exact results for a correlated percolation model abstract the equivalence between a correlated percolation model and the syozi model at t 0 has been used to calculate the exact percocation probabilities p as a function of bare p 0 and renormalized p occupation probability for such a percolation process on two dimensional lattices we also find that for both two and three dimensional lattices p p 0 p 0 c near critical point p 0 c where is the magnetic exponent of the corresponding spin model

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

exact cluster size distributions and mean cluster sizes for the q state bond correlated percolation model the author shows that the q state bond correlated percolation model qbcpm which is the percolation representation of the q state potts model qpm on the lattice without closed loops is equivalent to the bond random percolation model brpm on the same lattice using such results and exact results for the brpm on the linear and bethe lattices the author obtains exact cluster size distributions and the mean cluster size s for the qbcpm on the linear and bethe lattices the mean cluster sizes obtained from this method are the same as those obtained by more tedious exact calculations near the critical point the average number of m site clusters per site nm for the qbcpm on the linear and bethe lattices may be written in the scaling form for large values of m which is the geometrical basis for the scaling laws of critical exponents

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

on the oscillation of certain second order nonlinear differential equations abstract in this paper oscillation criteria for the nonlinear second order ordinary differential equation r t x t x t p 2 x t c t f x t 0 are given the results extend the integral averaging technique and include earlier results our methodology is somewhat different from that of previous authors

cluster monte carlo study of the q state potts model on hypercubic lattices abstract the q state potts model qpm may be related to many interesting mathematical problems and physical systems previous studies of the qpm have been restricted mainly to small q values using the connection between the qpm and the q state bond correlated percolation model qbcpm and swendsen and wang s cluster monte carlo simulation method in this paper we study the qpm on d dimensional hypercubic lattices with d being 3 4 5 and 6 and q being 2 3 4 8 16 32 64 128 256 512 1024 and 2048 for q 2 where phase transitions are second order a percolation monte carlo renormalization group method is used to determine more accurate critical points we find that in the studied space dimensions the critical points increase slowly with q as q increases the critical points for different d approach to each other such that when q 2048 the absolute values of the differences between such critical points are less than 1 1 we also discuss some related theoretical problems including implications of our results to the percolation theory of supercooled water

a comparison theorem for a piecewise lipschitz continuous hamiltonian and application to shape from shading problems the reconstruction from a shaded image of a lambertian and not self shadowing surface illuminated by a single distant pointwise light source may be written as a first order hamilton jacobi equation in this paper we continue the investigation begun in e rouy and a tourin into the uniqueness of the solution of this equation the approach is based on the viscosity solutions theory and the dynamic programming principle more precisely we concentrate here on the uniqueness of the viscosity solution of this equation in case the measured luminous intensity reflected by the surface is discontinuous along a smooth curve we prove a general comparison result for a piecewise lipschitz continuous hamiltonian and illustrate it by numerical experiments

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

exact multileg correlation functions for the dense phase of branching polymers in two dimensions we consider branching polymers on the planar square lattice with open boundary conditions and exactly calculate correlation functions of k polymer chains that connect two lattice sites with a large distance r apart for odd number of polymer chains k we find that besides the standard power law factor the leading term also has a logarithmic multiplier

numerical shape from shading for discontinuous photographic images the height u x y of a continuous lambertian surface of known albedo i e grayness is related to u x y information recoverable from a black and white flash photograph of the surface by the partial differential equation sqrt u 2 x u 2 y n 0 we review the notion of a unique viscosity solution for this equation when n is continuous and a recent unique extension of the viscosity solution when n is discontinuous we prove convergence to this extension for a wide class of the numerical algorithms that converge when n is continuous after discussing the properties of the extension and the order of error in the algorithms simulating the extension we point out warning signs which when observed in the numerical solution usually indicate that the surface is not continuous or that the viscosity solution or its extension does not correspond to the actual surface finally we discuss a method that in some of these cases allows us to correct the simulation and recover the actual surface again

oscillation criteria for certain forced second order nonlinear differential equations with delayed argument some new sufficient conditions for the oscillation criteria are given for the forced second order nonlinear differential equations with delayed argument in the form x t q t f x t t e t the results are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our methodology is somewhat different from that of previous authors our results generalize and extend some earlier results of sun 1 furthermore we are able to answer the question in recent paper 1 for the oscillation of bounded solutions of the above equation when f x x lsgn x 0 l 1

oscillation theorems for superlinear second order damped differential equations abstract in this paper we concerned with the oscillation of the second order superlinear differential equation of the form a t y t p t y t q t f y t 0 several new oscillation criteria are established under quite general assumptions our methodology is somewhat different from that of previous authors ch g philos oscillation theorems for linear differential equations of second order arch math basel 53 1989 482 492 ch g philos oscillation criteria for second order superlinear differential equations canad j math anal xli 1989 321 340 ch g philos i k purnaras oscillations in superlinear differential equations of second order j math anal appl 165 1992 1 11 y h yu oscillation criteria for second order nonlinear differential equations with damping acta math appl 16 1993 433 441 in chinese our results generalize and extend some earlier results of yu 1993 example is also given to illustrate the results

