ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES.

Size: px
Start display at page:

Download "ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES."

Transcription

1 ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. ANDREW SALCH 1. The Jacobia criterio for osigularity. You have probably oticed by ow that some poits o varieties are smooth i a sese somethig like that of a maifold from differetial geometry, while some poits are defiitely ot; for example, V (y 2 x 3 )(C) A 2 C has a kik or a corer (the right word is actually cusp ) at (0, 0), but the it of the taget lies of poits o the curve V (y 2 x 3 )(C) as you approach (0, 0) does ot deped o what directio alog the curve you approach (0, 0) from. O the other had, V (y 2 x 3 +x 2 )(C) A 2 C has a loop (the right word is actually ode ) at (0, 0), where the curve itersects itself, with two differet taget lies. Both this cusp ad this ode are sigular poits, which we ow defie: Defiitio 1.1. Let X A k be a variety of dimesio d, ad let x X. Choose a set of geerators ( f 1,, f m ) for the ideal I(X) k[x 1,, x ]. We say that X is osigular at x if the rak of the Jacobia matrix δ f 1 δ f 1 δx 1 δx 2 δ f 1 δx δx 1 δx 2 δx δx 1 δx 2 is d. If X is ot osigular a poit x X, the we say that x is a sigular poit of X. If X is osigular at every poit x X, the we just say that X is osigular. Nosigular varieties bear eough similarities to smooth maifolds that may of the techiques ad results from differetial topology which are used i the classificatio of smooth maifolds ca be imported (with some effort) ito algebraic geometry, where they apply to (at least some) osigular varieties (but usually ot sigular varieties, which are more like maifolds with sigularities). Observatio 1.2. If X is a affie variety, the set of sigular poits of X is give by the vaishig set of the determiats of various miors of the Jacobia matrix. Those determiats are all polyomials, so the set of sigular poits of X is the vaishig locus of some fiite set of polyomials, so the set of sigular poits of X is a closed subset of X. δx Date: Jauary

2 2 ANDREW SALCH Fidig sigular ad osigular poits o varieties is very computatioally approachable: Example 1.3. Suppose k is algebraically closed. I claim that the variety V (x 3 +y 3 xy+1)(k) A 2 k is osigular. The Jacobia matrix is [ 3x 2 y ] 3y 2 x ad (x, y) V (x 3 +y 3 xy+1)(k) is a sigular poit if ad oly if the Jacobia matrix is equal to [0 0] at (x, y). Suppose the characteristic of k is ot 3. Solvig the equatios 3x 2 y = 0 3y 2 x = 0, we get four solutios (0, 0), ( 1 3, 1 3 ), (ζ 1 3 3, 1 ζ2 3 3 ), 1 (ζ2 3 3, ζ ), with ζ 3 a primitive third root of uity i k. Now the questio is: are ay of these four poits i A 2 k actually i the variety? Pluggig each of them ito the equatio x3 + y 3 xy + 1 = 0, we get that oe of these four poits solve the equatio. Hece oe of the poits o the variety are sigular! (The same coclusio holds if the characteristic of k is 3, sice i that case, the Jacobia matrix becomes just [ y x], which is clearly zero oly at (0, 0), ad clearly (0, 0) is ot o the variety.) Exercise 1.4. (Nosigularity i a family of surfaces.) Let k be a algebraically closed field. (1) Let α, β k, ad suppose β 0. We write X α,β for the variety V (x 2 +y 2 +z 2 +αxyz+β)(k) A 3 k. Fid ecessary ad sufficiet coditios o α ad β for X α,β to be osigular. (2) Now let β 0, ad cosider the variety Y β = V (x 2 +y 2 +z 2 +wxyz+β)(k) A 4 k, i.e., we o loger thik of α as fixed, but we treat it as a variable like x ad y ad z! Each variety X α,β is a surface obtaied from the 3-dimesioal variety Y β by itersectig Y β with the hyperplae w = α. Show that Y β is osigular. Exercise 1.4 makes the poit that a variety ca be osigular (like Y β ) but still have closed subvarieties of smaller dimesio iside of it which do have sigularities (like some of the surfaces X α,β Y β ). 2. The local criterio for osigularity. Defiitio 2.1. Recall that, for a affie variety X, there is a oe-to-oe correspodece betwee poits of X ad maximal ideals i O(X). So, for each poit x X, we have a commutative local rig O(X) mx. We call the local rig O(X) mx the rig of germs of X at x. Now thik about smooth maifolds, from differetial topology, agai. A smooth maifold has the property that its dimesio is equal to the vector-space dimesio of its taget space at every poit. Oe could ask whether the Jacobia criterio for osigularity, our Defiitio 1.1, is equivalet to some local criterio for smoothess like that property of smooth maifolds. Ideed it is! It turs out that osigularity of X at x is equivalet to a certai algebraic property of the rig of germs of X at x:

