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### homework_3

Course: MATH 170, Fall 2008
School: UPenn
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170) 1 (Math Homework 3: Due October 2, 2008 Exercise 1: Prove there is no largest odd number (hint use proof by contradiction) Exercise 2: Find the greatest common divisor of 7920 and 8316 (hint 7920 = 24 32 5 11 and 8316 = 22 33 7 11) Exercise 3: Use Euclids Algorithm to nd the greatest common divisor of 308 and 238 Exercise 4: Use Euclids Algorithm to nd numbers x and y such 308 that x + 238 y =...

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170) 1 (Math Homework 3: Due October 2, 2008 Exercise 1: Prove there is no largest odd number (hint use proof by contradiction) Exercise 2: Find the greatest common divisor of 7920 and 8316 (hint 7920 = 24 32 5 11 and 8316 = 22 33 7 11) Exercise 3: Use Euclids Algorithm to nd the greatest common divisor of 308 and 238 Exercise 4: Use Euclids Algorithm to nd numbers x and y such 308 that x + 238 y = gcd(308, 238) Exercise 5: Heart of Mathematics Chapter 2.3 Exercise 21 Exercise 6: Heart of Mathematics Chapter 2.3 Exercise 23 Exercise 7: Heart of Mathematics Chapter 2.3 Exercise 24 Exercise 8: Heart o...

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UPenn - MATH - 170
1(Math 170) Homework 9:Due November 20, 2008Exercise 4: Heart of Mathematics Chapter 2.2 Exercise 7 Exercise 4: Heart of Mathematics Chapter 2.2 Exercise 8 Exercise 5: Heart of Mathematics Chapter 2.2 Exercise 10 Exercise 6: Heart of Mathematics
UPenn - MATH - 10
1Denition of Real NumbersThe goal of the presentation is to discuss dierent denitions of the real numbers. Your presentation should include at least An explanation of what the Cauchy Sequences are. An explanation of what Dedekind cuts are. A di
UPenn - MATH - 170
1Denition of Real NumbersThe goal of the presentation is to discuss dierent denitions of the real numbers. Your presentation should include at least An explanation of what the Cauchy Sequences are. An explanation of what Dedekind cuts are. A di
UPenn - MATH - 170
1Algebraic and Transcendental NumbersThe goal of the presentation is to discuss logarithms and exponentials. Your presentation should include at least A discussion of what an Algebraic Number is A discussion of what a Transcendental Number is A
UPenn - MATH - 170
1August Mobius and Mobius StripsThe goal of the presentation is to discuss August Mobius. Your presentation should include at least A discussion of who Mobius was and his life. An explanation of what a Mobius strip is. A discussion of what it m
UPenn - MATH - 170
1Primeality Tests and Finding PrimesThe goal of the presentation is to discuss methods to test if a number is prime and for nding prime numbers. Your presentation should include at least A discussion of the Sieve of Eratosthenes An explanation a
UPenn - MATH - 12
1John Horton ConwayThe goal of the presentation is to discuss John Conway. Your presentation should include at least A discussion his life. A Conway chained arrow notation. A discussion of what it means to be non-associative. A discussion of t
UPenn - MATH - 170
1John Horton ConwayThe goal of the presentation is to discuss John Conway. Your presentation should include at least A discussion his life. A Conway chained arrow notation. A discussion of what it means to be non-associative. A discussion of t
UPenn - MATH - 170
1Zenos ParadoxesThe goal of the presentation is to discuss Zenos Paradoxes. Your presentation should include at least A discussion of who Zeno was. A discussion of Zenos three paradoxes, including a discussion of what they are and how they are r
UPenn - MATH - 170
1Carl Friedrich GaussThe goal of the presentation is to discuss Carl Friedrich Gauss. Your presentation should include at least A discussion of who he is. A discussion of the type of polygons he was able to construct using just a compass and str
UPenn - MATH - 12
1Cellular AutomataThe goal of the presentation is to discuss Cellular Automata. Your presentation should include at least A discussion of what cellular automata is. A discussion of the game of life (by Conway) An explanation of a 1 dimensional
UPenn - MATH - 170
