Iterations of homographic functions and recurrence equations involving a homographic function

Jan Górowski; Adam Łomnicki

Displaying similar documents to “Iterations of homographic functions and recurrence equations involving a homographic function”

Propozycja wyzwalania twórczości matematycznej studentów przy pomocy pewnej nierówności funkcyjnej

Edyta Fiduk, Stanisław Siudut (2016)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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T. Szostok and Sz. Wąsowicz in (Szostok, Wąsowicz, 2011) studied the following functional inequality: $| F (y) - F (x) - (y - x) f (\frac{x + y}{2}) | \leq ε$ stemming from the Lagrange mean value theorem. They proved that the functon $f$ is affine, provided $f, F : ℝ \to ℝ$ satisfy the above inequality for all $x, y \in ℝ$ . The aim of our paper is to extend the results of (Szostok, Wąsowicz, 2011) to more general situations (for example, we change $ℝ$ to $ℂ$ or $ℍ)$ .

O ilości nierównoważnych metryk

Marcin Zieliński (2017)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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It is well known that there exist many metrics on a non-emptyset. In the case of $(X, ϱ)$ − a finite metric set, it can be easily shown that all the metrics on X are equivalent. This paper examines the number of non-equivalent metrics on uncountably infinite sets.

O sumowaniu szeregu $\sum_{n > a}^{n \leq b} τ (n) f (n)$ , gdzie τ(n) oznacza liczbę rozkładów liczby n na sumę kwadratów dwóch liczb całkowitych

W. Sierpiński (1907)

Prace Matematyczno-Fizyczne

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O liczbach ujemnych z perspektywy historycznej i dydaktycznej

Piotr Błaszczyk, Mirosława Sajka (2017)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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We identify two ways of introducing negative numbers. In the first one, a totally ordered set (L, $≺$ ) is presupposed, an element 0 in L is arbitrarily taken, and a number a is negative when a 0. In the second one, a negative number is defined by the formula a + (−a) = 0. From a mathematical perspective, the first method involves the idea of a totally ordered group (G,+, 0,<), while the second one considers the idea of the algebraic group (G,+, 0) alone. Through the analysis of source...

Fixed-point theorems for multi-valued functions and their applications to functional equations

Roman Węgrzyk

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IntroductionIn this paper we present some fixed point theorems for multi-valued functions (m.v. functions for short), generalizing well-known results (Chapter 2).Next, in Chapter 3, we consider continuous solutions of equation (3.1). First we characterize the continuous solutions of this equation, and then we give some theorems on extending continuous solutions.Chapter 4 contains theorems on the existence of continuous solutions when this equation is of order n, i.e. if the set of indices...

Wstęp do teorii funkcji rzeczywistych

Stefan Banach

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SPIS RZECZY PRZEDMOWA...................... III WSTĘP. Liczby rzeczywiste...... 1 1. Aksjomaty i definicje. 2. Zbiory liniowe. 3. Liczby nieskończone. ROZDZIAŁ I. Teoria zbiorów § 1. Algebra zbiorów....... 4 1. Działania na zbiorach. 2. Działania nieskończone. 3. Znakowanie logiczne. 4. Produkt zbiorów. Funkcje zdaniowe wielu zmiennych. 5. Interpretacja geometryczna kwantora. § 2. Odwzorowania zbiorów, pojęcie ciągu, produkt nieskończony zbiorów...... 14 1. Odwzorowanie (funkcja). 2....

Cechy równoboczności trójkątów

Jan Górowski, Adam Łomnicki (2012)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The paper presents some non-standard (in the sense: not yet presented in the literature), necessary and sufficient conditions for a triangle to be isosceles.

Przekształcenia wykresów funkcji

Antoni Chronowski (2006)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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This article is concerned with some topics of the transformationsof the graphs of functions. It is designated to serve as a text for thestudents of mathematics (prospective teachers) and the teachers of ma-thematics.In this paper we give the answer to the following problem:Given the graph of the function f:R Df

O różnych sposobach definiowania wartość bezwzględnej

Antoni Chronowski, Joanna Major, Zbigniew Powązka (2009)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this article we present some didactic ideas of introducing theabsolute value of a real number by means of functional equations and aninequality. The basic properties of the absolute value of a real number arewritten as functional equations and an inequality. The function "absolutevalue" is a solution of a suitable functional equation or an inequality.