3 ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. 3 Defiitio 2.2. Let A be a Noetheria local commutative rig with maximal ideal m ad residue field k. We say that A is regular if the Krull dimesio of A is equal to the k-vector space dimesio of m/m 2. Theorem 2.3. Let k be a algebraically closed field ad let X A k be a affie variety. Let x X. The X is osigular at x if ad oly if the rig of germs O(X) mx is regular. Proof. Suppose X is osigular at the poit x = (a 1,, a ) X A k. Let a X = (x 1 a 1,, x a ), ad we have a k-liear map θ : k[x 1,, x ] k ( δ f θ( f ) = (x),, δ f ) (x), δx 1 δx i.e., take all the partial derivatives, ad evaluate them at the poit x. Now θ(x i α i ) is the vector (0, 0,, 0, 1, 0,, 0), with 1 i the ith coordiate, so θ(x α ),, θ(x α ) is a k-liear basis for k. Furthermore, δ δx h (x i α i )(x j α j ) = 0 if h i, j x i α i if h i, h = j x j α j if h = i, h j 2(x i α i ) if h = i = j so ( δ δx h (x i α i )(x j α j ) ) (x) = 0 i all four cases. Hece θ vaishes o a x. Hece θ ax yields a k-liear isomorphism a x /a 2 x k. Now the rak of the Jacobia matrix of X is the dimesio of the image of I(X) k[x 1,, x ] uder the map θ, as a sub-k-vector-space of k, i.e., it is the dimesio of (a 2 x + I(X))/a2 x as a sub-k-vector-space of a x /a2 x. But, regardig m x as the maximal ideal of O(X) mx, we have m x = a x /(a x + I(X)). Hece dim k m x + rak of Jacobia = dim k mx /m2 x + dim k(a 2 x + I(X))/a2 x = dim k a x /(a 2 x + I(X)) + dim k(a 2 x + I(X))/a2 x = dim k a x /a 2 x =. Hece the rak of the Jacobia is equal to d, where d = dim(x), if ad oly if dim(x) = dim k m x, i.e., if ad oly if the local rig O(X) m x is regular. The local criterio for osigularity is itrisic (i.e., does ot deped o a choice of embeddig ito A k, or ito P k for that matter) ad makes perfect sese for quasi-projective ad quasi-affie varieties, ot just affie varieties, so we use it to defie: Defiitio 2.4. Let k be a algebraically closed field, let X be a variety over k, ad let x X be a poit. We say that X is osigular at x if the local rig O(X) mx is regular, ad we say that X is sigular at x if X is ot osigular at x. We say that X is osigular if X is osigular at x for every x X. The vector space m x really is the right otio of a taget space to X at x: elemets of m x are taget directios you ca move i, alog X, from the poit x. This is most trasparet i the special case X = A k ad x = (0, 0,, 0): the m x = (x 1, x 2,, x ), ad the k-vector space m x has k-liear basis x 1,, x, i.e., the directios you ca move

