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Title: Two Extensions to Finsler's Recurring Theorem

Abstract

Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R{sup n} : (i) f( {xi} ) >0 for all {xi}{ne} 0 with g( {xi} ) =0; (ii) f-{lambda} g>0 for some {lambda} element of R. We prove two extensions: 1. We admit a vector-valued quadratic form g:R{sup n{yields}}R{sup k} , for which we show that (i) implies that f-{lambda} . . . g>0 on an ( n-k+1)-dimensional subspace Y is subset of R{sup n} for some {lambda} element of R{sup k} . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R{sup n} by a real vector space X.

Authors:
 [1]
  1. Hohle Gasse 77, D-53177 Bonn (Germany)
Publication Date:
OSTI Identifier:
21067543
Resource Type:
Journal Article
Journal Name:
Applied Mathematics and Optimization
Additional Journal Information:
Journal Volume: 40; Journal Issue: 2; Other Information: DOI: 10.1007/s002459900121; Copyright (c) 1999 Springer-Verlag New York Inc.; Article Copyright (c) Inc. 1999 Springer-Verlag New York; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FUNCTIONS; MANY-DIMENSIONAL CALCULATIONS; MATHEMATICAL LOGIC; MATHEMATICAL SPACE; VECTORS

Citation Formats

Hamburger, C. Two Extensions to Finsler's Recurring Theorem. United States: N. p., 1999. Web. doi:10.1007/S002459900121.
Hamburger, C. Two Extensions to Finsler's Recurring Theorem. United States. doi:10.1007/S002459900121.
Hamburger, C. Wed . "Two Extensions to Finsler's Recurring Theorem". United States. doi:10.1007/S002459900121.
@article{osti_21067543,
title = {Two Extensions to Finsler's Recurring Theorem},
author = {Hamburger, C.},
abstractNote = {Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R{sup n} : (i) f( {xi} ) >0 for all {xi}{ne} 0 with g( {xi} ) =0; (ii) f-{lambda} g>0 for some {lambda} element of R. We prove two extensions: 1. We admit a vector-valued quadratic form g:R{sup n{yields}}R{sup k} , for which we show that (i) implies that f-{lambda} . . . g>0 on an ( n-k+1)-dimensional subspace Y is subset of R{sup n} for some {lambda} element of R{sup k} . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R{sup n} by a real vector space X.},
doi = {10.1007/S002459900121},
journal = {Applied Mathematics and Optimization},
issn = {0095-4616},
number = 2,
volume = 40,
place = {United States},
year = {1999},
month = {9}
}