Implementation of The Power of Two Learning Model Operating Materials For Algebrial Form Based on The Power Point of Class VII High Schools

This research is a classroom action research project that aims to improve the mathematics learning outcomes of students in class VII Junior High School 6 Samarinda by using Microsoft PowerPoint to facilitate cooperative learning type The Power of Two on algebraic action material. The research was conducted at Junior High School 6 Samarinda, with 40 students in class VII1 as research subjects nd the object of research is cooperative learning type The Power of Two. Worksheets, final cycle tests, and observation sheets were used as instruments. LKS consists of LKS 1, 2, 3, 4, 5, and 6, which are done in groups in classrooms, as well as assessments at the end of each cycle. The descriptive statistics of averages and proportions were used in the data analysis technique. The analysis and discussion revealed that the type of cooperative going to lean. The research and discussion revealed that the form of cooperative learning used in The Power of Two with Microsoft PowerPoint is a small group learning consisting of two people who collaborate. Cycle I has a basic value of 54.7 and a final value of 66.1 with a 20.8 % increase in the proportion, cycle II has a basic value of 66.1 and a final value of 76.5 with a 15.7 % increase in the proportion, and cycle III has a base value of 76.5 and a final value of 83 with an 8.5 % increase in the proportion. The metrics for the teacher's behavior in the first cycle of accessing are good, the second cycle is good, and the third cycle is really good. As a result, cooperative learning is used to perfect it.

New examples of Bernoulli algebraic actions

We give an example of a principal algebraic action of the non-commutative free group ${\mathbb {F}}$ of rank two by automorphisms of a connected compact abelian group for which there is an explicit measurable isomorphism with the full Bernoulli 3-shift action of ${\mathbb {F}}$ . The isomorphism is defined using homoclinic points, a method that has been used to construct symbolic covers of algebraic actions. To our knowledge, this is the first example of a Bernoulli algebraic action of ${\mathbb {F}}$ without an obvious independent generator. Our methods can be generalized to a large class of acting groups.

Pseudo-orbit tracing and algebraic actions of countable amenable groups

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved that they do. We provide examples showing that Chung and Li’s result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li’s algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.

Invariant ideals of commutative rings

Let R be a commutative Noetherian ring and G a group of elements acting on R as automorphisms. In this note, we are concerned with the structure of the lattice of invariant ideals of R. In particular we shall compute the Krull dimension of this lattice. Our group is an arbitrary group. There are none of the usual assumptions of some sort of algebraic action.