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Volume 16, Number 1
2 International Journal "Information Theories & Applications" Vol.16 / 2009 International Journal INFORMATION THEORIES & APPLICATIONS Volume 16 / 2009, Number 1 Editor in chief: Krassimir Markov (Bulgaria) International Editorial Staff Chairman: Victor Gladun (Ukraine) Adil Timofeev (Russia) Ilia Mitov (Bulgaria) Aleksey Voloshin (Ukraine) Juan Castellanos (Spain) Alexander Eremeev (Russia) Koen Vanhoof (Belgium) Alexander Kleshchev (Russia) Levon Aslanyan (Armenia) Alexander Palagin (Ukraine) Luis F. de Mingo (Spain) Alfredo Milani (Italy) Nikolay Zagoruiko (Russia) Anatoliy Krissilov (Ukraine) Peter Stanchev (Bulgaria) Anatoliy Shevchenko (Ukraine) Rumyana Kirkova (Bulgaria) Arkadij Zakrevskij (Belarus) Stefan Dodunekov (Bulgaria) Avram Eskenazi (Bulgaria) Tatyana Gavrilova (Russia) Boris Fedunov (Russia) Vasil Sgurev (Bulgaria) Constantine Gaindric (Moldavia) Vitaliy Lozovskiy (Ukraine) Eugenia Velikova-Bandova (Bulgaria) Vitaliy Velichko (Ukraine) Galina Rybina (Russia) Vladimir Donchenko (Ukraine) Gennady Lbov (Russia) Vladimir Jotsov (Bulgaria) Georgi Gluhchev (Bulgaria) Vladimir Lovitskii (GB) IJ ITA is official publisher of the scientific papers of the members of the ITHEA® International Scientific Society IJ ITA welcomes scientific papers connected with any information theory or its application. IJ ITA rules for preparing the manuscripts are compulsory. The rules for the papers for IJ ITA as well as the subscription fees are given on www.ithea.org . The camera-ready copy of the paper should be received by http://ij.ithea.org. Responsibility for papers published in IJ ITA belongs to authors. General Sponsor of IJ ITA is the Consortium FOI Bulgaria (www.foibg.com). International Journal “INFORMATION THEORIES & APPLICATIONS” Vol.16, Number 1, 2009 Printed in Bulgaria Edited by the Institute of Information Theories and Applications FOI ITHEA®, Bulgaria, in collaboration with the V.M.Glushkov Institute of Cybernetics of NAS, Ukraine, and the Institute of Mathematics and Informatics, BAS, Bulgaria. -
Hyperoperations and Nopt Structures
Hyperoperations and Nopt Structures Alister Wilson Abstract (Beta version) The concept of formal power towers by analogy to formal power series is introduced. Bracketing patterns for combining hyperoperations are pictured. Nopt structures are introduced by reference to Nept structures. Briefly speaking, Nept structures are a notation that help picturing the seed(m)-Ackermann number sequence by reference to exponential function and multitudinous nestings thereof. A systematic structure is observed and described. Keywords: Large numbers, formal power towers, Nopt structures. 1 Contents i Acknowledgements 3 ii List of Figures and Tables 3 I Introduction 4 II Philosophical Considerations 5 III Bracketing patterns and hyperoperations 8 3.1 Some Examples 8 3.2 Top-down versus bottom-up 9 3.3 Bracketing patterns and binary operations 10 3.4 Bracketing patterns with exponentiation and tetration 12 3.5 Bracketing and 4 consecutive hyperoperations 15 3.6 A quick look at the start of the Grzegorczyk hierarchy 17 3.7 Reconsidering top-down and bottom-up 18 IV Nopt Structures 20 4.1 Introduction to Nept and Nopt structures 20 4.2 Defining Nopts from Nepts 21 4.3 Seed Values: “n” and “theta ) n” 24 4.4 A method for generating Nopt structures 25 4.5 Magnitude inequalities inside Nopt structures 32 V Applying Nopt Structures 33 5.1 The gi-sequence and g-subscript towers 33 5.2 Nopt structures and Conway chained arrows 35 VI Glossary 39 VII Further Reading and Weblinks 42 2 i Acknowledgements I’d like to express my gratitude to Wikipedia for supplying an enormous range of high quality mathematics articles. -
An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics
STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters 10.1090/stml/087 An Introduction to Ramsey Theory STUDENT MATHEMATICAL LIBRARY Volume 87 An Introduction to Ramsey Theory Fast Functions, Infinity, and Metamathematics Matthew Katz Jan Reimann Mathematics Advanced Study Semesters Editorial Board Satyan L. Devadoss John Stillwell (Chair) Rosa Orellana Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 05D10, 03-01, 03E10, 03B10, 03B25, 03D20, 03H15. Jan Reimann was partially supported by NSF Grant DMS-1201263. For additional information and updates on this book, visit www.ams.org/bookpages/stml-87 Library of Congress Cataloging-in-Publication Data Names: Katz, Matthew, 1986– author. | Reimann, Jan, 1971– author. | Pennsylvania State University. Mathematics Advanced Study Semesters. Title: An introduction to Ramsey theory: Fast functions, infinity, and metamathemat- ics / Matthew Katz, Jan Reimann. Description: Providence, Rhode Island: American Mathematical Society, [2018] | Series: Student mathematical library; 87 | “Mathematics Advanced Study Semesters.” | Includes bibliographical references and index. Identifiers: LCCN 2018024651 | ISBN 9781470442903 (alk. paper) Subjects: LCSH: Ramsey theory. | Combinatorial analysis. | AMS: Combinatorics – Extremal combinatorics – Ramsey theory. msc | Mathematical logic and foundations – Instructional exposition (textbooks, tutorial papers, etc.). msc | Mathematical -