4 4 ANDREW SALCH i, iside A k, from the origi. But the same idea also works for ay poit o a variety. If X is osigular, the you ca take the k-liear dual of m x ad thik of it as a kid of space of differetial 1-forms o X, but there s a eve better way to set up the differetial forms o a osigular variety (the Kähler forms) ad build up a kid of vector calculus, de Rham cohomology, etc., o varieties; we will retur to this later o i the semester. 3. Completig the local rigs. Recall from commutative algebra that, if A is a commutative rig ad I is a ideal i A, the the I-adic completio of A is defied as the it  I = A/I A. Defiitio 3.1. Let X be a variety, ad let x be a poit i X. By the completed rig of germs of X at x, writte Ô(X) mx, we mea the m x -adic completio O(X) mx /m x of the rig of germs O(X) mx. If X, Y are varieties ad x X ad y Y, we say that x ad y are aalytically isomorphic if the completed rigs of germs are isomorphic as k-algebras: Ô(X) mx Ô(Y) my. Here is oe of my favorite theorems i pure algebra, which I will ot offer you a proof of, but we will make use of it: Theorem 3.2. (Cohe structure theorem.) Let k be a field, let be a oegative iteger, ad let A be a complete regular local rig of Krull dimesio. Suppose that A cotais a field as a subrig. The there exists a isomorphism of k-algebras A k[[x 1,, x ]]. Corollary 3.3. If X, Y are varieties of the same dimesio ad x X ad y Y are both regular poits, the x ad y are aalytically isomorphic. This is a strikig result: after completio, all regular poits o varieties of the same dimesio look the same. Oe cosequece is that, if you wat to uderstad ad classify some family of differet types of geometric sigularities of varieties, your first task is to classify their aalytic isomorphism types, i.e., to classify the complete local rigs at those sigular poits! Example 3.4. (Hartshore s Example ) Let k be algebraically closed. The variety V (y 2 x 3 x 2 )(k) has a sigularity at (0, 0), sice the Jacobia vaishes etirely there, ad (0, 0) is o this curve. The completed local rig at this poit is k[x, y]/(y 2 x 3 x 2 ). Cosult Hartshore for a argumet (it s log eough that I do t wat to reproduce it here!) that this rig is isomorphic to the completed local rig k[s, t]/(st). The crucial poit i this argumet is that, to write dow a isomorphism k[s, t]/(st) k[x, y]/(y 2 x 3 x 2 ), you eed to be able to sed the geerators s ad t i the domai to power series i the codomai, ot just polyomials: you ca t produce the desired isomorphism if you eed all but fiitely may coefficiets to be zero! So this isomorphism of local rigs exists oly after completio. The geometric meaig of this isomorphism is that the sigularity at (0, 0) o the curve y 2 = x 3 + x 2 is aalytically isomorphic to two lies crossig, i.e., the sigularity at (0, 0) i

5 ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. 5 the algebraic set V (st) (k) A 2 k (i.e., two lies i the affie plae which cross orthogoally at (0, 0)). Note that eve though V (y 2 x 3 x 2 )(k) is a variety, i.e., k[x, y]/(y 2 x 3 x 2 ) has o zero divisors, the completio of the rig of germs of V (y 2 x 3 x 2 )(k) at (0, 0) does have zero divisors! This ca happe at sigular poits. Example 3.5. (Two differet aalytic isomorphism types of curve sigularities.) Let k be algebraically closed. The variety X = V (y 2 x 3 )(k) has a sigularity at (0, 0), sice the Jacobia vaishes etirely there, ad (0, 0) is o this curve. The completed local rig at this poit is k[x, y]/(y 2 x 3 ). Notice what happes here: the maximal ideal i the rig of germs is (x, y), ad (x, y) 2 = (x 2, xy, y 2 ), but y 2 = x 3 (x, y) 3. This is importat: it meas that, writig m (0,0) for the maximal ideal i the rig of germs O(X) m(0,0), the quotiet m (0,0) /m (0,0) 3, has three k-liearly idepedet elemets (amely, y, x 2, xy) whose squares are zero. Compare this to the situatio (from Example 3.4) i k[s, t]/st completed at (0, 0): the maximal ideal (s, t) = m i the local rig (k[s, t]/st) (s,t) has the property that m/m 3 has oly two k-liearly idepedet elemets (amely, s 2, t 2 ) whose squares are zero. The dimesio of the space of square-zero elemets i m/m 3 is a isomorphism ivariat of a local rig, so this tells us that there caot possibly be a isomorphism betwee the local rigs O(X) m(0,0) ad (k[s, t]/st) (s,t). Eve better, it says that there caot possibly be a isomorphism betwee the completed local rigs Ô(X) m(0,0) ad (k[s, t]/st) (s,t) /(s, t)! Eve though it is possible i geeral for two o-isomorphic local rigs to become isomorphic after completio, i this case, it does t happe, sice m-adic completio (of Noetheria local rigs) does ot chage the fiite quotiets m i /m j.