1Cellular AutomataThe goal of the presentation is to discuss Cellular Automata. Your presentation should include at least A discussion of what cellular automata is. A discussion of the game of life (by Conway) An explanation of a 1 dimensional
UPenn - MATH - 170
1Blaise PascalThe goal of the presentation is to discuss Blaise. Your presentation should include at least A discussion of who he is and his religious/philosophical beliefs. A discussion of how him and Fermat discovered probability. A discussio
UPenn - MATH - 170
1Amalie Emmy NoetherThe goal of the presentation is to discuss Amalie Emmy Noether. Your presentation should include at least A discussion of who Amalie Emmy Noether was including a brief history of her life. A discussion of what an ascending/de
UPenn - MATH - 170
1Sir Isaac NewtonThe goal of the presentation is to discuss Isaac Newton. Your presentation should include at least A discussion of who Newton was. A discussion of his life. A discussion of his laws of gravity A brief discussion of his discove
UPenn - MATH - 170
1Peano ArithmeticThe goal of the presentation is to discuss Peano arithmetic. Your presentation should include at least A discussion of the the axioms of Peano arithmetic. Its relationship with induction (which axiom(s) allows proof by induction
UPenn - MATH - 170
1Vigenre cipherThe goal of the presentation is to explain and discuss the Vigenre cipher. Your presentation should include at least A discussion the history of the cipher. An explanation of how it is implemented (both in terms of the Vigenre tab
UPenn - MATH - 170
1Rene DescartesThe goal of the presentation is to discuss Rene Descartes. Your presentation should include at least A discussion of who he is. A discussion analytic geometry A discussion of the dualism In addition you should include at least on
UPenn - MATH - 12
1Bertrand RussellThe goal of the presentation is to discuss Blaise. Your presentation should include at least A discussion of who he. A discussion of Russells Paradox A discussion of the Principia Mathematica. In addition you should include at
UPenn - MATH - 170
1Bertrand RussellThe goal of the presentation is to discuss Blaise. Your presentation should include at least A discussion of who he. A discussion of Russells Paradox A discussion of the Principia Mathematica. In addition you should include at
UPenn - MATH - 170
1Seven Bridges of Knigsberg oThe goal of the presentation is to discuss Seven Bridges of Knigsberg. Your o presentation should include at least A discussion of the problem of the Seven Bridges of Knigsberg. o Which two elds did this problem help
UPenn - MATH - 170
1Leonhard EulerThe goal of the presentation is to discuss the Leonhard Euler. Your presentation should include at least A discussion of who he is (including what time period he lived in) A discussion of his major contributions, including the Eul
UPenn - MATH - 170
1(Math 170) Homework 4:Due October 9, 2008For a natural number n a multiplication table mod n consists is a grid such that The rows and columns are each labeled with numbers 0 to n 1 The number in the ath column and the bth row is a b mod n.
UPenn - MATH - 170
1(Math 170) Homework 7:Due November 6, 2008Exercise 1: What is 1000 base 10 equal to base 8? What is it equal to base 4? What is it equal to base 2? Exercise 2: Heart of Mathematics Chapter 2.6 Exercise 8 Exercise 3: Heart of Mathematics Chapter
UPenn - MATH - 170
15)ConsiderthediscretedynamicalsystemgivenbyPn+1=2Pn1Whataretheequilibriumsandaretheystableorunstable? Thefollowingisthegraphof y=x y=2x1a)Thestableequilibriumsare:x=0,x=1.Therearenounstableequilibriums. b)Thestableequilibriumis:x=1.Theunstablee
UPenn - MATH - 170
Math 170 Final Formula SheetDecember 6, 200811 MATHEMATICAL MODELS21Mathematical ModelsLet Mn+1 = f (Mn ) be a mathematical model (without the starting value specied). Denition 1.0.1. M is a equilibrium if M = f (M ). An equilibrium M i
UPenn - MATH - 170
1Instructions Write all answers in capital letters in the spaces provided on the next page! Do not remove this answer sheet from the rest of the exam. All problems are worth the same amount. Problems marked with (EC) are extra credit. There is
UPenn - MATH - 170
1 Math 170 Practice Final Exam 2NameSection Number2 (1) Suppose A is a symbol which represents ten and B is a symbol which represents eleven. What is A3B base 12 expressed base 4? (2) What is 443.21 base 5 expressed base 10? (3) What is 323.323
