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17 Pages

### Supplement

Course: MATH 367, Spring 2011
School: Middle East Technical...
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The Supplement: Long Exact Homology Sequence and Applications S1. Chain Complexes In the supplement, we will develop some of the building blocks for algebraic topology. As we go along, we will make brief comments [in brackets] indicating the connection between the algebraic machinery and the topological setting, but for best results here, please consult a text or attend lectures on algebraic topology. S1.1...

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Middle East Technical University - MATH - 367
Math 420, Point Set TopologySuggested exercises 1Problem 1. Consider the following collection of subsets of RB = cfw_B : R\Bis nite(a) Show that B is a basis for topology on R.(b) Determined if the generated topological space is Hausdor.Problem 2.
Middle East Technical University - MATH - 367
Math 420, Point Set TopologySuggested exercises 2Problem 1. Let be the uniform metric on R . Given x = cfw_xn R and (0, 1), letU (x, ) = (x1 + , x1 ) (xn + , xn ) . . .a. Show that U (x, ) is not equal to the -ball B (x, ).b. Show that U (x, ) is n
Middle East Technical University - MATH - 367
Math 420, Point Set TopologySuggested exercisesProblem 1. Let I and I be two topologies on X . Show that if X is acompact Hausdor space under both I and I , then either I = I or theyare not comparable.Problem 2. Is the interval [0, 1] compact in the
Middle East Technical University - MATH - 367
Abstract Algebra done ConcretelyDonu ArapuraFebruary 19, 2004Introduction: I wrote these notes for my math 453 class since I couldntnd a book that covered basic abstract algebra with the level and emphasis thatI wanted. Rather than spending a lot of
Middle East Technical University - MATH - 367
Elementary Abstract AlgebraW. Edwin ClarkDepartment of MathematicsUniversity of South Florida(Last revised: December 23, 2001)Copyright c 1998 by W. Edwin ClarkAll rights reserved.iiiPrefaceThis book is intended for a one semester introduction t
Middle East Technical University - MATH - 367
Incomplete Notes onFraleighs Abstract Algebra (4th ed.)Afra ZomorodianDecember 12, 2001Contents1 A Few Preliminaries1.1 Mathematics and Proofs . . . . .1.2 Sets and Equivalence Relations .1.3 Mathematical Induction . . . . .1.4 Complex and Matrix
Middle East Technical University - MATH - 367
ABSTRACT ALGEBRA:A STUDY GUIDEFOR BEGINNERSJohn A. BeachyNorthern Illinois University20062This is a supplement toAbstract Algebra, Third Editionby John A. Beachy and William D. BlairISBN 1577664344, Copyright 2005Waveland Press, Inc.4180 IL Ro
Middle East Technical University - MATH - 367
ABSTRACT ALGEBRA:A STUDY GUIDEFOR BEGINNERSJohn A. BeachyNorthern Illinois University2000iiThis is a supplement toAbstract Algebra, Second Editionby John A. Beachy and William D. BlairISBN 0881338664, Copyright 1996Waveland Press, Inc.P.O. Box
Middle East Technical University - MATH - 367
Jonathan BergknoHerstein SolutionsChapters 1 and 2Throughout, G is a group and p is a prime integer unless otherwise stated. A B denotes that A is asubgroup of B while AB denotes that A is a normal subgroup of B .H 1.3.14* (Fermats Little Theorem) P
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G. Evans, J. Blackledge and P. YardleyNumerical Methodsfor Partial DifferentialEquationsWith 55 FiguresSpringerContents1.B ackground Mathematics11.1 Introduction11.2 Vector and Matrix Norms31.3 Gerschgorin's Theorems71.4 Iterative Solution
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Math 480Spring 2011Homework IDue date, March 14th, 2011Chapter 3: The Taylor series methodChapter 4: Linear Multistep Methods-I: Construction and ConsistencyChapter 5 Linear Multistep MethodsII: Convergence and Zero-Stability1. pp. 41, Exercise 3.8
Middle East Technical University - MATH - 480
G1BINMIntroduction to Numerical Methods717 Iterative methods for matrixequations7.1The need for iterative methodsWe have seen that Gaussian elimination provides a method for nding the exact solution(if rounding errors can be avoided) of a system o
