a new conservation law constructed without using either lagrangians or hamiltonians a new conservation theorem is derived the conserved quantity is constructed in terms of a symmetry transformation vector of the equations of motion only without using either lagrangian or hamiltonian structures which may even fail to exist for the equations at hand one example and implications of the theorem on the structure of point symmetry transformations are presented

w symmetries of ito stochastic differential equations we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones

approximate symmetries and conservation laws for ito and stratonovich dynamical systems a new definition for the approximate symmetries of ito dynamical system is given determining systems of approximate symmetries for ito and stratonovich dynamical systems have been obtained it has been shown that approximate conservation laws can be found from the approximate symmetries of stochastic dynamical systems which do not arise from a hamiltonian the results have been applied to an example

block preconditioners with circulant blocks for general linear systems block preconditioner with circulant blocks bpcb has been used for solving linear systems with block toeplitz structure since 1992 r chan x jin a family of block preconditioners for block systems siam j sci statist comput 13 1992 1218 1235 in this new paper we use bpcbs to general linear systems with no block structure usually the bpcbs are constructed by partitioning a general matrix into a block matrix with blocks of the same size and then applying t chan s optimal circulant preconditioner t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 to each block these bpcbs can be viewed as a generalization of t chan s preconditioner it is well known that the optimal circulant preconditioner works well for solving some structured systems such as toeplitz systems by using the preconditioned conjugate gradient pcg method but it is usually not efficient for solving general linear systems unlike t chan s preconditioner bpcbs used here are efficient for solving some general linear systems by the pcg method several basic properties of bpcbs are studied the relations of the block partition with the cost per iteration and the convergence rate of the pcg method are discussed numerical tests are given to compare the cost of the pcg method with different bpcbs

the circulant operator in the banach algebra of matrices abstract we study an operator c which maps every n by n matrix a n to a circulant matrix c a n that minimizes the frobenius norm a n c n f over all n by n circulant matrices c n the circulant matrix c a n called the optimal circulant preconditioner has proved to be a good preconditioner for a general class of toeplitz systems in this paper we give different formulations of the operator discuss its algebraic and geometric properties and compute its operator norms in different banach algebras of matrices using these results we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of i c n 1 a n f over all nonsingular circulant matrices c n

random lie point symmetries of stochastic differential equations we study the invariance of stochastic differential equations under random diffeomorphisms and establish the determining equations for random lie point symmetries of stochastic differential equations both in ito and in stratonovich forms we also discuss relations with previous results in the literature

the group classification of a scalar stochastic differential equation lie point group classification of a scalar stochastic differential equation sde with one dimensional brownian motion is presented the admitted symmetry group can be zero one two or three dimensional if an equation admits a three dimensional symmetry group it can be transformed into the equation of brownian motion by a change of variables with non random time transformation such equations can be integrated by quadratures there are described drift coefficients for which sdes with constant diffusion coefficients admit two and three dimensional symmetry groups

the best circulant preconditioners for hermitian toeplitz systems ii the multiple zero case in 10 14 circulant type preconditioners have been proposed for ill conditioned hermitian toeplitz systems that are generated by nonnegative continuous functions with a zero of even order the proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions it was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems in 10 14 it was claimed that this result can be the case where the generating functions have multiple zeros the main aim of this paper is to give a complete convergence proof of the method in 10 14 for this class of generating functions

a note on preconditioned block toeplitz matrices in siam j sci statist comput 13 1992 pp 948 966 ku and kuo proposed and analysed a block circulant preconditioner r mn for solving a family of block toeplitz systems t mn v b for a special class of block matrices called the quadrantally symmetric toeplitz matrices they proved that the eigenvalues of r mn 1 t mn are clustered around one except at most o m n outliers with t mn generated by a two dimensional rational function the superior convergence rate of the preconditioned conjugate gradient pcg method is explained by the clustering property of the spectrum of r mn 1 t mn however in their analysis there is no discussion on the positive definiteness of the matrix t mn and the preconditioner r mn is assumed to be invertible in this paper we give some results on these two aspects under the assumptions in the above referenced paper we prove that if the generating function f x y of t mn is positive then t mn is posit

