type 2 interval fuzzy rule based systems in spatial analysis abstract in this paper we present a rule based system by using interval type 2 fuzzy sets it2 fs for solving problems of spatial analysis this system is encapsulated in a geographical information system gis by performing numerous tests for evaluating their reliability

optimization of type 2 fuzzy systems based on bio inspired methods a concise review a review of the optimization methods used in the design of type 2 fuzzy systems which are relatively novel models of imprecision has been considered in this work the fundamental focus of the work has been based on the basic reasons of the need for optimizing type 2 fuzzy systems for different areas of application recently bio inspired methods have emerged as powerful optimization algorithms for solving complex problems in the case of designing type 2 fuzzy systems for particular applications the use of bio inspired optimization methods have helped in the complex task of finding the appropriate parameter values and structure of the fuzzy systems in this review we consider the application of genetic algorithms particle swarm optimization and ant colony optimization as three different paradigms that help in the design of optimal type 2 fuzzy systems we also provide a comparison of the different optimization methods for the case of designing type 2 fuzzy systems

mach reflection for the two dimensional burgers equation abstract we study shock reflection for the two 2d burgers equation this model equation is an asymptotic limit of the euler equations and retains many of the features of the full equations a von neumann type analysis shows that the 2d burgers equation has detachment sonic and crocco points in complete analogy with gas dynamics numerical solutions support the detachment sonic criterion for transition from regular to mach reflection there is also strong numerical evidence that the reflected shock in the 2d burgers mach reflection forms a smooth wave near the mach stem as proposed by colella and henderson in their study of the euler equations

existence and regularity of solution of mixed boundary value problem of a busemann equation with nonlinear absorb term abstract the busemann equation is a classical equation coming from fluid dynamics the well posed problem and regularity of solution of busemann equation with nonlinear term are interesting and important the busemann equation is elliptic in y 0 and is degenerate at the line y 0 in r 2 with a special nonlinear absorb term we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain by means of elliptic regularization technique a delicate prior estimate and compact argument we show that the solution of mixed boundary value problem of busemann equation is smooth in the interior and lipschitz continuous up to the degenerate boundary on some conditions the result is better than the result of classical boundary degenerate elliptic equation

a class of singular gierer meinhardt systems of elliptic boundary value problems abstract we establish existence results for singular gierer meinhardt elliptic systems with zero dirichlet boundary conditions with the different range of growth rates of the source term we use the schauder fixed point theorem and upper lower solution methods to obtain the existence results we also study boundary estimates for certain singularities

optimal holder regularity for mixed boundary value problems abstract we continue our study of mixed boundary value problems lu 1 in mu 2 on u 3 on for second order elliptic operators l and first order oblique operators m a best possible estimate of the regularity of u is proved under weak hypotheses on the coefficients of l and m

on km algorithms for solving type 2 fuzzy set problems computing the centroid and performing type reduction for type 2 fuzzy sets and systems are operations that must be taken into consideration karnik mendel km algorithms are the standard ways to do these operations however because these algorithms are iterative much research has been conducted during the past decade about centroid and type reduction computations this tutorial paper focuses on the research that has been conducted to 1 improve the km algorithms 2 understand the km algorithms leading to further improved algorithms 3 eliminate the need for km algorithms 4 use the km algorithms to solve other nonfuzzy logic system problems and 5 use or not use km algorithms for general type 2 fuzzy sets and fuzzy logic systems