UPenn - MATH - 170
1 Math 170 Practice Final Exam 3NameSection Number2 (1) Suppose A is a symbol which represents ten and B is a symbol which represents eleven. What is AAA base 11 expressed base 6? (2) What is 555.55 base 6 expressed base 10? (3) What is 123.123
UPenn - MATH - 170
1 Math 170 Practice Final Exam 1NameSection Number2 (1) Suppose A is a symbol which represents ten and B is a symbol which represents eleven. What is B9A base 12 expressed base 7? (2) What is 123.45 base 8 expressed base 10? (3) What is 123.123
UPenn - MATH - 170
16) Consider the discrete dynamical system given byPn+1 = 2Pn - Pn2What are the equilibrium and are they stable or unstable? The following is the graph of y =x y=2x -x 247) Below is an outline of the Mandelbrot Set. The points marked are: A = -0
UPenn - MATH - 170
15) Consider the discrete dynamical system given byPn+1 = 2What are the equilibrium and are they stable or unstable from the left and right? The following is the graph of y =x y=216) Below is an outline of the Mandelbrot Set. The points marked a
UPenn - MATH - 170
15) Consider the discrete dynamical system given byPn+1 = Pn 2What are the equilibriums and are they stable or unstable? The following is the graph of y =x y= y=x 216) Below is an outline of the Mandelbrot Set. The points marked are: A = -1.25 i
UPenn - MATH - 170
1 Math 170 Homework 10 Due December 4, 2008Name TA: Section Number:2 (1) Suppose A is a symbol representing ten. What is AA1 base 11 expressed base 4? (2) What is 123.12 base 4 in base 10? (3) Express 102.102 base 3 as a fraction (in lowest terms
UPenn - MATH - 170
1Straightedge and CompassThe goal of the presentation is to discuss constructions with a strait edge and compass. Your presentation should include at least A discussion what a straightedge and compass are. A discussion of what a straightedge and
UPenn - M - 584
# &gt; restart;&gt; #In this worksheet we use maple to compute some Fourier transform and&gt; graph the results. We can define a function as a integral. The&gt; function defined in the next line is the Fourier transform of the&gt; characteristic function of th
UPenn - M - 584
# # &gt; restart;&gt; #This worksheet is intended to give you some feeling of how to use&gt; convolution to approximately compute derivatives. In the next command&gt; replace sin(x) with the function of your choice. Use a function for&gt; which maple can comp
UPenn - M - 584
# # &gt; restart;&gt; #Here we define several functions with support in the interval [-1,1].&gt; f0 := x-&gt;piecewise(x&gt;-1 and x&lt;1, 1, 0);&gt; f1 := x-&gt; piecewise(x&gt;-1 and x&lt;1, (1-x^2), 0);&gt; f2 := x-&gt; (f1(x)^2;&gt; #We plot the functions defined above. f0 has
UPenn - M - 584
# This routine uses the fast fourier transform to compute the DFT. The input is a # vector [fh[1], .fh[n] of samples and the length, n of this vector. The routine finds # the smallest power of 2 larger than n and computes a FFT of that length with
UPenn - M - 509
Math 509 Problem set 3, due February 2, 2006 Dr. EpsteinIn class we will discuss material on pages 44-58 of Fourier Analysis. The following are standard or review problems. Please be sure you know how to do them. You do not need to hand in solutions
UPenn - M - 584
Math 584, Problem set 4 due November 7, 2006 Dr. EpsteinReading: Read sections 5.1, 5.2, 5.3, and 6.1. You are free to use maple (or Mathematica, MATLAB etc.) to do these problems. Try running maple worksheets 4, and 5. If your browser does not auto
UPenn - M - 584
Math 584, Problem set 1 due September 26, 2006 Dr. EpsteinReading: Read chapters 1 and 2 of the textbook. You may use Maple (or Mathematica, MATLAB etc.) to do these problems. If you do, then please attach the output of the program to your solutions
UPenn - CIT - 597
RailsApr 10, 2009What is Rails? Rails is a framework for building web applications This involves: Getting information from the user (client), using HTML forms Doing validation Maintaining session information Managing a database Displayin
UPenn - CIT - 597
XPathApr 10, 2009What is XPath? XPath is a syntax used for selecting parts of an XML document The way XPath describes paths to elements is similar to the way an operating system describes paths to files XPath is almost a small programmin
UPenn - CIT - 594
