Volume XIV , issue 2 ( 2011 ) | back |

Milosav Marjanović - on the occasion of his 80th birthday | 57$-$70 |

**Abstract**

The article is dedicated to Professor Milosav Marjanović, academician and professor at Faculty of Mathematics, University of Belgrade and chief editor of the ``Teaching of Mathematics'', on the occasion of his 80th birthday. It contains a short curriculum vitae of M. Marjanović, an interview with him (titled ``An occasional talk with professor M. Marjanović'') and a review of his selected publications.

A universal sequence of continuous functions | 71$-$76 |

**Abstract**

We show that for each positive integer $k$ there is a sequence $F_n:\Bbb{R}^k \rightarrow\Bbb{R}$ of {ıt continuous} functions which represents via point-wise limits {ıt arbitrary} functions $G\:X^k\rightarrow \Bbb{R}$ defined on domains $X\subseteq \Bbb{R}$ of sizes not exceeding a standard cardinal characteristic of the continuum.

**Keywords:** Point-wise limit; continuum; continuous function.

Farkas' lemma of alternative | 77$-$86 |

**Abstract**

We will present Farkas' formulation of the theorem of alternative related to solvability of a system of linear inequalities and present one review of the proofs based on quite different ideas.

**Keywords:** Theorems of alternative; separating hyperplane; absence of arbitrage.

A contribution to the development of functional thinking related to convexity and one-dimensional motion | 87$-$96 |

**Abstract**

Mathematical concepts are defined precisely using the language of the branch of mathematics to which they belong. But their meaning can be enriched through different interpretations and those of them belonging to the real world situations, we call ``vivid'' mathematics. In contacts with Professor M. Marjanović, we investigated a case of ``vivid'' mathematics in some earlier papers and we continue to do so in this paper. Suppose that a liquid (water) flow has a constant inflow rate and that a vessel has the form of a surface of revolution, and suppose that this process begins at moment $t=0$ and ends at moment $t=T$. We study the dependence of the height $h(t)$ of the liquid level at the time $t$, which will be called the {ıt height filling function}. It is convex or concave depending on the way how the level of the liquid changes. This vivid interpretation holds in general, namely we prove that given a strictly increasing convex (concave) continuous function on $[0,T]$ satisfying certain conditions, there exists a vessel such that its height filling function is equal to the given function. This is a fact that seems to be new and we continue paying attention to it. In this way, we hope that we are providing a matter that can serve as a motivation and an illustration for a deeper understanding of basic concepts and ideas of the differential and integral calculus. It can also serve for a further development of functional thinking in teaching mathematics. We also consider a more general concept of one-dimensional motion, including changes in direction of motion and the difference between velocity and acceleration defined by the position and the path as functions of time. We indicate how one can apply this for studying the height filling function of a liquid flow, which can be considered now as a one-dimensional motion of a liquid along the axis of rotation of the vessel.

**Keywords:** Height filling function; monotone function; convex function; one-dimensional motion.

Some classical inequalities and their application to olympiad problems | 97$-$106 |

**Abstract**

The technique introduced by M. Marjanović in [10] is used to prove several classical inequalities. Examples of application are given which can be used for preparing students for mathematical competitions.

**Keywords:** Chebyshev's inequality; Karamata's inequality; Steffensen's inequality; Jensen's inequality.

Euler formula and maps on surfaces | 107$-$117 |

**Abstract**

This paper is addressed primarily to those high school students with an intensive interest in mathematics, who are often in search for some extra reading materials not being on school curriculum, as well as to their teachers. We have chosen to offer here a material elaborating how the Euler formula could be used to establish non-planarity of some graphs, and results about the colorings of maps on surfaces.

**Keywords:** Euler formula; planar and non-planar graphs; coloring of maps.

Let's get acquainted with mapping degree! | 119$-$136 |

**Abstract**

Given a continuous map $f : M\rightarrow N$ between oriented manifolds of the same dimension, the associated {ıt degree} $deg(f)$ is an integer which evaluates the number of times the domain manifold $M$ ``wraps around'' the range manifold $N$ under the mapping $f$. The mapping degree is met at almost every corner of mathematics. Some of its avatars, pseudonyms, or close relatives are ``winding number'', ``index of a vector field'', ``multiplicity of a zero'', ``Milnor number of a singularity'', ``degree of a variety'', ``incidence numbers of cells in a $CW$-complex'', etc. We review some examples and applications involving this important invariant. One of emerging guiding principles, useful for a mathematical student or teacher, is that the study of mathematical concepts which transcend the boundaries between different mathematical disciplines should receive a special attention in mathematical (self)education.

**Keywords:** Mapping degree; winding number.

Three manifestations of Morse theory in two dimensions | 137$-$145 |

**Abstract**

In this article we present main notions and ideas of Morse theory in two dimensions, adjusted to school teachers and their talented students. We count numbers of critical points of different types and obtain interesting results about plane curves, mountainous landscapes and planets. We also derive the Euler formula for polyhedra.

**Keywords:** Critical point; Morse function; polyhedron.

How to understand Grassmannians? | 147$-$157 |

**Abstract**

Grassmannians or Grassmann manifolds are very important manifolds in modern mathematics. They naturally appear in algebraic topology, differential geometry, analysis, combinatorics, mathematical physics, etc. Grassmannians have very rich geometrical, combinatorial and topological structure, so understanding them has been one of the central research themes in mathematics. They occur in many important constructions such as universal bundles, flag manifolds and others, hence studying their properties and finding their topological and geometrical invariants is still a very attractive question. In this article we offer a quick introduction into the geometry of Grassmannians suitable for readers without any previous exposure to these concepts.

**Keywords:** Grassmann manifold.