Composition-closed ℓ-groups of almost-piecewise-linear functions

In the present work, we shall construct some non-essential H-closed epireflections of W that are not comparable with any other known H-closed epireflections of W other than the divisible hull and the epicompletion. We show first that the free objects in any H-closed epireflective subcategory must be closed under composition (see Section 2 for a precise definition), and that any epic extension of a free W-object on n generators that is closed under composition is actually the free object on n generators in some H-closed epireflective subcategory of W. We then apply these results to certain ℓ-groups of almost-piecewise-linear Baire functions on R{double-struck}. By definition, a function f:R{double-struck}→R{double-struck} is almost-piecewise-linear if there is a finite point set S⊂R{double-struck} such that f is piecewise-linear on the complement of any neighborhood of S. © 2011 Elsevier B.V

On the Menger and almost Menger properties in locales

[EN] The Menger and the almost Menger properties are extended to locales. Regarding the former, the extension is conservative (meaning that a space is Menger if and only if it is Menger as a locale), and the latter is conservative for sober TD-spaces. Non-spatial Menger (and hence almost Menger) locales do exist, so that the extensions genuinely transcend the topological notions. We also consider projectively Menger locales, and show that, as in spaces, a locale is Menger precisely when it is Lindelöf and projectively Menger. Transference of these properties along localic maps (via direct image or pullback) is considered.The second-named author acknowledges funding from the National Research Foundation of South Africa under Grant 113829.Bayih, T.; Dube, T.; Ighedo, O. (2021). On the Menger and almost Menger properties in locales. Applied General Topology. 22(1):199-221. https://doi.org/10.4995/agt.2021.14915OJS199221221R. N. Ball and J. Walters-Wayland, C- and C*-quotients in pointfree topology, Dissert. Math. (Rozprawy Mat.) 412 (2002), 1-62. https://doi.org/10.4064/dm412-0-1B. Banaschewski and C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37 (1996), 579-589B. Banaschewski and A. Pultr, Variants of openness, Appl. Categ. Structures 2 (1994), 331-350. https://doi.org/10.1007/BF00873038M. Bonanzinga, F. Cammaroto and M. Matveev, Projective versions of selection principles, Topology Appl. 157 (2010), 874-893. https://doi.org/10.1016/j.topol.2009.12.004C. H. Dowker and D. Strauss, Paracompact frames and closed maps, in: Symposia Mathematica, Vol. XVI, pp. 93-116 (Convegno sulla Topologia Insiemistica e Generale, INDAM, Rome, 1973) Academic Press, London, 1975.C. H. Dowker and D. Strauss, Sums in the category of frames, Houston J. Math. 3 (1976), 17-32.T. Dube, M. M. Mugochi and I. Naidoo, Cech completeness in pointfree topology, Quaest. Math. 37 (2014), 49-65. https://doi.org/10.2989/16073606.2013.779986T. Dube, I. Naidoo and C. N. Ncube, Isocompactness in the category of locales, Appl. Categ. Structures 22 (2014), 727-739. https://doi.org/10.1007/s10485-013-9341-8M. J. Ferreira, J. Picado and S. M. Pinto, Remainders in pointfree topology, Topology Appl. 245 (2018), 21-45. https://doi.org/10.1016/j.topol.2018.06.007J. Gutiérrez García, I. Mozo Carollo and J. Picado, Normal semicontinuity and the Dedekind completion of pointfree function rings, Algebra Universalis 75 (2016), 301-330. https://doi.org/10.1007/s00012-016-0378-zW. He and M. Luo, Completely regular proper reflection of locales over a given locale, Proc. Amer. Math. Soc. 141 (2013), 403-408. https://doi.org/10.1090/S0002-9939-2012-11329-2P. T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982.D. Kocev, Menger-type covering properties of topological spaces, Filomat 29 (2015), 99-106. https://doi.org/10.2298/FIL1501099KJ. Madden and J. Vermeer, Lindelöf locales and realcompactness, Math. Proc. Camb. Phil. Soc. 99 (1986), 473-480. https://doi.org/10.1017/S0305004100064410J. Picado and A. Pultr, Frames and Locales: topology without points, Frontiers in Mathematics, Springer, Basel, 2012. https://doi.org/10.1007/978-3-0348-0154-6J. Picado and A. Pultr, Axiom $T_D$ and the Simmons sublocale theorem, Comment. Math. Univ. Carolin. 60 (2019), 701-715.J. Picado and A. Pultr, Notes on point-free topology, manuscript.T. Plewe, Sublocale lattices, J. Pure Appl. Algebra 168 (2002), 309-326. https://doi.org/10.1016/S0022-4049(01)00100-1V. Pták, Completeness and the open mapping theorem, Bull. Soc. Math. France 86 (1958), 41-74. https://doi.org/10.24033/bsmf.1498Y.-K. Song, Some remarks on almost Menger spaces and weakly Menger spaces, Publ. Inst. Math. (Beograd) (N.S.) 112 (2015), 193-198. https://doi.org/10.2298/PIM150513031SJ. J. C. Vermeulen, Proper maps of locales, J. Pure Appl. Algebra 92 (1994), 79-107. https://doi.org/10.1016/0022-4049(94)90047-

