Discrete Structures Lecture Notes
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Finite Sum – Product Logic
Theory and Applications of Categories, Vol. 8, No. 5, pp. 63–99. FINITE SUM – PRODUCT LOGIC J.R.B. COCKETT1 AND R.A.G. SEELY2 ABSTRACT. In this paper we describe a deductive system for categories with finite products and coproducts, prove decidability of equality of morphisms via cut elimina- tion, and prove a “Whitman theorem” for the free such categories over arbitrary base categories. This result provides a nice illustration of some basic techniques in categorical proof theory, and also seems to have slipped past unproved in previous work in this field. Furthermore, it suggests a type-theoretic approach to 2–player input–output games. Introduction In the late 1960’s Lambekintroduced the notion of a “deductive system”, by which he meant the presentation of a sequent calculus for a logic as a category, whose objects were formulas of the logic, and whose arrows were (equivalence classes of) sequent deriva- tions. He noticed that “doctrines” of categories corresponded under this construction to certain logics. The classic example of this was cartesian closed categories, which could then be regarded as the “proof theory” for the ∧ – ⇒ fragment of intuitionistic propo- sitional logic. (An excellent account of this may be found in the classic monograph [Lambek–Scott 1986].) Since his original work, many categorical doctrines have been given similar analyses, but it seems one simple case has been overlooked, viz. the doctrine of categories with finite products and coproducts (without any closed structure and with- out any extra assumptions concerning distributivity of the one over the other). We began looking at this case with the thought that it would provide a nice simple introduction to some techniques in categorical proof theory, particularly the idea of rewriting systems modulo equations, which we have found useful in investigating categorical structures with two tensor products (“linearly distributive categories” [Blute et al. -
Math Preliminaries
Preliminaries We establish here a few notational conventions used throughout the text. Arithmetic with ∞ We shall sometimes use the symbols “∞” and “−∞” in simple arithmetic expressions involving real numbers. The interpretation given to such ex- pressions is the usual, natural one; for example, for all real numbers x, we have −∞ < x < ∞, x + ∞ = ∞, x − ∞ = −∞, ∞ + ∞ = ∞, and (−∞) + (−∞) = −∞. Some such expressions have no sensible interpreta- tion (e.g., ∞ − ∞). Logarithms and exponentials We denote by log x the natural logarithm of x. The logarithm of x to the base b is denoted logb x. We denote by ex the usual exponential function, where e ≈ 2.71828 is the base of the natural logarithm. We may also write exp[x] instead of ex. Sets and relations We use the symbol ∅ to denote the empty set. For two sets A, B, we use the notation A ⊆ B to mean that A is a subset of B (with A possibly equal to B), and the notation A ( B to mean that A is a proper subset of B (i.e., A ⊆ B but A 6= B); further, A ∪ B denotes the union of A and B, A ∩ B the intersection of A and B, and A \ B the set of all elements of A that are not in B. For sets S1,...,Sn, we denote by S1 × · · · × Sn the Cartesian product xiv Preliminaries xv of S1,...,Sn, that is, the set of all n-tuples (a1, . , an), where ai ∈ Si for i = 1, . , n. We use the notation S×n to denote the Cartesian product of n copies of a set S, and for x ∈ S, we denote by x×n the element of S×n consisting of n copies of x. -
1. Directed Graphs Or Quivers What Is Category Theory? • Graph Theory on Steroids • Comic Book Mathematics • Abstract Nonsense • the Secret Dictionary
