DOCSLIB.ORG

Factorization Algebras in Quantum Field Theory Learning Seminar

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Factorization Algebras in Quantum Field Theory Learning Seminar FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR SUNGHYUK PARK Abstract. These are notes for \Factorization Algebras in Quantum Field Theory" learning seminar. We follow the two-volume book by Costello and Gwilliam [CG1, CG2] as well as Costello's book [C], almost verbatim in many parts. Contents 1. Free theories ([CG1] 2.1-2.5, 4.1-4.2)1 1.1. General framework of QFTs1 1.2. Divergence operator2 1.3. Divergence complex5 2. Examples of factorization algebras ([CG1] 4.5, 5.4, ··· )6 3. Interacting theories and renormalization ([C, CG2])7 4. Gauge theories and BV formalism ([C, CG2])7 4.1. Classical BV formalism7 4.2. Quantum BV formalism 11 Appendix A. Some background 12 A.1. Chevalley-Eilenberg complex and Koszul resolution 12 A.2. L1 algebras ([KS] 3.2) 12 References 13 1. Free theories ([CG1] 2.1-2.5, 4.1-4.2) In this section we'll see how (pre)factorization algebras naturally arise in quantum field theories. 1.1. General framework of QFTs. What is a quantum field theory? A space-time is a (not necessarily compact) manifold M. Often, fields are sections of a bundle E on M. Let E be the corresponding sheaf of sections. For instance, scalar fields, often denoted by φ, are sections of the trivial line bundle. Often, a quantum field theory is described by an action S, which is a function of fields. If this action is quadratic (kinetic term) and has no higher order terms (interaction terms), then this theory is called free. In path integral formalism, the action determines a (not mathematically well-defined) functional measure e−S=~dφ. Date: October 2019. 1 FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 2 Here, the Planck's constant ~ is a formal parameter that should be regarded \small". In the limit ~ ! 0, the theory becomes classical, and the physical fields satisfy the Euler-Lagrange equation (equation of motion) describing the critical locus of S. Observables O are, roughly, functions on the space of fields. More precisely, classical observables are functions on the critical locus of S, whereas the meaning of quantum observables will be described in later subsections. The expectation value of an observable O is often formally expressed as Z hOi = O e−S=~dφ. Because this expression itself is usually not well-defined in infinite dimensional case, we take another approach, which is described in the next subsection. So far we have described QFT from Lagrangian point of view, but there is also Hamiltonian point of view. Let's put the theory on M = Σ × R and look at the moduli space MΣ of germs of classical solutions on a space-like slice Σ (i.e. the space of classical boundary conditions). Whenever the equation of motion is given by an elliptic PDE, MΣ is symplectic. The classical field theory is then reduced to a classical mechanics where fields are functions f : R !MΣ: The classical observables are functions on MΣ, and it is equipped with a Poisson bracket. We'll see later in this seminar that deformation quantization replaces classical observables with quantum observables and the Poisson bracket with commutator in such a way that in the classical limit it recovers the classical picture. 1.2. Divergence operator. 1.2.1. Finite dimensional model. Instead of defining expectation values directly, we can define when two observables have the same expectation values. Let's consider a finite dimensional model of free scalar fields. Let Rn be the space of fields and consider Gaussian integrals of the form Z Z 1 X n f(x) exp − xiAijxj d x = f(x) !A x2Rn 2 x2Rn where A is a symmetric positive definite matrix and f 2 P (Rn), the space of polynomial functions on Rn. Let Vect(Rn) be the space of vector fields on Rn with polynomial coefficients. Definition 1 (p.25 of [CG1]). The divergence operator associated to the Gaussian measure !A is the linear map n n Div!A : Vect(R ) ! P (R ) satisfying LV !A = (Div!A V ) !A for any V 2 Vect(Rn). Applying the Cartan's formula LV = dιV +ιV d, we can express the divergence in coordinates. X @ 1 X Div f ! = d ιP @ exp − x A x dx ^ · · · ^ dx !A i A fi i ij j 1 n @xi @xi 