time finite element methods a unified framework for numerical discretizations of odes abstract we present a unified framework for the numerical discretization of odes based on time finite element methods we relate time finite element methods to runge kutta methods with infinitely many stages by means of the corresponding numerical quadrature we establish the relation between time finite element methods and partitioned runge kutta methods we also provide order estimates and superconvergence of the corresponding numerical methods in use of the simplifying assumptions we apply time finite methods to hamiltonian systems and investigate the conservation of energy and symplectic structure for the resulting numerical discretizations for hamiltonian systems we also construct new classes of symplectic integrators by combining different time finite element methods and verify the results by performing some numerical experiments

geometric invariants for liaison of space curves let k be an algebraically closed field and let s k x x x2 x3 for a curve cc pi the hartshorne rao module m c bneie h p3 xc n is a graded s module of finite length and up to duals and shifts is a complete invariant of liaison cf 12 1 given a graded s module m z m of finite length the action of s p p 6 1 between two consecutive components i e ql s hom m m gives rise to a degeneracy locus which can be thought of as lying in psi p namely let i p les irk4 l r and let v v s where s max r i v 5 p3 or else v 0 these loci are isomo hism invariants and are preserved under duals and shifts for m n c these loci are related to the geometry of c itself the main philosophy that emerges section 2 is that the degeneracy locus generally corresponds to those planes h in p3 which meet c nongenerically either containing a component of c or having cn n impose an unusually small number of conditions on some plane curves on h we conclude section 2 by applying these ideas to derive some necessary conditions for m c to have components in negative degrees the remaining sections give various apphcations of the techniques introduced in section 2 first in section 3 we classify sets of skew lines up to liaison that is if c and c consist of t 2 3 and t skew lines respectively then we ask when c can be linked to c the answer in general is never but the situation changes signi cantly if c lies on a quadric surface cf theorem 3 1 a large part of the answer can be read from the degeneracy locus v

ideals with stable betti numbers componentwise linear ideals were introduced earlier to generalize the result that the stanley reisner ideal i of a simplicial complex has a linear resolution if and only if its alexander dual is cohen macaulay it turns out that i is componentwise linear if and only if is sequentially cohen macaulay in this paper we discuss betti number properties of componentwise linear ideals let i be a graded ideal of a polynomial ring s and gin i the generic initial ideal of i with respect to the reverse lexicographic term order on s our main result is that i and gin i admit the same graded betti numbers if and only if i is componentwise linear for the proof of this fact we describe some properties of the betti diagram of a generic initial ideal combinatorial implications for shifted complexes will also be discussed

surfaces in p4 with extremal general hyperplane section abstract optimal upper bounds for the cohomology groups of space curves have been derived recently curves attaining all these bounds are called extremal curves this note is a step to analyze the corresponding problems for surfaces we state optimal upper bounds for the second and third cohomology groups of surfaces in p 4 and show that surfaces attaining all these bounds exist and must have an extremal curve as general hyperplane section surprisingly all the first cohomology groups of such surfaces vanish it follows that an extremal curve does not lift to a locally cohen macaulay surface unless the curve is arithmetically cohen macaulay

formal energy of symplectic scheme for hamiltonian systems and its applications ii abstract in this paper transforming the symplecticity of a difference scheme into the closedness of some differential 1 form and using the poincare lemma on closed differential forms the author deduces directly the existence of formal energy as a power series in stepsize of any symplectic scheme for hamiltonian systems theorem 1 and then proves its non conservativity of the original energy remark 1 according to theorem 1 the revertible symplectic schemes are characterized corollary 1 and 2 theorem 2 gives the formulae for calculation of the first three terms of formal energy then a few examples are displayed for the composition of two symplectic schemes the author obtains the valuable formulae of the first three terms of its formal energy theorem 3 finally as an application of theorem 3 two examples are given

a characterization of energy preserving methods and the construction of parallel integrators for hamiltonian systems high order energy preserving methods for hamiltonian systems are presented for this aim an energy preserving condition of continuous stage runge kutta methods is proved order conditions are simplified and parallelizable conditions are also given the computational cost of our high order methods is comparable to that of the average vector field method of order two

on efficiency and duality for multiobjective variational control problems with f ρ convexity abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective variational control problems wolfe and mond weir type duals are formulated under the generalized f convexity on the functions involved weak and strong duality theorems are proved

sufficiency and duality for multiobjective variational control problems with g invexity antczak introduced vector valued g invex functions in 2009 which is a new class of generalized convex functions for differentiable multiobjective programming problems in this paper we extend the vector valued g invex functions to multiobjective variational control problems by using the new concepts a number of sufficient optimality results and mond weir type duality results are obtained for multiobjective variational control programming problems

