# Linear Systems with Constant Coefficients

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Liner Systems with Constnt Coefficients Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system Thus, the coefficients ij re constnt, nd you cn see tht the equtions re liner in the vriles x,, x n nd their derivtives The reson for the term homogeneous will e cler when I ve written the system in mtrix form The primes on x,, x n denote differentition with respect to n independent vrile t The prolem is to solve for x,, x n in terms of t Write the system in mtrix form s Equivlently, x x n n n n nn x A x (A nonhomogeneous system would look like x A x+ ) It s possile to solve such system if you know the eigenvlues (nd possily the eigenvectors) for the coefficient mtrix n n n nn First, I ll do n exmple which shows tht you cn solve smll liner systems y rute force x x n Exmple Consider the system of differentil equtions dx dt x 9x 48x, dx dt x 6x +7x The ide is to solve for x nd x in terms of t One pproch is to use rute force Solve the first eqution for x, then differentite to find x : Plug these into second eqution: x 48 ( x 9x ), x 48 ( x 9x ) 48 ( x 9x ) 6x ( x 9x ), x +x 5x 0

2 This is constnt coefficient liner homogeneous eqution in x The chrcteristic eqution is m m 5 0 The roots re m 5 nd m 3 Therefore, Plug ck in to find x : x c e 5t +c e 3t x 48 ( x 9x ) ( ( 5c e 5t +3c e 3t ) 9(c e 5t +c e 3t ) ) 48 c e 5t 3 c e 3t The procedure works, ut it s cler tht the computtions would e pretty horrile for lrger systems To descrie etter pproch, look t the coefficient mtrix: A Find the eigenvlues: A λi 9 λ λ (λ+9)(λ 7)+6 48 λ +λ 5 This is the sme polynomil tht ppered in the exmple Since λ + λ 5 (λ+5)(λ 3), the eigenvlues re λ 5 nd λ 3 Thus, you don t need to go through the process of eliminting x nd isolting x You know tht x c e 5t +c e 3t once you know the eigenvlues of the coefficient mtrix You cn now finish the prolem s ove y plugging x ck in to solve for x This is etter thn rute force, ut it s still cumersome if the system hs more thn two vriles I cn improve things further y mking use of eigenvectors s well s eigenvlues Consider the system x A x Suppose λ is n eigenvlue of A with eigenvector v This mens tht Av λv I clim tht x ce λt v is solution to the eqution, where c is constnt To see this, plug it in: x cλe λt v ce λt (λv) ce λt (Av) A(ce λt v) A x To otin the generl solution to x A x, you should hve one ritrry constnt for ech differentition In this cse, you d expect n ritrry constnts This discussion should mke the following result plusile Suppose the mtrix A hs n independent eigenvectors v,, v n with corresponding eigenvlues λ,, λ n Then the generl solution to x A x is x c e λ t v + c n e λ nt v n

3 Exmple Solve dx dt dx dt x 9x 48x, x 6x +7x The mtrix form is x x The mtrix A x hs eigenvlues λ 5 nd λ 3 I need to find the eigenvectors Consider λ 5: 4 48 A+5I 6 3 The lst mtrix sys + 0, or Therefore, x Tke The eigenvector is (,) Now consider λ 3: A 3I The lst mtrix sys + 3 0, or 3 Therefore, 3 3 Tke The eigenvector is ( 3,) You cn check tht the vectors (,), ( 3,), re independent Hence, the solution is x x x c e 5t +c e 3t 3 Exmple Find the generl solution (x(t), y(t)) to the liner system The mtrix form is x Let y dx dt x+y dy dt 6x+y A x y

4 det(a xi) x 6 x x 3x 4 (x 4)(x+) The eigenvlues re x 4 nd x For x 4, I hve If (,) is n eigenvector, then So (, 3) is n eigenvector For x, I hve If (,) is n eigenvector, then So (, ) is n eigenvector The solution is A 4I A+I , , x c y e 4t +c 3 e t Exmple (Complex roots) Solve The chrcteristic polynomil is x 5x +5x, x 4x 3x 5 λ λ λ λ+5 The eigenvlues re λ ±i You cn check tht the eigenvectors re: +i λ i : i λ +i : Oserve tht the eigenvectors re conjugtes of one nother This is lwys true when you hve complex eigenvlue The eigenvector method gives the following complex solution: x x c e ( i)t +i +c e (+i)t i 4

