182 research outputs found
Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions
The main aim of this paper is to provide a novel approach to deriving
identities for the Bernstein polynomials using functional equations. We derive
various functional equations and differential equations using generating
functions. Applying these equations, we give new proofs for some standard
identities for the Bernstein basis functions, including formulas for sums,
alternating sums, recursion, subdivision, degree raising, differentiation and a
formula for the monomials in terms of the Bernstein basis functions. We also
derive many new identities for the Bernstein basis functions based on this
approach. Moreover, by applying the Laplace transform to the generating
functions for the Bernstein basis functions, we obtain some interesting series
representations for the Bernstein basis functions.Comment: 1
Automatic Estimation of Verified Floating-Point Round-Off Errors via Static Analysis
This paper introduces a static analysis technique for computing formally verified round-off error bounds of floating-point functional expressions. The technique is based on a denotational semantics that computes a symbolic estimation of floating-point round-o errors along with a proof certificate that ensures its correctness. The symbolic estimation can be evaluated on concrete inputs using rigorous enclosure methods to produce formally verified numerical error bounds. The proposed technique is implemented in the prototype research tool PRECiSA (Program Round-o Error Certifier via Static Analysis) and used in the verification of floating-point programs of interest to NASA
The Potential Energy Surface in Molecular Quantum Mechanics
The idea of a Potential Energy Surface (PES) forms the basis of almost all
accounts of the mechanisms of chemical reactions, and much of theoretical
molecular spectroscopy. It is assumed that, in principle, the PES can be
calculated by means of clamped-nuclei electronic structure calculations based
upon the Schr\"{o}dinger Coulomb Hamiltonian. This article is devoted to a
discussion of the origin of the idea, its development in the context of the Old
Quantum Theory, and its present status in the quantum mechanics of molecules.
It is argued that its present status must be regarded as uncertain.Comment: 18 pages, Proceedings of QSCP-XVII, Turku, Finland 201
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