1. Directed graphs or quivers What is category theory? • Graph theory on steroids • Comic book mathematics • Abstract nonsense • The secret dictionary Sets and classes: For S = fX j X2 = Xg, have S 2 S , S2 = S Directed graph or quiver: C = (C0;C1;@0 : C1 ! C0;@1 : C1 ! C0) Class C0 of objects, vertices, points, . Class C1 of morphisms, (directed) edges, arrows, . For x; y 2 C0, write C(x; y) := ff 2 C1 j @0f = x; @1f = yg f 2C1 tail, domain / @0f / @1f o head, codomain op Opposite or dual graph of C = (C0;C1;@0;@1) is C = (C0;C1;@1;@0) Graph homomorphism F : D ! C has object part F0 : D0 ! C0 and morphism part F1 : D1 ! C1 with @i ◦ F1(f) = F0 ◦ @i(f) for i = 0; 1. Graph isomorphism has bijective object and morphism parts. Poset (X; ≤): set X with reflexive, antisymmetric, transitive order ≤ Hasse diagram of poset (X; ≤): x ! y if y covers x, i.e., x 6= y and [x; y] = fx; yg, so x ≤ z ≤ y ) z = x or z = y. Hasse diagram of (N; ≤) is 0 / 1 / 2 / 3 / ::: Hasse diagram of (f1; 2; 3; 6g; j ) is 3 / 6 O O 1 / 2 1 2 2. Categories Category: Quiver C = (C0;C1;@0 : C1 ! C0;@1 : C1 ! C0) with: • composition: 8 x; y; z 2 C0 ; C(x; y) × C(y; z) ! C(x; z); (f; g) 7! g ◦ f • satisfying associativity: 8 x; y; z; t 2 C0 ; 8 (f; g; h) 2 C(x; y) × C(y; z) × C(z; t) ; h ◦ (g ◦ f) = (h ◦ g) ◦ f y iS qq <SSSS g qq << SSS f qqq h◦g < SSSS qq << SSS qq g◦f < SSS xqq << SS z Vo VV < x VVVV << VVVV < VVVV << h VVVV < h◦(g◦f)=(h◦g)◦f VVVV < VVV+ t • identities: 8 x; y; z 2 C0 ; 9 1y 2 C(y; y) : 8 f 2 C(x; y) ; 1y ◦ f = f and 8 g 2 C(y; z) ; g ◦ 1y = g f y o x MM MM 1y g MM MMM f MMM M& zo g y Example: N0 = fxg ; N1 = N ; 1x = 0 ; 8 m; n 2 N ; n◦m = m+n ; | one object, lots of arrows [monoid of natural numbers under addition] 4 x / x Equation: 3 + 5 = 4 + 4 Commuting diagram: 3 4 x / x 5 ( 1 if m ≤ n; Example: N1 = N ; 8 m; n 2 N ; jN(m; n)j = 0 otherwise | lots of objects, lots of arrows [poset (N; ≤) as a category] These two examples are small categories: have a set of morphisms. -
Relations in Categories
Relations in Categories Stefan Milius A thesis submitted to the Faculty of Graduate Studies in partial fulfilment of the requirements for the degree of Master of Arts Graduate Program in Mathematics and Statistics York University Toronto, Ontario June 15, 2000 Abstract This thesis investigates relations over a category C relative to an (E; M)-factori- zation system of C. In order to establish the 2-category Rel(C) of relations over C in the first part we discuss sufficient conditions for the associativity of horizontal composition of relations, and we investigate special classes of morphisms in Rel(C). Attention is particularly devoted to the notion of mapping as defined by Lawvere. We give a significantly simplified proof for the main result of Pavlovi´c,namely that C Map(Rel(C)) if and only if E RegEpi(C). This part also contains a proof' that the category Map(Rel(C))⊆ is finitely complete, and we present the results obtained by Kelly, some of them generalized, i. e., without the restrictive assumption that M Mono(C). The next part deals with factorization⊆ systems in Rel(C). The fact that each set-relation has a canonical image factorization is generalized and shown to yield an (E¯; M¯ )-factorization system in Rel(C) in case M Mono(C). The setting without this condition is studied, as well. We propose a⊆ weaker notion of factorization system for a 2-category, where the commutativity in the universal property of an (E; M)-factorization system is replaced by coherent 2-cells. In the last part certain limits and colimits in Rel(C) are investigated. -
Chapter 2 of Concrete Abstractions: an Introduction to Computer
Out of print; full text available for free at http://www.gustavus.edu/+max/concrete-abstractions.html CHAPTER TWO Recursion and Induction 2.1 Recursion We have used Scheme to write procedures that describe how certain computational processes can be carried out. All the procedures we've discussed so far generate processes of a ®xed size. For example, the process generated by the procedure square always does exactly one multiplication no matter how big or how small the number we're squaring is. Similarly, the procedure pinwheel generates a process that will do exactly the same number of stack and turn operations when we use it on a basic block as it will when we use it on a huge quilt that's 128 basic blocks long and 128 basic blocks wide. Furthermore, the size of the procedure (that is, the size of the procedure's