the motivic nearby fiber and degeneration of stable rationality we prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities our proof is based on motivic specialization techniques and the criterion of larsen and lunts for stable rationality in the grothendieck ring of varieties

iterated vanishing cycles convolution and a motivic analogue of a conjecture of steenbrink we prove a motivic analogue of steenbrink s conjecture 25 conjecture 2 2 on the hodge spectrum proved by m saito in 21 to achieve this we construct and compute motivic iterated vanishing cycles associated with two functions we are also led to introduce a more general version of the convolution operator appearing in the motivic thom sebastiani formula throughout the article we use the framework of relative equivariant grothendieck rings of varieties endowed with an algebraic torus action

on the hamiltonian structure of ermakov systems a canonical hamiltonian formalism is derived for a class of ermakov systems specified by several different frequency functions this class of systems comprises all known cases of hamiltonian ermakov systems and can always be reduced to quadratures the hamiltonian structure is explored to find exact solutions for the calogero system and for a non central potential with dynamic symmetry some generalizations of these systems possessing exact solutions are also identified and solved

eigenvalue estimates of an indefinite block triangular preconditioner for saddle point problems in this paper an indefinite block triangular preconditioner for symmetric saddle point problems is discussed the bounds for all the real eigenvalues of the preconditioned matrix are provided and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided the corresponding theoretical results of paper by bergamaschi 2012 are improved numerical experiments are reported to confirm our theoretical results

quantum mechanics as a multidimensional ermakov theory i time independent systems presents a multidimensional ermakov theory applicable to both separable and nonseparable time independent quantum mechanical systems the separability of the system being determined by the separability of the wavefunction in configuration space a consequence of the theory is the existence of a new exact invariant for multidimensional quantum systems in one dimension the author shows that this new invariant reduces to the well known ermakov lewis invariant two applications of the theory are given in the first case a new exact invariant is obtained for planar optical and particle channelling systems in the second case he obtains the ermakov invariants and electron ray path equations for the hydrogen atom

deformations of the riccati equation by using miura type transformations using some different miura type transformations a c integrable ordinary differential equation the riccati equation is deformed to some different s integrable models such as the 1 1 dimensional and 2 1 dimensional sinh gordon equations and mikhailov dodd bullough equations

new backlund transformations of the 2 1 dimensional bogoyavlenskii equation via localization of residual symmetries abstract the residual symmetry of the 2 1 dimensional bogoyavlenskii equation is obtained and localized to a lie point symmetry by introducing new dependent variables to enlarge the system then the corresponding finite transformation is obtained by using lie s first theorem furthermore by introducing more dependent variables the linear superposition of arbitrary number of residual symmetries is also localized and the corresponding finite transformations which are just n th backlund transformations of the 2 1 dimensional bogoyavlenskii equation are obtained in determinants form with some concrete solutions explicitly given

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

similarity solutions of the kadomtsev petviashvili equation using the direct method introduced by clarkson and kruskal ck 1989 recently the author obtains the similarity reductions of the kadomtsev petviashvili equation kpe some of them possess the same painleve properties as that of the boussinesq equation be obtained by ck however the similarity solutions of kpe may include some more than three arbitrary functions of time there exist some other types of similarity reductions of the kpe different from the similarity reductions of the be obtained by ck

robust preconditioners for optimal control with time periodic parabolic equation abstract based on a new approximation of a schur complement matrix and some preconditioning techniques considered in earlier papers three robust preconditioners are presented for solving the kkt systems arising from discretizing optimal control problem with time periodic parabolic equation the eigenvalue bounds of the preconditioned matrices and some new theoretical results are obtained moreover selection of quasi optimal parameter is studied and an explicit expression of the quasi optimal parameter is given numerical examples are given to illustrate the effectiveness of the three proposed optimized preconditioners

