DOCSLIB.ORG

Superior Mathematics from an Elementary Point of View ” Course Notes Jacopo D’Aurizio

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Superior Mathematics from an Elementary Point of View ” Course Notes Jacopo D’Aurizio ” Superior Mathematics from an Elementary point of view ” course notes Jacopo d’Aurizio To cite this version: Jacopo d’Aurizio. ” Superior Mathematics from an Elementary point of view ” course notes. Master. Italy. 2017. cel-01813978 HAL Id: cel-01813978 https://cel.archives-ouvertes.fr/cel-01813978 Submitted on 12 Jun 2018 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. \Superior Mathematics from an Elementary point of view" course notes Undergraduate course, 2017-2018, University of Pisa Jack D'Aurizio Contents 0 Introduction 2 1 Creative Telescoping and DFT 3 2 Convolutions and ballot problems 15 3 Chebyshev and Legendre polynomials 30 4 The glory of Fourier, Laplace, Feynman and Frullani 40 5 The Basel problem 60 6 Special functions and special products 70 7 The Cauchy-Schwarz inequality and beyond 97 8 Remarkable results in Linear Algebra 121 9 The Fundamental Theorem of Algebra 125 10 Quantitative forms of the Weierstrass approximation Theorem 133 11 Elliptic integrals and the AGM 137 12 Dilworth, Erdos-Szekeres, Brouwer and Borsuk-Ulam's Theorems 147 13 Continued fractions and elements of Diophantine Approximation 158 14 Symmetric functions and elements of Analytic Combinatorics 173 15 Spherical Trigonometry 183 0 Introduction This course has been designed to serve University students of the first and second year of Mathematics. The purpose of these notes is to give elements of both strategy and tactics in problem solving, by explaining ideas and techniques willing to be elementary and powerful at the same time. We will not focus on a single subject among Calculus, Algebra, Combinatorics or Geometry: we will just try to enlarge the \toolbox" of any professional mathematician wannabe, by starting from humble requirements: • knowledge of number sets (N; Z; Q; R; C) and their properties; • knowledge of mathematical terminology and notation; • knowledge of the main mathematical functions; • knowledge of the following concepts: limits, convergence, derivatives, Riemann integral; 1 CREATIVE TELESCOPING AND DFT • knowledge of basic Combinatorics and Arithmetics. A collection of problems in Analysis and Advanced integration techniques, kindly provided by Tolaso J. Kos and Zaid Alyafeai, are excellent sources of exercises to match with the study of these notes. These notes are distributed under a Creative Commons Share-Alike (CCSA) license. Personal use is allowed, distribution is allowed with the only contraint of making a proper mention of the author (Jack D'Aurizio). Commercial use, modification or inclusion in other works are not allowed. 1 Creative Telescoping and DFT It is soon evident, during the study of Mathematics, that the bijectivity of some function f does not grant that the explicit computation of f −1(y) is \just as easy" as the explicit computation of f(x). Some examples are related with the ease of multiplication, against the hardness of factorization; the possibility of computing derivatives in a algorithmic fashion, against the lack of a completely algorithmic way to find indefinite integrals; the determination of a Galois group of an irreducible polynomial over Q, against the difficult task of finding a polynomial having a given Galois group. In the present section we will outline two interesting techniques for solving (or getting arbitrarily close to an actual \solution") a peculiar inverse problem, that is the computation of series. Definition 1. We give the adjective telescopic to objects of the form a1 + a2 + ::: + an, where each ai can be written as bi − bi+1 for some sequence b1; b2; : : : ; bn; bn+1. With such assumption we have: a1 + a2 + ::: + an = (b1 − b2) + (b2 − b3) + ::: + (bn − bn+1) = b1 − bn+1: Essentially, every telescopic sum is simple to compute, just as any convergent series with terms of the form bi − bi+1. A peculiar example is provided by Mengoli series: the identity N N X 1 X 1 1 1 = − = 1 − n(n + 1) n n + 1 N + 1 n=1 n=1 grants we have X 1 = 