Advanced Complex Analysis
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Avoiding the Undefined by Underspecification
Avoiding the Undefined by Underspecification David Gries* and Fred B. Schneider** Computer Science Department, Cornell University Ithaca, New York 14853 USA Abstract. We use the appeal of simplicity and an aversion to com- plexity in selecting a method for handling partial functions in logic. We conclude that avoiding the undefined by using underspecification is the preferred choice. 1 Introduction Everything should be made as simple as possible, but not any simpler. --Albert Einstein The first volume of Springer-Verlag's Lecture Notes in Computer Science ap- peared in 1973, almost 22 years ago. The field has learned a lot about computing since then, and in doing so it has leaned on and supported advances in other fields. In this paper, we discuss an aspect of one of these fields that has been explored by computer scientists, handling undefined terms in formal logic. Logicians are concerned mainly with studying logic --not using it. Some computer scientists have discovered, by necessity, that logic is actually a useful tool. Their attempts to use logic have led to new insights --new axiomatizations, new proof formats, new meta-logical notions like relative completeness, and new concerns. We, the authors, are computer scientists whose interests have forced us to become users of formal logic. We use it in our work on language definition and in the formal development of programs. This paper is born of that experience. The operation of a computer is nothing more than uninterpreted symbol ma- nipulation. Bits are moved and altered electronically, without regard for what they denote. It iswe who provide the interpretation, saying that one bit string represents money and another someone's name, or that one transformation rep- resents addition and another alphabetization. -
A Quick Algebra Review
A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation. -
Undefinedness and Soft Typing in Formal Mathematics
Undefinedness and Soft Typing in Formal Mathematics PhD Research Proposal by Jonas Betzendahl January 7, 2020 Supervisors: Prof. Dr. Michael Kohlhase Dr. Florian Rabe Abstract During my PhD, I want to contribute towards more natural reasoning in formal systems for mathematics (i.e. closer resembling that of actual mathematicians), with special focus on two topics on the precipice between type systems and logics in formal systems: undefinedness and soft types. Undefined terms are common in mathematics as practised on paper or blackboard by math- ematicians, yet few of the many systems for formalising mathematics that exist today deal with it in a principled way or allow explicit reasoning with undefined terms because allowing for this often results in undecidability. Soft types are a way of allowing the user of a formal system to also incorporate information about mathematical objects into the type system that had to be proven or computed after their creation. This approach is equally a closer match for mathematics as performed by mathematicians and has had promising results in the past. The MMT system constitutes a natural fit for this endeavour due to its rapid prototyping ca- pabilities and existing infrastructure. However, both of the aspects above necessitate a stronger support for automated reasoning than currently available. Hence, a further goal of mine is to extend the MMT framework with additional capabilities for automated and interactive theorem proving, both for reasoning in and outside the domains of undefinedness and soft types. 2 Contents 1 Introduction & Motivation 4 2 State of the Art 5 2.1 Undefinedness in Mathematics . -
The Riemann Mapping Theorem Christopher J. Bishop
The Riemann Mapping Theorem Christopher J. Bishop C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary: 30C35, Secondary: 30C85, 30C62 Key words and phrases. numerical conformal mappings, Schwarz-Christoffel formula, hyperbolic 3-manifolds, Sullivan’s theorem, convex hulls, quasiconformal mappings, quasisymmetric mappings, medial axis, CRDT algorithm The author is partially supported by NSF Grant DMS 04-05578. Abstract. These are informal notes based on lectures I am giving in MAT 626 (Topics in Complex Analysis: the Riemann mapping theorem) during Fall 2008 at Stony Brook. We will start with brief introduction to conformal mapping focusing on the Schwarz-Christoffel formula and how to compute the