Training Multilayer Perceptrons Via Minimization of Sum of Ridge Functions

Motivated by the problem of training multilayer perceptrons in neural networks, we consider the problem of minimizing E(x)= _i=1 ⁿ f _i ( _i x), where _i R ^s , 1in, and each f _i ( _i x) is a ridge function. We show that when n is small the problem of minimizing E can be treated as one of minimizing univariate functions, and we use the gradient algorithms for minimizing E when n is moderately large. For large n, we present the online gradient algorithms and especially show the monotonicity and weak convergence of the algorithms.  

Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions

By extending Wendlands meshless Galerkin methods using RBFs, we develop mixed methods for solving fourth-order elliptic and parabolic problems by using RBFs. Similar error estimates as classical mixed finite element methods are proved. AMS subject classification 35G15, 65N12  

Perturbation from symmetry for indefinite semilinear elliptic equations

We prove the existence of an unbounded sequence of solutions for an elliptic equation of the form \({-\Delta u=\lambda u + a(x)g(u)+f(x), u\in H^1_0(\Omega)}\), where \({\lambda \in \mathbb{R}, g(\cdot)}\) is subcritical and superlinear at infinity, and a(x) changes sign in Ω; moreover, g( − s) =  − g(s) \({\forall s}\). The proof uses Rabinowitz’s perturbation method applied to a suitably truncated problem; subsequent energy and Morse index estimates allow us to recover the original problem. We consider the case of \({\Omega\subset \mathbb{R}^N}\) bounded as well as \({\Omega=\mathbb{R}^N, \, N\geqslant 3}\).  

Application of Faà di Bruno's formula in characterization of inverse relations

It is shown that a pair of inverse relations can be constructed by the suitable application of Faà di Bruno's formula for finding higher order derivatives of composite functions. Various examples illustrating the applications of inverse relations are presented.  

Ternary cyclotomic polynomials with an optimally large set of coefficients

Ternary cyclotomic polynomials are polynomials of the form , where are odd primes and the product is taken over all primitive -th roots of unity . We show that for every there exists an infinite family of polynomials such that the set of coefficients of each of these polynomials coincides with the set of integers in the interval . It is known that no larger range is possible even if gaps in the range are permitted.

  

Canonical vector heights on K3 surfaces with Picard number three-- An argument for nonexistence

In this paper, we investigate a K3 surface with Picard number three and present evidence that strongly suggests a canonical vector height cannot exist on this surface.

  

On nonlinear oscillations in a suspension bridge system

In this paper, we study nonlinear oscillations in a suspension bridge system governed by two coupled nonlinear partial differential equations. By applying the Leray-Schauder degree theory, it is proved that the suspension bridge system has at least two solutions, one is a near-equilibrium oscillation, and the other is a large amplitude oscillation.

  

Flat Cyclotomic Polynomials of Order Three

A cyclotomic polynomial _n is said to be of order 3 if n = pqrfor three distinct odd primes p, q, and r. We establish theexistence of an infinite family of such polynomials whose coefficientsdo not exceed 1 in modulus. 2000 Mathematics Subject Classification11B83, 11C08.  

On an optimality property of Ramanujan sums

We evaluate , where the is taken over sequences satisfying . In particular we show that it is attained by taking for all , which reduces the summation over to a Ramanujan sum .

  

Correlation in Distribution of Two Linear Sequences in the Residue System of a Prime Modulus

In this paper we furnish the answer to the following problem. Given an odd prime p, and integers a and b coprime with p, estimate the number of solutions n, 0≤ n < p, to the inequality , where a₀ and b₀ are arbitrary integers, {x} denotes the fractional part of x, and 0 < γ < 1. As an application of our estimate we give a new bound for coefficients of ternary cyclotomic polynomials , where p, q, and r are distinct odd primes and the product is taken over all primitive pqr-th roots of unity ρ. 2000 Mathematics Subject Classification: Primary–11B50, 11B83