Nowe cechy równobocznych trójkątów

Michał Biesiada, Jan Górowski, Adam Łomnicki (2014)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper the authors formulate and prove several conditions fortriangle to be equilateral. These conditions are associated with Gergonne,Nagel and Torricelli points and were obtained by composing and solvingthe so-called ‘enforcement tasks’. Both, the method and the results, can beused to trigger some mathematical student activities at different levels ofeducation or even some teacher activities.

Different ways of solving quadratic equations

Barbara Pieronkiewicz, James Tanton (2019)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper we explore different ways of solving quadratic equations. Our main goal is to review traditional textbooks methods and offer an alternative, often side-stepped method based on the area model. We conclude that whereas traditional methods offer effective algorithms that quickly lead to the desired results, alternative methods may enhance meaningful and joyful learning.

Prime counting function π

Jan Górowski, Adam Łomnicki (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The aim of this paper is to derive new explicit formulas for thefunction π, where π(x) denotes the number of primes not exceeding x. Some justifications and generalisations of the formulas obtained by Willans (1964),Minac (1991) and Kaddoura and Abdul-Nabi (2012) are also obtained.

O nieskończonych ciągach liczb parami względnie pierwszych raz jeszcze

Jan Górowski, Adam Łomnicki (2012)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In the paper the authors, using general formulas, determine and describe a class of infinite sequences of natural numbers, pairs of which are relatively prime. This result was obtained using the theory of Lucas sequences and recurrent equations of the second degree.

O związku między istnieniem granicy $l i m_{Δ x = 0} (Δ ³ f (x) / Δ x ³)$ a ciągłością funkcyi f(x)

Stefan Mazurkiewicz (1915)

Prace Matematyczno-Fizyczne

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Rola portów morskich w rozwoju cruisingu na Karaibach

Renata Rettinger, Anna Urbańska (2012)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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Sea tourism brings huge profits to the Caribbean. The cruise market is set mainly for tourists coming from North America. Low travel costs, high income and the long-lasting tradition of sailing excursions are the main reasons. More and more European tourists can also be seen in the Caribbean thanks to numerous air connections. The cruise routes follow the most attractive ports. The ongoing competition causes the increase of quality of passenger’s service and goods. The investment of...

Potential of tourism services in the Irkutsk region

Tatyana Kirillova (2012)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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O badaniach nad kształtowaniem się u studentów matematyki podstawowych pojęć analizy matematycznej

Zbigniew Powązka (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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This paper is a reflection on the process of forming the basic concepts of mathematical analysis in students. These studies were conducted in the years 2002-2011 and they coincided with the period of major educational reforms in our country. These reforms had an impact on the level of Mathematics teaching in various schools and on anticipatory preparing of young people to study this subject. Thus, this work is an attempt to summarize the results of the already described various stages...

Sur les rapports entre l’existence des intégrales $\int_{0}^{1} f (x, y) d x$ , $\int_{0}^{1} f (x, y) d y$ et $\int_{0}^{1} d x \int_{0}^{1} f (x, y) d y$

Wacław Sierpiński (1920)

Fundamenta Mathematicae

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Le but de cette note est de démontrer que la réponse au problème (posée par Stanisław Ruziewicz) suivant: L'existence (pour une function bornée f(x,y), définie dans le carré 0 ≤ x ≤ 1, 0 ≤ y ≤ 1) des intégrales au sens de Lebesgue: ∫_0^1f(x,y)dx pour 0 ≤ y ≤ 1 ∫_0^1f(x,y)dy pour 0 ≤ x ≤ 1 entraîne-t-elle toujours l'existence de l'intégrale (au sens de Lebesgue) ∫_0^1 dx∫_0^1f(x,y)dy ? est négative, si l'on admet l'hypothèse du continu.

