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The measure of drought duration strongly depends on the definition considered. In meteorology, dryness is habitually measured by means of fixed thresholds (e.g. 0.1 or 1mm usually define dry spells) or climatic mean values (as is the case of the standardised precipitation index), but this also depends on the aggregation time interval considered. However, robust measurements of drought duration are required for analysing the statistical significance of possible changes. Herein we climatically classified the drought duration around the world according to its similarity to the voids of the Cantor set. Dryness time structure can be concisely measured by the n index (from the regular or irregular alternation of dry or wet spells), which is closely related to the Gini index and to a Cantor-based exponent. This enables the world’s climates to be classified into six large types based on a new measure of drought duration. To conclude, outcomes provide the ability to determine when droughts start and finish. We performed the dry-spell analysis using the full global gridded daily Multi-Source Weighted-Ensemble Precipitation (MSWEP) dataset. The MSWEP combines gauge-, satellite-, and reanalysis-based data to provide reliable precipitation estimates. The study period comprises the years 1979–2016 (total of 45 165 d), and a spatial resolution of 0.5º, with a total of 259 197 grid points. The dataset is publicly available at https://doi.org/10.5281/zenodo.3247041 (Monjo et al., 2019)., Unión Europea. Horizonte 2020, Ministerio de Ciencia e Innovación (MICINN), Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Dedicado a María Teresa Lozano Imízcoz tras 27 años de fructífera colaboración, This is a survey article discussing the author's work in a series of several publications on the relationship between 3-manifolds and wild knots in the 3-sphere and strings in R3 given by branched coverings. He includes an introduction to ordinary combinatorial branched coverings and a parallel introduction to the general topological branched coverings defined by R. H. Fox. Among other things one may conclude that every closed, oriented 3-manifold is a 3-fold covering of the 3-sphere branched over a wild knot. The paper ends with a brief discussion of two open problems., Colciencias, MTM, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

[EN] In this work, we analyze the effect of the distribution of transparent Fresnel regions over the focusing profile of Soret Zone Plates (SZP) based on binary sequences. It is shown that this effect becomes very significant in those fields where directional transducers are employed, such as microwaves or acoustics. A thorough analysis of both the SZP transmission efficiency and the focusing enhancement factor is presented. Moreover, experimental measurements are also carried out for a particular type of binary sequence, the Cantor ternary set, validating the theoretical model and demonstrating that the distribution of transparent Fresnel regions becomes a critical parameter in applications requiring directional emitters.

[EN] In this work, we analyze the effect of the distribution of transparent Fresnel regions over the focusing profile of Soret Zone Plates (SZP) based on binary sequences. It is shown that this effect becomes very significant in those fields where directional transducers are employed, such as microwaves or acoustics. A thorough analysis of both the SZP transmission efficiency and the focusing enhancement factor is presented. Moreover, experimental measurements are also carried out for a particular type of binary sequence, the Cantor ternary set, validating the theoretical model and demonstrating that the distribution of transparent Fresnel regions becomes a critical parameter in applications requiring directional emitters.

[EN] In this work, we analyze the effect of the distribution of transparent Fresnel regions over the focusing profile of Soret Zone Plates (SZP) based on binary sequences. It is shown that this effect becomes very significant in those fields where directional transducers are employed, such as microwaves or acoustics. A thorough analysis of both the SZP transmission efficiency and the focusing enhancement factor is presented. Moreover, experimental measurements are also carried out for a particular type of binary sequence, the Cantor ternary set, validating the theoretical model and demonstrating that the distribution of transparent Fresnel regions becomes a critical parameter in applications requiring directional emitters.

In this thesis I hope to provide a clear and concise introduction to Feigenbaum scaling accessible to undergraduate students. This is accompanied by a description of how to obtain numerical results by various means. A more intricate approach drawing from renormalization theory as well as a short consideration of some of the topological properties will also be presented. I was furthermore trying to put great emphasis on diagrams throughout the text to make the contents more comprehensible and intuitive.