unknown parameters. In later chapters we will fill in some of the details of results and proofs in geometric function theory and survey various numerical methods for computing conformal maps, including a method of my own using ideas from hyperbolic and computational geometry. Contents Chapter 1. Introduction to conformal mapping 1 1. Conformal and holomorphic maps 1 2. M¨obius transformations 16 3. The Schwarz-Christoffel Formula 20 4. Crowding 27 5. Power series of Schwarz-Christoffel maps 29 6. Harmonic measure and Brownian motion 39 7. The quasiconformal distance between polygons 48 8. Schwarz-Christoffel iterations and Davis’s method 56 Chapter 2. The Riemann mapping theorem 67 1. The hyperbolic metric 67 2. Schwarz’s lemma 69 3. The Poisson integral formula 71 4. A proof of Riemann’s theorem 73 5. Koebe’s method 74 6. -
Complex Analysis Class 24: Wednesday April 2
Complex Analysis Math 214 Spring 2014 Fowler 307 MWF 3:00pm - 3:55pm c 2014 Ron Buckmire http://faculty.oxy.edu/ron/math/312/14/ Class 24: Wednesday April 2 TITLE Classifying Singularities using Laurent Series CURRENT READING Zill & Shanahan, §6.2-6.3 HOMEWORK Zill & Shanahan, §6.2 3, 15, 20, 24 33*. §6.3 7, 8, 9, 10. SUMMARY We shall be introduced to Laurent Series and learn how to use them to classify different various kinds of singularities (locations where complex functions are no longer analytic). Classifying Singularities There are basically three types of singularities (points where f(z) is not analytic) in the complex plane. Isolated Singularity An isolated singularity of a function f(z) is a point z0 such that f(z) is analytic on the punctured disc 0 < |z − z0| <rbut is undefined at z = z0. We usually call isolated singularities poles. An example is z = i for the function z/(z − i). Removable Singularity A removable singularity is a point z0 where the function f(z0) appears to be undefined but if we assign f(z0) the value w0 with the knowledge that lim f(z)=w0 then we can say that we z→z0 have “removed” the singularity. An example would be the point z = 0 for f(z) = sin(z)/z. Branch Singularity A branch singularity is a point z0 through which all possible branch cuts of a multi-valued function can be drawn to produce a single-valued function. An example of such a point would be the point z = 0 for Log (z). -
Control Systems
ECE 380: Control Systems Course Notes: Winter 2014 Prof. Shreyas Sundaram Department of Electrical and Computer Engineering University of Waterloo ii c Shreyas Sundaram Acknowledgments Parts of these course notes are loosely based on lecture notes by Professors Daniel Liberzon, Sean Meyn, and Mark Spong (University of Illinois), on notes by Professors Daniel Davison and Daniel Miller (University of Waterloo), and on parts of the textbook Feedback Control of Dynamic Systems (5th edition) by Franklin, Powell and Emami-Naeini. I claim credit for all typos and mistakes in the notes. The LATEX template for The Not So Short Introduction to LATEX 2" by T. Oetiker et al. was used to typeset portions of these notes. Shreyas Sundaram University of Waterloo c Shreyas Sundaram iv c Shreyas Sundaram Contents 1 Introduction 1 1.1 Dynamical Systems . .1 1.2 What is Control Theory? . .2 1.3 Outline of the Course . .4 2 Review of Complex Numbers 5 3 Review of Laplace Transforms 9 3.1 The Laplace Transform . .9 3.2 The Inverse Laplace Transform . 13 3.2.1 Partial Fraction Expansion . 13 3.3 The Final Value Theorem . 15 4 Linear Time-Invariant Systems 17 4.1 Linearity, Time-Invariance and Causality . 17 4.2 Transfer Functions . 18 4.2.1 Obtaining the transfer function of a differential equation model . 20 4.3 Frequency Response . 21 5 Bode Plots 25 5.1 Rules for Drawing Bode Plots . 26 5.1.1 Bode Plot for Ko ....................... 27 5.1.2 Bode Plot for sq ....................... 28 s −1 s 5.1.3 Bode Plot for ( p + 1) and ( z + 1) . -
Division by Zero in Logic and Computing Jan Bergstra
