an immersed boundary method with direct forcing for the simulation of particulate flows we present an improved method for computing incompressible viscous flow around suspended rigid particles using a fixed and uniform computational grid the main idea is to incorporate peskin s regularized delta function approach acta numerica 11 2002 1 into a direct formulation of the fluid solid interaction force in order to allow for a smooth transfer between eulerian and lagrangian representations while at the same time avoiding strong restrictions of the time step this technique was implemented in a finite difference and fractional step context a variety of two and three dimensional simulations are presented ranging from the flow around a single cylinder to the sedimentation of 1000 spherical particles the accuracy and efficiency of the current method are clearly demonstrated

three step iterations for variational inequalities and nonexpansive mappings in this paper we introduce a new three step iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality using the technique of updating the solution we show that the sequence converges strongly to a common element of two sets under some control conditions results proved in this paper may be viewed as an improvement and refinement of the recent results of noor and huang m aslam noor z huang three step methods for nonexpansive mappings and variational inequalities appl math comput in press and yao and yao y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings appl math comput in press

types of arcs in a fuzzy graph the concept of connectivity plays an important role in both theory and applications of fuzzy graphs depending on the strength of an arc this paper classifies arcs of a fuzzy graph into three types namely a strong b strong and d arcs the advantage of this type of classification is that it helps in understanding the basic structure of a fuzzy graph completely we analyze the relation between strong paths and strongest paths in a fuzzy graph and obtain characterizations for fuzzy bridges fuzzy trees and fuzzy cycles using the concept of a strong b strong and d arcs an arc of a fuzzy tree is a strong if and only if it is an arc of its unique maximum spanning tree also we identify different types of arcs in complete fuzzy graphs

strong convergence of a general iterative method for variational inequality problems and fixed point problems in hilbert spaces abstract in this paper we introduce a new iterative scheme to investigate the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods our results improve and extend the recent ones announced by chen et al j m chen l j zhang t g fan viscosity approximation methods for nonexpansive mappings and monotone mappings doi 10 1016 j jmaa 2006 12 088 iiduka and tahakshi h iiduka w takahashi strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings nonlinear anal 61 2005 341 350 yao and yao y h yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings appl math comput doi 10 1016 j amc 2006 08 062 and many others

two extragradient methods for generalized mixed equilibrium problems nonexpansive mappings and monotone mappings in this paper we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of an infinite a finite family of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone lipschitz continuous mapping we obtain some weak convergence theorems for the sequences generated by these processes in hilbert spaces the results in this paper generalize extend and unify some well known weak convergence theorems in the literature

effects of the computational time step on numerical solutions of turbulent flow effects of large computational time steps on the computed turbulence were investigated using a fully implicit method in turbulent channel flow computations the largest computational time step in wall units which led to accurate prediction of turbulence statistics was determined turbulence fluctuations could not be sustained if the computational time step was near or larger than the kolmogorov time scale

a relaxed extragradient like method for a generalized mixed equilibrium problem a general system of generalized equilibria and a fixed point problem abstract very recently takahashi and takahashi s takahashi w takahashi strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a hilbert space nonlinear anal 69 2008 1025 1033 suggested and analyzed an iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem of a nonexpansive mapping in a hilbert space in this paper based on takahashi takahashi s iterative method and well known extragradient method we introduce a relaxed extragradient like method for finding a common solution of a generalized mixed equilibrium problem a general system of generalized equilibria and a fixed point problem of a strictly pseudocontractive mapping in a hilbert space and then obtain a strong convergence theorem utilizing this theorem we establish some new strong convergence results in fixed point problems variational inequalities mixed equilibrium problems and systems of generalized equilibria

strong convergence theorems for strongly nonexpansive sequences the aim of this paper is to prove a strong convergence theorem for a pair of sequences of nonexpansive mappings in a hilbert space where one of them is a strongly nonexpansive sequence and provide some applications of the theorem

a new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization abstract in this paper we introduce a new iterative scheme for finding a common element of the set of solutions of an equilibrium problem the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for inverse strongly monotone mappings in a hilbert space under suitable conditions some strong convergence theorems for approximating a common element of the above three sets are obtained as applications at the end of the paper we utilize our results to study the optimization problem and some convergence problem for strictly pseudocontractive mappings the results presented in the paper extend and improve some recent results of yao and yao y y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings appl math comput 186 2 2007 1551 1558 plubtieng and punpaeng s plubtieng r punpaeng a new iterative method for equilibrium problems and fixed point problems of nonlinear mappings and monotone mappings appl math comput 2007 doi 10 1016 j amc 2007 07 075 s takahashi and w takahashi s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2006 506 515 su shang and qin y f su m j shang x l qin an iterative method of solution for equilibrium and optimization problems nonlinear anal 2007 doi 10 1016 j na 2007 08 045 and chang cho and kim s s chang y j cho j k kim approximation methods of solutions for equilibrium problem in hilbert spaces dynam systems appl in print