In this work, we show a generalization to the ternary Cantor set based on the beta-expantion of a number, futhermore, we present that, under appropriate hypotheses, this extension also corresponds to a constructive way of the definition of the ternary Cantor set . Finally, we prove that these sets that are, in effect, Cantor sets., En el presente artículo, se expone una generalización del conjunto ternario de Cantor basada en la beta-expansión de un número; además, se presenta que, bajo hipótesis adecuadas, esta extensión también corresponde a la manera constructiva de la definición del conjunto ternario de Cantor. Finalmente, se demuestra que los conjuntos que se obtienen son, en efecto, conjuntos de Cantor.

In this thesis I hope to provide a clear and concise introduction to Feigenbaum scaling accessible to undergraduate students. This is accompanied by a description of how to obtain numerical results by various means. A more intricate approach drawing from renormalization theory as well as a short consideration of some of the topological properties will also be presented. I was furthermore trying to put great emphasis on diagrams throughout the text to make the contents more comprehensible and intuitive.

[EN] In this work, we demonstrate the application of Cantor fractal lenses in acoustics. The Cantor Zone Plate (CZP), previously introduced in optics, is designed from a conventional Fresnel Zone Plate (FZP) using a binary sequence governed by the distribution of a generalized Cantor set. The CZP maintains its main focus at the same focal distance than its associated FZP, providing a softer multi-foci focusing profile which is very useful in certain ultrasound therapeutic applications. Experimental measurements are in good agreement with the theoretical model, demonstrating that CZPs are suitable for the ultrasound field.

[EN] In this work, we demonstrate the application of Cantor fractal lenses in acoustics. The Cantor Zone Plate (CZP), previously introduced in optics, is designed from a conventional Fresnel Zone Plate (FZP) using a binary sequence governed by the distribution of a generalized Cantor set. The CZP maintains its main focus at the same focal distance than its associated FZP, providing a softer multi-foci focusing profile which is very useful in certain ultrasound therapeutic applications. Experimental measurements are in good agreement with the theoretical model, demonstrating that CZPs are suitable for the ultrasound field.

The method of electromotive force (emf) for calculation of antenna impedance is applicable to a very limited number of antenna configurations such as monopole and dipole antenna. In more general antenna structures, especially when complicated geometry such as fractals is involved numerical techniques are essential. There are Electromagnetic (EM) simulation softwares such as G-NEC, XFDTD for the modelling studies in which is essential an understanding of the fundamental theory behind these numerical techniques in creating the antenna models. In this paper, we compare the radiation from dipole antenna and the antenna with shape of the first iteration of Cantor set. The simulation of this problem has been done by COMSOL Multi-physics software.

