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Generalizing Zeckendorf's Theorem to F-Decompositions
GENERALIZING ZECKENDORF’S THEOREM TO f-DECOMPOSITIONS PHILIPPE DEMONTIGNY, THAO DO, ARCHIT KULKARNI, STEVEN J. MILLER, DAVID MOON, AND UMANG VARMA ABSTRACT. A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers fFng, where F1 = 1, F2 = 2 and Fn+1 = Fn + Fn−1. For general recurrences fGng with non-negative coefficients, there is a notion of a legal decomposition which again leads to a unique representation, and the number of sum- mands in the representations of uniformly randomly chosen m 2 [Gn;Gn+1) converges to a normal distribution as n ! 1. We consider the converse question: given a notion of legal decomposition, is it possible to con- struct a sequence fang such that every positive integer can be decomposed as a sum of terms from the sequence? We encode a notion of legal decomposition as a function f : N0 ! N0 and say that if an is in an “f-decomposition”, then the decomposition cannot contain the f(n) terms immediately before an in the sequence; special choices of f yield many well known decompositions (including base-b, Zeckendorf and factorial). We prove that for any f : N0 ! N0, there exists a sequence 1 fangn=0 such that every positive integer has a unique f-decomposition using fang. Further, if f is periodic, then the unique increasing sequence fang that corresponds to f satisfies a linear recurrence relation. Previous research only handled recurrence relations with no negative coefficients. We find a function f that yields a sequence that cannot be described by such a recurrence relation. -
Hyper-Dual Numbers for Exact Second-Derivative Calculations
49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition AIAA 2011-886 4 - 7 January 2011, Orlando, Florida The Development of Hyper-Dual Numbers for Exact Second-Derivative Calculations Jeffrey A. Fike∗ andJuanJ.Alonso† Department of Aeronautics and Astronautics Stanford University, Stanford, CA 94305 The complex-step approximation for the calculation of derivative information has two significant advantages: the formulation does not suffer from subtractive cancellation er- rors and it can provide exact derivatives without the need to search for an optimal step size. However, when used for the calculation of second derivatives that may be required for approximation and optimization methods, these advantages vanish. In this work, we develop a novel calculation method that can be used to obtain first (gradient) and sec- ond (Hessian) derivatives and that retains all the advantages of the complex-step method (accuracy and step-size independence). In order to accomplish this task, a new number system which we have named hyper-dual numbers and the corresponding arithmetic have been developed. The properties of this number system are derived and explored, and the formulation for derivative calculations is presented. Hyper-dual number arithmetic can be applied to arbitrarily complex software and allows the derivative calculations to be free from both truncation and subtractive cancellation errors. A numerical implementation on an unstructured, parallel, unsteady Reynolds-Averaged Navier Stokes (URANS) solver, -
A Robust Software Watermarking for Copyright Protection
computers & security 28 (2009) 395–409 available at www.sciencedirect.com journal homepage: www.elsevier.com/locate/cose A robust software watermarking for copyright protection Ibrahim Kamela,*, Qutaiba Albluwib aDepartment of Computer Engineering Sharjah University, P.O. Box 27272, United Arab Emirates bSchool of Computing, Queens University, Canada article info abstract Article history: This paper advocates protecting software copyright through hiding watermarks in various Received 18 August 2008 data structures used by the code, e.g., Bþ-trees, R-trees, linked lists, etc. Prior proposals Received in revised form hide the watermarks in dummy data structures, e.g., linked lists and graphs that are 13 January 2009 created, solely for this reason, during the execution of the hosting software. This makes Accepted 30 January 2009 them vulnerable to subtractive attacks, because the attacker can remove the dummy data structures without altering the functionality or the semantic of the software program. We Keywords: argue that hiding watermarks in one or more data structures that are used by the program Information security would make the watermark more robust because disturbing the watermark would affect R-trees the semantic and the functionality of the underlying software. The challenge is that the Watermarking insertion of the watermark should have a minimal effect on the operations and perfor- Multimedia indexes mance of the data structure. Spatial data structures This paper proposes a novel method for watermarking R-tree data structure and its variants. Copyright protection The proposed watermarking technique does not change the values of the stored data objects. It takes advantage of the redundancy in the order of entries inside the R-tree nodes. -