a simple and efficient direct forcing immersed boundary framework for fluid structure interactions a direct forcing immersed boundary framework is presented for the simple and efficient simulation of strongly coupled fluid structure interactions the immersed boundary method developed by yang and balaras j yang e balaras an embedded boundary formulation for large eddy simulation of turbulent flows interacting with moving boundaries j comput phys 215 1 2006 12 40 is greatly simplified by eliminating several complicated geometric procedures without sacrificing the overall accuracy the fluid structure coupling scheme of yang et al j yang s preidikman e balaras a strongly coupled embedded boundary method for fluid structure interactions of elastically mounted rigid bodies j fluids struct 24 2008 167 182 is also significantly expedited by moving the fluid solver out of the predictor corrector iterative loop without altering the strong coupling property central to these improvements are the reformulation of the field extension strategy and the evaluation of fluid force and moment exerted on the immersed bodies by taking advantage of the direct forcing idea in a fractional step method several cases with prescribed motions are examined first to validate the simplified field extension approach then a variety of strongly coupled fluid structure interaction problems including vortex induced vibrations of a circular cylinder transverse and rotational galloping of rectangular bodies and fluttering and tumbling of rectangular plates are computed the excellent agreement between the present results and the reference data from experiments and other simulations demonstrates the accuracy simplicity and efficiency of the new method and its applicability in a wide range of complicated fluid structure interaction problems

strong convergence theorems based on a new modified extragradient method for variational inequality problems and fixed point problems in banach spaces in this paper we introduce a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of a general variational inequality and the set of common fixed points of an infinite family of k i strict pseudocontractions in a banach space we obtain some strong convergence theorems under suitable conditions the results obtained in this paper improve and extend the recent ones announced by many others

algorithms for approximating minimization problems in hilbert spaces in this paper we study the following minimization problem minx fix s w m2 12 x 2 h x where b is a bounded linear operator m 0 is some constant h is a potential function for c f fix t is the set of fixed points of nonexpansive mapping s and w is the solution set of an equilibrium problem this paper introduces two new algorithms one implicit and one explicit that can be used to find the solution of the above minimization problem

hybrid noor iteration method for nonexpansive mappings in hilbert spaces in this paper hybrid noor iteration method was introduced for nonexpansive mappings in hilbert spaces the sufficient and necessary conditions that the iteration sequences converge strongly to a fixed point of a nonexpansive mapping are obtained in hilbert spaces

a hybrid algorithm for pseudo contractive mappings abstract in this paper we suggest and analyze a hybrid algorithm for pseudo contractive mappings in hilbert spaces further we prove the strong convergence of the proposed iterative algorithm for lipschitz pseudo contractive mappings in hilbert spaces

strong convergence of krasnoselskii and mann s type sequences for one parameter nonexpansive semigroups without bochner integrals abstract in this paper we prove krasnoselskii and mann s type convergence theorems for nonexpansive semigroups without using bochner integral and without assuming the strict convexity of banach spaces one of our main results is the following let c be a compact convex subset of a banach space e and let t t t 0 be a one parameter strongly continuous semigroup of nonexpansive mappings on c let t n be a sequence in 0 satisfying lim inf n t n lim sup n t n and lim n t n 1 t n 0 let 0 1 define a sequence x n in c by x 1 c and x n 1 t t n x n 1 x n for n n then x n converges strongly to a common fixed point of t t t 0

convergence theorems for a countable family of lipschitzian mappings this paper is concerned with convergence of an approximating common fixed point sequence of countable lipschitzian mappings in a uniformly convex banach space we introduce a new condition for a class of mappings to obtain several weak and strong convergence theorems this new condition is implied by many previous known conditions introduced by many authors we also apply our results for a class of nonexpansive mappings and asymptotically nonexpansive mappings and we immediately obtain convergence theorems proved by song chen kimura takahashi tan xu and many others

variational inequalities with generalized monotone operators we investigate the variational inequality with pseudomonotone operators in the sense of karamardian in banach spaces new existence results which extend many known results in infinite dimensional spaces are derived under rather weak assumptions new uniqueness results which also seem to be new even in finite dimensional spaces are also derived in particular new existence and uniqueness results for the complementarity problem with pseudomonotone operators in banach spaces are obtained also existence and uniqueness results for minimization problems with pseudoconvex functions in banach spaces are obtained

approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method in this paper we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real hilbert space furthermore we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters

sharp interface immersed boundary level set method for wave body interactions a sharp interface cartesian grid method for the large eddy simulation of two phase turbulent flows interacting with moving bodies is presented the overall approach uses a sharp interface immersed boundary formulation and a level set ghost fluid method for solid fluid and fluid fluid interface treatments respectively a four step fractional step method is used for velocity pressure coupling and a lagrangian dynamic smagorinsky subgrid scale model is adopted for large eddy simulations a simple contact angle boundary condition treatment that conforms to the immersed boundary formulation is developed a variety of test cases of different scales ranging from bubble dynamics water entry and exit landslide generated waves to ship hydrodynamics are performed for validation extensions for high reynolds number ship flows using wall layer models are also considered

a dffd simulation method combined with the spectral element method for solid fluid interaction problems a 2d simulation method for a rigid body moving in an incompressible viscous fluid is proposed it combines one of the immersed boundary methods the dffd direct forcing fictitious domain method with the spectral element method the former is employed for efficiently capturing the two way fsi fluid structure interaction and the geometric flexibility of the latter is utilized for any possibly co existing stationary and complicated solid or flow boundary a pseudo body force is imposed within the solid domain to enforce the rigid body motion and a lagrangian mesh composed of triangular elements is employed for tracing the rigid body in particular a so called sub cell scheme is proposed to smooth the discontinuity at the fluid solid interface and to execute integrations involving eulerian variables over the moving solid domain the accuracy of the proposed method is verified through an observed agreement of the simulation results of some typical flows with analytical solutions or existing literatures the direct forcing fictitious domain method and the spectral element method are combined direct interpolation of eulerian variables onto lagrangian mesh is not necessary non fluctuating pressure fields are obtained with a smoothed pseudo body force the proposed method possesses a spectral accuracy both small and large solid to fluid density ratios are applicable