Fractals is a relatively new mathematical topic which received thorough treatment only starting with 1960's. Fractals can be observed everywhere in nature and in day-to-day life. To give a few examples, common fractals are the spiral cactus, the romanesco broccoli, human brain and the outline of the Swedish map. Fractal dimension is a dimension which need not take integer values. In fractal geometry, a fractal dimension is a ratio providing an index of the complexity of fractal pattern with regard to how the local geometry changes with the scale at which it is measured. In recent years, fractal analysis is used increasingly in many areas of engineering and technology. Among others, fractal analysis is used in signal and image compression, computer and video design, neuroscience and fractal based cancer modelling and diagnosing. This study consists of two main parts. The first part of the study aims to understand the appearance of an irregular Cantor set generated by the chaotic dynamical system generated by the logistic function on the unit interval [0,1]. In order to understand this irregular Cantor set, we studied the topological properties of the Cantor Middle-thirds set and the generalised Cantor sets, all of which have zero length. The necessity to compare these sets with regard to their size led us to the second part of this paper, namely the dimension studies of fractals. More complex fractals were presented in the second part, three definitions of dimension were introduced. The fractal dimension of the irregular Cantor set generated by the logistic mapping was estimated and we found that the Hausdorff dimension has the widest scope and greatest flexibility in the fractal studies., Fraktaler är ett relativt nytt ämne inom matematik som fick sitt stora genomslag först efter 60-talet. En fraktal är ett självliknande mönster med struktur i alla skalor. Några vardagliga exempel på fraktaler är spiralkaktus, romanescobroccoli, mänskliga hjärnan, blodkärlen och Sveriges fastlandskust. Bråktalsdimension är en typ av dimension där dimensionsindexet tillåts att anta alla icke-negativa reella tal. Inom fraktalgeometri kan dimensionsindexet betraktas som ett komplexitetsindex av mönstret med avseende på hur den lokala geometrin förändras beroende på vilken skala mönstret betraktas i. Under det senaste decenniet har fraktalanalysen använts alltmer flitigt inom tekniska och vetenskapliga tillämpningar. Bland annat har fraktalanalysen använts i signal- och bildkompression, dator- och videoformgivning, neurovetenskap och fraktalbaserad cancerdiagnos. Denna studie består av två huvuddelar. Den första delen fokuserar på att förstår hur en fraktal kan uppstå i ett kaotiskt dynamiskt system. För att vara mer specifik studerades den logistiska funktionen och hur denna ickelinjära avbildning genererar en oregelbunden Cantormängd på intervalet [0,1]. Vidare, för att förstå den oregelbundna Cantormängden studerades Cantormängden (eng. the Cantor Middle-Thirds set) och de generaliserade Cantormängderna, vilka alla har noll längd. För att kunna jämföra de olika Cantormängderna med avseende på storlek, leds denna studie vidare till dimensionsanalys av fraktaler som är huvudämnet i den andra delen av denna studie. Olika topologiska fraktaler presenterades, tre olika definitioner av dimension introducerades, bland annat lådräkningsdimensionen och Hausdorffdimensionen. Slutligen approximerades dimensionen av den oregelbundna Cantormängden med hjälp av Hausdorffdimensionen. Denna studie demonstrerar att Hausdorffdimensionen har större omfattning och mer flexibilitet för fraktalstudier.

Fractals is a relatively new mathematical topic which received thorough treatment only starting with 1960's. Fractals can be observed everywhere in nature and in day-to-day life. To give a few examples, common fractals are the spiral cactus, the romanesco broccoli, human brain and the outline of the Swedish map. Fractal dimension is a dimension which need not take integer values. In fractal geometry, a fractal dimension is a ratio providing an index of the complexity of fractal pattern with regard to how the local geometry changes with the scale at which it is measured. In recent years, fractal analysis is used increasingly in many areas of engineering and technology. Among others, fractal analysis is used in signal and image compression, computer and video design, neuroscience and fractal based cancer modelling and diagnosing. This study consists of two main parts. The first part of the study aims to understand the appearance of an irregular Cantor set generated by the chaotic dynamical system generated by the logistic function on the unit interval [0,1]. In order to understand this irregular Cantor set, we studied the topological properties of the Cantor Middle-thirds set and the generalised Cantor sets, all of which have zero length. The necessity to compare these sets with regard to their size led us to the second part of this paper, namely the dimension studies of fractals. More complex fractals were presented in the second part, three definitions of dimension were introduced. The fractal dimension of the irregular Cantor set generated by the logistic mapping was estimated and we found that the Hausdorff dimension has the widest scope and greatest flexibility in the fractal studies., Fraktaler är ett relativt nytt ämne inom matematik som fick sitt stora genomslag först efter 60-talet. En fraktal är ett självliknande mönster med struktur i alla skalor. Några vardagliga exempel på fraktaler är spiralkaktus, romanescobroccoli, mänskliga hjärnan, blodkärlen och Sveriges fastlandskust. Bråktalsdimension är en typ av dimension där dimensionsindexet tillåts att anta alla icke-negativa reella tal. Inom fraktalgeometri kan dimensionsindexet betraktas som ett komplexitetsindex av mönstret med avseende på hur den lokala geometrin förändras beroende på vilken skala mönstret betraktas i. Under det senaste decenniet har fraktalanalysen använts alltmer flitigt inom tekniska och vetenskapliga tillämpningar. Bland annat har fraktalanalysen använts i signal- och bildkompression, dator- och videoformgivning, neurovetenskap och fraktalbaserad cancerdiagnos. Denna studie består av två huvuddelar. Den första delen fokuserar på att förstår hur en fraktal kan uppstå i ett kaotiskt dynamiskt system. För att vara mer specifik studerades den logistiska funktionen och hur denna ickelinjära avbildning genererar en oregelbunden Cantormängd på intervalet [0,1]. Vidare, för att förstå den oregelbundna Cantormängden studerades Cantormängden (eng. the Cantor Middle-Thirds set) och de generaliserade Cantormängderna, vilka alla har noll längd. För att kunna jämföra de olika Cantormängderna med avseende på storlek, leds denna studie vidare till dimensionsanalys av fraktaler som är huvudämnet i den andra delen av denna studie. Olika topologiska fraktaler presenterades, tre olika definitioner av dimension introducerades, bland annat lådräkningsdimensionen och Hausdorffdimensionen. Slutligen approximerades dimensionen av den oregelbundna Cantormängden med hjälp av Hausdorffdimensionen. Denna studie demonstrerar att Hausdorffdimensionen har större omfattning och mer flexibilitet för fraktalstudier.