CHAPTER 5. Theory and Solution Using Matrix Techniques

CHAPTER 5. Theory and Solution Using Matrix Techniques A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL

More information

Math 203A, Solution Set 8.

Math 203A, Solution Set 8. Math 20A, Solutio Set 8 Problem 1 Give four geeral lies i P, show that there are exactly 2 lies which itersect all four of them Aswer: Recall that the space of lies i P is parametrized by the Grassmaia

More information

Chapter 2. Periodic points of toral. automorphisms. 2.1 General introduction

Chapter 2. Periodic points of toral. automorphisms. 2.1 General introduction Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad

More information

Lecture 4: Grassmannians, Finite and Affine Morphisms

Lecture 4: Grassmannians, Finite and Affine Morphisms 18.725 Algebraic Geometry I Lecture 4 Lecture 4: Grassmaias, Fiite ad Affie Morphisms Remarks o last time 1. Last time, we proved the Noether ormalizatio lemma: If A is a fiitely geerated k-algebra, the,

More information

6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.

6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer. 6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio

More information

MATH 205 HOMEWORK #2 OFFICIAL SOLUTION. (f + g)(x) = f(x) + g(x) = f( x) g( x) = (f + g)( x)

MATH 205 HOMEWORK #2 OFFICIAL SOLUTION. (f + g)(x) = f(x) + g(x) = f( x) g( x) = (f + g)( x) MATH 205 HOMEWORK #2 OFFICIAL SOLUTION Problem 2: Do problems 7-9 o page 40 of Hoffma & Kuze. (7) We will prove this by cotradictio. Suppose that W 1 is ot cotaied i W 2 ad W 2 is ot cotaied i W 1. The

More information

b i u x i U a i j u x i u x j

b i u x i U a i j u x i u x j M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here

More information

Inverse Matrix. A meaning that matrix B is an inverse of matrix A.

Inverse Matrix. A meaning that matrix B is an inverse of matrix A. Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix

More information

CHAPTER I: Vector Spaces

CHAPTER I: Vector Spaces CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig

More information

M A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O

M A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #4 CORRECTION Algebra I 1 4 / 1 0 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O P r o f e s s o r : D r o r B a r - N a t a Correctio Homework Assigmet

More information

, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)

, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx) Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

A brief introduction to linear algebra

A brief introduction to linear algebra CHAPTER 6 A brief itroductio to liear algebra 1. Vector spaces ad liear maps I what follows, fix K 2{Q, R, C}. More geerally, K ca be ay field. 1.1. Vector spaces. Motivated by our ituitio of addig ad

More information

(for homogeneous primes P ) defining global complex algebraic geometry. Definition: (a) A subset V CP n is algebraic if there is a homogeneous

(for homogeneous primes P ) defining global complex algebraic geometry. Definition: (a) A subset V CP n is algebraic if there is a homogeneous Math 6130 Notes. Fall 2002. 4. Projective Varieties ad their Sheaves of Regular Fuctios. These are the geometric objects associated to the graded domais: C[x 0,x 1,..., x ]/P (for homogeeous primes P )

More information

Week 5-6: The Binomial Coefficients

Week 5-6: The Binomial Coefficients Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers

More information

The Method of Least Squares. To understand least squares fitting of data.

The Method of Least Squares. To understand least squares fitting of data. The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve

More information

Math 61CM - Solutions to homework 3

Math 61CM - Solutions to homework 3 Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig

More information

11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4.