potential theory for regular and mach reflection of a shock at a wedge if a plane shock hits a wedge a self similar pattern of reflected shocks travels outward as the shock moves forward in time the nature of the pattern is explored for weak incident shocks strength b and small wedge angles 2 w through potential theory a number of different scalings some study of mixed equations and matching asymptotics for the different scalings the self similar equations are of mixed type a linearization gives a linear mixed flow valid away from a sonic curve near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used the parameter c1 2w 1 b ranges from 0 to here is the polytropic constant and c1 is the sound speed behind the incident shock for 2 regular reflection weak or strong can occur and the whole pattern is reconstructed to lowest order in shock strength for 1 2 mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle with an open elliptic problem and matched the case 0 can be solved for 1 2 2 or even larger the flow behind a mach reflection may be transonic and further investigation must be made to determine what happens the basic pattern of reflection is an almost semi circular shock issuing for regular reflection from the reflection point on the wedge and for mach reflection matched with a local interaction flow assuming their nature choosing the least entropy generation the weak regular reflection will occur for sufficiently large von neumann paradox an accumulation point of vorticity occurs on the wedge above the leading point 1994 john wiley sons inc

regularity of solutions to regular shock reflection for potential flow the shock reflection problem is one of the most important problems in mathematical fluid dynamics since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete however most of the fundamental issues for shock reflection have not been understood including the regularity and transition of different patterns of shock reflection configurations therefore it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection on the other hand for a regular reflection configuration the potential flow governs the exact behavior of the solution in c 1 1 across the pseudo sonic circle even starting from the full euler flow that is both of the nonlinear systems are actually the same in a physically significant region near the pseudo sonic circle thus it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo sonic circle the transonic boundary from the elliptic to hyperbolic region and at the point where the pseudo sonic circle the degenerate elliptic curve meets the reflected shock a free boundary connecting the elliptic to hyperbolic region in this paper we study the regularity of solutions to regular shock reflection for potential flow in particular we prove that the c 1 1 regularity is optimal for the solution across the pseudo sonic circle and at the point where the pseudo sonic circle meets the reflected shock we also obtain the c 2 regularity of the solution up to the pseudo sonic circle in the pseudo subsonic region the problem involves two types of transonic flow one is a continuous transition through the pseudo sonic circle from the pseudo supersonic region to the pseudo subsonic region the other a jump transition through the transonic shock as a free boundary from another pseudo supersonic region to the pseudo subsonic region the techniques and ideas developed in this paper will be useful to other regularity problems for nonlinear degenerate equations involving similar difficulties

on constructing parsimonious type 2 fuzzy logic systems via influential rule selection type 2 fuzzy systems are increasing in popularity and there are many examples of successful applications while many techniques have been proposed for creating parsimonious type 1 fuzzy systems there is a lack of such techniques for type 2 systems the essential problem is to reduce the number of rules while maintaining the system s approximation performance in this paper four novel indexes for ranking the relative contribution of type 2 fuzzy rules are proposed which are termed r values c values omega1 values and omega2 values the r values of type 2 fuzzy rules are obtained by applying a qr decomposition pivoting algorithm to the firing strength matrices of the trained fuzzy model the c values rank rules based on the effects of rule consequents while the omega1 values and omega2 values consider both the rule base structure via firing strength matrices and the output contribution of fuzzy rule consequents two procedures for utilizing these indexes in fuzzy rule selection termed ldquoforward selectionrdquo and ldquobackward eliminationrdquo are described experiments are presented which demonstrate that by using the proposed methodology the most influential type 2 fuzzy rules can be effectively retained in order to construct parsimonious type 2 fuzzy models

the application of explicit nystrom methods to singular second order differential equations abstract an important class of singular second order initial value problems is y 2 x y f x y 0 0 x x f y 0 a y 0 0 this class includes for example the well known singular equations of emden and liouville the purpos of this paper is to show the interesting result that explicit nystrom methods existing for the direct integration of special second order regular initial value problems can be used for the integration of this class of singular initial value problems and the methods show their proper respective orders of convergence this is justified mathematically and demonstrated computationally

free boundary and mixed boundary value problems for quasilinear elliptic equations tangential oblique derivative and degenerate dirichlet boundary problems abstract we present holder estimates and holder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector does not vanish at any point on the boundary we also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation

a singular quasilinear anisotropic elliptic boundary value problem ii we consider the problem uauxx ubuyy p x 0 with a 0 b 0 on a smooth convex bounded region in r2 with dirichlet boundary conditions we show that if the positive function p is uniformly bounded away from zero then the problem has a classical solution

