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About 6,944 papers for “Elementary proof”

95.32 an elementary proof

Robert M. Young · The Mathematical Gazette · 2011

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Cited by 0

Proof in elementary geometry

Frank Davis · The Mathematical Gazette · 2004

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Elementary proof of Funahashi's theorem

Mitsuo Izuki, Takahiro Noi, Yoshihiro Sawano et al. · Constructive Mathematical Analysis · 2024

Funahashi established that the space of two-layer feedforward neural networks is dense in the space of all continuous functions defined over compact sets in $n$-dimensional Euclidean space. The purpose of this short survey is to reexamine the proof of Theorem 1 in Funahashi \cite{Funahashi}. The Tietze extension theorem, whose proof is contained in the appendix, will be used. This paper is based on harmonic analysis, real analysis, and Fourier analysis. However, the audience in this paper is supposed to be researchers who do not specialize in these fields of mathematics. Some fundamental facts that are used in this paper without proofs will be collected after we present some notation in this paper.

Cited by 1Open access

Elementary proof of QAOA convergence

Binkowski, Lennart, Koßmann, Gereon, Ziegler, Timo et al. · Leibniz Universität Hannover · 2024

The quantum alternating operator ansatz (QAOA) and its predecessor, the quantum approximate optimization algorithm, are one of the most widely used quantum algorithms for solving combinatorial optimization problems. However, as there is yet no rigorous proof of convergence for the QAOA, we provide one in this paper. The proof involves retracing the connection between the quantum adiabatic algorithm and the QAOA, and naturally suggests a refined definition of the ‘phase separator’ and ‘mixer’ keywords.

Cited by 0Open access

Elementary proof of QAOA convergence

Lennart Binkowski, Gereon Koßmann, Timo Ziegler et al. · New Journal of Physics · 2024

Abstract The quantum alternating operator ansatz (QAOA) and its predecessor, the quantum approximate optimization algorithm, are one of the most widely used quantum algorithms for solving combinatorial optimization problems. However, as there is yet no rigorous proof of convergence for the QAOA, we provide one in this paper. The proof involves retracing the connection between the quantum adiabatic algorithm and the QAOA, and naturally suggests a refined definition of the ‘phase separator’ and ‘mixer’ keywords.

Cited by 10Open access

The most elementary proof that ?

Nick Lord · The Mathematical Gazette · 2016

Here we present a simplification of one of the standard proofs that . We then look at extensions of the new approach, and add comments on the nature of the simplification (which relates to Step 1 below) and finally on the literature.As in most proofs of the result, we shall actually prove that ; becausethis is equivalent to the result sought.

Cited by 0

Elementary Proof of Beal’s Conjecture

Kamal Barghout · Preprints.org · 2025

The 6 variable general equation of Beal’s conjecture equation xa + by = zc, where x,y,z,a,b , and c are positive integers, and a,b,c≥3, is identified as an identity made by expansion of powers of binomials of integers c and y ; where x, y and z have common prime factor. Here, a proof of the conjecture is presented in two folds: First, powers of binomials of integers and expand to all integer solutions of Beal’s equation if they have common prime factor. Second, powers of binomials of coprime integers x, y expand in two terms such that if two of the three terms of the equation are perfect powers the third one is not a perfect power.

Cited by 0Open access

Elementary Proof of Beal’s Conjecture

Kamal Barghout · Preprints.org · 2025

The 6 variable general equation of Beal’s conjecture equation xa + by = zc, where x,y,z,a,b , and c are positive integers, and a,b,c≥3, is identified as an identity made by expansion of powers of binomials of integers c and y ; where x, y and z have common prime factor. Here, a proof of the conjecture is presented in two folds: First, powers of binomials of integers and expand to all integer solutions of Beal’s equation if they have common prime factor. Second, powers of binomials of coprime integers x, y expand in two terms such that if two of the three terms of the equation are perfect powers the third one is not a perfect power.

Cited by 0Open access

An elementary proof of Gorny's inequality

C. Fabry · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1987

Synopsis Gorny's inequality provides upper bounds for the sup-norms ∥ f ( k ) of a function f over an interval [a, b] for k = 1, …, n − 1, assuming the sup-norms of f and f (n) to be given. We present a simple proof of that inequality and obtain sharper estimates of the constants contained in that inequality, compared with the original verison of Gorny.

Cited by 1