This work presents the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-CASAM-N), which enables most efficient computation of exactly determined expressions arbitrarily high-order sensitivities generic nonlinear system responses with respect to model parameters, uncertain boundaries, and internal interfaces in model’s phase space. The mathematical fram...

In this paper, the multiple solutions of Nth-order fuzzy differential equations by the equivalent integral forms are considered. Also, an Existence and uniqueness theorem of solution of Nth-order fuzzy differential equations is proved under Nth-order generalized differentiability in Banach space.

We consider with a new point of view the notion of nth powers in connection with the k-abelian equivalence of words. For a fixed natural number k, words u and v are k-abelian equivalent if every factor of length at most k occurs in u as many times as in v. The usual abelian equivalence coincides with 1-abelian equivalence. Usually k-abelian squares are defined as words w for which there exist n...

Even though the theory of orthogonal polynomials on the unit circle, also known as the theory of Szegő polynomials, is very extensive, it is less known than the theory of orthogonal polynomials on the real line. One reason for this may be that “beautiful” examples on the theory of Szegő polynomials are scarce. This is in contrast to the wonderful examples of Jacobi, Laguerrer and Hermite polyno...

We introduce the notion of well-founded recursive order-sorted equational logic (OS) theories modulo axioms. Such theories de ne functions by well-founded recursion and are inherently terminating. Moreover, for well-founded recursive theories important properties such as con uence and su cient completeness are modular for so-called fair extensions. This enables us to incrementally check these p...

(1) inf{‖f ‖p : f ∈ C [0, 1], f(ti) = ei, i = 1, . . . , m}, where ‖ · ‖p is the usual L [0, 1] norm, 1 ≤ p ≤ ∞, and to characterize f for which the above infimum is attained. If m ≤ n, then this problem has a simple solution. There are algebraic polynomials of degree at most n−1 satisfying the interpolation data, and any of these polynomials obviously solve our problem. If m > n then, in gener...

This paper introduces an analysis technique, commutativity analysis, for automatically parallelizing computations that manipulate dynamic, pointer-based data structures. Commutativity analysis views computations as composed of operations on objects. It then analyzes the program to discover when operations commute, i.e. leave the objects in the same state regardless of the order in which they ex...

After introducing the concept of commutativity for continuous-time linear time-varying systems, the related literature and the results obtained so far are presented. For a simple introduction of the commutativity of discrete-time linear time-varying systems, the problem is formulated for first-order systems. Finally, explicit necessary and sufficient conditions for the commutativity of first-or...

A ring satisfying x = x is necessarily commutative. We consider a variety of weaker forms of this condition and show that many but not all of them imply commutativity. We also present a variety of elementary proofs of the fact that x = x implies commutativity.