a stable fluid structure interaction solver for low density rigid bodies using the immersed boundary projection method dispersion of low density rigid particles with complex geometries is ubiquitous in both natural and industrial environments we show that while explicit methods for coupling the incompressible navier stokes equations and newton s equations of motion are often sufficient to solve for the motion of cylindrical particles with low density ratios for more complex particles such as a body with a protrusion they become unstable we present an implicit formulation of the coupling between rigid body dynamics and fluid dynamics within the framework of the immersed boundary projection method similarly to previous work on this method the resulting matrix equation in the present approach is solved using a block lu decomposition each step of the block lu decomposition is modified to incorporate the rigid body dynamics we show that our method achieves second order accuracy in space and first order in time third order for practical settings only with a small additional computational cost to the original method our implicit coupling yields stable solution for density ratios as low as 10 4 we also consider the influence of fictitious fluid located inside the rigid bodies on the accuracy and stability of our method

strong convergence to common fixed points of families of nonexpansive mappings abstract we introduce iteration schemes for families of nonexpansive mappings in hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

an immersed boundary finite volume method for simulations of flow in complex geometries abstract a new immersed boundary method for simulating flows over or inside complex geometries is developed by introducing a mass source sink as well as a momentum forcing the present method is based on a finite volume approach on a staggered mesh together with a fractional step method both momentum forcing and mass source are applied on the body surface or inside the body to satisfy the no slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary in the immersed boundary method the choice of an accurate interpolation scheme satisfying the no slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary therefore a stable second order interpolation scheme for evaluating the momentum forcing on the body surface or inside the body is proposed three different flow problems decaying vortices and flows over a cylinder and a sphere are simulated using the immersed boundary method proposed in this study and the results agree very well with previous numerical and experimental results verifying the accuracy of the present method

strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings abstract in this paper we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping in a hilbert space then we show that the sequence converges strongly to a common element of two sets using this result we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly monotone mapping

viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problems abstract in this paper we first introduce a general iterative scheme by halpern approximation method for finding a common element of the set of solutions of an equilibrium problem the set of solutions of the variational inequality for an infinite family of i inverse strongly monotone mappings and the set of common fixed points for a family of infinitely nonexpansive mappings in a hilbert space then we extend the halpern iterative scheme to a general viscosity approximation scheme by a simple and different method as applications we utilize our results to investigate some convergence problems for strictly pseudocontractive mappings and maximal monotone operators the results obtained in this paper extend and improve the recent ones announced by many others

three step methods for nonexpansive mappings and variational inequalities in this paper we suggest and analyze three step iteration methods for finding the common element of the set of fixed points of a nonexpansive mappings and the set of solutions of the variational inequalities we also study the convergence criteria of three step iterative method under some mild conditions our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results

a smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations the effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods the problem with this representation is that the volume force exhibits non physical oscillations in moving boundary simulations a smoothing technique for discrete delta functions has been developed in this paper to suppress the non physical oscillations in the volume forces we have found that the non physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions it has been shown that the smoothed discrete delta functions constructed in this paper have one order higher derivative than the regular ones moreover not only the smoothed discrete delta functions satisfy the first two discrete moment conditions but also their derivatives satisfy one order higher moment condition than the regular ones the smoothed discrete delta functions are tested by three test cases a one dimensional heat equation with a moving singular force a two dimensional flow past an oscillating cylinder and the vortex induced vibration of a cylinder the numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations

euler index in uncertain graph abstract as the system becomes more complex in practical application of graph theory different types of uncertainty are frequently encountered in an uncertain graph whether two vertices of the graph are joined cannot be completely determined within the framework of uncertainty theory the concept of euler index of uncertain graph is proposed a method to calculate euler index of uncertain graph is also given what s more the euler index of uncertain cycle and uncertain graph with blocks can be obtained in a simple way

convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems abstract the purpose of this work is to introduce a hybrid projection method for finding a common element of the set of a generalized equilibrium problem the set of solutions to a variational inequality and the set of fixed points of a strict pseudo contraction in a real hilbert space

on modified iterative method for nonexpansive mappings and monotone mappings abstract in this paper we introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for inverse strongly monotone mappings we show that the sequence converges strongly to a common element of the above two sets under some parameters controlling conditions

strong convergence theorem for uniformly l lipschitzian asymptotically pseudocontractive mapping in real banach space abstract let e be a real banach space let k be a nonempty closed and convex subset of e t k k a uniformly l lipschitzian asymptotically pseudocontractive mapping with sequence k n n 0 1 lim n k n 1 such that f t let n n 0 0 1 be such that n 0 n n 0 n 2 and n 0 n k n 1 suppose x n n 0 is iteratively defined by x n 1 1 n x n n t n x n n 0 and suppose there exists a strictly increasing continuous function 0 0 0 0 such that t n x x j x x k n x x 2 x x x k it is proved that x n n 0 converges strongly to x f t it is also proved that the sequence of iteration x n defined by x n 1 a n x n b n t n x n c n u n n 0 where u n n 0 is a bounded sequence in k and a n n 0 b n n 0 c n n 0 are sequences in 0 1 satisfying appropriate conditions converges strongly to a fixed point of t

