387 research outputs found
Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators
In a previous paper by the first two authors, a tube formula for fractal
sprays was obtained which also applies to a certain class of self-similar
fractals. The proof of this formula uses distributional techniques and requires
fairly strong conditions on the geometry of the tiling (specifically, the inner
tube formula for each generator of the fractal spray is required to be
polynomial). Now we extend and strengthen the tube formula by removing the
conditions on the geometry of the generators, and also by giving a proof which
holds pointwise, rather than distributionally.
Hence, our results for fractal sprays extend to higher dimensions the
pointwise tube formula for (1-dimensional) fractal strings obtained earlier by
Lapidus and van Frankenhuijsen.
Our pointwise tube formulas are expressed as a sum of the residues of the
"tubular zeta function" of the fractal spray in . This sum ranges
over the complex dimensions of the spray, that is, over the poles of the
geometric zeta function of the underlying fractal string and the integers
. The resulting "fractal tube formulas" are applied to the important
special case of self-similar tilings, but are also illustrated in other
geometrically natural situations. Our tube formulas may also be seen as fractal
analogues of the classical Steiner formula.Comment: 43 pages, 13 figures. To appear: Advances in Mathematic
Courbure discrÚte : théorie et applications
International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor
Geometric Dilation and Halving Distance
Let us consider the network of streets of a city represented by a geometric graph G in the plane. The vertices of G represent the crossroads and the edges represent the streets. The latter do not have to be straight line segments, they may be curved. If one wants to drive from a place p to some other place q, normally the length of the shortest path along streets, d_G(p,q), is bigger than the airline distance (Euclidean distance) |pq|. The (relative) DETOUR is defined as delta_G(p,q) := d_G(p,q)/|pq|. The supremum of all these ratios is called the GEOMETRIC DILATION of G. It measures the quality of the network. A small dilation value guarantees that there is no bigger detour between any two points. Given a finite point set S, we would like to know the smallest possible dilation of any graph that contains the given points on its edges. We call this infimum the DILATION of S and denote it by delta(S). The main results of this thesis are - a general upper bound to the dilation of any finite point set S, delta(S) - a lower bound for a specific set P, delta(P)>(1+10^(-11))pi/2, which approximately equals 1.571 In order to achieve these results, we first consider closed curves. Their dilation depends on the HALVING PAIRS, pairs of points which divide the closed curve in two parts of equal length. In particular the distance between the two points is essential, the HALVING DISTANCE. A transformation technique based on halving pairs, the HALVING PAIR TRANSFORMATION, and the curve formed by the midpoints of the halving pairs, the MIDPOINT CURVE, help us to derive lower bounds to dilation. For constructing graphs of small dilation, we use ZINDLER CURVES. These are closed curves of constant halving distance. To give a structured overview, the mathematical apparatus for deriving the main results of this thesis includes - upper bound: * the construction of certain Zindler curves to generate a periodic graph of small dilation * an embedding argument based on a number theoretical result by Dirichlet - lower bound: * the formulation and analysis of the halving pair transformation * a stability result for the dilation of closed curves based on this transformation and the midpoint curve * the application of a disk-packing result In addition, this thesis contains - a detailed analysis of the dilation of closed curves - a collection of inequalities, which relate halving distance to other important quantities from convex geometry, and their proofs; including four new inequalities - the rediscovery of Zindler curves and a compact presentation of their properties - a proof of the applied disk packing result.Geometrische Dilation und Halbierungsabstand Man kann das von den StraĂen einer Stadt gebildete Netzwerk durch einen geometrischen Graphen in der Ebene darstellen. Die Knoten dieses Graphen reprĂ€sentieren die Kreuzungen und die Kanten sind die StraĂen. Letztere mĂŒssen nicht geradlinig sein, sondern können beliebig gekrĂŒmmt sein. Wenn man nun von einem Ort p zu einem anderen Ort q fahren möchte, dann ist normalerweise die LĂ€nge des kĂŒrzesten Pfades ĂŒber StraĂen, d_G(p,q), lĂ€nger als der Luftlinienabstand (euklidischer Abstand) |pq|. Der (relative) UMWEG (DETOUR) ist definiert als