an upper bound of the bezout number for piecewise algebraic curves a piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function based on the discussion of the number of the zeros of homogeneous trigonometric splines with different smoothness and the common points of two piecewise algebraic curves over a star partition a better upper bound of bezout number of two piecewise algebraic curves over arbitrary triangulation is found moreover upper bounds of the bezout number b n m r n r for piecewise algebraic curves over several special partitions such as rectangular partition type 1 triangulation and type 2 triangulation are obtained

dimensions of spline spaces over t meshes a t mesh is basically a rectangular grid that allows t junctions in this paper we propose a method based on bezier nets to calculate the dimension of a spline function space over a t mesh when the order of the smoothness is less than half of the degree of the spline functions a dimension formula is derived which involves only the topological quantities of the t mesh the construction of basis functions is briefly discussed furthermore the dimension formulae for t meshes after mesh operations such as edge insertion and mesh merging are also obtained

cayley bacharach theorems and conjectures a theorem of pappus of alexandria proved in the fourth century a d began a long development in algebraic geometry in its changing expressions one can see reflected the changing concerns of the field from synthetic geometry to projective plane curves to riemann surfaces to the modern development of schemes and duality we survey this development historically and use it to motivate a brief treatment of a part of duality theory we then explain one of the modern developments arising from it a series of conjectures about the linear conditions imposed by a set of points in projective space on the forms that vanish on them we give a proof of the conjectures in a new

lattice boltzmann simulation of pattern formation under cross diffusion in this paper a lattice boltzmann model for cross reaction diffusion system is proposed in order to recover the cross diffusion equations by lattice boltzmann method we transform the equations into a complex reaction diffusion equation this complex reaction diffusion equation is recovered with higher order accuracy of the truncation error based on this model two typical reaction diffusion systems with cross diffusion predator prey system and inhomogeneous brusselator model are simulated the numerical results show that this model can be used to simulate the cross reaction diffusion systems

convergence of a generalized pulse spectrum technique gpst for inverse problems of 1 d diffusion equations in space time domain etude de la convergence d un algorithme iteratif de resolution des problemes inverses pour les equations de diffusion a une dimension dans un domaine d espace temps

cayley bacharach theorem of piecewise algebraic curves the piecewise algebraic curve determined by a bivariate spline function is a generalization of the classical algebraic curve in this paper by using bezout s theorem and nother type theorem of piecewise algebraic curves the cayley bacharach theorem and hilbert function of c0 piecewise algebraic curves are presented

a numerical algorithm for solving inverse problems of two dimensional wave equations abstract the new iterative numerical algorithm of the pulse spectrum technique pst is extended and developed to solve inverse problems of two dimensional linear wave equations it has the practical advantages of having the necessary data measured on a portion of the boundary only and no geometric limitation on the testing objects numerical simulations on several simple examples are carried out to test the feasibility and to study the general characteristics of this technique without the real measurement data it is found that pst does give excellent results even with a very coarse computational grid and it is as robust as in the one dimensional case

lagrange interpolation by bivariate splines on cross cut partitions the piecewise algebraic curve is a generalization of the classical algebraic curve in this paper we give the nother type theorem of piecewise algebraic curves on cross cut partitions by using interpolation along a piecewise algebraic curve we also present a method to construct lagrange interpolation sets for bivariate spline spaces on cross cut partitions

generalized pulse spectrum technique the generalized pulse spectrum technique gpst is a versatile and efficient iterative numerical algorithm for solving multiparameter inverse problems of a system of nonlinear partial differential equations here it is used to determine bulk modulus shear modulus and density variation from seismic data numerical simulations of simple but nontrivial examples are carried out on coarse uniform grids and adaptive grids to test and to study the general characteristics of gpst without the real measurements seismograms gpst does give reasonably good results it can be made more efficient by implementing parallelism automatic adaptive grids and methods of domain decomposition into the numerical algorithm

a widely convergent generalized pulse spectrum technique for the inversion of two dimensional acoustic wave equation an algorithm of the generalized pulse spectrum technique gpst is used for the inverse problems of two dimensional acoustic wave equation and a skill of saving computational cost is designed in choosing regularizing parameters to overcome the defect of the local convergence of the gpst a widely convergent gpst wcgpst is constructed numerical simulations are carried out to test the feasibility and to study the general characteristics of the wcgpst it is found that the wcgpst is not only as robust as the standard gpst but also possessing widely convergent region

