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Set Theory, Including the Axiom of Choice) Plus the Negation of CH
ANNALS OF ~,IATltEMATICAL LOGIC - Volume 2, No. 2 (1970) pp. 143--178 INTERNAL COHEN EXTENSIONS D.A.MARTIN and R.M.SOLOVAY ;Tte RockeJ~'ller University and University (.~t CatiJbrnia. Berkeley Received 2 l)ecemt)er 1969 Introduction Cohen [ !, 2] has shown that tile continuum hypothesis (CH) cannot be proved in Zermelo-Fraenkel set theory. Levy and Solovay [9] have subsequently shown that CH cannot be proved even if one assumes the existence of a measurable cardinal. Their argument in tact shows that no large cardinal axiom of the kind present;y being considered by set theorists can yield a proof of CH (or of its negation, of course). Indeed, many set theorists - including the authors - suspect that C1t is false. But if we reject CH we admit Gurselves to be in a state of ignorance about a great many questions which CH resolves. While CH is a power- full assertion, its negation is in many ways quite weak. Sierpinski [ 1 5 ] deduces propcsitions there called C l - C82 from CH. We know of none of these propositions which is decided by the negation of CH and only one of them (C78) which is decided if one assumes in addition that a measurable cardinal exists. Among the many simple questions easily decided by CH and which cannot be decided in ZF (Zerme!o-Fraenkel set theory, including the axiom of choice) plus the negation of CH are tile following: Is every set of real numbers of cardinality less than tha't of the continuum of Lebesgue measure zero'? Is 2 ~0 < 2 ~ 1 ? Is there a non-trivial measure defined on all sets of real numbers? CIhis third question could be decided in ZF + not CH only in the unlikely event t Tile second author received support from a Sloan Foundation fellowship and tile National Science Foundation Grant (GP-8746). -
Geometric Integration Theory Contents
Steven G. Krantz Harold R. Parks Geometric Integration Theory Contents Preface v 1 Basics 1 1.1 Smooth Functions . 1 1.2Measures.............................. 6 1.2.1 Lebesgue Measure . 11 1.3Integration............................. 14 1.3.1 Measurable Functions . 14 1.3.2 The Integral . 17 1.3.3 Lebesgue Spaces . 23 1.3.4 Product Measures and the Fubini–Tonelli Theorem . 25 1.4 The Exterior Algebra . 27 1.5 The Hausdorff Distance and Steiner Symmetrization . 30 1.6 Borel and Suslin Sets . 41 2 Carath´eodory’s Construction and Lower-Dimensional Mea- sures 53 2.1 The Basic Definition . 53 2.1.1 Hausdorff Measure and Spherical Measure . 55 2.1.2 A Measure Based on Parallelepipeds . 57 2.1.3 Projections and Convexity . 57 2.1.4 Other Geometric Measures . 59 2.1.5 Summary . 61 2.2 The Densities of a Measure . 64 2.3 A One-Dimensional Example . 66 2.4 Carath´eodory’s Construction and Mappings . 67 2.5 The Concept of Hausdorff Dimension . 70 2.6 Some Cantor Set Examples . 73 i ii CONTENTS 2.6.1 Basic Examples . 73 2.6.2 Some Generalized Cantor Sets . 76 2.6.3 Cantor Sets in Higher Dimensions . 78 3 Invariant Measures and the Construction of Haar Measure 81 3.1 The Fundamental Theorem . 82 3.2 Haar Measure for the Orthogonal Group and the Grassmanian 90 3.2.1 Remarks on the Manifold Structure of G(N,M).... 94 4 Covering Theorems and the Differentiation of Integrals 97 4.1 Wiener’s Covering Lemma and its Variants . -
Notes on Partial Differential Equations John K. Hunter
Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous functions 1 1.3. H¨olderspaces 2 1.4. Lp spaces 3 1.5. Compactness 6 1.6. Averages 7 1.7. Convolutions 7 1.8. Derivatives and multi-index notation 8 1.9. Mollifiers 10 1.10. Boundaries of open sets 12 1.11. Change of variables 16 1.12. Divergence theorem 16 Chapter 2. Laplace's equation 19 2.1. Mean value theorem 20 2.2. Derivative estimates and analyticity 23 2.3. Maximum principle 26 2.4. Harnack's inequality 31 2.5. Green's identities 32 2.6. Fundamental solution 33 2.7. The Newtonian potential 34 2.8. Singular integral operators 43 Chapter 3. Sobolev spaces 47 3.1. Weak derivatives 47 3.2. Examples 47 3.3. Distributions 50 3.4. Properties of weak derivatives 53 3.5. Sobolev spaces 56 3.6. Approximation of Sobolev functions 57 3.7. Sobolev embedding: p < n 57 3.8. Sobolev embedding: p > n 66 3.9. Boundary values of Sobolev functions 69 3.10. Compactness results 71 3.11. Sobolev functions on Ω ⊂ Rn 73 3.A. Lipschitz functions 75 3.B. Absolutely continuous functions 76 3.C. Functions of bounded variation 78 3.D. Borel measures on R 80 v vi CONTENTS 3.E. Radon measures on R 82 3.F. Lebesgue-Stieltjes measures 83 3.G. Integration 84 3.H. Summary 86 Chapter 4. -
Shape Analysis, Lebesgue Integration and Absolute Continuity Connections
NISTIR 8217 Shape Analysis, Lebesgue Integration and Absolute Continuity Connections Javier Bernal This publication is available free of charge from: https://doi.org/10.6028/NIST.IR.8217 NISTIR 8217 Shape Analysis, Lebesgue Integration and Absolute Continuity Connections Javier Bernal Applied and Computational Mathematics Division Information Technology Laboratory This publication is available free of charge from: https://doi.org/10.6028/NIST.IR.8217 July 2018 INCLUDES UPDATES AS OF 07-18-2018; SEE APPENDIX U.S. Department of Commerce Wilbur L. Ross, Jr., Secretary National Institute of Standards and Technology Walter Copan, NIST Director and Undersecretary of Commerce for Standards and Technology ______________________________________________________________________________________________________ This Shape Analysis, Lebesgue Integration and publication Absolute Continuity Connections Javier Bernal is National Institute of Standards and Technology, available Gaithersburg, MD 20899, USA free of Abstract charge As shape analysis of the form presented in Srivastava and Klassen’s textbook “Functional and Shape Data Analysis” is intricately related to Lebesgue integration and absolute continuity, it is advantageous from: to have a good grasp of the latter two notions. Accordingly, in these notes we review basic concepts and results about Lebesgue integration https://doi.org/10.6028/NIST.IR.8217 and absolute continuity. In particular, we review fundamental results connecting them to each other and to the kind of shape analysis, or more generally, functional data analysis presented in the aforeme- tioned textbook, in the process shedding light on important aspects of all three notions. Many well-known results, especially most results about Lebesgue integration and some results about absolute conti- nuity, are presented without proofs. -
On the Structures of Generating Iterated Function Systems of Cantor Sets
ON THE STRUCTURES OF GENERATING ITERATED FUNCTION SYSTEMS OF CANTOR SETS DE-JUN FENG AND YANG WANG Abstract. A generating IFS of a Cantor set F is an IFS whose attractor is F . For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in R under the open set condition (OSC). If dimH F < 1 we prove that all generating IFSs of the set must have logarithmically commensurable contraction factors. From this Logarithmic Commensurability Theorem we derive a structure theorem for the semi-group of generating IFSs of F under the OSC. We also examine the impact of geometry on the structures of the semi-groups. Several examples will be given to illustrate the difficulty of the problem we study. 1. Introduction N d In this paper, a family of contractive affine maps Φ = fφjgj=1 in R is called an iterated function system (IFS). According to Hutchinson [12], there is a unique non-empty compact d SN F = FΦ ⊂ R , which is called the attractor of Φ, such that F = j=1 φj(F ). Furthermore, FΦ is called a self-similar set if Φ consists of similitudes. It is well known that the standard middle-third Cantor set C is the attractor of the iterated function system (IFS) fφ0; φ1g where 1 1 2 (1.1) φ (x) = x; φ (x) = x + : 0 3 1 3 3 A natural question is: Is it possible to express C as the attractor of another IFS? Surprisingly, the general question whether the attractor of an IFS can be expressed as the attractor of another IFS, which seems a rather fundamental question in fractal geometry, has 1991 Mathematics Subject Classification. -