delta_G(p,q) := d_G(p,q)/|pq|. Das Supremum all dieser BrĂŒche wird GEOMETRISCHE DILATION (GEOMETRIC DILATION) von G genannt. Es ist ein MaĂ fĂŒr die QualitĂ€t des StraĂennetzes. Ein kleiner Dilationswert garantiert, dass es keinen gröĂeren Umweg zwischen beliebigen zwei Punkten gibt. FĂŒr eine gegebene endliche Punktmenge S wĂŒrden wir nun gerne bestimmen, was der kleinste Dilationswert ist, den wir mit einem Graphen erreichen können, der die gegebenen Punkte auf seinen Kanten enthĂ€lt. Dieses Infimum nennen wir die DILATION von S und schreiben kurz delta(S). Die Haupt-Ergebnisse dieser Arbeit sind - eine allgemeine obere Schranke fĂŒr die Dilation jeder beliebigen endlichen Punktmenge S: delta(S) - eine untere Schranke fĂŒr eine bestimmte Menge P: delta(P)>(1+10^(-11))pi/2, was ungefĂ€hr der Zahl 1.571 entspricht Um diese Ergebnisse zu erreichen, betrachten wir zunĂ€chst geschlossene Kurven. Ihre Dilation hĂ€ngt von sogenannten HALBIERUNGSPAAREN (HALVING PAIRS) ab. Das sind Punktpaare, die die geschlossene Kurve in zwei Teile gleicher LĂ€nge teilen. Besonders der Abstand der beiden Punkte ist von Bedeutung, der HALBIERUNGSABSTAND (HALVING DISTANCE). Eine auf den Halbierungspaaren aufbauende Transformation, die HALBIERUNGSPAARTRANSFORMATION (HALVING PAIR TRANSFORMATION), und die von den Mittelpunkten der Halbierungspaare gebildete Kurve, die MITTELPUNKTKURVE (MIDPOINT CURVE), helfen uns untere Dilationsschranken herzuleiten. Zur Konstruktion von Graphen mit kleiner Dilation benutzen wir ZINDLERKURVEN (ZINDLER CURVES). Dies sind geschlossene Kurven mit konstantem Halbierungspaarabstand. Die mathematischen Hilfsmittel, mit deren Hilfe wir schlieĂlich die Hauptresultate beweisen, sind unter anderem - obere Schranke: * die Konstruktion von bestimmten Zindlerkurven, mit denen periodische Graphen kleiner Dilation gebildet werden können * ein Einbettungsargument, das einen zahlentheoretischen Satz von Dirichlet benutzt - untere Schranke: * die Definition und Analyse der Halbierungspaartransformation * ein StabilitĂ€tsresultat fĂŒr die Dilation geschlossener Kurven, das auf dieser Transformation und der Mittelpunktkurve basiert * die Anwendung eines Kreispackungssatzes ZusĂ€tzlich enthĂ€lt diese Dissertation - eine detaillierte Analyse der Dilation geschlossener Kurven - eine Sammlung von Ungleichungen, die den Halbierungsabstand zu anderen wichtigen GröĂen der Konvexgeometrie in Beziehung setzen, und ihre Beweise; inklusive vier neuer Ungleichungen - die Wiederentdeckung von Zindlerkurven und eine kompakte Darstellung ihrer Eigenschaften - einen Beweis des angewendeten Kreispackungssatzes
Geometric algorithms for algebraic curves and surfaces
This work presents novel geometric algorithms dealing with algebraic curves and surfaces of arbitrary degree. These algorithms are exact and complete â they return the mathematically true result for all input instances. Efficiency is achieved by cutting back expensive symbolic computation and favoring combinatorial and adaptive numerical methods instead, without spoiling exactness in the overall result. We present an algorithm for computing planar arrangements induced by real algebraic curves. We show its efficiency both in theory by a complexity analysis, as well as in practice by experimental comparison with related methods. For the latter, our solution has been implemented in the context of the Cgal library. The results show that it constitutes the best current exact implementation available for arrangements as well as for the related problem of computing the topology of one algebraic curve. The algorithm is also applied to related problems, such as arrangements of rotated curves, and arrangments embedded on a parameterized surface. In R3, we propose a new method to compute an isotopic triangulation of an algebraic surface. This triangulation is based on a stratification of the surface, which reveals topological and geometric information. Our implementation is the first for this problem that makes consequent use of numerical methods, and still yields the exact topology of the surface.Diese Arbeit stellt neue Algorithmen fĂŒr algebraische Kurven und FlĂ€chen von beliebigem Grad vor. Diese Algorithmen liefern fĂŒr alle Eingaben das mathematisch korrekte Ergebnis. Wir erreichen Effizienz, indem wir aufwendige symbolische Berechnungen weitesgehend vermeiden, und stattdessen kombinatorische und adaptive numerische Methoden einsetzen, ohne die Exaktheit des Resultats zu zerstören. Der Hauptbeitrag ist ein Algorithmus zur Berechnung von planaren Arrangements, die durch reelle algebraische Kurven induziert sind. Wir weisen die Effizienz des Verfahrens sowohl theoretisch durch eine KomplexitĂ€tsanalyse, als auch praktisch durch experimentelle Vergleiche nach. Dazu haben wir unser Verfahren im Rahmen der Softwarebibliothek Cgal implementiert. Die Resultate belegen, dass wir die zur Zeit beste verfĂŒgbare exakte Software bereitstellen. Der Algorithmus wird zur Arrangementberechnung rotierter Kurven, oder fĂŒr Arrangements auf parametrisierten OberflĂ€chen eingesetzt. Im R3 geben wir ein neues Verfahren zur Berechnung einer isotopen Triangulierung einer algebraischen OberflĂ€che an. Diese Triangulierung basiert auf einer Stratifizierung der OberflĂ€che, die topologische und geometrische Informationen berechnet. Unsere Implementierung ist die erste fĂŒr dieses Problem, welche numerische Methoden konsequent einsetzt, und dennoch die exakte Topologie der OberflĂ€che liefert