lie symmetries nonlinear equations of motion and new ermakov systems it is shown that a lagrangian of the form l 1 1 rho 2 omega 2 t rho 2 g t f k t rho said to be in factored form yields an equation of motion that is equivalent to the most general equation derivable via noether s theorem from the unfactored lagrangian l 1 2 rho 2 p rho t in view of this equivalence the theory of extended lie groups is applied to the factored nonlinear equation of motion p omega 2 t rho g t f k t rho to obtain its lie symmetries the latter are obtained when g t and k t initially arbitrary are determined in terms of a function x t which satisfies the auxiliary equation x omega 2 t x k x3 it is then possible with the auxiliary equation and the equation of motion to form a coupled pair of nonlinear equations an ermakov system whose first integral is not invariant under the action of the symmetry group in contrast to previous ermakov systems

explicit solutions of the generalized kdv equations with higher order nonlinearity by constructing auxiliary functions and equations we find some new explicit solutions of the generalized kdv equations with higher order nonlinearity in particular we get the peaked solitary wave solutions of the generalized kdv equations

solutions of a 2 1 dimensional dispersive long wave equation a special type of multisoliton solution with a particular dispersion relation is obtained for wu zhang equation which describes 2 1 dimensional dispersive long waves by the standard weiss tabor carnvale painleve truncation expansion using a nonstandard truncation of a modified conte s invariant painleve expansion two different types of soliton solutions without any dispersive relation is found two types of periodic wave solutions expressed by jacobi elliptic functions are found by the truncations of a special extended painleve expansion the soliton solutions are special cases of the corresponding periodic solutions

nonlinear superposition higher order nonlinear equations and classical linear invariants we find a class of nonlinear ordinary differential equations in of any order m 2 whose solutions are given by the nonlinear superposition law t x t r d t dt where dmr d m f r r and x t satisfies a special self adjoint linear equation of order m the coefficients of the self adjoint equation in x which are identical to those of the nonlinear equations in can be deduced from the well known invariants of the classical theory of linear differential equations

on motivic zeta functions and the motivic nearby fiber we collect some properties of the motivic zeta functions and the motivic nearby fiber defined by denef and loeser in particular we calculate the relative dual of the motivic nearby fiber we give a candidate for a nearby cycle morphism on the level of grothendieck groups of varieties using the motivic nearby fiber

ermakov systems velocity dependent potentials and nonlinear superposition we derive new ermakov systems with velocity dependent potentials the extended ermakov system presented contains all known one dimensional cases and many new systems these ermakov systems lead to a nonlinear superposition law for the solutions

equivariant motivic integration and proof of the integral identity conjecture for regular functions we develop denef loeser s motivic integration to an equivariant version and use it to prove the full integral identity conjecture for regular functions in comparison with hartmann s work the equivariant grothendieck ring defined in this article is more elementary and it yields the application to the conjecture

the modified kdv equation with variable coefficients exact uni bi variable travelling wave like solutions in this paper the modified korteweg de vries mkdv equation with variable coefficients vc mkdv equation is investigated via two kinds of approaches and symbolic computation on the one hand we firstly reduce the vc mkdv equation to a second order nonlinear nonhomogeneous ode using travelling wave like similarity transformation and then we obtain its many types of exact fractional solutions with one travelling wave like variable by applying some fractional transformations to the obtained nonlinear ode on the other hand we reduce the vc mkdv equation to two nonlinear pdes with variable coefficients using the anti tangent and anti hypertangent function transformations respectively and then we given its many types of exact solutions with two different travelling wave like variables by studying the obtained nonlinear pde with variable coefficients c 2008 elsevier inc all rights reserved

coupled kdv equations derived from two layer fluids some types of coupled korteweg de vries kdv equations are derived from a two layer fluid system in the derivation procedure an unreasonable y average trick usually adopted in the literature is removed the derived models are classified by means of the painleve test three types of function and multiple soliton solutions of the models are explicitly given via the exact solutions of the usual kdv equation it is also discovered that a non painleve integrable coupled kdv system can have multiple soliton solutions

an iteration method for solving the linear system ax b abstract in this paper based on a convergence splitting of the matrix a we present an inner outer iteration method for solving the linear system a x b we analyze the overall convergence of this method without any other restriction on its parameters moreover we give the accelerated inner outer iteration method and discuss how to apply the inner outer iterations as a preconditioner for the krylov subspace methods the inner outer iteration method can also be used for the solution of a x b c finally several numerical examples are given to validate the performance of our proposed algorithms

