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The Computational Significance of Hausdorff's Maximal Chain Principle [chapter]

Peter Schuster, Daniel Wessel
2020 Lecture Notes in Computer Science  
As a fairly frequent form of the Axiom of Choice about relatively simple structures (posets), Hausdorff's Maximal Chain Principle appears to be little amenable to computational interpretation.  ...  We obtain the latter, which is nothing but the desired computational core of Hausdorff's principle, by passing from maximal chains to paths of finite binary trees of an adequate inductively generated class  ...  in this paper are those of the authors and do not necessarily reflect the views of those foundations.).  ... 
doi:10.1007/978-3-030-51466-2_21 fatcat:m7ios446cfeahni63eorjtsaem

Singular Cardinals [chapter]

Menachem Kojman
2012 Handbook of the History of Logic  
is significant.  ...  claim to significance.  ...  ., 15 almost free group, 23 Axiom of Choice, 6  ... 
doi:10.1016/b978-0-444-51621-3.50007-4 fatcat:fr3pdoy3nbdn5pc4pz52reng24

Moment problems and the causal set approach to quantum gravity

Avner Ash, Patrick McDonald
2003 Journal of Mathematical Physics  
The transition probabilities of these chains satisfy a general covariance principle, a causality principle, and a renormalizability condition.  ...  We study a collection of discrete Markov chains related to the causal set approach to modeling discrete theories of quantum gravity.  ...  As discussed in [1] , these systems can be realized as Markov chains whose transition probabilities are required to satisfy a discrete covariance principle and a discrete causality principle.  ... 
doi:10.1063/1.1519668 fatcat:7hhl533y4rgprcgpljcpjsyzny

Entropies in investigation of dynamical systems and their application to digital image analysis

Igor Soloviev, Natalia Ampilova
2018 Journal of Measurements in Engineering  
The dynamics of a system may be studied by analyzing the phase portrait of a system obtained as a digital image.  ...  We also describe the results of applications of described methods to images of biomedical preparations.  ...  Acknowledgements The work was partially supported by the grant Erasmus + International Mobility (KA107) / National Technical University of Athens -St. Petersburg State University.  ... 
doi:10.21595/jme.2018.19891 fatcat:nbrr6gxrhbagfcntlhnvbye27e

Right simple subsemigroups and right subgroups of compact convergence semigroups

Phoebe Ho, Shing S. So
2000 International Journal of Mathematics and Mathematical Sciences  
In the discussion of maximal right simple subsemigroups and maximal right subgroups of semigroups, generalization of the results that no two maximal right simple subsemigroups and maximal right subgroups  ...  It was shown in Roy and So (1998) that, among topological semigroups, compact right simple or left cancellative semigroups are in fact right groups, and the closure of a right simple subsemigroup of a  ...  By the Hausdorff maximal principle, there is a maximal chainof . Let M = ∪Ꮿ. Lemma 2. 13 . 13 Let S be a compact convergence semigroup and Z be a right zero subsemigroup of S.  ... 
doi:10.1155/s0161171200003604 fatcat:iixhaglg55h2bbvpfe2qwq36ry

Transport Constitutive Relations, Quantum Diffusion and Periodic Reactions [chapter]

Jiří J. Mareš, Jaroslav Šesták, Pavel Hubík
2010 Hot Topics in Thermal Analysis and Calorimetry  
Starting from the classical Einstein-von Smoluchowski theory of Brownian movement and making use of a significant fact that classical and quantum Brownian processes are essentially indistinguishable in  ...  attains, with appreciable accuracy, the value of Planck's quantum _.  ...  Acknowledgements The work was partially supported by the Grant Agencies of the Czech Republic, Contract nos. 202/03/0410 and 401/02/0579 and of the Academy of Sciences, Contract no. A1010404.  ... 
doi:10.1007/978-90-481-2882-2_14 fatcat:ahv5fdlwrnehdphdhrhip7wlba

Set Theory from Cantor to Cohen [chapter]

Akihiro Kanamori
2012 Handbook of the History of Logic  
The whole transfinite landscape can be viewed as having been articulated by Cantor in significant part to solve the Continuum Problem.  ...  into the theory of large cardinals and the investigation of definable sets of reals as transmuted into descriptive set theory.  ...  ACKNOWLEDGEMENTS This is a revision with significant changes of the author's The Mathematical Development of Set Theory from Cantor to Cohen, The Bulletin of Symbolic Logic, volume 2, 1996, pages 1-71,  ... 
doi:10.1016/b978-0-444-51621-3.50001-3 fatcat:ogdmpb6mlvgxbdvoh3jqq6umqe

