Cheap Non-Standard Analysis and Computability
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
Proofs, Sets, Functions, and More: Fundamentals of Mathematical Reasoning, with Engaging Examples from Algebra, Number Theory, and Analysis
Proofs, Sets, Functions, and More: Fundamentals of Mathematical Reasoning, with Engaging Examples from Algebra, Number Theory, and Analysis Mike Krebs James Pommersheim Anthony Shaheen FOR THE INSTRUCTOR TO READ i For the instructor to read We should put a section in the front of the book that organizes the organization of the book. It would be the instructor section that would have: { flow chart that shows which sections are prereqs for what sections. We can start making this now so we don't have to remember the flow later. { main organization and objects in each chapter { What a Cfu is and how to use it { Why we have the proofcomment formatting and what it is. { Applications sections and what they are { Other things that need to be pointed out. IDEA: Seperate each of the above into subsections that are labeled for ease of reading but not shown in the table of contents in the front of the book. ||||||||| main organization examples: ||||||| || In a course such as this, the student comes in contact with many abstract concepts, such as that of a set, a function, and an equivalence class of an equivalence relation. What is the best way to learn this material. We have come up with several rules that we want to follow in this book. Themes of the book: 1. The book has a few central mathematical objects that are used throughout the book. 2. Each central mathematical object from theme #1 must be a fundamental object in mathematics that appears in many areas of mathematics and its applications. -
Cameron E. Freer
Cameron E. Freer 955 Massachusetts Ave. #256 email: [email protected] Cambridge, MA 02139-3233 www: http://cfreer.org/ USA phone: +1-716-713-8228 EMPLOYMENT Massachusetts Institute of Technology (ACADEMIA) Research Scientist, Department of Brain and Cognitive Sciences. 2019 – Present Research scientist in the Probabilistic Computing Project. Postdoctoral Associate, Department of Brain and Cognitive Sciences. 2013 – 2015 Research postdoc. Postdoctoral mentors: Joshua B. Tenenbaum and Vikash K. Mansinghka Postdoctoral Fellow, Computer Science and Artificial Intelligence Laboratory. 2011 – 2013 Research postdoc. Postdoctoral mentors: Joshua B. Tenenbaum and Leslie P.Kaelbling Instructor in Pure Mathematics, Department of Mathematics. 2008 – 2010 Research and teaching postdoc. Postdoctoral mentors: Hartley Rogers, Jr. and Michael Sipser University of Hawai‘i at Manoa¯ Junior Researcher, Department of Mathematics. 2010 – 2011 Research and teaching postdoc. Postdoctoral mentor: Bjørn Kjos-Hanssen EMPLOYMENT Borelian Corporation (INDUSTRY) Founder and Chief Scientist. 2016 – Present Remine Chief Scientist. 2017 – 2018 Gamalon Labs Research Scientist. 2013 – 2016 Analog Devices Lyric Labs Visiting Fellow. 2013 – 2014 Hibernia Atlantic Advisory Board Member. 2011 – 2012 EDUCATION Harvard University Ph.D. in Mathematics, 2008. Dissertation: Models with high Scott rank. Advisor: Gerald E. Sacks University of Chicago S.B. in Mathematics with Honors, 2003. Last updated: November 2020 Cameron E. Freer 2 RESEARCH INTERESTS Interactions of randomness and computation, including the foundations of probabilistic comput- ing, efficient samplers and testing methods for probabilistic inference, and the mathematics of random structures. PUBLICATIONS N. L. Ackerman, C. E. Freer, and R. Patel, On computable aspects of algebraic and definable closure, Journal of Logic and Computation, to appear. F. A. -
Grade 7/8 Math Circles the Scale of Numbers Introduction
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Introduction Last week we quickly took a look at scientific notation, which is one way we can write down really big numbers. We can also use scientific notation to write very small numbers. 1 × 103 = 1; 000 1 × 102 = 100 1 × 101 = 10 1 × 100 = 1 1 × 10−1 = 0:1 1 × 10−2 = 0:01 1 × 10−3 = 0:001 As you can see above, every time the value of the exponent decreases, the number gets smaller by a factor of 10. This pattern continues even into negative exponent values! Another way of picturing negative exponents is as a division by a positive exponent. 1 10−6 = = 0:000001 106 In this lesson we will be looking at some famous, interesting, or important small numbers, and begin slowly working our way up to the biggest numbers ever used in mathematics! Obviously we can come up with any arbitrary number that is either extremely small or extremely large, but the purpose of this lesson is to only look at numbers with some kind of mathematical or scientific significance. 1 Extremely Small Numbers 1. Zero • Zero or `0' is the number that represents nothingness. It is the number with the smallest magnitude. • Zero only began being used as a number around the year 500. Before this, ancient mathematicians struggled with the concept of `nothing' being `something'. 2. Planck's Constant This is the smallest number that we will be looking at today other than zero. -
The Evolution of Numbers