buchsbaum rim sheaves and their multiple sections abstract this paper begins by introducing and characterizing buchsbaum rim sheaves on z proj r where r is a graded gorenstein k algebra they are reflexive sheaves arising as the sheafification of kernels of sufficiently general maps between free r modules then we study multiple sections of a buchsbaum rim sheaf b i e we consider morphisms p b of sheaves on z dropping rank in the expected codimension where h0 z p is a free r module the main purpose of this paper is to study properties of schemes associated to the degeneracy locus s of it turns out that s is often not equidimensional let x denote the top dimensional part of s in this paper we measure the difference between x and s compute their cohomology modules and describe ring theoretic properties of their coordinate rings moreover we produce graded free resolutions of x and s which are in general minimal among the applications we show how one can embed a subscheme into an arithmetically gorenstein subscheme of the same dimension and prove that zero loci of sections of the dual of a null correlation bundle are arithmetically buchsbaum

construction of symplectic partitioned runge kutta methods with continuous stage hamiltonian systems as one of the most important class of dynamical systems are associated with a well known geometric structure called symplecticity symplectic numerical algorithms which preserve such a structure are therefore of interest in this article we study the construction of symplectic partitioned runge kutta methods with continuous stage this construction of symplectic methods mainly relies upon the expansion of orthogonal polynomials and the simplifying assumptions for partitioned runge kutta type methods by using suitable quadrature formulae it also provides a new and simple way to construct symplectic partitioned runge kutta methods in classical sense

a note on the continuous stage runge kutta nystrom formulation of hamiltonian boundary value methods hbvms abstract in recent years the class of energy conserving methods named hamiltonian boundary value methods hbvms has been devised for numerically solving hamiltonian problems in this short note we study their natural formulation as continuous stage runge kutta nystrom methods which allows a deeper insight in the methods

duality for multiobjective variational problems abstract wolfe and mond weir type duals for multiobjective variational problems are formulated under convexity assumptions on the functions involved weak and converse duality theorems are proved to relate properly efficient solutions of the primal and dual problems a close relationship between these variational problems and nonlinear multiobjective programming problems is also indicated

on liaison arithmetical buchsbaum curves and monomial curves in p3 in the context of liaison by 181 and 191 for curves in ip3 the property of being the zero scheme of a section of a rank two bundle on ip3 is connected to that of being ideally the intersection of three surfaces an immediate consequence of this property is that the homogeneous ideal i c of a curve c c p3 is generated by precisely three elements if and only if c is ideally the intersection of three suraces and c is arithmetically cohen macaulay non complete intersection furthermore the characterization of the equivalence classes defined by liaison among curves in p3 yields the well known statement that a curve c is in the liaison class of a complete intersection if and only if c is arithmetically cohen macaulay see e g 2 10 13 or 141 it seems that a cayley see 7 p 1521 in 1847 was the first who posed the problem to describe this liaison class of a complete intersection now in this paper we will begin to investigate the next simple case that is the liaison classes which are characterized by a finite dimensional vector space of dimension l using the theory of buchsbaum rings see section 2 this means we will study liaison among arithmetical buchsbaum curves in p3 first from the point of view of local algebra we get the following more general result

multiple structures with arbitrarily large projective dimension supported on linear subspaces abstract let k be an algebraically closed field there has been much interest in characterizing multiple structures in p k n defined on a linear subspace of small codimension under additional assumptions e g cohen macaulay we show that no such finite characterization of multiple structures is possible if one only assumes unmixedness specifically we prove that for any positive integers h e 2 with h e 2 2 and p 5 there is a homogeneous ideal i in a polynomial ring over k such that 1 the height of i is h 2 the hilbert samuel multiplicity of r i is e 3 the projective dimension of r i is at least p and 4 the ideal i is primary to a linear prime x 1 x h this result is in stark contrast to manolache s characterization of cohen macaulay multiple structures in codimension 2 and multiplicity at most 4 and also to engheta s characterization of unmixed ideals of height 2 and multiplicity 2

on efficiency and duality for multiobjective programs abstract for a multiobjective nonlinear program which involved inequality and equality constraints wolfe mond weir and general mond weir type duals are formulated and the concept of efficiency pareto optimum is used to state some duality results under generalized f convexity assumptions

extended degree functions and monomial modules the arithmetic degree the smallest extended degree and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not cohen macaulay we compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule this leads to various bounds and to counterexamples to a conjecture of gunston and vasconcelos respectively particular attention is given to the class of sequentially cohen macaulay modules the results in this case lead to an algorithm that computes the smallest extended degree