m preconditioner for m matrices linear systems with m matrices often appear in a wide variety of areas of biological physical and social sciences see a berman r plemmons nonnegative matrices in mathematical sciences siam press philadelphia 1994 in this paper we propose a preconditioner for solving the system with symmetric nonsingular m matrix by the preconditioned conjugate gradient pcg method we show that our proconditioner increases the convergence rate of the pcg method and reduces the operation cost numerical results are also given

the generalized superoptimal preconditioner abstract for any given n by n matrix a a specific circulant preconditioner t f a introduced by tyrtyshnikov e tyrtyshnikov optimal and super optimal circulant preconditioners siam j matrix anal appl 13 1992 459 473 is defined to be the solution of min c i c 1 a f over all n by n nonsingular circulant matrices c the preconditioner t f a called the superoptimal circulant preconditioner has been proved to be a good preconditioner for a large class of structured systems in this paper we study this preconditioner in the general case by using the moore penrose inverse we give a formula for the superoptimal preconditioner and discuss the stability properties of this preconditioner a spectral relation between the optimal and superoptimal preconditioned matrices in the general case is also given

a remark on symmetry of stochastic dynamical systems and their conserved quantities the symmetry properties of stochastic dynamical systems described by a stochastic differential equation of stratonovich type and related conserved quantities are discussed extending previous results by misawa new conserved quantities are given by applying symmetry operators to known conserved quantities some detailed examples are presented

lie point symmetries and stochastic differential equations we discuss lie point symmetries of stochastic ordinary differential equations and the interrelations between these and analogous symmetries of the associated fokker planck equation for the probability measure

cusp development in hele shaw flow with a free surface free surface flow in a hele shaw cell is considered it has been known for some time that when the fluid region is contracting a finite time blow up can occur in which a cusp forms in the free surface the solution does not exist beyond the time of blow up examples are presented of this blow up and also of a new process in which a cusp of a different type forms in the free surface but after which the solution continues to exist after a transformation of the dependent variable the pressure the new cusps are related to permissible singularities in the free boundary of the obstacle problem

on parametric gevrey asymptotics for some nonlinear initial value cauchy problems abstract we study a nonlinear initial value cauchy problem depending upon a complex perturbation parameter with vanishing initial data at complex time t 0 and whose coefficients depend analytically on t near the origin in c 2 and are bounded holomorphic on some horizontal strip in c w r t the space variable this problem is assumed to be non kowalevskian in time t therefore analytic solutions at t 0 cannot be expected in general nevertheless we are able to construct a family of actual holomorphic solutions defined on a common bounded open sector with vertex at 0 in time and on the given strip above in space when the complex parameter belongs to a suitably chosen set of open bounded sectors whose union form a covering of some neighborhood of 0 in c these solutions are achieved by means of laplace and fourier inverse transforms of some common depending function on c r analytic near the origin and with exponential growth on some unbounded sectors with appropriate bisecting directions in the first variable and exponential decay in the second when the perturbation parameter belongs to moreover these solutions satisfy the remarkable property that the difference between any two of them is exponentially flat for some integer order w r t with the help of the classical ramis sibuya theorem we obtain the existence of a formal series generally divergent in which is the common gevrey asymptotic expansion of the built up actual solutions considered above

long time existence for a slightly perturbed vortex sheet consider a flat two dimensional vortex sheet perturbed initially by a small analytic disturbance by a formal perturbation analysis moore derived an approximate differential equation for the evolution of the vortex sheet we present a simplified derivation of moore s approximate equation and analyze errors in the approximation the result is used to prove existence of smooth solutions for long time if the initial perturbation is of size and is analytic in a strip m existence of a smooth solution of birkhoff s equation is shown for time t k2p if is sufficiently small with 1 as 0 for the particular case of sinusoidal data of wave length and amplitude e moore s analysis and independent numerical results show singularity development at time tc log o log log our results prove existence for t log if is sufficiently small with k 1 as 0 thus our existence results are nearly optimal

a circulant preconditioner for the systems of lmf based ode codes in this paper a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations odes is investigated most ode codes based on implicit formulas at each integration step need the solution of one or more unsymmetric linear systems that are often large and sparse here the boundary value methods a class of implicit methods for the numerical integration of odes based on linear multistep formulas are considered more in detail for initial value problems theoretical and practical arguments are given to show that the block circulant preconditioner can give fast preconditioned iterations for various classes of differential problems moreover the p circulants a recently introduced circulant approximation for unsymmetric toeplitz matrices are shown to be more suitable sometimes than other circulant matrices for the underlying block preconditioner

existence and uniqueness for a class of nonlinear higher order partial differential equations in the complex plane we prove existence and uniqueness results for nonlinear third order partial differential equations of the form