THÈSE DOCTEUR DE L'université DU LUXEMBOURG EN INFORMATIQUE ET DOCTEUR DE L'université LILLE1 EN INFORMATIQUE Malika Mehd
PhD-FSTC-2011-17 Faculté des Sciences, de la Technologie et de Faculté des Sciences et Technologies de Lille la Communication THÈSE Soutenue le 20/10/2011 à Luxembourg En vue de l’obtention du grade académique de DOCTEUR DE L’UNIVERSITÉ DU LUXEMBOURG EN INFORMATIQUE ET DOCTEUR DE L’UNIVERSITÉ LILLE1 EN INFORMATIQUE par Malika Mehdi née le 10 septembre 1981 à Bejaia PARALLEL HYBRID OPTIMIZATION METHODS FOR PERMUTATION BASED PROBLEMS Jury de thèse Prof. Dr. Pascal Bouvry, directeur de thèse Professeur, Université du Luxembourg Prof. Dr. El-Ghazali Talbi, directeur de thèse Professeur, Université Lille1 Prof. Dr. Nouredine Melab, co-directeur de thèse Professeur, Université Lille1 Prof. Dr. Raymond Bisdorff, président Professeur, Université du Luxembourg Prof. Dr. Enrique Alba, membre Professeur, Université de Malaga Dr. Didier El-Baz, membre HDR, LAAS-CNRS, Toulouse Abstract Solving efficiently large benchmarks of NP-hard permutation-based problems re- quires the development of hybrid methods combining different classes of optimiza- tion methods. Indeed, it is now acknowledged that such methods perform better than traditional optimization methods when used separately. The key challenge is how to find connections between the divergent search strategies used in each class of methods in order to build efficient hybridization strategies. Genetic algorithms (GAs) are very popular population-based metaheuristics based on stochastic evo- lutionary operators. The hybridization of GAs with tree-based exact methods such as Branch-and-Bound is a promising research trend. B&B algorithms are based on an implicit enumeration of the solution space represented as a tree. Our hybridization approach consists in providing a common solution and search space coding and associated search operators enabling an efficient cooperation between the two methods. -
A Nestable Vectorized Templated Dual Number Library for C++11
cppduals: a nestable vectorized templated dual number library for C++11 Michael Tesch1 1 Department of Chemistry, Technische Universität München, 85747 Garching, Germany DOI: 10.21105/joss.01487 Software • Review Summary • Repository • Archive Mathematical algorithms in the field of optimization often require the simultaneous com- Submitted: 13 May 2019 putation of a function and its derivative. The derivative of many functions can be found Published: 05 November 2019 automatically, a process referred to as automatic differentiation. Dual numbers, close rela- License tives of the complex numbers, are of particular use in automatic differentiation. This library Authors of papers retain provides an extremely fast implementation of dual numbers for C++, duals::dual<>, which, copyright and release the work when replacing scalar types, can be used to automatically calculate a derivative. under a Creative Commons A real function’s value can be made to carry the derivative of the function with respect to Attribution 4.0 International License (CC-BY). a real argument by replacing the real argument with a dual number having a unit dual part. This property is recursive: replacing the real part of a dual number with more dual numbers results in the dual part’s dual part holding the function’s second derivative. The dual<> type in this library allows this nesting (although we note here that it may not be the fastest solution for calculating higher order derivatives.) There are a large number of automatic differentiation libraries and classes for C++: adolc (Walther, 2012), FAD (Aubert & Di Césaré, 2002), autodiff (Leal, 2019), ceres (Agarwal, Mierle, & others, n.d.), AuDi (Izzo, Biscani, Sánchez, Müller, & Heddes, 2019), to name a few, with another 30-some listed at (autodiff.org Bücker & Hovland, 2019). -
Application of Dual Quaternions on Selected Problems