The infinite plays a significant role in many of Franz Kafka's stories. In this note we show that a version of the middle-thirds Cantor set construction appears in Kafka's Great Wall of China. His description of the Wall's construction mirrors an iterated system of functions whose limit set is the Cantor set; we present the relevant mathematical details and a close reading of the text of the story to illustrate this metaphor.

In this paper, the Weierstrass-Mandelbrot function on the Cantor set is presented with emphasis on possible applications in science and engineering. An asymptotic estimation of its one-sided Fourier transform, in accordance with the simulation results, is analytically derived. Moreover, a time-frequency analysis of the Weierstrass-Mandelbrot function is provided by the numerical computation of its continuous wavelet transform.

An overview is provided of several recent results, constructions and publications relating dynamical systems, wavelets, transfer operators satisfying covariance commutation relations associated to non-invertible dynamics, defining generalizations of crossed product operator algebras to non-invertible dynamics or actions by semigroups, ideals in the corresponding crossed product type algebras and commutants of elements and subsets in the algebras and in their representations. Some open directions and open problems on this rich interplay motivated by these constructions and results are also indicated.

[EN] This work presents a new virtual laboratory, OPHTALMIC, developed with MATLAB GUI for using in Optics and Optometry courses as a computer tool for studying the focusing properties of multifocal diffractive both on unconventional structures both periodic and aperiodic. This virtual laboratory enables students to quickly and easily analyze the influence of the different parameters of construction and find the best solution for a user., [ES] En este trabajo presentamos un nuevo laboratorio virtual, OPHTALMIC, desarrollado en MATLAB GUI (Interfaz Gráfica de Usuario) para su uso en cursos de Óptica y Optometría como una herramienta informática para estudiar las propiedades de focalización de lentes oftálmicas multifocales basadas en estructuras no convencionales, tanto periódicas como aperiódicas. A modo de ejemplo, este laboratorio virtual permite a los estudiantes analizar rápida y fácilmente la in fluencia de los diferentes parámetros de construcción y de este modo encontrar el diseño de la lente oftálmica que más se adecue a las necesidades del usuario.

[EN] This work presents a new virtual laboratory, OPHTALMIC, developed with MATLAB GUI for using in Optics and Optometry courses as a computer tool for studying the focusing properties of multifocal diffractive both on unconventional structures both periodic and aperiodic. This virtual laboratory enables students to quickly and easily analyze the influence of the different parameters of construction and find the best solution for a user., [ES] En este trabajo presentamos un nuevo laboratorio virtual, OPHTALMIC, desarrollado en MATLAB GUI (Interfaz Gráfica de Usuario) para su uso en cursos de Óptica y Optometría como una herramienta informática para estudiar las propiedades de focalización de lentes oftálmicas multifocales basadas en estructuras no convencionales, tanto periódicas como aperiódicas. A modo de ejemplo, este laboratorio virtual permite a los estudiantes analizar rápida y fácilmente la in fluencia de los diferentes parámetros de construcción y de este modo encontrar el diseño de la lente oftálmica que más se adecue a las necesidades del usuario.

