Current Issue : July-September Volume : 2022 Issue Number : 3 Articles : 5 Articles
This paper presents a graphical procedure, using an unmarked straightedge and compass only , for trisecting an arbitrary acute angle. The procedure, when applied to the 30˚ angle that has been “proven” to be not trisectable, produced a construction having the identical angular relationship with Archimedes’ Construction, in which the required trisection angle was found to be exactly one-third of the given angle (or E'MA = 1/3E'CG = 10˚), as shown in Figure 1(D) and Figure 1(E) and Section 4 PROOF in this paper. Hence, based on this identical angular relationship between the construction presented and Archimedes’ Construction, one can only conclude that geometric requirements for arriving at an exact trisection have been met, notwithstanding the theoretical proofs of Wantzel, Dudley, and others....
In this paper, first, we give a definition of Besicovitch almost periodic functions by using the Bohr property and the Bochner property, respectively; study some basic properties of Besicovitch almost periodic functions, including composition theorem; and prove the equivalence of the Bohr definition and the Bochner definition. Then, using the contraction fixed point theorem, we study the existence and uniqueness of Besicovitch almost periodic solutions for a class of abstract semi-linear delay differential equations. Even if the equation we consider degenerates into ordinary differential equations, our result is new....
The finite field q has q elements, where q = pk for prime p and k ∈ . Then [ ] q x is a unique factorization domain and its polynomials can be bijectively associated with their unique (up to order) factorizations into irreducibles. Such a factorization for a polynomial of degree n can be viewed as conforming to a specific template if we agree that factors with higher degree will be written before those with lower degree, and factors of equal degree can be written in any order. For example, a polynomial f (x) of degree n may factor into irreducibles and be written as (a)(b)(c), where deg a ≥ deg b ≥ deg c . Clearly, the various partitions of n correspond to the templates available for these canonical factorizations and we identify the templates with the possible partitions. So if f (x) is itself irreducible over q , it would belong to the template [n], and if f (x) split over q , it would belong to the template [1,1, ,1] . Our goal is to calculate the cardinalities of the sets of polynomials corresponding to available templates for general q and n. With this information, we characterize the associated probabilities that a randomly selected member of [ ] q x belongs to a given template. Software to facilitate the investigation of various cases is available upon request from the authors....
In this work, a highly efficient algorithm is developed for solving the parabolic partial differential equation (PDE) with the nonlocal condition. For this purpose, we employ orthogonal Chelyshkov polynomials as the basis. The convergence analysis of the proposed scheme is derived. Numerical experiments are carried out to explain the efficiency and precision of the proposed scheme. Furthermore, the reliability of the scheme is verified by comparisons with assured existing methods....
The principles of convexity and symmetry are inextricably linked. Because of the considerable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions (IV-Fs) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs). Then, we obtain Hermite–Hadamard (-Hermite–Hadamard–Fejér (--Fejér type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudoorder relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results....
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