5 e t ( (c +c )+i(c c ))cost+((c +c )+i(c c ))sint (c +c )cost i(c c )sint Note tht the constnts occur in the comintions c +c nd i(c c ) Something like this will lwys hppen in the complex cse Set d c +c nd d i(c c ) The solution is x e t ( d +d )cost+(d +d )sint d cost d sint x In fct, if you re given initil conditions for x nd x, the new constnts d nd d will turn out to e rel numers You cn get picture of the solution curves for system x f( x) even if you cn t solve it y sketching the direction field Suppose you hve two-vrile liner system x x y c d y This is equivlent to the equtions Then dx dt x+y nd dy dt cx+dy dy dx dy dt dx dt cx+dy x+y Tht is, the expression on the right gives the slope of the solution curve t the point (x,y) To sketch the direction field, pick set of smple points for exmple, the points on grid At ech point (x,y), drw the vector (x+y,cx+dy) strting t the point (x,y) The collection of vectors is the direction field You cn pproximte the solution curves y sketching in curves which re tngent to the direction field Exmple Sketch the direction field for x x y, y x+y I ve computed the vectors for 9 points: x y x y x+y vector 0 (0, ) 0 (, ) 0 (, 0) 0 (, ) (0, 0) 0 (, ) 0 (, 0) 0 (, ) 0 (0, ) 5

6 Thus, from the second line of the tle, I d drw the vector (, ) strting t the point (,0) Here s sketch of the vectors: y x While it s possile to plot fields this wy, it s very tedious You cn use softwre to plot fields quickly Here is the sme field s plotted y Mthemtic: The first picture shows the field s it would e if you plotted it y hnd As you cn see, the vectors overlp ech other, mking the picture it ugly The second picture is the wy Mthemtic drws the field y defult: The vectors lengths re scled so tht the vectors don t overlp In susequent exmples, I ll dopt the second lterntive when I disply direction field picture The rrows in the pictures show the direction of incresing t on the solution curves You cn see from these pictures tht the solution curves for this system pper to spirl out from the origin Exmple (A comprtment model) Two tnks hold 50 gllons of liquid ech The first tnk strts with 5 pounds of dissolved slt, while the second strts with pure wter Pure wter flows into the first tnk t 3 gllons per minute; the well-stirred micture flows into tnk t 4 gllons per minute The mixture in tnk is pumped ck into tnk t gllon per minute, nd lso drins out t 3 gllons per minute Find the mount of slt in ech tnk fter t minutes Let x e the numer of pounds of slt dissolved in the first tnk t time t nd let y e the numer of pounds of slt dissolved in the second tnk t time t The rte equtions re Simplify: ( dx dt 3 gl )( 0 ls ) ( + gl )( ) ( yls 4 gl )( ) xls, min gl min 50gl min 50gl ( dy dt 4 gl )( ) ( xls gl )( ) ( yls 3 gl )( ) yls min 50gl min 50gl min 50gl x 008x+00y, y 008x 008y Next, find the chrcteristic polynomil: 008 λ λ λ +06λ+0048 (λ+004)(λ+0) 6

7 The eigenvlues re λ 004, λ 0 Consider λ 004: A+004I This sys 0, so Therefore, Set The eigenvector is (,) Now consider λ 0: A+0I This sys + 0, so Therefore, Set The eigenvector is (,) The solution is x c e 004t When t 0, x 5 nd y 0 Plug in: 5 c 0 +c +c e 0t c Solving for the constnts, I otin c 5, c 5 Thus, x 5e 004t +5e 0t 5e 004t +5e 0t 5e 004t 5e 0t The direction field for the system is shown in the first picture In the second picture, I ve sketched in some solution curves c 7