text) is a good indicator of the size of the processes it generates: Small procedures generate small processes and large procedures generate large processes. On the other hand, there are procedures of a ®xed size that generate computa- tional processes of varying sizes, depending on the values of their parameters, using a technique called recursion. To illustrate this, the following is a small, ®xed-size procedure for making paper chains that can still make chains of arbitrary lengthÐ it has a parameter n for the desired length. You'll need a bunch of long, thin strips of paper and some way of joining the ends of a strip to make a loop. You can use tape, a stapler, or if you use slitted strips of cardstock that look like this , you can just slip the slits together. -
What's in a Name? the Matrix As an Introduction to Mathematics
St. John Fisher College Fisher Digital Publications Mathematical and Computing Sciences Faculty/Staff Publications Mathematical and Computing Sciences 9-2008 What's in a Name? The Matrix as an Introduction to Mathematics Kris H. Green St. John Fisher College, [email protected] Follow this and additional works at: https://fisherpub.sjfc.edu/math_facpub Part of the Mathematics Commons How has open access to Fisher Digital Publications benefited ou?y Publication Information Green, Kris H. (2008). "What's in a Name? The Matrix as an Introduction to Mathematics." Math Horizons 16.1, 18-21. Please note that the Publication Information provides general citation information and may not be appropriate for your discipline. To receive help in creating a citation based on your discipline, please visit http://libguides.sjfc.edu/citations. This document is posted at https://fisherpub.sjfc.edu/math_facpub/12 and is brought to you for free and open access by Fisher Digital Publications at St. John Fisher College. For more information, please contact [email protected]. What's in a Name? The Matrix as an Introduction to Mathematics Abstract In lieu of an abstract, here is the article's first paragraph: In my classes on the nature of scientific thought, I have often used the movie The Matrix to illustrate the nature of evidence and how it shapes the reality we perceive (or think we perceive). As a mathematician, I usually field questions elatedr to the movie whenever the subject of linear algebra arises, since this field is the study of matrices and their properties. So it is natural to ask, why does the movie title reference a mathematical object? Disciplines Mathematics Comments Article copyright 2008 by Math Horizons. -
The Matrix As an Introduction to Mathematics
St. John Fisher College Fisher Digital Publications Mathematical and Computing Sciences Faculty/Staff Publications Mathematical and Computing Sciences 2012 What's in a Name? The Matrix as an Introduction to Mathematics Kris H. Green St. John Fisher College, [email protected] Follow this and additional works at: https://fisherpub.sjfc.edu/math_facpub Part of the Mathematics Commons How has open access to Fisher Digital Publications benefited ou?y Publication Information Green, Kris H. (2012). "What's in a Name? The Matrix as an Introduction to Mathematics." Mathematics in Popular Culture: Essays on Appearances in Film, Fiction, Games, Television and Other Media , 44-54. Please note that the Publication Information provides general citation information and may not be appropriate for your discipline. To receive help in creating a citation based on your discipline, please visit http://libguides.sjfc.edu/citations. This document is posted at https://fisherpub.sjfc.edu/math_facpub/18 and is brought to you for free and open access by Fisher Digital Publications at St. John Fisher College. For more information, please contact [email protected]. What's in a Name? The Matrix as an Introduction to Mathematics Abstract In my classes on the nature of scientific thought, I have often used the movie The Matrix (1999) to illustrate how evidence shapes the reality we perceive (or think we perceive). As a mathematician and self-confessed science fiction fan, I usually field questionselated r to the movie whenever the subject of linear algebra arises, since this field is the study of matrices and their properties. So it is natural to ask, why does the movie title reference a mathematical object? Of course, there are many possible explanations for this, each of which probably contributed a little to the naming decision. -
