An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function @math on the reals, and for any positive continuous function @math , it has a @math solution with @math for all @math . Lee A. Rubel provided an explicit example of such a polynomial DAE. Other examples of universal DAE have later been proposed by other authors. However, Rubel's DAE has a unique solution, even with a finite number of conditions of the form @math . The question whether one can require the solution that approximates @math to be the unique solution for a given initial data is a well known open problem [Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we solve it and show that Rubel's statement holds for polynomial ordinary differential equations (ODEs), and since polynomial ODEs have a unique solution given an initial data, this positively answers Rubel's open problem. More precisely, we show that there exists a polynomial ODE such that for any @math and @math there exists some initial condition that yields a solution that is @math -close to @math at all times. In particular, the solution to the ODE is necessarily analytic, and we show that the initial condition is computable from the target function and error function.
First, let us mention that Rubel's universal differential equation has been extended in several papers. In particular, Duffin proved in @cite_8 that implicit universal differential equations with simpler expressions exists, such as @math for any @math The idea of @cite_8 is basically to replace the @math function @math of Rub81 by some piecewise polynomial of fixed degree, that is to say by splines. Duffin also proves that considering trigonometric polynomials for function @math leads to the universal differential equation @math This is done at the price of approximating function @math respectively by splines or trigonometric splines solutions which are @math (and @math can be taken arbitrary big) but not @math as in @cite_18 . Article @cite_14 proposes another universal differential equation whose construction is based on Jacobian elliptic functions. Notice that @cite_14 is also correcting some statements of @cite_8 .
@cite_18 @cite_14 @cite_8
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Abstract Fourth-order differential equations such as 16y′my′2 - 32ymyny′ + 17y03 = 0 are developed. It is shown that the equation is “universal” in the sense that any continuous function can be approximated with arbitrary accuracy over the whole x axis by a solution y(x) of the equation. This solution is a piecewise polynomial of degree 9 and of class C4. I construct a new universal differential equation (B), in the sense of Rubel. That is, its solutions approximate to arbitrary accuracy any continuous function on any interval of the real line. An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper-surfaces is introduced in this paper. The basic idea is to approximate the intrinsic weighted distance by the Euclidean weighted distance computed in a band surrounding the implicit hyper-surface in the embedding space, thereby performing all the computations in a Cartesian grid with classical and efficient numerics. Based on work on geodesics on Riemannian manifolds with boundaries, we bound the error between the two distance functions. We show that this error is of the same order as the theoretical numerical error in computationally optimal, Hamilton‐Jacobi-based, algorithms for computing distance functions in Cartesian grids. Therefore, we can use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on implicit hyper-surfaces a computationally efficient technique. The approach can be extended to solve a more general class of Hamilton‐Jacobi equations defined on the implicit surface, following the same idea of approximating their solutions by the solutions in the embedding Euclidean space. The framework here introduced thereby allows for the computations to be performed on a Cartesian grid with computationally optimal algorithms, in spite of the fact that the distance and Hamilton‐Jacobi equations are intrinsic to the implicit hyper-surface. c ∞ 2001 Academic Press Recently, Frumkin (9) pointed out that none of the well-known algorithms that transform an integer matrix into Smith (16) or Hermite (12) normal form is known to be polynomially bounded in its running time. In fact, Blankinship (3) noticed--as an empirical fact--that intermediate numbersmaybecome quite large during standard calculations Of these canonical forms. Here we present new algorithms in which both the number of algebraic operations and the number of (binary) digits of all intermediate numbers are bounded by polynomials in the length of the input data (assumed to be encoded in binary). These algorithms also find the multiplier-matrices K, U' and K' such thatAK and U'AK' are the Hermite and Smith normal forms of the given matrix A. This provides the first proof that multipliers with small enough entries exist. 1. Introduction. Every nonsingular integer matrix can be transformed into a lower triangular integer matrix using elementary column operations. This was shown by Hermite ((12), Theorem 1 below). Smith ((16), Theorem 3 below) proved that any integer matrix can be diagonalized using elementary row and column operations. The Smith and Hermite normal forms play an important role in the study of rational matrices (calculating their characteristic equations), polynomial matrices (determining the latent roots), algebraic group theory (Newman 15)), system theory (Heymann and Thorpe (13)) and integer programming (Garfi!akel and Nemhauser (10)). Algorithms that compute Smith andHermite normal forms of an integer matrix are given (among others) by Barnette and Pace (1), Bodewig (5), Bradley (7), Frumkin (9) and Hu (14). The methods of Hu, Bodewig and Bradley are based on the explicit calculation of the greatestcommon divisor (GCD) and a set of multiplierswhereas other algorithms ((1)) perform GCD calculations implicitly. As Frumkin (9) pointed out, none of these algorithms is known to be polynomial. In transforming an integer matrix into Smith or Hermite normal form using known techniques, the number of digits of intermediate numbers does not appear to be bounded by a polynomial in the length of the input data as was pointed out by Blankinship (3), (4-1 and Frumkin (9).