A Survey of Recursive Analysis and Moore's Notion of Real Computation
A Survey of Recursive Analysis and Moore’s Notion of Real Computation Walid Gomaa INRIA Nancy Grand-Est Research Centre, France, Faculty of Engineering, Alexandria University, Egypt [email protected] Abstract The theory of analog computation aims at modeling computational systems that evolve in a continuous manner. Unlike the situation with the discrete setting there is no unified theory of analog computation. There are several proposed theories, some of them seem quite orthogonal. Some theories can be considered as generalizations of the Turing machine theory and classical recursion theory. Among such are recursive analysis and Moore’s class of recursive real functions. Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. Real computation in this context is viewed as effective (in the sense of Turing machine theory) convergence of sequences of rational numbers. In 1996 Moore introduced a function algebra that captures his notion of real computation; it consists of some basic functions and their closure under composition, integration and zero- finding. Though this class is inherently unphysical, much work have been directed at stratifying, restricting, and comparing it with other theories of real computation such as recursive analysis and the GPAC. In this article we give a detailed exposition of recursive analysis and Moore’s class and the relationships between them. 1 Introduction Analog computation is a computational paradigm that attempts to model systems whose internal states are continuous rather than discrete. Unlike the case of discrete computation, which has enjoyed a kind of uniformity and conceptual universality, there have been several approaches to analog computations some of which are not even comparable. -
0.1 Axt, Loop Program, and Grzegorczyk Hier- Archies
1 0.1 Axt, Loop Program, and Grzegorczyk Hier- archies Computable functions can have some quite complex definitions. For example, a loop programmable function might be given via a loop program that has depth of nesting of the loop-end pair, say, equal to 200. Now this is complex! Or a function might be given via an arbitrarily complex sequence of primitive recursions, with the restriction that the computed function is majorized by some known function, for all values of the input (for the concept of majorization see Subsection on the Ackermann function.). But does such definitional|and therefore, \static"|complexity have any bearing on the computational|dynamic|complexity of the function? We will see that it does, and we will connect definitional and computational complexities quantitatively. Our study will be restricted to the class PR that we will subdivide into an infinite sequence of increasingly more inclusive subclasses, Si. A so-called hierarchy of classes of functions. 0.1.0.1 Definition. A sequence (Si)i≥0 of subsets of PR is a primitive recur- sive hierarchy provided all of the following hold: (1) Si ⊆ Si+1, for all i ≥ 0 S (2) PR = i≥0 Si. The hierarchy is proper or nontrivial iff Si 6= Si+1, for all but finitely many i. If f 2 Si then we say that its level in the hierarchy is ≤ i. If f 2 Si+1 − Si, then its level is equal to i + 1. The first hierarchy that we will define is due to Axt and Heinermann [[5] and [1]]. -
New Computational Paradigmsnew Computational Paradigms
Lecture Notes in Computer Science 3526 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen University of Dortmund, Germany Madhu Sudan Massachusetts Institute of Technology, MA, USA Demetri Terzopoulos New York University, NY, USA Doug Tygar University of California, Berkeley, CA, USA Moshe Y. Vardi Rice University, Houston, TX, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany S. Barry Cooper Benedikt Löwe Leen Torenvliet (Eds.) New Computational Paradigms First Conference on Computability in Europe, CiE 2005 Amsterdam, The Netherlands, June 8-12, 2005 Proceedings 13 Volume Editors S. Barry Cooper University of Leeds School of Mathematics Leeds, LS2 9JT, UK E-mail: [email protected] Benedikt Löwe Leen Torenvliet Universiteit van Amsterdam Institute for Logic, Language and Computation Plantage Muidergracht 24, 1018 TV Amsterdam, NL E-mail: {bloewe,leen}@science.uva.nl Library of Congress Control Number: 2005926498 CR Subject Classification (1998): F.1.1-2, F.2.1-2, F.4.1, G.1.0, I.2.6, J.3 ISSN 0302-9743 ISBN-10 3-540-26179-6 Springer Berlin Heidelberg New York ISBN-13 978-3-540-26179-7 Springer Berlin Heidelberg New York This work is subject to copyright. -
Hierarchies of Recursive Functions