Middle East Technical University - MATH - 480
Crash course in MATLABc Tobin A. Driscoll. All rights reserved. June 23, 2006The purpose of this document is to give a medium-length introduction to the essentials of MATLAB and how to use it well. Im aiming for a document thats somewhere between a twop
Middle East Technical University - MATH - 480
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MATH480Numerical Solution of Differential EquationsMIDTERM EXAMApril, 27st,20111. (10 poinis) Given hat r is a positi'e real number. consider the jineartwo-step method'n_2x. -J 4,- : h &quot; * -Af .*f &quot;)1U&quot;-zQInDetermine the set of all cu such th
Middle East Technical University - MATH - 480
F1.4ZH2 Numerical Solution of PDEs1.4Page 7Taylor series and dierence operatorsWe can use Taylor series expansions in one of the variables to see how well Finite Dierence(FD) approximations work. Consider the approximation (u(x + x, t) u(x, t)/x to u
Middle East Technical University - MATH - 480
F1.4ZH2 Numerical Solution of PDEs2.6.1Page 23LTE analysis of the -methodThe PDE ut = uxx is approximated on a uniform grid of size x = 1/N in space and t intime, with approximate solutionnwj u(xj , tn ) ,xj = x0 + j x , tn = nt,where u(x, t) is
Middle East Technical University - MATH - 480
F1.4ZH2 Numerical Solution of PDEsPage 15LTEs (continued)The rst term on the right of the LTE for the FTCS scheme is referred to as the leading termof the local truncation error (LTE): for this scheme it is rst order accurate in time and 2ndorder in
Middle East Technical University - MATH - 373
Middle East Technical University - MATH - 373
Middle East Technical University - MATH - 373
Middle East Technical University - MATH - 373
Problems in Geometry (1)1. In a triangle ABC with orthocenter H and circumcircle O prove that&lt; BAH =&lt; OAC .12. Let ABC be a triangle with orthocenter H in which AH, BH, CH meet BC, CA, ABin D, E, F respectively. Prove thatHA HD = HB HE = HC HF.23.
Middle East Technical University - MATH - 373
Problems in Geometry (6)1. What is the locus of a point whereof the power with respect to a xed circle is aconstant ?2. Compute the power of the point P (x0 , y0 ) with respect to the circle x2 + y 2 +2ax +2by +c = 0. Write down the equation of the ra
Middle East Technical University - MATH - 373
Problems in Geometry (7)1. Let be an ellipse with foci F, F . If the tangent lines at A, B intersect in P ,prove that P F AB i F AB . 12. Let be an ellipse with foci F, F . Let t, t be the tangents to which intersect in P .If H, H are respectively the
Middle East Technical University - MATH - 373
Problems in Geometry (2)1. Given a rectangle ABCD prove that |P A|2 + |P C |2 = |P B |2 + |P D|2 for any point P.2. Given an acute angled positively oriented triangle ABC, let O and H be the circumcenter and the orthocenter of ABC, respectively. Let AH
Middle East Technical University - MATH - 373
Problems in Geometry (3)1. Given a triangle ABC, let A be the midpoint of [B, C ] and consider Y CA cfw_C, A,Z AB cfw_A, B such that BY and CZ meet on AA . Prove that Y Z is parallel to BC.12. Given a triangle ABC, consider P BC cfw_B, C , Q CA cfw_C,
Middle East Technical University - MATH - 373
Problems in Geometry (4)1. Consider a quadrangle ABCD with points P, Q, R, S, T on BD, AB, AD, CB, CDrespectively. Prove that if P, Q, R are collinear and P, S, T are collinear, then the lines1QS, RT, AC are concurrent.2. The following is a problem b
Middle East Technical University - MATH - 373
Problems in Geometry (5)1. Given a circle , consider points P, C, D and let [A, B ] be a diameter of . Provethat the Simson line of P with respect to the triangle ACD is perpendicular to the Simsonline of P with respect to the triangle BCD.12. Given a
Middle East Technical University - MATH - 373
Problems in Geometry (8)1. (A) What is the image of the line 2x + y = 2 under Tr[1,3] ?(B) What is the image of the line x = 1 under Rot(0, 0), /4) ?(C) What is the image of the line x + y = 1 under Refk , where k is the line y = 2x ?12. (A) Let P, Q
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Middle East Technical University - MATH - 420
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