on mann implicit iterations for strongly accretive and strongly pseudo contractive mappings abstract for a lipschitz strongly accretive map considered by chidume in c e chidume picard iteration for strongly accretive and strongly pseudo contractive lipschitz maps ictp preprint no ic2000098 c e chidume iterative algorithms for nonexpansive mappings and some of their generalizations nonlinear analysis and applications to v lakshmikantam on his 80th birthday vols 1 and 2 kluwer acad publ dordrecht 2003 pp 383 429 it is known that a classical picard type iteration process converges strongly to a zero of the operator he also proved that the rate of convergence in this case is at least as fast as a geometric progression in this paper we study the mann implicit iteration sequence for strongly accretive and strongly pseudo contractive mappings we showed that this implicit scheme gives better convergence rate estimate presented results improve the corresponding results of c e chidume picard iteration for strongly accretive and strongly pseudo contractive lipschitz maps ictp preprint no ic2000098 c e chidume iterative algorithms for nonexpansive mappings and some of their generalizations nonlinear analysis and applications to v lakshmikantam on his 80th birthday vols 1 and 2 kluwer acad publ dordrecht 2003 pp 383 429 l liu approximation of fixed points of a strictly pseudo contractive mapping proc am math soc 125 2 1997 1363 1366 w r sastry g v r babu approximation of fixed points of strictly pseudo contractive mappings on arbitrary closed convex sets in a banach space proc amer math soc 128 2000 2907 2909 y song r chen viscosity approximative methods to cesaro means for non expansive mappings appl math comput 186 2 2007 1120 1128

weak convergence theorems for a countable family of lipschitzian mappings this paper is concerned with convergence of an approximating common fixed point sequence of countable lipschitzian mappings in a uniformly convex banach space we also establish weak convergence theorems for finding a common element of the set of fixed points the set of solutions of an equilibrium problem and the set of solutions of a variational inequality with an appropriate setting we obtain and improve the corresponding results recently proved by moudafi a moudafi weak convergence theorems for nonexpansive mappings and equilibrium problems j nonlinear convex anal 9 2008 37 43 tada takahashi a tada and w takahashi weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem j optim theory appl 133 2007 359 370 and plubtieng kumam s plubtieng and p kumam weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings j comput appl math 2008 doi 10 1016 j cam 2008 05 045 some of our results are established with weaker assumptions

a non iterative direct forcing immersed boundary method for strongly coupled fluid solid interactions a non iterative direct forcing immersed boundary method is presented for the strongly coupled simulations of fluid solid interactions while it retains many advantages of the immersed boundary framework by yang and stern 2012 30 especially the simplified field extension strategy for moving boundary treatment and the pointwise integration of hydrodynamic force using the momentum forcing term the present approach improves upon the previous method in several aspects including optimized computational cost for a strong coupling scheme reduced algorithm complexity for a straightforward implementation and enhanced numerical stability for low density ratio problems central to these improvements is a simple intermediate step in which the velocity fields around solid bodies are predicted on temporary non inertial reference frames attached to moving solid bodies in a one to one manner this step enables the explicit inverse of the implicit equations for rigid body dynamics thus rendering unnecessary the previous predictor corrector scheme for iteratively adjusting the displacements and velocities of the immersed bodies until reaching a convergence in addition a simple generalized procedure is developed to obtain the interpolation coefficients in a local reconstruction stencil explicitly from the geometric relationship for verification and validation the vortex induced vibration of a circular cylinder and the rotational galloping of a rectangular body are considered first then several particulate flow problems including settling and buoyant particles of low density ratios a settling particle in a small container and the kissing drafting tumbling problem of two settling particles are studied the agreement between the present results and the reference data in the literature is excellent an overall second order accuracy of the algorithm is verified in two systematic grid convergence tests the present idea can be easily applied to similar methods for achieving a strong coupling scheme on top of a weak one with a nominal increase in the computational cost details of the algorithm are provided to facilitate its implementation in other solvers using non boundary conforming grids

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces abstract in this paper we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space then we prove a strong convergence theorem which is connected with combettes and hirstoaga s result p l combettes s a hirstoaga equilibrium programming in hilbert spaces j nonlinear convex anal 6 2005 117 136 and wittmann s result r wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491 using this result we obtain two corollaries which improve and extend their results

plane augmentation of plane graphs to meet parity constraints abstract a plane topological graph g v e is a graph drawn in the plane whose vertices are points in the plane and whose edges are simple curves that do not intersect except at their endpoints given a plane topological graph g v e and a set cg of parity constraints in which every vertex has assigned a parity constraint on its degree either even or odd we say that g is topologically augmentable to meet cg if there exists a set e of new edges disjoint with e such that g v e e is noncrossing and meets all parity constraints in this paper we prove that the problem of deciding if a plane topological graph is topologically augmentable to meet parity constraints is np complete even if the set of vertices that must change their parities is v or the set of vertices with odd degree in particular deciding if a plane topological graph can be augmented to a eulerian plane topological graph is np complete analogous complexity results are obtained when the augmentation must be done by a plane topological perfect matching between the vertices not meeting their parities we extend these hardness results to planar graphs when the augmented graph must be planar and to plane geometric graphs plane topological graphs whose edges are straight line segments in addition when it is required that the augmentation is made by a plane geometric perfect matching between the vertices not meeting their parities we also prove that this augmentation problem is np complete for plane geometric paths for the particular family of maximal outerplane graphs we characterize maximal outerplane graphs that are topological augmentable to satisfy a set of parity constraints we also provide a polynomial time algorithm that decides if a maximal outerplane graph is topologically augmentable to meet parity constraints and if so produces a set of edges with minimum cardinality

approximation of common fixed points of a countable family of nonexpansive mappings in a banach space in this paper to find a common fixed point of a family of nonexpansive mappings we introduce a halpern type iterative sequence then we prove that such a sequence converges strongly to a common fixed point of nonexpansive mappings moreover we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings and the problem of finding a zero of an accretive operator

