2 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS


 Brianna O’Connor’
 1 years ago
 Views:
Transcription
1 2 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS 1. The group rig k[g] The mai idea is that represetatios of a group G over a field k are the same as modules over the group rig k[g]. First I defied both terms Represetatios of groups. Defiitio 1.1. A represetatio of a group G over a field k is defied to be a group homomorphism where V is a vector space over k. ρ : G Aut k (V ) Here Aut k (V ) is the group of kliear automorphisms of V. This also writte as GL k (V ). This is the group of uits of the rig Ed k (V )= Hom k (V, V ) which, as I explaied before, is a rig with additio defied poitwise ad multiplicatio give by compositio. If dim k (V )=d the Aut k (V ) = Aut k (k d )=GL d (k) which ca also be described as the group of uits of the rig Mat d (k) or as: GL d (k) ={A Mat d (k) det(a) 0} d = dim k (V ) is called the dimesio of the represetatio ρ examples. Example 1.2. The first example I gave was the trivial represetatio. This is usually defied to be the oe dimesioal represetatio V = k with trivial actio of the group G (which ca be arbitrary). Trivial actio meas that ρ(σ) = 1 = id V for all σ G. I the ext example, I poited out that the group G eeds to be writte multiplicatively o matter what. Example 1.3. Let G = Z/3. Writte multiplicatively, the elemets are 1, σ, σ 2. Let k = R ad let V = R 2 with ρ(σ) defied to be rotatio by 120 =2π/3. I.e., ( ) 1/2 3/2 ρ(σ) = 3/2 1/2 Example 1.4. Suppose that E is a field extesio of k ad G = Gal(E/k). The G acts o E by kliear trasformatios. This gives a represetatio: ρ : G Aut k (E) Note that this map is a iclusio by defiitio of Galois group.
2 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS axioms. I a elemetary discussio of group represetatios I would write a list of axioms as a defiitio. However, they are just logwided explaatios of what it meas for ρ : G Aut k (V ) to be a group homomorphism. The oly advatage is that you do t eed to assume that ρ(σ) is a automorphism. Here are the axioms. (I switched the order of (2) ad (3) i the lecture.) (1) ρ(1) = 1 1v = v v V (2) ρ(στ) =ρ(σ)ρ(τ) σ, τ G (στ)v = σ(τv) v V (3) ρ(σ) is kliear σ G σ(av + bw) =aσv + bσw v, w V, a, b k The first two coditios say that ρ is a actio of G o V. Actios are usually writte by juxtapositio: σv := ρ(σ)(v) The third coditio says that the actio is kliear. So, together, the axioms say that a represetatio of G is a kliear actio of G o a vector space V Modules over k[g]. The group rig k[g] is defied to be the set of all fiite k liear combiatios of elemets of G: a σ σ where α σ k for all σ G ad a σ = 0 for almost all σ. For example, R[Z/3] is the set of all liear combiatios x + yσ + zσ 2 where x, y, z R. I.e., R[Z/3] = R 3. I geeral k[g] is a vector space over k with G as a basis. Multiplicatio i k[g] is give by ( )( ( ) aσ σ bτ τ) = cλ λ where c λ G ca be give i three differet ways: c λ = στ=λ a σ b τ = Propositio 1.5. k[g] is a kalgebra. a σ b σ 1 λ = τ G a λτ 1b τ This is straightforward ad tedious. So, I did t prove it. But I did explai what it meas ad why it is importat. Recall that a algebra over k is a rig which cotais k i its ceter. The ceter Z(R) of a (ocommutative) rig R is defied to be the set of elemets of R which commute with all the other elemets: Z(R) is a subrig of R. Z(R) := {x R xy = yx y R}
3 4 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS The ceter is importat for the followig reaso. Suppose that M is a (left) Rmodule. The each elemet r R acts o M by left multiplicatio λ r λ r : M M, λ r (x) =rx This is a homomorphism of Z(R)modules sice: λ r (ax) =rax = arx = aλ r (x) a Z(R) Thus the actio of R o M gives a rig homomorphism: ρ : R Ed Z(R) (M) Gettig back to k[g], suppose that M is a k[g]module. The the actio of k[g] o M is kliear sice k is i the ceter of k[g]. So, we get a rig homomorphism ρ : k[g] Ed k (M) This restricts to a group homomorphism ρ G : G Aut k (M) I poited out that, i geeral, ay rig homomorphism φ : R S will iduce a group homomorphism U(R) U(S) where U(R) is the group of uits of R. Ad I poited out earlier that Aut k (M) is the group of uits of Ed k (M). G is cotaied i the group of uits of k[g]. (A iterestig related questio is: Which fiite groups occur as groups of uits of rigs?) This discussio shows that a k[g]module M gives, by restrictio, a represetatio of the group G o the kvector space M. Coversely, suppose that ρ : G Aut k (V ) is a group represetatio. The we ca exted ρ to a rig homomorphism ρ : k[g] Ed k (V ) by the simple formula ( ) ρ aσ σ = a σ ρ(σ) Whe we say that a represetatio of a group G is the same as a k[g]module we are talkig about this correspodece. The vector space V is also called a Gmodule. So, it would be more accurate to say that a Gmodule is the same as a k[g]module. Corollary 1.6. (1) Ay group represetatio ρ : G Aut k (V ) exteds uiquely to a rig homomorphism ρ : k[g] Ed k (V ) makig V ito a k[g]module.
4 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS 5 (2) For ay k[g]module M, the actio of k[g] o M restricts to give a group represetatio G Aut k (M). (3) These two operatios are iverse to each other i the sese that ρ is the restrictio of ρ ad a actio of the rig k[g] is the uique extesio of its restrictio to G. There are some coceptual differeces betwee the group represetatio ad the correspodig k[g]module. For example, the module might ot be faithful eve if the group represetatio is: Defiitio 1.7. A group represetatio ρ : G Aut k (V ) is called faithful if oly the trivial elemet of G acts as the idetity o V. I.e., if the kerel of ρ is trivial. A Rmodule M is called faithful if the aihilator of M is zero. (a(m) ={r R rx =0 x M}). These two defiitios do ot agree. For example, take the represetatio ρ : Z/3 GL 2 (R) which we discussed earlier. This is faithful. But the extesio to a rig homomorphism ρ : R[Z/3] Mat 2 (R) is ot a moomorphism sice 1 + σ + σ 2 is i its kerel Semisimplicity of k[g]. The mai theorem about k[g] is the followig. Theorem 1.8 (Maschke). If G is a fiite group of order G = ad k is a field with char k (or char k =0) the k[g] is semisimple. Istead of sayig char k is either 0 or a prime ot dividig, I will say that 1/ k. By the Wedderbur structure theorem we get the followig. Corollary 1.9. If 1/ G k the k[g] = Mat d1 (D i ) Mat db (D b ) where D i are fiite dimesioal divisio algebras over k. Example R[Z/3] = R C I geeral, if G is abelia, the the umbers d i must all be 1 ad D i must be fiite field extesios of k.
5 6 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS homomorphisms. I order to prove Maschke s theorem, we eed to talk about homomorphisms of Gmodules. We ca defie these to be the same as homomorphisms of k[g]modules. The the followig is a propositio. (Or, we ca take the followig as the defiitio of a G module homomorphism, i which case the propositio is that Gmodule homomorphisms are the same as homomorphisms of k[g]modules.) Propositio Suppose that V, W are k[g]modules. The a k liear mappig φ : V W is a homomorphism of k[g]modules if ad oly if it commutes with the actio of G. I.e., if for all σ G. σ(φ(v)) = φ(σv) Proof. Ay homomorphism of k[g]modules will commute with the actio of k[g] ad therefore with the actio of G k[g]. Coversely, if φ : V W commutes with the actio of G the, for ay a σ σ k[g], we have ( ) φ a σ σv = a σ φ(σv) = ( ) a σ σφ(v) = a σ σ φ(v) So, φ is a homomorphism of k[g]modules. We also have the followig Propositio/Defiitio of a Gsubmodule. Propositio A subset W of a Gmodule V over k is a k[g] submodule (ad we call it a Gsubmodule) if ad oly if (1) W is a vector subspace of V ad (2) W is ivariat uder the actio of G. I.e., σw W for all σ G. Proof of Maschke s Theorem. Suppose that V is a fiitely geerated Gmodule ad W is ay Gsubmodule of V. The we wat to show that W is a direct summad of V. This is oe of the characterizatios of semisimple modules. This will prove that all f.g. k[g]modules are semisimple ad therefore k[g] is a semisimple rig. Sice W is a submodule of V, it is i particular a vector subspace of V. So, there is a liear projectio map φ : V W so that φ W = id W. If φ is a homomorphism of Gmodules, the V = W ker φ ad W would split from V. So, we would be doe. If φ is ot a G homomorphism, we ca make it ito a Ghomomorphism by averagig over the group, i.e., by replacig it with ψ = 1 λσ 1 φ λ σ.
