convex fuzzy mapping and operations of convex fuzzy mappings convex and concave fuzzy mappings are considered in this paper based on the seminal work of nanda and kar on the definition of convex fuzzy mapping the operations of convex fuzzy mapping are established some of the important concepts such as positively homogeneous infimal convolution right scalar multiplication and convex hull are introduced and the corresponding theorems are demonstrated by using the parametric representation of fuzzy numbers

convergent finite difference solvers for viscosity solutions of the elliptic monge ampere equation in dimensions two and higher the elliptic monge ampere equation is a fully nonlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail novel solution methods are required for stability and convergence to the weak viscosity solution in this article we build a wide stencil finite difference discretization for the monge ampere equation the scheme is monotone so the barles souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation solutions of the scheme are found using a damped newton s method we prove convergence of newton s method and provide a systematic method to determine a starting point for the newton iteration computational results are presented in two and three dimensions which demonstrates the speed and accuracy of the method on a number of exact solutions which range in regularity from smooth to nondifferentiable

on integration of fuzzy mappings abstract in this paper we introduce the concept of an integral of a fuzzy mapping this generalises the integral introduced by matloka 10 and also extends riemann stieltjes integral over a closed interval to fuzzy mappings

mathematical modeling of different types of instabilities in time fractional reaction diffusion systems abstract in this article we investigate possible scenarios of pattern formations in reaction diffusion systems with time fractional derivatives linear stability analysis is performed for different values of derivative orders results of qualitative analysis are confirmed by numerical simulations of specific partial differential equations most attention is paid to two models a fractional order reaction diffusion system with bonhoeffer van der pol kinetics and to the brusselator model

an efficient numerical method for the solution of the l_2 optimal mass transfer problem in this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal l 2 mass transport mapping in contrast to the integration of a time dependent partial differential equation proposed in s angenent s haker and a tannenbaum siam j math anal 35 2003 pp 61 97 we employ in the present work a direct variational method the efficacy of the approach is demonstrated on both real and synthetic data

pseudolinear fuzzy mappings two classes of fuzzy mappings called pseudolinear and pseudolinear fuzzy mappings are introduced by relaxing the definitions of pseudo convex and pseudo invex fuzzy mappings first some characterizations of pseudolinear and pseudolinear fuzzy mappings are obtained then characterizations of the solution sets of pseudolinear and pseudolinear fuzzy programs are derived

effect of bacterial memory dependent growth by using fractional derivatives reaction diffusion chemotactic model in this paper numerical solutions of a reaction diffusion chemotactic model of fractional orders for bacterial growth will be present a new solution is constructed in power series the fractional derivatives are described in the caputo sense we compare the experimental result obtained with those obtained by simulation of the chemotactic model without fractional derivatives the results show that the solution continuously depends on the time fractional derivative the resulting solutions spread faster than the classical solutions and may exhibit asymmetry depending on the fractional derivative used we present results of numerical simulations to illustrate the method and investigate properties of numerical solutions the adomian s decomposition method adm is used to find the approximate solution of fractional reaction diffusion chemotactic model numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional order

duality theory in fuzzy mathematical programming problems with fuzzy coefficients in this paper the notions of subgradient subdifferential and differential with respect to convex fuzzy mappings are investigated which provides the basis for the fuzzy extremum problem theory we consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and or sufficient optimality conditions furthermore the concept of saddle points and minimax theorems under fuzzy environment is discussed the results obtained are used to formulate the lagrangian dual of fuzzy programming under certain fuzzy convexity assumptions kkt conditions for fuzzy programming are derived and the perturbed convex fuzzy programming is considered finally these results are applied to fuzzy linear programming and fuzzy quadratic programming

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

approximate solution of space and time fractional higher order phase field equation abstract this paper is concerned with a class of space and time fractional partial differential equation stfde with riesz derivative in space and caputo in time the proposed stfde is considered as a generalization of a sixth order partial phase field equation we describe the application of the optimal homotopy analysis method oham to obtain an approximate solution for the suggested fractional initial value problem an averaged squared residual error function is defined and used to determine the optimal convergence control parameter two numerical examples are studied considering periodic and non periodic initial conditions to justify the efficiency and the accuracy of the adopted iterative approach the dependence of the solution on the order of the fractional derivative in space and time and model parameters is investigated