The Essence of Mathematics Through Elementary Problems
"It is increasingly clear that the shapes of reality â whether of the natural world, or of the built environment â are in some profound sense mathematical. Therefore it would benefit students and educated adults to understand what makes mathematics itself âtickâ, and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the first part, and explore the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside. The Essence of Mathematics consists of a sequence of 270 problems â with commentary and full solutions. The reader is assumed to have a reasonable grasp of school mathematics. More importantly, s/he should want to understand something of mathematics beyond the classroom, and be willing to engage with (and to reflect upon) challenging problems that highlight the essence of the discipline. The book consists of six chapters of increasing sophistication (Mental Skills; Arithmetic; Word Problems; Algebra; Geometry; Infinity), with interleaved commentary. The content will appeal to students considering further study of mathematics at university, teachers of mathematics at age 14-18, and anyone who wants to see what this kind of elementary content has to tell us about how mathematics really works.
Collection of abstracts of the 24th European Workshop on Computational Geometry
International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop
Dancing with the Shadows of Wellbeing: An Exploration of Participatory Action Research Processes as a Catalyst for Transformation of Staff Wellbeing
My abiding concern with human and planetary wellbeing and an emerging interest in the potential of Participatory Action Research as a method of engaging with restorative, life-enhancing ways of being are central to the research reported here.
Chomsky (2003), Kelsey (2002), Roddick (2001), and Stiglitz (2003) are amongst many authors who argue that the way in which we shape and are shaped by our relationships with one another has contributed to an intolerable, inhumane and unsustainable compromise of human and planetary wellbeing. Through a Participatory Action Research (PAR) project with staff from Te Ra, a community based on the holistic ideals of Rudolf Steiner, and thus an organisation explicitly committed to holistic wellbeing of people and planet, we sought to explore the challenges to such wellbeing. While our mutual attention was focused on enhancing staff wellbeing at their place of employment, my wider attention was also given to an investigation of the value of a critical analysis to the wider political and economic context in which this organisation works to meet the aspirations of this community. My deep attention has also focused on the potential for PAR to make a contribution to the transformational aspirations of critical theorists who are concerned to uncover and transform aspects of society that inhibit justice and wellbeing of people and planet. My metaphor of choice, to allow me the engagement in all three spheres simultaneously, is the metaphor of Dance. In my work with the staff of Te Ra, our intent was to 'dance with the Shadows', alluding to Jungian references to hidden aspects of ourselves and this community, to discover if un-wellness and disconnection from self and others could be transformed into flourishing relationships and wellbeing in the organisation.
McNiff (2000) proposes that PAR has the potential to generate living theories that redefine the main purpose of organisation theory in terms of human wellbeing. Throughout this research project, principles of PAR are woven in with work of critical organisational theorists, psychologists and anthropologists. The already established ideas of reflection, observation, reflexivity, and action are choreographed with the less often considered ideas of those aspects of the research relationship that may inhibit mutuality. While this organisation is explicitly and deeply committed to underpinning all that is aspired to with a relational ethic, the impact that instrumental practices associated with an intensifying neo-liberal economic external environment have not left this organisation untouched.