equilibrium programming using proximal like algorithms we compute constrained equilibria satisfying an optimality condition important examples include convex programming saddle problems noncooperative games and variational inequalities under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization in the main algorithm each stage of computation requires two proximal steps possibly using bregman functions one step serves to predict the next point the other helps to correct the new prediction to enhance practical applicability we tolerate numerical errors

on connectivity of the cartesian product of two graphs in this paper sufficient conditions for the cartesian product of two graphs to be maximum edge connected maximum point connected super edge connected or super point connected are presented as a corollary some graphs with these properties are obtained furthermore the relationship between super point connectivity and super edge connectivity on the cartesian product of graphs is pointed out

convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications in this paper we consider the problem of convergence of an iterative algorithm for a system of generalized variational inequalities and a nonexpansive mapping strong convergence theorems are established in the framework of real banach spaces

one fluid formulation for fluid structure interaction with free surface abstract a simple and efficient computational framework is presented for the simulation of fluid structure interaction problems involving rigid body and multiphase flows in the context of hydrodynamics unlike existing publications this method does not solve the general motion of rigid bodies in the lagrangian form of newton s law derived from the distributed lagrange multiplier treatment of the rigid body a new set of governing equations is presented on the fully eulerian one fluid formulation to solve the problem numerically the complex problem is separated into three parts balance of the momentum and mass dynamic problem evolving of the heaviside function by the external velocity geometric problem and rigid motion projection kinematic problem the conservation of mass and momentum is guaranteed by the multiphase fluid solver the water air and structure coupling is accomplished by the smeared interface a new way of initialisation and convection of the rigid heaviside function is designed for an arbitrary shape to deal with rigid velocity vector a linear least square method is proposed the excellent agreement between the numerical experiment and the reference data from experiments demonstrate the validity and applicability of the new methodology

uncertain multi product newsboy problem with chance constraint a multi product newsboy problem with uncertain demand and uncertain storage space is investigated under a warehousing chance constraint it is assumed that indeterminacy may appear in demand and storage space due to rough estimation of their distributions by decision maker uncertainty theory provides a new tool to deal with the human uncertainty we develop an uncertain expected value programming model uevpm based on uncertainty theory uncertain variables functions from uncertainty space to the set of real numbers are used to describe subjective estimation the objective function is to maximize the expected profit of newsboy at a predetermined confidence level furthermore we convert the chance constrained uncertain programming model to its equivalent deterministic form so as to be solved by classic integer programming method finally three numerical examples are given to illustrate the potential usefulness of the study

weak and strong convergence theorems for countable lipschitzian mappings and its applications abstract we use mann s iteration and the hybrid method in mathematical programming to obtain weak and strong convergence to common fixed points of a countable family of lipschitzian mappings finally we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings

on m strong fuzzy graphs the cartesian product and disjoint sum of graphs play a prominent role and have numerous interesting algebraic properties in this note we consider operations on fuzzy graphs under which m strong property is preserved if g1 and g2 are m strong fuzzy graphs then we prove that g1 g2 g1 g2 and g1 g2 are also m strong but g1 g2 need not be m strong if g1 g2 is m strong then we show that at least one factor must be m strong we show that the product of a m strong fuzzy graph g1 with a non m strong fuzzy graph g2 remains m strong if and only if g2 satisfies special condition for any fuzzy graph g gcc is the smallest m strong fuzzy graph that contains g and g gcc if and only if g is m strong we further show that m strong fuzzy graph g is a fuzzy tree if and only if the support g is a tree

some new algorithms for solving mixed equilibrium problems in this paper we suggest and analyze two projection methods one implicit and one explicit for finding a particular solution of a mixed equilibrium problem in a real hilbert space furthermore we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem

a ghost cell immersed boundary method for flow in complex geometry an efficient ghost cell immersed boundary method gcibm for simulating turbulent flows in complex geometries is presented a boundary condition is enforced through a ghost cell method the reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second order accuracy of the base solver both dirichlet and neumann boundary conditions can be treated the current ghost cell treatment is both suitable for staggered and non staggered cartesian grids the accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface numerical results are compared with experimental data and boundary fitted grid results the method is further extended to an existing ocean model mitgcm to simulate geophysical flow over a three dimensional bump the method is easily implemented as evidenced by our use of several existing codes

iterative algorithms for a general system of generalized nonlinear mixed composite type equilibria in this paper we consider and study a general system of generalized nonlinear mixed composite type equilibria in hilbert spaces first we prove the existence and uniqueness of the solution for this system of generalized nonlinear mixed composite type equilibria second the mann iterative method with errors is extended to develop some new iterative algorithms for finding approximate solutions for this system of generalized nonlinear mixed composite type equilibria we also derive the strong convergence of the sequences generated by these iterative algorithms in hilbert spaces

a new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings in this paper we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many nonexpansive mappings the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for a inverse strongly monotone mapping in hilbert spaces we prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions this main result improve and extend plubtieng and punpaeng s corresponding result s plubtieng r punpaeng a new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings applied mathematics and computation 197 2008 548 558 using this theorem we obtain three corollaries