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

the variational iteration method for solving riesz fractional partial differential equations in this paper the variational iteration method is applied to obtain the solution for space fractional partial differential equations where the space fractional derivative is in the riesz sense on the basis of the properties and definition of the fractional derivative the iterative technique is carried out in a straightforward manner without the need for transforms or numerical approximations examples demonstrate that the series solution obtained shows agreement with the exact solutions of the problems solved

generalized convex fuzzy mappings and fuzzy variational like inequality in this paper we introduce the concepts of some generalized convex fuzzy mappings from r n to the set of fuzzy numbers under the condition of upper or lower semicontinuity a new criterion be obtained for the existence of a fuzzy preinvex mapping a fuzzy preinvex mapping can be achieved when the invex subset of r n satisfies certain conditions the relations between generalized convex fuzzy mappings are discussed and some properties are obtained that relate the concepts of fuzzy pseudoinvex and fuzzy prequasiinvex mapping fuzzy invex and fuzzy preinvex mapping fuzzy preinvex and fuzzy prequasiinvex mapping we also discuss the relationship between the fuzzy variational like inequality the fuzzy weak variational like inequality and fuzzy optimization problems and an application example of a fuzzy variational like inequality the fuzzy variational like inequality representation of a fuzzy transportation equilibrium problem

properties of b vex fuzzy mappings and applications to fuzzy optimization convexity plays a key role in operations research and fuzzy optimization theory the concept of b vex and logarithmic b vex for fuzzy mappings is introduced by relaxing the definition of convexity of a fuzzy mapping most of the basic properties of b vex fuzzy mapping are discussed and established for the nondifferentiable case necessary and sufficient conditions for b vex fuzzy mapping are presented several important results are given for nonlinear fuzzy optimization problems assuming that objective and constraint functions are b vex fuzzy mappings

convergence framework for the second boundary value problem for the monge ampere equation it is well known that the quadratic cost optimal transportation problem is formally equivalent to the second boundary value problem for the monge ampere equation viscosity solutions are a powerfu

convexity and semicontinuity of fuzzy mappings in this paper a criterion for the convex fuzzy mapping is obtained under the condition of upper and lower semicontinuity respectively an upper lower semicontinuous fuzzy mapping is proved which convexity is equivalent to weak convexity or b vexity satisfying a special condition

φ1 φ2 convex fuzzy mappings abstract the concepts of 1 2 convexity 1 b vexity and 1 convexity for fuzzy mappings are introduced through the so called fuzzy max order among fuzzy numbers we show that the class of 1 2 convex fuzzy mappings 1 b vex fuzzy mappings and 1 convex fuzzy mappings includes many well known classes of fuzzy mappings such as convex fuzzy mappings and preinvex fuzzy mappings as its subclasses we also give a characterization for 1 b vex fuzzy mappings by proposing the concept of 1 b vex sets in addition some basic properties of fuzzy mappings from the standpoint of convex analysis are discussed

a least squares method for optimal transport using the monge ampere equation in this article we introduce a novel numerical method to solve the problem of optimal transport and the related elliptic monge ampere equation it is one of the few numerical algorithms capable of solving this problem efficiently with the proper transport boundary condition the computation time scales well with the grid size and has the additional advantage that the target domain may be nonconvex we present the method and several numerical experiments

generalization of preinvex and b vex fuzzy mappings a class of fuzzy mappings called b preinvex and strictly b preinvex fuzzy mappings is introduced by relaxing the definition of preinvexity of a fuzzy mapping similarly based on a notion of differentiability of fuzzy mappings different from the usual one the class of pseudo b vex b invex and pseudo b invex fuzzy mappings is defined as a generalization of pseudo convex invex and pseudo invex fuzzy mappings we prove that a strictly b vex or b preinvex fuzzy mapping has at most one global minimum point and that the class of b vex fuzzy mappings forms a subset of the class of quasi convex fuzzy mappings b vex resp b preinvex fuzzy mappings satisfy most of the basic properties of convex resp preinvex fuzzy mappings in addition characterizations and sufficient optimality conditions are obtained for b vex and b preinvex fuzzy mappings

on the level convergence of a sequence of fuzzy numbers abstract in this paper we first give a simplified proof of the existence theorem of supremum and infimum in fuzzy number space e 1 established by wu and wu j math anal appl 210 1997 499 and improve the expressions of the supremum and infimum as a straightforward corollary of this result we obtain a necessary and sufficient condition under which for a bounded sequence of fuzzy numbers u n the pair of functions sup n u n and sup n u n can determine a fuzzy number secondly we give a necessary and sufficient condition for a sequence of fuzzy numbers u n to be levelwise convergent in e 1 and generalize some important theorems in real number spaces to fuzzy number spaces finally we prove the existence of supremum and infimum for the level continuous fuzzy valued function on a closed interval and give a necessary and sufficient condition under which its supremum and infimum can be attained

turing pattern dynamics in an activator inhibitor system with superdiffusion the fractional operator is introduced to an activator inhibitor system to describe species anomalous superdiffusion the effects of the superdiffusive exponent on pattern formation and pattern selection are studied our linear stability analysis shows that the wave number of the turing pattern increases with the superdiffusive exponent a weakly nonlinear analysis yields a system of amplitude equations and the analysis of these amplitude equations predicts parameter regimes where hexagons stripes and their coexistence are expected numerical simulations of the activator inhibitor model near the stability boundaries confirm our analytical results since diffusion rate manifests in both diffusion constant and diffusion exponent we numerically explore their interactions on the emergence of turing patterns when the activator and inhibitor have different superdiffusive exponents we find that the critical ratio of the diffusion rate of the inhibitor to the activator required for the formation of the turing pattern increases monotonically with the superdiffusive exponent we conclude that small ratio than unity of anomalous diffusion exponent between the inhibitor and activator is more likely to promote the emergence of the turing pattern relative to the normal diffusion