8 The solution curve picture is referred to s the phse portrit The eigenvectors (,) nd (,) hve slopes nd, respectively These pper s the two lines (liner solutions) Consider the liner system x Ax Suppose it hs hs conjugte complex eigenvlues λ, λ with eigenvectors v, v, respectively This yields solutions e λt v, e λ t v If +i is complex numer, re(+i) ((+i)+( i)) ((+i)+(+i) ), im(+i) ((+i) ( i)) ((+i) (+i) ) I ll pply this to e λt v, using the fct tht ( e λt v ) e λ t v Then re ( e λt v ) im ( e λt v ) ( e λt v +e λ t v ), ( e λt v e λ t v ) The point is tht since the terms on the right re independent solutions, so re the terms on the left The terms on the left, however, re rel solutions Here is wht this mens If liner system hs pir of complex conjugte eigenvlues, find the eigenvector solution for one of them (the e λt v ove) Then tke the rel nd imginry prts to otin two independent rel solutions Exmple Solve the system x x y, y x+y Set A The eigenvlues re λ ±i Consider λ +i: A (+i)i i i i The lst mtrix sys i 0, so i The eigenvectors re i 8 i

9 Tke This yields the eigenvector (i,) Write down the complex solution e (+i)t i e t e it i e t i (cost+isint) e t sint + icost cost + isint Tke the rel nd imginry prts: The generl solution is ree t sint + icost e t sint, cost + isint cost ime t sint + icost e t cost cost + isint sint x c e t sint +c cost e t cost sint The eigenvector method produces solution to (constnt coefficient homogeneous) liner system whenever there re enough eigenvectors There might not e enough eigenvectors if the chrcteristic polynomil hs repeted roots I ll consider the cse of repeted roots with multiplicity two or three (ie doule or triple roots) The generl cse cn e hndled y using the exponentil of mtrix Consider the following liner system: x A x Suppose λ is n eigenvlue of A of multiplicity, nd v is n eigenvector for λ e λt v is one solution; I wnt to find second independent solution Recll tht the constnt coefficient eqution (D 3) y 0 hd independent solutions e 3x nd xe 3x By nlogy, it s resonle to guess solution of the form Here w is constnt vector Plug the guess into x A x: x te λt w x te λt λ w +e λt w A(te λt w) Compre terms in te λt nd e λt on the left nd right: A w λ w nd w 0 While it s true tht te λt 0 0 is solution, it s not very useful solution I ll try gin, this time using x te λt w +e λt w Then Note tht Hence, x te λt λ w +e λt w +λe λt w A x te λt A w +e λt A w te λt λ w +e λt w +λe λt w te λt A w +e λt A w 9

10 Equte coefficients in e λt, te λt : A w λ w so (A λi) w 0, A w w +λ w so (A λi) w w In other words, w is n eigenvector, nd w is vector which is mpped y A λi to the eigenvector w is clled generlized eigenvector Exmple Solve x 3 8 x 5 3 λ 8 5 λ (λ+3)(λ 5)+6 λ λ+ (λ ) Therefore, λ is n eigenvlue of multiplicity Now 4 8 A I 4 The lst mtrix sys + 0, or Therefore, Tke The eigenvector is (,) This gives solution Next, I ll try to find vector w such tht Write w (c,d) The eqution ecomes e t (A I) w c d Row reduce: The lst mtrix sys tht c+d, so c d + In this sitution, I my tke d 0; doing so ( ) produces w,0 This work genertes the solution te t +e t 0 0

11 The generl solution is ( ) x c e t +c te t +e t 0 The first picture shows the direction field; the second shows the phse portrit, with some typicl solution curves This kind of phse portrit is clled n improper node Exmple Solve the system x 0 x The eigenvlues re λ nd λ (doule) I ll do λ first A I The lst mtrix implies tht 0 nd 0, so the eigenvectors re For λ, c c c A I The lst mtrix implies tht 0 nd c, so the eigenvectors re c 0 c c c 0 I ll use v (0,,) Next, I find generlized eigenvector w (,,c ) It must stisfy (A I) w v