APPLICATION OF DUAL QUATERNIONS ON SELECTED PROBLEMS Jitka Prošková Dissertation thesis Supervisor: Doc. RNDr. Miroslav Láviˇcka, Ph.D. Plze ˇn, 2017 APLIKACE DUÁLNÍCH KVATERNIONU˚ NA VYBRANÉ PROBLÉMY Jitka Prošková Dizertaˇcní práce Školitel: Doc. RNDr. Miroslav Láviˇcka, Ph.D. Plze ˇn, 2017 Acknowledgement I would like to thank all the people who have supported me during my studies. Especially many thanks belong to my family for their moral and material support and my advisor doc. RNDr. Miroslav Láviˇcka, Ph.D. for his guidance. I hereby declare that this Ph.D. thesis is completely my own work and that I used only the cited sources. Plzeˇn, July 20, 2017, ........................... i Annotation In recent years, the study of quaternions has become an active research area of applied geometry, mainly due to an elegant and efficient possibility to represent using them ro- tations in three dimensional space. Thanks to their distinguished properties, quaternions are often used in computer graphics, inverse kinematics robotics or physics. Furthermore, dual quaternions are ordered pairs of quaternions. They are especially suitable for de- scribing rigid transformations, i.e., compositions of rotations and translations. It means that this structure can be considered as a very efficient tool for solving mathematical problems originated for instance in kinematics, bioinformatics or geodesy, i.e., whenever the motion of a rigid body defined as a continuous set of displacements is investigated. The main goal of this thesis is to provide a theoretical analysis and practical applications of the dual quaternions on the selected problems originated in geometric modelling and other sciences or various branches of technical practise. -
Version 13.0 Release Notes
Release Notes The tool of thought for expert programming Version 13.0 Dyalog is a trademark of Dyalog Limited Copyright 1982-2011 by Dyalog Limited. All rights reserved. Version 13.0 First Edition April 2011 No part of this publication may be reproduced in any form by any means without the prior written permission of Dyalog Limited. Dyalog Limited makes no representations or warranties with respect to the contents hereof and specifically disclaims any implied warranties of merchantability or fitness for any particular purpose. Dyalog Limited reserves the right to revise this publication without notification. TRADEMARKS: SQAPL is copyright of Insight Systems ApS. UNIX is a registered trademark of The Open Group. Windows, Windows Vista, Visual Basic and Excel are trademarks of Microsoft Corporation. All other trademarks and copyrights are acknowledged. iii Contents C H A P T E R 1 Introduction .................................................................................... 1 Summary........................................................................................................................... 1 System Requirements ....................................................................................................... 2 Microsoft Windows .................................................................................................... 2 Microsoft .Net Interface .............................................................................................. 2 Unix and Linux .......................................................................................................... -
Connes on the Role of Hyperreals in Mathematics
Found Sci DOI 10.1007/s10699-012-9316-5 Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics Vladimir Kanovei · Mikhail G. Katz · Thomas Mormann © Springer Science+Business Media Dordrecht 2012 Abstract We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in func- tional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “vir- tual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnera- ble to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes’ magnum opus) V. -
A Ternary Arithmetic and Logic
Proceedings of the World Congress on Engineering 2010 Vol I WCE 2010, June 30 - July 2, 2010, London, U.K. A Ternary Arithmetic and Logic Ion Profeanu which supports radical ontological dualism between the two Abstract—This paper is only a chapter, not very detailed, of eternal principles, Good and Evil, which oppose each other a larger work aimed at developing a theoretical tool to in the course of history, in an endless confrontation. A key investigate first electromagnetic fields but not only that, (an element of Manichean doctrine is the non-omnipotence of imaginative researcher might use the same tool in very unusual the power of God, denying the infinite perfection of divinity areas of research) with the stated aim of providing a new perspective in understanding older or recent research in "free that has they say a dual nature, consisting of two equal but energy". I read somewhere that devices which generate "free opposite sides (Good-Bad). I confess that this kind of energy" works by laws and principles that can not be explained dualism, which I think is harmful, made me to seek another within the framework of classical physics, and that is why they numeral system and another logic by means of which I will are kept far away from public eye. So in the absence of an praise and bring honor owed to God's name; and because adequate theory to explain these phenomena, these devices can God is One Being in three personal dimensions (the Father, not reach the design tables of some plants in order to produce them in greater number. -