a least squares method for the inverse reflector problem in arbitrary orthogonal coordinates abstract in this article we solve the inverse reflector problem for a light source emitting a parallel light bundle and a target in the far field of the reflector by use of a least squares method we derive the monge ampere equation expressing conservation of energy while assuming an arbitrary coordinate system we generalize a cartesian coordinate least squares method presented earlier by prins et al 13 to arbitrary orthogonal coordinate systems this generalized least squares method provides us the freedom to choose a coordinate system suitable for the shape of the light source this results in significantly increased numerical accuracy decrease of errors by factors up to 104 is reported we present the generalized least squares method and compare its numerical results with the cartesian version for a disk shaped light source

a numerical method for the elliptic monge ampere equation with transport boundary conditions the problem of optimal mass transport arises in numerous applications including image registration mesh generation reflector design and astrophysics one approach to solving this problem is via the monge ampere equation while recent years have seen much work in the development of numerical methods for solving this equation very little has been done on the implementation of the transport boundary condition in this paper we propose a method for solving the transport problem by iteratively solving a monge ampere equation with neumann boundary conditions to enable mappings between variable densities we extend an earlier discretization of the equation to allow for right hand sides that depend on gradients of the solution b d froese and a m oberman siam j numer anal 49 2011 pp 1692 1714 this discretization provably converges to the viscosity solution the resulting system is solved efficiently with newton s method we provide several challenging computational examples that demonstra

numerical solution of the optimal transportation problem using the monge ampere equation a numerical method for the solution of the elliptic monge ampere partial differential equation with boundary conditions corresponding to the optimal transportation ot problem is presented a local representation of the ot boundary conditions is combined with a finite difference scheme for the monge ampere equation email protected s method is implemented leading to a fast solver comparable to solving the laplace equation on the same grid several times theoretical justification for the method is given by a convergence proof in the companion paper 4 solutions are computed with densities supported on non convex and disconnected domains computational examples demonstrate robust performance on singular solutions and fast computational times

designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport we study here the problem of determining a system of two refractive interfaces transforming a plane wavefront of a given shape and radiation intensity into a coherent output plane wavefront with prescribed output position shape and intensity such interfaces can be refracting surfaces of two different lenses or of one lens in geometrical optics approximation the analytic formulation of this problem in both cases requires construction of maps with controlled jacobian though this jacobian can be expressed as a second order partial differential equation of monge ampere type for a scalar function defining one of the refracting surfaces its analysis is not straightforward in this paper we use a geometric approach for reformulating the problem in certain associated measures and defining weak solutions existence and uniqueness of weak solutions in lipschitz classes for both cases are established by variational methods our results show in particular that two types of interfaces exist in each case for the same data one of these types always consists of two interfaces one of which is concave or convex and the second convex or concave while the interfaces of the second type may be neither convex nor concave the availability of a design with convex concave lenses is particularly important for fabrication the truly geometric nature of this problem permits its statement and investigation in mathbb r n 1 n geqq 1

fast finite difference solvers for singular solutions of the elliptic monge ampere equation the elliptic monge ampere equation is a fully nonlinear partial differential equation which originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail in this article we build a finite difference solver for the monge ampere equation which converges even for singular solutions regularity results are used to select a priori between a stable provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain this allows singular solutions to be computed using a stable method and regular solutions to be computed more accurately the resulting nonlinear equations are then solved by newton s method computational results in two and three dimensions validate the claims of accuracy and solution speed a computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities

fuzzy continuous function and its properties abstract in this paper we introduce the concepts of fuzzy distance of fuzzy numbers fuzzy limit of fuzzy numbers fuzzy function and fuzzy limit of a fuzzy function and fuzzy continuous function on the set of fuzzy numbers and give some of their elementary properties we also discuss some important theorems of fuzzy numbers and the fuzzy continuous function on the m closed interval

the sinkhorn algorithm parabolic optimal transport and geometric monge ampere equations we show that the discrete sinkhorn algorithm as applied in the setting of optimal transport on a compact manifold converges to the solution of a fully non linear parabolic pde of monge ampere type in a large scale limit the latter evolution equation has previously appeared in different contexts e g on the torus it can be be identified with the ricci flow this leads to algorithmic approximations of the potential of the optimal transport map as well as the optimal transport distance with explicit bounds on the arithmetic complexity of the construction and the approximation errors as applications we obtain explicit schemes of nearly linear complexity at each iteration for optimal transport on the torus and the two sphere as well as the far field antenna problem connections to quasi monte carlo methods are exploited

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

fuzzy linear spaces over valued fields abstract the concept of fuzzy linear spaces was introduced by nanda 3 the purpose of the present paper is to introduce some further concepts such as convex fuzzy sets etc in fuzzy linear spaces over any valued field

convex fuzzy mappings abstract in this note the concept of convexity and logarithmic convexity for fuzzy mappings are introduced and some applications to optimization are discussed