SearchingSearching an array of integersIf an array is not sorted, there is no better algorithm than linear search for finding an element in itstaticfinalintNONE=1;/notalegalindex staticintlinearSearch(inttarget,int[]a){ for(intp=0;p&lt;a.length;p+
UPenn - CIT - 591
Everything You Ever Wanted To Know About Java O-OApr 10, 2009What is a class?A class is primarily a description of objects, or instances, of that class A class contains one or more constructors to create objects A class is a type A type
UPenn - CIT - 594
State MachinesApr 10, 2009What is a state machine? A state machine is a different way of thinking about computation A state machine has some number of states, and transitions between those states Transitions occur because of inputs A pure
UPenn - CIT - 594
A Binary Tree ADTFields The definition of a binary tree pretty much requires the following fields: Objectvalue; BinaryTreeleftChild; BinaryTreerightChild; I also wanted to have this additional field: BinaryTreeparent; Should these fields b
UPenn - CIT - 591
RecursionApr 10, 2009Definitions IA recursive definition is a definition in which the thing being defined occurs as part of its own definition Example: An atom is a name or a number A list consists of: An open parenthesis, &quot;(&quot; Zero or m
UPenn - CIT - 597
DOMApr 10, 2009SAX and DOMSAX and DOM are standards for XML parsers-program APIs to read and interpret XML files DOM is a W3C standard SAX is an ad-hoc (but very popular) standard There are various implementations available Java imple
UPenn - CIT - 597
More JavaScriptApr 10, 2009Browser support JavaScript works on almost all browsers Internet Explorer uses JScript (referred to in menus as Active Scripting), which is Microsofts dialect of JavaScript Older browsers dont support some of the n
UPenn - CIT - 591
ArraysA problem with simple variables One variable holds one value The value may change over time, but at any given time, a variable holds a single value If you want to keep track of many values, you need many variables All of these variables
UPenn - CIT - 591
State MachinesApr 10, 2009What is a state machine? A state machine is a different way of thinking about computation A state machine has some number of states, and transitions between those states Transitions occur because of inputs A pure
UPenn - CIT - 597
&lt;?xml version=&quot;1.0&quot; encoding=&quot;UTF-8&quot;?&gt; &lt;Error&gt;&lt;Code&gt;NoSuchKey&lt;/Code&gt;&lt;Message&gt;The specified key does not exist.&lt;/Message&gt;&lt;Key&gt;4ef63fcf58f1ab753365b9d97a3437ac45e082e8.ppt&lt;/Key&gt;&lt;RequestId&gt;D F69F396087F7AD8&lt;/RequestId&gt;&lt;HostId&gt;avMP9wNCqTnFBeRSd6/yIudFtAy
UPenn - TCOM - 370
The following figure shows the MAC frame structure for IEEE 802.3:
UPenn - CIS - 500
CIS 500 Software Foundations Midterm I Review questionsOctober 10, 2007The following problems should give you a general idea of the style and coverage of the exam questions. Note that this is not a sample exam: in particular, we have not tried to
UPenn - CIS - 500
CIS 500 Software Foundations Final Exam Review questionsDecember 11, 2007The following problems should give you a general idea of the style and coverage of the exam questions from the last part of the course. Note that this is not a sample exam:
UPenn - CIS - 500
CIS 500 Software Foundations Midterm I, Review QuestionsUntyped lambda-calculus1. (2 points) We have seen that a linear expression like x. y. x y x is shorthand for an abstract syntax tree that can be drawn like this: x y applyxr rrr apply M
UPenn - CIS - 500
1 Exercise Show the derivations of the following typing judgments: 1. if false then 0 else succ 0 : Nat 2. succ (if iszero 0 then succ 0 else pred 0) : Nat 1 Solution This solution uses the prooftree package. 0 : Nat 1. false : Bool T-False 0 : Nat T
UPenn - CIS - 500
CIS 500 Software Foundations Final Exam Review questions with answersDecember 11, 2007Work each of the review problems yourself before looking at the answers given here. If your answer diers from ours, make sure you understand why.SubtypingThe
UPenn - CIS - 500
CIS 500 Software Foundations Midterm II Answer keyNovember 8, 2006Instructions This is a closed-book exam. You have 80 minutes to answer all of the questions. The entire exam is worth 80 points for students in section 002 and 90 points for stud