[EN] This work presents a new virtual laboratory, OPHTALMIC, developed with MATLAB GUI for using in Optics and Optometry courses as a computer tool for studying the focusing properties of multifocal diffractive both on unconventional structures both periodic and aperiodic. This virtual laboratory enables students to quickly and easily analyze the influence of the different parameters of construction and find the best solution for a user., [ES] En este trabajo presentamos un nuevo laboratorio virtual, OPHTALMIC, desarrollado en MATLAB GUI (Interfaz Gráfica de Usuario) para su uso en cursos de Óptica y Optometría como una herramienta informática para estudiar las propiedades de focalización de lentes oftálmicas multifocales basadas en estructuras no convencionales, tanto periódicas como aperiódicas. A modo de ejemplo, este laboratorio virtual permite a los estudiantes analizar rápida y fácilmente la in fluencia de los diferentes parámetros de construcción y de este modo encontrar el diseño de la lente oftálmica que más se adecue a las necesidades del usuario.

[EN] This work presents a new virtual laboratory, OPHTALMIC, developed with MATLAB GUI for using in Optics and Optometry courses as a computer tool for studying the focusing properties of multifocal diffractive both on unconventional structures both periodic and aperiodic. This virtual laboratory enables students to quickly and easily analyze the influence of the different parameters of construction and find the best solution for a user., [ES] En este trabajo presentamos un nuevo laboratorio virtual, OPHTALMIC, desarrollado en MATLAB GUI (Interfaz Gráfica de Usuario) para su uso en cursos de Óptica y Optometría como una herramienta informática para estudiar las propiedades de focalización de lentes oftálmicas multifocales basadas en estructuras no convencionales, tanto periódicas como aperiódicas. A modo de ejemplo, este laboratorio virtual permite a los estudiantes analizar rápida y fácilmente la in fluencia de los diferentes parámetros de construcción y de este modo encontrar el diseño de la lente oftálmica que más se adecue a las necesidades del usuario.

[EN] This work presents a new virtual laboratory, OPHTALMIC, developed with MATLAB GUI for using in Optics and Optometry courses as a computer tool for studying the focusing properties of multifocal diffractive both on unconventional structures both periodic and aperiodic. This virtual laboratory enables students to quickly and easily analyze the influence of the different parameters of construction and find the best solution for a user., [ES] En este trabajo presentamos un nuevo laboratorio virtual, OPHTALMIC, desarrollado en MATLAB GUI (Interfaz Gráfica de Usuario) para su uso en cursos de Óptica y Optometría como una herramienta informática para estudiar las propiedades de focalización de lentes oftálmicas multifocales basadas en estructuras no convencionales, tanto periódicas como aperiódicas. A modo de ejemplo, este laboratorio virtual permite a los estudiantes analizar rápida y fácilmente la in fluencia de los diferentes parámetros de construcción y de este modo encontrar el diseño de la lente oftálmica que más se adecue a las necesidades del usuario.

[EN] This work presents a new virtual laboratory, OPHTALMIC, developed with MATLAB GUI for using in Optics and Optometry courses as a computer tool for studying the focusing properties of multifocal diffractive both on unconventional structures both periodic and aperiodic. This virtual laboratory enables students to quickly and easily analyze the influence of the different parameters of construction and find the best solution for a user., [ES] En este trabajo presentamos un nuevo laboratorio virtual, OPHTALMIC, desarrollado en MATLAB GUI (Interfaz Gráfica de Usuario) para su uso en cursos de Óptica y Optometría como una herramienta informática para estudiar las propiedades de focalización de lentes oftálmicas multifocales basadas en estructuras no convencionales, tanto periódicas como aperiódicas. A modo de ejemplo, este laboratorio virtual permite a los estudiantes analizar rápida y fácilmente la in fluencia de los diferentes parámetros de construcción y de este modo encontrar el diseño de la lente oftálmica que más se adecue a las necesidades del usuario.

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in [13], and we present several equivalent formulations of the problem.

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in [13], and we present several equivalent formulations of the problem.

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in [13], and we present several equivalent formulations of the problem.