an embedded boundary formulation for large eddy simulation of turbulent flows interacting with moving boundaries a non boundary conforming formulation for simulating complex turbulent flows with dynamically moving boundaries on fixed cartesian grids is proposed the underlying finite difference solver for the filtered incompressible navier stokes equations is based on a second order fractional step method on a staggered grid to satisfy the boundary conditions on an arbitrary immersed interface the velocity field at the grid points near the interface is reconstructed using momentum forcing without smearing the sharp interface the concept of field extension is also introduced to treat the points emerging from a moving solid body to the fluid laminar flow cases and large eddy simulations les are presented to demonstrate the formal accuracy and range of applicability of the method in particular simulations of laminar flow induced by the harmonic in line oscillation of a circular cylinder in quiescent fluid and from a transversely oscillating cylinder in a free stream are presented and compared to reference simulations and experiments les of turbulent flow over a traveling wavy wall and transitional flow through a bileaflet prosthetic heart valve are also shown all results are in very good agreement with reference results in the literature

proteus a direct forcing method in the simulations of particulate flows a new and efficient direct numerical method for the simulation of particulate flows is introduced the method combines desired elements of the immersed boundary method the direct forcing method and the lattice boltzmann method adding a forcing term in the momentum equation enforces the no slip condition on the boundary of a moving particle by applying the direct forcing scheme proteus eliminates the need for the determination of free parameters such as the stiffness coefficient in the penalty scheme or the two relaxation parameters in the adaptive forcing scheme the method presents a significant improvement over the previously introduced immersed boundary lattice boltzmann method ib lbm where the forcing term was computed using a penalty method and a user defined parameter the method allows the enforcement of the rigid body motion of a particle in a more efficient way compared to the bounce back scheme used in the conventional lbm the direct forcing method provides a smoother computational boundary for particles and is capable of achieving results at higher reynolds number flows by using a set of lagrangian points to track the boundary of a particle proteus eliminates any need for the determination of the boundary nodes that are prescribed by the bounce back scheme at every time step it also makes computations for particles of irregular shapes simpler and more efficient proteus has been developed in two as well as three dimensions this new method has been validated by comparing its results with those from experimental measurements for a single sphere settling in an enclosure under gravity as a demonstration of the efficiency and capabilities of the present method the settling of a large number 1232 of spherical particles is simulated in a narrow box under two different boundary conditions it is found that when the no slip boundary condition is imposed at the front and rear sides of the box the particles motion is significantly hindered under the periodic boundary conditions the particles move faster the simulations show that the sedimentation characteristics in a box with periodic boundary conditions at the two sides are very close to those found in the sedimentation of two dimensional circular particles

an improved immersed boundary method with direct forcing for the simulation of particle laden flows an efficient approach for the simulation of finite size particles with interface resolution was presented by uhlmann m uhlmann an immersed boundary method with direct forcing for the simulation of particulate flows j comput phys 209 2005 448 476 the present paper proposes several enhancements of this method which considerably improve the results and extend the range of applicability an important step is a simple low cost iterative procedure for the euler lagrange coupling yielding a substantially better imposition of boundary conditions at the interface even for large time steps furthermore it is known that the basic method is restricted to ratios of particle density and fluid density larger than some critical value above 1 hence excluding for example non buoyant particles this can be remedied by an efficient integration step for the artificial flow field inside the particles to extend the accessible density range down to 0 3 this paper also shows that the basic scheme is inconsistent when moving surfaces are allowed to approach closer than twice the step size a remedy is developed based on excluding from the force computation all surface markers whose stencil overlaps with the stencil of a marker located on the surface of a collision partner the resulting algorithm is throughly validated and is demonstrated to substantially improve upon the original method

immersed boundary method for flow around an arbitrarily moving body for the simulation of flow around an arbitrarily moving body an immersed boundary method is developed in a non inertial reference frame that is fixed to the body the navier stokes equation is formulated in a conservative form such that the force terms due to the rotation and the translational and rotational accelerations are included in the nonlinear term in order to satisfy the no slip condition on the body surface momentum forcing and mass source sink are applied on the body surface or inside the body the numerical method is based on a finite volume approach on a staggered mesh together with a fractional step method the present numerical method is applied to both the forced motion and fluid structure interaction problems in the latter we solve fully coupled navier stokes and dynamic equations for the moving body without introducing any iteration four different flow problems are tested and the results are in excellent agreements with previous numerical and experimental ones

immersed boundary methods for general finite difference and finite volume navier stokes solvers we present an immersed boundary algorithm for incompressible flows with complex boundaries suitable for cartesian or curvilinear grid system the key stages of any immersed boundary technique are the interpolation of a velocity field given on a mesh onto a general boundary a line in 2d a surface in 3d and the spreading of a force field from the immersed boundary to the neighboring mesh points to enforce the desired boundary conditions on the immersed boundary points we propose a technique that uses the reproducing kernel particle method w k liu s jun y f zhang reproducing kernel particle methods int j numer methods fluids 20 8 1995 1081 1106 for the interpolation and spreading unlike other methods presented in the literature the one proposed here has the property that the integrals of the force field and of its moment on the grid are conserved independent of the grid topology uniform or non uniform cartesian or curvilinear the technique is easy to implement and is able to maintain the order of the original underlying spatial discretization applications to two and three dimensional flows in cartesian and non cartesian grid system with uniform and non uniform meshes are presented

a stable partitioned fsi algorithm for rigid bodies and incompressible flow part i model problem analysis a stable partitioned algorithm is developed for fluid structure interaction fsi problems involving viscous incompressible flow and rigid bodies this added mass partitioned amp algorithm remains stable without sub iterations for light and even zero mass rigid bodies when added mass and viscous added damping effects are large the scheme is based on a generalized robin interface condition for the fluid pressure that includes terms involving the linear acceleration and angular acceleration of the rigid body added mass effects are handled in the robin condition by inclusion of a boundary integral term that depends on the pressure added damping effects due to the viscous shear forces on the body are treated by inclusion of added damping tensors that are derived through a linearization of the integrals defining the force and torque added damping effects may be important at low reynolds number or for example in the case of a rotating cylinder or rotating sphere when the rotational moments of inertia are small in this first part of a two part series the properties of the amp scheme are motivated and evaluated through the development and analysis of some model problems the analysis shows when and why the traditional partitioned scheme becomes unstable due to either added mass or added damping effects the analysis also identifies the proper form of the added damping which depends on the discrete time step and the grid spacing normal to the rigid body the results of the analysis are confirmed with numerical simulations that also demonstrate a second order accurate implementation of the amp scheme