The Evolution of Numbers Counting Numbers Counting Numbers: {1, 2, 3, …} We use numbers to count: 1, 2, 3, 4, etc You can have "3 friends" A field can have "6 cows" Whole Numbers Whole numbers are the counting numbers plus zero. Whole Numbers: {0, 1, 2, 3, …} Negative Numbers We can count forward: 1, 2, 3, 4, ...... but what if we count backward: 3, 2, 1, 0, ... what happens next? The answer is: negative numbers: {…, -3, -2, -1} A negative number is any number less than zero. Integers If we include the negative numbers with the whole numbers, we have a new set of numbers that are called integers: {…, -3, -2, -1, 0, 1, 2, 3, …} The Integers include zero, the counting numbers, and the negative counting numbers, to make a list of numbers that stretch in either direction indefinitely. Rational Numbers A rational number is a number that can be written as a simple fraction (i.e. as a ratio). 2.5 is rational, because it can be written as the ratio 5/2 7 is rational, because it can be written as the ratio 7/1 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3 More formally we say: A rational number is a number that can be written in the form p/q where p and q are integers and q is not equal to zero. Example: If p is 3 and q is 2, then: p/q = 3/2 = 1.5 is a rational number Rational Numbers include: all integers all fractions Irrational Numbers An irrational number is a number that cannot be written as a simple fraction. -
Control Number: FD-00133
Math Released Set 2015 Algebra 1 PBA Item #13 Two Real Numbers Defined M44105 Prompt Rubric Task is worth a total of 3 points. M44105 Rubric Score Description 3 Student response includes the following 3 elements. • Reasoning component = 3 points o Correct identification of a as rational and b as irrational o Correct identification that the product is irrational o Correct reasoning used to determine rational and irrational numbers Sample Student Response: A rational number can be written as a ratio. In other words, a number that can be written as a simple fraction. a = 0.444444444444... can be written as 4 . Thus, a is a 9 rational number. All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction. b = 0.354355435554... cannot be written as a fraction, so it is irrational. The product of an irrational number and a nonzero rational number is always irrational, so the product of a and b is irrational. You can also see it is irrational with my calculations: 4 (.354355435554...)= .15749... 9 .15749... is irrational. 2 Student response includes 2 of the 3 elements. 1 Student response includes 1 of the 3 elements. 0 Student response is incorrect or irrelevant. Anchor Set A1 – A8 A1 Score Point 3 Annotations Anchor Paper 1 Score Point 3 This response receives full credit. The student includes each of the three required elements: • Correct identification of a as rational and b as irrational (The number represented by a is rational . The number represented by b would be irrational). -
The Exponential Function
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 5-2006 The Exponential Function Shawn A. Mousel University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Mousel, Shawn A., "The Exponential Function" (2006). MAT Exam Expository Papers. 26. https://digitalcommons.unl.edu/mathmidexppap/26 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. The Exponential Function Expository Paper Shawn A. Mousel In partial fulfillment of the requirements for the Masters of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor May 2006 Mousel – MAT Expository Paper - 1 One of the basic principles studied in mathematics is the observation of relationships between two connected quantities. A function is this connecting relationship, typically expressed in a formula that describes how one element from the domain is related to exactly one element located in the range (Lial & Miller, 1975). An exponential function is a function with the basic form f (x) = ax , where a (a fixed base that is a real, positive number) is greater than zero and not equal to 1. The exponential function is not to be confused with the polynomial functions, such as x 2. One way to recognize the difference between the two functions is by the name of the function. -
Simple Statements, Large Numbers
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2007 Simple Statements, Large Numbers Shana Streeks University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Streeks, Shana, "Simple Statements, Large Numbers" (2007). MAT Exam Expository Papers. 41. https://digitalcommons.unl.edu/mathmidexppap/41 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Master of Arts in Teaching (MAT) Masters Exam Shana Streeks In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Gordon Woodward, Advisor July 2007 Simple Statements, Large Numbers Shana Streeks July 2007 Page 1 Streeks Simple Statements, Large Numbers Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, as defined by Wikipedia (2007). Large numbers typically refer to large positive integers, or more generally, large positive real numbers, but may also be used in other contexts. Very large numbers often occur in fields such as mathematics, cosmology, and cryptography. Sometimes people refer to numbers as being “astronomically large”. However, it is easy to mathematically define numbers that are much larger than those even in astronomy. We are familiar with the large magnitudes, such as million or billion. -
CURRICULUM VITAE Contact Department of Mathematics
CURRICULUM VITAE DAMIR D. DZHAFAROV Contact Department of Mathematics University of Connecticut 341 Mansfield Road Storrs, Connecticut 06269-1009 U.S.A. Telephone: +1 860-486-3120 E-mail: [email protected] Homepage: damir.math.uconn.edu Personal Date of birth: February 2, 1983. Place of birth: Prague, Czech Republic. Citizenships: Czech Republic, United States. Research interests Mathematical logic and foundations, specifically computability theory, reverse math- ematics, effective combinatorics, computable analysis, and interactions of logic with philosophy and cognitive science. Employment Regular positions. Associate Professor of Mathematics, University of Connecticut, 2019–present. Affiliate Associate Professor of Philosophy, University of Connecticut, 2019–present. Assistant Professor of Mathematics, University of Connecticut, 2013–2019. NSF Postdoctoral Fellow, University of California, Berkeley, 2012–2013. NSF Visiting Assistant Professor, University of Notre Dame, 2011–2012. Visiting positions. Fulbright Research Scholar, Charles University, Prgue, 2020–2021. Visiting Scholar, Institute Henri Poincare, Paris, February 2020. Visiting Instructor, University of Chicago, Summer 2019. Visiting Postdoctoral Fellow, National University of Singapore, Summer 2011. 