11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4. 11. FINITE FIELDS 11.1. A Field With 4 Elemets Probably the oly fiite fields which you ll kow about at this stage are the fields of itegers modulo a prime p, deoted by Z p. But there are others. Now although

More information

4 The Sperner property.

4 The Sperner property. 4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,

More information

PAPER : IIT-JAM 2010

PAPER : IIT-JAM 2010 MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

Algebraic Geometry I

Algebraic Geometry I 6.859/15.083 Iteger Programmig ad Combiatorial Optimizatio Fall 2009 Lecture 14: Algebraic Geometry I Today... 0/1-iteger programmig ad systems of polyomial equatios The divisio algorithm for polyomials

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

Machine Learning Theory Tübingen University, WS 2016/2017 Lecture 11

Machine Learning Theory Tübingen University, WS 2016/2017 Lecture 11 Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract We will itroduce the otio of reproducig kerels ad associated Reproducig Kerel Hilbert Spaces (RKHS). We will cosider couple

More information

LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS

LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS I the previous sectio we used the techique of adjoiig cells i order to costruct CW approximatios for arbitrary spaces Here we will see that the same techique

More information

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 2012 C1 Note. C1 Notes (Edexcel) Copyright   - For AS, A2 notes and IGCSE / GCSE worksheets 1 September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

More information

Properties and Tests of Zeros of Polynomial Functions

Properties and Tests of Zeros of Polynomial Functions Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by

More information

Complex Analysis Spring 2001 Homework I Solution

Complex Analysis Spring 2001 Homework I Solution Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle

More information

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.

Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j. Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α

More information

LECTURE 12: THE DEGREE OF A MAP AND THE CELLULAR BOUNDARIES

LECTURE 12: THE DEGREE OF A MAP AND THE CELLULAR BOUNDARIES LECTURE 12: THE DEGREE OF A MAP AD THE CELLULAR BOUDARIES I this lecture we will study the (homological) degree of self-maps of spheres, a otio which geeralizes the usual degree of a polyomial. We will

More information

Math 210A Homework 1

Math 210A Homework 1 Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called

More information

Orthogonal transformations

Orthogonal transformations Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear

More information

1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y

1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:

More information

Determinants of order 2 and 3 were defined in Chapter 2 by the formulae (5.1)

Determinants of order 2 and 3 were defined in Chapter 2 by the formulae (5.1) 5. Determiats 5.. Itroductio 5.2. Motivatio for the Choice of Axioms for a Determiat Fuctios 5.3. A Set of Axioms for a Determiat Fuctio 5.4. The Determiat of a Diagoal Matrix 5.5. The Determiat of a Upper

More information

Stanford Math Circle January 21, Complex Numbers

Stanford Math Circle January 21, Complex Numbers Staford Math Circle Jauary, 007 Some History Tatiaa Shubi (shubi@mathsjsuedu) Complex Numbers Let us try to solve the equatio x = 5x + x = is a obvious solutio Also, x 5x = ( x )( x + x + ) = 0 yields

More information

lim za n n = z lim a n n.

lim za n n = z lim a n n. Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget

More information

Lesson 10: Limits and Continuity

Lesson 10: Limits and Continuity www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

More information

Complex Numbers Solutions

Complex Numbers Solutions Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i

More information

Lemma Let f(x) K[x] be a separable polynomial of degree n. Then the Galois group is a subgroup of S n, the permutations of the roots.

Lemma Let f(x) K[x] be a separable polynomial of degree n. Then the Galois group is a subgroup of S n, the permutations of the roots. 15 Cubics, Quartics ad Polygos It is iterestig to chase through the argumets of 14 ad see how this affects solvig polyomial equatios i specific examples We make a global assumptio that the characteristic

More information

(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:

(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row: Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

Question 1: The magnetic case

Question 1: The magnetic case September 6, 018 Corell Uiversity, Departmet of Physics PHYS 337, Advace E&M, HW # 4, due: 9/19/018, 11:15 AM Questio 1: The magetic case I class, we skipped over some details, so here you are asked to

More information

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations

Section 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?