many new kinds exact solutions to 2 1 dimensional burgers equation and klein gordon equation used a new method with symbolic computation abstract in this paper we obtained many new exact solutions including 2 soliton solutions complexiton solutions which possess various combination of trigonometric periodic hyperbolic function jacobi elliptic function and rational function solutions periodic solutions and rational solutions to 2 1 dimensional burgers equations and cubic klein gordon equation by introducing two sub equations with different variables we should point out that this method could be extended in many aspect if we introduce n sub equations with different variables for solving nonlinear partial differential equations we could get n soliton solutions to nonlinear partial differential equations

an efficient preconditioned variant of the pss preconditioner for generalized saddle point problems abstract for the generalized saddle point problems from the stokes equations we propose an efficient preconditioned variant of the positive semidefinite and skew hermitian pvpss preconditioner the new preconditioner is established by adopting matrix preconditioning strategy and relaxation technique for the pss one derived by pan et al 2006 compared with the pss one the pvpss preconditioner is much closer to the coefficient matrix and easier to be implemented if proper preconditioning matrices are adopted we prove the convergence of the pvpss iteration method under some restrictions and discuss the spectral properties of the pvpss preconditioned matrix meanwhile the implementation and a practical way to choose the parameter of the pvpss preconditioner are discussed comparisons between the pvpss preconditioner and some existing ones are also given numerical experiments are carried out to illustrate that the proposed preconditioner is effective for the generalized saddle point problems and outperforms several other commonly used ones

analysis on block diagonal and triangular preconditioners for a pml system of an electromagnetic scattering problem abstract we shall propose several block triangular preconditioners for a pml system of an electromagnetic wave scattering problem and analyze the spectral behavior of the preconditioned systems when the pml system is discretized by edge element methods it results in a discrete system with its stiffness matrix being complex symmetric but indefinite which can be formulated into a real symmetric but indefinite saddle point system in order to preserve the symmetry of the coefficient matrix we present block triangular preconditioners with two sided preconditioning for the discrete pml system we will estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively on the other hand one may also like to apply some iteration methods for nonsymmetric linear systems in applications although the discrete systems are symmetric to this end we propose a block triangular preconditioner to precondition the systems only from one side and analyze the spectrum of the preconditioned systems in addition we have also established a spectral estimate of the preconditioned system by an effective preconditioner that was recently developed in literature numerical experiments are presented to demonstrate the effectiveness and robustness of these new preconditioners and our theoretical predictions on the spectral bounds of the preconditioned systems

an inexact relaxed dpss preconditioner for saddle point problem based on the relaxed deteriorated positive definite and skew hermitian splitting dpss preconditioner in this paper we proposed a class of relaxed deteriorated positive definite and skew hermitian splitting rdpss preconditioner for solving the saddle point problem the proposed rdpss preconditioner is a technical modification of the deteriorated positive definite and skew hermitian splitting dpss preconditioner 36 the pss preconditioner is a straightforward application of the positive definite and skew hermitian splitting pss iteration method for solving non hermitian positive definite linear systems initially established by bai et al 37 numerical results have shown that the proposed rdpss preconditioner is advantageous over the existing dpss preconditioner

a variant of the hss preconditioner for complex symmetric indefinite linear systems abstract using the equivalent block two by two real linear systems we establish a new variant of the hermitian and skew hermitian splitting hss preconditioner for a class of complex symmetric indefinite linear systems the new preconditioner is not only a better approximation to the block two by two real coefficient matrix than the well known hss preconditioner but also resulting in an unconditional convergent fixed point iteration the quasi optimal parameter which minimizes an upper bound of the spectral radius of the iteration matrix is analyzed eigen properties and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed finally two numerical examples are provided to show the efficiency of the new preconditioner

compactifications of subvarieties of tori we study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non archimedean amoebas these compactifications have many nice properties for example any k boundary divisors intersect in codimension k we consider some examples including m 0 n subset overline m 0 n and more generally log canonical models of complements of hyperplane arrangements and compact quotients of grassmannians by a maximal torus