2 X @ 1 X = fk exp − xiAijxj dx1 ^ · · · ^ dxn @xk 2 k FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 3 X @ X X @fi ) Div f = − f A x + !A i @x i ij j @x i i;j i i By Stokes' theorem, Z Z (Div!A V ) !A = d(ιV !A) = 0 for all V 2 Vect(Rn). Lemma 1 (Lemma 2.1.1 of [CG1]). The quotient map n n ∼ P (R ) ! P (R )=Im Div!A = R is the map sending a function f to its expectation value R n f !A R hfiA := R n !A R n Proof. It suffices to show that Im Div!A has codimension 1. By changing basis of R , we may assume that A = I. In this basis it is easy to see that we have a splitting n ∼ P (R ) = Im Div!A ⊕ R n where R ⊂ P (R ) denotes constant polynomials. 1.2.2. Free scalar field theory. Now let's study infinite dimensional case, the free scalar field theory on a Riemannian manifold (M; g). The space of fields is C1(M). Set Z 1 2 S(φ) = φ(∆g + m )φ dvolg 2 M To define the corresponding divergence operator, we should first define the space of polynomial 1 functions and of polynomial vector fields. Observe that every element f 2 Cc (U) ⊂ Dc(U) defines a linear functional on C1(U) by Z φ 7! fφ dvolg: U 1 Similarly each monomial f1 ··· fn defines a function on C (U) by Z φ 7! f1(x1)φ(x1) ··· fn(xn)φ(xn) dvolg(x1) ^ · · · ^ dvolg(xn): U n 1 1 Hence as the space of polynomial functions, we can take Pe(C (U)) = Sym Cc (U), or rather its completion 1 M 1 n P (C (U)) = Cc (U )Sn : n≥0 1 The subscript indicates coinvariants (i.e. quotients). Similarly we define Vectg c(C (U)) = 1 1 Pe(C (U)) ⊗ Cc (U), or rather its completion 1 M 1 n+1 Vectc(C (U)) = Cc (U )Sn n≥0 where Sn acts on the first n factors. FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 4 Definition 2 (Definition 2.2.1 in [CG1]). The divergence operator associated to the quadratic form Z 1 2 S(φ) = φ(∆ + m )φ dvolg 2 U is the linear map 1 1 Div : Vectc(C (U)) ! P (C (U)) defined by1 n Z @ 2 X f1 ··· fn 7! −f1 ··· fn(∆ + m )φ + f1 ··· fbi ··· fn fi φ dvolg @φ i=1 U and more generally n Z 2 X F (x1; ··· ; xn+1) 7! −(∆n+1 +m )F (x1; ··· ; xn+1)+ F (x1; ··· ; xi; ··· ; xn; xi)dvolg: i=1 xi2U Definition 3 (Definition 2.2.3 in [CG1]). For an open subset U ⊂ M, define the quantum observables of a free field theory to be H0(Obsq(U)) = P (C1(U))=Im Div: In next subsection we'll see that the map Div naturally extends to a cochain complex of quantum observables for which the 0-th cohomology is what we have just defined. 1.2.3. Prefactorization algebra structure on observables. The space of polynomial functions forms a commutative algebra, but Im Div is not an ideal, and H0(Obsq(U)) is not an algebra. Still, H0(Obsq(U)) is a prefactorization algebra. Definition 4. A prefactorization algebra F on M is an assignment U 7! F(U) of vector spaces to each open subset, along with linear maps (\structure maps") ⊗1≤i≤nF(Ui) !F(V ) whenever U1; ··· ;Un ⊂ V are disjoint. The structure maps should satisfy obvious compati- bility conditions. In other words, it is a functor (of multicategories) from DisjM to Vect. In general Vect can be replaced by any symmetric monoidal category, or even a multicategory. Lemma 2 (Lemma 2.3.1 in [CG1]). If U1;U2 are disjoint open subsets of V ⊂ M, then we have a map 1 1 1 P (C (U1)) ⊗ P (C (U2)) ! P (C (V )) obtained by inclusion followed by product. This map does descends to a map2 0 q 0 q 0 q H (Obs (U1)) ⊗ H (Obs (U2)) ! H (Obs (V )): Proof. We have the following diagram : 1 @ @φ denotes the constant-coefficient vector field given by infinitesimal translation in the direction φ in 1 @ R C (U). Hence it is linear in φ, which may be a bit confusing because of the notation. That @φ f = U fφ dvolg is clear if we consider Dirac delta functions. 2Note that the product map itself does not descend to the level of observables. FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 5 1 1 q12 0 q 0 q ker q12 P (C (U1)) ⊗ P (C (U1)) H (Obs (U1)) ⊗ H (Obs (U2)) ? q Im Div P (C1(V )) H0(Obsq(V )) 0 q It suffices to show that the image of ker q12 in H (Obs (V )) is zero, or equivalently, that its 1 image in P (C (V )) lies in Im DivV . Note that ker q12 is the image of the following map. 1⊗Div +Div ⊗1 1 1 1 1 U2 U1 1 1 P (C (U1))⊗Vectc(C (U2))⊕Vectc(C (U1))⊗P (C (U2)) −−−−−−−−−−−! P (C (U1))⊗P (C (U2)) 1 1 Let F 2 P (C (U1)) and X 2 Vectc(C (U2)).