6 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS 7 First, I claim that ψ W = id W. To see this take ay w W. The σw W. So, φ(σw) =σw ad ψ(w) = 1 σ 1 φ(σw) = 1 σ 1 (σw) =w Next I claim that ψ is a homomorphism of Gmodules. To show this take ay τ G ad v V. The ψ(τv) = 1 σ 1 φ(στv) = 1 αφ(βv) αβ=τ = 1 τσ 1 φ(σv) =τψ(v) So, ψ gives a splittig of V as required R[Z/3]. I gave a logwided explaatio of Example 1.10 usig the uiversal property of the group rig k[g]. I these otes, I will just summarize this property i oe equatio. If R is ay kalgebra ad U(R) is the group of uits of R, the: Hom kalg (k[g],r) = Hom grp (G, U(R)) The isomorphism is give by restrictio ad liear extesio. The isomorphism R[Z/3] = R C is give by the mappig: φ : Z/3 R C which seds the geerator σ to (1,ω) where ω is a primitive third root of uity. Sice (1, 0), (1,ω), (1, ω) are liearly idepedet over R, the liear extesio φ of φ is a isomorphism of Ralgebras group rigs over C. We will specialize to the case k = C. I that case, there are o fiite dimesioal divisio algebras over C (Part C, Theorem 3.12). So, we get oly matrix algebras: Corollary If G is ay fiite group the I particular, = G = d 2 i. C[G] = Mat d1 (C) Mat db (C) Example If G is a fiite abelia group of order the C[G] = C. Example Take G = S 3, the symmetric group o 3 letters. Sice this group is oabelia, the umbers d i caot all be equal to 1. But
7 8 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS the oly way that 6 ca be writte as a sum of squares, ot all 1, is 6 = Therefore, C[S 3 ] = C C Mat 2 (C) This ca be viewed as a subalgebra of Mat 4 (C) give by Each star ( ) represets a idepedet complex variable. I this descriptio, it is easy to visualize what are the simple factors Mat di (C) give by the Wedderbur structure theorem. But what are the correspodig factors of the group rig C[G]? 1.4. idempotets. Suppose that R = R 1 R 2 R 3 is a product of three subrigs. The the uity of R decomposes as 1 = (1, 1, 1). This ca be writte as a sum of uit vectors: 1 = (1, 0, 0) + (0, 1, 0) + (0, 0, 1) = e 1 + e 2 + e 3 This is a decompositio of uity (1) as a sum of cetral, orthogoal idempotets e i. Recall that idempotet meas that e 2 i = e i for all i. Also, 0 is ot cosidered to be a idempotet. Orthogoal meas that e i e j = 0 if i j. Cetral meas that e i Z(R). Theorem A rig R ca be writte as a product of b subrigs R 1,R 2,,R b iff 1 R ca be writte as a sum of b cetral, orthogoal idempotets ad, i that case, R i = e i R. A cetral idempotet e is called primitive if it caot be writte as a sum of two cetral orthogoal idempotets. Corollary The umber of factors R i = e i R is maximal iff each e i is primitive. So, the problem is to write uity 1 C[G] as a sum of primitive, cetral ( orthogoal) idempotets. We will derive a formula for this decompositio usig characters.