More information

SEMIGROUPS OF VALUATIONS DOMINATING LOCAL DOMAINS

SEMIGROUPS OF VALUATIONS DOMINATING LOCAL DOMAINS SEMIGROUPS OF VALUATIONS DOMINATING LOCAL DOMAINS STEVEN DALE CUTKOSKY Let (R, m R ) be a equicharacteristic local domai, with quotiet field K. Suppose that ν is a valuatio of K with valuatio rig (V, m

More information

1 Last time: similar and diagonalizable matrices

1 Last time: similar and diagonalizable matrices Last time: similar ad diagoalizable matrices Let be a positive iteger Suppose A is a matrix, v R, ad λ R Recall that v a eigevector for A with eigevalue λ if v ad Av λv, or equivaletly if v is a ozero

More information

Brief Review of Functions of Several Variables

Brief Review of Functions of Several Variables Brief Review of Fuctios of Several Variables Differetiatio Differetiatio Recall, a fuctio f : R R is differetiable at x R if ( ) ( ) lim f x f x 0 exists df ( x) Whe this limit exists we call it or f(

More information

sin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =

sin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n = 60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece

More information

Notes The Incremental Motion Model:

Notes The Incremental Motion Model: The Icremetal Motio Model: The Jacobia Matrix I the forward kiematics model, we saw that it was possible to relate joit agles θ, to the cofiguratio of the robot ed effector T I this sectio, we will see

More information

8. Applications To Linear Differential Equations

8. Applications To Linear Differential Equations 8. Applicatios To Liear Differetial Equatios 8.. Itroductio 8.. Review Of Results Cocerig Liear Differetial Equatios Of First Ad Secod Orders 8.3. Eercises 8.4. Liear Differetial Equatios Of Order N 8.5.

More information

Recitation 4: Lagrange Multipliers and Integration

Recitation 4: Lagrange Multipliers and Integration Math 1c TA: Padraic Bartlett Recitatio 4: Lagrage Multipliers ad Itegratio Week 4 Caltech 211 1 Radom Questio Hey! So, this radom questio is pretty tightly tied to today s lecture ad the cocept of cotet

More information

Solving Algebraic Equations in Commutative Rings Hideyuki Matsumura Nagoya University

Solving Algebraic Equations in Commutative Rings Hideyuki Matsumura Nagoya University s 0 Itroductio Solvig Algebraic Equatios i Commutative Rigs Hideyuki Matsumura Nagoya Uiversity Let A be a commutative Rig, ad let f(x) = ( f 1,..., f,) be a vector of polyomials i X 1,, X with coefficiets

More information

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

More information

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

ECON 3150/4150, Spring term Lecture 3

ECON 3150/4150, Spring term Lecture 3 Itroductio Fidig the best fit by regressio Residuals ad R-sq Regressio ad causality Summary ad ext step ECON 3150/4150, Sprig term 2014. Lecture 3 Ragar Nymoe Uiversity of Oslo 21 Jauary 2014 1 / 30 Itroductio

More information

Algebra of Least Squares

Algebra of Least Squares October 19, 2018 Algebra of Least Squares Geometry of Least Squares Recall that out data is like a table [Y X] where Y collects observatios o the depedet variable Y ad X collects observatios o the k-dimesioal

More information

MATH10212 Linear Algebra B Proof Problems

MATH10212 Linear Algebra B Proof Problems MATH22 Liear Algebra Proof Problems 5 Jue 26 Each problem requests a proof of a simple statemet Problems placed lower i the list may use the results of previous oes Matrices ermiats If a b R the matrix

More information

Linear Elliptic PDE s Elliptic partial differential equations frequently arise out of conservation statements of the form

Linear Elliptic PDE s Elliptic partial differential equations frequently arise out of conservation statements of the form Liear Elliptic PDE s Elliptic partial differetial equatios frequetly arise out of coservatio statemets of the form B F d B Sdx B cotaied i bouded ope set U R. Here F, S deote respectively, the flux desity

More information

Some examples of vector spaces

Some examples of vector spaces Roberto s Notes o Liear Algebra Chapter 11: Vector spaces Sectio 2 Some examples of vector spaces What you eed to kow already: The te axioms eeded to idetify a vector space. What you ca lear here: Some

