An approach to analyzing structures by using beam elements is developed with adaptive displacement interpolation functions. First, the element stiffness matrix and equivalent nodal loads are derived on the basis of the equilibrium between nodal forces and section forces rather than the compatibility between nodal deformations and section deformations, which avoids discretization errors caused b...

In this paper x, r are real numbers. Let A be a set. We say that A is symmetrical if and only if: (Def. 1) For every complex number x such that x ∈ A holds −x ∈ A. Let us note that there exists a subset of C which is symmetrical. Let us observe that there exists a subset of R which is symmetrical. In the sequel A denotes a symmetrical subset of C. Let R be a binary relation. We say that R has s...

The allowed patterns of a map on a one-dimensional interval are those permutations that are realized by the relative order of the elements in its orbits. The set of allowed patterns is completely determined by the minimal patterns that are not allowed. These are called basic forbidden patterns. In this paper we study basic forbidden patterns of several functions. We show that the logistic map L...

It is shown that all the provably total functions of Basic Arithmetic , a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which is sound, is introduced. This realizability is similar to Kleene’s recursive realizability, except that recursive functions are restricted to primitive recurs...

We consider the von Kármán nonlinearity and the Casimir force to develop reduced-order models for prestressed clamped rectangular and circular electrostatically actuated microplates. Reduced-order models are derived by taking flexural vibration mode shapes as basis functions for the transverse displacement. The in-plane displacement vector is decomposed as the sum of displacements for irrotatio...

With the exception of q-hypergeometric summation, the use of computer algebra packages implementing Zeilberger’s “holonomic systems approach” in a broader mathematical sense is less common in the field of q-series and basic hypergeometric functions. A major objective of this article is to popularize the usage of such tools also in these domains. Concrete case studies showing software in action ...

In this article we introduce some definitions concerning measurable functions and prove related properties. In this paper k is a natural number, r is a real number, i is an integer, and q is a rational number. The subset Z − of R is defined by: (Def. 1) r ∈ Z − iff there exists k such that r = −k. Let us note that Z − is non empty. The following three propositions are true: (1) N ≈ Z −. Z is a ...