8 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS Ceter of C[G]. Before I move o to characters, I wat to prove oe last thig about the group rig C[G]. Theorem The umber of factors b i the decompositio C[G] b = Mat di (C) is equal to the umber of cojugacy classes of elemets of G. i=1 For, example, the group S 3 has three cojugacy classes: the idetity {1}, the traspositios {(12), (23), (13)} ad the 3cycles {(123), (132)}. I order to prove this we ote that b is the dimesio of the ceter of the right had side. Ay cetral elemet of Mat di (C) is a scalar multiple of the uit matrix which we are callig e i (the ith primitive cetral idempotet). Therefore: Lemma The ceter of b i=1 Mat d i (C) is the vector subspace spaed by the primitive cetral idempotets e 1,,e b. I particular it is bdimesioal. So, it suffices to show that the dimesio of the ceter of C[G] is equal to the umber of cojugacy classes of elemets of G. (If G is abelia, this is clearly true.) Defiitio A class fuctio o G is a fuctio f : G X so that f takes the same value o cojugate elemets. I.e., for all σ, τ G. Usually, X = C. f(τστ 1 )=f(σ) For example, ay fuctio o a abelia group is a class fuctio. Lemma For ay field k, the ceter of k[g] is the set of all a σσ so that a σ is a class fuctio o G. So, Z(k[G]) = k c where c is the umber of cojugacy classes of elemets of G. Proof. If a σσ is cetral the a σ σ = τ a σ στ 1 = a σ τστ 1 The coefficiet of τστ 1 o both sides must agree. So a τστ 1 = a σ I.e., a σ is a class fuctio. The coverse is also clear. These two lemmas clearly imply Theorem 1.18 (which ca ow be stated as: b = c if k = C).
The Field of Complex Numbers
The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that
More informationLecture 5: Span, linear independence, bases, and dimension
Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More information+ 1= x + 1. These 4 elements form a field.
Itroductio to fiite fields II Fiite field of p elemets F Because we are iterested i doig computer thigs it would be useful for us to costruct fields havig elemets. Let s costruct a field of elemets; we
More informationThe Field Q of Rational Numbers
Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees
More informationMeasurable Functions
Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationif A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,
Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σalgebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More information8.3 POLAR FORM AND DEMOIVRE S THEOREM
SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationLinear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant
MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationSum and Product Rules. Combinatorics. Some Subtler Examples
Combiatorics Sum ad Product Rules Problem: How to cout without coutig. How do you figure out how may thigs there are with a certai property without actually eumeratig all of them. Sometimes this requires
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationAdvanced Probability Theory
Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σalgebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio
More informationLecture 7: Borel Sets and Lebesgue Measure
EE50: Probability Foudatios for Electrical Egieers JulyNovember 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationTHE LEAST SQUARES REGRESSION LINE and R 2
THE LEAST SQUARES REGRESSION LINE ad R M358K I. Recall from p. 36 that the least squares regressio lie of y o x is the lie that makes the sum of the squares of the vertical distaces of the data poits from
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationDivide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016
CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationA CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet
More information
Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett
(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1
More informationFORMALITY QUANTIZATION OF LIE BIALGEBRAS FROM E 2
QUANTIZATION OF LIE BIALGEBRAS FROM E 2 FORMALITY PAVEL SAFRONOV 1. Outlie 1.1. Statemet. Let k be a field of characteristic zero. The goal of this talk is to prove the followig statemet: Theorem. Let
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationMATH 361 Homework 9. Royden Royden Royden
MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informationrepresented by 4! different arrangements of boxes, divide by 4! to get ways
Problem Set #6 solutios A juggler colors idetical jugglig balls red, white, ad blue (a I how may ways ca this be doe if each color is used at least oce? Let us preemptively color oe ball i each color,
More informationPermutations, the Parity Theorem, and Determinants
1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits
More information. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2
4. Basic feasible solutios ad vertices of polyhedra Due to the fudametal theorem of Liear Programmig, to solve ay LP it suffices to cosider the vertices (fiitely may) of the polyhedro P of the feasible
More information1.3 Binomial Coefficients
18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More information2.7 Sequences, Sequences of Sets
2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationSolutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork
Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationSearching Algorithm Efficiencies
Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay
More informationDivide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015
CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationFactors of sums of powers of binomial coefficients
ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationFourier Series and the Wave Equation Part 2
Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationB1. Fourier Analysis of Discrete Time Signals
B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)
More informationJoint Probability Distributions and Random Samples
STAT5 Sprig 204 Lecture Notes Chapter 5 February, 204 Joit Probability Distributios ad Radom Samples 5. Joitly Distributed Radom Variables Chapter Overview Joitly distributed rv Joit mass fuctio, margial
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationMATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. I  Matrices, Vectors, Determinants, and Linear Algebra  Tadao ODA
MATRICES, VECTORS, DETERMINANTS, AND LINEAR ALGEBRA Tadao Tohoku Uiversity, Japa Keywords: matrix, determiat, liear equatio, Cramer s rule, eigevalue, Jorda caoical form, symmetric matrix, vector space,
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationMath Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:
Math 355  Discrete Math 4.14.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationAlternatives To Pearson s and Spearman s Correlation Coefficients
Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alteratives
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationNotes on exponential generating functions and structures.
Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a elemet set, (2) to fid for each the
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationTagore Engineering College Department of Electrical and Electronics Engineering EC 2314 Digital Signal Processing University Question Paper PartA
Tagore Egieerig College Departmet of Electrical ad Electroics Egieerig EC 34 Digital Sigal Processig Uiversity Questio Paper PartA UitI. Defie samplig theorem?. What is kow as Aliasig? 3. What is LTI
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationTHE UNLIKELY UNION OF PARTITIONS AND DIVISORS
THE UNLIKELY UNION OF PARTITIONS AND DIVISORS Abdulkadir Hasse, Thomas J. Osler, Mathematics Departmet ad Tirupathi R. Chadrupatla, Mechaical Egieerig Rowa Uiversity Glassboro, NJ 828 I the multiplicative
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationf(x + T ) = f(x), for all x. The period of the function f(t) is the interval between two successive repetitions.
Fourier Series. Itroductio Whe the Frech mathematicia Joseph Fourier (76883) was tryig to study the flow of heat i a metal plate, he had the idea of expressig the heat source as a ifiite series of sie
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationANALYZING GALOIS GROUP FOR CUBIC, QUARTIC AND QUINTIC POLYNOMIAL
ANALYZING GALOIS GROUP FOR CUBIC QUARTIC AND QUINTIC POLYNOMIAL A M.Sc. DISSERTATION SUBMITTED BY ARCHANA MISHRA 0MA09 Uder the supervisio of Prof. K. C. PATI May 0 DEPARTMENT OF MATHEMATICS NATIONAL INSTITUTE
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More information1 n. n > dt. t < n 1 + n=1
Math 05 otes C. Pomerace The harmoic sum The harmoic sum is the sum of recirocals of the ositive itegers. We kow from calculus that it diverges, this is usually doe by the itegral test. There s a more
More informationDEFINITION OF INVERSE MATRIX
Lecture. Iverse matrix. To be read to the music of Back To You by Brya dams DEFINITION OF INVERSE TRIX Defiitio. Let is a square matrix. Some matrix B if it exists) is said to be iverse to if B B I where
More informationAn example of nonquenched convergence in the conditional central limit theorem for partial sums of a linear process
A example of oqueched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationMetric, Normed, and Topological Spaces
Chapter 13 Metric, Normed, ad Topological Spaces A metric space is a set X that has a otio of the distace d(x, y) betwee every pair of poits x, y X. A fudametal example is R with the absolutevalue metric
More informationAnalysis Notes (only a draft, and the first one!)
Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationOn the L p conjecture for locally compact groups
Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/0302376, ublished olie 2007080 DOI 0.007/s0003007993x Archiv der Mathematik O the L cojecture for locally comact
More information1. Hilbert spaces. 1.1 Definitions Vector spaces
. Hilbert spaces. Defiitios.. Vector spaces Defiitio. Vector space (*9&)8& "(9/). A vector space over a field F is a set V that has the structure of a additive group. Moreover, a product F V V, deoted
More informationSection 9.2 Series and Convergence
Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives
More information