finite volume element approximation of an inhomogeneous brusselator model with cross diffusion this paper is concerned with the study of pattern formation for an inhomogeneous brusselator model with cross diffusion modeling an autocatalytic chemical reaction taking place in a three dimensional domain for the spatial discretization of the problem we develop a novel finite volume element fve method associated to a piecewise linear finite element approximation of the cross diffusion system we study the main properties of the unique equilibrium of the related dynamical system a rigorous linear stability analysis around the spatially homogeneous steady state is provided and we address in detail the formation of turing patterns driven by the cross diffusion effect in addition we focus on the spatial accuracy of the fve method and a series of numerical simulations confirm the expected behavior of the solutions in particular we show that depending on the spatial dimension the magnitude of the cross diffusion influences the selection of spatial patterns

the maximum number and its distribution of singular points for parametric piecewise algebraic curves abstract the piecewise algebraic curve as the zero set of a bivariate spline function is a generalization of the classical algebraic curve based on the previous method presented by lai et al 2009 we show that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial systems in this article we give a method to compute the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve this method also produces necessary and sufficient conditions of its parameters must be satisfied an illustrated example shows that our method is flexible

multivariate weak spline function space in this paper b net method for studying multivariate weak spline is discussed and the dimension of w21 i1 1 is presented several b splines on type 1 triangulation are also obtained

the correspondence between multivariate spline ideals and piecewise algebraic varieties as a piecewise polynomial with a certain smoothness the spline plays an important role in computational geometry the algebraic variety is the most important subject in classical algebraic geometry as the zero set of multivariate splines the piecewise algebraic variety is a generalization of the algebraic variety in this paper the correspondence between piecewise algebraic varieties and spline ideals is discussed furthermore hilbert s nullstellensatz for the piecewise algebraic variety is also studied

a wavelet multiscale homotopy method for the inverse problem of two dimensional acoustic wave equation abstract by introducing the wavelet multiscale method and the homotopy method to the inversion process for the velocity estimation problem of two dimensional acoustic wave equation in seismic prospecting a joint inversion method wavelet multiscale homotopy method is designed which is stable globally convergent and has the ability of noise suppression the results of numerical simulations and noise suppression tests indicate our method s effectiveness

a widely convergent generalized pulse spectrum technique for the coefficient inverse problem of differential equations abstract a new algorithm of the generalized pulse spectrum technique gpst is constructed for the solution of coefficient inverse problems of differential equations and a skill of saving computational cost is designed in choosing regularizing parameters to overcome the defect of the local convergence of the gpst a widely convergent gpst is illustrated numerical simulations are carried out for solving the inverse problem of 1 d seismic prospection to test the feasibility without real measurement data the computational results show the method s effectiveness

structure and dimension of multivariate spline space of lower degree on arbitrary triangulation in this paper we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules on the base of analyzing the algebraic and geometric results about singularity of s21 ms we build the structure of triangulation and give some useful geometric conditions such that s 1 space is singular and we obtain an algebraic condition which is necessary and sufficient for the singularity of s 1 spaces as well as their dimension formulae moreover the structure matrix of spline spaces over any given partition is defined which has been used to discuss the structure of s31 and s21 spaces over arbitrary triangulation and to prove the nonsingularity of s31 spaces this partially settles a conjecture on the singularity of spline spaces in wang et al multivariate spline and its applications kluwer press dordrecht 2002 academic press beijing 1994 in chinese meanwhile the dimension formulae of s31 s21 spaces and the dimension formulae of s 1 1 spaces are also given in this paper

recent researches on multivariate spline and piecewise algebraic variety the multivariate splines as piecewise polynomials have become useful tools for dealing with computational geometry computer graphics computer aided geometrical design and image processing it is well known that the classical algebraic variety in algebraic geometry is to study geometrical properties of the common intersection of surfaces represented by multivariate polynomials recently the surfaces are mainly represented by multivariate piecewise polynomials i e multivariate splines so the piecewise algebraic variety defined as the common intersection of surfaces represented by multivariate splines is a new topic in algebraic geometry moreover the piecewise algebraic variety will be also important in computational geometry computer graphics computer aided geometrical design and image processing the purpose of this paper is to introduce some recent researches on multivariate spline piecewise algebraic variety curve and their applications

turing pattern formation in the brusselator model with superdiffusion the effect of superdiffusion on pattern formation and pattern selection in the brusselator model is studied our linear stability analysis shows in particular that unlike the case of normal diffusion the turing instability can occur even when diffusion of the inhibitor is slower than that of the initiator a weakly nonlinear analysis yields a system of amplitude equations analysis of which predicts parameter regimes where hexagons stripes and their coexistence are expected numerical computations of the original brusselator model near the stability boundaries confirm the results of the analysis in addition further from the stability boundaries we find a regime of self replicating spots