1: n(n + 1) n≥1 The following generalization is also straightforward: Lemma 2. X 1 1 8k 2 +; = : N n(n + 1) · ::: · (n + k) k · k! n≥1 In a forthcoming section we will also see a proof not relying on telescoping, but on properties of Euler's Beta function. The first technical issue we meet in this framework is related to the fact that recognizing a contribution of the form bi − bi+1 in the general term of a series is not always easy, just like in the continuous analogue: if the task is to R b 0 find a f(x) dx, it is not always easy to devise a function g such that f(x) = g (x). Here there are some non-trivial examples: Lemma 3. X 1 π arctan = : 1 + n + n2 4 n≥1 Page 3 / 195 Proof. If we use \backwards" the sum/subtraction formulas for the tangent function, we have that tan(x) ± tan(y) tan(x ± y) = 1 ∓ tan(x) tan(y) implies: 1 1 ! 1 1 n − n+1 1 arctan − arctan = arctan 1 = arctan 2 n n + 1 1 + n(n+1) 1 + n + n π so the given series is telescopic and it converges to arctan(1) = 4 . Exercise 4. The sequence of Fibonacci numbers fFngn≥0 is defined through F0 = 0;F1 = 1 and Fn+2 = Fn+1 + Fn for any n ≥ 0. Show that: X (−1)n+1 p arctan = arctan( 5 − 2): Fn+1(Fn + Fn+2) n≥1 Exercise 5. Show that: X sinh 1 π X 1 π π arctan = ; arctan = − arctan tanh : cosh(2n) 2 8n2 4 4 n2Z n≥1 Exercise 6. Prove that: N X 1 2n N + 1 2N + 2 = ; 4n n 22N+1 N + 1 n=0 for instance by considering that by De Moivre's formula we have Z π 1 2n 1 2n An = n = cos (x) dx 4 n 2π −π 2n 1 P2n 2n (2n−j)ix −jix R π kix since cos(x) = 4n j=0 j e e and −π e dx = 2π · δ(k), so the only non-vanishing contribution is related to the j = n term, and N X 1 2n 1 Z π 1 − cos2N+2(x) N + 1 2N + 2 = dx = (2N + 2) A = 4n n 2π 2 N+1 22N+1 N + 1 n=0 −π sin (x) R dx follows from the integration by parts formula ( sin2 x = − cot x). Prove also that X 2k 1 = 1 k (k + 1)4k k≥0 by recognizing in the main term a telescopic contribution. Give a probabilistic interpretation to the proved identities, by considering random paths on a infinite grid (Z × Z) where only unit movements towards North or East are allowed. We now outline the first (really) interesting idea, namely creative telescoping: even if we are not able to write the main term of a series in the bi − bi+1 form, it is not unlikely there is an accurate approximation of the main term that can be represented in such a telescopic form. By subtracting the accurate telescopic approximation from the main term, the original problem boils down to computing/approximating a series that is likely to converge faster than the Page 4 / 195 1 CREATIVE TELESCOPING AND DFT original one, and the same approximation-by-telescopic-series trick can be performed again. For instance, we might employ creative telescoping for producing very accurate approximations of the series X 1 ζ(2) = n2 n≥1 that will be the main character of a forthcoming section. 1 1 In particular, for any n > 1 the term n2 is quite close to the telescopic term 2 1 : n − 4 1 4 1 1 = = 2 − 2 1 n − 4 (2n − 1)(2n + 1) 2n − 1 2n + 1 1 1 1 and we have n2 − 2 1 = − (2n−1)n2(2n+1) , so: n − 4 X 1 X 1 X 1 ζ(2) = 1 + = 1 + − n2 n2 − 1 (2n − 1)n2(2n + 1) n≥2 n≥2 4 n≥2 2 X 1 = 1 + − 3 (2n − 1)n2(2n + 1) n≥2 5 2 gives us ζ(2) < 3 (that we will prove to be equivalent to π < 10), and the magenta \residual series" can be manipu- lated in the same fashion (by extracting the first term, approximating the main term with a telescopic contribution, considering the residual series) or simply bounded above by: X 1 X 1 3 22 < = ζ(2) − (2n − 1)n2(2n + 1) (2n − 1) n − 1 n + 1 (2n + 1) 2 9 n≥2 n≥2 2 2 74 74 5 from which the lower bound ζ(2) > 45 follows. We may notice that the difference between 45 and 3 is already pretty small. In this framework the iteration of creative telescoping leads to two interesting consequences: the identity X 1 X 3 ζ(2) = = ; n2 n22n n≥1 n≥1 n providing a remarkable acceleration of the series defining ζ(2), and Stirling's inequality: nn p 1 nn p 1 2πn exp ≤ n! ≤ 2πn exp e 12n + 1 e 12n further details will be disclosed soon. It is also possible to employ creative telescoping for proving that: X 1 5 X (−1)n+1 ζ(3) = = n3 2 n32n n≥1 n≥1 n a key identity in Apery's proof of ζ(3) 62 Q.