sharp error bounds of some krylov subspace methods for non hermitian linear systems1project 19601036 supported by the national natural science foundation of china 1 the convergence of the krylov subspace methods e g full orthogonal method fom and generalized minimal residual method gmres etc for solving large non hermitian linear systems is studied in a unified and detailed way when the coefficient matrix is defective in particular when its spectrum lies in the open right left half plane or is on the real axis related theoretical error bounds are established which reveal some intrinsic relationships between the convergence properties and the eigen characteristics of the coefficient matrix these results not only generalize all the known ones for the diagonalizable matrices in the literature but also sharp the corresponding estimates in jia acta mathematica sinica new series 14 1998 507 518

fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations abstract the discretizations of two and three dimensional spatial fractional diffusion equations with the shifted finite difference formulas of the grunwald letnikov type can result in discrete linear systems whose coefficient matrices are of the form d t where d is a nonnegative diagonal matrix and t is a block toeplitz with toeplitz block matrix or a block toeplitz with each block being block toeplitz with toeplitz block matrix for these discrete spatial fractional diffusion matrices we construct diagonal and block circulant with circulant block splitting preconditioner for the two dimensional case and diagonal and block circulant with each block being block circulant with circulant block splitting preconditioner for the three dimensional case to further accelerate the convergence rates of krylov subspace iteration methods and we analyze the eigenvalue distributions for the corresponding preconditioned matrices theoretical results show that except for a small number of outliners the eigenvalues of the preconditioned matrices are located within a complex disk centered at 1 with the radius being exactly less than 1 and numerical experiments demonstrate that these structured preconditioners can significantly improve the convergence behavior of the krylov subspace iteration methods moreover this approach is superior to the geometric multigrid method and the preconditioned conjugate gradient methods incorporated with the approximate inverse circulant plus diagonal preconditioners in both iteration counts and computing times

proofs of the integral identity conjecture over algebraically closed fields recently it has become well known that the conjectural integral identity is of crucial importance in the motivic donaldson thomas invariants theory for noncommutative calabi yau threefolds the purpose of this article is to consider different versions of the identity for regular functions and formal functions and to give them the positive answer for the algebraically closed ground fields technically the result on the motivic milnor fiber by hrushovski loeser using hrushovski kazhdan s motivic integration and nicaise s computations on motivic integrals on special formal schemes are main tools

symmetries and exact solutions of the sharma tass olver equation abstract the simple symmetry reduction procedure is repeatedly used to obtain infinitely many symmetries as well as exact solutions of the sharma tass olver sto equation some sets of exact solutions such as singular rational complexitons and both analytical and singular rational negaton positons with new soliton fission and fusion phenomena are obtained

a class of efficient parameterized shift splitting preconditioners for block two by two linear systems abstract based on the effective constructing an equivalent block two by two linear systems we establish a class of efficient parameterized shift splitting eps preconditioners for solving block two by two linear systems theoretical analysis shows that the eps iteration method is unconditionally convergent and then we study the spectral properties of the corresponding preconditioned matrix by removing one of the shift term of the eps preconditioners two classes of local eps preconditioners are presented and the eigenvalues distribution of the preconditioned matrices are also studied numerical experiments are provided to demonstrate the feasibility and effectiveness of the proposed preconditioners

interactions among periodic waves and solitary waves of the n 1 dimensional sine gordon field exact solutions of the n 1 dimensional sine gordon field equation are studied with help of those of the cubic nonlinear klein gordon fields the mapping relations among the sine gordon field equation and the cubic nonlinear klein gordon fields are pure algebraic by solving the cubic nonlinear klein gordon equations many new types of exact explicit solutions such as the periodic periodic interaction waves periodic kink interaction waves periodic perturbed snake shape solitary waves etc are displayed both analytically and graphically

on 2 1 dimensional ermakov systems ermakov type systems in 2 1 dimensions are introduced multiwave solutions of a 2 1 dimensional pinney equation and a modulated 2 1 dimensional sine gordon equation are thereby constructed