Division by Zero in Logic and Computing Jan Bergstra To cite this version: Jan Bergstra. Division by Zero in Logic and Computing. 2021. hal-03184956v2 HAL Id: hal-03184956 https://hal.archives-ouvertes.fr/hal-03184956v2 Preprint submitted on 19 Apr 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. DIVISION BY ZERO IN LOGIC AND COMPUTING JAN A. BERGSTRA Abstract. The phenomenon of division by zero is considered from the per- spectives of logic and informatics respectively. Division rather than multi- plicative inverse is taken as the point of departure. A classification of views on division by zero is proposed: principled, physics based principled, quasi- principled, curiosity driven, pragmatic, and ad hoc. A survey is provided of different perspectives on the value of 1=0 with for each view an assessment view from the perspectives of logic and computing. No attempt is made to survey the long and diverse history of the subject. 1. Introduction In the context of rational numbers the constants 0 and 1 and the operations of addition ( + ) and subtraction ( − ) as well as multiplication ( · ) and division ( = ) play a key role. When starting with a binary primitive for subtraction unary opposite is an abbreviation as follows: −x = 0 − x, and given a two-place division function unary inverse is an abbreviation as follows: x−1 = 1=x. -
Half-Plane Capacity and Conformal Radius
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 3, March 2014, Pages 931–938 S 0002-9939(2013)11811-3 Article electronically published on December 4, 2013 HALF-PLANE CAPACITY AND CONFORMAL RADIUS STEFFEN ROHDE AND CARTO WONG (Communicated by Jeremy Tyson) Abstract. In this note, we show that the half-plane capacity of a subset of the upper half-plane is comparable to a simple geometric quantity, namely the euclidean area of the hyperbolic neighborhood of radius one of this set. This is achieved by proving a similar estimate for the conformal radius of a subdomain of the unit disc and by establishing a simple relation between these two quantities. 1. Introduction and results Let H = {z ∈ C:Imz>0} be the upper half-plane. A bounded subset A ⊂ H is called a hull if H \ A is a simply connected region. The half-plane capacity of a hull A is the quantity hcap(A) := lim z [gA(z) − z] , z→∞ where gA : H \ A → H is the unique conformal map satisfying the hydrodynamic 1 →∞ normalization g(z)=z + O( z )asz . It appears frequently in connection with the Schramm-Loewner Evolution SLE, since it serves as the conformally natural parameter in the chordal Loewner equation; see [1]. In the study of SLE, one often needs estimates of hcap(A) in terms of geometric properties of A. The definition of hcap in terms of conformal maps (or in terms of Brownian motion as in [1]) does not immediately yield such estimates. The purpose of this note is to provide a geometric quantity that is comparable to hcap(A), via a simple relation between half-plane capacity and conformal radius. -
4.10 Conformal Mapping Methods for Interfacial Dynamics
4.10 CONFORMAL MAPPING METHODS FOR INTERFACIAL DYNAMICS Martin Z. Bazant1 and Darren Crowdy2 1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA 2Department of Mathematics, Imperial College, London, UK Microstructural evolution is typically beyond the reach of mathematical analysis, but in two dimensions certain problems become tractable by complex analysis. Via the analogy between the geometry of the plane and the algebra of complex numbers, moving free boundary problems may be elegantly formu- lated in terms of conformal maps. For over half a century, conformal mapping has been applied to continuous interfacial dynamics, primarily in models of viscous fingering and solidification. Current developments in materials science include models of void electro-migration in metals, brittle fracture, and vis- cous sintering. Recently, conformal-map dynamics has also been formulated for stochastic problems, such as diffusion-limited aggregation and dielectric breakdown, which has re-invigorated the subject of fractal pattern formation. Although restricted to relatively simple models, conformal-map dynam- ics offers unique advantages over other numerical methods discussed in this chapter (such as the Level–Set Method) and in Chapter 9 (such as the phase field method). By absorbing all geometrical complexity into a time-dependent conformal map, it is possible to transform a moving free boundary problem to a simple, static domain, such as a circle or square, which obviates the need for front tracking. Conformal mapping also allows the exact representation of very complicated domains, which are not easily discretized, even by the most sophisticated adaptive meshes. Above all, however, conformal mapping offers analytical insights for otherwise intractable problems. -
The Chazy XII Equation and Schwarz Triangle Functions
