linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

hidden integrable hierarchies of akns type a method for constructing integrable hierarchies by restricting akns flows on manifolds of finite codimension in the space of independent parameters is provided two particular types of hierachies are characterized one is given by nonlinear differential equations with coordinate dependent coefficients and the other is related to schrodinger spectral problems with energy dependent potentials of even degree

zero sets of functions and hidden hierarchies of kdv type the zero sets of kdv functions are characterized in terms of the stratification of the infinite grassmannian it is shown that these sets are related to integrable hierarchies arising from schrodinger equations with energy dependent potentials

a uniform mesh finite difference method for a class of singular two point boundary value problems a new finite difference method based on uniform mesh is given for the weakly singular two point boundary value problem x alpha y f x y y 0 a y 1 b 0 alpha 1 under quite general conditions on f and f we show that our present method based on uniform mesh provides o h 2 convergent approximations for all alpha in 0 1 our method is based on one evaluation of f and for alpha 0 it reduces to the classical second order method for y f x y

a novel numerical approach and its convergence for numerical solution of nonlinear doubly singular boundary value problems this paper deals with two new recursive numerical schemes for the solution of a class of doubly singular two point boundary value problems with different types of boundary conditions we first transform the original problem into an equivalent integral equation to overcome the singular behaviour at the origin and then establish recursive schemes by employing homotopy perturbation method for the solution of the resultant equation we establish the convergence results of the new numerical methods four numerical examples are considered to demonstrate the efficiency and accuracy of the proposed methods the present methods are shown their advantage over some existing methods chawla and katti 1982 pandey and singh 2007 pandey and singh 2004 caglar et al 2009 kumar and aziz 2004 khuri and sayfy 2010 developed for the solution of singular two point boundary value problems moreover the new methods are used to obtain the approximate solutions of the singular boundary value problems arising in various physical problems

singular sector of the kadomtsev petviashvili hierarchy operators of nonzero index and associated integrable systems integrable hierarchies associated with the singular sector of the kadomtsev petviashvili kp hierarchy or equivalently with operators of nonzero index are studied they arise as the restriction of the standard kp hierarchy to submanifolds of finite codimension in the space of independent variables for higher index these hierarchies represent themselves as families of multidimensional equations with multidimensional constraints the dressing method is used to construct these hierarchies hidden korteweg de vries boussinesq and hidden gelfand dikii hierarchies are considered too

linear equations over noncommutative graded rings abstract we call a graded connected algebra r effectively coherent if for every linear equation over r with homogeneous coefficients of degrees at most d the degrees of generators of its module of solutions are bounded by some function d d for commutative polynomial rings this property has been established by hermann in 1926 we establish the same property for several classes of noncommutative algebras including the most common class of rings in noncommutative projective geometry that is strongly noetherian rings which includes noetherian pi algebras and sklyanin algebras we extensively study so called universally coherent algebras that is such that the function d d is bounded by 2d for d 0 for example finitely presented monomial algebras belong to this class as well as many algebras with finite groebner basis of relations

diagonal subalgebras of bigraded algebras and embeddings of blow ups of projective spaces let v be closed subscheme of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i defined by a homogeneous ideal i a k x 1 x n and let x be the n 1 fold obtained by blowing up inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i along v if one embeds x in some projective space one is led to consider the subalgebra k i e c of a for some positive integers c and e the aim of this paper is to study ring theoretic properties of k i e c this is achieved by developing a theory which enables us to describe the local cohomology of certain modules over generalized segre products of bigraded algebras these results are applied to the study of the cohen macaulay property of the homogeneous coordinate ring of the blow up of the projective space along a complete intersection we also study the koszul property of diagonal subalgebras of bigraded standard k algebras

the approach to the dispersionless kp hierarchy the dispersionless limit of the scalar nonlocal problem is derived it is given by a special class of nonlinear first order equations a quasiclassical version of the dressing method is presented it is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of beltrami type equations the universal whitham hierarchy and in particular the dispersionless kp hierarchy turn out to be rings of symmetries for the quasiclassical problem pacs number 02 30 uk

topological landau ginzburg theory with a rational potential and the dispersionless kp hierarchy based on the dispersionless kp dkp theory we study a topological landau ginzburg lg theory characterized by a rational potential writing the dkp hierarchy in a general form treating all the primaries in an equal basis we find that the hierarchy naturally includes the dispersionless continuous limit of toda hierarchy and its generalizations having a finite number of primaries several flat solutions of the topological lg theory are obtained in this formulation and are identified with those discussed by dubrovin we explicitly construct gravitational descendants for all the primary fields giving a residue formulae for the 3 point functions of the fields we show that these 3 point functions satisfy the topological recursion relation the string equation is obtained as the generalized hodograph solutions of the dkp hierarchy which show that all the gravitational effects to the constitutive equations 2 point functions can be renormalized into the coupling constants in the small phase space

on the convergence of finite difference methods for weakly regular singular boundary value problems the second order finite difference methods m 1 based on a non uniform mesh and m 2 based on an uniform mesh developed by chawla and katti finite difference methods and their convergence for a class of singular two point boundary value problems numer math 39 1982 341 350 for weakly regular singular boundary value problems p x y f x y 0

on the convergence of a finite difference method for a class of singular boundary value problems arising in physiology using chawla s identity bit 29 1989 566 a finite difference method based on uniform mesh is described for a class of singular boundary value problems p x y p x f x y 0 x 1 y 0 a y 1 y 1 or y 0 0 y 1 y 1 with p x xb0 g x b0 0 and it is shown that the method is of second order accuracy under quite general conditions on p x and f x y this work also extends the method developed by chawla et al bit 26 1986 326 for p x xb0 b0 1 to a general class of function p x xb0 g x b0 0 numerical examples for general function p x verify the order of the convergence of the method and two physiological problems have also been solved

