Solutions to Homework 6. (b) (This was not part of Exercise 1, but should have been) Let A and B be groups with A B. Then B A.
|
|
- Wilfrid Harrington
- 5 years ago
- Views:
Transcription
1 MTH 4- Astrct Alger II S7 Solutions to Homework 6 Exercise () Let A, B n C e groups with A B n B C Show tht A C () (This ws not prt of Exercise, ut shoul hve een) Let A n B e groups with A B Then B A () Since A B n B C, there exist isomorphism f A B n g B C By Exercise 7 on Homework 3 g f A C is homomorphism Since f n g re isomorphism, they re ijections Hence y Lemm A23(c), g f is ijection Thus g f is n isomorphism from A to C Hence A C () Since A B, there exist n isomorphism f A B Then f is ijection n so y Lemm A26(c) f hs n inverse, tht is function g B A with f g = i B n g = i A By Lemm A26(,), we know tht g is - n onto, so g is ijection We will now show tht g is homomorphism Note tht for ny B we hve ( ) f(g()) = (f g)() = i B () = Let, B n put ( ) = g() n = g( ) By ( ) pplie to n we hve ( ) f() = n f( ) = We compute f( ) =f()f( ) = f is homomorphism ( ) =f(g( )) ( ) pplie to We prove tht f( ) = f(g( )) Since f is - we infer g( ) = ( ) = g()g( ) Thus g is homomorphism We lrey prove tht g is ijection, so g is n isomorphism Hence B A Exercise 2 Let A n B e groups with A B Show tht A is elin if n only if B is elin Since A B there exists n isomorphism φ A B : Suppose first tht A is elin Let, B Since φ is n isomorphism, φ is onto So there exist, c A with ( ) φ() = n φ(c) =
2 Since φ is isomorphism, φ is homomorphism We hve = φ()φ(c) ( ) = φ(c) φ is homomorphism = φ(c) A is elin = φ(c)φ() φ is homomorphism = ( ) Thus = for ll, B n B is elin : Suppose next tht B is elin I will give two proofs tht A is elin Proof : By Exercise () we know tht B A Hence y the lrey proven forwr irection of this exercise (pplie with B n A interchnge) we conclue tht A is elin Proof 2: Let, c A Then φ(c) = φ()φ(c) φ is homomorphism = φ(c)φ() B is elin = φ(c) φ is homomorphism We prove tht φ(c) = φ(c) Since φ is n isomorphism, we know tht φ is - So c = c for ll, c A n A is elin Exercise 3 Let φ A B e n isomorphism of groups Let C A n put D = φ(c) Show tht () C D () D B (c) A/C B/D () Define φ C C B, c φ(c) By 93(c), φ C is n homomorphism n since φ is -, so is φ C Hence y 65, Im φ C is sugroup of B isomorphic to C We hve n so () hols Im φ C = {φ C (c) c C} = {φ(c) c C} = φ(c) = D () Since φ is n onto homomorphism, 96 () shows tht φ(c) B Since D = φ(c), () hols (c) Let π B B/D, D e the nturl homomorphism By 94 π is n onto homomorphism Put α = π φ By Homework 3#7 α is homomorphism Since π n φ re onto, α is onto, see Lemm A23() Thus 2