Load more

Recommended publications
  • Algorithmic Factorization of Polynomials Over Number Fields
    Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3).
    [Show full text]
  • Renormalization and Effective Field Theory
    Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society surv-170-costello-cov.indd 1 1/28/11 8:15 AM http://dx.doi.org/10.1090/surv/170 Renormalization and Effective Field Theory Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 81T13, 81T15, 81T17, 81T18, 81T20, 81T70. The author was partially supported by NSF grant 0706954 and an Alfred P. Sloan Fellowship. For additional information and updates on this book, visit www.ams.org/bookpages/surv-170 Library of Congress Cataloging-in-Publication Data Costello, Kevin. Renormalization and effective fieldtheory/KevinCostello. p. cm. — (Mathematical surveys and monographs ; v. 170) Includes bibliographical references. ISBN 978-0-8218-5288-0 (alk. paper) 1. Renormalization (Physics) 2. Quantum field theory. I. Title. QC174.17.R46C67 2011 530.143—dc22 2010047463 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA.
    [Show full text]
  • Subclass Discriminant Nonnegative Matrix Factorization for Facial Image Analysis
    Pattern Recognition 45 (2012) 4080–4091 Contents lists available at SciVerse ScienceDirect Pattern Recognition journal homepage: www.elsevier.com/locate/pr Subclass discriminant Nonnegative Matrix Factorization for facial image analysis Symeon Nikitidis b,a, Anastasios Tefas b, Nikos Nikolaidis b,a, Ioannis Pitas b,a,n a Informatics and Telematics Institute, Center for Research and Technology, Hellas, Greece b Department of Informatics, Aristotle University of Thessaloniki, Greece article info abstract Article history: Nonnegative Matrix Factorization (NMF) is among the most popular subspace methods, widely used in Received 4 October 2011 a variety of image processing problems. Recently, a discriminant NMF method that incorporates Linear Received in revised form Discriminant Analysis inspired criteria has been proposed, which achieves an efficient decomposition of 21 March 2012 the provided data to its discriminant parts, thus enhancing classification performance. However, this Accepted 26 April 2012 approach possesses certain limitations, since it assumes that the underlying data distribution is Available online 16 May 2012 unimodal, which is often unrealistic. To remedy this limitation, we regard that data inside each class Keywords: have a multimodal distribution, thus forming clusters and use criteria inspired by Clustering based Nonnegative Matrix Factorization Discriminant Analysis. The proposed method incorporates appropriate discriminant constraints in the Subclass discriminant analysis NMF decomposition cost function in order to address the problem of finding discriminant projections Multiplicative updates that enhance class separability in the reduced dimensional projection space, while taking into account Facial expression recognition Face recognition subclass information. The developed algorithm has been applied for both facial expression and face recognition on three popular databases.