rational ermakov systems of fuchsian type studies a subclass of a class of systems of coupled nonlinear oscillators that have well behaved singularities which can be studied using an exact linearization this allows the author to classify such systems and to isolate those with special properties

the generalized modified shift splitting preconditioners for nonsymmetric saddle point problems for a nonsymmetric saddle point problem the modified shift splitting mss preconditioner has been proposed by zhou et al by replacing the parameter in 2 2 block in the mss preconditioner by another parameter a generalized mss gmss preconditioner is established in this paper which results in a fixed point iteration called the gmss iteration method we provide the convergent and semi convergent analysis of the gmss iteration method which show that this method is convergence and semi convergence if the related parameters satisfy suitable restrictions meanwhile the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analyzed in detail finally numerical examples show that the gmss method is more feasibility and robustness than the mss uzawa hss and pu sts methods as a solver and the gmss preconditioner outperforms the gsor uzawa hss mss and lmss preconditioners for the gmres method for solving both the nonsingular and the singular saddle point problems with nonsymmetric positive definite and symmetric dominant 1 1 parts

pre symmetric approach and applications to indefinite non symmetric problems a class of pre symmetric minimal residual method to solve large scale system of linear algebraic equations with indefinite symmetrizable coefficient matrix is presented in this paper briefly called as psminres method the psminres method is established by first transforming the non symmetric system of equations into symmetric system of equations by the pre symmetric technique and then utilizing the minimal residual minres method to solve the symmetric system of equations and getting a new approximation to the original system of linear equations theoretical analysis and numerical computations show that the pre symmetric technique is feasible and behaviors of the psminres are superior to some krylov subspace methods which are usual adopted for solving general non symmetric linear systems such as cgs gmres etc

geometric criteria for tame ramification we prove an a campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field k as a first application we relate the orders of the tame monodromy eigenvalues on the l adic cohomology of a k curve to the geometry of a relatively minimal sncd model and we show that the semi stable reduction theorem and saito s criterion for cohomological tameness are immediate consequences of this result as a second application we compute the error term in the trace formula for smooth and proper k varieties we see that the validity of the trace formula would imply a partial generalization of saito s criterion to arbitrary dimension

motivations and realizations of krylov subspace methods for large sparse linear systems we briefly introduce typical and important direct and iterative methods for solving systems of linear equations concretely describe their fundamental characteristics in viewpoints of both theory and applications and clearly clarify the substantial differences among these methods in particular the motivations of searching the solution of a linear system in a krylov subspace are described and the algorithmic realizations of the generalized minimal residual gmres method are shown and several classes of state of the art algebraic preconditioners are briefly reviewed all this is useful for correctly deeply and completely understanding the application scopes theoretical properties and numerical behaviors of these methods and is also helpful in designing new methods for solving systems of linear equations

the relations among a special type of solutions in some d 1 dimensional nonlinear equations among a special type of solutions of some kinds of nonlinear equations in d 1 dimensions such as the cubic nonlinear klein gordon sine gordon double sine gordon ginzburg landau equations etc some mapping relations exist that allow one to find a solution of one system from a solution of another furthermore one can find some new solutions from a known one within the same system some new n kinklike solutions are also presented

a note on parameterized block triangular preconditioners for generalized saddle point problems in this paper based on the results presented by jiang et al m q jiang y cao l q yao on parameterized block triangular preconditioners for generalized saddle point problems appl math comput 216 2010 1777 1789 we consider an indefinite parameterized block triangular preconditioner for symmetric saddle point problems the eigenvalue bound of the preconditioned matrix is discussed some presented results are improved numerical experiments are given to confirm our theoretical results

parameterized generalized shift splitting preconditioners for nonsymmetric saddle point problems abstract to solve nonsymmetric saddle point problems the parameterized generalized shift splitting pgss preconditioner is presented and analyzed the corresponding pgss iteration method can be applied not only to the nonsingular saddle point problems but also to the singular ones the convergence and semi convergence of the pgss iteration method are discussed carefully meanwhile the spectral properties of the preconditioned matrix and the strategy of the choices of the parameters are given numerical experiments further demonstrate that the pgss iteration method and the pgss preconditioner are efficient and have better performance than some existing iteration methods and newly proposed preconditioners respectively for solving both the nonsingular and singular nonsymmetric saddle point problems

