A neural network with mixed order weights, n neurons and a modified Hebbian learning rule can learn any function f : {−1, 1}n → R and reproduce its output as the network’s energy function. The network weights are equal to Walsh coefficients, the fixed point attractors are local maxima in the function, and partial sums across the weights of the network calculate averages for hyperplanes through ...

In this paper, we investigate the order and the hyper-order of solutions of the linear differential equation

Hyper-tubes consisting of hyper-cubes of n-dimensions were designed and formulas for substructures of vary dimensions established.

In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation f ′′ +Q ( e ) f ′ + (A1e a1z +A2e 2) f = 0, where n > 2 is an integer, Aj (z) ( 6≡ 0) (j = 1, 2) are entire functions with max {σ (Aj) : j = 1, 2} < 1, Q (z) = qmz + · · · + q1z + q0 is a nonconstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove tha...

A novel design of reversed domains in a short-range ordered manner by placing randomly oriented basic units on a square lattice is propose and demonstrated. The corresponding reciprocal space is characterized not only by discrete diffraction peaks, but also by the continuous RVs distribution as homocentric rings. Then continuous nonlinear process is available as long as its wave vector mismatch...

Abstract Hyper-singular integral equations are often applied in the frequency-domain wave diffraction/radiation analyses of marine structures with thin plates or shell sub-structures. Their numerical solutions require higher-order derivatives free-surface Green function featuring hyper-singularity, and hence corresponding evaluation is very challenging. To circumvent associated difficulties, th...