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UCSD >> MATH >> 130a (Fall, 2008)
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UCSD >> MATH >> 130a (Fall, 2008)
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UCSD >> MATH >> 130a (Fall, 2008)
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UCSD >> MATH >> 130a (Fall, 2008)
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UCSD >> MATH >> 130a (Fall, 2008)
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UCSD >> MATH >> 130a (Fall, 2008)
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UCSD >> MATH >> 130a (Fall, 2008)
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UCSD >> MATH >> 130a (Fall, 2008)
C u q y C i U C U G Ed ud E q E Gd Q u q C q G E R q bd G i G y R E q Ed ud d f i C U G G q U G E i i 44rYmYPhx4ht4r74&rP4WSH4rrP4r7Por\'p Gd C f i C SSe4eU PkD4Pv\'7r1d F7Pm5r\'14esrPD i fp Q u q u id G b u Ip i u I y qd q E G q U G E ...
UCSD >> MATH >> 130a (Fall, 2008)
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UCSD >> MATH >> 130a (Fall, 2008)
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UCSD >> MATH >> 130a (Fall, 2008)
Q U E Q T g d Q T b P U A P C H d E Y d g Y C i Y d U G Y d C a TP C s d uFVViRVVlDme`8iFVu8RQ G 58ig x G p #x G p Vuu8 7 x G p 88n 0p x iPY y x s sq ftvutrp nYC QC g d Q UEPYC Q d H C U x QC 8`FVk7d `uFVkRoFVQ us kV x G h...
UCSD >> MATH >> 130a (Fall, 2008)
@Vw4m@5@HE1sHVGHFd4d k ve 5YH4rVaY5yfwyHr5RY G w Gs G C E C w y m o l s G l y s y E l G E s y YHoPH5yHY4r5Y4sH\"5wE k H5G fHY5f@5w k H o o y o o y y w w e w w G y o G G o y w w e k wThG d k ve k He ...
UCSD >> MATH >> 130b (Winter, 2008)
Homework 7 solutions 7.5.3 We have x + k(x2 4)x + x = 1 with k 1. This is very similar to the van der Pol equation and so we will mimic the analysis done for that. First we parametrize by letting F (x) = 1 x3 4x so that if we let w = x+kF (x), the...
UCSD >> MATH >> 130b (Winter, 2008)
Math 130B Prof. Rabin Homework Assignment 6 due February 19, 2008 (1) Consider the system (a chemical reaction model) x = (1/8) x + x2 y, y = (1/2) x2 y. Show that the set {(x, y) : x 1/16, 0 y 128, x + y 130} is positively invariant and ...
UCSD >> MATH >> 130b (Winter, 2008)
Homework 2 solutions 6.6.1 To check if the system is reversible we replace y and x by y and x respectively (i.e., replace t by t and y by y, note that y thus becomes ()()y = y); if it reduces to what we started with then it is reversible. For ou...
UCSD >> MATH >> 130b (Winter, 2008)
Homework 1 solutions 6.4.1 The xed points will occur when x = 0 and y = 0. So at the xed point we have that [either x = 0 or x + y = 3] and [either y = 0 or x + y = 2]. This leads to four possible combinations x = 0, y = 0 gives (x , y ) = (0, 0);...
UCSD >> MATH >> 130b (Winter, 2008)
Solutions to Midterm 1 By H akan Nordgren As always, the sketches are all in a separate pdf. Problem 1: Consider the non-linear system x y = x2 + y = xy 1. Locate the equilibrium points and use linearisation to determine their type. 2. Draw the null...
UCSD >> MATH >> 130b (Winter, 2008)
Solutions to Homework 8 By H akan Nordgren Problem 12.1: Draw the phase-portrait of the following system. x = y x2 y = x. Solution: The x-nullcline is the curve C1 = {(x, y) R2 : y = x2 }, and the y-nullcline is the curve C2 = {(x, y) R2 : x = ...
UCSD >> MATH >> 130b (Winter, 2008)
Solutions to Homework 5 By H akan Nordgren REMARK: I am still into x notation. Also, please note that these solutions are not complete. If you have any suggestions about how to make them more solid, please feel free to email me. Problem 10.4: Consid...
UCSD >> MATH >> 130b (Winter, 2008)
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UCSD >> MATH >> 130b (Winter, 2008)
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UCSD >> MATH >> 130b (Winter, 2008)
e Xv}H}HvAo}v}H{5oASxS6}!dScx! k k G e e c}6oA}AAv}fvAo}vmSruS6SP}XvxI8}\"Pvh}Ao96fv} e ...