A study of localic subspaces, separation, and variants of normality and their duals.

198 p.As in classical topology, in localic topology one often needs to restrict to locales satisfyinga certain degree of separation. In fact, the study of separation in the category of localesconstitutes a non-trivial and important piece of the theory. For instance, it is sometimesimpossible to give an exact counterpart of a classical axiom, while other times a singleproperty for spaces yields multiple non-equivalent localic versions.The main goal of this thesis is to investigate several classes of separated locales and theirconnections with different classes of sublocales, that is, the regular subobjects in the categoryof locales.In particular, we introduce a new diagonal separation and show that it is, in a certainsense, dual to Isbell¿s (strong) Hausdorff property. The duality between suplattices andpreframes, and that between normality and extremal disconnectedness, turn out to be ofspecial interest in this context.Regarding higher separation, we introduce cardinal generalizations of normality andtheir duals (e.g., properties concerning extensions of disjoint families of cozero elements),and give characterizations via suitable insertion or extension results.The lower separation property known as the TD-axiom, also plays an important role inthe thesis. Namely, we investigate the TD-duality between the category of TD-spaces and acertain (non-full) subcategory of the category of locales, identifying the regular subobjects inthe localic side, and provide several applications in point-free topology.Tal como na topologia clássica, também na topologia dos locales (reticulados locais) éfrequente termos que nos restringir a locales que satisfaçam um certo grau de separação.De facto, o estudo de axiomas de separação na categoria dos locales constitui um aspectonão trivial e relevante da teoria. Por exemplo, em alguns casos é impossível termos acontrapartida exacta de um axioma clássico, enquanto noutros casos uma única propriedadepara espaços topológicos produz, na categoria dos locales, diversas versões não equivalentesentre si.O objectivo principal desta tese é investigar várias classes de locales separados e suasconexões com diferentes classes de sublocales (os subobjetos regulares na categoria doslocales).Em particular, introduzimos uma nova propriedade de separação diagonal e mostramosque se trata, em certo sentido, de uma propriedade dual do axioma (forte) de Hausdorffintroduzido por Isbell. As dualidades entre semi-reticulados e reticulados pré-locais, e entrenormalidade e desconexão extrema, acabam por ter um papel relevante neste contexto.Relativamente a axiomas de separação fortes, introduzimos generalizações de normalidade,em função de um cardinal arbitrário, e suas duais (por exemplo, propriedadesenvolvendo extensões de famílias disjuntas de elementos co-zero), e apresentamos caracterizaçõesem termos de propriedades de inserção ou extensão de funções.O axioma TD, uma propriedade de separação muito fraca, também desempenha umpapel importante nesta tese. Especificamente, investigamos a dualidade TD entre a categoriados espaços topológicos TD e uma determinada subcategoria (não plena) da categoria doslocales, identificando os subobjetos regulares na subcategoria de locales, e apresentamosvárias aplicações à topologia sem pontos.Tal y como ocurre en topología clásica, en topología locálica frecuentemente uno tiene querestringir su atención a locales que cumplen cierto grado de separación. De hecho, el estudiode la separación en la categoría de locales es un aspecto no trivial y relevante de la teoría. Enalgunos casos, es imposible dar una contrapartida exacta a un axioma clásico, mientras queen otros casos, una sola propiedad produce multitud de versiones locálicas no equivalentesentre sí.El principal objetivo de esta tesis es investigar varias clases de locales separados y susrelaciones con diferentes clases de sublocales, esto es, los subobjetos regulares en la categoríade locales.En particular, introducimos una nueva separación diagonal, y probamos que es, en ciertosentido, dual al axioma Hausdorff (fuerte) de Isbell. En este contexto, la dualidad entreretículos completos y premarcos, y aquella entre la normalidad y la desconexión extremaresultan ser de especial interés.En cuanto a la separación más fuerte, introducimos generalizaciones cardinales de lanormalidad y sus duales (por ejemplo, propiedades que consisten en la extensión de familiasdisjuntas de elementos cozero), y damos caracterizaciones de las mismas en términos deteoremas de extensión o inserción.Ciertas propiedades de separación más débiles, especialmente el axioma TD, tambiéndesempeñan un papel importante en esta tesis. Específicamente, investigamos la dualidad TDentre la categoría de espacios topológicos TD y cierta subcategoría (no plena) de la categoríade locales, identificando los subobjetos regulares en la categoría de locales, y proporcionamosalgunas aplicaciones en la topología sin puntos