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

the best circulant preconditioners for hermitian toeplitz systems in this paper we propose a new family of circulant preconditioners for ill conditioned hermitian toeplitz systems a x b the preconditioners are constructed by convolving the generating function f of a with the generalized jackson kernels for an n by n toeplitz matrix a the construction of the preconditioners requires only the entries of a and does not require the explicit knowledge of f when f is a nonnegative continuous function with a zero of order 2p the condition number of a is known to grow as o n2p we show however that our preconditioner is positive definite and the spectrum of the preconditioned matrix is uniformly bounded except for at most 2p 1 outliers moreover the smallest eigenvalue is uniformly bounded away from zero hence the conjugate gradient method when applied to solving the preconditioned system converges linearly the total complexity of solving the system is therefore of o n log n operations in the case when f is positive we show that the convergence is superlinear numerical results are included to illustrate the effectiveness of our new circulant preconditioners

resonance theory for schrodinger operators resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods a general theory is developed with new and weaker conditions allowing for perturbations of threshold eigenvalues and relaxed fermi golden rule the exponential decay rate of resonances is addressed its uniqueness in the time dependent picture is shown in certain cases the relation to the existence of meromorphic continuation of the properly weighted green s function to time dependent resonance is further elucidated by giving an equivalent time dependent asymptotic expansion of the solutions of the schrodinger equation

new conserved quantities derived from symmetry for stochastic dynamical systems recently the author has proposed an elementary theory of conserved quantities and symmetry for stochastic dynamical systems described by stochastic differential equations of stratonovich type within the framework a new method for deriving conserved quantities from symmetry is developed in a similar manner to that of hojman s work 1992 which gives a new conservation law constructed without using either lagrangians or hamiltonians for deterministic dynamical systems some examples of the conserved quantities obtained through the new method are given

a generalization of t chan s preconditioner abstract in this paper we propose a new preconditioner for solving linear systems by the preconditioned conjugate gradient pcg method the preconditioner can be thought of as a generalization of the well known t chan s preconditioner for hermitian positive definite matrix we observe that our preconditioner is also hermitian positive definite the operation cost and convergence of the pcg method are discussed numerical experiments have been performed on structured problems to show the competitiveness of this preconditioner

a survey and some extensions of t chan s preconditioner abstract for any given n by n matrix an t chan s circulant preconditioner cf an proposed by t chan t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 is defined to be the solution of min c n a n c n f over all n by n circulant matrices cn the cf an called the optimal circulant preconditioner by t chan has been proved to be a good preconditioner for a large class of structured systems a generalization of t chan s circulant preconditioner was given by huckle t huckle circulant and skew circulant matrices for solving toeplitz matrix problems siam j matrix anal appl 13 1992 767 777 which can be used for solving some general linear systems in this paper we review some old and develop some new properties of t chan s circulant preconditioner

a note on t chan s preconditioner abstract a matrix is said to be stable if the real part of all the eigenvalues are negative in this note for some class of matrices we show that t chan s preconditioner is stable by using this result we prove the invertibility of some preconditioners proposed in numerical odes recently

galilei invariant gauge symmetries in fokker planck dynamics with logarithmic diffusion and drift terms a tensorial approach to galilean invariance is utilized together with lie symmetries of differential equations in order to derive equations of fokker planck type containing a logarithmic diffusion tensor and drift term the formalism is based on the projection from an extended by one space like dimension minkowski manifold to the usual newtonian spacetime so that non relativistic models are described by manifestly covariant lagrangians in this paper we obtain the fokker planck equations from the euler lagrange equations with the extended manifold by using a specific choice of the gauge condition we work in 1 1 spacetime and carry out the analysis for both abelian and non abelian symmetries

