polygons we calculate the average area for positive values of the pressure for large pressures the area has the asymptotic behaviour langle a rangle a max 1 k j tilde p 2 mathcal o rho tilde p where tilde p pn gg 1 and 1 the constant k j is found to be the same for both types of polygons we argue that self avoiding polygons should exhibit the same asymptotic behavior for self avoiding polygons our predictions are in good agreement with exact enumeration data for j 0 and monte carlo simulations for j 0 we also study polygons where self intersections are allowed verifying numerically that the asymptotic behavior described above continues to hold

universal features of the shapes of percolation clusters and lattice animals the shapes of percolation clusters and lattice animals are investigated the universal quantities delta d and sd which were introduced to measure the average asymmetry and degree of prolate or oblateness respectively of long chain polymers are here computed in an in expansion for percolation clusters and for lattice animals delta d is computed to o in while sd is computed to o 1 the clusters are shown to be on average anisotropic and prolate but less so than polymers percolation clusters and lattice animals have identical shapes above eight dimensions below d 8 animals are slightly more anisotropic than percolation clusters

on g invariant norms abstract a result of r mathias and horn cf linear algebra appl 142 1990 63 on the representation of the unitarily invariant norm is extended in the context of eaton triples and of real semisimple lie algebras the representation is related to a function criteria for being a norm is given in terms of and the longest element of the underlying finite reflection group in particular for the real simple lie algebras is a norm on its cartan subspace for a given nonzero if and only if 0 where 0 is the longest element of the weyl group this is the case if the weyl group contains id some related results are obtained

phase transitions in pressurized semiflexible polymer rings we propose and study a model for the equilibrium statistical mechanics of a pressurized semiflexible polymer ring in two dimensions the hamiltonian has a term which couples to the algebraic or signed area of the ring and a term which accounts for bending semiflexibility the model allows for self intersections using a combination of monte carlo simulations flory type scaling theory mean field approximations and lattice enumeration techniques we obtain a phase diagram in which collapsed and inflated phases are separated by a continuous transition the scaling properties of the averaged area as a function of the number of units of the ring are derived for large pressures the asymptotic behavior of the area is calculated for both continuum and lattice versions of the model for small pressures the area is obtained through a known mapping onto the quantum mechanical problem of an electron moving in a magnetic field the simulation data agree well with the analytic and mean field results

on the asymmetry of a random walk in the presence of a field any ensemble of random walks with symmetric transition probabilities will have symmetric properties however any single realization of such a random walk may be asymmetric in an earlier paper weiss and weissman developed a measure of asymmetry and applied it to random walks in the absence of a field showing that the degree of asymmetry in the diffusion limit is independent of time and that the most probable degree of asymmetry corresponds to the maximum possible we show in the present paper how the presence of a symmetric field can change this result both in making the degree of asymmetry depend on time and driving the random walk toward a more symmetric state

the shapes of ideal five junction comb polymers in two and three dimensions abstract this work investigates a variety of properties of eleven and fourteen branch five junction comb polymers in the ideal regime in two and three dimensions a method based on the kirchhoff matrix eigenvalue spectrum for arbitrary tree branched polymers is used to compute shape properties and a scheme originally proposed by benhamous 2004 is used to produce an exact equation for the form factor of the fourteen branch comb polymers a monte carlo growth algorithm is also employed to compute the same properties it is found that the values obtained by all of these methods are in fine agreement with each other and available theory

affine images of riemannian manifolds we describe all affine maps from a riemannian manifold to a metric space and all possible image spaces

new lower bounds on the self avoiding walk connective constant we give an elementary new method for obtaining rigorous lower bounds on the connective constant for self avoiding walks on the hypercubic lattice d the method is based on loop erasure and restoration and does not require exact enumeration data our bounds are best for highd and in fact agree with the first four terms of the 1 d expansion for the connective constant the bounds are the best to date for dimensionsd 3 but do not produce good results in two dimensions ford 3 4 5 and 6 respectively our lower bound is within 2 4 0 43 0 12 and 0 044 of the value estimated by series extrapolation

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

supersymmetry approach in the field theory of ergodicity breaking transitions abstract the supersymmetry susy self consistent approximation for the model of non equilibrium thermodynamic system with quenched disorder is derived from the dynamical action calculated by means of generalized second legendre transformation technique the equations for adiabatic and isothermal susceptibilities memory and field induced parameters are obtained on the basis of asymptotic analysis of dynamic dyson equations it is shown that the marginal stability condition that defines the critical point is governed by fluctuations violating fluctuation dissipation theorem fdt the temperature of ergodicity breaking transition is calculated as a function of quenched disorder intensities transformation of superfields related to the mapping between an instanton process and the corresponding causal solution is discussed

integrated semigroups and related partial differential equations abstract we investigate the relationship between abstract linear evolution equations of heat wave and schrodinger types in terms of well posedness in banach spaces more precisely we study our operators as generators of integrated semigroups and integrated cosine functions as applications we consider in a banach space context an abstract laplacian which generalizes the ordinary laplacian in thelp r n spaces we obtain optimal results when compared to the classical situation where the generation results were obtained by 17 using completely different methods our main tools are the boundary value theorem for holomorphic semigroups and the abstract weierstrass formula related partial differential equations have been considered previously by bragg and dettman