a fourth order finite difference method based on uniform mesh for singular two point boundary value problems we present a new fourth order finite difference method based on uniform mesh for the weakly singular two point boundary value problem x y f x y 0 x 1 y 0 a y 1 b 0 1 our method provides o h4 convergent approximations for all 0 1 for 0 it reduces to the well known fourth order method of numerov for y x y

limit matrices for the toda flow and periodic flags for loop groups sato s theory of infinite dimensional grassmannians has been applied to explain the geometry of the k p equation s djkm it has been used as a tool to study blow up behaviors and to regularize the solutions near the blow up a vm2 the point is that realizing the k p flow as a holomorphic flow of planes enables one to follow what happens to the limiting planes as the equation in the original bad coordinates blows up the blow up behaviors are characterized by the various strata the orbit of planes visits in the grassmannian in this paper such ideas are applied to the n periodic toda flow on periodic jacobi matrices which translates into a flow on the space of n periodic flags of planes in the grassmannians indeed here the n periodic toda flow amounts to n coupled kp equations with special interactions between time flows u t how such matrices blow up has been studied in fi f1 ha a vm1 for arbitrary lie algebras and kac moody lie algebras whereas this paper focusses on regularizing the flow near the blow up locus that is on finding the boundary of isospectral sets if n periodic jacobi matrices

quasiclassical limit of coupled kdv equations riemann invariants and multi hamiltonian structure abstract we consider the quasiclassical limit of the first nontrivial flow in coupled kdv hierarchy written down in riemann invariants it assumes the extremely simple form r i k 1 n r k 2r i r x i i 1 n up to certain natural equivalence this turns out to be the only hydrodynamic system with n 1 compatible purely local hamiltonian structures

existence uniqueness results for a class of singular boundary value problems ii abstract in this paper we establish existence uniqueness of solution of a class of singular boundary value problem p x y x q x f x y for 0 x b and y 0 a 1 y b 1 y b 1 where p x satisfies i p x 0 in 0 b ii p x c 1 0 r and for some r b iii x p x p x is analytic in z z r and q x satisfies i q x 0 in 0 b ii q x l 1 0 b and for some r b iii x 2 q x p x is analytic in z z r with quite general conditions on f x y region for multiple solutions have also been determined

spline finite difference methods for singular two point boundary value problems in this paper we discuss the construction of a spline function for a class of singular two point boundary value problemx x u f x u u 0 a u 1 b 0 1 or 1 2 the boundary conditions may also be of the formu 0 0 u 1 b three point finite difference methods using the above splines are obtained for the solution of the boundary value problem these methods are of second order and are illustrated by four numerical examples

wavelet galerkin method for solving nonlinear singular boundary value problems arising in physiology abstract in this study wavelet galerkin method has been developed for solving nonlinear singular boundary value problems associated with physiology science the quasilinearization technique is applied to reduce the given nonlinear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using compactly supported cubic b spline wavelets and galerkin method these wavelets are used as testing and weighting functions the method is computationally attractive and applications are demonstrated through illustrative examples

the toda lattice dynkin diagrams singularities and abelian varieties the periodic l 1 particle toda lattices coresponding to extended dynkin diagrams have l 1 polynomial constants of the motion namely as many as there are dots in the dynkin diagram for most values of the constants their intersection defines as l dimensional affine invariant manifold which completes into a complex algebraic torus abelian variety by glueing on a divisor entirely specified by the extended dynkin diagram therefore the global geometry of the complex invariant tori such as polarization dimension of certain linear systems divisor equivalences etc is entirely given by the extended dynkin diagram in particular an explicit geometric description for the case of three particle toda lattices is given which describes the divisors their singularities and how the various divisors intersect

bihamiltonian hierarchies in 2d topological field theory at one loop approximation we compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2d topological field theory the bihamiltonian structure of the hierarchy is given by a classical w algebra we compute the central charge of this algebra we also express the generating function of elliptic gromov witten invariants via tau function of the isomonodromy deformation problem arising in the theory of wdvv equations of associativity

singular sector of the burgers hopf hierarchy and deformations of hyperelliptic curves we discuss the structure of shock singularities of the burgers hopf hierarchy it is shown that the set of singular solutions defines a stratification of the affine space of the flow parameters in the hierarchy the stratification is associated with the birkhoff decomposition of the grassmannian given by the set of linear spaces spanned by the hierarchy we then construct integrable hierarchy on each stratum and demonstrate that it describes a deformation of a hyperelliptic curve parametrizing the stratum the hierarchy is called the hidden burgers hopf hierarchy and we found the riemann invariant form and the hodograph solution

the numerical solution of non linear singular boundary value problems arising in physiology a class of non linear singular ordinary differential equations is solved by a new method based on non polynomial cubic spline we use the quesilinearization technique to reduce the given non linear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using non polynomial cubic spline approximation the resulting system of algebraic equations is solved by using a tri diagonal solver computational results are provided to demonstrate the viability of the new method

finite difference methods and their convergence for a class of singular two point boundary value problems we discuss the construction of three point finite difference approximations and their convergence for the class of singular two point boundary value problems x y f x y y 0 a y 1 b 0 1 we first establish a certain identity based on general non uniform mesh from which various methods can be derived to obtain a method having order two for all 0 1 we investigate three possibilities by employing an appropriate non uniform mesh over 0 1 we obtain a methodm1 based on just one evaluation off for uniform mesh we obtain two methodsm2 andm3 each based on three evaluations off for 0 m1 andm2 both reduce to the classical second order method based on one evaluation off these three methods are investigated theiro h2 convergence established and illustrated by numerical examples