single hidden layer fuzzy recurrent wavelet neural network when a fuzzy wavelet neural network fwnn has a large number of neurons in its consequent part it may not be able to effectively follow fast variations in the process this paper aims to develop a single hidden layer fuzzy recurrent wavelet neural network slfrwnn for the function approximation and identification of dynamic systems in the proposed framework the consequent part of each fuzzy rule is developed by a single neuron with the capability of storing the past data of the network to guarantee the convergence and to speed up the process of the on line training algorithm the optimal learning rate olr is introduced based on lyapunov stability theory the modifications allow the slfrwnn to be much faster than fwnn and hence it is more appropriate in real time applications the efficiency of the modified model is investigated using three different types of wavelet families in approximating a benchmark piecewise function furthermore two nonlinear dynamic plants are considered to demonstrate the feasibility of the slfrwnn in the system identification the results indicate that slfrwnn achieves higher accuracy and needs a lower number of parameters than the other models

a proof of existence of perturbed steady transonic shocks via a free boundary problem we prove the existence of a solution of a free boundary problem for the transonic small disturbance equation the free boundary is the position of a transonic shock dividing two regions of smooth flow assuming inviscid irrotational flow as modeled by the transonic small disturbance equation the equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region to study the stability of a uniform planar transonic shock we consider perturbation by a steady c1 upstream disturbance if the upstream disturbance is small in a c1 sense then there is a steady solution in which the downstream flow and the transonic shock are holder continuous perturbations of the uniform configuration this result provides a new use of inviscid perturbation techniques to demonstrate in two dimensions the stability of transonic shock waves of the type that appear for example over the wing of an airplane along an airfoil or as bow shocks in a flow with a supersonic free stream velocity 2000 john wiley sons inc

shock diffraction by convex cornered wedges for the nonlinear wave system we are concerned with rigorous mathematical analysis of shock diffraction by two dimensional convex cornered wedges in compressible fluid flow through the nonlinear wave system this shock diffraction problem can be formulated as a boundary value problem for second order nonlinear partial differential equations of mixed elliptic hyperbolic type in an unbounded domain it can be further reformulated as a free boundary problem for nonlinear degenerate elliptic equations of second order with a degenerate oblique derivative boundary condition we establish a global theory of existence and optimal regularity for this shock diffraction problem to achieve this we develop several mathematical ideas and techniques which are also useful for other related problems involving similar analytical difficulties

riemann problems for the two dimensional unsteady transonic small disturbance equation we study a two parameter family of riemann problems for the unsteady transonic small disturbance utsd equation also called the two dimensional burgers equation which is used to model the transition from regular to mach reflection for weak shock waves the related initial value problem consists of oblique shock data in the upper half plane with two parameters a and b corresponding to the slopes of the initial shock waves the study of quasi steady solutions leads to a problem that changes type when written in self similar coordinates the problem is hyperbolic in the region where the flow is supersonic and elliptic where the flow is subsonic in this paper we give a complete description of the flow in the hyperbolic region by resolving the hyperbolic wave interactions in the form of quasi one dimensional riemann problems in the region of physical space where the flow is subsonic we pose the related free boundary problems and discuss the behavior of the subsonic solution using results from our previo

the application of runge kutta schemes to singular initial value problems a theory for explicit runge kutta schemes applied to the initial value problem for a first order system of differential equations with a singularity of the first kind is developed it is shown that in general the order of convergence is at most two but that the usual order up to a logarithmic term can be obtained for level three and four schemes applied to specific problems 1 introduction in this paper we examine the application of explicit runge kutta schemes to the numerical solution of the problem

nystrom methods and singular second order differential equations chawla jain and subramanian studied the application of nystrom methods to a class of singular initial value problems following their approach we generalize this class by applying the nystrom method to the initial value problem for an equation of the form y p t y q t y t 0 email protected 0 1 where p has a certain specified type of singularity and q is sufficiently differentiable and then we determine the order of convergence this is followed by computational evidence