Multicultural education as policy and praxis in Finland: Heading in a problematic direction?

Peer reviewe

Proceedings of the tenth international conference Models in developing mathematics education: September 11 - 17, 2009, Dresden, Saxony, Germany

This volume contains the papers presented at the International Conference on “Models in Developing Mathematics Education” held from September 11-17, 2009 at The University of Applied Sciences, Dresden, Germany. The Conference was organized jointly by The University of Applied Sciences and The Mathematics Education into the 21st Century Project - a non-commercial international educational project founded in 1986. The Mathematics Education into the 21st Century Project is dedicated to the improvement of mathematics education world-wide through the publication and dissemination of innovative ideas. Many prominent mathematics educators have supported and contributed to the project, including the late Hans Freudental, Andrejs Dunkels and Hilary Shuard, as well as Bruce Meserve and Marilyn Suydam, Alan Osborne and Margaret Kasten, Mogens Niss, Tibor Nemetz, Ubi D’Ambrosio, Brian Wilson, Tatsuro Miwa, Henry Pollack, Werner Blum, Roberto Baldino, Waclaw Zawadowski, and many others throughout the world. Information on our project and its future work can be found on Our Project Home Page http://math.unipa.it/~grim/21project.htm It has been our pleasure to edit all of the papers for these Proceedings. Not all papers are about research in mathematics education, a number of them report on innovative experiences in the classroom and on new technology. We believe that “mathematics education” is fundamentally a “practicum” and in order to be “successful” all new materials, new ideas and new research must be tested and implemented in the classroom, the real “chalk face” of our discipline, and of our profession as mathematics educators. These Proceedings begin with a Plenary Paper and then the contributions of the Principal Authors in alphabetical name order. We sincerely thank all of the contributors for their time and creative effort. It is clear from the variety and quality of the papers that the conference has attracted many innovative mathematics educators from around the world. These Proceedings will therefore be useful in reviewing past work and looking ahead to the future

Scottish space sector and innovation: a PERIpatetic study of an emerging innovation system and the roles of innovation intermediaries