Load more

Recommended publications
  • Euler Product Asymptotics on Dirichlet L-Functions 3
    EULER PRODUCT ASYMPTOTICS ON DIRICHLET L-FUNCTIONS IKUYA KANEKO Abstract. We derive the asymptotic behaviour of partial Euler products for Dirichlet L-functions L(s,χ) in the critical strip upon assuming only the Generalised Riemann Hypothesis (GRH). Capturing the behaviour of the partial Euler products on the critical line is called the Deep Riemann Hypothesis (DRH) on the ground of its importance. Our result manifests the positional relation of the GRH and DRH. If χ is a non-principal character, we observe the √2 phenomenon, which asserts that the extra factor √2 emerges in the Euler product asymptotics on the central point s = 1/2. We also establish the connection between the behaviour of the partial Euler products and the size of the remainder term in the prime number theorem for arithmetic progressions. Contents 1. Introduction 1 2. Prelude to the asymptotics 5 3. Partial Euler products for Dirichlet L-functions 6 4. Logarithmical approach to Euler products 7 5. Observing the size of E(x; q,a) 8 6. Appendix 9 Acknowledgement 11 References 12 1. Introduction This paper was motivated by the beautiful and profound work of Ramanujan on asymptotics for the partial Euler product of the classical Riemann zeta function ζ(s). We handle the family of Dirichlet L-functions ∞ L(s,χ)= χ(n)n−s = (1 χ(p)p−s)−1 with s> 1, − ℜ 1 p X Y arXiv:1902.04203v1 [math.NT] 12 Feb 2019 and prove the asymptotic behaviour of partial Euler products (1.1) (1 χ(p)p−s)−1 − 6 pYx in the critical strip 0 < s < 1, assuming the Generalised Riemann Hypothesis (GRH) for this family.
    [Show full text]
  • Accessing Bernoulli-Numbers by Matrix-Operations Gottfried Helms 3'2006 Version 2.3
    Accessing Bernoulli-Numbers by Matrix-Operations Gottfried Helms 3'2006 Version 2.3 Accessing Bernoulli-Numbers by Matrixoperations ........................................................................ 2 1. Introduction....................................................................................................................................................................... 2 2. A common equation of recursion (containing a significant error)..................................................................................... 3 3. Two versions B m and B p of bernoulli-numbers? ............................................................................................................... 4 4. Computation of bernoulli-numbers by matrixinversion of (P-I) ....................................................................................... 6 5. J contains the eigenvalues, and G m resp. G p contain the eigenvectors of P z resp. P s ......................................................... 7 6. The Binomial-Matrix and the Matrixexponential.............................................................................................................. 8 7. Bernoulli-vectors and the Matrixexponential.................................................................................................................... 8 8. The structure of the remaining coefficients in the matrices G m - and G p........................................................................... 9 9. The original problem of Jacob Bernoulli: "Powersums" - from G
    [Show full text]
  • CC Geometry H
    CC Geometry H Aim #8: How do we find perimeter and area of polygons in the Cartesian Plane? Do Now: Find the area of the triangle below using decomposition and then using the shoelace formula (also known as Green's theorem). 1) Given rectangle ABCD, a. Identify the vertices. b. Find the exact perimeter. c. Find the area using the area formula. d. List the vertices starting with A moving counterclockwise and apply the shoelace formula. Does the method work for quadrilaterals? 2) Calculate the area using the shoelace formula and then find the perimeter (nearest hundredth). 3) Find the perimeter (to the nearest hundredth) and the area of the quadrilateral with vertices A(-3,4), B(4,6), C(2,-3), and D(-4,-4). 4) A textbook has a picture of a triangle with vertices (3,6) and (5,2). Something happened in printing the book and the coordinates of the third vertex are listed as (-1, ). The answers in the back of the book give the area of the triangle as 6 square units. What is the y-coordinate of this missing vertex? 5) Find the area of the pentagon with vertices A(5,8), B(4,-3), C(-1,-2), D(-2,4), and E(2,6). 6) Show that the shoelace formula used on the trapezoid shown confirms the traditional formula for the area of a trapezoid: 1 (b1 + b2)h 2 D (x3,y) C (x2,y) A B (x ,0) (0,0) 1 7) Find the area and perimeter (exact and nearest hundredth) of the hexagon shown.