SET THEORY FROM CANTOR TO COHEN [chapter]

Akihiro Kanamori
2009 Philosophy of Mathematics  
The whole transfinite landscape can be viewed as having been articulated by Cantor in significant part to solve the Continuum Problem.  ...  into the theory of large cardinals and the investigation of definable sets of reals as transmuted into descriptive set theory.  ...  ACKNOWLEDGEMENTS This is a revision with significant changes of the author's The Mathematical Development of Set Theory from Cantor to Cohen, The Bulletin of Symbolic Logic, volume 2, 1996, pages 1-71,  ... 
doi:10.1016/b978-0-444-51555-1.50014-6 fatcat:vnj2vdlpofdp7pg37tyiemdkiq

The Fractal Dimension of SAT Formulas [chapter]

Carlos Ansótegui, Maria Luisa Bonet, Jesús Giráldez-Cru, Jordi Levy
2014 Lecture Notes in Computer Science  
Recently, there have been some attempts to analyze the structure of these formulas in terms of complex networks, with the long-term aim of explaining the success of these SAT solving techniques, and possibly  ...  We also show that this dimension is not affected by the addition of learnt clauses.  ...  At every step, for every possible node c, we compute the number of unburned nodes covered by the circle of center c and radius r, then select the node c that maximizes this number, and burn the new covered  ... 
doi:10.1007/978-3-319-08587-6_8 fatcat:buzy72ifgjdpnf5mna3qkcyedq

Contemporary Infinitesimalist Theories of Continua and their late 19th- and early 20th-century forerunners [article]

Philip Ehrlich
2018 arXiv   pre-print
The purpose of this paper is to provide a historical overview of some of the contemporary infinitesimalist alternatives to the Cantor-Dedekind theory of continua.  ...  and some of their relations to the contemporary ones.  ...  The idea of writing this historical overview grew out of our (2007).  ... 
arXiv:1808.03345v3 fatcat:yopietg43fgshfmtqdbid2h7f4

Hausdorff dimension in probability theory

Patrick Billingsley
1960 Illinois Journal of Mathematics  
It is the main purpose of the present paper to compute the dimensions of various sets where the strong law of large num- bers is violated, under the assumption that {2,} is a Markov chain. 3.  ...  Then dim’ M is Hausdorff’s original definition [10] of the frac- tional dimension of a subset M of the line. (Of course Hausdorff’s defini- tion applies in any metric space.)  ... 
doi:10.1215/ijm/1255455863 fatcat:nlrndoasrvacrnb3kcxsxv6nle

Making abstract interpretations complete

Roberto Giacobazzi, Francesco Ranzato, Francesca Scozzari
2000 Journal of the ACM  
of information accumulated in abstract computations.  ...  significant results, but paying for heavier order-theoretic details. 2 In abstract interpretation literature, unfortunately, there is not uniformity on the terminology related to completeness issues.  ...  Let us point out that a chain Y in a poset P is maximal (with respect to set inclusion) whenever for any other chain YЈ in P, Y ʕ YЈ implies Y ϭ YЈ. HAUSDORFF'S MAXIMAL PRINCIPLE.  ... 
doi:10.1145/333979.333989 fatcat:uiwdkf7ptja5fg2cdzj2k43sum

Toward a Theory of Chaos [article]

A. Sengupta
2004 arXiv   pre-print
convergence of a net of functions in an extended multifunction space, and the topological theory of convergence.  ...  solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively.  ...  Financial assistance during the initial stages of this work from the National Board for Higher Mathematics is also acknowledged.  ... 
arXiv:nlin/0408044v1 fatcat:igx7fsbyores7diqfmqaig6g2a

TOWARD A THEORY OF CHAOS

A. SENGUPTA
2003 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering  
of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively.  ...  convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000] , and the topological theory of convergence.  ...  Financial assistance during the initial stages of this work from the National Board for Higher Mathematics is also acknowledged.  ... 
doi:10.1142/s021812740300851x fatcat:gythnivbqvhr3pps5sr7y6g6xq

Algebraic Topology 2.0 [article]

Göran Fors
2012 arXiv   pre-print
This article presents an algebraization of Hausdorff's century old definition of the category of topological spaces as well as some useful algebraic topological consequences thereof.  ...  Thereby synchronizing the category of topological spaces with the structures within the contemporary category of simplicial complexes as well as with the structures within the algebraic categories.  ...  the unique homogeneous maximal ideal of "·", i.e., the irrelevant ideal of "·".) 1.  ... 
arXiv:1212.6169v1 fatcat:4iavje527bbx7h6hiedv6q2dby
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