liaison classes of modules abstract we propose a concept of module liaison that extends gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative gorenstein ring analyzing the resulting equivalence classes we show that several results known for gorenstein liaison are still true in the more general case of module liaison in particular we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules as a consequence we show that the intermediate cohomology modules and properties like being perfect cohen macaulay buchsbaum or surjective buchsbaum are preserved in even module liaison classes furthermore we prove that the module liaison class of a complete intersection of codimension one consists of precisely all perfect modules of codimension one

duality and sufficiency in control problems with invexity recently mond chandra and husain 3 established duality results for a variational problem under more general convexity conditions called invex than mond and hanson 4 in a similar fashion the results of mond and hanson s for duality in control problems are extended to more general convex or invex functions it is also shown that for invex functions the necessary conditions for optimality in the control problem are also sufficient

ribbons and their canonical embeddings we study double structures on the projective line and on certain other varieties with a view to having a nice family of degenerations of curves and k3 surfaces of given genus and clifford index our main interest is in the canonical embeddings of these objects with a view toward green s conjecture on the free resolutions of canonical curves we give the canonical embeddings explicitly and exhibit an approach to determining a minimal free resolution

discontinuous galerkin methods for hamiltonian odes and pdes in this article we present a unified framework of discontinuous galerkin dg discretizations for hamiltonian odes and pdes we show that with appropriate numerical fluxes the numerical algorithms deduced from dg discretizations can be combined with the symplectic methods in time to derive the multi symplectic prk schemes the resulting numerical discretizations are applied to the linear and nonlinear schrodinger equations some conservative properties of the numerical schemes are investigated and confirmed in the numerical experiments

kam theorem of symplectic algorithms for hamiltonian systems in this paper we prove that an analog of the celebrated kam theorem holds for symplectic algorithms which channel and scovel 1990 feng kang 1991 and sanz serna and calvo 1994 suggested a few years ago the main results consist of the existence of invariant tori with a smooth foliation structure of a symplectic numerical algorithm when it applies to a generic integrable hamiltonian system if the system is analytic and the time step size of the algorithm is s ufficiently small this existence result also implies that the algorithm when it is applied to a generic integrable system possesses n independent smooth invariant functions which are in involution and well defined on the set filled by the invariant tori in the sense of whitney the invariant tori are just the level sets of these functions some quantitative results about the numerical invariant tori of the algorithm approximating the exact ones of the system are also given

non degenerate curves with maximal hartshorne rao module extending results for space curves we establish bounds for the cohomology of a non degenerate curve in projective n space as a consequence for any given n we determine all possible pairs d g where d is the degree and g is the arithmetic genus of the curve furthermore we show that curves attaining our bounds always exist and describe properties of these extremal curves in particular we determine the hartshorne rao module the generic initial ideal and the graded betti numbers of an extremal curve

uniqueness of varieties of minimal degree containing a given scheme we prove that if x p n has dimension k and it is r buchsbaum with r max codim x k 0 then x is contained in at most one variety of minimal degree and dimension k 1

liaison equivalence classes we will work in ip n 2 over an algebraically closed field the notion of liaison among subvarieties of ip was introduced in 3 we will consider here liaison in codimension two generalizing results in 4 recall therefore that two subschemes v 1 v 2 of ip that are locally cohenmacaulay and equidimensional of codimension two are said to be linked algebraically via the complete intersection x of two hypersurfaces denoted v l v z via x if d e v j x v r o v x t j they are said to be linked geometrically if i 1 and v2 in the above definition have no components in common according to 4 theorem 1 7 the two notions of linkage generate the same equivalence relation among those subschemes v which are also generically locally complete intersections we will hence consider only algebraic linkage

on the genus and hartshorne rao module of projective curves in this paper optimal upper bounds for the genus and the dimension of the graded components of the hartshorne rao module of curves in projective n space are established this generalizes earlier work by hartshorne h and martin deschamps and perrin mdp special emphasis is put on curves in bf p 4 the first main result is a so called restriction theorem it says that a non degenerate curve of degree d geq 4 in bf p 4 over a field of characteristic zero has a non degenerate general hyperplane section if and only if it does not contain a planar curve of degree d 1 see th 1 3 then using methods of brodmann and nagel bounds for the genus and hartshorne rao module of curves in bf p n with non degenerate general hyperplane section are derived it is shown that these bounds are best possible in a very strict sense coupling these bounds with the restriction theorem gives the second main result for curves in bf p 4 then curves of maximal genus are investigated the betti numbers of their minimal free resolutions are computed and a description of all reduced curves of maximal genus in bf p n of degree geq n 2 is given finally all pairs d g of integers which really occur as the degree d and genus g of a non degenerate curve in bf p 4 are described