a free boundary problem for a quasi linear degenerate elliptic equation regular reflection of weak shocks we prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance utsd model for shock reflection by a wedge in weak regular reflection the state immediately behind the reflected shock is supersonic and constant the flow becomes subsonic further downstream the equation in self similar coordinates is degenerate at the sonic line the reflected shock becomes transonic and begins to curve there its position is the solution to a free boundary problem for the degenerate equation using the rankine hugoniot conditions along the reflected shock we derive an evolution equation for the transonic shock and an oblique derivative boundary condition at the unknown shock position by regularizing the degenerate problem we construct uniform bounds we apply local compactness arguments to extract a limit that solves the problem the solution is smooth in the interior and continuous up to the degenerate boundary this work completes a stage in our program to construct self similar solutions of two dimensional riemann problems in a series of papers we developed techniques for solving the degenerate elliptic equations that arise in self similar reductions of hyperbolic conservation laws in other papers especially in joint work with gary lieberman we developed techniques for solving free boundary problems of the type that arise from rankine hugoniot relations for the first time in this paper we combine these approaches and show that they are compatible although our construction is limited to a finite part of the unbounded subsonic region it suggests that this approach has the potential to solve a variety of problems in weak shock reflection including mach and von neumann reflection in the utsd equation and the analogous problems for the unsteady full potential equation 2002 john wiley sons inc

on the existence of positive solutions of quasilinear elliptic boundary value problems abstract we establish the existence of positive solutions to a class of quasilinear anisotropic problems which have either sublinear or superlinear nonlinearity with a b nonnegative constants and positive constants one example is if b a 0 1 has a solution on the other hand if b a 1 superlinear case then there exists a 0 such that for 0 1 has at least one solution and for no solution exists

aggregation using the linguistic weighted average and interval type 2 fuzzy sets the focus of this paper is the linguistic weighted average lwa where the weights are always words modeled as interval type 2 fuzzy sets it2 fss and the attributes may also but do not have to be words modeled as it2 fss consequently the output of the lwa is an it2 fs the lwa can be viewed as a generalization of the fuzzy weighted average fwa where the type 1 fuzzy inputs are replaced by it2 fss this paper presents the theory algorithms and an application of the lwa it is shown that finding the lwa can be decomposed into finding two fwas since the lwa can model more uncertainties it should have wide applications in distributed and hierarchical decision making

existence and stability of perturbed transonic shocks for compressible steady potential flows abstract we establish an existence theorem for perturbed transonic steady potential flows where the boundary of the subsonic region consists of a transonic shock and thus the governing equation becomes a free boundary problem we establish gradient estimates and use the fixed point theorem of caristi s which does not require a compactness of the map to establish the existence result we provide new techniques to handle the gradients and show that the flow is strictly elliptic inside of the subsonic region

enhanced karnik mendel algorithms the karnik mendel km algorithms are iterative procedures widely used in fuzzy logic theory they are known to converge monotonically and superexponentially fast however several usually two to six iterations are still needed before convergence occurs methods to reduce their computational cost are proposed in this paper extensive simulations show that on average the enhanced km algorithms can save about two iterations which corresponds to more than a 39 reduction in computation time an additional at least 23 computational cost can be saved if no sorting of the inputs is needed

local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations we consider solutions and subsolutions and supersolutions of the boundary value problem au x u du diju u x u du 0 in 2 s x diu y x u g x on3fi for a lipschitz domain s a positive definite matrix valued function u j and a vector field s which points uniformly into q without making any continuity assumptions on the known functions we prove harnack and holder estimates for u near 3i2 in addition we bound the l norm of u near 3i2 in terms of an appropriate lp norm and the known functions our approach is based on that for the corresponding interior estimates of trudinger this paper is concerned with analogs for solutions of oblique boundary value problems of the results in 10 our results will extend previous work of lieberman and trudinger 5 and of nadirashvili 9 specifically we consider second order differential operators q of the form

