Homework Assignments for netMA348SU13

30dec11 and 20may13
\begin{document} \maketitle \section{Homework 1} Chapter 1 of the D'Angelo-West text is meant to be an inventory of your mathematical readiness for this course. You are not expected to know all of this material in this chapter. If you did, perhaps you should not take this course. But you are expected to understand the explanations. Chapter 1 ends with 56 problems, all of which you should read with with a pencil and pad of paper at your side. In preparation for this homework you should \textit{skim} Chapter 1 so see what's where. Submit a PDF of a document in which you classify each of the 56 problems according to the following rubric. As review for the midterm and final, we will spend time in class [respectively online in the Moodle and the Virtual Office] going over the most popular problems submitted. \begin{itemize} \item [N] Which problems are \textbf{not arranged} in the order in which their topics appear in the text of Chapter 1? \item [I] Which of the problems look \textbf{interesting} enough for you to invest time in understanding a solution? \item [P] Which of the \textbf{practice problems}, marked (-), can you do right now \item [R] Which of the \textbf{regular problems}, not those with a (!), would you be unabe to do even with some studying? \item (You may list the problems in order and mark each of them with an one or more letter \textbf{N,I,P,R} as appropriate.) \itme Submit a PDF document in any format. No reduction of credit for the manner in which the PDF was created for this homework. \end{itemize} \section{Homework 2} \begin{itemize} \item D'Angelo-West 1.13 (write up the solution done in class [resp. in the Moodle]), 1.20, 1.21, and 1.28. \item Solve each problem first on scratch paper. \item Enter the statement and solution in your Journal. \item Any solution to a problem in this course which does not state the question, either explicitly or implicitly, receives at most 3/4 credit. \item Bring your journal to in class lab [Does not apply online.] \item Preferred Format (lesser formats are still acceptable for this homework.) \item Compose a single LaTeX document for all problems, in texWins or otherwise. \item Print out a copy to bring to class for correction. [Does not apply online.] \item Upload its .pdf file to the Moodle. \item Upload its .tex file to the Moodle as well for stylistic correction. \end{itemize} \subsection{Lesser acceptible formats.} \begin{itemize} \item Note, most important is to get the homework in on time (1/2 of credit). \item For your submission to receive credit it must be \begin{itemize} \item Easily Readable, use complete sentences, justify your answers. See Advice. \item Immutable (hence PDF or a screen print, but NO .txt, .doc, or .docx files} \item Only .pdf files will be returned with corrections and comments. \end{itemize} \item A screen print of a legible handwritten homework converted to a PDF. You can insert the screen print into MS Word and "print it" to the electronic PDF format. \item A document prepared with the Equations Editor in MS Word, and then printed to a PDF. \item A document prepared in texPad, with pictures roughly aligned, and composed on the desktop, then made into a screen print. Convert the screen print into a PDF to submit. \end{itemize} \subsection{Advanced Composition Component} \begin{itemize} \item Each problem solved must be stated, preferably in your own words. \item Use complete sentences, but you should use mathematical notation to shorten the discourse. \item If you are unable to compose text in LaTeX, you may use a wordprocessor with mathematical formulas and equations as figures. We recomment MS Word 2010, whose Equation Editor understands LaTeX equation commands. A good way to learn LaTeX. \item If you are unable to use any of the above enumerated lesser, alternate forms of submission, consult with the instructor. There are extenuating circumstances, and an oral report in the Virtual Office may be an extraordinary, allowed format. \end{itemize} \section{Homework 3} These are the sections of the chapter your should \textit{study}, and to ask questions if you do not understand the text. You should \textit{skim} the rest of the chapter to know where things are discussed. \subsection{Arithmetic series} Sections 3.7 and 3.10 treat the sum of consecutive numbers in two different ways, by induction, and by discovery. The discovery method depends on a trick. The induction method requires knowledge of the answer before you can prove its generality. A (finite) sequence of integers for which successive differences are constant, $a_{n+1}-a_n = d $, is called an \textit{ arithmetic sequence}. The sum of an arithmetic sequence, called a (finite) \textit{ arithmetic series}, may be discovered by Gauss's trick. Know how to do this, and given the answer you discovered, prove it by induction. Put this solution into your Journal. \subsection{Geometric series} Similarly, sections 3.13 and 3.14 deal with (finite) \textit{ geometric series}. Be sure a similar analyis of the simplest case, $\sum_{i=1}^n x^i$ is in your Journal. \subsection{Series of powers} Do not confuse a geometric series with a (finite) sum of the powers of consecutive numbers, $S_e^n := \sum_{i=1}^n i^e $. You should have inductive proofs for the following cases in your Journal: \begin{itemize} \item $S_0^n = n$ , this is easy! \item $e=1$, see 3.7 \item $e=2$, see 3.21 \item $e=3$, see Problem 3.16 \end{itemize} \subsection{Assignemd Problems} Solutions in LaTeX format of the five problems 3.16, 3.17, 3.19, 3.28, 3.29 are due as stated in the syllabus. Lesser format may still be used, but for reduced credit. \section{Homework 4} There is no separate HW unit number 4. This is the Midterm exam. \section{Homework 5} To do the assigned problems you may have to read the textbook, unless you remember the definitions from elsewhere. The vocabulary inludes \textit{ relatively prime, greatest common divisor, long division, Euclid's algorithm, Diophantine equations.} The chief tool you are to use in solving these problems is the Prime Decomposition Theorem, whenever possible. \subsection{Practical definitions} Here are some practical definitions to help in doing the problems. \begin{itemize} \item \textbf{ prime decomposition:} Write $pc(a)$ for $a = a_1 a_2 ... a_n$ as a product of prime factors, each $1 < a_j \le a$. \item \textbf{ divides, divisible:} $d|a$ if $(\exists q)(a = dq)$ \item \textbf{ divides, divisible:} $d|a$ if the $pc(d)$ is part or all of $pc(a)$. \item \textbf{ long division :} We write $divmod(a,d)=(q,r)$ for $a = dq + r, 0\le r < d$. Thus, the long division of \textit{ dividend}$a$ by \textit{ divisor} $d$ results in the\textit{ quotient} $q$ and \textit{ remainder} $r$. \texttt{ divmod( , )} is the Python function that does long division. \item \textbf{ greatest common divisor :} $d = gcd(a,b)$ if it is the largest among all divisors of both $a$ and $b$. \item \textbf{ relatively prime:} Two integers have no common prime factor iff $gcd(a,b)=1$ iff none of the $a_i$ is one of the $b_j$ and vice versa, are said to be relatively prime. \item \textbf{ greatest common divisor :} $d = gcd(a,b)$ is also the largest product of primes that cancel out in their fraction, $\frac{a}{b} = \frac{da'}{db'}=\frac{a'}{b'}$ where $(a',b')=1$. \item \textbf{ Euclid's Algorithm :} For positive integers $(a,b)$ the repeated subtraction of the lesser from the greater terminates at zero. The last positive integer is the $gcd(a,b)$. (This is a theorem). \item \textbf{Diophantine equation :} has the form $ax + by = c$ where $a,b,c$ are given integers, and $x,y$ are unknown integers, if they exist. \end{itemize} \subsection{Homework assigned for HW 5.} D'Angelo-West 6.8, 6.9, 6.17, 6.18, 6.28, see repeated here Bring a printout of your work to class for peer editing. \subsection{Homework assigned for HW 6} D'Angelo-West 7.8, 7.9, 7.11, 7.18, 7.24 Bring a printout of your work to class for peer editing. [Does not apply online.]

Current Homework Scoring Policy is this:

\begin{itemize} \item On time homework count 1/2 of total. \item A correct, complete and properly composed solution counts for 1/2 to 3/4 more. \item Effective editing of another student's homework, if assigned, doubles your score. \item If revision of your faulty work is advised, you may earn up to full credit of the original homework. \end{itemize} The reason for this peculiar scoring is that it conforms with the General Education Advanced Composition policy and with the purpose of formative assessment. Some comments on the assigned problems. \begin{itemize} \item Note the solution to 7.1 is in the class notes. \item Problems 7.8 and 7.18 require ingenuity applied to the definitions. \item Problem 7.11 requres you to read the text on the definition of an equivalence relation. \item Problem 7.24 reviews the notion of injective functions as well. \item Problems 7.28 and 7.30 concern divisibility test for decimals. You should definitely put the solution of this into your journals. But we will do this in class. \end{itemize} \end{document}