UCSD >> MATH >> 130b (Winter, 2008)
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UCSD >> MATH >> 130b (Winter, 2008)
! h p U Vb V ab h Vb h U h V ` P2wdHdYdqqY0x v u P!6 6gng0ymd0wu v Pgd d ck 4n02YH!Ha ` b r e p u x a b x d k p h p X V ` t t p e p u Vb g0Dd0x v qq!iq6H02sP2p u P vP Pk P! H6md0wu x h u p t r e Ub h ` p X ` h p U u V ` ...
UCSD >> MATH >> 130b (Winter, 2008)
Homework 5 solutions 7.3.1 1 a) The Jacobian at the origin is 1 1 , which has a characteristic equation 2 1 2 + 2. So the eigenvalues are 1 + i and 1 i which would make this point an unstable spiral. b) We have rr = xx + y y = x x y x(x2 + 5y 2 ...
UCSD >> MATH >> 130b (Winter, 2008)
2 vertices 1, 2 3 vertices 1, 3 3 vertices 2, 3 4 vertices 1, 4 4 vertices 3, 4 4 vertices 2, 3 5 vertices 1, 5 1.5 1.5 1.75 1.80902 1.75 1.80902 1.80902 1.5 0.690983 1 1 0 0 0 0 1.25 0.690983 1.25 1.5 1 0.690983 0.5 0 0 0 5 ver...
UCSD >> MATH >> 130b (Winter, 2008)
...
UCSD >> MATH >> 130b (Winter, 2008)
Using discrepancy to control singular values for nonnegative matrices Steve Butler Abstract We will consider two parameters which can be associated with a nonnegative matrix: the second largest singular value of the normalized matrix, and the discre...
UCSD >> MATH >> 130b (Winter, 2008)
A property of positive semidenite matrices Steve Butler Recall that a matrix S is positive denite if for all x = 0 x Sx > 0 and positive semidenite if x Sx 0. For symmetric matrices being positive denite is equivalent to having all eigenvalues posi...
UCSD >> MATH >> 130b (Winter, 2008)
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UCSD >> MATH >> 130b (Winter, 2008)
Homework 5 Due November 26 * 5.2.4, 5.2.5, 5.2.6, 5.2.7, 5.2.8 (the same problems as last time, but this time follow the books instructions: use Tr and Det to classify the xed point, and sketch the phase portrait.) * 5.2.11 * Solve each of the thre...
UCSD >> MATH >> 130b (Winter, 2008)
Student comments Math 10A, Winter 2007 The following are the student comments about my teaching from CAPE (Course And Professor Evaluation) forms for the Math 10A (beginning calculus) course I taught in Winter 2007 at UC San Diego. These comments ar...
UCSD >> MATH >> 130b (Winter, 2008)
Jumping Sequences Steve Butler Ron Graham Nan Zang Abstract An integer sequence a(n) is called a jump sequence if a(1) = 1 and 1 a(n) < n for n 2. Such a sequence has the property that ak (n) = a(a( (a(n) ) goes to 1 in nitely many steps and ...
UCSD >> MATH >> 130b (Winter, 2008)
Midterm 1 solutions (1) Consider the dierential equation x = x4 . (a) If we specify the value of x(0), is the solution unique? How do you know? Solution: Since f (x) = x4 and f (x) = 4x3 are continuous in some interval around 0 then the existence an...
UCSD >> MATH >> 130b (Winter, 2008)
Generalizing some results to the normalized Laplacian Steve Butler September 6, 2006 1 Introduction Our graph terminology is standard, any undened terms can be found in standard graph theory texts, such as West [9]. We assume that our graphs are s...
UCSD >> MATH >> 130b (Winter, 2008)
Anti-covers of double edge coverings of graphs Steve Butler May 5, 2007 1 Introduction Spectral graph theory looks at the connections between the structure of a graph and the eigenvalues of various matrices associated with the graph. By examining ...
UCSD >> MATH >> 130b (Winter, 2008)
Tangent Line Transformations Steven Butler Steven Butler (butler@math.byu.edu) graduated from Brigham Young University with bachelors degrees in mathematics and economics in 2001. He is currently nishing his Masters degree in mathematics at BYU and ...
UCSD >> MATH >> 130b (Winter, 2008)
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UCSD >> MATH >> 130b (Winter, 2008)
...