subsonic solutions for compressible transonic potential flows abstract we establish an existence theorem for transonic isentropic potential flows where the subsonic region is bounded by the sonic line and thus the governing equation may become degenerate on the boundary partly or entirely it has been conjectured by experiments and numerical studies that the self similar multidimensional flow changes its type namely hyperbolic far from the origin supersonic region and elliptic near the origin subsonic region furthermore the potential equation has a different nonlinearity compared to other transonic problems such as the unsteady transonic small disturbance equation the nonlinear wave equation and the pressure gradient equation namely the coefficients of the potential equation depend on the gradients while others are independent of the gradients we provide techniques to handle the gradients establish interior and boundary gradient estimates for the potential flow in a convex region and answer the conjecture that is the flow is strictly elliptic and the region is subsonic

analytical solution methods for the fuzzy weighted average for the fuzzy weighted average fwa despite various discrete solution algorithms and their improvements attempts at analytical solutions are very rare this paper provides an analytical solution method for the fwa based on the conclusions of the karnik mendel km algorithm compared with the two current popular kinds of a cut based computational methods for the fwa mathematical programming transformations and direct iterate computations our method is precise and has a concise structure efficient computation process and sound theoretical proofs we propose two algorithms for computing the analytical solution of the fwa two numerical examples illustrate our proposed approach

instability of strong regular reflection and counterexamples to the detachment criterion we consider a particular instance of reflection of shock waves in self similar compressible flow we prove that local self similar regular reflection rr cannot always be extended into a global flow therefore the detachment criterion is not universally correct more precisely consider the following angle condition the tangent of the strong type reflected shock meets the opposite wall at a sharp or right downstream side angle in cases where the condition is violated and the weak type reflected shock is transonic we show that global rr does not exist combined with earlier work we have shown that none of the classical criteria for rr rightarrow mr transition is universally correct a new criterion is proposed moreover we have shown that strong type rr is unstable in the sense that global rr cannot persist under perturbations to one side this yields a definite answer to the weak strong problem because earlier work shows stability of weak rr in the same sense

an efficient centroid type reduction strategy for general type 2 fuzzy logic system in this paper an efficient centroid type reduction strategy for general type 2 fuzzy set is introduced this strategy makes use of the result of a plane representation and performs the centroid type reduction on each a plane simulations show that it usually needs only several resolution of a value such that the defuzzified value converges to a real value consequently comparing with the exhaustive computation approach this approach can tremendously decrease the computation complexity from exponential into linear

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

study on enhanced karnik mendel algorithms initialization explanations and computation improvements computing the centroid of an interval type 2 fuzzy set is an important operation in a type 2 fuzzy logic system and is usually implemented by karnik mendel km iterative algorithms by connecting km algorithms and continuous km algorithms together this paper gives theoretical explanations on the initialization methods of km and enhanced karnik mendel ekm algorithms proposes exact methods for centroid computation of an interval type 2 fuzzy set and extends the enhanced karnik mendel ekm algorithms to three different forms of weighted ekm wekm algorithms it shows that ekm algorithms become a special case of the wekm algorithms when the weights of the latter are constant value it also shows that in general the weighted ekm algorithms have smaller absolute error and faster convergence speed than the ekm algorithms which make them very attractive for real time applications of fuzzy logic system four numerical examples are used to illustrate and analyze the performance of wekm algorithms

stability of transonic shock fronts in two dimensional euler systems we study the stability of stationary transonic shock fronts under two dimensional perturbation in gas dynamics the motion of the gas is described by the full euler system the system is hyperbolic ahead of the shock front and is a hyperbolic elliptic composed system behind the shock front the stability of the shock front and the downstream flow under two dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic elliptic composed system we develop a method to deal with boundary value problems for such systems the crucial point is to decompose the system to a canonical form in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients by several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem our result indicates the stability of the transonic shock front and the flow field behind the shock