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 095, 24 pages The Chazy XII Equation and Schwarz Triangle Functions Oksana BIHUN and Sarbarish CHAKRAVARTY Department of Mathematics, University of Colorado, Colorado Springs, CO 80918, USA E-mail: [email protected], [email protected] Received June 21, 2017, in final form December 12, 2017; Published online December 25, 2017 https://doi.org/10.3842/SIGMA.2017.095 Abstract. Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120{348] showed that the Chazy XII equation y000 − 2yy00 + 3y02 = K(6y0 − y2)2, K 2 C, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of K, can be expressed as y = a1w1 +a2w2 +a3w3 where wi solve the generalized Darboux{Halphen system. This relationship holds only for certain values of the coefficients (a1; a2; a3) and the Darboux{Halphen parameters (α; β; γ), which are enumerated in Table2. Consequently, the Chazy XII solution y(z) is parametrized by a particular class of Schwarz triangle functions S(α; β; γ; z) which are used to represent the solutions wi of the Darboux{Halphen system. The paper only considers the case where α + β +γ < 1. The associated triangle functions are related among themselves via rational maps that are derived from the classical algebraic transformations of hypergeometric functions. The Chazy XII equation is also shown to be equivalent to a Ramanujan-type differential system for a triple (P;^ Q;^ R^). -
Conjugacy Classification of Quaternionic M¨Obius Transformations
CONJUGACY CLASSIFICATION OF QUATERNIONIC MOBIUS¨ TRANSFORMATIONS JOHN R. PARKER AND IAN SHORT Abstract. It is well known that the dynamics and conjugacy class of a complex M¨obius transformation can be determined from a simple rational function of the coef- ficients of the transformation. We study the group of quaternionic M¨obius transforma- tions and identify simple rational functions of the coefficients of the transformations that determine dynamics and conjugacy. 1. Introduction The motivation behind this paper is the classical theory of complex M¨obius trans- formations. A complex M¨obiustransformation f is a conformal map of the extended complex plane of the form f(z) = (az + b)(cz + d)−1, where a, b, c and d are complex numbers such that the quantity σ = ad − bc is not zero (we sometimes write σ = σf in order to avoid ambiguity). There is a homomorphism from SL(2, C) to the group a b of complex M¨obiustransformations which takes the matrix ( c d ) to the map f. This homomorphism is surjective because the coefficients of f can always be scaled so that σ = 1. The collection of all complex M¨obius transformations for which σ takes the value 1 forms a group which can be identified with PSL(2, C). A member f of PSL(2, C) is simple if it is conjugate in PSL(2, C) to an element of PSL(2, R). The map f is k–simple if it may be expressed as the composite of k simple transformations but no fewer. Let τ = a + d (and likewise we write τ = τf where necessary). -
Conformal Mapping Solution of Laplace's Equation on a Polygon
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 14 (1986) 227-249 227 North-Holland Conformal mapping solution of Laplace’s equation on a polygon with oblique derivative boundary conditions Lloyd N. TREFETHEN * Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. Ruth J. WILLIAMS ** Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, U.S.A. Received 6 July 1984 Abstract: We consider Laplace’s equation in a polygonal domain together with the boundary conditions that along each side, the derivative in the direction at a specified oblique angle from the normal should be zero. First we prove that solutions to this problem can always be constructed by taking the real part of an analytic function that maps the domain onto another region with straight sides oriented according to the angles given in the boundary conditions. Then we show that this procedure can be carried out successfully in practice by the numerical calculation of Schwarz-Christoffel transformations. The method is illustrated by application to a Hall effect problem in electronics, and to a reflected Brownian motion problem motivated by queueing theory. Keywor& Laplace equation, conformal mapping, Schwarz-Christoffel map, oblique derivative, Hall effect, Brownian motion, queueing theory. Mathematics Subject CIassi/ication: 3OC30, 35525, 60K25, 65EO5, 65N99. 1. The oblique derivative problem and conformal mapping Let 52 by a polygonal domain in the complex plane C, by which we mean a possibly unbounded simply connected open subset of C whose boundary aa consists of a finite number of straight lines, rays, and line segments.