on the definition of an admitted lie group for stochastic differential equations with multi brownian motion the definition of an admitted lie group of transformations for stochastic differential equations has been already presented for equations with one dimensional brownian motion the transformation of the dependent variables involves time as well and it has been proven that brownian motion is transformed to brownian motion in this paper we will discuss this concept for stochastic differential equations involving multi dimensional brownian motion and present applications to a variety of stochastic differential equations

a nonlinear approximation for vortex sheet evolution and singularity formation abstract the evolution of a vortex sheet in two dimensional incompressible inviscid flow is governed by the integro differential equation of birkhoff rott we derive a simple approximation for vortex sheet evolution consisting of a system of four first order differential equations this approximate system has the advantage of involving only local operators the errors in the approximation are shown to be relatively small even if the sheet has infinite curvature at a point for the approximate equations exact similarity solutions exhibiting singularity formation are constructed

existence uniqueness analyticity and borel summability for boussinesq equations through borel summation methods we analyze the boussinesq equations for coupled fluid velocity and temperature fields 1 ut u p u u ae2 f t u we prove that an equivalent system of integral equations in the borel variable p r dual to 1 t has a unique solution in a class of exponentially bounded functions implying the existence of a classical solution to 1 in a complex t region that includes a real positive time axis segment for analytic initial data and forcing it is shown that the solution is borel summable implying that formal series in powers of t is gevrey 1 asymptotic and within the time interval of existence the solution remains analytic with the same analyticity strip width as the initial data and forcing we also determine conditions on the integral equation solution that improve the estimate for existence time

some stability properties of t chan s preconditioner abstract a matrix is said to be stable if the real parts of all the eigenvalues are negative in this paper for any matrix a n we give some sufficient and necessary conditions for the stability of t chan s preconditioner c u a n

zero viscosity limit for analytic solutions of the navier stokes equation on a half space ii construction of the navier stokes solution this is the second of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in either 2d or 3d under the assumption of analytic initial data we construct solutions of navier stokes for a short time which is independent of the viscosity the navier stokes solution is constructed through a composite asymptotic expansion involving the solutions of the euler and prandtl equations which were constructed in the first paper plus an error term this shows that the navier stokes solution goes to an euler solution outside a boundary layer and to a solution of the prandtl equations within the boundary layer the error term is written as a sum of first order euler and prandtl corrections plus a further error term the equation for the error term is weakly nonlinear its linear part is the time dependent stokes equation this error equation is solved by inversion of the stokes equation through expressing the solution as a regular euler like part plus a boundary layer prandtl like part the main technical tool in this analysis is the abstract cauchy kowalewski theorem

lie point symmetries of stratonovich stochastic differential equations this paper considers lie point symmetries of stochastic differential equations sdes in stratonovich form first we derive the determining equations of the random symmetries which correspond to lie group transformations acting in the space of the independent variable time the dependent variables and the brownian motion then the determining equations of the random symmetries are used to obtain the determining equations of the deterministic symmetries which correspond to lie group transformations acting in the space of the independent variable and dependent variables in the case of the deterministic symmetries the transformation of the brownian motion is induced by the random time change it is proved that the corresponding ito and stratonovich sdes admit the same random symmetries and the same deterministic symmetries

symmetries of systems of stochastic differential equations with diffusion matrices of full rank lie point symmetries of a system of stochastic differential equations sdes with diffusion matrices of full rank are considered it is proved that the maximal dimension of a symmetry group admitted by a system of n sdes is n 2 in addition such systems cannot admit symmetry operators whose coefficients are proportional to a nonconstant coefficient of proportionality these results are applied to compute the lie group classification of a system of two sdes the classification is obtained with the help of non equivalent realizations of real lie algebras by fiber preserving vector fields in 1 2 variables possibilities of using symmetries for integration of sdes by quadratures are discussed

design and analysis of toeplitz preconditioners the solution of symmetric positive definite toeplitz systems ax b by the preconditioned conjugate gradient pcg method was recently proposed by strang 1986 and analyzed by chan 1989 and chan and strang 1989 the convergence rate of the pcg method depends on the choice of preconditioners for the given toeplitz matrices the authors present a general approach to the design of toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions all resulting preconditioners can therefore be inverted via various fast transform algorithms with o n log n operations for a wide class of problems the pcg method converges in a finite number of iterations independent of n so that the computational complexity for solving these toeplitz systems is o n log n