a numerical study of riemann problems for the two dimensional unsteady transonic small disturbance equation we study a two parameter family of riemann problems for the unsteady transonic small disturbance utsd equation also called the two dimensional burgers equation the two parameters a and b which define oblique shock initial data correspond to the slopes of the initial shock waves in the upper half plane for each a and b the three constant states in the upper half plane satisfy the rankine hugoniot conditions across the shocks this leads to a two parameter family of oblique shock interaction problems in this paper we present a numerical study of global solution behavior for the values of a and b in a previously obtained bifurcation diagram our study supplements the related theoretical results and conjectures recently obtained by s canic and b l keyfitz we employ a high resolution numerical method which reveals fine solution structures our findings confirm theoretical results and conjectures about the solution patterns and deepen the understanding of the structure of several intricate wave in

a linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition abstract the problem of shock reflection by a wedge which the flow is dominated by the unsteady potential flow equation is a important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math 57 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle which between incident shock and wedge is small we can see that weak regular reflection as the disturbance of normal reflection as in shuxing chen linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with above viewpoint we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques delicate a priori estimate and compact arguments we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and lipschitz continuous up to the degenerate boundary

a global subsonic solution to an interacting transonic shock for the self similar nonlinear wave equation abstract we present the existence of the subsonic solution to a two dimensional riemann problem governed by a self similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle by utilizing the barrier methods and iterative methods we show the well posedness of the transonic shock in the entire subsonic region and thus establish the global solution this result does not rely on any smallness of riemann data

global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone in this paper we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body the flow is assumed to be polytropic isentropic and described by a steady potential equation theoretically as indicated in r courant k o friedrichs supersonic flow and shock waves interscience publishers inc new york 1948 it follows from the rankine hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface which correspond to the supersonic shock and the transonic shock respectively in the references shuxing chen zhouping xin huicheng yin global shock wave for the supersonic flow past a perturbed cone comm math phys 228 2002 47 84 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone polytropic case preprint 2006 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone isothermal case pacific j math 233 2 2007 257 289 and zhouping xin huicheng yin global multidimensional shock wave for the steady supersonic flow past a three dimensional curved cone anal appl 4 2 2006 101 132 the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow at present for the supersonic symmetric incoming flow we will study the global transonic shock problem when the pressure at infinity is appropriately large

enhanced centroid flow algorithm for computing the centroid of general type 2 fuzzy sets recently a centroid flow cf algorithm has been proposed to compute the centroid of a type 2 fuzzy set a this algorithm utilizes the karnik mendel km or the enhanced km ekm algorithm only at the 0 level of a and then lets its result flow upward to the 1 level of a it avoids having to apply the km ekm algorithms at every level which significantly improves its computational efficiency however the cf algorithm approximation errors will gradually accumulate as the algorithm flows upward and in some cases this can cause the centroid of the 1 level of a to differ from its theoretical value this paper introduces an improved version of the cf algorithm which is called enhanced cf algorithm that reduces such accumulative errors by half and therefore greatly improves the computational accuracy

a mixed boundary value problem for chaplygin s hodograph equation abstract in this paper we will prove existence uniqueness and regularity of a classical solution to a mixed boundary value problem for chaplygin s hodograph equation which is degenerate elliptic on a part of the boundary this problem is derived from the study of detached bow shock ahead of a straight ramp in uniform supersonic flows in the hodograph plane the proof depends on perron s method and some techniques from linear elliptic equations

an elliptic problem arising from the unsteady transonic small disturbance equation abstract we prove a theorem on existence of a weak solution of the dirichlet problem for a quasilinear elliptic equation with a degeneracy on one part of the boundary the degeneracy is of a type keldysh type associated with singular behavior blow up of a derivative at the boundary we define an associated operator which is continuous pseudo monotone and coercive and show that a weak solution displaying singular behavior at the boundary exists