More information

Generating Functions for Laguerre Type Polynomials. Group Theoretic method

Generating Functions for Laguerre Type Polynomials. Group Theoretic method It. Joural of Math. Aalysis, Vol. 4, 2010, o. 48, 257-266 Geeratig Fuctios for Laguerre Type Polyomials α of Two Variables L ( xy, ) by Usig Group Theoretic method Ajay K. Shula* ad Sriata K. Meher** *Departmet

More information

Chapter 6 Infinite Series

Chapter 6 Infinite Series Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat

More information

TENSOR PRODUCTS AND PARTIAL TRACES

TENSOR PRODUCTS AND PARTIAL TRACES Lecture 2 TENSOR PRODUCTS AND PARTIAL TRACES Stéphae ATTAL Abstract This lecture cocers special aspects of Operator Theory which are of much use i Quatum Mechaics, i particular i the theory of Quatum Ope

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=

More information

Support vector machine revisited

Support vector machine revisited 6.867 Machie learig, lecture 8 (Jaakkola) 1 Lecture topics: Support vector machie ad kerels Kerel optimizatio, selectio Support vector machie revisited Our task here is to first tur the support vector

More information

2 Geometric interpretation of complex numbers

2 Geometric interpretation of complex numbers 2 Geometric iterpretatio of complex umbers 2.1 Defiitio I will start fially with a precise defiitio, assumig that such mathematical object as vector space R 2 is well familiar to the studets. Recall that

More information

Chapter 1. Complex Numbers. Dr. Pulak Sahoo

Chapter 1. Complex Numbers. Dr. Pulak Sahoo Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler

More information

Math 312 Lecture Notes One Dimensional Maps

Math 312 Lecture Notes One Dimensional Maps Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,

More information

10-701/ Machine Learning Mid-term Exam Solution

10-701/ Machine Learning Mid-term Exam Solution 0-70/5-78 Machie Learig Mid-term Exam Solutio Your Name: Your Adrew ID: True or False (Give oe setece explaatio) (20%). (F) For a cotiuous radom variable x ad its probability distributio fuctio p(x), it

More information

NBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian?

NBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian? NBHM QUESTION 7 NBHM QUESTION 7 NBHM QUESTION 7 Sectio : Algebra Q Let G be a group of order Which of the followig coditios imply that G is abelia? 5 36 Q Which of the followig subgroups are ecesarily

More information

MATHEMATICS Code No. 13 INSTRUCTIONS

MATHEMATICS Code No. 13 INSTRUCTIONS DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO COMBINED COMPETITIVE (PRELIMINARY) EXAMINATION, 0 Serial No. MATHEMATICS Code No. A Time Allowed : Two Hours Maimum Marks : 00 INSTRUCTIONS. IMMEDIATELY

More information

Symmetrization maps and differential operators. 1. Symmetrization maps

Symmetrization maps and differential operators. 1. Symmetrization maps July 26, 2011 Symmetrizatio maps ad differetial operators Paul Garrett garrett@math.um.edu http://www.math.um.edu/ garrett/ The symmetrizatio map s : Sg Ug is a liear surjectio from the symmetric algebra

More information

1 Approximating Integrals using Taylor Polynomials

1 Approximating Integrals using Taylor Polynomials Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................

More information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

Chapter Vectors

Chapter Vectors Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it

More information

The Borel hierarchy classifies subsets of the reals by their topological complexity. Another approach is to classify them by size.

The Borel hierarchy classifies subsets of the reals by their topological complexity. Another approach is to classify them by size. Lecture 7: Measure ad Category The Borel hierarchy classifies subsets of the reals by their topological complexity. Aother approach is to classify them by size. Filters ad Ideals The most commo measure

More information

Chapter 6 Principles of Data Reduction

Chapter 6 Principles of Data Reduction Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a

More information

PRELIM PROBLEM SOLUTIONS

PRELIM PROBLEM SOLUTIONS PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems

More information

Introduction to Optimization Techniques. How to Solve Equations

Introduction to Optimization Techniques. How to Solve Equations Itroductio to Optimizatio Techiques How to Solve Equatios Iterative Methods of Optimizatio Iterative methods of optimizatio Solutio of the oliear equatios resultig form a optimizatio problem is usually