random lie symmetries of ito stochastic differential equations lie point symmetries of stochastic differential equations sdes which include transformations depending on the brownian motion are considered the corresponding lie group transformations acting in the space of the independent variable time the dependent variables and the brownian motion preserve the differential form of sdes and the properties of the brownian motion such symmetries are called random symmetries the focus of the paper is a definition of the random symmetries and their properties it is shown that the random symmetries form a lie algebra we also investigate relations between the random symmetries and the conserved quantities of sdes for several stochastic differential equations we find their random symmetries and compare them with the deterministic symmetries which correspond to lie group transformations in the space of the independent and dependent variables

circulant preconditioned wr bvm methods for ode systems we consider the solution of a system of ordinary differential equations odes by waveform relaxation wr iterations in conjunction with boundary value methods bvms the wr method is a continuous in time analogue of the stationary method and it iterates with functions in each wr iteration we use bvms to discretize systems of odes bvms are relatively new ode solvers based on linear multistep formulae in this paper we discuss the use of the generalized minimal residual gmres method with block circulant circulant block preconditioners for solving the linear systems arising from the application of bvms in each wr iteration these preconditioners are effective in speeding up the convergence rate of the gmres method numerical experiments are presented to illustrate the effectiveness of our methods

convergence analysis of superoptimal pcg algorithm for toeplitz systems with a fisher hartwig singularity recently lu and hurvich y lu c hurvich on the complexity of the preconditioned conjugate gradient algorithm for solving toeplitz systems with a fisher hartwig singularity siam j matrix anal appl 27 2005 638 653 used the preconditioned conjugate gradient method with the optimal circulant preconditioner proposed in chan t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 for solving the toeplitz system t n f x b where the generating function f is given by f 1 e i 2d h with d e 7 7 0 the function h is positive continuous on and differentiable on 0 in this paper we will use the superoptimal circulant preconditioner proposed by tyrtyshnikov e tyrtyshnikov optimal and superoptimal circulant preconditioners siam j matrix anal appl 13 1992 459 473 to solve the same problem when 0 d 1 2 our convergence analysis shows that the number of iterations is bounded by o log 3 n and therefore the complexity of our algorithm is o n log 4 n we notice that the numerical performance of superoptimal preconditioner is almost the same as that of the optimal preconditioner

borel summability of the heat equation with variable coefficients abstract we prove borel summability in nonsingular directions of solutions of partial differential equations of the form u t a z u z z where a z is a quartic polynomial and the initial condition is analytic in the special case a z z we obtain the detailed resurgent structure even in such a simple case the structure of the singular manifolds is quite intricate

on gevrey solutions of threefold singular nonlinear partial differential equations abstract we study gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain we deal with both fuchsian and irregular singularities and allow the presence of a singular perturbation parameter by means of the borel laplace summation method we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their gevrey asymptotic expansion in the perturbation parameter this result rests on the malgrange sibuya theorem and it requires to prove that the difference between two neighboring solutions is exponentially small what in this case involves an asymptotic estimate for a particular dirichlet like series

optimal preconditioners for functions of matrices abstract the optimal preconditioner c u a of a given matrix a was proposed in 1988 by t chan 6 since then it has been proved to be efficient for solving a large class of structured systems in this paper we construct the optimal preconditioners for different functions of matrices more precisely let f be a function of matrices from c n n to c n n given a c n n there are two possible optimal preconditioners for f a c u f a and f c u a in the paper we study properties of both c u f a and f c u a for different functions of matrices numerical experiments are given to illustrate the efficiency of the optimal preconditioners when they are used to solve f a x b

on maximal lie point symmetry groups admitted by scalar stochastic differential equations it is proved that the lie point symmetry group admitted by a scalar stochastic differential equation sde of order n 3 is at most n 2 dimensional this result supplements those for first and second order sdes maximal dimensionality can be used to carry out lie group classifications lie point group classification of the third order sdes is provided