| The limit of the increments of the Hölder means of asymptotically arithmetic sequences | 1$-$14 |
Abstract
We call a sequence of real numbers, $\{a_n\}_{n\geq1}$, an asymptotically arithmetic sequence, if its increment $a_{n+1}-a_{n}$ approaches a real number $d$, as $n\toınfty$. For each $pın[-ınfty,ınfty]$, we compute the limit of the increment $H_p(a_1,\dots,a_n,a_{n+1})-H_p(a_1,\dots,a_n)$, of the $p$-Hölder mean sequence, $\{H_p(a_1,\dots,a_n)\}_{n\geq1}$, of an asymptotically arithmetic sequence $\{a_n\}_{n\geq1}$, with positive terms. Moreover, for $p\leq-1$, we not only show that this limit is $0$, but we also compute the rate with which the increment approaches zero.
Keywords: Hölder means; Stolz-Ces\`{a}ro theorem; D'Alembert theorem; Lagrange Mean Value theorem; Lalescu sequence.
MSC Subject Classification: 97I30
MathEduc Subject Classification: I35
| Two views on entropy in dynamical systems | 15$-$26 |
Abstract
Banach's fixed point theorem is a part of standard curriculum of several university courses. It is also an example of a discrete dynamical system that is very regular -- in the limit, the orbit of each point ``ends'' at a single fixed point. This is the starting point for this article. We begin by analyzing how small changes in the assumptions of this theorem affect the regularity of the system. We then discuss how the concept of regularity and chaos can be formalized. With this goal in mind, we talk about topological entropy. We give definitions and some examples of topological and polynomial entropy in dynamical systems. We also explain two ways of looking at these dynamical invariants. We also consider points that are in a sense the opposite to fixed points, namely wandering points and at the end we explain the role of wandering points in measuring the complexity of a dynamical system.
Keywords: Topological entropy; dynamical system; fixed point; wandering point.
MSC Subject Classification: 97I99
MathEduc Subject Classification: I95
| A chain of eight inequalities involving means of two arguments | 27$-$32 |
Abstract
For two positive real numbers $a$ and $b$, let $H:=H(a,b)$, $G:=G(a,b)$, $A:=A(a,b)$ and $Q:Q(a,b)$ be the harmonic mean, the geometric mean, the arithmetic mean and the quadratic mean of $a$ and $b$, respectively. In this short note, we prove the following interesting chain involving eight inequalities: $G\le\sqrt{QH}\le\sqrt{AG}\le\frac{A+G}2\le\frac{Q+H}2\le\sqrt{\frac{A^2+G^2}2}\le\sqrt{\frac{Q^2+H^2}2}\le\frac{Q+G}2\le A$, where equality holds in each of these inequalities if and only if $a=b$. Some remarks, in particular connected with Muirhead's inequality and two questions related to the similar form of chain of inequalities, are also given.
Keywords: Harmonic mean; geometric mean, arithmetic mean; quadratic mean; H-G-A-Q inequality; Muirhead's inequality.
MSC Subject Classification: 97H30
MathEduc Subject Classification: H34
| Measuring conceptual knowledge of basic algebraic concepts | 33$-$51 |
Abstract
Basic algebra is often the first step that enables school students to enter the world of mathematics. Concepts such as relations, equations, and polynomials are considered basic algebraic concepts. Understanding these basic concepts determines the further progress and development of mathematical competencies. After all, some educational systems insist on developing procedural knowledge in mathematics, which is why these and many other fundamental concepts remain underdeveloped. In this paper, we present the research results at two mathematics faculties in the Western Balkans on students' conceptual knowledge of basic algebraic ideas at the beginning of their studies. We also discuss possible explanations of the results.
Keywords: Conceptual knowledge; procedural knowledge; mathematics education; algebraic concepts.
MSC Subject Classification: 97G40, 97D60
MathEduc Subject Classification: G45, D65
| The distance from a point to a line or a plane in coordinate geometry: A revisit with three elementary solutions | 52$-$58 |
Abstract
The popular problem of the distance from a point to a line in two-dimensional coordinate geometry has been presented in many textbooks, books, and articles with multiple solutions. In this paper, we revisit this problem by presenting three interesting elementary solutions. In addition, we discuss some applications of these solutions in solving some related three-dimensional coordinate geometry problems. These elementary solutions to the two-dimensional and three-dimensional geometry problems can be used suitably in teaching mathematics at high school or junior college level.
Keywords: Distance from a point to a line; distance from a point to a plane; coordinate geometry; elementary solutions.
MSC Subject Classification: 97G70
MathEduc Subject Classification: G14, G44, G74