Nearly Monotone Neural Approximation with Quadratic Activation Function

Estimates on the Approximation of 3-monotone Functions by 3-monotone Quadratic Splines

The period function for second-order quadratic ODEs is monotone∗

Very little is known about the period function for large families of centers. In one of the pioneering works on this problem, Chicone [?] conjectured that all the centers encountered in the family of second-order differential equations ẍ = V (x, ẋ), being V a quadratic polynomial, should have a monotone period function. Chicone solved some of the cases but some others remain still unsolved. In ...

Nearly Comonotone Approximation

We discuss the degree of approximation by polynomials of a function f that is piecewise monotone in ?1; 1]. We would like to approximate f by polynomials which are comonotone with it. We show that by relaxing the requirement for comonotonicity in small neighborhoods of the points where changes in monotonicity occur and near the endpoints, we can achieve a higher degree of approximation. We show...

Nearly Comonotone Approximation Ii

When we approximate a continuous function f which changes its monotonicity nitely many, say s times, in ?1; 1], we wish sometimes that the approximating polynomials follow these changes in monotonicity. However, it is well known that this requirement restricts very much the degree of approximation that the polynomials can achieve, namely, only the rate of ! 2 (f; 1=n) and even this not with a c...

Monotone Spaces and Nearly Lipschitz Maps

A metric space (X, d) is called c-monotone if there is a linear order < on X and c > 0 such that d(x, y) 6 c d(x, z) for all x < y < z in X. A brief account of investigation of monotone spaces including applications is presented. 1 Monotone and σ-Monotone Spaces. In [6] I investigated existence of sets in Euclidean spaces that have large Hausdorff dimension and yet host no continuous finite Bor...