More information

MIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS

MIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will

More information

1. By using truth tables prove that, for all statements P and Q, the statement

1. By using truth tables prove that, for all statements P and Q, the statement Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3

More information

( 1) n (4x + 1) n. n=0

( 1) n (4x + 1) n. n=0 Problem 1 (10.6, #). Fid the radius of covergece for the series: ( 1) (4x + 1). For what values of x does the series coverge absolutely, ad for what values of x does the series coverge coditioally? Solutio.

More information

A REMARK ON A PROBLEM OF KLEE

A REMARK ON A PROBLEM OF KLEE C O L L O Q U I U M M A T H E M A T I C U M VOL. 71 1996 NO. 1 A REMARK ON A PROBLEM OF KLEE BY N. J. K A L T O N (COLUMBIA, MISSOURI) AND N. T. P E C K (URBANA, ILLINOIS) This paper treats a property

More information

On the Linear Complexity of Feedback Registers

On the Linear Complexity of Feedback Registers O the Liear Complexity of Feedback Registers A. H. Cha M. Goresky A. Klapper Northeaster Uiversity Abstract I this paper, we study sequeces geerated by arbitrary feedback registers (ot ecessarily feedback

More information

PROBLEM SET I (Suggested Solutions)

PROBLEM SET I (Suggested Solutions) Eco3-Fall3 PROBLE SET I (Suggested Solutios). a) Cosider the followig: x x = x The quadratic form = T x x is the required oe i matrix form. Similarly, for the followig parts: x 5 b) x = = x c) x x x x

More information

Random Models. Tusheng Zhang. February 14, 2013

Random Models. Tusheng Zhang. February 14, 2013 Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the

More information

The multiplicative structure of finite field and a construction of LRC

The multiplicative structure of finite field and a construction of LRC IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio

More information

4.3 Growth Rates of Solutions to Recurrences

4.3 Growth Rates of Solutions to Recurrences 4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.

More information

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

More information

Fundamental Concepts: Surfaces and Curves

Fundamental Concepts: Surfaces and Curves UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat

More information

Geometry of LS. LECTURE 3 GEOMETRY OF LS, PROPERTIES OF σ 2, PARTITIONED REGRESSION, GOODNESS OF FIT

Geometry of LS. LECTURE 3 GEOMETRY OF LS, PROPERTIES OF σ 2, PARTITIONED REGRESSION, GOODNESS OF FIT OCTOBER 7, 2016 LECTURE 3 GEOMETRY OF LS, PROPERTIES OF σ 2, PARTITIONED REGRESSION, GOODNESS OF FIT Geometry of LS We ca thik of y ad the colums of X as members of the -dimesioal Euclidea space R Oe ca

More information

a for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a

a for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a Math E-2b Lecture #8 Notes This week is all about determiats. We ll discuss how to defie them, how to calculate them, lear the allimportat property kow as multiliearity, ad show that a square matrix A

More information

Lecture 23: Minimal sufficiency

Lecture 23: Minimal sufficiency Lecture 23: Miimal sufficiecy Maximal reductio without loss of iformatio There are may sufficiet statistics for a give problem. I fact, X (the whole data set) is sufficiet. If T is a sufficiet statistic

More information

n outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n,

n outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n, CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 9 Variace Questio: At each time step, I flip a fair coi. If it comes up Heads, I walk oe step to the right; if it comes up Tails, I walk oe

More information

Kinetics of Complex Reactions

Kinetics of Complex Reactions Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet

More information

Mathematical Induction

Mathematical Induction Mathematical Iductio Itroductio Mathematical iductio, or just iductio, is a proof techique. Suppose that for every atural umber, P() is a statemet. We wish to show that all statemets P() are true. I a

More information

Subject: Differential Equations & Mathematical Modeling-III

Subject: Differential Equations & Mathematical Modeling-III Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso

More information

In number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.

In number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play. Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers

More information

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014. Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

More information

REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS

REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x]

More information