791 research outputs found
Semialgebraic Solutions of Linear Equations with Continuous Semialgebraic Coefficients
Starting from the studies proposed by Charles Fefferman and Janos Kollar in
"Continuous Solutions of Linear Equations", we generalise and prove, by using
Fefferman's techniques, a part of the result shown by Kollar. In particular,
considered a system of linear equation with continuous and semialgebraic
coefficient, we find that a continuous and semialgebraic solution exists if and
only if there is a continuous solution and a semialgebraic one (they may
possibly be different), under the hypothesis that the projection of a solution
on the fibers of the singular affine bundle associated to the system is
discontinuous at most at isolated points.Comment: 15 page
A simple and direct method for generating travelling wave solutions for nonlinear equations
We propose a simple and direct method for generating travelling wave
solutions for nonlinear integrable equations. We illustrate how nontrivial
solutions for the KdV, the mKdV and the Boussinesq equations can be obtained
from simple solutions of linear equations. We describe how using this method, a
soliton solution of the KdV equation can yield soliton solutions for the mKdV
as well as the Boussinesq equations. Similarly, starting with cnoidal solutions
of the KdV equation, we can obtain the corresponding solutions for the mKdV as
well as the Boussinesq equations. Simple solutions of linear equations can also
lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve
some new families of KdV equations and show how soliton solutions are also
obtained for the higher order equations of the KdV hierarchy using this method.Comment: RevTex, 15 pages, 3 figures; version with new section and references,
to appear in Annals of Physic
Conditional bounds for small prime solutions of linear equations
Let a 1, a 2, a 3 be non-zero integers with gcd(a 1 a 2, a 3)=1 and let b be an arbitrary integer satisfying gcd (b, a i, a j) =1 for i≠j and b≡a 1+a 2+a 3 (mod 2). In a previous paper [3] which completely settled a problem of A. Baker, the 2nd and 3rd authors proved that if a 1, a 2, a 3 are not all of the same sign, then the equation a 1 p 1+a 2 p 2+a 3 p 3=b has a solution in primes p j satisfying {Mathematical expression} where A>0 is an absolute constant. In this paper, under the Generalized Riemann Hypothesis, the authors obtain a more precise bound for the solutions p j . In particular they obtain A0. An immediate consquence of the main result is that the Linnik's courtant is less than or equal to 2. © 1992 Springer-Verlag.postprin
Mean values of Dirichlet polynomials and applications to linear equations with prime variables
We prove a new mean-value theorem for Dirichlet polynomials with coefficients
given by the von Mangoldt function. We then use our theorem to derive new
estimates for certain exponential sums over primes. The latter have
applications to additive problems with prime variables. In particular, we are
able to improve on a recent result of J.Y. Liu and K.M. Tsang on the size of
the solutions of linear equations with prime variables
The geometry of basic, approximate, and minimum-norm solutions of linear equations
AbstractThe basic solutions of the linear equations Ax = b are the solutions of subsystems corresponding to maximal nonsingular submatrices of A. The convex hull of the basic solutions is denoted by C = C(A, b). Given 1 ≤ p ≤ ∞, the lp-approximate solutions of Ax = b, denoted x{p}, are minimizers of ∥Ax − b∥p. Given M ∈ Dm, the set of positive diagonal m × m matrices, the solutions of minx ∥M(Ax − b)∥p are called scaledlp-approximate solutions. For 1 ≤ p1, p2 ≤ ∞, the minimum-lp2-norm lp1-approximate solutions are denoted x{p1}{p2}. Main results: 1.(1) If A ∈ Rm × nm, then C contains all [some] minimum lp-norm solutions, for 1 ≤ p < ∞ [p = ∞].2.(2) For general A and any 1 ≤ p1, p2 < ∞ the set C contains all x{p1}{p2}.3.(3) The set of scaled lp-approximate solutions, with M ranging over Dm, is the same for all 1 < p < ∞.4.(4) The set of scaled least-squares solutions has the same closure as the set of solutions of minx f (|Ax − b|), where f:Rm+ → R ranges over all strictly isotone functions
Lie point symmetries and ODEs passing the Painlev\'e test
The Lie point symmetries of ordinary differential equations (ODEs) that are
candidates for having the Painlev\'e property are explored for ODEs of order . Among the 6 ODEs identifying the Painlev\'e transcendents only
, and have nontrivial symmetry algebras and that only
for very special values of the parameters. In those cases the transcendents can
be expressed in terms of simpler functions, i.e. elementary functions,
solutions of linear equations, elliptic functions or Painlev\'e transcendents
occurring at lower order. For higher order or higher degree ODEs that pass the
Painlev\'e test only very partial classifications have been published. We
consider many examples that exist in the literature and show how their symmetry
groups help to identify those that may define genuinely new transcendents
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