Sets, Functions
Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: • real numbers, • complex numbers, C • integers, • All students in our class Defining Sets Sets can be defined directly: e.g. {1,2,4,8,16,32,…}, {CSC1130,CSC2110,…} Order, number of occurence are not important. e.g. {A,B,C} = {C,B,A} = {A,A,B,C,B} A set can be an element of another set. {1,{2},{3,{4}}} Defining Sets by Predicates The set of elements, x, in A such that P(x) is true. {}x APx| ( ) The set of prime numbers: Commonly Used Sets • N = {0, 1, 2, 3, …}, the set of natural numbers • Z = {…, -2, -1, 0, 1, 2, …}, the set of integers • Z+ = {1, 2, 3, …}, the set of positive integers • Q = {p/q | p Z, q Z, and q ≠ 0}, the set of rational numbers • R, the set of real numbers Special Sets • Empty Set (null set): a set that has no elements, denoted by ф or {}. • Example: The set of all positive integers that are greater than their squares is an empty set. • Singleton set: a set with one element • Compare: ф and {ф} – Ф: an empty set. Think of this as an empty folder – {ф}: a set with one element. The element is an empty set. Think of this as an folder with an empty folder in it. Venn Diagrams • Represent sets graphically • The universal set U, which contains all the objects under consideration, is represented by a rectangle. -
Mathematical Analysis of the Accordion Grating Illusion: a Differential Geometry Approach to Introduce the 3D Aperture Problem
Neural Networks 24 (2011) 1093–1101 Contents lists available at SciVerse ScienceDirect Neural Networks journal homepage: www.elsevier.com/locate/neunet Mathematical analysis of the Accordion Grating illusion: A differential geometry approach to introduce the 3D aperture problem Arash Yazdanbakhsh a,b,∗, Simone Gori c,d a Cognitive and Neural Systems Department, Boston University, MA, 02215, United States b Neurobiology Department, Harvard Medical School, Boston, MA, 02115, United States c Universita' degli studi di Padova, Dipartimento di Psicologia Generale, Via Venezia 8, 35131 Padova, Italy d Developmental Neuropsychology Unit, Scientific Institute ``E. Medea'', Bosisio Parini, Lecco, Italy article info a b s t r a c t Keywords: When an observer moves towards a square-wave grating display, a non-rigid distortion of the pattern Visual illusion occurs in which the stripes bulge and expand perpendicularly to their orientation; these effects reverse Motion when the observer moves away. Such distortions present a new problem beyond the classical aperture Projection line Line of sight problem faced by visual motion detectors, one we describe as a 3D aperture problem as it incorporates Accordion Grating depth signals. We applied differential geometry to obtain a closed form solution to characterize the fluid Aperture problem distortion of the stripes. Our solution replicates the perceptual distortions and enabled us to design a Differential geometry nulling experiment to distinguish our 3D aperture solution from other candidate mechanisms (see Gori et al. (in this issue)). We suggest that our approach may generalize to other motion illusions visible in 2D displays. ' 2011 Elsevier Ltd. All rights reserved. 1. Introduction need for line-ends or intersections along the lines to explain the illusion. -
Graph Theory