3 ( ) Im α = B/D Let A Then Thus Kerα α() = e B/D efinition of Kerα (π φ)() = e B/D efinition of α π(φ()) = e B/D efinition of φ() Kerπ efinition of Kerπ φ() D 94 φ() φ(c) efinition of D φ() = φ(c) for some c C efinition of φ(c) = c for some c C φ is - C ( ) Kerα = C By the First Isomorphism Theorem n so y ( ) n ( ): A/Kerα Im α A/C B/D Exercise 4 Let G e group, N G n H G Suppose tht G/N (Z 24, +), tht H = 8 n tht N is o Show tht H N = {e} n H (Z 8, +) Oserve tht the function i H H H, h h is homomorphism with Ker i H = {e} n Im i H = H Hence the First isomorphism Theorem shows tht ( ) H H/{e} Since H is power of 2 n N is o we hve gc ( H, N ) = Thus Exercise 3 on Homework 4 shows tht H N = {e} By the Secon Isomorphism Theorem we hve H/H N HN/N Thus ( ) H/{e} HN/N By the Corresponence Theorem HN/N is sugroup of G/N Since G/N Z 24 n HN/N G/N, we conclue from Exercise 3() tht 3
4 ( ) HN/N K for some K Z 24 By 99(c) (+) K Z l for some integer l From ( ), ( ), ( ), (+) n Exercise we hve H Z l In prticulr, l = Z l = H = 8 n so H Z 8 Exercise 5 Let G e group, N norml sugroup of G n efine Show tht is well-efine ction of G on N G N N, (g, n) gng Since N G, Lemm 86(f) shows tht gng N for ll g G n n N Thus is well-efine Let, G n n N e n = ege = ege = n n so (ct i) hols ( n) = (n ) = ()n( ) 48 = ()n() = () n So lso (ct ii) hols n is well efine ction Exercise 6 Let G e group cting on the set I Let J e suset of I Put Show tht St G (J) is sugroup of G Let g G By efinition of St G (J) we hve St G (J) = {g G gj = j for ll j J} ( ) g St G (J) if n only if gj = j for ll j J Let, St G (J) Then y ( ): ( ) j = j n j = j for ll j J We will now verify the three conitions of the Sugroup Proposition: (i) By (ct i) ej = j for ll j J n so ( ) gives (ii) for ll j J n so y ( ) ()j e St G (J) ct ii = (j) ( ) = j ( ) = j St G (J) 4
5 (iii) Let j J By ( ) j = j n so y 27(c) j = j Thus ( ) implies tht St G (J) We verifie the three conitions of the Sugroup Proposition n we conclue tht St G (J) is sugroup of G Exercise 7 Let G = GL 2 (R) Recll from Exmple 22(3) tht G cts on R 2 Compute St G (J) for J = {( 0 )}, J = {( )} n J = {( 0 ), ( )} Let A = [ c ] G Suppose first tht J = {( 0 )} Then A St G(J) if n only if if n only if n if n only if c =, 0 0 = c 0 = n c = 0 Note tht [ 0 ] is invertile if n only if 0 Hence St G (J) = R, R 0 Suppose next J = {( )} Then A St G(J) if n only if if n only if n if n only if Oserve tht c So [ ] is invertile if n only if Hence =, + = c + = n c = et = ( ) ( ) = + = 5
6 St G (J) =, R, Suppose finlly tht J = {( 0 ), ( )} Then A St G(J) if n only if c 0 = 0 n c = From the clcultion in the previous two cses this hols if n only if =, c = 0, =, c = n so if n only if =, = 0, c = 0 n = 0, tht is if n only if A = 0 0 Hence St G (J) = 0 0 6
Math 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationAlgebraic systems Semi groups and monoids Groups. Subgroups and homomorphisms Cosets Lagrange s theorem. Ring & Fields (Definitions and examples)
Prepred y Dr. A.R.VIJAYALAKSHMI Algeric systems Semi groups nd monoids Groups Sugroups nd homomorphms Cosets Lgrnge s theorem Ring & Fields (Definitions nd exmples Stndrd Nottions. N :Set of ll nturl numers.