an improved direct forcing immersed boundary method for finite difference applications a modified immersed boundary method is developed using the direct forcing concept an improved bilinear interpolation extrapolation algorithm is implemented for more accurate boundary forcing expressions and easier implementation detailed discussions of the method are presented on the stability velocity interpolation on the immersed boundary direct forcing extrapolation to the grid points resolution of the immersed boundary points and internal treatment the method can achieve second order accurate solutions the method is then applied to a finite difference scheme to compute flow over a stationary cylinder an oscillating cylinder and a stationary sphere the accuracy of the computational results is verified using numerous computational and experimental results in the literature

algorithms with strong convergence for a system of nonlinear variational inequalities in banach spaces abstract in this paper a general system of nonlinear variational inequality problem in banach spaces was considered which includes some existing problems as special cases for solving this nonlinear variational inequality problem we construct two methods which were inspired and motivated by korpelevich s extragradient method furthermore we prove that the suggested algorithms converge strongly to some solutions of the studied variational inequality

a new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings abstract in this paper we introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for inverse strongly monotone mappings we show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions this main theorem extends a recent result of yao and yao y yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings applied mathematics and computation 186 2 2007 1551 1558

strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in this paper we show strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in hilbert spaces by the hybrid method in the mathematical programming

strong convergence theorems for a countable family of quasi lipschitzian mappings and its applications abstract we use the hybrid method in mathematical programming to obtain strong convergence to common fixed points of a countable family of quasi lipschitzian mappings as a consequence several convergence theorems for quasi nonexpansive mappings and asymptotically strict pseudo contractions are deduced we also establish strong convergence of iterative sequences for finding a common element of the set of fixed point the set of solutions of an equilibrium problem and the set of solutions of a variational inequality with an appropriate setting we obtain the corresponding results due to tada takahashi and nakajo shimoji takahashi

iterative methods for common fixed points for a countable family of nonexpansive mappings in uniformly convex spaces abstract let e be a uniformly convex real banach space with a uniformly g teaux differentiable norm let k be a closed convex and nonempty subset of e let t i i 1 be a family of nonexpansive self mappings of k for arbitrary fixed 0 1 define a family of nonexpansive maps s i n 1 by s i 1 i t i where i is the identity map of k let f i 1 f t i 0 it is proved that an iterative sequence x n defined by x 0 k x n 1 n u i 1 i t n s i x n n 0 converges strongly to a common fixed point of the family t i i 1 where n and i t n are sequences in 0 1 satisfying appropriate conditions in each of the following cases a e l p 1 p and b at least one of the t i s is demicompact

weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations abstract let h be a real hilbert space k a nonempty closed convex subset of h and t k k a lipschitz pseudocontraction with a nonempty fixed point set weak and strong convergence theorems for the iterative approximation of fixed points of t are proved furthermore if t h h is a lipschitz monotone operator and f r t where r t denotes the range of t weak and strong convergence theorems for the iterative approximation of solutions of the operator equation tx f are proved

strong conservative form of the incompressible navier stokes equations in a rotating frame with a solution procedure the navier stokes equations in a rotating frame of reference have been formulated in the so called strong conservative form i e without the traditional source terms viz the coriolis and centrifugal forces these equations have been coupled with the continuity equation by using the modified artificial compressibility method in order to develop an implicit numerical scheme that has third order accuracy in space and second order accuracy in time this scheme uses the roe fluxes and the muscl extrapolation techniques to obtain the desired accuracies in space and the backward euler formula to obtain the desired time accuracy the flux jacobians and their eigensystem are presented which are required in the development of the numerical scheme the resulting scheme was used to solve the ekman boundary layer problems with a no slip and b applied surface stress boundary conditions and excellent agreement between computed and exact solutions has been obtained supporting the new formulation of the governing equations as well as the solution procedure

direct numerical simulation of particulate flows with an overset grid method abstract we evaluate an efficient overset grid method for two dimensional and three dimensional particulate flows for small numbers of particles at finite reynolds number the rigid particles are discretised using moving overset grids overlaid on a cartesian background grid this allows for strongly enforced boundary conditions and local grid refinement at particle surfaces thereby accurately capturing the viscous boundary layer at modest computational cost the incompressible navier stokes equations are solved with a fractional step scheme which is second order accurate in space and time while the fluid solid coupling is achieved with a partitioned approach including multiple sub iterations to increase stability for light rigid bodies through a series of benchmark studies we demonstrate the accuracy and efficiency of this approach compared to other boundary conformal and static grid methods in the literature in particular we find that fully resolving boundary layers at particle surfaces is crucial to obtain accurate solutions to many common test cases with our approach we are able to compute accurate solutions using as little as one third the number of grid points as uniform grid computations in the literature a detailed convergence study shows a 13 fold decrease in cpu time over a uniform grid test case whilst maintaining comparable solution accuracy