12 Tht is, 0 0 c Solving this system yields, c + I cn tke c 0, so, nd w c 0 The solution is x c e t +c e t 0 +c te t 0 +e t 0 I ll give rief description of the sitution for n eigenvlue λ of multiplicity 3 First, if there re three independent eigenvectors u, v, w, the solution is x c e λt u+c e λt v +c e λt w Suppose there is one independent eigenvector, sy u One solution is Find generlized eigenvector v y solving A second solution is e λt u (A λi) v u te λt u+e λt v Next, otin nother generlized eigenvector w y solving A third independent solution is (A λi) w v t e λt u+te λt v +e λt w Finlly, comine the solutions to otin the generl solution The only other possiility is tht there re two independent eigenvectors u nd v These give solutions Find generlized eigenvector w y solving e λt u nd e λt v (A λi) w u+ v The constnts nd re chosen so tht the eqution is solvle w yields the solution te λt ( u+ v)+e λt w The est wy of explining why this works involves something clled the Jordn cnonicl form for mtrices It s lso possile to circumvent these techniclities y using the exponentil of mtrix c 05 y Bruce Ikeng

### dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

### dy ky, dt where proportionality constant k may be positive or negative

Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.

### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

### 2. VECTORS AND MATRICES IN 3 DIMENSIONS

2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

### 1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

### The Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5

The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### Lecture 2 : Propositions DRAFT

CS/Mth 240: Introduction to Discrete Mthemtics 1/20/2010 Lecture 2 : Propositions Instructor: Dieter vn Melkeeek Scrie: Dlior Zelený DRAFT Lst time we nlyzed vrious mze solving lgorithms in order to illustrte

### Best Approximation. Chapter The General Case

Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

### Farey 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

### Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

### STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

### Matrix Eigenvalues and Eigenvectors September 13, 2017

Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues

### NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

### 5: The Definite Integral

5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

### expression simply by forming an OR of the ANDs of all input variables for which the output is

2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output

### Homework 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.

### Homework 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)

### 1 Error Analysis of Simple Rules for Numerical Integration

cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

### 8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

### Math 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 =

### CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

### Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

### u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + \$

### 4.1. Probability Density Functions

STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

### Wave Equation on a Two Dimensional Rectangle

Wve Eqution on Two Dimensionl Rectngle In these notes we re concerned with ppliction of the method of seprtion of vriles pplied to the wve eqution in two dimensionl rectngle. Thus we consider u tt = c

### 63. 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 =

### MATH 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.

### 2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

### Topic 1 Notes Jeremy Orloff

Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble

### Math 0230 Calculus 2 Lectures

Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

### Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

### Preview 11/1/2017. Greedy Algorithms. Coin Change. Coin Change. Coin Change. Coin Change. Greedy algorithms. Greedy Algorithms

Preview Greed Algorithms Greed Algorithms Coin Chnge Huffmn Code Greed lgorithms end to e simple nd strightforwrd. Are often used to solve optimiztion prolems. Alws mke the choice tht looks est t the moment,

### CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes

### a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.

Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting

### Project 6: Minigoals Towards Simplifying and Rewriting Expressions

MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

### 5.5 The Substitution Rule

5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

### Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

### 12.1 Nondeterminism Nondeterministic Finite Automata. a a b ε. CS125 Lecture 12 Fall 2016

CS125 Lecture 12 Fll 2016 12.1 Nondeterminism The ide of nondeterministic computtions is to llow our lgorithms to mke guesses, nd only require tht they ccept when the guesses re correct. For exmple, simple

### MT Integral equations

MT58 - Integrl equtions Introduction Integrl equtions occur in vriety of pplictions, often eing otined from differentil eqution. The reson for doing this is tht it my mke solution of the prolem esier or,

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### The Wave Equation I. MA 436 Kurt Bryan

1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

### Math 8 Winter 2015 Applications of Integration

Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

### Math 360: A primitive integral and elementary functions

Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

### BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS

BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible

### Lecture 1. Functional series. Pointwise and uniform convergence.

1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

### USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

### 3 Regular expressions

3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll

### Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

### Grammar. Languages. Content 5/10/16. Automata and Languages. Regular Languages. Regular Languages

5//6 Grmmr Automt nd Lnguges Regulr Grmmr Context-free Grmmr Context-sensitive Grmmr Prof. Mohmed Hmd Softwre Engineering L. The University of Aizu Jpn Regulr Lnguges Context Free Lnguges Context Sensitive

### A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

### 10. AREAS BETWEEN CURVES

. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

### Mapping the delta function and other Radon measures

Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

### Chapter 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

### PDE Notes. Paul Carnig. January ODE s vs PDE s 1

PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................

