In this paper, a discontinuity in beams whose intensity is adjusted by the spring stiffness factor is modeled using a torsional spring. Adapting two analyses in strong and weak forms for discontinuous beams, the improved governing differential equations and the modified stiffness matrix are derived respectively. In the strong form, two different solution methods have been presented to make an a...

We discuss three forms of convergence in distribution which are stronger than the normal weak convergence. They have the advantage that they are non-topological in nature and are inherited by discontinuous functions of the original random variables—clearly an improvement on ‘normal’ weak convergence. We give necessary and sufficient conditions for the three types of convergence and go on to giv...

We prove that in a rosy theory, the n-uniqueness of a strong type p implies the strong n-uniqueness of p. In addition we study some of the boundary properties of p. In [5], given a strong type p = tp(c/A) in a stable theory T , if T has (≤ n)-uniqueness while p does not have (n + 1)-uniqueness then, over an infinite Morley sequence of the type of some finite tuple c′ ∈ acl(cA) over A witnessing...

In this paper, we will obtain the strong type and weak type estimates for vector-valued analogs of intrinsic square functions in the generalized weighted Morrey spacesM w (l2). We study the boundedness of intrinsic square functions including the Lusin area integral, the Littlewood-Paley g-function and gλ -function, and their multilinear commutators on vector-valued generalized weighted Morrey s...

We discuss the properties of strictly typed languages on the one hand and soft typing of the other and identify disadvantages of these approaches to type checking in the context of powerful type languages. To overcome the problems we develop an approach that combines ideas of strict and soft typing. This approach is based on the concept of complete typing that is guaranteed to accept every well...

When using existing ACL2 datatype frameworks, many theorems require type hypotheses. These hypotheses slow down the theorem prover, are tedious to write, and are easy to forget. We describe a principled approach to types that provides strong type safety and execution efficiency while avoiding type hypotheses, and we present a library that automates this approach. Using this approach, types help...