a necessary and sufficient condition for duality in multiobjective variational problems abstract in this paper we move forward in the study of duality and efficiency in multiobjective variational problems we introduce new classes of pseudoinvex functions and prove that not only it is a sufficient condition to establish duality results but it is also necessary moreover these functions are characterized in order that all kuhn tucker or fritz john points are efficient solutions recent papers are improved we provide an example to show this improvement and illustrate these classes of functions and results

on the cohen macaulay type of the general hypersurface section of a curve a theorem of strano 24 states that if c is a reduced irreducible curve in lp 3 lying on no quadric surface and if a general hyperplane section c n h of c is a complete intersection then c itself must be a complete intersection this was extended by re 22 to the case where c is a locally cohen macaulay curve in lw it was then shown by huneke and ulrich 14 that if c is reduced and connected not lying on a quadric hypersurface and if c a h is merely gorenstein then the same is true of c finally the first author 18 extended the techniques of 14 to show that if h is replaced by a general hypersurface of degree d then similar results hold recall that one definition of a gorenstein scheme is arithmetically cohenmacaulay with cohen macaulay type one in other words in a minimal free resolution

symmetric integrators based on continuous stage runge kutta nystrom methods for reversible systems abstract in this paper we study symmetric integrators for solving second order ordinary differential equations on the basis of the notion of continuous stage runge kutta nystrom methods the construction of such methods heavily relies on the legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions new families of symmetric integrators as illustrative examples are presented for comparing the numerical behaviors of the presented methods some numerical experiments are also reported

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

high order symplectic integrators based on continuous stage runge kutta nystrom methods abstract on the basis of the previous work by tang and zhang 37 in this paper we present a more effective way to construct high order symplectic integrators for solving second order hamiltonian equations instead of analyzing order conditions step by step as shown in the previous work the new technique of this paper is using legendre expansions to deal with the simplifying assumptions for order conditions with the new technique high order symplectic integrators can be conveniently devised by truncating an orthogonal series

regularity of prime ideals we answer several natural questions which arise from a recent paper of mccullough and peeva providing counterexamples to the eisenbud goto regularity conjecture we give counterexamples using rees algebras and also construct counterexamples that do not rely on the mayr meyer construction furthermore examples of prime ideals for which the difference between the maximal degree of a minimal generator and the maximal degree of a minimal first syzygy can be made arbitrarily large are given using a result of ananyan hochster we show that there exists an upper bound on regularity of prime ideals in terms of the multiplicity alone

symmetric duality with invexity in variational problems abstract weak and strong duality results are established under invexity hypotheses for symmetric dual variational problems without positivity constraints self dual problems and static symmetric dual programs are included as special cases

on the projective dimension and the unmixed part of three cubics let r be a polynomial ring over a field in an unspecified number of variables we prove that if j r is an ideal generated by three cubic forms and the unmixed part of j contains a quadric then the projective dimension of r j is at most 4 to this end we show that if k r is a three generated ideal of height two and l r an ideal linked to the unmixed part of k then the projective dimension of r k is bounded above by the projective dimension of r l plus one

the non lefschetz locus abstract we study the weak lefschetz property of artinian gorenstein algebras and in particular of artinian complete intersections in codimension four and higher it is an open problem whether all complete intersections have the weak lefschetz property for a given artinian gorenstein algebra a we ask what linear forms are lefschetz elements for this particular algebra i e which linear forms l give maximal rank for all the multiplication maps l a i a i 1 this is a zariski open set and its complement is the non lefschetz locus for monomial complete intersections we completely describe the non lefschetz locus for general complete intersections of codimension three and four we prove that the non lefschetz locus has the expected codimension which in particular means that it is empty in a large family of examples for general gorenstein algebras of codimension three with a given hilbert function we prove that the non lefschetz locus has the expected codimension if the first difference of the hilbert function is of decreasing type for completeness we also give a full description of the non lefschetz locus for artinian algebras of codimension two

symplecticity preserving continuous stage runge kutta nystrom methods abstract in this paper we develop continuous stage runge kutta nystrom csrkn methods for numerical integration of second order ordinary differential equations odes written in the form q f t q numerous odes in such form can be reduced to first order odes with the separable form of hamiltonian systems and symplecticity preserving discretizations of these systems are of interest for the sake of designing symplectic csrkn methods we explore the sufficient conditions for symplecticity and we show a simple way to derive symplectic rkn type integrators by using legendre polynomial expansion numerical results show the efficiency of the presented methods