Leibniz Universitat¨ Hannover Institut f¨urtheoretische Informatik Bachelor Thesis Hierarchies of recursive functions Maurice Chandoo February 21, 2013 Erkl¨arung Hiermit versichere ich, dass ich diese Arbeit selbst¨andigverfasst habe und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. i Dedicated to my dear grandmother Anna-Maria Kniejska ii Contents 1 Preface 1 2 Class of primitve recursive functions PR 2 2.1 Primitive Recursion class PR ..................2 2.2 Course-of-value Recursion class PRcov .............4 2.3 Nested Recursion class PRnes ..................5 2.4 Recursive Depth . .6 2.5 Recursive Relations . .6 2.6 Equivalence of PR and PRcov .................. 10 2.7 Equivalence of PR and PRnes .................. 12 2.8 Loop programs . 14 2.9 Grzegorczyk Hierarchy . 19 2.10 Recursive depth Hierarchy . 22 2.11 Turing Machine Simulation . 24 3 Multiple and µ-recursion 27 3.1 Multiple Recursion class MR .................. 27 3.2 µ-Recursion class µR ....................... 32 3.3 Synopsis . 34 List of literature 35 iii 1 Preface I would like to give a more or less exhaustive overview on different kinds of recursive functions and hierarchies which characterize these functions by their computational power. For instance, it will become apparent that primitve recursion is more than sufficiently powerful for expressing algorithms for the majority of known decidable ”real world” problems. This comes along with the hunch that it is quite difficult to come up with a suitable problem of which the characteristic function is not primitive recursive therefore being a possible candidate for exploiting the more powerful multiple recursion. At the end the Turing complete µ-recursion is introduced. -
Computability and Complexity in Analysis Schloß Dagstuhl, November 11–16, 2001
Dagstuhl-Seminar 01461 on Computability and Complexity in Analysis Schloß Dagstuhl, November 11–16, 2001 Organizers: Vasco Brattka (Hagen), Peter Hertling (Hagen), Mariko Yasugi (Kyoto), Ning Zhong (Cincinnati). Preface This meeting was the third Dagstuhl seminar which was concerned with the theory of computability and complexity over the real numbers. This theory, which is built on the Turing machine model, was initiated by Turing, Grzegorczyk, Lacombe, Banach and Mazur, and has seen a rapid growth in recent years. Recent monographs are by Pour- El/Richards, Ko, and Weihrauch. Computability theory and complexity theory are two central areas of research in the- oretical computer science. Until recently, most work in these areas concentrated on prob- lems over discrete structures. In the last years, though, there has been an enormous growth of computability theory and complexity theory over the real numbers and other continuous structures. One of the reasons for this phenomenon is that more and more practical computation problems over the real numbers are being dealt with by computer scientists, for example, in computational geometry, in the modelling of dynamical and hybrid systems, but also in classical problems from numerical mathematics. The scien- tists working on these questions come from different fields, such as theoretical computer science, domain theory, logic, constructive mathematics, computer arithmetic, numerical mathematics, analysis etc. This Dagstuhl seminar provided a unique opportunity for 46 participants from such diverse areas and 12 different countries to meet and exchange ideas and knowledge. One of the topics of interest was foundational work concerning the various models and approaches for defining or describing computability and complexity over the real numbers. -
Curriculum Vitae
Roussanka Loukanova Curriculum Vitae Roussanka Loukanova September 30, 2021 Contents I CV R´esum´e3 1 Education and Degrees3 2 Academic Qualification4 3 Pedagogic Training4 4 Employment and Association4 4.1 Present...........................................4 4.2 Former Positions......................................4 4.3 Visiting Positions and Fellowships............................5 II Scientific Profile6 5 Expertise 6 6 Experience with Software Packages7 III Pedagogic Profile7 7 Experience in Teaching7 7.1 Courses Taught and Developed..............................8 8 Experience as a Supervisor9 8.1 Supervision of Bachelor and Master's Theses......................9 8.2 Supervision of Ph.D. Students...............................9 8.3 Participation in Evaluation Committees of M.Sc. and Ph.D. Theses.........9 9 Training in Teaching and Learning in Higher Education9 10 Some Pedagogical Work and Teaching Materials 10 IV Grants 10 V Scholarly Activities 13 1 / 27 Roussanka Loukanova 11 Scientific Organizations, Committees 13 11.1 Organization Committees................................. 13 11.2 Reviewing and Member of Program Committees.................... 15 11.2.1 Area Supervisory Committee........................... 15 11.2.2 Program Committees............................... 15 12 Other International Engagements 16 12.1 Tutorials........................................... 16 12.2 Some Conference and Seminar Participation, Talks, Presentations........... 16 VI Scientific Works 18 13 Work in Progress 18 14 List of Publications 19 14.1 Publications -
Fractional Mathematical Operators and Their Computational Approximation