Hausdorff Measure
Hausdorff Measure Jimmy Briggs and Tim Tyree December 3, 2016 1 1 Introduction In this report, we explore the the measurement of arbitrary subsets of the metric space (X; ρ); a topological space X along with its distance function ρ. We introduce Hausdorff Measure as a natural way of assigning sizes to these sets, especially those of smaller \dimension" than X: After an exploration of the salient properties of Hausdorff Measure, we proceed to a definition of Hausdorff dimension, a separate idea of size that allows us a more robust comparison between rough subsets of X. Many of the theorems in this report will be summarized in a proof sketch or shown by visual example. For a more rigorous treatment of the same material, we redirect the reader to Gerald B. Folland's Real Analysis: Modern techniques and their applications. Chapter 11 of the 1999 edition served as our primary reference. 2 Hausdorff Measure 2.1 Measuring low-dimensional subsets of X The need for Hausdorff Measure arises from the need to know the size of lower-dimensional subsets of a metric space. This idea is not as exotic as it may sound. In a high school Geometry course, we learn formulas for objects of various dimension embedded in R3: In Figure 1 we see the line segment, the circle, and the sphere, each with it's own idea of size. We call these length, area, and volume, respectively. Figure 1: low-dimensional subsets of R3: 2 4 3 2r πr 3 πr Note that while only volume measures something of the same dimension as the host space, R3; length, and area can still be of interest to us, especially 2 in applications. -
Georg Cantor English Version
GEORG CANTOR (March 3, 1845 – January 6, 1918) by HEINZ KLAUS STRICK, Germany There is hardly another mathematician whose reputation among his contemporary colleagues reflected such a wide disparity of opinion: for some, GEORG FERDINAND LUDWIG PHILIPP CANTOR was a corruptor of youth (KRONECKER), while for others, he was an exceptionally gifted mathematical researcher (DAVID HILBERT 1925: Let no one be allowed to drive us from the paradise that CANTOR created for us.) GEORG CANTOR’s father was a successful merchant and stockbroker in St. Petersburg, where he lived with his family, which included six children, in the large German colony until he was forced by ill health to move to the milder climate of Germany. In Russia, GEORG was instructed by private tutors. He then attended secondary schools in Wiesbaden and Darmstadt. After he had completed his schooling with excellent grades, particularly in mathematics, his father acceded to his son’s request to pursue mathematical studies in Zurich. GEORG CANTOR could equally well have chosen a career as a violinist, in which case he would have continued the tradition of his two grandmothers, both of whom were active as respected professional musicians in St. Petersburg. When in 1863 his father died, CANTOR transferred to Berlin, where he attended lectures by KARL WEIERSTRASS, ERNST EDUARD KUMMER, and LEOPOLD KRONECKER. On completing his doctorate in 1867 with a dissertation on a topic in number theory, CANTOR did not obtain a permanent academic position. He taught for a while at a girls’ school and at an institution for training teachers, all the while working on his habilitation thesis, which led to a teaching position at the university in Halle. -
Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions
Journal of Applied Mathematics and Computation, 2020, 4(4), 123-129 https://www.hillpublisher.com/journals/JAMC/ ISSN Online: 2576-0653 ISSN Print: 2576-0645 Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions Srijanani Anurag Prasad Department of Mathematics and Statistics, Indian Institute of Technology Tirupati, India. How to cite this paper: Srijanani Anurag Prasad. (2020) Riemann-Liouville Frac- Abstract tional Calculus of Blancmange Curve and Riemann-Liouville fractional calculus of Blancmange Curve and Cantor Func- Cantor Functions. Journal of Applied Ma- thematics and Computation, 4(4), 123-129. tions are studied in this paper. In this paper, Blancmange Curve and Cantor func- DOI: 10.26855/jamc.2020.12.003 tion defined on the interval is shown to be Fractal Interpolation Functions with appropriate interpolation points and parameters. Then, using the properties of Received: September 15, 2020 Fractal Interpolation Function, the Riemann-Liouville fractional integral of Accepted: October 10, 2020 Published: October 22, 2020 Blancmange Curve and Cantor function are described to be Fractal Interpolation Function passing through a different set of points. Finally, using the conditions for *Corresponding author: Srijanani the fractional derivative of order ν of a FIF, it is shown that the fractional deriva- Anurag Prasad, Department of Mathe- tive of Blancmange Curve and Cantor function is not a FIF for any value of ν. matics, Indian Institute of Technology Tirupati, India. Email: [email protected] Keywords Fractal, Interpolation, Iterated Function System, fractional integral, fractional de- rivative, Blancmange Curve, Cantor function 1. Introduction Fractal geometry is a subject in which irregular and complex functions and structures are researched. -
Fractal Geometry and Applications in Forest Science
ACKNOWLEDGMENTS Egolfs V. Bakuzis, Professor Emeritus at the University of Minnesota, College of Natural Resources, collected most of the information upon which this review is based. We express our sincere appreciation for his investment of time and energy in collecting these articles and books, in organizing the diverse material collected, and in sacrificing his personal research time to have weekly meetings with one of us (N.L.) to discuss the relevance and importance of each refer- enced paper and many not included here. Besides his interdisciplinary ap- proach to the scientific literature, his extensive knowledge of forest ecosystems and his early interest in nonlinear dynamics have helped us greatly. We express appreciation to Kevin Nimerfro for generating Diagrams 1, 3, 4, 5, and the cover using the programming package Mathematica. Craig Loehle and Boris Zeide provided review comments that significantly improved the paper. Funded by cooperative agreement #23-91-21, USDA Forest Service, North Central Forest Experiment Station, St. Paul, Minnesota. Yg._. t NAVE A THREE--PART QUE_.gTION,, F_-ACHPARToF:WHICH HA# "THREEPAP,T_.<.,EACFi PART" Of:: F_.AC.HPART oF wHIct4 HA.5 __ "1t4REE MORE PARTS... t_! c_4a EL o. EP-.ACTAL G EOPAgTI_YCoh_FERENCE I G;:_.4-A.-Ti_E AT THB Reprinted courtesy of Omni magazine, June 1994. VoL 16, No. 9. CONTENTS i_ Introduction ....................................................................................................... I 2° Description of Fractals .................................................................................... -
Fractals Lindenmayer Systems
FRACTALS LINDENMAYER SYSTEMS November 22, 2013 Rolf Pfeifer Rudolf M. Füchslin RECAP HIDDEN MARKOV MODELS What Letter Is Written Here? What Letter Is Written Here? What Letter Is Written Here? The Idea Behind Hidden Markov Models First letter: Maybe „a“, maybe „q“ Second letter: Maybe „r“ or „v“ or „u“ Take the most probable combination as a guess! Hidden Markov Models Sometimes, you don‘t see the states, but only a mapping of the states. A main task is then to derive, from the visible mapped sequence of states, the actual underlying sequence of „hidden“ states. HMM: A Fundamental Question What you see are the observables. But what are the actual states behind the observables? What is the most probable sequence of states leading to a given sequence of observations? The Viterbi-Algorithm We are looking for indices M1,M2,...MT, such that P(qM1,...qMT) = Pmax,T is maximal. 