The role of visualisation in the process of generalisation

Lidia Zaręba (2015)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In my paper, I discuss the results of individual observations of pupils aged 13-14. These children undertook an attempt to solve problems which were designed in order to bring about an inductive type of generalizing. The main aim of the study was to classify typical methods of proceeding, which represent the pupils’ ways of reasoning and to determine what influence a particular method of proceeding may have on the final outcome of their work. These results indicate the general pupils’...

Wprowadzenie funkcji trygonometrycznych przez szeregi Eisensteina

Maciej Klakla, Renata Liszka, Vladimir Mityushev (2009)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In the paper, we describe how to introduce the trigonometric functions using their functional characteristics and the Eisenstein series.

Wyniki badań nad stanem nauczania statystyki opisowej w Polsce

Zofia Aleksandra Tomaszewska (2006)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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This article presents the results of the study of teaching descriptive statistics in Poland nowadays. After a short review of the history of teaching this branch of school mathematics in the second part of the 20th century, I made an attempt at diagnosing the stage of teacher’s preparation for teaching descriptive statistics. The diagnosis was based on the analysis of: - the survey conducted among working teachers; - teaching standards for mathematics at universities; - curriculum and...

Współczesne problemy i praktyka kształcenia matematycznego studentów

Antoni Pardała (2012)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The subject matter of this paper relates to one of the modern research trends in teaching mathematics: the problems of quality of mathematical education of pupils and students in the twenty-first century in different countries and cultural traditions. This analysis throws light on the literature, research results, and observations from the Polish perspective of practice, and outlines the intentions of the range. The work also discusses some theoretical bases for the concept of improving...

Tourism development in small cities on the example of Pistoia in Tuscany

Małgorzata Bajgier-Kowalska, Renata Rettinger (2014)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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The dynamic development of tourism in many regions of the world causes competition boost which, in turn, leads to a variety of changes in tourist offers. Tourism is an alternative for other types of economic activity, this relates to spatial units of diverse area, including cities. The tourist function can become the mechanism of local development for cities and the protection of the cultural and natural heritage. The main goal of the article is to present the tourist potential of a...

Musielak-Orlicz algebras

Hudzik, Henryk

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Around the Wilson's theorem

Jan Górowski, Adam Łomnicki (2013)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In this paper some known conditions and new congruences characterising prime numbers are given. Some of them are obtained by the generalised Wilson theorem given by Gauss. The elementary proof of this theorem is also presented.

O elementach logiki erotetycznej w edukacji matematycznej

Tadeusz Rams (2012)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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Asking questions is one of the most common activities of mathematics teachers at various levels of education. Moreover, the way the teachers formulate questions has profound impact on the results of teaching, especially on their pupils' reasoning process. Erotetic logic is frequently presented in a very formal manner. In this paper, we restrict our attention to these aspects of erotetic logic that can turn it into a useful framework to describe various didactic situations. In particular,...

Innowacyjność w hotelarstwie z uwzględnieniem adaptacji obiektów zabytkowych na bazę noclegową

Kamila Ziółkowska-Weiss (2012)

Annales Universitatis Paedagogicae Cracoviensis Studia Geographica

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Nowadays, tourism is a crucial sector of national economy, and for many countries and regions it is one of the goals of economic activity, at the same time being a means of social and economic revival. Therefore, in order to professionally present the tourist potential and offer, and to attract tourists, tourist products should be organized, promoted and introduced in a relevant way. It is necessary to bear in mind that tourism is a service sector, a sector in which different kinds of...

A remark on $C D_{0} (K)$ -spaces.

Alpay, Safak, Ercan, Zafer (2006)

Sibirskij Matematicheskij Zhurnal

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Актуальные проблемы обучения математике в школе и вузе в свете идей Л. С. Выготского

Людмила Боженкова, Марина Егупова (2016)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The article gives a brief overview of the scientific results of the III International Conference held in Moscow State Pedagogical University in November 2016. The main directions of the development of ideas of L. S. Vygotsky, presented in the reports of leading Russian and foreign scientists – participants of the conference.

Применение методов математической статистики в научных исследованиях

Владимир В. Афанасьев (2012)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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In the paper, the author presents the basic concepts and theorems concerning methods of mathematical statistics and its application in the domain of empirical research. Several examples presented in the paper can be useful as patterns of using statistical tool in didactic research.