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Higher Order Whitney Forms
 in Geometrical Methods in Computational Electromagnetics
, 2000
"... 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 interpo ..."
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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 Whitneyforms by local augmentation. This paves the way for an adaptive pversion approach to discrete differential forms.
The Failure of Rolle's Theorem in InfiniteDimensional Banach Spaces
, 2001
"... We prove the following new characterization of C p (Lipschitz) smoothness in Banach spaces. An infinitedimensional 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 poi ..."
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We prove the following new characterization of C p (Lipschitz) smoothness in Banach spaces. An infinitedimensional 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 selfmappings of starlike bodies in all infinitedimensional spaces. 1.
The Constructive Implicit Function Theorem and Applications in Mechanics
 Chaos, Solitons and Fractals
, 1997
"... 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 F ..."
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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...
Modeling the Statistics of Linear Beamformers
, 2008
"... 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 o ..."
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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