on variational inclusion and common fixed point problems in hilbert spaces with applications abstract the purpose of this paper is to introduce a general iterative method for finding a common element of the solution set of quasi variational inclusion problems and of the common fixed point set of an infinite family of nonexpansive mappings in the framework hilbert spaces strong convergence of the sequences generated by the purposed iterative scheme is obtained

an adaptive reconstruction for lagrangian direct forcing immersed boundary methods abstract lagrangian direct forcing immersed boundary ib methods have been receiving increased attention due to their robustness in complex fluid structure interaction problems they are very sensitive however on the selection of the lagrangian grid which is typically used to define a solid or flexible body immersed in a fluid flow in the present work we propose a cost efficient solution to this problem without compromising accuracy central to our approach is the use of isoparametric mapping to bridge the relative resolution requirements of lagrangian ib and eulerian grids with this approach the density of surface lagrangian markers which is essential to properly enforce boundary conditions is adapted dynamically based on the characteristics of the underlying eulerian grid the markers are not stored and the lagrangian data structure is not modified the proposed scheme is implemented in the framework of a moving least squares reconstruction formulation but it can be adapted to any lagrangian direct forcing formulation the accuracy and robustness of the approach is demonstrated in a variety of test cases of increasing complexity

strong convergence of modified mann iterations for asymptotically nonexpansive mappings and semigroups abstract the mann iterations for nonexpansive mappings have in general only weak convergence in a hilbert space we modify an iterative method of mann s type introduced by nakajo and takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 for nonexpansive mappings and prove strong convergence of our modified mann s iteration processes for asymptotically nonexpansive mappings and semigroups

operations on fuzzy graphs abstract we define the operations of cartesian product composition union and join on fuzzy subgraphs of graphs g1 and g2 if the graph g is formed from g1 and g2 by one of these operations we determine necessary and sufficient conditions for an arbitrary fuzzy subgraph of g also to be formed by the same operation from fuzzy subgraphs of g1 and g2

a pre conditioned implicit direct forcing based immersed boundary method for incompressible viscous flows a novel immersed boundary ib method based on an implicit direct forcing idf scheme is developed for incompressible viscous flows the key idea for the present idf method is to use a block lu decomposition technique in momentum equations with taylor series expansion to construct the implicit ib forcing in a recurrence form which imposes more accurate no slip boundary conditions on the ib surface to accelerate the ib forcing convergence during the iterative procedure a pre conditioner matrix is introduced in the recurrence formulation of the ib forcing a jacobi type parameter is determined in the pre conditioner matrix by minimizing the frobenius norm of the matrix function representing the difference between the ib forcing solution matrix and the pre conditioner matrix in addition the pre conditioning parameter is restricted due to the numerical stability in the recurrence formulation consequently the present pre conditioned idf pidf enables accurate calculation of the ib forcing within a few iterations we perform numerical simulations of two dimensional flows around a circular cylinder and three dimensional flows around a sphere for low and moderate reynolds numbers the result shows that pidf yields a better imposition of no slip boundary conditions on the ib surfaces for low reynolds number with a fairly larger time step than ib methods with different direct forcing schemes due to the implicit treatment of the diffusion term for determining the ib forcing finally we demonstrate the robustness of the present pidf scheme by numerical simulations of flow around a circular array of cylinders flows around a falling sphere and two sedimenting spheres in gravity

a discrete forcing immersed boundary method for the fluid structure interaction of an elastic slender body we present an immersed boundary ib method for the simulation of flow around an elastic slender body the present method is based on the discrete forcing ib method for a stationary rigid body proposed by kim kim and choi 2001 25 the discrete forcing approach is used to relieve the limitation on the computational time step size the incompressible navier stokes equations are implicitly coupled with the dynamic equation for an elastic slender body motion the first is solved in the eulerian coordinate and the latter is described in the lagrangian coordinate the elastic slender body is modeled as a thin and flexible solid and is segmented by finite number of thin blocks each block is moved by external and internal forces such as the hydrodynamic elastic and buoyancy forces where the hydrodynamic force is obtained directly from the discrete forcing used in the ib method all the spatial derivative terms are discretized with the second order central difference scheme the present method is applied to three different fluid structure interaction problems flows around a flexible filament a flapping flag in a free stream and a flexible flapping wing in normal hovering respectively computations are performed at maximum cfl numbers of 0 75 1 the results obtained agree very well with those from previous studies

combined immersed boundary finite difference methods for three dimensional complex flow simulations a second order accurate highly efficient method is developed for simulating unsteady three dimensional incompressible flows in complex geometries this is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given surface not coinciding with the computational grid the governing equations therefore can be discretized and solved on a regular mesh thus retaining the advantages and the efficiency of the standard solution procedures two different forcings are tested showing that while the quality of the results is essentially the same in both cases the efficiency of the calculation strongly depends on the particular expression a major issue is the interpolation of the forcing over the grid that determines the accuracy of the scheme this ranges from zeroth order for the most commonly used interpolations up to second order for an ad hoc velocity interpolation the present scheme has been used to simulate several flows whose results have been validated by experiments and other results available in the literature finally in the last example we show the flow inside an ic piston cylinder assembly at high reynolds number to our knowledge this is the first example in which the immersed boundary technique is applied to a full three dimensional complex flow with moving boundaries and with a reynolds number high enough to require a subgrid scale turbulence model

weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings in this paper we introduce an iterative process for finding the common element of the set of common fixed points of a countable family of nonexpansive mappings and the set of solutions of the variational inequality problem for an a inverse strongly monotone mapping we obtain a weak convergence theorem for a sequence generated by this process moreover we apply our result to the problem for finding a common element of the set of equilibrium problems and the set of solutions of the variational inequality problem of a monotone mapping