The calculus of differential forms can be used to devise a unified description of discrete differential forms of any order and polynomial degree on simplicial meshes in any spatial dimension. A general formula for suitable degrees of freedom is also available. Fundamental properties of nodal interpolation can be established easily. It turns out that higher order spaces, including variants with locally varying polynomial order, emerge from the usual Whitney-forms by local augmentation. This paves the way for an adaptive p-version approach to discrete differential forms.

This article deals with the regulation of water flow in open-channels modelled by Saint-Venant equations. By means of a Riemann invariants approach, we deduce stabilizing control laws for a single horizontal reach without friction. The approach is extended to general networks of canals and is illustrated with a simple case study: two reaches in cascade. To prove control stability of the canal network, we use a consequence of a previous result from Li Ta-tsien concerning the existence and decay of classical solutions of hyperbolic systems. An extension of the control law is proposed for channels with slope and friction.

We examine some ways of proving the Implicit Function Theorem and the Inverse Function Theorem within Bishop's constructive mathematics. Section 2 contains a new, entirely constructive proof of the Implicit Function Theorem. The paper ends with some comments on the application of the Implicit Function Theorem in classical mechanics. 1 Introduction In this paper, which is written entirely within the framework of constructive mathematics #BISH# erected by the late Errett Bishop #2#, we examine a standard proof of the Implicit Function Theorem and give a completely new proof. As far as understanding constructive mathematics goes, the reader need only be aware that when working constructively,weinterpret #existence" strictly as #computability". To do so, we need to be careful about our logic. For example, when we prove a disjunction P Q; we need to either produce a proof of P or produce a proof of Q; it is not enough, constructively, to show that : #:P :Q#:To understand this better, con...

.<F4.039e+05> The dynamic programming (DP) technique rests on a very simple idea, the principle of optimality due to Bellman. This principle is instrumental in solving numerous problems of optimal control. The control law minimizes a cost functional and is determined by using the optimality principle. However, applicability of the optimality principle requires that the cost functional satisfies the property called "matrix dynamic programming (MDP) property." A simple definition of this property will be provided and functionals having it will be considered.<F4.601e+05> Key words.<F4.039e+05> dynamic programming, determinants, monotonicity, target tracking<F4.601e+05> AMS subject classifications.<F4.039e+05> 49L20, 15A15, 15A45, 15A69<F4.601e+05> PII.<F4.039e+05> S0895479895288334<F5.406e+05> 1. Introduction.<F4.753e+05> The DP technique rests on a very simple idea, the principle of optimality due to Bellman [1]. This principle simply asserts that if<F4.491e+05> #<F4.162e+05> #<F4.753e+...

We prove the following new characterization of C p (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a C p smooth (Lipschitz) bump function if and only if it has another C p smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in the interior of the support of f (that is, f does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. The "twisted tube" method we use in the proof is interesting in itself, as it provides other useful characterizations of C p smoothness related to the existence of a certain kind of deleting di#eomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces. 1.

Given the increasing use of adaptive arrays and multiple antenna transceivers in wireless communications systems, a common problem arises in how to e ciently model these kinds of systems. This paper focuses on the performance of minimum mean square error (MMSE) linear beamformers over a xed number of received training samples. We obtain a closed form solution for the beamformer statistics in the presence of Gaussian random noise and interference. We also demonstrate an e cient way to model related performance statistics such as the signal to interference and noise ratio (SINR), the mean square error (MSE) and other statistics of interest. A surprising result is that these statistics can be modeled using only three receiver parameters, the maximum obtainable SINR, the number of antennas and the length of the training sequence. As a result, high level simulations that require performance metrics, such as MAC layer simulations, need only provide these three parameters, in order to sample from the SINR or MSE random variables. This reduces the need for computationally expensive network simulations. 1