Algorithmic Combinatorial Game Theory∗
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory∗ Erik D. Demaine† Robert A. Hearn‡ Abstract Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer. 1 Introduction Many classic games are known to be computationally intractable (assuming P 6= NP): one-player puzzles are often NP-complete (as in Minesweeper) or PSPACE-complete (as in Rush Hour), and two-player games are often PSPACE-complete (as in Othello) or EXPTIME-complete (as in Check- ers, Chess, and Go). Surprisingly, many seemingly simple puzzles and games are also hard. Other results are positive, proving that some games can be played optimally in polynomial time. In some cases, particularly with one-player puzzles, the computationally tractable games are still interesting for humans to play. We begin by reviewing some basics of Combinatorial Game Theory in Section 2, which gives tools for designing algorithms, followed by reviewing the relatively new theory of Constraint Logic in Section 3, which gives tools for proving hardness. -
Vector Bundles and Brill–Noether Theory
MSRI Series Volume 28, 1995 Vector Bundles and Brill{Noether Theory SHIGERU MUKAI Abstract. After a quick review of the Picard variety and Brill–Noether theory, we generalize them to holomorphic rank-two vector bundles of canonical determinant over a compact Riemann surface. We propose sev- eral problems of Brill–Noether type for such bundles and announce some of our results concerning the Brill–Noether loci and Fano threefolds. For example, the locus of rank-two bundles of canonical determinant with five linearly independent global sections on a non-tetragonal curve of genus 7 is a smooth Fano threefold of genus 7. As a natural generalization of line bundles, vector bundles have two important roles in algebraic geometry. One is the moduli space. The moduli of vector bun- dles gives connections among different types of varieties, and sometimes yields new varieties that are difficult to describe by other means. The other is the linear system. In the same way as the classical construction of a map to a pro- jective space, a vector bundle gives rise to a rational map to a Grassmannian if it is generically generated by its global sections. In this article, we shall de- scribe some results for which vector bundles play such roles. They are obtained from an attempt to generalize Brill–Noether theory of special divisors, reviewed in Section 2, to vector bundles. Our main subject is rank-two vector bundles with canonical determinant on a curve C with as many global sections as pos- sible: especially their moduli and the Grassmannian embeddings of C by them (Section 4). -
COMPSCI 575/MATH 513 Combinatorics and Graph Theory
COMPSCI 575/MATH 513 Combinatorics and Graph Theory Lecture #34: Partisan Games (from Conway, On Numbers and Games and Berlekamp, Conway, and Guy, Winning Ways) David Mix Barrington 9 December 2016 Partisan Games • Conway's Game Theory • Hackenbush and Domineering • Four Types of Games and an Order • Some Games are Numbers • Values of Numbers • Single-Stalk Hackenbush • Some Domineering Examples Conway’s Game Theory • J. H. Conway introduced his combinatorial game theory in his 1976 book On Numbers and Games or ONAG. Researchers in the area are sometimes called onagers. • Another resource is the book Winning Ways by Berlekamp, Conway, and Guy. Conway’s Game Theory • Games, like everything else in the theory, are defined recursively. A game consists of a set of left options, each a game, and a set of right options, each a game. • The base of the recursion is the zero game, with no options for either player. • Last time we saw non-partisan games, where each player had the same options from each position. Today we look at partisan games. Hackenbush • Hackenbush is a game where the position is a diagram with red and blue edges, connected in at least one place to the “ground”. • A move is to delete an edge, a blue one for Left and a red one for Right. • Edges disconnected from the ground disappear. As usual, a player who cannot move loses. Hackenbush • From this first position, Right is going to win, because Left cannot prevent him from killing both the ground supports. It doesn’t matter who moves first.