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationMath 220A Homework 2 Solutions
Mth 22A Homework 2 Solutions Jim Agler. Let G be n open set in C. ()Show tht the product rule for nd holds for products of C z z functions on G. (b) Show tht if f is nlytic on G, then 2 z z f(z) 2 f (z)
More informationTHE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS. Circa 1870, G. Zolotarev observed that the Legendre symbol ( a p
THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS PETE L CLARK Circ 1870, Zolotrev observed tht the Legendre symbol ( p ) cn be interpreted s the sign of multipliction by viewed s permuttion of the set Z/pZ
More informationMath 211A Homework. Edward Burkard. = tan (2x + z)
Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
More informationSturm-Liouville Theory
LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationIf we have a function f(x) which is well-defined for some a x b, its integral over those two values is defined as
Y. D. Chong (26) MH28: Complex Methos for the Sciences 2. Integrls If we hve function f(x) which is well-efine for some x, its integrl over those two vlues is efine s N ( ) f(x) = lim x f(x n ) where x
More informationMath 581 Problem Set 9
Math 581 Prolem Set 9 1. Let m and n e relatively prime positive integers. (a) Prove that Z/mnZ = Z/mZ Z/nZ as RINGS. (Hint: First Isomorphism Theorem) Proof: Define ϕz Z/mZ Z/nZ y ϕ(x) = ([x] m, [x] n
More informationRectangular group congruences on an epigroup
cholrs Journl of Engineering nd Technology (JET) ch J Eng Tech, 015; 3(9):73-736 cholrs Acdemic nd cientific Pulisher (An Interntionl Pulisher for Acdemic nd cientific Resources) wwwsspulishercom IN 31-435X
More informationMath 6455 Oct 10, Differential Geometry I Fall 2006, Georgia Tech
Mth 6455 Oct 10, 2006 1 Differentil Geometry I Fll 2006, Georgi Tech Lecture Notes 12 Riemnnin Metrics 0.1 Definition If M is smooth mnifold then by Riemnnin metric g on M we men smooth ssignment of n
More information5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship
5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is
More information#A42 INTEGERS 11 (2011) ON THE CONDITIONED BINOMIAL COEFFICIENTS
#A42 INTEGERS 11 (2011 ON THE CONDITIONED BINOMIAL COEFFICIENTS Liqun To Shool of Mthemtil Sienes, Luoyng Norml University, Luoyng, Chin lqto@lynuedun Reeived: 12/24/10, Revised: 5/11/11, Aepted: 5/16/11,
More informationComputing with finite semigroups: part I
Computing with finite semigroups: prt I J. D. Mitchell School of Mthemtics nd Sttistics, University of St Andrews Novemer 20th, 2015 J. D. Mitchell (St Andrews) Novemer 20th, 2015 1 / 34 Wht is this tlk
More informationSurface Integrals of Vector Fields
Mth 32B iscussion ession Week 7 Notes Februry 21 nd 23, 2017 In lst week s notes we introduced surfce integrls, integrting sclr-vlued functions over prmetrized surfces. As with our previous integrls, we
More informationLecture 09: Myhill-Nerode Theorem
CS 373: Theory of Computtion Mdhusudn Prthsrthy Lecture 09: Myhill-Nerode Theorem 16 Ferury 2010 In this lecture, we will see tht every lnguge hs unique miniml DFA We will see this fct from two perspectives
More informationREPRESENTATION THEORY OF PSL 2 (q)
REPRESENTATION THEORY OF PSL (q) YAQIAO LI Following re notes from book [1]. The im is to show the qusirndomness of PSL (q), i.e., the group hs no low dimensionl representtion. 1. Representtion Theory
More informationCIRCULAR COLOURING THE PLANE
CIRCULAR COLOURING THE PLANE MATT DEVOS, JAVAD EBRAHIMI, MOHAMMAD GHEBLEH, LUIS GODDYN, BOJAN MOHAR, AND REZA NASERASR Astrct. The unit distnce grph R is the grph with vertex set R 2 in which two vertices
More informationConvert the NFA into DFA
Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationHomework Solution - Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.
More informationHomework Problem Set 1 Solutions
Chemistry 460 Dr. Jen M. Stnr Homework Problem Set 1 Solutions 1. Determine the outcomes of operting the following opertors on the functions liste. In these functions, is constnt..) opertor: / ; function:
More informationS. S. Dragomir. 2, we have the inequality. b a
Bull Koren Mth Soc 005 No pp 3 30 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Abstrct Compnions of Ostrowski s integrl ineulity for bsolutely
More informationON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI
Mth. J. Okym Univ. 44(2002), 51 56 ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI Koru MOTOSE Let t(g) be the nilpotency index of the rdicl J(KG) of group lgebr KG of finite p-solvble group
More informationthan 1. It means in particular that the function is decreasing and approaching the x-
6 Preclculus Review Grph the functions ) (/) ) log y = b y = Solution () The function y = is n eponentil function with bse smller thn It mens in prticulr tht the function is decresing nd pproching the
More informationPresentation Problems 5
Presenttion Problems 5 21-355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).