1 2 DAMIR D. DZHAFAROV Education Ph.D. in Mathematics, University of Chicago, June 2011. (Advisors: Robert I. Soare, Denis R. Hirschfeldt, Antonio Montalb´an.) M.S. in Mathematics, University of Chicago, June 2006. B.S. in Mathematics, Purdue University, May 2005. Professional memberships 1. American Mathematical Society. 2. Jednota ˇcesky´ch matematik˚u a fyzik˚u. 3. Association for Symbolic Logic. 4. Association for Computability in Europe. 5. University of Connecticut Logic Group. 6. Connecticut Institute for Brain and Cognitive Science. Grants and awards 1. U.S. Fulbright Scholar, 2020–2021. 2. NSF Focused Research Group Grant DMS-1854355 (PI, 379 450 USD), 2019– 2022. -
The Notion Of" Unimaginable Numbers" in Computational Number Theory
Beyond Knuth’s notation for “Unimaginable Numbers” within computational number theory Antonino Leonardis1 - Gianfranco d’Atri2 - Fabio Caldarola3 1 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] 2 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy 3 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] Abstract Literature considers under the name unimaginable numbers any positive in- teger going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathemati- cal definition (it is indeed used in many sources without a proper definition). This simply means that research in this topic must always consider shortened representations, usually involving recursion, to even being able to describe such numbers. One of the most known methodologies to conceive such numbers is using hyper-operations, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. arXiv:1901.05372v2 [cs.LO] 12 Mar 2019 The most important notations to represent such hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work’s introduction. Within this work we will give an axiomatic setup for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorith- mic nature of representations and the increased computation capabilities of 1 computers give the perfect field to develop further the topic, exploring some possibilities to effectively operate with such big numbers. -
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. -
Standards Chapter 10: Exponents and Scientific Notation Key Terms
Chapter 10: Exponents and Scientific Notation Standards Common Core: 8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical Essential Questions Students will… expressions. How can you use exponents to Write expressions using write numbers? integer exponents. 8.EE.3: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate How can you use inductive Evaluate expressions very large or very small quantities, and to express reasoning to observe patterns involving integer exponents. how many times as much one is than the other. and write general rules involving properties of Multiply powers with the 8.EE.4: Perform operations with numbers expressed exponents? same base. in scientific notation, including problems where both decimal and scientific notation are used. Use How can you divide two Find a power of a power. scientific notation and choose units of appropriate powers that have the same size for measurements of very large or very small base? Find a power of a product. quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been How can you evaluate a Divide powers with the same generated by technology. nonzero number with an base. exponent of zero? How can you evaluate a nonzero Simplify expressions number with a negative integer involving the quotient of Key Terms exponent? powers. A power is a product of repeated factors. How can you read numbers Evaluate expressions that are written in scientific involving numbers with zero The base of a power is the common factor. -
E6-132-29.Pdf
HISTORY OF MATHEMATICS – The Number Concept and Number Systems - John Stillwell THE NUMBER CONCEPT AND NUMBER SYSTEMS John Stillwell Department of Mathematics, University of San Francisco, U.S.A. School of Mathematical Sciences, Monash University Australia. Keywords: History of mathematics, number theory, foundations of mathematics, complex numbers, quaternions, octonions, geometry. Contents 1. Introduction 2. Arithmetic 3. Length and area 4. Algebra and geometry 5. Real numbers 6. Imaginary numbers 7. Geometry of complex numbers 8. Algebra of complex numbers 9. Quaternions 10. Geometry of quaternions 11. Octonions 12. Incidence geometry Glossary Bibliography Biographical Sketch Summary A survey of the number concept, from its prehistoric origins to its many applications in mathematics today. The emphasis is on how the number concept was expanded to meet the demands of arithmetic, geometry and algebra. 1. Introduction The first numbers we all meet are the positive integers 1, 2, 3, 4, … We use them for countingUNESCO – that is, for measuring the size– of collectionsEOLSS – and no doubt integers were first invented for that purpose. Counting seems to be a simple process, but it leads to more complex processes,SAMPLE such as addition (if CHAPTERSI have 19 sheep and you have 26 sheep, how many do we have altogether?) and multiplication ( if seven people each have 13 sheep, how many do they have altogether?). Addition leads in turn to the idea of subtraction, which raises questions that have no answers in the positive integers. For example, what is the result of subtracting 7 from 5? To answer such questions we introduce negative integers and thus make an extension of the system of positive integers.