We introduce the notion of a 2-absorbing prime fuzzy ideal which is a generalization of prime fuzzy ideal of S. In fact, our analysis demonstrates that if & micro; stands as a fuzzy ideal of S and P (0) does not equal 1, then P is a 2-absorbing prime fuzzy ideal of S only when the cardinality of the image set Im(Q) equals 2. Furthermore, the set P & lowast;= {x is an element of S|P(x) = P (0)} qualifies as a prime ideal of S.
. This article aims to study the long-term behavior of a piecewise linear difference equation system, focusing on solutions that converge to equilibrium points and 4-cycles under initial conditions located in the third quadrant. The system of equations studied is a continuation of previous research. The key results show that this system has a unique equilibrium point and two sets of 4-cycles. Additionally, it identifies the initial conditions that clearly affect the convergence to the equilibrium point or 4-cycles in the third quadrant. These findings help expand the understanding of the role of parameters and initial conditions in difference equation systems.
In 1976, Benkoski gave the probability of choosing any k integers being relatively r-prime for rk > 1. A lattice point in Z(k) is said to be r-primitive if its coordinates are relatively r-prime. In this note, based on the work of Walfisz and Landau, we establish some asymptotic formulas on the number of r-primitive lattice points in a k-dimensional sphere and ellipse.
In this article, we derive series representations of the arc-sine and arccosine functions that yield remarkably accurate approximations for these functions in terms of rational expressions. The computational advantages and applications to electrical engineering of such representations are highlighted, along with the pedagogical benefits of covering such approaches in undergraduate courses.
In this paper we will discuss our perspectives concerning some soft set properties in Boolean near rings and apply the concept of soft sets to them. Additionally, we discuss the idea of soft intersection Boolean near rings and look at some of their characteristics.
In this article, we present examples in which elementary methods related to complex variables can be used to determine the functions to which various trigonometric series converge. The benefits of supplementing coverage of infinite series with this topic in second-semester calculus courses are discussed.
A sunlet is a cycle with a pendant edge attached at each vertex of the cycle. For the bipartite toroidal grid graphs C-2n (sic) C-2n, factorizations into sunlets are given by homomorphisms from disjoint unions of s copies of a sunlet for s is an element of {1, n, n(2)}, n >= 3, such that edges are mapped bijectively.
. Both the classes of R-coneat injective modules and its superclass, pure Baer injective modules, are shown to be preenveloping. The former class is contained in another one, namely, self coneat injectives, i.e. modules M into which every homomorphism, whose kernel contains the annihilator of an element in M and whose domain is a coneat left ideal of R, can be extended to a homomorphism R -> M. Characterizations of certain types of rings are given using properties of the above modules. For instance, a commutative ring R is von Neumann regular precisely when all self coneat injective R-modules are quasi injective.
. In this work, we introduce a new class of modules called as a generalization of e*-extending modules. We gave examples of those concepts and studied their properties as the submodule of purely e*-extending and the submodule of e*- supplement extending; quotient, direct sum, and image homomorphism of purely e*extending and e*- supplement extending. Also, study the relationship to other types, such as e*-uniform modules and e*-closed submodules
In this paper, the concept of interval-valued intuitionistic fuzzy deductive systems (IVIF deductive systems) of Hilbert algebras is introduced. The relationship between deductive systems and IVIF deductive systems is studied in terms of upper and lower-level subsets. We also find a relationship between an IVIF deductive system and its fuzzy deductive system. The homomorphic inverse image of IVIF deductive systems in Hilbert algebras is studied, and some related properties are investigated. Equivalence relations on IVIF deductive systems are discussed.
The aim of this paper is to study some nonlinear elliptic problems with data in L1(ohm) in variable Lebesgue spaces. The existence of entropy solutions is established and an improved regularity result in Musielak-Orlicz spaces is given.
For an algebraically closed field k of characteristic 2, it is proven that a pointed Hopf algebra are isomorphic to the group algebra of odd prime dimension p. Also we show that a pointed Hopf Algebra H of dimension pq for distinct odd primes p, q is isomorphic to either a Taft algebra or a group algebra if p = q and H is isomorphic to a group algebra if p not equal q.
This article continues in a vein explored by the first author, E.Rarity, and J.Z. Schroeder, in which the smallest self-dual embeddable graphs in a pseudosurface were shown to have 7 vertices and 13 edges. We show here that the next-smallest self-dual embeddable graphs in a pseudosurface have 7 vertices and 14 edges, and we also establish a set of 11 candidate graphs for analysis; we establish the possible pseudosurfaces in which they can be embedded, which must have zero Euler characteristic. We conduct a computer search, which implicitly uses homology theory, to find that 7 of the 11 candidate graphs have self-dual embeddings in pseudosurfaces with pinchpoints, and we include embeddings in the pinched projective plane and in the twice pinched sphere with 2 distinct pinchpoints. We explore some properties of these graph embeddings, including examples of self-dual embeddings of the same graphs in the same pseudosurfaces that are not equivalent. We close with a brief discussion of ideas for further investigation.
Let R be a finite commutative ring with non-zero unity. The total graph of R, T Gamma(R), is the simple, undirected graph with all elements of R as vertices and two vertices, x and y, adjacent if x + y is a zero divisor of R. We classify all of the rings whose total graphs have low book thickness. We then show that there are only a finite number of rings for arbitrarily large book thicknesses.
For an inverse sequence {[0, 1], fi} with surjective, upper semi-continuous set-valued bonding functions, we show that if, for each i > 1, the graph of fiis a polygonal graph and the inverse sequence satisfies a dimension-restricting condition, then, for each n > 1, the Mahavier product (or partial graph) G '(f(1), ... , f(n)) is a polygonal graph. We also establish a type of converse to this implication.
We define geometric models for flat twisted K-homology and twisted K-homology with coefficients in R/7. We show that these models are equivalent.
The first derivative test and the second derivative test are commonly used to determine the relative extrema of single variable functions. For multivariable functions, however, most calculus books give only the second partials test for relative extrema. In this paper, we establish a first partials test for multivariable functions and provide several examples to illustrate the applications of this test.
. We introduce the notion of an unfair sequential game of perfect information as a general form of the EN-gammon, which is a concrete model of our discussion. We will discuss the unfairness of the EN-gammon when it is played for an odd number of times. We recall the notion of the EN-gammon, which is a backgammon variant and it follows exactly all the rules of the standard backgammon with the exception that both players using the entangled numbers of the dice for their turns, respectively. That is, the player who rolls the sixsided dice, plays with the face-up numbers showing on the dice and the opponent plays with the face-down numbers of the same rolling of those dice at his/her turn, respectively. In this game, only one of the players (by convention) rolls the dice till end of the game.
