91 research outputs found
Hilbert Mathematics Versus Gödel Mathematics. IV. The New Approach of Hilbert Mathematics Easily Resolving the Most Difficult Problems of Gödel Mathematics
The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional âdimensionâ allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the âdistance between finiteness and infinityâ, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special âflatâ case of Hilbert mathematics. The following four essential problems are considered for the idea to be elucidated: Fermatâs last theorem proved by Andrew Wiles; PoincarĂ©âs conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved âmachine-likelyâ by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermatâs last theorem is shown as a Gödel insoluble statement by means of Yabloâs paradox. Thus, Wilesâs proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in âFermat arithmeticâ introduced by âepochĂ© to infinityâ (following the pattern of Husserlâs original âepochĂ© to realityâ) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a âWittgenstein ladderâ. PoincarĂ©âs conjecture can be reinterpreted physically by Minkowski space and thus reduced to the ânonstandard homeomorphismâ of a bit of information mathematically. Perelmanâs proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of PoincarĂ©âs conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions
Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities
This is an extensive (published) survey on CR geometry, whose major themes
are: formal analytic reflection principle; generic properties of Systems of
(CR) vector fields; pairs of foliations and conjugate reflection identities;
Sussmann's orbit theorem; local and global aspects of holomorphic extension of
CR functions; Tumanov's solution of Bishop's equation in Hoelder classes with
optimal loss of smoothness; wedge-extendability on C^2,a generic submanifolds
of C^n consisting of a single CR orbit; propagation of CR extendability and
edge-of-the-wedge theorem; Painlev\'{e} problem; metrically thin singularities
of CR functions; geometrically removable singularities for solutions of the
induced d-barre. Selected theorems are fully proved, while surveyed results are
put in the right place in the architecture.Comment: 283 pages ; 33 illustrations ; 16 open problems
http://www.hindawi.com/journals/imrs
Novel Modeling and Simulation Concepts for Power Distribution Networks
L'abstract Ăš presente nell'allegato / the abstract is in the attachmen
Organising Equal Freedom:From antagonism to agonism
This thesis is a theoretical and empirical study of prefigurative, pluralist, radically horizontal intra and inter personal processes of subjectification. The research context is Greek radical worker co-operatives abiding to pluralist radically horizontal organizing practices and prefigurative politics (autonomous, and social and solidarity economy). My methodological, conceptual, and empirical contributions aimed to align and support this contextâs situated process for radically horizontal organizing, that is âco-formation toward synthesisâ (Gr. "syndiamorfosi pros synthesiâ) assembly organizing, and address a key situated challenge, that of informal hierarchies. Through an intersubjective interpretative phenomenological lens, I employed a Militant Research -situated, co-produced, movement-led- methodological framework. This framework also entailed developing this thesis through practices of pluralist radical horizontality. Within academic discussions on selfhood, equality, freedom, and prefiguration, I developed the sensitising concepts of agonistic self-creation for intra personal processes, and agonistic empathy for inter personal processes of pluralist radical horizontality. To support these intra and inter personal processes further, I followed a fictocritical approach to empirical data. This approach allowed me to blend fact, fiction, theory, critique, and literary methods in single narratives. I developed four such narratives. Finally, I proposed seven lenses for observing movements in intra and inter personal processes, and Heraclitusâ agonistic metaphor of the farmersâ drink kykeon for visualizing these movements
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