### MTH 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

### x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :

Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given

### Math 113 Exam 2 Practice

Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

### Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

### The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

### Calculus 2: Integration. Differentiation. Integration

Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

### 13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

### State Minimization for DFAs

Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid

### 4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

### Arithmetic & Algebra. NCTM National Conference, 2017

NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

### MTH 122 Calculus II Essex County College Division of Mathematics and Physics 1 Lecture Notes #10 Sakai Web Project Material

MTH 22 Clculus II Essex County College Division of Mthemtics nd Physics Lecture Notes # Ski Web Project Mteril Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry

### COSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature

COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III - Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting

### Lecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014

Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t Urbn-Chmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method

### The Trapezoidal Rule

_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

### Triangles The following examples explore aspects of triangles:

Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

### Quadratic Equations. Brahmagupta gave. Solving of quadratic equations in general form is often credited to ancient Indian mathematicians.

9 Qudrtic Equtions Qudrtic epression nd qudrtic eqution Pure nd dfected qudrtic equtions Solution of qudrtic eqution y * Fctoristion method * Completing the squre method * Formul method * Grphicl method

### If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du

Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find nti-derivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible

### Orthogonal Polynomials and Least-Squares Approximations to Functions

Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny

### Lecture V. Introduction to Space Groups Charles H. Lake

Lecture V. Introduction to Spce Groups 2003. Chrles H. Lke Outline:. Introduction B. Trnsltionl symmetry C. Nomenclture nd symols used with spce groups D. The spce groups E. Derivtion nd discussion of

### MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008

MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul

### Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,

### Solutions to Problems in Merzbacher, Quantum Mechanics, Third Edition. Chapter 7

Solutions to Problems in Merzbcher, Quntum Mechnics, Third Edition Homer Reid April 5, 200 Chpter 7 Before strting on these problems I found it useful to review how the WKB pproimtion works in the first

### The Leaning Tower of Pingala

The Lening Tower of Pingl Richrd K. Guy Deprtment of Mthemtics & Sttistics, The University of Clgry. July, 06 As Leibniz hs told us, from 0 nd we cn get everything: Multiply the previous line by nd dd

### arxiv: v2 [math.nt] 2 Feb 2015

rxiv:407666v [mthnt] Fe 05 Integer Powers of Complex Tridigonl Anti-Tridigonl Mtrices Htice Kür Duru &Durmuş Bozkurt Deprtment of Mthemtics, Science Fculty of Selçuk University Jnury, 08 Astrct In this

Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition

### DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive

### ECON 331 Lecture Notes: Ch 4 and Ch 5

Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve

### ECO 317 Economics of Uncertainty Fall Term 2007 Notes for lectures 4. Stochastic Dominance

Generl structure ECO 37 Economics of Uncertinty Fll Term 007 Notes for lectures 4. Stochstic Dominnce Here we suppose tht the consequences re welth mounts denoted by W, which cn tke on ny vlue between

### (0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35

7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know

### Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

### Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

### Distance And Velocity

Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl

### 3.4 Numerical integration

3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,

### Integrals along Curves.

Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the

### Line and Surface Integrals: An Intuitive Understanding

Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of

### Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

### Section 14.3 Arc Length and Curvature

Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

### 7 - Continuous random variables

7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin

### Motion. Acceleration. Part 2: Constant Acceleration. October Lab Phyiscs. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.

Motion ccelertion Prt : Constnt ccelertion ccelertion ccelertion ccelertion is the rte of chnge of elocity. = - o t = Δ Δt ccelertion = = - o t chnge of elocity elpsed time ccelertion is ector, lthough

### Ordinary Differential Equations- Boundary Value Problem

Ordinry Differentil Equtions- Boundry Vlue Problem Shooting method Runge Kutt method Computer-bsed solutions o BVPFD subroutine (Fortrn IMSL subroutine tht Solves (prmeterized) system of differentil equtions

### This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

### The solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr

Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the e-vectors nd e-vlues

### THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS

THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS CARLOS SUERO, MAURICIO ALMANZAR CONTENTS 1 Introduction 1 2 Proof of Gussin Qudrture 6 3 Iterted 2-Dimensionl Gussin Qudrture 20 4