More informationChapter Five - Eigenvalues, Eigenfunctions, and All That
Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl
More informationThe Modified Heinz s Inequality
Journl of Applied Mthemtics nd Physics, 03,, 65-70 Pulished Online Novemer 03 (http://wwwscirporg/journl/jmp) http://dxdoiorg/0436/jmp03500 The Modified Heinz s Inequlity Tkshi Yoshino Mthemticl Institute,
More informationTRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS
TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity
More informationON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS
ON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS A. BAZSÓ Astrct. Depending on the prity of the positive integer n the lternting power sum T k n = k + k + + k...+ 1 n 1 n 1 + k. cn e extended to polynomil
More informationTwo formulas for the Euler ϕ-function
Two formulas for the Euler ϕ-function Robert Frieman A multiplication formula for ϕ(n) The first formula we want to prove is the following: Theorem 1. If n 1 an n 2 are relatively prime positive integers,
More informationHandout: Natural deduction for first order logic
MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes
More informationLECTURE 2: ARTIN SYMBOL, ARTIN MAP, ARTIN RECIPROCITY LAW AND FINITENESS OF GENERALIZED IDEAL CLASS GROUP
Clss Field Theory Study Seminr Jnury 25 2017 LECTURE 2: ARTIN SYMBOL, ARTIN MAP, ARTIN RECIPROCITY LAW AND FINITENESS OF GENERALIZED IDEAL CLASS GROUP YIFAN WU Plese send typos nd comments to wuyifn@umich.edu
More informationVidyalankar S.E. Sem. III [CMPN] Discrete Structures Prelim Question Paper Solution
S.E. Sem. III [CMPN] Discrete Structures Prelim Question Pper Solution 1. () (i) Disjoint set wo sets re si to be isjoint if they hve no elements in common. Exmple : A = {0, 4, 7, 9} n B = {3, 17, 15}
More informationImprovements of some Integral Inequalities of H. Gauchman involving Taylor s Remainder
Divulgciones Mtemátics Vol. 11 No. 2(2003), pp. 115 120 Improvements of some Integrl Inequlities of H. Guchmn involving Tylor s Reminder Mejor de lguns Desigulddes Integrles de H. Guchmn que involucrn
More informationChapter 4. Lebesgue Integration
4.2. Lebesgue Integrtion 1 Chpter 4. Lebesgue Integrtion Section 4.2. Lebesgue Integrtion Note. Simple functions ply the sme role to Lebesgue integrls s step functions ply to Riemnn integrtion. Definition.
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More information5.3 The Fundamental Theorem of Calculus
CHAPTER 5. THE DEFINITE INTEGRAL 35 5.3 The Funmentl Theorem of Clculus Emple. Let f(t) t +. () Fin the re of the region below f(t), bove the t-is, n between t n t. (You my wnt to look up the re formul
More informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More informationOn Implicative and Strong Implicative Filters of Lattice Wajsberg Algebras
Glol Journl of Mthemtil Sienes: Theory nd Prtil. ISSN 974-32 Volume 9, Numer 3 (27), pp. 387-397 Interntionl Reserh Pulition House http://www.irphouse.om On Implitive nd Strong Implitive Filters of Lttie
More informationS. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:
FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationClosure Properties of Regular Languages
Closure Properties of Regulr Lnguges Regulr lnguges re closed under mny set opertions. Let L 1 nd L 2 e regulr lnguges. (1) L 1 L 2 (the union) is regulr. (2) L 1 L 2 (the conctention) is regulr. (3) L
More information1. On some properties of definite integrals. We prove
This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.
More informationad = cb (1) cf = ed (2) adf = cbf (3) cf b = edb (4)
10 Most proofs re left s reding exercises. Definition 10.1. Z = Z {0}. Definition 10.2. Let be the binry reltion defined on Z Z by, b c, d iff d = cb. Theorem 10.3. is n equivlence reltion on Z Z. Proof.