    [Show full text]
  • Inside the Perimeter Is Published by Perimeter Institute for Theoretical Physics
    the Perimeter fall/winter 2014 Skateboarding Physicist Seeks a Unified Theory of Self The Black Hole that Birthed the Big Bang The Beauty of Truth: A Chat with Savas Dimopoulos Subir Sachdev's Superconductivity Puzzles Editor Natasha Waxman [email protected] Contributing Authors Graphic Design Niayesh Afshordi Gabriela Secara Erin Bow Mike Brown Photographers & Artists Phil Froklage Tibra Ali Colin Hunter Justin Bishop Robert B. Mann Amanda Ferneyhough Razieh Pourhasan Liz Goheen Natasha Waxman Alioscia Hamma Jim McDonnell Copy Editors Gabriela Secara Tenille Bonoguore Tegan Sitler Erin Bow Mike Brown Colin Hunter Inside the Perimeter is published by Perimeter Institute for Theoretical Physics. www.perimeterinstitute.ca To subscribe, email us at [email protected]. 31 Caroline Street North, Waterloo, Ontario, Canada p: 519.569.7600 f: 519.569.7611 02 IN THIS ISSUE 04/ Young at Heart, Neil Turok 06/ Skateboarding Physicist Seeks a Unified Theory of Self,Colin Hunter 10/ Inspired by the Beauty of Math: A Chat with Kevin Costello, Colin Hunter 12/ The Black Hole that Birthed the Big Bang, Niayesh Afshordi, Robert B. Mann, and Razieh Pourhasan 14/ Is the Universe a Bubble?, Colin Hunter 15/ Probing Nature’s Building Blocks, Phil Froklage 16/ The Beauty of Truth: A Chat with Savas Dimopoulos, Colin Hunter 18/ Conference Reports 22/ Back to the Classroom, Erin Bow 24/ Finding the Door, Erin Bow 26/ "Bright Minds in Their Life’s Prime", Colin Hunter 28/ Anthology: The Portraits of Alioscia Hamma, Natasha Waxman 34/ Superconductivity Puzzles, Colin Hunter 36/ Particles 39/ Donor Profile: Amy Doofenbaker, Colin Hunter 40/ From the Black Hole Bistro, Erin Bow 42/ PI Kids are Asking, Erin Bow 03 neil’s notes Young at Heart n the cover of this issue, on the initial singularity from which everything the lip of a halfpipe, teeters emerged.
    [Show full text]
  • Finite Fields: Further Properties
    Chapter 4 Finite fields: further properties 8 Roots of unity in finite fields In this section, we will generalize the concept of roots of unity (well-known for complex numbers) to the finite field setting, by considering the splitting field of the polynomial xn − 1. This has links with irreducible polynomials, and provides an effective way of obtaining primitive elements and hence representing finite fields. Definition 8.1 Let n ∈ N. The splitting field of xn − 1 over a field K is called the nth cyclotomic field over K and denoted by K(n). The roots of xn − 1 in K(n) are called the nth roots of unity over K and the set of all these roots is denoted by E(n). The following result, concerning the properties of E(n), holds for an arbitrary (not just a finite!) field K. Theorem 8.2 Let n ∈ N and K a field of characteristic p (where p may take the value 0 in this theorem). Then (i) If p ∤ n, then E(n) is a cyclic group of order n with respect to multiplication in K(n). (ii) If p | n, write n = mpe with positive integers m and e and p ∤ m. Then K(n) = K(m), E(n) = E(m) and the roots of xn − 1 are the m elements of E(m), each occurring with multiplicity pe. Proof. (i) The n = 1 case is trivial. For n ≥ 2, observe that xn − 1 and its derivative nxn−1 have no common roots; thus xn −1 cannot have multiple roots and hence E(n) has n elements.
    [Show full text]
  • Prime Factorization
    Prime Factorization Prime Number ­ A number with only two factors: ____ and itself Circle the prime numbers listed below 25 30 2 5 1 9 14 61 Composite Number ­ A number that has more than 2 factors List five examples of composite numbers What kind of number is 0? What kind of number is 1? Every human has a unique fingerprint. Similarly, every COMPOSITE number has a unique "factorprint" called __________________________ Prime Factorization ­ the factorization of a composite number into ____________ factors You can use a _________________ to find the prime factorization of any composite number. Ask yourself, "what Factorization two whole numbers 24 could I multiply together to equal the given number?" If the number is prime, do not put 1 x the number. Once you have all prime numbers, you are finished. Write your answer in exponential form. 24 Expanded Form (written as a multiplication of prime numbers) _______________________ Exponential Form (written with exponents) ________________________ Prime Factorization Ask yourself, "what two 36 numbers could I multiply together to equal the given number?" If the number is prime, do not put 1 x the number. Once you have all prime numbers, you are finished. Write your answer in both expanded and exponential forms. Prime Factorization Ask yourself, "what two 68 numbers could I multiply together to equal the given number?" If the number is prime, do not put 1 x the number. Once you have all prime numbers, you are finished. Write your answer in both expanded and exponential forms. Prime Factorization Ask yourself, "what two 120 numbers could I multiply together to equal the given number?" If the number is prime, do not put 1 x the number.