    [Show full text]
  • Rational Root Theorem and Synthetic Division Worksheet
    Rational Root Theorem And Synthetic Division Worksheet Executory Geoffry approximate milkily or stodged tattily when Chaddie is unpastoral. Abaxial and Armenoid Garth nitrogenizes manneristically and gunges his Kaliyuga tremendously and availingly. Ivory-towered Wallas sometimes window his chechakoes recollectedly and ceres so underarm! In this section, we are discuss the variety of tools for writing polynomial functions and solving polynomial equations. This section we can find a quick foray into math help, use cookies to find all wikis and dirty test these theorems. Using synthetic division and rational root theorems. First look into factoring polynomials. How is my work scored? By the Factor Theorem, these zeros have factors associated with them. Rational Root Theorem Displaying top worksheets found for faith concept. Then determine the list of a synthetic division because if the! 23 Obj Students will that long division and synthetic division to divide polynomials. Solution because we can solve the original deed as follows. Work examples Homework: Pg. The resulting polynomial is now reduced to a quadratic equation so we can stop with the synthetic division and solve for the remaining zeros by either factoring or the quadratic formula. C Use the Rational Root Theorem to cellar the nostril of an possible rational roots it. For understanding the theorem and their uses cookies off or zero positive and rational root theorem and synthetic division worksheet if the. Very subtle but accuracy and synthetic division because if a root theorems and update to do not. Zero Theorem in party list further possible fractions can. End Encrypted Data After Losing Private Key? Rational Root Theorem Worksheet Kalmia.
    [Show full text]
  • Of the Riemann Hypothesis
    A Simple Counterexample to Havil's \Reformulation" of the Riemann Hypothesis Jonathan Sondow 209 West 97th Street New York, NY 10025 [email protected] The Riemann Hypothesis (RH) is \the greatest mystery in mathematics" [3]. It is a conjecture about the Riemann zeta function. The zeta function allows us to pass from knowledge of the integers to knowledge of the primes. In his book Gamma: Exploring Euler's Constant [4, p. 207], Julian Havil claims that the following is \a tantalizingly simple reformulation of the Rie- mann Hypothesis." Havil's Conjecture. If 1 1 X (−1)n X (−1)n cos(b ln n) = 0 and sin(b ln n) = 0 na na n=1 n=1 for some pair of real numbers a and b, then a = 1=2. In this note, we first state the RH and explain its connection with Havil's Conjecture. Then we show that the pair of real numbers a = 1 and b = 2π=ln 2 is a counterexample to Havil's Conjecture, but not to the RH. Finally, we prove that Havil's Conjecture becomes a true reformulation of the RH if his conclusion \then a = 1=2" is weakened to \then a = 1=2 or a = 1." The Riemann Hypothesis In 1859 Riemann published a short paper On the number of primes less than a given quantity [6], his only one on number theory. Writing s for a complex variable, he assumes initially that its real 1 part <(s) is greater than 1, and he begins with Euler's product-sum formula 1 Y 1 X 1 = (<(s) > 1): 1 ns p 1 − n=1 ps Here the product is over all primes p.
    [Show full text]
  • Session 9: Pigeon Hole Principle and Ramsey Theory - Handout
    Session 9: Pigeon Hole Principle and Ramsey Theory - Handout “Suppose aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack.” image: Frank P. Ramsey (1903-1926) quote: Paul Erd¨os (1913-1996) The following dots are in general position (i.e. no three lie on a straight line). 1) If it exists, find a convex quadrilateral in this cluster of dots. 2) If it exists, find a convex pentagon in this cluster of dots. 3) If it exists, find a convex hexagon in this cluster of dots. 4) If it exists, find a convex heptagon in this cluster of dots. 5) If it exists, find a convex octagon in this cluster of dots. 6) Draw 8 points in general position in such a way that the points do not contain a convex pentagon. 7) In my family there are 2 adults and 3 children. When our family arrives home how many of us must enter the house in order to ensure there is at least one adult in the house? 8) Show that among any collection of 7 natural numbers there must be two whose sum or difference is divisible by 10. 9) Suppose a party has six people.