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 4356371, 11 pages http://dx.doi.org/10.1155/2016/4356371 Research Article Fractional Mathematical Operators and Their Computational Approximation José Crespo and Francisco Javier Montáns Escuela Tecnica´ Superior de Ingenier´ıa Aeronautica´ y del Espacio, Universidad Politecnica´ de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid, Spain Correspondence should be addressed to Francisco Javier Montans;´ [email protected] Received 27 January 2016; Revised 30 July 2016; Accepted 9 August 2016 Academic Editor: Manuel Doblare´ Copyright © 2016 J. Crespo and F. J. Montans.´ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Usual applied mathematics employs three fundamental arithmetical operators: addition, multiplication, and exponentiation. However, for example, transcendental numbers are said not to be attainable via algebraic combination with these fundamental operators. At the same time, simulation and modelling frequently have to rely on expensive numerical approximations of the exact solution. The main purpose of this article is to analyze new fractional arithmetical operators, explore some of their properties, and devise ways of computing them. These new operators may bring new possibilities, for example, in approximation theory and in obtaining closed forms of those approximations and solutions. We show some simple demonstrative examples. 1. Introduction parameters [2, 3] rendering linear equations. We will also see possible applications of the presented operators in constitu- Science mostly hinges on mathematics which has been built tive modelling. on three fundamental operations: the addition, the multipli- Consider the classification described in Table 1. -
The Grzegorczyk Hierarchy
Chapter 4 The Grzegorczyk Hierarchy 4.1 Reprise The primitive recursive functions are defined as the closure of R and C over a basic stock of functions. Both the logician’s diagonal argument and the double- recursion diagonal arguments involve functions that can “simulate” n nested primitive recursions when given an argument n. Intuitively, the situation is that each primitive recursive function is allowed only finitely many nested primitive recursions, so functions that require arbitrarily many (without bound) are not primitive recursive. This situation calls out for greater attention to which functions can be de- fined with a given, fixed number of primitive recursion operations. In other words, it makes sense to look at the hierarchy of increasing classes En defined in terms of increasing numbers of primitive recursions, where each class is closed under composition. Every primitive recursive function will be at some finite level of the hierarchy: [ P rim = Ex x A natural idea would be to start with the basic functions closed under com- position and then to add in successively more complex functions like addition, multiplication, exponentiation, etc., closing under composition each time. But composition is a little weak. Recall that some slow-growing functions like cutoff subtraction required the R schema for their definitions even though they don’t really use it to generate faster growth. We would like to distinguish growth- generating applications of R from applications that don’t generate growth. We can get around the difficulty by closing each class under bounded re- cursion as well as composition. An application of R is bounded in a class of functions if the function resulting from the application is bounded everywhere by some other function already established to be in the class. -
Solving for the Analytic Piecewise Extension of Tetration and the Super-Logarithm
Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm by Andrew Robbins Abstract An overview of previous extensions of tetration is presented. Specific conditions for differentiability and piecewise continuity are shown. This leads to a way of generating approximations of the super-logarithm. These approximations are shown to converge to a function that satisfies two basic properties of extensions of tetration. Table of Contents Introduction 2 Background 4 Extensions 6 Results 10 Generalization 20 Conclusion 21 Appendices 22 Copyright 2005 Andrew Robbins 1 Introduction Of the operators in the sequence: addition, multiplication, exponentiation, and tetration, only the first three are well-defined analytic operations. Tetration, also known as the hyper4 operator, power towers, and iterated exponentiation, has only been “nicely” defined for integer y in y x . Although extensions of tetration to real y have been made, those extensions have not followed simply from the properties of hyper-operations. The particular form of this property as it pertains to tetration is: Property 1. Iterated exponential property y−1 y x = x( x ) for all real y. Tetration has two inverses, the super-root, and the super-logarithm: = y = = z x tet x ( y ) twr y ( x ) = = −1 x srt y ( z ) twr y ( z ) = = −1 y slogx ( z ) tet x ( z ) When y is fixed, the function twr y ( x ) is called an order y power tower of x [14]. When x is fixed, the function tet x ( y ) is called a base x tetrational function of y. In general, it is pronounced: x tetra y. Inverting a power tower gives the super-root, and inverting a tetrational function gives the super-logarithm, More detail about these inverses can be found in [12], [14] and [15].