UCSD >> MATH >> 130b (Winter, 2008)
Homework 4 solutions 5.2.1 a) Starting with x = 4x y and y = 2x + y we have x= x y = 4x y 2x + y = 4 1 21 =A x y = Ax. The characteristic polynomial is then given by det(A I) = det 4 1 2 1 = (4 )(1 ) (2)(1) = 2 5 + 6. The eigenvalue...
UCSD >> MATH >> 130b (Winter, 2008)
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UCSD >> MATH >> 130b (Winter, 2008)
TANGENT LINE TRANSFORMATIONS: OR THERE AND BACK AGAIN STEVEN K. BUTLER Introduction In the rst semester of Calculus students learn to take a curve and nd all of the curves tangent lines. Now consider the converse problem, if given all of the curves ...
UCSD >> MATH >> 130b (Winter, 2008)
Homework 4 Due February 5 * 7.2.12, 7.2.13 (all parameters are positive), 7.2.14, 7.2.15, 7.2.17. * Show that the Jacobian matrix for a 2d gradient system is symmetric. Show that a 22 real symmetric matrix has real eigenvalues. Therefore, no xed poi...
UCSD >> MATH >> 130b (Winter, 2008)
Student name: Student PID: MATH 10A (Butler) Midterm 2, 5 March 2007 This test is closed book and closed notes, with the exception that you are allowed one 8 1 11 page of handwritten notes. You may use any shortcuts for derivatives 2 unless explicitl...
UCSD >> MATH >> 130b (Winter, 2008)
Hat Guessing Games Steven Butler Mohammad T. Hajiaghayi Tom Leighton Robert D. Kleinberg Abstract Hat problems have become a popular topic in recreational mathematics. In a typical hat problem, each of n players tries to guess the color of the ha...
UCSD >> MATH >> 130b (Winter, 2008)
Lecture notes from IPCO Summer School 2005 Fast Randomized Algorithms for Partitioning, Sparsication, and Solving Linear Systems Daniel A. Spielman Contents 1 Disclaimer 2 Overview 3 Solving Linear Systems 3.1 3.2 3.3 Direct Methods . . . . . . . . ...
UCSD >> MATH >> 130b (Winter, 2008)
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UCSD >> MATH >> 130b (Winter, 2008)
...
UCSD >> MATH >> 130b (Winter, 2008)
Homework 3 solutions 3.7.2 a) The graphs of r(x) = 2x3 (1 + x2 )2 and k(x) = 2x3 x2 1 are shown below. Note that since the derivatives of these functions are r (x) = 2x2 (3 x2 ) (1 + x2 )3 and k (x) = 2x2 (x2 3) (x2 1)2 respectively, it is simp...
UCSD >> MATH >> 130b (Winter, 2008)
Homework 6 solutions (1) (a) We have that the characteristic equation for the matrix is det 2 2 2 1 = (2 )(1 ) 4 = 2 6 = ( 3)( + 2), so that the eigenvalues are 3 and 2. To nd an eigenvector for 3 we can take a column of A + 2I, so 2 , and ...
UCSD >> MATH >> 130b (Winter, 2008)
BOUNDING THE NUMBER OF GRAPHS CONTAINING VERY LONG INDUCED PATHS by Steven Kay Butler A thesis submitted to the faculty of Brigham Young University in partial fulllment of the requirements for the degree of Master of Science Department of Mathema...
UCSD >> MATH >> 130b (Winter, 2008)
Homework 8 solutions 7.6.19 a) By setting = t we have that x= d2 (x) d2 (x) d2 (x) = 2 = 2x . = d(t2 ) d( /)2 d( )2 So the Dung equation x + x + x3 = 0 becomes 2 x + x + x3 = 0. b) We let x(, ) = x0 ( ) + x1 ( ) + 2 x2 ( ) + O(3 ) and = 1 + 1 +...
UCSD >> MATH >> 130b (Winter, 2008)
The Clique Number of the Graph of Pairwise Sums and Products is 3 or 4 Jacob Fox, Yeu-Whai Kathy Lin, and Matthew Thibault licht@mit.edu, ykl@mit.edu, matt tbo@mit.edu Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 0213...
UCSD >> MATH >> 130b (Winter, 2008)
Math 130A Prof. Rabin Homework #0 due October 4, 2007 These problems will help you review some concepts from Math 20D and 20F. They will not be graded, but you should be prepared to discuss them at the rst section meeting. (1) Solve by separation ...
UCSD >> MATH >> 130b (Winter, 2008)
UNIVERSITY OF CALIFORNIA, SAN DIEGO Eigenvalues and Structures of Graphs A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Steven Kay Butler Committee in charge: Professor Prof...