    [Show full text]
  • THE S.V.D. of the POISSON KERNEL 1. Introduction This Paper
    THE S.V.D. OF THE POISSON KERNEL GILES AUCHMUTY Abstract. The solution of the Dirichlet problem for the Laplacian on bounded regions N Ω in R , N 2 is generally described via a boundary integral operator with the Poisson kernel. Under≥ mild regularity conditions on the boundary, this operator is a compact 2 2 2 linear transformation of L (∂Ω,dσ) to LH (Ω) - the Bergman space of L harmonic functions on Ω. This paper describes the singular value decomposition (SVD)− of this operator and related results. The singular functions and singular values are constructed using Steklov eigenvalues and eigenfunctions of the biharmonic operator on Ω. These allow a spectral representation of the Bergman harmonic projection and the identification 2 of an orthonormal basis of the real harmonic Bergman space LH (Ω). A reproducing 2 2 kernel for LH (Ω) and an orthonormal basis of the space L (∂Ω,dσ) also are found. This enables the description of optimal finite rank approximations of the Poisson kernel with error estimates. Explicit spectral formulae for the normal derivatives of eigenfunctions for the Dirichlet Laplacian on ∂Ω are obtained and used to identify a constant in an inequality of Hassell and Tao. 1. Introduction This paper describes representation results for harmonic functions on a bounded region Ω in RN ; N 2. In particular a spectral representation of the Poisson kernel for solutions of the Dirichlet≥ problem for Laplace’s equation with L2 boundary data is described. This leads to an explicit description of the Reproducing− Kernel (RK) for 2 the harmonic Bergman space LH (Ω) and the singular value decomposition (SVD) of the Poisson kernel for the Laplacian.
    [Show full text]
  • Genius Manual I
    Genius Manual i Genius Manual Genius Manual ii Copyright © 1997-2016 Jiríˇ (George) Lebl Copyright © 2004 Kai Willadsen Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License (GFDL), Version 1.1 or any later version published by the Free Software Foundation with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find a copy of the GFDL at this link or in the file COPYING-DOCS distributed with this manual. This manual is part of a collection of GNOME manuals distributed under the GFDL. If you want to distribute this manual separately from the collection, you can do so by adding a copy of the license to the manual, as described in section 6 of the license. Many of the names used by companies to distinguish their products and services are claimed as trademarks. Where those names appear in any GNOME documentation, and the members of the GNOME Documentation Project are made aware of those trademarks, then the names are in capital letters or initial capital letters. DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT ARE PROVIDED UNDER THE TERMS OF THE GNU FREE DOCUMENTATION LICENSE WITH THE FURTHER UNDERSTANDING THAT: 1. DOCUMENT IS PROVIDED ON AN "AS IS" BASIS, WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, WITHOUT LIMITATION, WARRANTIES THAT THE DOCUMENT OR MODIFIED VERSION OF THE DOCUMENT IS FREE OF DEFECTS MERCHANTABLE, FIT FOR A PARTICULAR PURPOSE OR NON-INFRINGING. THE ENTIRE RISK AS TO THE QUALITY, ACCURACY, AND PERFORMANCE OF THE DOCUMENT OR MODIFIED VERSION OF THE DOCUMENT IS WITH YOU.
    [Show full text]
  • Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities
    NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M4 GEOMETRY Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities Student Outcomes . Students find the perimeter of a triangle or quadrilateral in the coordinate plane given a description by inequalities. Students find the area of a triangle or quadrilateral in the coordinate plane given a description by inequalities by employing Green’s theorem. Lesson Notes In previous lessons, students found the area of polygons in the plane using the “shoelace” method. In this lesson, we give a name to this method—Green’s theorem. Students will draw polygons described by a system of inequalities, find the perimeter of the polygon, and use Green’s theorem to find the area. Classwork Opening Exercises (5 minutes) The opening exercises are designed to review key concepts of graphing inequalities. The teacher should assign them independently and circulate to assess understanding. Opening Exercises Graph the following: a. b. Lesson 11: Perimeters and Areas of Polygonal Regions Defined by Systems of Inequalities 135 Date: 8/28/14 This work is licensed under a © 2014 Common Core, Inc. Some rights reserved. commoncore.org Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 11 M4 GEOMETRY d. c. Example 1 (10 minutes) Example 1 A parallelogram with base of length and height can be situated in the coordinate plane as shown. Verify that the shoelace formula gives the area of the parallelogram as . What is the area of a parallelogram? Base height . The distance from the -axis to the top left vertex is some number .