It took commitment, courage and resources to identify and engage with the Shadows masked by intrinsic and extrinsic pressures and processes that these research participants were experiencing. Engaging in PAR processes allowed us dance 'up close and personal' with their aspirations to begin transforming what was not well, while recognising and reinforcing the organisation's existing strong philosophical and spiritual foundations that emphasized individual freedom and collective responsibility for wellbeing of all.
Based on the significant transformations achieved during this project we posit that PAR provides a collaborative opportunity for academics and practitioners to 'dance with the Shadows' of individuals and communities to make a significant contribution to the development of sustainable relationships in workplaces where human and planetary wellbeing is the priority
The Essence of Mathematics Through Elementary Problems
"It is increasingly clear that the shapes of reality â whether of the natural world, or of the built environment â are in some profound sense mathematical. Therefore it would benefit students and educated adults to understand what makes mathematics itself âtickâ, and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the first part, and explore the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside.
The Essence of Mathematics consists of a sequence of 270 problems â with commentary and full solutions. The reader is assumed to have a reasonable grasp of school mathematics. More importantly, s/he should want to understand something of mathematics beyond the classroom, and be willing to engage with (and to reflect upon) challenging problems that highlight the essence of the discipline.
The book consists of six chapters of increasing sophistication (Mental Skills; Arithmetic; Word Problems; Algebra; Geometry; Infinity), with interleaved commentary. The content will appeal to students considering further study of mathematics at university, teachers of mathematics at age 14-18, and anyone who wants to see what this kind of elementary content has to tell us about how mathematics really works.
Ahlfors circle maps and total reality: from Riemann to Rohlin
This is a prejudiced survey on the Ahlfors (extremal) function and the weaker
{\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e.
those (branched) maps effecting the conformal representation upon the disc of a
{\it compact bordered Riemann surface}. The theory in question has some
well-known intersection with real algebraic geometry, especially Klein's
ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a
gallery of pictures quite pleasant to visit of which we have attempted to trace
the simplest representatives. This drifted us toward some electrodynamic
motions along real circuits of dividing curves perhaps reminiscent of Kepler's
planetary motions along ellipses. The ultimate origin of circle maps is of
course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass.
Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found
in Klein (what we failed to assess on printed evidence), the pivotal
contribution belongs to Ahlfors 1950 supplying an existence-proof of circle
maps, as well as an analysis of an allied function-theoretic extremal problem.
Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree
controls than available in Ahlfors' era. Accordingly, our partisan belief is
that much remains to be clarified regarding the foundation and optimal control
of Ahlfors circle maps. The game of sharp estimation may look narrow-minded
"Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to
contemplate how conformal and algebraic geometry are fighting together for the
soul of Riemann surfaces. A second part explores the connection with Hilbert's
16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by
including now Rohlin's theory (v.2
Topics in Programming Languages, a Philosophical Analysis through the case of Prolog
[EN]Programming languages seldom find proper anchorage in philosophy of logic, language and science. is more, philosophy of language seems to be restricted to natural languages and linguistics, and even philosophy of logic is rarely framed into programming languages topics. The logic programming paradigm and Prolog are, thus, the most adequate paradigm and programming language to work on this subject, combining natural language processing and linguistics, logic programming and constriction methodology on both algorithms and procedures, on an overall philosophizing declarative status. Not only this, but the dimension of the Fifth Generation Computer system related to strong Al wherein Prolog took a major role. and its historical frame in the very crucial dialectic between procedural and declarative paradigms, structuralist and empiricist biases, serves, in exemplar form, to treat straight ahead philosophy of logic, language and science in the contemporaneous age as well.
In recounting Prolog's philosophical, mechanical and algorithmic harbingers, the opportunity is open to various routes. We herein shall exemplify some:
- the mechanical-computational background explored by Pascal, Leibniz, Boole, Jacquard, Babbage, Konrad Zuse, until reaching to the ACE (Alan Turing) and EDVAC (von Neumann), offering the backbone in computer architecture, and the work of Turing, Church, Gödel, Kleene, von Neumann, Shannon, and others on computability, in parallel lines, throughly studied in detail, permit us to interpret ahead the evolving realm of programming languages. The proper line from lambda-calculus, to the Algol-family, the declarative and procedural split with the C language and Prolog, and the ensuing branching and programming languages explosion and further delimitation, are thereupon inspected as to relate them with the proper syntax, semantics and philosophical élan of logic programming and Prolog
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