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The GelfondSchnirelman Method In Prime Number Theory
 Canad. J. Math
"... The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for t ..."
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The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's #function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1. Lower bounds for arithmetic functions Let #(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that ##.
From Euler, Ritz, and Galerkin to Modern Computing ∗
"... Abstract. The socalled Ritz–Galerkin method is one of the most fundamental tools of modern computing. Its origins lie in Hilbert’s “direct ” approach to the variational calculus of Euler– Lagrange and in the thesis of Walther Ritz, who died 100 years ago at the age of 31 after a long battle with tu ..."
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Abstract. The socalled Ritz–Galerkin method is one of the most fundamental tools of modern computing. Its origins lie in Hilbert’s “direct ” approach to the variational calculus of Euler– Lagrange and in the thesis of Walther Ritz, who died 100 years ago at the age of 31 after a long battle with tuberculosis. The thesis was submitted in 1902 in Göttingen, during a period of dramatic developments in physics. Ritz tried to explain the phenomenon of Balmer series in spectroscopy using eigenvalue problems of partial differential equations on rectangular domains. While this physical model quickly turned out to be completely obsolete, his mathematics later enabled him to solve difficult problems in applied sciences. He thereby revolutionized the variational calculus and became one of the fathers of modern computational mathematics. We will see in this article that the path leading to modern computational methods and theory involved a long struggle over three centuries requiring the efforts of many great mathematicians. Key words. Walther Ritz, variational calculus, finite element method
ACTIVE MASS UNDER PRESSURE
, 2005
"... After a historical introduction to Poisson’s equation for Newtonian gravity, its analog for static gravitational fields in Einstein’s theory is reviewed. It appears that the pressure contribution to the active mass density in Einstein’s theory might also be noticeable at the Newtonian level. A form ..."
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After a historical introduction to Poisson’s equation for Newtonian gravity, its analog for static gravitational fields in Einstein’s theory is reviewed. It appears that the pressure contribution to the active mass density in Einstein’s theory might also be noticeable at the Newtonian level. A form of its surprising appearance, first noticed by Richard Chase Tolman, was discussed half a century ago in the Hamburg Relativity Seminar and is resolved here. 1
EXISTENCE AND STABILITY FOR A NONLOCAL ISOPERIMETRIC MODEL OF CHARGED LIQUID DROPS
"... Abstract. We consider a variational problem related to the shape of charged liquid drops at equilibrium. We show that this problem never admits global minimizers with respect to L 1 perturbations preserving the volume. This leads us to study it in more regular classes of competitors, for which we sh ..."
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Abstract. We consider a variational problem related to the shape of charged liquid drops at equilibrium. We show that this problem never admits global minimizers with respect to L 1 perturbations preserving the volume. This leads us to study it in more regular classes of competitors, for which we show existence of minimizers. We then prove that the ball is the unique solution for sufficiently small charges. 1.