On Finite Local Rings with Clique Number Four

We study the algebraic structure of rings [Formula: see text] whose zero-divisor graph [Formula: see text]has clique number four. Furthermore, we give complete characterizations of all the finite commutative local rings with clique number 4.

THE EFFECT OF VIRTUAL LEARNING THROUGH ONLINE LEARNING SYSTEM (SPADA) ON STUDENTS' LEARNING OUTCOMES IN ALGEBRAIC STRUCTURE COURSE IN DEPARTMENT OF MATHEMATICS EDUCATION, MUHAMMADIYAH UNIVERSITY OF MAKASSAR

This study aims to determine the effect of treatment on student learning outcomes in the algebraic structure course at the Department of Mathematics Education of the University of Muhammadiyah Makassar in virtual learning through the online learning system (SPADA). This study is a quasi-experimental design with a post-test-only control group whose results were analyzed using descriptive analysis and inferential analysis (mann whitney test). The population is the class of 2019 who is programming an algebraic structure course consisting of 3 classes, then a sample consisting of 2 classes was selected, namely, one experimental class taught through SPADA, and one control class taught in addition to using SPADA, namely, zoom meeting, google meet, email, and whatsApp groups. The instrument used is a test of student learning outcomes. The results obtained are that the average learning outcomes of students who are taught through SPADA are higher than the average learning outcomes of students who are taught other than through SPADA, but the results of the Mann Whitney test show that there is no significant difference from the learning outcomes of students who have been taught through SPADA or other than SPADA. Researchers suggest that providing virtual learning not only relies on one learning media but also on the synergy between one media and another, which will further optimize virtual learning.

Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities.

Alternative Entropy Measures and Generalized Khinchin–Shannon Inequalities

The Khinchin–Shannon generalized inequalities for entropy measures in Information Theory, are a paradigm which can be used to test the Synergy of the distributions of probabilities of occurrence in physical systems. The rich algebraic structure associated with the introduction of escort probabilities seems to be essential for deriving these inequalities for the two-parameter Sharma–Mittal set of entropy measures. We also emphasize the derivation of these inequalities for the special cases of one-parameter Havrda–Charvat’s, Rényi’s and Landsberg–Vedral’s entropy measures.

A note on almost prime submodule of CSM module over principal ideal domain

An almost prime submodule is a generalization of prime submodule introduced in 2011 by Khashan. This algebraic structure was brought from an algebraic structure in ring theory, prime ideal, and almost prime ideal. This paper aims to construct similar properties of prime ideal and almost prime ideal from ring theory to module theory. The problem that we want to eliminate is the multiplication operation, which is missing in module theory. We use the definition of module annihilator to bridge the gap. This article gives some properties of the prime submodule and almost prime submodule of CMS module over a principal ideal domain. A CSM module is a module that every cyclic submodule. One of the results is that the idempotent submodule is an almost prime submodule.

The Manhattan and Lorentz Mirror Models: A Result on the Cylinder with Low Density of Mirrors

We study the Manhattan and Lorentz mirror models on an infinite cylinder of finite even width n, with the mirror probability p satisfying $$p<Cn^{-1}$$ p < C n - 1 , C a constant. We show that the maximum height along the cylinder reached by a walker is order $$p^{-2}$$ p - 2 . We observe an algebraic structure, which helps organise our argument. The models on the cylinder can be thought of as Markov chains on the Brauer (in the Mirror case) or Walled Brauer (in the Manhattan case) algebra, with the transfer matrix given by multiplication by an element of the algebra.