1. Initialization ()ib 1 i i k1 1(i ) 0 2. Recursion (1 t T-1) t1(j ) max( t ( i ) a i j ) b j k i t1 t1(j ) i : t ( i ) a i j max. 3. Termination Pimax,TT max( ( )) qmax,T q i: T ( i ) max. 4. Backtracking MMt t11() t Efficiency of the Viterbi Algorithm • The brute force approach takes O(TNT) steps. This is even for N = 2 and T = 100 difficult to do. • The Viterbi – algorithm in contrast takes only O(TN2) which is easy to do with todays computational means. Applications of HMM • Analysis of handwriting. • Speech analysis. • Construction of models for prediction. -
Chapter 6 Introduction to Calculus
RS - Ch 6 - Intro to Calculus Chapter 6 Introduction to Calculus 1 Archimedes of Syracuse (c. 287 BC – c. 212 BC ) Bhaskara II (1114 – 1185) 6.0 Calculus • Calculus is the mathematics of change. •Two major branches: Differential calculus and Integral calculus, which are related by the Fundamental Theorem of Calculus. • Differential calculus determines varying rates of change. It is applied to problems involving acceleration of moving objects (from a flywheel to the space shuttle), rates of growth and decay, optimal values, etc. • Integration is the "inverse" (or opposite) of differentiation. It measures accumulations over periods of change. Integration can find volumes and lengths of curves, measure forces and work, etc. Older branch: Archimedes (c. 287−212 BC) worked on it. • Applications in science, economics, finance, engineering, etc. 2 1 RS - Ch 6 - Intro to Calculus 6.0 Calculus: Early History • The foundations of calculus are generally attributed to Newton and Leibniz, though Bhaskara II is believed to have also laid the basis of it. The Western roots go back to Wallis, Fermat, Descartes and Barrow. • Q: How close can two numbers be without being the same number? Or, equivalent question, by considering the difference of two numbers: How small can a number be without being zero? • Fermat’s and Newton’s answer: The infinitessimal, a positive quantity, smaller than any non-zero real number. • With this concept differential calculus developed, by studying ratios in which both numerator and denominator go to zero simultaneously. 3 6.1 Comparative Statics Comparative statics: It is the study of different equilibrium states associated with different sets of values of parameters and exogenous variables. -
History and Mythology of Set Theory
Chapter udf History and Mythology of Set Theory This chapter includes the historical prelude from Tim Button's Open Set Theory text. set.1 Infinitesimals and Differentiation his:set:infinitesimals: Newton and Leibniz discovered the calculus (independently) at the end of the sec 17th century. A particularly important application of the calculus was differ- entiation. Roughly speaking, differentiation aims to give a notion of the \rate of change", or gradient, of a function at a point. Here is a vivid way to illustrate the idea. Consider the function f(x) = 2 x =4 + 1=2, depicted in black below: f(x) 5 4 3 2 1 x 1 2 3 4 1 Suppose we want to find the gradient of the function at c = 1=2. We start by drawing a triangle whose hypotenuse approximates the gradient at that point, perhaps the red triangle above. When β is the base length of our triangle, its height is f(1=2 + β) − f(1=2), so that the gradient of the hypotenuse is: f(1=2 + β) − f(1=2) : β So the gradient of our red triangle, with base length 3, is exactly 1. The hypotenuse of a smaller triangle, the blue triangle with base length 2, gives a better approximation; its gradient is 3=4. A yet smaller triangle, the green triangle with base length 1, gives a yet better approximation; with gradient 1=2. Ever-smaller triangles give us ever-better approximations. So we might say something like this: the hypotenuse of a triangle with an infinitesimal base length gives us the gradient at c = 1=2 itself.