More informationMath Advanced Calculus II
Mth 452 - Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused
More informationFree groups, Lecture 2, part 1
Free groups, Lecture 2, prt 1 Olg Khrlmpovich NYC, Sep. 2 1 / 22 Theorem Every sugroup H F of free group F is free. Given finite numer of genertors of H we cn compute its sis. 2 / 22 Schreir s grph The
More informationLecture 5: Burnside s Lemma and the Pólya Enumeration Theorem
Mth 6 Professor: Pric Brtlett Lecture 5: Burnsie s Lemm n the Póly Enumertion Theorem Weeks 8-9 UCSB 205 We finishe our Möbius function nlysis with question bout seshell necklces: Question. Over the weeken,
More informationRevision Sheet. (a) Give a regular expression for each of the following languages:
Theoreticl Computer Science (Bridging Course) Dr. G. D. Tipldi F. Bonirdi Winter Semester 2014/2015 Revision Sheet University of Freiurg Deprtment of Computer Science Question 1 (Finite Automt, 8 + 6 points)
More informationSection 6.3 The Fundamental Theorem, Part I
Section 6.3 The Funmentl Theorem, Prt I (3//8) Overview: The Funmentl Theorem of Clculus shows tht ifferentition n integrtion re, in sense, inverse opertions. It is presente in two prts. We previewe Prt
More informationCSCI 340: Computational Models. Kleene s Theorem. Department of Computer Science
CSCI 340: Computtionl Models Kleene s Theorem Chpter 7 Deprtment of Computer Science Unifiction In 1954, Kleene presented (nd proved) theorem which (in our version) sttes tht if lnguge cn e defined y ny
More informationIntroduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationALGEBRAIC DISTANCES IN VARIOUS ALGEBRAIC CONE METRIC SPACES. Kamal Fallahi, Ghasem Soleimani Rad, and Stojan Radenović
PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 104(118) (2018), 89 99 DOI: https://oi.org/10.2298/pim1818089f ALGEBRAIC DISTANCES IN VARIOUS ALGEBRAIC CONE METRIC SPACES Kml Fllhi, Ghsem
More informationIntroduction to Complex Variables Class Notes Instructor: Louis Block
Introuction to omplex Vribles lss Notes Instructor: Louis Block Definition 1. (n remrk) We consier the complex plne consisting of ll z = (x, y) = x + iy, where x n y re rel. We write x = Rez (the rel prt
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationA Note on Heredity for Terraced Matrices 1
Generl Mthemtics Vol. 16, No. 1 (2008), 5-9 A Note on Heredity for Terrced Mtrices 1 H. Crwford Rhly, Jr. In Memory of Myrt Nylor Rhly (1917-2006) Abstrct A terrced mtrix M is lower tringulr infinite mtrix
More informationMinimal DFA. minimal DFA for L starting from any other
Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA
More informationChapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)
C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,
More information1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.
York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech
More informationZachary Scherr Math 503 HW 5 Due Friday, Feb 26
Zachary Scherr Math 503 HW 5 Due Friay, Feb 26 1 Reaing 1. Rea Chapter 9 of Dummit an Foote 2 Problems 1. 9.1.13 Solution: We alreay know that if R is any commutative ring, then R[x]/(x r = R for any r
More informationLecture 8: Abstract Algebra
Mth 94 Professor: Pri Brtlett Leture 8: Astrt Alger Week 8 UCSB 2015 This is the eighth week of the Mthemtis Sujet Test GRE prep ourse; here, we run very rough-n-tumle review of strt lger! As lwys, this
More informationFrobenius numbers of generalized Fibonacci semigroups
Frobenius numbers of generlized Fiboncci semigroups Gretchen L. Mtthews 1 Deprtment of Mthemticl Sciences, Clemson University, Clemson, SC 29634-0975, USA gmtthe@clemson.edu Received:, Accepted:, Published:
More informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984-820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810-193 Aveiro, Portugl
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationMatrix & Vector Basic Linear Algebra & Calculus
Mtrix & Vector Bsic Liner lgebr & lculus Wht is mtrix? rectngulr rry of numbers (we will concentrte on rel numbers). nxm mtrix hs n rows n m columns M x4 M M M M M M M M M M M M 4 4 4 First row Secon row
More informationMath 61CM - Solutions to homework 9
Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationHomework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)
More informationAPPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line
APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente
More information9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes
The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationThe usual algebraic operations +,, (or ), on real numbers can then be extended to operations on complex numbers in a natural way: ( 2) i = 1
Mth50 Introduction to Differentil Equtions Brief Review of Complex Numbers Complex Numbers No rel number stisfies the eqution x =, since the squre of ny rel number hs to be non-negtive. By introducing
More information(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer
Divisibility In this note we introduce the notion of divisibility for two integers nd b then we discuss the division lgorithm. First we give forml definition nd note some properties of the division opertion.