UCSD >> MATH >> 130b (Winter, 2008)
Midterm 2 solutions (1) Consider the system x = x2 y, where a is a real parameter. (a) For what critical value a = ac does a bifurcation occur? Solution: Looking for bifurcations we can rst look for xed points. This will occur when x2 y = 0 and x ...
UCSD >> MATH >> 130b (Winter, 2008)
Spectral graph theory: Three common spectra Steve Butler September 2006 Abstract In this rst talk we will introduce three of the most commonly used types of matrices in spectral graph theory. They are the adjacency matrix, the combinatorial Laplacia...
UCSD >> MATH >> 130b (Winter, 2008)
Teaching Statement Steven Kay Butler Teaching requires a balancing between giving students enough knowledge and interest so that they have the abilities and interest to learn the material but not losing them in minutiae by trying to explain too much ...
UCSD >> MATH >> 130b (Winter, 2008)
A new discrepancy denition for hypergraphs Steve Butler 1 Discrepancy on simple graphs For an undirected graph we let A denote the adjacency matrix (i.e., given vertices u and v we have that Auv = 1 if u and v are adjacent and 0 otherwise), and D ...
UCSD >> MATH >> 130b (Winter, 2008)
...
UCSD >> MATH >> 130b (Winter, 2008)
Student name: Student PID: MATH 10A (Butler) Midterm I, 29 January 2007 This test is closed book and closed notes, with the exception that you are allowed 1 one 8 2 11 page of handwritten notes. No calculator is allowed for this test. When answering ...
UCSD >> MATH >> 130b (Winter, 2008)
Enumerating (multiplex) juggling sequences Steve Butler Ron Graham Abstract We consider the problem of enumerating periodic -juggling sequences of length n for multiplex juggling, where is the initial state (or landing schedule) of the balls. We rs...
UCSD >> MATH >> 130b (Winter, 2008)
How to play the majority game with a liar 1 Steve Butler a, , Jia Mao b,2 , Ron Graham b,3 a Dept. b Dept. of Mathematics, UC San Diego, La Jolla, CA 92093-0112, USA of Computer Science and Engineering, UC San Diego, La Jolla, CA 92093-0404, USA ...
UCSD >> MATH >> 132a (Spring, 2008)
Lecture 12: Poissons formula. We will now derive a formula for the solution of the Dirichlet problem for the unit disc: uxx + uyy = 0, for x2 + y 2 < 1, u = h(), when x2 + y 2 = 1, and is the angular variable in polar coordinates in the plane. It i...
UCSD >> MATH >> 132a (Spring, 2008)
Lecture 10: Diusion equation. We will now derive the fundamental solution for the diusion equation in three space dimensions: (10.1) We claim that (10.2) u(x, t) = 1 (4kt)3/2 exp |x y|2 f (y) dy = 4kt S3 (x y, t)f (y) dy, ut = k u = k uxx + uyy + ...
UCSD >> MATH >> 132a (Spring, 2008)
Lecture 9: 9.1 Invariance of the wave operator under Lorentz transformations and 9.3 The wave equation with a Source. 6.1 Invariance of the Laplacian under rotations. A rotation or orthogonal transformation Rn x Qx Rn , is multiplication by an orth...
UCSD >> MATH >> 132a (Spring, 2008)
Lecture 11: 6.1 Laplace equation. Laplace equation: (11.1) u = 0. A function satisfying Laplace equation is called harmonic. The inhomogeneous version of Laplace equation is usually called Poissons equation: (11.2) u = f. Dirichlet problem in a dom...
UCSD >> MATH >> 132a (Spring, 2008)
Lecture 20: 12.3 Fourier transform. Lemma. If S set (x) = (x/)/n , then f (x) (x) dx = f (x) (x) dx f (0) (x) dx 0, for f S Proof. Since |f (x) (x)| supy |f (y)| |(x)| the lemma follows from the theorem of Dominated converge. It is also easy...
UCSD >> MATH >> 132a (Spring, 2008)
Math 132A Partial Differential Equations Lecture 1: Introduction. Notation Let u (t, x) denote a function of time t and n space variables x = (x1 , . . . , xn ) or x, y, z. u Let xiu = , or ux , uy , uz be the partial derivative of u with respect to ...
UCSD >> MATH >> 132a (Spring, 2008)
Lecture 17: 12.1 Distributions. If f is a distribution and u is a smooth function then the product uf is dened by uf, = f, u . Multiplication of distributions is however not always dened. E.g. we cant multiply (x) with itself. In fact, if (x) = (x/...
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