    [Show full text]
  • Performance and Difficulties of Students in Formulating and Solving Quadratic Equations with One Unknown* Makbule Gozde Didisa Gaziosmanpasa University
    ISSN 1303-0485 • eISSN 2148-7561 DOI 10.12738/estp.2015.4.2743 Received | December 1, 2014 Copyright © 2015 EDAM • http://www.estp.com.tr Accepted | April 17, 2015 Educational Sciences: Theory & Practice • 2015 August • 15(4) • 1137-1150 OnlineFirst | August 24, 2015 Performance and Difficulties of Students in Formulating and Solving Quadratic Equations with One Unknown* Makbule Gozde Didisa Gaziosmanpasa University Ayhan Kursat Erbasb Middle East Technical University Abstract This study attempts to investigate the performance of tenth-grade students in solving quadratic equations with one unknown, using symbolic equation and word-problem representations. The participants were 217 tenth-grade students, from three different public high schools. Data was collected through an open-ended questionnaire comprising eight symbolic equations and four word problems; furthermore, semi-structured interviews were conducted with sixteen of the students. In the data analysis, the percentage of the students’ correct, incorrect, blank, and incomplete responses was determined to obtain an overview of student performance in solving symbolic equations and word problems. In addition, the students’ written responses and interview data were qualitatively analyzed to determine the nature of the students’ difficulties in formulating and solving quadratic equations. The findings revealed that although students have difficulties in solving both symbolic quadratic equations and quadratic word problems, they performed better in the context of symbolic equations compared with word problems. Student difficulties in solving symbolic problems were mainly associated with arithmetic and algebraic manipulation errors. In the word problems, however, students had difficulties comprehending the context and were therefore unable to formulate the equation to be solved.
    [Show full text]
  • Algebraic Number Theory Summary of Notes
    Algebraic Number Theory summary of notes Robin Chapman 3 May 2000, revised 28 March 2004, corrected 4 January 2005 This is a summary of the 1999–2000 course on algebraic number the- ory. Proofs will generally be sketched rather than presented in detail. Also, examples will be very thin on the ground. I first learnt algebraic number theory from Stewart and Tall’s textbook Algebraic Number Theory (Chapman & Hall, 1979) (latest edition retitled Algebraic Number Theory and Fermat’s Last Theorem (A. K. Peters, 2002)) and these notes owe much to this book. I am indebted to Artur Costa Steiner for pointing out an error in an earlier version. 1 Algebraic numbers and integers We say that α ∈ C is an algebraic number if f(α) = 0 for some monic polynomial f ∈ Q[X]. We say that β ∈ C is an algebraic integer if g(α) = 0 for some monic polynomial g ∈ Z[X]. We let A and B denote the sets of algebraic numbers and algebraic integers respectively. Clearly B ⊆ A, Z ⊆ B and Q ⊆ A. Lemma 1.1 Let α ∈ A. Then there is β ∈ B and a nonzero m ∈ Z with α = β/m. Proof There is a monic polynomial f ∈ Q[X] with f(α) = 0. Let m be the product of the denominators of the coefficients of f. Then g = mf ∈ Z[X]. Pn j Write g = j=0 ajX . Then an = m. Now n n−1 X n−1+j j h(X) = m g(X/m) = m ajX j=0 1 is monic with integer coefficients (the only slightly problematical coefficient n −1 n−1 is that of X which equals m Am = 1).
    [Show full text]
  • DISCRIMINANTS in TOWERS Let a Be a Dedekind Domain with Fraction
    DISCRIMINANTS IN TOWERS JOSEPH RABINOFF Let A be a Dedekind domain with fraction field F, let K=F be a finite separable ex- tension field, and let B be the integral closure of A in K. In this note, we will define the discriminant ideal B=A and the relative ideal norm NB=A(b). The goal is to prove the formula D [L:K] C=A = NB=A C=B B=A , D D ·D where C is the integral closure of B in a finite separable extension field L=K. See Theo- rem 6.1. The main tool we will use is localizations, and in some sense the main purpose of this note is to demonstrate the utility of localizations in algebraic number theory via the discriminants in towers formula. Our treatment is self-contained in that it only uses results from Samuel’s Algebraic Theory of Numbers, cited as [Samuel]. Remark. All finite extensions of a perfect field are separable, so one can replace “Let K=F be a separable extension” by “suppose F is perfect” here and throughout. Note that Samuel generally assumes the base has characteristic zero when it suffices to assume that an extension is separable. We will use the more general fact, while quoting [Samuel] for the proof. 1. Notation and review. Here we fix some notations and recall some facts proved in [Samuel]. Let K=F be a finite field extension of degree n, and let x1,..., xn K. We define 2 n D x1,..., xn det TrK=F xi x j .