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in [13], and we present several equivalent formulations of the problem.

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in [13], and we present several equivalent formulations of the problem.

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in [13], and we present several equivalent formulations of the problem.

We answer a question by Judith Packer about the irreducibility of the wavelet representation associated to the Cantor set. We prove that if the QMF filter does not have constant absolute value, then the wavelet representation is reducible.

The Freudenthal compactification of a 3-manifold M is a compactification M′ such that the end space e(M)=M′∖M is a totally disconnected compact set. The topological uniformization conjecture states that the Freudenthal compactification of the universal covering of a connected, closed 3-manifold is S3 with its end space tamely embedded. A 3-manifold M has the knot engulfing property if every contractible polyhedral simple closed curve in M lies in some polyhedral 3-cell in M. A classical result by Bing (1958) is that the only simply connected closed 3-manifold with the knot engulfing property is S3. The author proves a generalization of Bing's result: A simply connected 3-manifold M has the knot engulfing property if and only if the Freudenthal compactification M′ of M is homeomorphic to S3 with the end space e(M) tamely embedded. Thus the topological uniformization conjecture can be rephrased as stating that the universal covering of a connected, closed 3-manifold has the knot engulfing property., Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics. In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared. A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics. In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared. A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.

[EN] Up to homeomorphism, there are 9 topologies on a three point set {a, b, c}. Among the resulting topological spaces we have the so called Davey space, where the only non-trivial open set is, let us say, {a}. This is an interesting topological space to the extent that every topological space can be embedded in a product of Davey spaces. In this note we will consider the problem of obtaining the Davey space as a quotient R/G, where G is a suitable homeomorphism group. The present work can be regarded as a follow-up to some previous work done by one of the authors and Bernd Wegner.

[EN] Up to homeomorphism, there are 9 topologies on a three point set {a, b, c}. Among the resulting topological spaces we have the so called Davey space, where the only non-trivial open set is, let us say, {a}. This is an interesting topological space to the extent that every topological space can be embedded in a product of Davey spaces. In this note we will consider the problem of obtaining the Davey space as a quotient R/G, where G is a suitable homeomorphism group. The present work can be regarded as a follow-up to some previous work done by one of the authors and Bernd Wegner.

[EN] Up to homeomorphism, there are 9 topologies on a three point set {a, b, c}. Among the resulting topological spaces we have the so called Davey space, where the only non-trivial open set is, let us say, {a}. This is an interesting topological space to the extent that every topological space can be embedded in a product of Davey spaces. In this note we will consider the problem of obtaining the Davey space as a quotient R/G, where G is a suitable homeomorphism group. The present work can be regarded as a follow-up to some previous work done by one of the authors and Bernd Wegner.

[EN] Up to homeomorphism, there are 9 topologies on a three point set {a, b, c}. Among the resulting topological spaces we have the so called Davey space, where the only non-trivial open set is, let us say, {a}. This is an interesting topological space to the extent that every topological space can be embedded in a product of Davey spaces. In this note we will consider the problem of obtaining the Davey space as a quotient R/G, where G is a suitable homeomorphism group. The present work can be regarded as a follow-up to some previous work done by one of the authors and Bernd Wegner.

Diplomová práce se zabývá historií fraktální geometrie a popisuje vývoj nauky o fraktálech. Po počátečním seznámení se základními pojmy jsou popsány jednotlivé druhy fraktálů a jejich typické příklady. Dále jsou uvedeny oblasti, ve kterých je možno se s fraktály setkat mimo obor počítačové grafiky. Práce seznamuje s praktickým využitím fraktální geometrie. V textu jsou uvedeny v současné době známé programy a softwarové balíky vhodné pro zobrazování fraktálů a jsou popsány jejich možnosti. Praktickou část diplomové práce tvoří slajdy, demonstrační program a plakát. Elektronické slajdy představují osnovou využitelnou pro přednášky o problematice fraktální geometrie. Program slouží k demonstraci vybraných druhů fraktálů. Plakát je grafickým shrnutím výsledků práce., This Master's thesis deals with history of Fractal geometry and describes the fractal science development. In the begining there are essential Fractal science terms explained. Then description of fractal types and typical or most known examples of them are mentioned. Fractal knowledge application besides computer graphics area is discussed. Thesis informs about fractal geometry practical usage. Few present software packages or more programs which can be used for making fractal pictures are described in this work. Some of theirs capabilities are described. Thesis' practical part consists of slides, demonstrational program and poster. Electronical slides represents brief scheme usable for fractal geometry realm lectures. Program generates selected fractal types. Thesis results are projected on poster.