an iterative numerical algorithm for solving multi parameter inverse problems of evolutional partial differential equations abstract the iterative numerical algorithm of the pulse spectrum technique pst is extended and developed to solve the multi parameter inverse problems of one dimensional evolutional partial differential equations wave equations or diffusion equations it has the practical advantages of universality economy of programing economy of data acquisition and economy of computing costs without the real measurement data numerical simulations are carried out only for the two parameter inverse problems of a one dimensional linear wave equation to test the feasibility and to study the general characteristics of the pst it is found that the pst does give excellent results and it is as robust as in the single parameter case

existence of spatial patterns in a predator prey model with self and cross diffusion in this work we investigate the spatiotemporal dynamics of reaction diffusion equations subject to cross diffusion in the frame of a two dimensional ratio dependent predator prey model the conditions for diffusion driven instability are obtained and the turing space in the parameters space is achieved moreover the criteria for local and global asymptotic stability of the unique positive homogeneous steady state without diffusion are discussed numerical simulations are carried out in order to validate the feasibility of the obtained analytical findings different types of spatial patterns through diffusion driven instability of the proposed model are portrayed and analysed lastly the paper finishes with an external discussion of biological relevance of the analysis regarding cross diffusion and pattern issues

multivariate spline and algebraic geometry abstract the purpose of this survey is to emphasize the special relationship between multivariate spline and algebraic geometry we will not only point out the algebraic geometric method of multivariate spline but also the algebraic geometric background and essence of multivariate spline especially we have made an introduction to the so called piecewise algebraic curve piecewise algebraic variety and some of their properties

three dimensional multi relaxation time mrt lattice boltzmann models for multiphase flow in this paper three dimensional 3d multi relaxation time mrt lattice boltzmann lb models for multiphase flow are presented in contrast to the bhatnagar gross krook bgk model a widely employed kinetic model in mrt models the rates of relaxation processes owing to collisions of particle populations may be independently adjusted as a result the mrt models offer a significant improvement in numerical stability of the lb method for simulating fluids with lower viscosities we show through the chapman enskog multiscale analysis that the continuum limit behavior of 3d mrt lb models corresponds to that of the macroscopic dynamical equations for multiphase flow we extend the 3d mrt lb models developed to represent multiphase flow with reduced compressibility effects the multiphase models are evaluated by verifying the laplace young relation for static drops and the frequency of oscillations of drops the results show satisfactory agreement with available data and significant gains in numerical stability

roughness and cavitations effects on electro osmotic flows in rough microchannels using the lattice poisson boltzmann methods this paper investigates the effects of roughness and cavitations in microchannels on the electro osmotic flow behaviors using the lattice poisson boltzmann methods which combined one lattice evolution method for solving the non linear poisson boltzmann equation for electric potential distribution with the other lattice evolution method for solving the navier stokes equations for fluid flow the boundary conditions are correctly treated for consistency between the both the results show that for the electro osmotic flows in homogeneously charged rough channels the flow rate does not vary with the roughness height or the interval space monotonically the flow rate varies slightly with the roughness height or even increases a little when the roughness is very small and then decreases when the roughness height is larger than 5 channel width the flow rate decreases first and then increase with the roughness interval space an interval space at twice roughness width makes the flow rate minimum for the heterogeneously charged rough channel the flow rate increases with the roughness surface potential at a super linear rate for the electro osmotic flows in microchannels with cavitations the flow rate change little with the cavitations depth when the depth value is very low and decreases sharply when the depth is greater than 3 channel width the flow rate trends to be a constant when the cavitations are very deep the flow rate decreases with the cavitations width but increases with the cavitations interval

multivariate spline space over cross cut partition in this paper we mainly study the generator bases of the prime module determined by the global conformality condition of multivariate spline over star cross cut partition the results can help us to get the dimension and construct the spline bases for multivariate spline space over cross cut partition quasi cross cut partition and super cross cut partition

lattice boltzmann method for non newtonian power law fluids we study an ad hoc extension of the lattice boltzmann method that allows the simulation of non newtonian fluids described by generalized newtonian models we extensively test the accuracy of the method for the case of shear thinning and shear thickening truncated power law fluids in the parallel plate geometry and show that the relative error compared to analytical solutions decays approximately linear with the lattice resolution finally we also tested the method in the reentrant flow geometry in which the shear rate is no longer a scalar and the presence of two singular points requires high accuracy in order to obtain satisfactory resolution in the local stress near these points in this geometry we also found excellent agreement with the solutions obtained by standard finite element methods and the agreement improves with higher lattice resolution