percolation fractals and the finite size scaling of existence probability abstract based on the connection between phase transitions of thermodynamic systems and percolation transitions we obtain the geometrical meaning of critical exponents using finite size scaling of the percolation probability p and the existence probability e p for percolating clusters and percolating subgraphs respectively where e p is the ratio of the total probability weight for percolating subgraphs to the total probability weight for all subgraphs we use a histogram monte carlo simulation method to calculate e p for a q state bond correlated percolation model qbcpm which is corresponding to the q state potts model on the square lattice with various linear dimensions we find that for a given q e p has very good scaling behavior but e p for different q has different scaling functions

oscillation criteria for second order nonlinear differential equations with damping abstract for a second order nonlinear differential equation with a damping term we obtain new oscillation criteria without an assumption that has been required for related results reported in the literature over the last two decades two approaches refining the standard integral averaging technique suggested in the paper can be used to derive a variety of simpler oscillation theorems for different classes of nonlinear differential equations and both sets of oscillation criteria established in the paper are of independent interest several examples are provided to illustrate the relevance of the new theorems

a pde based approach to nondominated sorting nondominated sorting is a fundamental combinatorial problem in multiobjective optimization and is equivalent to the longest chain problem in combinatorics and random growth models for crystals in materials science in a previous work siam j math anal 46 2014 pp 603 638 we showed that nondominated sorting has a continuum limit that corresponds to solving a hamilton jacobi equation in this work we present and analyze a fast numerical scheme for this hamilton jacobi equation and show how it can be used to design a fast algorithm for approximate nondominated sorting

oscillations in superlinear differential equations of second order abstract a new oscillation criterion is given for general superlinear ordinary differential equations of second order of the form x t a t f x t 0 where a c t 0 f c r with yf y 0 for y 0 and 1 1 f y dy and f is continuously differentiable on r 0 with f y 0 for all y 0 the coefficient a is not assumed to be eventually nonnegative and the oscillation cirterion obtained involves the average behavior of the integral of a in the special case of the differential equation x t a t x t sgn x t 0 1 this criterion improves a recent oscillation result due to wong oscillation theorems for second order nonlinear differential equations proc amer math soc 106 1989 1069 1077

scaling functions of the q state potts model on planar lattices abstract based on the subgraph expansion of the q state potts model qpm in an external field it has been shown that the qpm is corresponding to a q state bond correlated percolation model qbcpm the histogram monte carlo simulation method proposed by hu is used to calculate the existence probability e p g p q and the percolation probability p g p q of the qbcpm on the honeycomb the kagome and the plane triangular lattices with various linear dimensions from e p g p q and p g p q we obtain scaling functions of the qpm and qbcpm we find that as q or the coordination number of the lattices increases the widths of the scaling functions also increase the implication of this study is discussed

exit time problems in optimal control and vanishing viscosity method the authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first order hamilton jacobi hj equation up to the boundary this equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary when these solutions are equal they can be obtained through the vanishing viscosity method finally when the hj equation has a continuous solution it is proved to be the value function for the first exit time of the domain it is also the vanishing viscosity limit arising in particular in some large deviations problems

local shading analysis local analysis of image shading in the absence of prior knowledge about the viewed scene may be used to provide information about the scene the following has been proved every image point has the same image intensity and first and second derivatives as the image of some point on a lambertian surface with principal curvatures of equal magnitude further if the principal curvatures are assumed to be equal there is a unique combination of image formation parameters up to a mirror reversal that will produce a particular set of image intensity and first and second derivatives a solution for the unique combination of surface orientation etc is presented this solution has been extended to natural imagery by using general position and regional constraints to obtain estimates of the following surface orientation at each image point the qualitative type of the surface i e whether the surface is planar cylindrical convex concave or saddle the illuminant direction within a region algorithms to recover illuminant direction and estimate surface orientation have been evaluated on both natural and synthesized images and have been found to produce useful information about the scene

linear shape from shading in many situations the reflectance function of a surface is approximately linear and there is an effielent closed form solution to the shape from shading problem when boundary conditions e g edges singular points are not available good estimates of shape may still be extracted by using the assumption of general viewing position an improved method for estimating the illuminant direction is also presented

geometrical factor and thermal properties of the potts model based on the connection between the q state potts model qpm and a q state bond correlated percolation model qbcpm we define a non percolating geometrical factor g and percolating geometrical factor gp which depend only on q and geometrical properties of the system the thermal properties of the qpm could be derived simply from g and g the idea is confirmed by an exact calculation for one dimensional qpm and may be extended to other ising like spin models our formulation also provides a geometrical meaning for the finite size scaling and broadening at first order phase transitions of the spin model

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

shape from shading viscosity solutions and edges this article deals with the so called shape from shading problem which arises when recovering a shape from a single image the general case of a distribution of light sources illuminating a lambertian surface is considered this involves original definitions of three types of edges mainly the apparent contours the grazing light edges and the shadow edges the elevation of the shape is expressed in terms of viscosity solution of a first order hamilton jacobi equation with various boundary conditions on these edges various existence and uniqueness results are presented