More informationScience and Technology RMUTT Journal
Science nd Technolog RMUTT Journl Vol7 No (017) : 00 05 wwwscirmuttcth/stj Online ISSN 9-1547 Possile Solutions of the Diophntine Eqution x Pinut Pungjump Deprtment of Mthemtics nd Sttistics, Fcult of
More informationCosets and Normal Subgroups
Cosets and Normal Subgroups (Last Updated: November 3, 2017) These notes are derived primarily from Abstract Algebra, Theory and Applications by Thomas Judson (16ed). Most of this material is drawn from
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationModel Reduction of Finite State Machines by Contraction
Model Reduction of Finite Stte Mchines y Contrction Alessndro Giu Dip. di Ingegneri Elettric ed Elettronic, Università di Cgliri, Pizz d Armi, 09123 Cgliri, Itly Phone: +39-070-675-5892 Fx: +39-070-675-5900
More informationMyhill-Nerode Theorem
Overview Myhill-Nerode Theorem Correspondence etween DA s nd MN reltions Cnonicl DA for L Computing cnonicl DFA Myhill-Nerode Theorem Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute
More informationFourier series. Preliminary material on inner products. Suppose V is vector space over C and (, )
Fourier series. Preliminry mteril on inner products. Suppose V is vector spce over C nd (, ) is Hermitin inner product on V. This mens, by definition, tht (, ) : V V C nd tht the following four conditions
More informationPrinciples of Real Analysis I Fall VI. Riemann Integration
21-355 Principles of Rel Anlysis I Fll 2004 A. Definitions VI. Riemnn Integrtion Let, b R with < b be given. By prtition of [, b] we men finite set P [, b] with, b P. The set of ll prtitions of [, b] will
More informationMTH3101 Spring 2017 HW Assignment 6: Chap. 5: Sec. 65, #6-8; Sec. 68, #5, 7; Sec. 72, #8; Sec. 73, #5, 6. The due date for this assignment is 4/06/17.
MTH30 Spring 07 HW Assignment 6: Chp. 5: Sec. 65, #6-8; Sec. 68, #5, 7; Sec. 7, #8; Sec. 73, #5, 6. The due dte for this ssignment is 4/06/7. Sec. 65: #6. Wht is the lrgest circle within which the Mclurin
More informationLECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for
ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More
More informationA negative answer to a question of Wilke on varieties of!-languages
A negtive nswer to question of Wilke on vrieties of!-lnguges Jen-Eric Pin () Astrct. In recent pper, Wilke sked whether the oolen comintions of!-lnguges of the form! L, for L in given +-vriety of lnguges,
More informationSynchronizing Automata with Random Inputs
Synchronizing Automt with Rndom Inputs Vldimir Gusev Url Federl University, Ekterinurg, Russi 9 August, 14 Vldimir Gusev (UrFU) Synchronizing Automt with Rndom Input 9 August, 14 1 / 13 Introduction Synchronizing
More informationHOMOLOGY AND COHOMOLOGY. 1. Introduction
HOMOLOGY AND COHOMOLOGY ELLEARD FELIX WEBSTER HEFFERN 1. Introduction We have been introduced to the idea of homology, which derives from a chain complex of singular or simplicial chain groups together
More informationRegular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15
Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More information1.1 Definition. A monoid is a set M together with a map. 1.3 Definition. A monoid is commutative if x y = y x for all x, y M.
1 Monoids and groups 1.1 Definition. A monoid is a set M together with a map M M M, (x, y) x y such that (i) (x y) z = x (y z) x, y, z M (associativity); (ii) e M such that x e = e x = x for all x M (e
More informationSection 15 Factor-group computation and simple groups
Section 15 Factor-group computation and simple groups Instructor: Yifan Yang Fall 2006 Outline Factor-group computation Simple groups The problem Problem Given a factor group G/H, find an isomorphic group
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More information