1 Graph Theory “Begin at the beginning,” the King said, gravely, “and go on till you come to the end; then stop.” — Lewis Carroll, Alice in Wonderland The Pregolya River passes through a city once known as K¨onigsberg. In the 1700s seven bridges were situated across this river in a manner similar to what you see in Figure 1.1. The city’s residents enjoyed strolling on these bridges, but, as hard as they tried, no residentof the city was ever able to walk a route that crossed each of these bridges exactly once. The Swiss mathematician Leonhard Euler learned of this frustrating phenomenon, and in 1736 he wrote an article [98] about it. His work on the “K¨onigsberg Bridge Problem” is considered by many to be the beginning of the field of graph theory. FIGURE 1.1. The bridges in K¨onigsberg. J.M. Harris et al., Combinatorics and Graph Theory , DOI: 10.1007/978-0-387-79711-3 1, °c Springer Science+Business Media, LLC 2008 2 1. Graph Theory At first, the usefulness of Euler’s ideas and of “graph theory” itself was found only in solving puzzles and in analyzing games and other recreations. In the mid 1800s, however, people began to realize that graphs could be used to model many things that were of interest in society. For instance, the “Four Color Map Conjec- ture,” introduced by DeMorgan in 1852, was a famous problem that was seem- ingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions are colored differently. -
Arxiv:1811.04966V3 [Math.NT]
Descartes’ rule of signs, Newton polygons, and polynomials over hyperfields Matthew Baker and Oliver Lorscheid Abstract. In this note, we develop a theory of multiplicities of roots for polynomials over hyperfields and use this to provide a unified and conceptual proof of both Descartes’ rule of signs and Newton’s “polygon rule”. Introduction Given a real polynomial p ∈ R[T ], Descartes’ rule of signs provides an upper bound for the number of positive (resp. negative) real roots of p in terms of the signs of the coeffi- cients of p. Specifically, the number of positive real roots of p (counting multiplicities) is bounded above by the number of sign changes in the coefficients of p(T ), and the number of negative roots is bounded above by the number of sign changes in the coefficients of p(−T ). Another classical “rule”, which is less well known to mathematicians in general but is used quite often in number theory, is Newton’s polygon rule. This rule concerns polynomi- als over fields equipped with a valuation, which is a function v : K → R ∪{∞} satisfying • v(a) = ∞ if and only if a = 0 • v(ab) = v(a) + v(b) • v(a + b) > min{v(a),v(b)}, with equality if v(a) 6= v(b) for all a,b ∈ K. An example is the p-adic valuation vp on Q, where p is a prime number, given by the formula vp(s/t) = ordp(s) − ordp(t), where ordp(n) is the maximum power of p dividing a nonzero integer n. Another example is the T -adic valuation v on k(T), for any field k, given by v ( f /g) = arXiv:1811.04966v3 [math.NT] 26 May 2020 T T ordT ( f )−ordT (g), where ordT ( f ) is the maximum power of T dividing a nonzero polyno- mial f ∈ k[T]. -
The Structure of the 3-Separations of 3-Connected Matroids
THE STRUCTURE OF THE 3{SEPARATIONS OF 3{CONNECTED MATROIDS JAMES OXLEY, CHARLES SEMPLE, AND GEOFF WHITTLE Abstract. Tutte defined a k{separation of a matroid M to be a partition (A; B) of the ground set of M such that |A|; |B|≥k and r(A)+r(B) − r(M) <k. If, for all m<n, the matroid M has no m{separations, then M is n{connected. Earlier, Whitney showed that (A; B) is a 1{separation of M if and only if A is a union of 2{connected components of M.WhenM is 2{connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2{separations. When M is 3{connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3{ separations of M. 1. Introduction One of Tutte’s many important contributions to matroid theory was the introduction of the general theory of separations and connectivity [10] de- fined in the abstract. The structure of the 1–separations in a matroid is elementary. They induce a partition of the ground set which in turn induces a decomposition of the matroid into 2–connected components [11]. Cun- ningham and Edmonds [1] considered the structure of 2–separations in a matroid. They showed that a 2–connected matroid M can be decomposed into a set of 3–connected matroids with the property that M can be built from these 3–connected matroids via a canonical operation known as 2–sum. Moreover, there is a labelled tree that gives a precise description of the way that M is built from the 3–connected pieces.