This thesis seeks a more effective understanding of Open Innovation (OI) and the available strategies for its development within (geographically–bound) sectoral systems of innovation (GSSIs). Theoretically, it draws upon the competing intellectual traditions (from innovation studies and from science and technology studies) with their different presumptions, which alternatively favour either macro-level positivist or micro-level interpretativist perspectives. These divides prevent a more holistic theoretical understanding of OI, and present a challenge to practitioners, who struggle to operationalise the theories’ insights. Hence, this thesis proposes a novel Practical Epistemology for Researching Innovation, i.e. the PERIpatetic Approach, which aims to integrate multiple theoretical and empirical perspectives for a flexible, problem-driven academic enquiry. This new framework for participatory action research is based on “abductive” theory development, which uses bottom-up empirical engagement to identify emergent challenges to state-of-the-art understanding. The research methodology put forward for this approach is centred on strategic ethnography of innovation, which combines multi-sited mixed-method research design, with constructive embeddedness in the field. The empirical focus of this thesis is on the emergence of the (New) Space Sector in Scotland - mainly made up of small-to-medium-sized enterprises (SMEs). Here, miniaturisation and cheapening of core technologies and increased access to space data has been driving significant sectoral growth and diversification – which is of interest to entrepreneurs and politicians alike. I approached the field by being embedded within an active intermediary, which wanted to understand and respond to these trends and opportunities. Consequently, this thesis analyses the modelling of OI between macro-level trends and micro-level practices, through a focus on the activities and organisational behaviour of a network of SMEs and opportunities to support them through the work of innovation intermediaries. In its first part, the thesis analyses the UK/Scottish innovation policy in the Space Sector, exposing the dispersion of public investment, which is creating divergent clusters. These clusters attempt to integrate through the concept of “Agile Space” into a collaborative “Living Laboratory”, constructing new markets and developing products. Applying social network analysis and outlining a new concept of innovation moments, I focus on the structures at play within this integrative framing, identifying processes of organisational learning which develop structural absorptive capacity. Thus, I form an integrated multi-level perspective on a (geographically–bound) sectoral system of innovation (MLP-GSSI), which can be applied to other OI contexts and can be adapted for analysing other aspects of complex innovation systems. In the thesis’ second part, the analysis seeks to redress the lack of systemic understanding of the central role of innovation intermediaries, by developing new classification and prototypology of their interventions. To validate and operationalise this new model, I apply it to the network of innovation intermediaries in the Scottish (New) Space Sector. I further contextualise this insight through a detailed case study of two large investments in innovation intermediation in similarly positioned Space Sectors - examining the tension between business development and R&D support for OI-driven smart specialisation

Japonisme in Polish Pictorial Arts (1885 – 1939)

This thesis chronicles the development of Polish Japonisme between 1885 and 1939. It focuses mainly on painting and graphic arts, and selected aspects of photography, design and architecture. Appropriation from Japanese sources triggered the articulation of new visual and conceptual languages which helped forge new art and art educational paradigms that would define the modern age. Starting with Polish fin-de-siècle Japonisme, it examines the role of Western European artistic centres, mainly Paris, in the initial dissemination of Japonisme in Poland, and considers the exceptional case of Julian Fałat, who had first-hand experience of Japan. The second phase of Polish Japonisme (1901-1918) was nourished on local, mostly Cracovian, infrastructure put in place by the ‘godfather’ of Polish Japonisme Feliks Manggha Jasieński. His pro-Japonisme agency is discussed at length. Considerable attention is given to the political incentive provided by the outbreak of the Russo-Japanese war in 1904, which rendered Japan as Poland’s ally against its Russian oppressor. The first two decades of the 20th century are regarded as the ‘Renaissance’ of Japonisme in Poland, and it is this part of the thesis that explores Japanese inspirations as manifested in the genres of portraiture, still life, landscape, representations of flora and fauna, erotic imagery, and caricature. Japonisme in graphic and applied graphic arts, including the poster, is also discussed. The existence of the taste for Japanese art in the West after 1918 is less readily acknowledged than that of the preceding decades. The third phase of Polish Japonisme (1919-1939) helps challenge the tacit conviction that Japanese art stopped functioning as an inspirational force around 1918. This part of the thesis examines the nationalisation of heretofore private resources of Japanese art in Cracow and Warsaw, and the inauguration of official cultural exchange between Poland and Japan. Polish Japonisme within École de Paris, both before 1918 and thereafter, inspired mainly by the painting of Foujita Tsuguharu, is an entirely new contribution to the field. Although Japanese inspirations frequently appeared in Polish painting of the interwar period, it was the graphic arts that became most receptive to the Japanese aesthetic at that time. The thesis includes a case study of Leon Wyczółkowski’s interbellum Japonisme, and interprets it as patriotic transpositions of the work of Hiroshige and the Japanese genre of meisho-e. Japonisme in Polish design and architecture is addressed only in the context of the creation of Polish national style in design (1901-1939). Art schools in Britain and America became important centres for Japonisme at the beginning of the 20th century. The thesis considers the case of Cracow Academy of Fine Arts, which due to radical changes introduced by its new director Julian Fałat, became an important centre for the dissemination of the taste for Japanese art in Poland

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book