numerical study on incomplete orthogonal factorization preconditioners we design analyse and test a class of incomplete orthogonal factorization preconditioners constructed from givens rotations incorporating some dropping strategies and updating tricks for the solution of large sparse systems of linear equations comprehensive accounts about how the preconditioners are coded what storage is required and how the computation is executed for a given accuracy are presented a number of numerical experiments show that these preconditioners are competitive with standard incomplete triangular factorization preconditioners when they are applied to accelerate krylov subspace iteration methods such as gmres and bicgstab

analysis on inexact block diagonal preconditioners for elliptic pde constrained optimization problems abstract by using the galerkin finite element method the distributed control problems can be discretized into a saddle point problem with a coefficient matrix of block three by three which can be reduced to a linear system with lower order we first introduce a class of inexact block diagonal preconditioners and estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively based on the cholesky decomposition of the known matrices we also analyze a lower triangular preconditioner to accelerate the minimal residual method for the reduced linear system and discuss its real and complex eigenvalues respectively moreover these bounds do not rely on the regularization parameter and the eigenvalues of the matrices in the discrete system numerical experiments are also presented to demonstrate the effectiveness and robustness of the two new preconditioners

on preconditioned iterative methods for unsteady incompressible navier stokes equations when the implicit finite difference scheme and linearization technique in conjunction with the artificial compressibility method are employed to discretize the unsteady incompressible navier stokes equations in each pseudo time step a structured system of linear equations is obtained and needs to be solved numerically in this paper preconditioning techniques are applied to solve such linear systems numerical example is given to illustrate the effectiveness of the proposed method we also find that a combination of the linearization technique with the preconditioned gmres method is quite efficient for the implicit finite difference discretization of the incompressible navier stokes equations

motivic degree zero donaldson thomas invariants given a smooth complex threefold x we define the virtual motive operatorname hilb n x operatorname vir of the hilbert scheme of n points on x in the case when x is calabi yau operatorname hilb n x operatorname vir gives a motivic refinement of the n point degree zero donaldson thomas invariant of x the key example is x 3 where the hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety and its virtual motive is defined in terms of the denef loeser motivic nearby fiber a crucial technical result asserts that if a function is equivariant with respect to a suitable torus action its motivic nearby fiber is simply given by the motivic class of a general fiber this allows us to compute the generating function of the virtual motives operatorname hilb n mathbb c 3 operatorname vir via a direct computation involving the motivic class of the commuting variety we then give a formula for the generating function for arbitrary x as a motivic exponential generalizing known results in lower dimensions the weight polynomial specialization leads to a product formula in terms of deformed macmahon functions analogous to gottsche s formula for the poincare polynomials of the hilbert schemes of points on surfaces

higher order special self adjoint equations and particle dynamics we exhibit a remarkable connection between a hierarchy of higher order special self adjoint ordinary differential equations and the description of motion of a cluster of particles in classical mechanics the cluster is assumed to consist of equal mass particles all moving in one dimension in a perturbation schema based on the first order equation of motion of the center of mass point the time evolution of the moments of order m 1 is governed by the solution of a special self adjoint equation of order m a similar connection exists for the moments of a wave packet in quantum mechanics

classification scheme for two dimensional ermakov type systems and generalizations we describe a classification scheme for certain pairs of coupled nonlinear second order ordinary differential equations having a first integral which can be obtained by an ermakov type elimination procedure the classification produces a hierarchy of systems having different levels of generality different cases within the same level are distinguished by the integrability of a certain one form which is related to a nonlinear superposition law between the two equations the study of these one forms and the classification are made systematic by using the general reduction theorem of pfaffian forms the paper contains illustrations of the way previous examples fit into the classification scheme and a discussion of its generalization to higher dimensions