    [Show full text]
  • 1 Factorization of Polynomials
    AM 106/206: Applied Algebra Prof. Salil Vadhan Lecture Notes 18 November 11, 2010 1 Factorization of Polynomials • Reading: Gallian Ch. 16 • Throughout F is a field, and we consider polynomials in F [x]. • Def: For f(x); g(x) 2 F [x], not both zero, the greatest common divisor of f(x) and g(x) is the monic polynomial h(x) of largest degree such that h(x) divides both f(x) and g(x). • Euclidean Algorithm for Polynomials: Given two polynomials f(x) and g(x) of degree at most n, not both zero, their greatest common divisor h(x), can be computed using at most n + 1 divisions of polynomials of degree at most n. Moreover, using O(n) operations on polynomials of degree at most n, we can also find polynomials s(x) and t(x) such that h(x) = s(x)f(x) + t(x)g(x). Proof: analogous to integers, using repeated division. Euclid(f; g): 1. Assume WLOG deg(f) ≥ deg(g) > 0. 2. Set i = 1, f1 = f, f2 = g. 3. Repeat until fi+1 = 0: (a) Compute fi+2 = fi mod fi+1 (i.e. fi+2 is the remainder when fi is divided by fi+1). (b) Increment i. 4. Output fi divided by its leading coefficient (to make it monic). Here the complexity analysis is simpler than for integers: note that the degree of fi+2 is strictly smaller than that of fi, so fn+2 is of degree zero, and fn+3 = 0. Thus we do at most n divisions.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach Topologie
    Mathematisches Forschungsinstitut Oberwolfach Report No. 42/2014 DOI: 10.4171/OWR/2014/42 Topologie Organised by Thomas Schick, G¨ottingen Peter Teichner, Bonn Nathalie Wahl, Copenhagen Michael Weiss, M¨unster 14 September – 20 September 2014 Abstract. The Oberwolfach conference “Topologie” is one of only a few opportunities for researchers from many different areas in algebraic and geo- metric topology to meet and exchange ideas. The program covered new de- velopments in fields such as automorphisms of manifolds, applications of alge- braic topology to differential geometry, quantum field theories, combinatorial methods in low-dimensional topology, abstract and applied homotopy theory and applications of L2-cohomology. We heard about new results describ- ing the cohomology of the automorphism spaces of some smooth manifolds, progress on spaces of positive scalar curvature metrics, a variant of the Segal conjecture without completion, advances in classifying topological quantum field theories, and a new undecidability result in combinatorial group theory, to mention some examples. As a special attraction, the conference featured a series of three talks by Dani Wise on the combinatorics of CAT(0)-cube complexes and applications to 3-manifold topology. Mathematics Subject Classification (2010): 55-xx, 57-xx, 18Axx, 18Bxx. Introduction by the Organisers This conference was the third topology conference in Oberwolfach organized by Thomas Schick, Peter Teichner, Nathalie Wahl and Michael Weiss. About 50 mathematicians participated, working in many different areas of algebraic and geometric topology. 2354 Oberwolfach Report 42/2014 The talks were of three types. There were 14 regular one-hour talks, 3 one-hour talks by keynote speaker Dani Wise and a “gong show” where 12 young speak- ers had the opportunity to present their research in 10 minutes each, including question time.
    [Show full text]
  • Notes on Factors, Prime Numbers, and Prime Factorization
    Notes on Factors, Prime Numbers, and Prime Factorization Factors are the numbers that multiply together to get another number. A Product is the number produced by multiplying two factors. All numbers have 1 and itself as factors. A number whose only factors are 1 and itself is a prime number. Prime numbers have exactly two factors. The smallest 168 prime numbers (all the prime numbers under 1000) are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 There are infinitely many prime numbers. Another way of saying this is that the sequence of prime numbers never ends. The number 1 is not considered a prime.
    [Show full text]