    [Show full text]
  • Abel's Lemma on Summation by Parts and Basic Hypergeometric Series
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Advances in Applied Mathematics 39 (2007) 490–514 www.elsevier.com/locate/yaama Abel’s lemma on summation by parts and basic hypergeometric series ✩ Wenchang Chu ∗ Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PR China Received 21 May 2006; accepted 9 February 2007 Available online 6 June 2007 Abstract Basic hypergeometric series identities are revisited systematically by means of Abel’s lemma on summa- tion by parts. Several new formulae and transformations are also established. The author is convinced that Abel’s lemma on summation by parts is a natural choice in dealing with basic hypergeometric series. © 2007 Elsevier Inc. All rights reserved. MSC: primary 33D15; secondary 05A30 Keywords: Abel’s lemma on summation by parts; Basic hypergeometric series In 1826, Abel [1] (see Bromwich [7, §20] and Knopp [22, §43] also) found the following ingenious lemma on summation by parts. For two arbitrary sequences {ak}k0 and {bk}k0,if we denote the partial sums by n An = ak where n = 0, 1, 2,... k=0 then for two natural numbers m and n with m n, there holds the relation: ✩ The work carried out during the visit to Center for Combinatorics, Nankai University (2005). * Current address: Dipartimento di Matematica, Università degli Studi di Lecce, Lecce-Arnesano PO Box 193, 73100 Lecce, Italy. Fax: 39 0832 297594. E-mail address: [email protected]. 0196-8858/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.aam.2007.02.001 W.
    [Show full text]
  • The Parameterized Complexity of Finding Point Sets with Hereditary Properties
    The Parameterized Complexity of Finding Point Sets with Hereditary Properties David Eppstein1 Computer Science Department, University of California, Irvine, USA [email protected] Daniel Lokshtanov2 Department of Informatics, University of Bergen, Norway [email protected] Abstract We consider problems where the input is a set of points in the plane and an integer k, and the task is to find a subset S of the input points of size k such that S satisfies some property. We focus on properties that depend only on the order type of the points and are monotone under point removals. We show that not all such problems are fixed-parameter tractable parameterized by k, by exhibiting a property defined by three forbidden patterns for which finding a k-point subset with the property is W[1]-complete and (assuming the exponential time hypothesis) cannot be solved in time no(k/ log k). However, we show that problems of this type are fixed-parameter tractable for all properties that include all collinear point sets, properties that exclude at least one convex polygon, and properties defined by a single forbidden pattern. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms Keywords and phrases parameterized complexity, fixed-parameter tractability, point set pattern matching, largest pattern-avoiding subset, order type 1 Introduction In this work, we study the parameterized complexity of finding subsets of planar point sets having hereditary properties, by analogy to past work on hereditary properties of graphs. In graph theory, a hereditary properties of graphs is a property closed under induced subgraphs.
    [Show full text]
  • Perfectly Matched Layers and High Order Difference Methods for Wave
    Dedicated to my father Mr. Ambrose Duru (1939 – 2001) List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I K. Duru and G. Kreiss, (2012). A Well–posed and discretely stable perfectly matched layer for elastic wave equations in second order formulation, Commun. Comput. Phys., 11, 1643–1672 (DOI:10.4208/cicp.120210.240511a). contributions: The author of this thesis initiated this project and performed all numerical experiments. The manuscript was prepared in close collaboration between the authors II G. Kreiss and K. Duru, (2012). Discrete stability of perfectly matched layers for anisotropic wave equations in first and second order formulation, (Submitted). contributions: The author of this thesis initiated this project and performed all numerical experiments. The manuscript was prepared in close collaboration between the authors. III K. Duru and G. Kreiss, (2012). On the accuracy and stability of the perfectly matched layers in transient waveguides, J. of Sc. Comput., DOI: 10.1007/s10915-012-9594-7. In press. contributions: The author of this thesis initiated this project performed all numerical experiments and had the responsibility of writing the paper. The remaining time was spent between the author and his advisor correcting misconceptions, improving the texts and the theory. IV K. Duru and G. Kreiss, (2012). Boundary waves and stability of the perfectly matched layer. Technical report 2012-007, Department of Information Technology, Uppsala University, (Submitted) contributions: The author of this thesis initiated this project and had the responsibility of writing the paper. The remaining time was spent between the author and his advisor correcting misconceptions, improving the texts and the theory.
    [Show full text]