seminormality and cohomology of projective varieties let x be a reduced closed subscheme of p proj r y k a field and r k x a r 2 2 let a be the homogeneous co ordinate ring of x in 2 3 51 the various authors have attempted to determine whether a is cohenmacaulay or seminormal or more generally to study the seminormalization or cohen macaulification of a the papers mentioned above work primarily with unions of straight lines in s corollary 5 91 it is proved for example that if x is a connected union of lines which have linearly independent directions at each intersection point then a is seminormal if and only if a is cohen macaulay the aim of this paper is to give a better understanding of the seminormalization of a and the relationship between seminormality and cohenmacaulayness hinted at by the above mentioned corollary 5 9 this we do in section 2 by interpreting the seminormalization of a in terms of sheaf cohomology we are then able to give a reasonable generalization our corollary 3 5 of s corollary 5 91 and also to resolve several computational problems left open in 2 31 and 4 the hilbert function of a graded ring a will be denoted by ha that is ha i dim ai if x proj a the hilbert polynomial of a is denoted hp x in some computations of section 6

optimality conditions and duality for a class of continuous time programming problems with nonlinear operator equality and inequality constraints abstract using a minmax approach we establish saddle point optimality conditions and lagrangian duality relations under generalized convexity hypotheses for a class of continuous time programming problems with nonlinear operator equality and inequality constraints we also discuss the implications of these results for differentiable problems

curves of degree two and ropes on a line their ideals and even liaison classes abstract in this paper we investigate degree two curves of arbitrary codimension this requires to study also ropes supported on a line after establishing several characterizations of ropes supported on a line we describe their homogeneous ideals and hartshorne rao modules in particular we characterize the arithmetically buchsbaum ropes then we describe the even gorenstein liaison classes of the non arithmetically buchsbaum non degenerate ropes generalizing a well know result of migliore on double lines of codimension two the result relies on the explicit description of certain arithmetically gorenstein curves with maximal tangent spaces at each point as a consequence we can decide if two curves of degree two belong to the same gorenstein liaison class finally we show as an application how ropes can be used in order to construct quasi extremal curves

sufficient optimality criteria and duality for multiobjective variational control problems with v invexity abstract the sufficient optimality criteria for a class of multiobjective variational control problems under the v invexity assumption are presented duality results are proved under a variety of v invexity assumptions multiobjective fractional variational control problems are also studied

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

a simple framework for the derivation and analysis of effective one step methods for odes in this paper we provide a simple framework to derive and analyse a class of one step methods that may be conceived as a generalization of the class of gauss methods the framework consists in coupling two simple tools firstly a local fourier expansion of the continuous problem is truncated after a finite number of terms and secondly the coefficients of the expansion are computed by a suitable quadrature formula different choices of the basis lead to different classes of methods even though we shall here consider only the case of an orthonormal polynomial basis from which a large subclass of runge kutta methods can be derived the obtained results are then applied to prove in a simplified way the order and stability properties of hamiltonian bvms hbvms a recently introduced class of energy preserving methods for canonical hamiltonian systems see 2 and references therein a few numerical tests are also included in order to confirm the effectiveness of the methods resulting from our analysis

construction of runge kutta type methods for solving ordinary differential equations in this paper we study runge kutta methods with continuous stage introduced by butcher in 1987 by setting the coefficients of this family of methods and choosing the appropriate numerical integration we derive new classes of runge kutta methods furthermore we extend the w transformation technique by permitting w to be a non square matrix this allows us to construct more high order implicit runge kutta methods with some geometric properties specially we provide runge kutta methods with continuous stage which are conjugate symplectic symmetric or energy preserving for solving hamiltonian systems we also present some numerical experiments to verify our results

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

non degenerate projective curves with very degenerate hyperplane section in this paper we study the hilbert scheme of non degenerate locally cohen macaulay projective curves with general hyperplane section spanning a linear space of dimension 2 and minimal hilbert function the main result is that those curves are almost always the general element of a generically smooth component hn d g of the corresponding hilbert scheme moreover we show that the curves with maximal cohomology almost always correspond to smooth points of hn d g

sufficiency and duality for multiobjective variational control problems with generalized f α ρ θ v convexity abstract zalmai introduced generalized f v convex functions a new class of functions that unifies several concepts of generalized convex functions in this article the generalized f v convex functions are extended to variational control problem by utilizing the new concepts we obtain sufficient optimality conditions and prove wolfe type and mond weir type duality results for multiobjective variational control programming problem