Variational formulas of higher order mean curvatures

In this paper,we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional M2p of a submanifold M n in a general Riemannian manifold N n+m for p = 0,1,...,[n 2 ].As an example,we prove that closed complex submanifolds in complex projective spaces are critical points of the functional M2p,called relatively 2p-minimal submanifolds,for all p.At last,we discuss the relations between relatively 2p-minimal submanifolds and austere submanifolds in real space forms,as well as a special variational problem.     

Index formulas for higher order Loewner vector fields

Let be the Cauchy-Riemann operator and f be a Cn real-valued function in a neighborhood of 0 in R² in which for all z≠0. In such cases, is known as a Loewner vector field due to its connection with Loewner's conjecture that the index of such a vector field is bounded above by n. The n=2 case of Loewner's conjecture implies Carathéodory's conjecture that any C²-immersion of S² into R³ must have at least two umbilics. Recent work of F. Xavier produced a formula for computing the index of Loewner vector fields when n=2 using data about the Hessian of f. In this paper, we extend this result and establish an index formula for for all n?2. Structurally, our index formula provides a defect term, which contains geometric data extracted from Hessian-like objects associated with higher order derivatives of f.     

Reduction formulas for higher order derivations and a hypergeometric identity

Aequationes mathematicae - A derivation (of order 1) satisfies the reduction formula $$f(x^k) = kx^{k-1}f(x)$$ for any integer k. In this article we find corresponding reduction formulas for...     

Explicit formulas and recurrence relations for higher order Eulerian polynomials

In the paper, the authors present explicit formulas, nonlinear ordinary differential equations, and recurrence relations for Eulerian polynomials, higher order Eulerian polynomials, and their generating functions in terms of the Stirling numbers of the second kind.     

Variational time discretizations of higher order and higher regularity

We consider a family of variational time discretizations that are generalizations of discontinuous Galerkin (dG) and continuous Galerkin–Petrov (cGP) methods. The family is characterized by two parameters. One describes the polynomial ansatz order while the other one is associated with the global smoothness that is ensured by higher order collocation conditions at both ends of the subintervals. Applied to Dahlquist’s stability problem, the presented methods provide the same stability properties as dG or cGP methods. Provided that suitable quadrature rules of Hermite type are used to evaluate the integrals in the variational conditions, the variational time discretization methods are connected to special collocation methods. For this case, we present error estimates, numerical experiments, and a computationally cheap postprocessing that allows to increase both the accuracy and the global smoothness by one order.

     

On the order of convergence of Voronovskaja-type formulas

In this work general quantitative estimates of Voronovskaja-type formulas in the space of functions of several variables that are twice continuously differentiable and in the space of functions with weighted second-order derivatives are established.     

Generalized higher order spt-functions

We give a new generalization of the spt-function of G.E. Andrews, namely $\operatorname {Spt}_{j}(n)$ , and give its combinatorial interpretation in terms of successive lower-Durfee squares. We then generalize the higher order spt-function $\operatorname {spt}_{k}(n)$ , due to F.G. Garvan, to ${}_{j\!}\operatorname {spt}_{k}(n)$ , thus providing a two-fold generalization of $\operatorname {spt}(n)$ , and give its combinatorial interpretation. Lastly, we show how the positivity of _j spt _k (n) can be used to generalize Garvan’s inequality between rank and crank moments to the moments of j-rank and (j+1)-rank.     

Hierarchies of higher order kernels

Summary Recent literature on functional estimation has shown the importance of kernels with vanishing moments although no general framework was given to build kernels of increasing order apart from some specific methods based on moment relationships. The purpose of the present paper is to develop such a framework and to show how to build higher order kernels with nice properties and to solve optimization problems about kernels. The proofs given here, unlike standard variational arguments, explain why some hierarchies of kernels do have optimality properties. Applications are given to functional estimation in a general context. In the last section special attention is paid to density estimates based on kernels of order (m, r), i.e., kernels of orderr for estimation of derivatives of orderm. Convergence theorems are easily derived from interpretation by means of projections inL ² spaces.     

Hypoellipticity and higher order Levi conditions

We study the C^∞$C^{\infty }$-hypoellipticity for a class of double characteristic operators with symplectic characteristic manifold, in the case the classical condition of minimal loss of derivatives is violated.     

A geometric construction of iterative formulas of order three

In this paper, we consider a geometric construction for improving the order of convergence of iterative formulas of order two. Using this approach, new third-order modifications of Newton’s method are derived. A comparison with other existing methods is given.     

Higher order conservation laws and a higher order Noether's theorem

Consider a general variational problem of a functional whose domain of definition consists of integral manifolds of an exterior differential system. In particular, this induces classical variational problems with constraints. With the assumption of existence of enough admissable variations the Euler-Lagrange equations associated to this problem are obtained. By studying a spectral sequence associated to the infinite prolongation of them, we extend the classical notion of infinitesimal Noether symmetries to what we shall call the “higher order Noether symmetries,” and a higher order Noether's theorem identifying the higher order conservation laws and the higher order Noether symmetries is obtained. These in turn are isomorphic to the solution space of certain linear differential operator. From these we also get a systematic method of computing the higher order conservation laws of certain determined PDE systems.     

A method for obtaining iterative formulas of order three

An improved method for the order of convergence of iterative formulas of order two is given. Using this method, new third-order modifications of Newton’s method are derived. A comparison with other methods is given.     

Constructibility in higher order arithmetics

Summary We define and investigate constructibility in higher order arithmetics. In particular we get an interpretation ofn-order arithmetic inn-order arithmetic without the scheme of choice such that and the property to be a well-ordering are absolute in it and such that this interpretation is minimal among such interpretations.     

On higher order Stickelberger-type theorems

We discuss an explicit refinement of Rubin?s integral version of Stark?s conjecture. We prove that this refinement is a consequence of the relevant case of the Equivariant Tamagawa Number Conjecture of Burns and Flach, hence obtaining a full proof in several important cases. We also derive several explicit consequences of this refinement concerning the annihilation as Galois modules of ideal class groups by explicit elements constructed from the values of higher order derivatives of Dirichlet L-functions. We finally describe the relation between our approach and those found in recent work of Emmons and Popescu and of Buckingham.