In this note we clarify the relationship between the local and global definitions of dual pairs in Poisson geometry. It turns out that these are not equivalent. For the passage from local to global one needs a connected fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also provide examples illustrating the necessity of ...

Abstract. J.-L. Loday introduced the concept of coherent unit actions on a regular operad and showed that such actions give Hopf algebra structures on the free algebras. Hopf algebras obtained this way include the Hopf algebras of shuffles, quasi-shuffles and planar rooted trees. We characterize coherent unit actions on binary quadratic regular operads in terms of linear equations of the genera...

We summarize a recently developed integral equation approach to tackling the long-time existence problem for smooth solution v(x, t) to the 3-D Navier-Stokes equation in the context of a periodic box problem with smooth time-independent forcing and initial condition v0. Using an inverse Laplace transform of v̂(k, t) − v̂0 in 1/t, we arrive at an integral equation for Û(k, p), where p is inverse-L...

The cubic spline interpolation method, the Runge–Kutta method, and the Newton–Raphson method are extended to dual versions (developed in the context of dual numbers). This extension allows the calculation of the derivatives of complicated compositions of functions which are not necessarily defined by a closed form expression. The code for the algorithms has been written in Fortran and some exam...

Conservative linear equations arise in many areas of application, including continuum mechanics or high-frequency geometrical optics approximations. This kind of equation admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In this paper, we study the convergence of a class of finite-difference numerical schemes and introduce an app...

By J. Douglas, Jr., R. B. Kellogg, and R. S. Varga We consider the iterative solution of the system of linear equations (1 ) (Xi + X-, + • ■ ■ + Xn)z = /, n 2; 2, where each Xj, 1 | j | n, is a Hermitian and positive definite N X N matrix. If n = 2, the iterative methods of Peaceman-Rachford [1, Chapter 7], or D'yakonov [2] and Kellogg [3], may be used to solve (1). In this paper these methods ...

A new duality theory is developed for a class of stochastic programs in which the probability distribution is not necessarily discrete. This provides a new framework for problems which are not necessarily bounded, are not required to have relatively complete recourse, and do not satisfy the typical Slater condition of strict feasibility. These problems instead satisfy a different constraint qua...

We consider the class of linear programs that can be formulated with infinitely many variables and constraints but where each constraint has only finitely many variables. This class includes virtually all infinite horizon planning problems modeled as infinite stage linear programs. Examples include infinite horizon production planning under time-varying demands and equipment replacement under t...

Value functions propagated from initial or terminal costs and constraints by way of a differential inclusion, or more broadly through a Lagrangian that may take on∞, are studied in the case where convexity persists in the state argument. Such value functions, themselves taking on ∞, are shown to satisfy a subgradient form of the Hamilton-Jacobi equation which strongly supports properties of loc...

We consider the system of m linear equations in n integer variables Ax = d and give sufficient conditions for the uniqueness of its integer solution x ∈ {−1,1}n by reformulating the problem as a linear program. Uniqueness characterizations of ordinary linear programming solutions are utilized to obtain sufficient uniqueness conditions such as the intersection of the kernel of A and the dual con...