Diplomová práce se zabývá historií fraktální geometrie a popisuje vývoj nauky o fraktálech. Po počátečním seznámení se základními pojmy jsou popsány jednotlivé druhy fraktálů a jejich typické příklady. Dále jsou uvedeny oblasti, ve kterých je možno se s fraktály setkat mimo obor počítačové grafiky. Práce seznamuje s praktickým využitím fraktální geometrie. V textu jsou uvedeny v současné době známé programy a softwarové balíky vhodné pro zobrazování fraktálů a jsou popsány jejich možnosti. Praktickou část diplomové práce tvoří slajdy, demonstrační program a plakát. Elektronické slajdy představují osnovou využitelnou pro přednášky o problematice fraktální geometrie. Program slouží k demonstraci vybraných druhů fraktálů. Plakát je grafickým shrnutím výsledků práce., This Master's thesis deals with history of Fractal geometry and describes the fractal science development. In the begining there are essential Fractal science terms explained. Then description of fractal types and typical or most known examples of them are mentioned. Fractal knowledge application besides computer graphics area is discussed. Thesis informs about fractal geometry practical usage. Few present software packages or more programs which can be used for making fractal pictures are described in this work. Some of theirs capabilities are described. Thesis' practical part consists of slides, demonstrational program and poster. Electronical slides represents brief scheme usable for fractal geometry realm lectures. Program generates selected fractal types. Thesis results are projected on poster.

Many specialists working in the field of the fractional calculus and its applications simply replace the integer differentiation and integration operators by their non-integer generalizations and do not give any serious justifications for this replacement. What kind of "Physics" lies in this mathematical replacement? Is it possible to justify this replacement or not for the given type of fractal and find the proper physical meaning? These or other similar questions are not discussed properly in the current papers related to this subject. In this paper new approach that relates to the procedure of the averaging of smooth functions on a fractal set with fractional integrals is suggested. This approach contains the previous one as a partial case and gives new solutions when the microscopic function entering into the structural-factor does not have finite value at N ≫ 1 (N is number of self-similar objects). The approach was tested on the spatial Cantor set having M bars with different symmetry. There are cases when the averaging procedure leads to the power-law exponent that does not coincide with the fractal dimension of the self-similar object averaged. These new results will help researches to understand more clearly the meaning of the fractional integral. The limits of applicability of this approach and class of fractal are specified. © 2013 Versita Warsaw and Springer-Verlag Wien.

Many specialists working in the field of the fractional calculus and its applications simply replace the integer differentiation and integration operators by their non-integer generalizations and do not give any serious justifications for this replacement. What kind of "Physics" lies in this mathematical replacement? Is it possible to justify this replacement or not for the given type of fractal and find the proper physical meaning? These or other similar questions are not discussed properly in the current papers related to this subject. In this paper new approach that relates to the procedure of the averaging of smooth functions on a fractal set with fractional integrals is suggested. This approach contains the previous one as a partial case and gives new solutions when the microscopic function entering into the structural-factor does not have finite value at N ≫ 1 (N is number of self-similar objects). The approach was tested on the spatial Cantor set having M bars with different symmetry. There are cases when the averaging procedure leads to the power-law exponent that does not coincide with the fractal dimension of the self-similar object averaged. These new results will help researches to understand more clearly the meaning of the fractional integral. The limits of applicability of this approach and class of fractal are specified. © 2013 Versita Warsaw and Springer-Verlag Wien.

Diplomová práce se zabývá historií fraktální geometrie a popisuje vývoj nauky o fraktálech. Po počátečním seznámení se základními pojmy jsou popsány jednotlivé druhy fraktálů a jejich typické příklady. Dále jsou uvedeny oblasti, ve kterých je možno se s fraktály setkat mimo obor počítačové grafiky. Práce seznamuje s praktickým využitím fraktální geometrie. V textu jsou uvedeny v současné době známé programy a softwarové balíky vhodné pro zobrazování fraktálů a jsou popsány jejich možnosti. Praktickou část diplomové práce tvoří slajdy, demonstrační program a plakát. Elektronické slajdy představují osnovou využitelnou pro přednášky o problematice fraktální geometrie. Program slouží k demonstraci vybraných druhů fraktálů. Plakát je grafickým shrnutím výsledků práce., This Master's thesis deals with history of Fractal geometry and describes the fractal science development. In the begining there are essential Fractal science terms explained. Then description of fractal types and typical or most known examples of them are mentioned. Fractal knowledge application besides computer graphics area is discussed. Thesis informs about fractal geometry practical usage. Few present software packages or more programs which can be used for making fractal pictures are described in this work. Some of theirs capabilities are described. Thesis' practical part consists of slides, demonstrational program and poster. Electronical slides represents brief scheme usable for fractal geometry realm lectures. Program generates selected fractal types. Thesis results are projected on poster.

Diplomová práce se zabývá historií fraktální geometrie a popisuje vývoj nauky o fraktálech. Po počátečním seznámení se základními pojmy jsou popsány jednotlivé druhy fraktálů a jejich typické příklady. Dále jsou uvedeny oblasti, ve kterých je možno se s fraktály setkat mimo obor počítačové grafiky. Práce seznamuje s praktickým využitím fraktální geometrie. V textu jsou uvedeny v současné době známé programy a softwarové balíky vhodné pro zobrazování fraktálů a jsou popsány jejich možnosti. Praktickou část diplomové práce tvoří slajdy, demonstrační program a plakát. Elektronické slajdy představují osnovou využitelnou pro přednášky o problematice fraktální geometrie. Program slouží k demonstraci vybraných druhů fraktálů. Plakát je grafickým shrnutím výsledků práce., This Master's thesis deals with history of Fractal geometry and describes the fractal science development. In the begining there are essential Fractal science terms explained. Then description of fractal types and typical or most known examples of them are mentioned. Fractal knowledge application besides computer graphics area is discussed. Thesis informs about fractal geometry practical usage. Few present software packages or more programs which can be used for making fractal pictures are described in this work. Some of theirs capabilities are described. Thesis' practical part consists of slides, demonstrational program and poster. Electronical slides represents brief scheme usable for fractal geometry realm lectures. Program generates selected fractal types. Thesis results are projected on poster.

Diplomová práce se zabývá historií fraktální geometrie a popisuje vývoj nauky o fraktálech. Po počátečním seznámení se základními pojmy jsou popsány jednotlivé druhy fraktálů a jejich typické příklady. Dále jsou uvedeny oblasti, ve kterých je možno se s fraktály setkat mimo obor počítačové grafiky. Práce seznamuje s praktickým využitím fraktální geometrie. V textu jsou uvedeny v současné době známé programy a softwarové balíky vhodné pro zobrazování fraktálů a jsou popsány jejich možnosti. Praktickou část diplomové práce tvoří slajdy, demonstrační program a plakát. Elektronické slajdy představují osnovou využitelnou pro přednášky o problematice fraktální geometrie. Program slouží k demonstraci vybraných druhů fraktálů. Plakát je grafickým shrnutím výsledků práce., This Master's thesis deals with history of Fractal geometry and describes the fractal science development. In the begining there are essential Fractal science terms explained. Then description of fractal types and typical or most known examples of them are mentioned. Fractal knowledge application besides computer graphics area is discussed. Thesis informs about fractal geometry practical usage. Few present software packages or more programs which can be used for making fractal pictures are described in this work. Some of theirs capabilities are described. Thesis' practical part consists of slides, demonstrational program and poster. Electronical slides represents brief scheme usable for fractal geometry realm lectures. Program generates selected fractal types. Thesis results are projected on poster.